From free gauge theories to strings Carmen Núñez I.A.F.E. – Physics Dept.-UBA Buenos Aires 10 Years of AdS/CFT strings@ar, December 19, 2007 Based on Work in progress.
Download ReportTranscript From free gauge theories to strings Carmen Núñez I.A.F.E. – Physics Dept.-UBA Buenos Aires 10 Years of AdS/CFT strings@ar, December 19, 2007 Based on Work in progress.
Slide 1
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 2
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 3
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 4
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 5
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 6
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 7
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 8
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 9
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 10
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 11
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 12
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 13
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 14
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 15
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 16
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 17
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 18
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 19
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 20
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 21
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 22
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 23
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 24
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 25
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 26
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 27
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 28
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 29
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 30
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 2
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 3
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 4
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 5
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 6
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 7
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 8
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 9
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 10
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 11
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 12
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 13
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 14
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 15
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 16
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 17
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 18
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 19
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 20
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 21
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 22
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 23
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 24
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 25
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 26
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 27
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 28
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 29
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.
Slide 30
From free gauge theories
to strings
Carmen Núñez
I.A.F.E. – Physics Dept.-UBA
Buenos Aires
10 Years of AdS/CFT
strings@ar, December 19, 2007
Based on
Work in progress in collaboration with
M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)
R. Gopakumar, Phys.Rev.D70(2004)025009, 025010,
Phys.Rev.D72 (2005) 066008
O. Aharony, Z. Komargodski and S. Razamat,
JHEP 0701 (2007) 063
J. David and R. Gopakumar, JHEP 0701 (2007) 063
O. Aharony, J. David, R. Gopakumar, Z. Komargodski
and S. Razamat, Phys.Rev.D75 (2007) 106006
Outline
Brief review of a proposal by R. Gopakumar to obtain the
string theory dual of large N free gauge theories.
Resulting integrand on moduli space has the right properties
to be that of a string theory.
Worldsheet vs. spacetime OPE in several examples
Future work
After 10 years…
Many examples known how to find closed string dual of gauge
theories which can be realized as world-volume theories of Dbranes in some decoupling limit.
Dual string theory is a standard closed string theory, living in a
warped higher dimensional space.
Strongly coupled gauge theory
weakly curved string
background
gravity approx. may be used.
In general, (weakly coupled gauge theories) dual string theory is
complicated, and not necessarily has geometrical interpretation.
Free gauge theories?
It is interesting to ask what is the string theory dual of the
simplest large N gauge theory: free gauge theory
Free large N gauge theories as a laboratory for understanding
the gauge/string correspondence (making this picture precise is
essential to obtain a string dual to realistic gauge theories.)
As limit of interacting gauge theories (not just N2 copies
of a free U(1) theory). Have topological expansion in powers
of 1/N2. In this limit gs ~ 1/N.
Useful starting point for perturbation theory in (perturbative
Feynman amplitudes are given in terms of free field diagrams).
General expectations
At least in the context of string theory on AdS5 S5 , free
field theory related to tensionless limit.
For 4D free conformal gauge theories one expects that any
geometrical intepretation should have an AdS5 factor.
Peculiar properties needed of w-sh theory: free correlators
terminate at finite order of 1/N expansion dual w-sh
correlators get contributions upto given maximal genus
What exactly is the string dual?
How exactly does a large N field theory reorganize itself into a
dual closed string theory?
Can we systematically construct the closed string theory starting
from the field theory?
Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M.
Kruczenski, B. Sundborg, G. Bonelli …
Gauge-string duality
General expectation is
O1 (k1 ) On (kn ) g
V (k , ) V
1
1
1
M g,n
Oi:
Vi :
Gauge invariant operators
Vertex operators of dual string theory
Can we recast the left hand side
into the form we expect from
the right hand side?
n
(k n , n )
ws
Gopakumar’s proposal I
Simple way to organize different Feynman diagram
contributions to given n-point function so that the net sum can
be written as an integral over the moduli space of an npunctured Riemann surface.
1. Skeleton diagram
Write gauge theory
amplitudes in Schwinger
parametrised form gluing
together homotopically
equivalent propagators
2
1
1 | xi x j |
d e
2
0
| xi x j |
Gopakumar’s prescription II
2.
Map Schwinger parameters to the moduli space of a
Riemann surface with holes Mg,n R+n
CONCRETE PROPOSAL: Identify the Schwinger
lij ij
parameters with Strebel lengths: Line integrals
between the zeroes of certain meromorphic
quadratic differentials (Strebel differentials)
zj
lij
( z )dz
zi
# independent for maximally connected Feynman graph of genus g for
n-point function (6g 6 + 3n = 6g 6 + 2n + n) =
= # real moduli for genus g Riemann surface with n punctures +
additional n moduli parametrize R+n
= # Strebel lengths lij
Gopakumar’s prescription III
3.
Integrate over the parameters of the holes.
Integral over (with sum over different graphs) can be converted into
integral over Mg,n R+n
Thus potentially a world-sheet n-point correlation function.
This procedure translates any Feynman diagram to a correlation
function on the string world-sheet.
The dictionary lij ij
For every Strebel differential there is a critical graph whose
vertices are the zeroes of the differential and along whose
zj
edges
lij ( z )dz is real
zi
For generically simple zeroes the vertices of critical graph are cubic.
Each of the n faces of critical graph contains only one double pole
Critical graph can be identified with dual of reduced Feynman graph
How can we check this hypothesis?
We don’t know how to quantize string theory in the
highly curved AdS backgrounds that would presumably
be dual to the free limit of conformal field theory.
Few modest checks
1. Two and three point functions give expected correlators in AdS.
E. g. Planar three point function
3
Ji
Tr
( xi )
Gg{ J i } ( x1 , x2 , x3 )
i 1
g 0
can be recast as a product of three bulk-boundary
propagators for scalars in AdS
x1
t /2
= J (d-2)/2
K ( x, z; t )
2
[t ( x z ) ]
dt
0
d
1
0
2
Gg{ Ji0} ( x1 , x2 , x3 )
x3
x2
t
3
d
d
z K i ( xi , z; t )
i 1
Probably special to 2- and 3- point functions
The Y four point function
2. Consider 4-point correlation functions of the form
Gg{ J i } ( x1 , x2 , x3 , x4 ) Tr J1 ( x1 )Tr J 2 ( x2 )Tr J 3 ( x3 )Tr J ( x4 )
with J = J1 + J2 + J3. Mapping gives
G({4J)i } ( x1 , x2 , x3 , x4 )
{Ji }
2
d
G
( xi )
( , )
with = (l1, l2, l3).
Explicit expression for the candidate worldsheet correlator
J. David and R. Gopakumar, JHEP 0701 (2007) 063
g
Prediction for string dual
{Ji }
{ xi }
G
(1 | | | 1 |)1/ 2
( , ) C ( J i )
| || 1 |
(1 | | | 1 |) J1 1/ 2 (1 | | | 1 |) J 2 1/ 2 (1 | | | 1 |) J 3 1/ 2
[ x12 (1 | | | 1 |) x22 (1 | | | 1 |) x32 (1 | | | 1 |)]J
The dependence on || and |1- | is what one expects of a
correlation function of local operators inserted at 0, 1, and .
G{{xJii}} ( , ) VxJ1 1 (0)VxJ2 2 (1)VxJ3 3 ()VxJ4 ( , )
Obeys crossing symmetry:
Consistent with locality: all
terms in OPE (when 0)
WS
G{{xJii}} (1 ,1 ) G{{xJii}} (, )
G{{xJii}} (
1
,
1
) | |4 G{{xJii}} ( , )
G{{xJii}} ( , ) C{{hJ,ih,}xi } h
h ,h
h
with hh
Worldsheet vs. spacetime OPE
Consider four point function of single trace operators
O1 ( x1 )O2 ( x2 )O3 ( x3 )O4 ( x4 )
As x1 x2 , OPE contains other gauge invariant operators
O1 ( x1 )O2 ( x2 ) C12k ( x1 x2 )Ok ( x2 )
k
UV in bdary spacetime
IR in bulk spacetime
UV on worldsheet
EXPECTATION: As x1 x2 , worldsheet correlator gets dominant
contribution from z 0
lij ij : when two ST positions collide, corresponding ij .
This corresponds to region of moduli space where vertices collide.
Worldsheet vs. spacetime OPE
(continued)
In free field theory, often correlators in which two operators do not
have any Wick contractions with each other, e.g.
Tr (1 )( x1 )Tr (1 2 )( x2 )Tr ( 2 3 )( x3 )Tr ( 3 )( x4 )
has contribution only from
x2
x3
x1
x4
Absence of ST OPE should be reflected in corresponding WS OPE
EXPECTATION: The strongest way in which this could happen is if
the corresponding vertex operators also do not have a WS OPE
The four point function
Consider correlator in free field theory with three adjoint scalar
fields X, Y, Z
Tr ( X 2 ( x1 ))Tr ( X 2Y ( x2 ))Tr (YZ 2 ( x3 ))Tr ( Z 2 ( x4 ))
x2
x3
x1
x4
The string theory amplitude has support only for
negative real values of the modular parameter.
The square and the whale diagrams
Consider the field theory amplitudes
Tr X 2 ( x1 )Tr ( XZ )(x2 )Tr Z 2 ( x3 )Tr ( XZ )(x4 )
Tr X 2 ( x1 )Tr ( X 2Y 2 )(x2 )Tr (Y 2 Z 2 )(x3 )Tr Z 2 ( x4 )
x2
x3
x1
x1
x2
x3
x4
There are no solutions for large
The solution can be obtained numerically, and it is always real
and 0< <1 for the square and localizes on small region of
complex plane for the whale.
x4
LOCALIZATION
The region of moduli space that these diagrams cover precisely
excludes the possibility of taking a worldsheet OPE
b/corresponding vertex operators (e.g. 1 when localized on
the negative real axis).
Pattern behind localization (or absence) in free field diagrams is
such that localization occurs only in those diagrams in which there
is no contraction between two pairs of vertices.
There is no worldsheet OPE exactly when there is no spacetime OPE.
Realization of EXPECTATION
LOCALIZATION (continued)
Localization on the worldsheet is compatible with properties of a
local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z.
Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)
Tr (1J1 ( x1 ))Tr ( 2J 2 3j ( x2 ))Tr ( 2J 2 4J 3 ( x3 ))Tr ( 3j 4J 3 1J1 ( x4 ))
It has contribution from the “broom” diagram.
x2
x3
x4
x1
In the limit j 0, reduces to Pi diagram which
shows localization. ADGKR showed localized
worldsheet correlators correspond to a limit of
the field theory correlation functions which is
governed by saddle point in Schwinger
parameter space
GENERAL LESSONS
The expansion in the position of the saddle point corresponds to an
expansion in the length of one or more small edges in the critical
graph of the corresponding Strebel differential.
Confirmation of expectation: localization of worldsheet correlators
appears to be correlated with absence of non-trivial ST OPE
QUESTIONS:
What is the criterion for localization of general free field diagram?
What is the subspace on which it localizes?
What does this tell us about the WS theory?
The square and the whale from the
The square with a small edge.
Strebel differential
p2
( z c) 4
2
( z )dz 2 2
dz
4 z ( z 1) 2 ( z ) 2
2
c
, 1
1
0
p2 ( z c) 2 ( z c ) 2 2
( z )dz 2 2
dz
2
2
4 z ( z 1) ( z )
2
c = c(0) /2
= (0) + a(li) 2
Graphical deformation of Strebel graph allows to determine phase
of and thus allows to identify potential delocalized diagrams.
Constructing Mg,n
There is a systematic way of constructing Mg,n from the ribbon
graph (familiar from open SFT):
When k edges meet at a vertex they form angles 2/k with each
other.
1
0
one face
two bivalent vertices
two single valued vertices
one zero
two faces with two edges
two faces with one edge
Deformation of the
1
1
0
0
cannot delocalize
1
0
might delocalize
The square with one diagonal
Deforming the to get the square with one diagonal
2
might delocalize
The diagram with two diagonals
1
0
1 = k 2 , k
Blow up n-fold zero moving appropriate number of lines along their
central direction allows to identify potentially delocalized diagrams
Conclusions
WS duals to free large N gauge theories exhibit interesting
behavior
Adding few contractions to field theory diagram or small edges to
dual graph, delocalizes correlators and allows to relate ST with WS
OPE. Fruitful approach to extract general features of WS theory.
We obtained graphical method to identify potential delocalization.
Future Work
Allows to obtain new worldsheet correlators which can be
studied and lead to better understanding of
the worldsheet CFT.
More diagrams have to be studied in order to
extract general properties of the worldsheet duals to
free large N gauge theories.