Transcript Diapositiva 1 - Istituto Nazionale di Fisica Nucleare

Testing AdS/CFT at string loops
Gianluca Grignani
Perugia University & INFN
2006 PRIN MEETING
Alessandria, December 15-16
References
G. Grignani, M. Orselli, B. Ramadanovich, G.W. Semenoff and D. Young,
“Divergence cancellation and loop corrections in string field theory on a
plane wave background”, JHEP 0512:017,2005 e-Print Archive:hep-th/0508126
G. Grignani, M. Orselli, B. Ramadanovich, G.W. Semenoff and D. Young,
“AdS/CFT versus string loops”, JHEP 06:040,2006,
e-Print Archive:hep-th/0605080
G. De Risi, G. Grignani, M. Orselli, G.W. Semenoff,
“DLCQ string spectrum from N=2 SYM theory”, JHEP411:053,2004
e-Print Archive:hep-th/0409315
D. Astolfi, V. Forini, G. Grignani, G.W. Semenoff,
“Finite size corrections and integrability of N =2 SYM and DLCQ strings
on a pp-wave”, e-Print Archive:hep-th/0606193
AdS/CFT
Maximally supersymmetric Yang-Mills theory on 4D flat space with gauge
group SU(N) and coupling constant gYM
is exactly equivalent to
type IIB superstring theory on background AdS5XS5 with N units of 5-form
flux through S5 . The radius of curvature is
‘t Hooft coupling
and
String state
Energy
gauge invariant
composite operator
conformal dimension of
composite operator
This is a weak coupling – strong coupling duality
•The Penrose limit of AdS5XS5 geometry gives a pp-wave space
transverse SO(4)xSO(4) symmetry
Free strings are exactly solvable in the light-cone gauge
light-cone
Hamiltonian
level matching condition
•The analogous limit of Super Yang-Mills can be taken (BMN)
detailed
matching of BMN operators in the large N, planar limit and free string states
planar limit
non-planar corrections
free strings
string interactions
(ongoing) matching interactions in light-cone string field theory on a ppwave background and Yang-Mills theory computations
Non planar corrections in Yang-Mills Theory
String loop corrections to the energy of the 2-oscillator (2-impurity) states
Symmetric traceless:
Trace:
In N=4 SYM there are 6 scalar fields
Select a combination of them
which is charged under a U(1) subgroup of the R Symmetry
Dual BMN operator:
Has not been computed using string theory!
The only available formulation of string theory in which
interactions can be computed is
pp-wave light-cone string field theory
Because of the presence of the RR-field, the only available formulation
of string theory in which interactions can be computed is
pp-wave light-cone string field theory
M.Spradlin and A.Volovich, hep-th/0204146, hep-th/0206073, hep-th/0310033 (review)
C.S.Chu, V.Khoze, M.Petrini, R.Russo and A.Tanzini, hep-th/0208146
Y.He, J.Schwarz, M.Spradlin and A.Volovich hep-th/0211198
A.Pankiewicz, hep-th/0208209
A.Pankiewicz and B.Stefanski, hep-th/0210246
R.Roiban, M.Spradlin and A.Volovich hep-th/0211220
S.Dobashi, H.Shimada and T.Toneya, hep-th/0209251
P.Di Vecchia, J.L.Petersen, M.Petrini, R.Russo and A.Tanzini, hep-th/0304025
J.Gomis, S.Moriyana and J.W.Park, hep-th/0301250
A.Pankiewicz, hep-th/0304232
P.Gutjahr and A.Pankiewicz, hep-th/0407098
S.Dobashi and T.Toneya, hep-th/0408058, hep-th/0406225
S. Lee and R. Russo, hep-th/0409261
G.Grignani, M.Orselli, B. Ramadanovich, G.W.Semenoff and D. Young, hep-th/0508126
G.Grignani, M.Orselli, B. Ramadanovich, G.W.Semenoff and D. Young, hep-th/0605080
Plan of the talk
1. Light-cone String Field Theory (SFT)
2. The cubic vertex:
Spradlin and Volovich (SVPS) vertex
Di Vecchia et al. (DPPRT) vertex
Dobashi and Yoneya (DY) vertex
3. Contact terms
4. Perturbation theory
5. A little history of the results
6. Main results:
1) one-loop correction to the energy of 2-oscillator states for each vertex.
2) cancellation of divergences for the trace state.
7. It’s just a factor of 2 !
8. Conclusions
Light-cone String Field Theory
string field operator  : fundamental object in light-cone string field theory
creates or annihilates complete strings
m-string Hilbert space
 is a functional of x+, p+ and the worldsheet coordinates
where XI()=XI(,=0) and likewise for other fields.
In the momentum-space representation  is a functional of
where  is – i times the momentum conjugate to , i.e.:
in the momentum-space representation
The light-cone string field theory dynamics of  is governed by the non-relativistic
Schroedinger equation
light-cone string field
theory Hamiltonian
In terms of the string coupling the light-cone SFT Hamiltonian has the expansion
In the free string theory limit it should be equal to the Hamiltonian coming from the
string theory -model
We will use perturbation theory to compute the energy of the states.
free strings
Consider the bosonic case and the Fourier modes of XI() and PI() for =0, xn
and pn
we write the free light-cone Hamiltonian
in terms of these Fourier modes
where
The free pp-wave Hamiltonian written in terms of these Fourier modes becomes
where
the eigenfunctions of the Schroedinger equation are products of an
infinite number of momentum eigenfunctions Nn(pn) where Nn is
the excitation number of the nth oscillator with frequency n/.
being a momentum eigenstates
a coherent state
the string field becomes
to quantize the string field we promote the
to operators acting on the string
Fock space where it creates or destroys a complete string with excitation number
{Nn} at =0
and
The superalgebra generators are promoted to operators acting on the SFT Hilbert
space. For example, the free Hamiltonian becomes
where
Turning on interactions: the cubic vertex
The corrections to superalgebra generators, once interactions are turned on, are
obtained following some guiding principles.
1. The interaction should couple the string worldsheet in a continuos way
=2(1+2)
x+
2

=21
3
1
=0
x+=0
The interaction vertex for the scattering of 3 strings is constructed with a
-functional enforcing worldsheet continuity.
In pp-wave superstring however the situation is slightly more complicated but the
basic principle governing the interaction is very simple:
2. The superalgebra has to be realized in the full interacting theory
This is the essential difference with the bosonic string case and modifies the form
of the vertex.
Supercharges that square to the Hamiltonian receive corrections when adding
interactions
The picture remains geometric but in addition to a delta-functional enforcing
continuity in superspace one has to insert local operators at the interaction point.
These operators represent functional generalizations of derivative couplings
prefactors
There are two different set of superalgebra generators.
1. Kinematical generators :
They are not corrected by interactions. The symmetries they generate are not
affected by adding higher order terms to the action
These generators remain quadratic in the string field  in the
interacting field theory and act diagonally on Hm
2. Dynamical generators:
They receive corrections in the presence of interactions and couple different
number of strings
quartic
in the string field
cubic
in the string field
the corrections are such that
,
and
still satisfy the superalgebra
The requirement that the superalgebra is satisfied in the interacting theory now
gives rise to two kind of constraints:
Kinematical constraints
Dynamical constraints
(anti)-commutation relations of
kinematical and dynamical generators
anti-commutation relations of
dynamical generators alone
leads to continuity condition
in superspace
requires insertion of interaction
point operators: the prefactors
To solve the constraints we use perturbation theory.
Example:
different from flat space
In the plane wave geometry transverse momentum is not a good quantum
number due to the confining harmonic potential. The expansion in gs, however,
implies the same kinematical constraint as in flat space
must be translationally invariant
the relation is the same as in flat space, many of the techniques developed
there can be used in the plane wave case as well
In momentum space the conservation of transverse momentum by the interaction
will be implemented by a -functional
Analogously, from
and since
and
P+ is a good quantum number and one concludes that the cubic interaction
contains also
Three string vertex in plane wave light-cone SFT
and
must be proportional to
Dynamical constraint: the supersymmetry algebra
Expanding the Hamiltonian and the supercharges at the order gs: the dynamical
constraint on
and
is
the same as in flat space
the constraints are solved by inserting prefactors
into the ansatz for
and
do not depend on
the string field 
where
to determine the explicit form of the vertex it is essential to use a number basis
eigenstates of
rather then momentum eigenstates
write explicitly  and perform
the momentum integral
To identify
and
it is sufficient to find their matrix elements between
two incoming strings and one outgoing string.
It is convenient to express
Hilbert space
not as an operator but as a state in the 3-string
and similarly for
|3i and h 3’| are related
by worldsheet time-reversal
We can write
prefactor
kinematical part of the vertex, common to all dynamical generators
and similar expressions for the fermionic part |Ebi
we get
performing the Gaussian integrals we find
One can proceed analogously for the fermionic case
Neumann matrices
by constructing the fermionic analog of the bosonic wave function one arrives at
the solution
General formulas for the kinematical part of the (SV) vertex
and similarly for
Two possible solutions for the prefactors.
a) PSSV , the prefactor is odd under Z2 symmetry.
The purely bosonic part of the prefactor is given by
The fermionic part of the prefactor was derived by Pankiewicz and Stephanski
and contains up to 8 fermionic and bosonic creation operators
b) DPPRT , the prefactor is even under Z2 symmetry.
A very simple way to realize the susy algebra is by acting with the free parts of
the Hamiltonian and dynamical charges
The purely bosonic part of the prefactor is given by
The fermionic part contains at most 2 bosonic or fermionic creation operators
c) Holographic SFT (Dobashi-Yoneya, Lee-Russo)
The purely bosonic part of the prefactor is given by
breaks maximally the Z2 symmetry
• when restricted to the zero-modes provides the resultant supergravity effective
theory in the pp-wave background.
• gives the correct 3-point correlation functions on the SYM side derived by
perturbation theory
The only linear combination of the two vertices with these properties is the
one which equally weights them.
For all the vertices: the dynamical constraints do not fix the overall
normalization of the vertex, this can depend on  and . In flat space the J- I
generator is also dynamical imposes further constraints on the vertex and uniquely
fixes the normalization. As J- I is not part of the plane wave superalgebra an overall
function f(,) remains unfixed in each vertex
Contact Terms
We want to compute the energy of string states using a quantum mechanical
perturbation theory.
The dynamical generators are expanded in terms of gs. Up to the O(gs2) we have
The susy algebra up to the O(gs2)
Necessarily there is an
the so-called contact term
Light-cone string field perturbation theory on a single string in a 2 oscillator state |ni
uses QM perturbation theory to compute the correction to its light-cone energy
The string join/split points
of each vertex coincide
In type IIA/B superstring on Minkowski space-time, it is known that the contact terms are
necessary to cancel certain singularities in the integrations over parameters of light-cone SFT
diagrams. In the conformal field theory they are also seen to arise as additional contributions
needed to cancel certain singular surface terms which arise in the integration of correlators of
vertex operators over the modular parameters of Riemann surfaces.
J.Greensite and F.R.Klinkhamer NPB 304 (1988)
Summary of results
String loop corrections to the energy of the 2-impurity (2-oscillator) bosonic states
Symmetric traceless:
Trace:
Dual string state:
Yang-Mills Prediction
A little history
The gauge theory result is
Roiban, Spradlin and Volovich, hep-th/0211220
It matches! But obtained from
Reflection symmetry of the one-loop light-cone string diagram
P.Gutjhar and A. Pankiewicz, hep-th/0407098, no ½
??
The gauge theory result is
P.Gutjhar and A. Pankiewicz, hep-th/0407098
It does not match!
• Expansion in half integer powers of ’
• Uses truncation to 2-impurities
• Uses Spradlin-Volovich vertex
• Fixes the pre-factor f=1
• Sets Q(4)=0
Results: for any vertex
1. We confirmed Gutjhar and Pankiewicz result, the natural expansion parameter
is
2. In the trace state |[1, 1]i
the leading order is
and it diverges
Each term in the equation above diverges individually. With the ½ of the
“reflection symmetry” no cancelation of divergences
no 1/2
3. We provided a general proof for this result just using the supersymmetry algebra
Then using
with Q(4)=0
•
The energy correction vanishes for a supersymmetric state Q(2)|ni=0
•
Divergences are canceled in this formula and the relative factor
between the two terms in the starting equation is fixed.
•
Q(2)|ni is proportional to the square root of the energy n
Results: for each vertex
1. Spradlin and Volovich vertex
We confirmed Gutjahr-Pankiewicz result
2. Di Vecchia et al. vertex
3. Dobashi-Yoneya vertex
Divergences cancel and it works at 1-loop!
Best matching of this quantity so far
DY vertex is an improvement over its predecessors
• No half integer powers up to ’7/2
• Uses truncation to 2-impurities
• Fixes the pre-factor f=4/3
• Sets Q(4)=0
4. Some experimental results, it’s just a factor of 2 !
Introducing a factor of 2 in front of the contact term
Both vertices match the gauge theory result up to two loops !
• Use truncation to 2-impurities
• Fix the pre-factor f to suitable values
• Set Q(4)=0
Conclusions
1. Dobashi Yoneya vertex seems to be the most promising
2. Possible sources of discrepancies
a) number of impurities in the intermediate states.
b) Q(4) ≠ 0
a contact term which does not diverge?
3. The problem of matching non-planar YM and string
interactions in the context of AdS/CFT is still open