Transcript Yangian Symmetry in Yang

Yangian Symmetry in
Yang-Mills Theories
S. G. Rajeev
Seminar at Cornell University
Dec 8th 2004
Work in Collaboration with
Abhishek Agarwal hep-th/0405116,hep-th/0409180
• Also earlier work with Herbert Lee,
Teoman Turgut and Govind Krishnaswamy
Yangian symmetry was originally proposed for
Yang-Mills theories in
L. Dolan, C.R.Nappi and E. Witten
hep-th/0308089,0401243
We extend the idea to one and two loops of the
quantum theory.
What is a Yangian?
D. Bernard and A. Leclair hep-th/9205064
It is a deformation of the associative algebra defined by the
commutation relations
[(Qm ) ab ; (Qn ) cd ] = ±bc (Qm + n ) ad ¡ ±da (Qm + n ) cb;
The most familiar deformation is the Kac—Moody algebra
[(Qm ) ab ; (Qn ) cd ] = ±bc (Qm + n ) ad ¡ ±da (Qm + n ) cb + km±bc ±da ±(m + n)
But there is another one if we restrict the range of the
indices to non-negative values. To understand this,
first note that the above algebra is generated by
Q0 ; Q1 :
Finite Presentation
These satisfy the obvious relations
£ 0 a
¤
0
c
(Q ) b ; (Q ) d = ±bc (Q0 ) ad ¡ ±da (Q0 ) cb
£ 0 a
¤
1
c
(Q ) b ; (Q ) d = ±bc (Q1 ) ad ¡ ±da (Q1 ) cb:
The higher generators are given by repeated commutators of
these. However since there are several ways of doing that
certain consistency relations need to be imposed:
£ 0 a £ 1 c
¤¤ £ 1 a £ 0 c
¤¤
1
e
1
e
(Q ) b ; (Q ) d ; (Q ) f ¡ (Q ) b ; (Q ) d ; (Q ) f = 0:
These `Serre relations’ give a presentation of the algebra in terms of
a finite number of generators and relations.
The Co-product
Equally important to representation theory is the co-product
¢ (Q0 ) ab (u) = (Q0 ) ab (u) I + I
¢ (Q1 ) ab (u) = (Q1 ) ab (u) I + I
(Q0 ) ab (u)
(Q1 ) ab (u):
T his is a homomorphism of t he above product :
¢ ([(Qm ) ab ; (Qn ) cd ]) = [¢ ((Qm ) ab ); ¢ ((Qn ) cd )]
The co-product allows us to form new representations by taking tensor
products of old ones. Physically it is analogous to the rules for `addition
of angular momentum’. Note that the order of composition doesn’t
matter as the co-product is co-commutative.
Lack of Co-Commutativity
It is possible now to modify this structure so that the co-product
is no longer co-commutative:
¢ (Q1 ) ab = (Q1 ) ab
¢ (Q0 ) ab = (Q0 ) ab I + I
I + I (Q1 ) ab + (Q0 ) ac
(Q0 ) ab
(Q0 ) cb:
This is the rule for addition of certain
non-local conserved charges of quantum
spin chains and matrix models. There is a
preferred order for combining spins (and
matrices) so that the rule for addition
of charges need not be co-commutative.
Can the multiplication law be changed so that this new coproduct is still a homomorphism? No change is needed in
0
0
0
1
the relations [Q ; Q ]; [Q ; Q ]:
The Terrific Relations
There is a modification of the Serre relations which are
preserved by this new co-product:
£ 0 a £ 1 c
¤¤ £ 1 a £ 0 c
¤¤
1
e
1
e
(Q ) b ; (Q ) d ; (Q ) f ¡ (Q ) b ; (Q ) d ; (Q ) f =
h
i
p
(Q0 ) cb; (Q0 ) cf (Q0 ) ep (Q0 ) ed ¡ (Q0 ) cp (Q0 ) f (Q0 ) ed
³
´
p
p
¡ ±de (Q0 ) af (Q0 ) cp (Q0 ) b ¡ (Q0 ) ap (Q0 ) f (Q0 ) cb
¡ 0 a 0 e 0 p
¢
p
c
0
a
0
0
e
+ ±f (Q ) d (Q ) p (Q ) b ¡ (Q ) p (Q ) d (Q ) b
The Hopf algebra defined by these relations is the Yangian.
It is a true `quantum group’: neither commutative nor cocommutative. A new kind of symmetry that explains the integrability of
many quantum systems: spin chains and matrix models.
Yangian Symmetry of Quantum
Spin Chains
A `spin chain’ is a sequence of L spins arranged on a line with the last
one connected to the first. Each `spin’ can take N possible values. A
typical hamiltonian (XXX Heisenberg chain)
X would be
H =
Pl ;l + 1 :
l
Here Pk ;l is t he operat or t hat int erchanges t he spins at sit es k; l.
There is a Yangian symmetry in this system that explains the exact
solvability of these spin chains by the celebrated Bethe ansatz.
`Cut and Paste’ Operators
T he st at e of a spin chain is given by a sequence ji 1 ; ¢¢¢i L > .
It is useful t o de¯ne t he operat ors
£ ij 11 ¢¢¢¢¢¢ij a jk1 ¢¢¢kc > =
c¡
X
b
b
±jk1d ¢¢¢±jk d + b¡ 1 jk1 ¢¢¢kd¡ 1 i 1 ¢¢¢i a kd+ b ¢¢¢kc >
b
d= 1
They check if the lower sequence appears in the beginning of the list
of spin states; if it does, it is cut out and replaced by the upper
sequence. (Recall that by cyclic symmetry we can bring any spin to the
beginning of the list.) Otherwise we get zero. Rather like the `cut and
paste’ function of a text editor.
The Heisenberg Hamiltonian is
H= £ ij ji :
More complicated hamiltonians can be written as linear
combinations of these operators.
The Commutation Relations of the `cut and paste’
Operators
These operators satisfy an interesting Lie algebra
C.W.H.Lee and S. G. Rajeev Phys. Rev. Lett. 80,2285-2288(1998)
[£ IJ ; £ KL ] = gJI KL PM £ PM
where the structure constants have a graphical
interpretation.
There is a sophisticated theory explaining the integrability of the
Heisenberg spin chain, in terms of transfer matrices and Yang-Baxter
relations. It was found that there is an underlying Yangian symmetry.
The Generators of the Yangian for the
Heisenberg Spin Chain
There is an obvious unitary symmetry in the Heisenberg
spin chain with the conserved quantity
(Q0 ) ij = £ ij
It is less obvious that there is another symmetry
X
1
i
(Q ) j =
£ ikII kj ;
I
the sum being over all possible sequences. This follows by expanding
the transfer matrix around the point at infinity in the spectral
parameter.
The Serre relations follow from the fact the transfer matrix of the spin
chain satisfies the co-product rules; but they can also be verified
directly using the commutation relations of the `cut and paste’
operators.
Matrix Models and Spin Chains
S. G. Rajeev and C.W.H. Lee Nuclear Physics B, 529, 656-688(1998).
A matrix model is a quantum system whose degrees of freedom are
matrices. The basic operators satisfy the canonical commutation
relations [a¯ ; ayj ° ] = ±j ±®° ±¯ ;
i ® ±
i
±
The hamiltonian is a unitary invariant operator
such as
1
1
H = hij ayj ai + hki jl 2 ayi ayj ak al
N
N
The states that survive the large N limit are
1 ayi 1 ayi 2 ¢¢¢ayi n j0 > = ji ¢¢¢i >
n
1
n
N
2
Note that these states are in one-one correspondence with the
states of a quantum spin chain.
`Cut and Paste’ Operators in Large N Matrix Models
£
I
J
= p
1
N jI j+ jJ j¡ 2
Tr
¡
ayi 1
¢¢¢ayi j I
¢
j
aj j J j ¢¢¢aj 1 :
The effect of these operators on the states above is
exactly those of the cut and paste operators on spin
chains. Thus there is an equivalence between the large N
limit of matrix models and quantum spin chains. Certain
matrix models go over to integrable spin chains. For
example, the Heisenberg spin chain corresponds to
¡ yi yj
¢
1
H =
Tr a a ai aj
N
Thus these matrix models can be solved at least in the large N
limit by the Bethe ansatz. Several such examples were given in
C.W.H.Lee and S. G. Rajeev Phys. Rev. Lett. 80,2285(1998)
Yangian Symmetry in Matrix Models
Using the equivalence of matrix models to spin chains, it should be
possible to translate the Yangian symmetries into the language of
matrix models. Matrix models are more general objects than spin
chains, since the equivalence is only true at large N. Also they are
prototypes of Yang-Mills theories.
(Q0 ) ij = £ ij = Trayi aj
(Q1 ) ij
X
=
I
£ ikII kj =
1
N
jI j+ jJ j+ 1
2
Trai I k aj I¹ k :
Is there a further deformation of the Yangian which is also a
symmetry of the finite N matrix model?
Matrix Approach to String Theory
T. Banks, W. Fishler, S.H. Shenker and L. Susskind, Phys. Rev.
D55, 5112(1995); R. Dijkgraaf, E. Verlinde, H. Verlinde hepth/9703030; N. Kim and J.Plefka hep-th/0207034
One of the approaches to string theory is through the large N
limit of matrix models. That matrix models can be integrable and
have hidden symmetries suggest that string theory might be more
tractable than it looks at first sight. For example, string theory in
flat space is expected to be equivalent to the matrix model with
lagrangian
L = Tr X_2 + Tr[X i ; X i ]2 + ¢¢¢
The dots representing SUSY completion.
Mass deformation of Matrix Models
However this theory is hard to study since it doesn’t
have a minimum for its potential: there is a degeneracy
which must be lifted by some quantum effect. If we
add the `mass term’ we get a theory that can be
studied perturbatively:
1
1
¡ M 2 Tr(X a ) 2 ¡ M 2 (X i ) 2 + ¢¢¢
2
4
This should represent string theory in a
plane wave background:
ds2 = dx 2 + ¹ 2 [(x a ) 2 +
1 (x i ) 2 ](dx + ) 2
4
Maximal Super-Yang-Mills Theory
Understanding Yang-Mills theories is the great challenge for theoretical
physics. The Yang-Mills theory with the best chance of being
integrable is the maximally supersymmetric one, with a set of four
fermions and six scalars for each gauge boson. We don’t yet know
what it means for such a theory to be integrable. But certain limiting
cases are integrable. And these have Yangian symmetries.
Maldacena has conjecutured that this theory is equivalent to a string
theory in the AdS background. We don’t yet know how to formulate
such a string theory. However both major approaches (sigma models
and matrix models) lead to theories with Yangian symmetries.
Yangian (more generally Hopf,) symmetries could be key to proving such an
equivalence. Much like the use of current algebra in proving the Bose-Fermi
correspondence in two dimensional field theory.(Polchinski, Roiban..sigma model
approach to AdS string)
The Dilatation Operator of N=4 SYM
Although N=4 SUSY YM has zero beta function (no coupling constant
renormalization) its gauge invariant obsevables have anomalous dimension.
In fact the anomalous dimensions form an infinite dimensional matrix which
can be computed in perturbation theory.
An example of t he kind of operat ors one can st udy are
t r©i 1 ¢¢¢©i a
where ©i are t he four scalars of t he t heory. At one loop t hey only mix wit h
each ot her. Remarkably t he mixing mat rix is t he hamilt onian of t heSU(4)
Heisenberg spin chain in t he large N limit .
N. Beisert, C. Kristjansen, M. Staudacher hep-th/0303060;N. B., M.S.
hep-th/0307042;J.A. Minahan, K. Zarembo hep-th/0212208; V.A.
Kazakov, A. Marshakov, J.M.,K. Z. hep-th/0402207 ;
Integrability of N=4 SUSY YM
The analogue of the mass spectrum in a conformal field theory is the set
of eigenvalues of its dilatation operator: i.e., the anamolous dimension
matrix. At least at one loop the dilatation operator can be diagonalized
by the Bethe ansatz. There are indications that it persists to higher
loops.
The integrability is explained by the Yangian symmetry.
In our papers S.G.R.and Abhishek Agarwal hep-th/0405116,0409180 we
construct the Yangian generators directly in terms of the scalar field
variables (matrix variables) and show that the Serre relations are satisied at
large N using the `cut and paste’operators. Also we construct deformations
to the Yangian charges that extend the symmetry to the two-loop dilatation
operator. What happens beyond that is not yet known.
Will Integrability survive to Realistic Theories?
Most realistic systems in nature are not integrable,but we find
that studying a limiting case that is integrable is usually a
good starting point. N=4 SUSY-YM could be like the Kepler
problem while QCD is like celestial mechanics.
The Birkoff procedure in mechanics allows us to extend conserved
quantities to any perturbation of a classical system order by order in
perturbation theory. It is only for integrable systems that this procedure
converges. That there are such perturbations in some regions of the
phase space was eventually established by the KAM theorem. Realistic
theories could have some sectors are integrable and others that are not.
(QCD dilatation dynamics appears to be this way.)
Dilatation Dynamics of QCD
The full anomalous dimension matrix at one loop of QCD has just
been calculated:N. Beisert, G. Ferretti, R. Heise, K. Zarembo hepth/0412029.
It is an SU(2,2) spin chain but not integrable. (For N=4 SYM
we would get an SU(2,2|4) integrable spin chain.)
Nevertheless they are able to determine the ground state
(the operator with the smallest anomalous dimension) as
well as the low lying excitations using a Bether ansatz:
`quasi-integrable’ system.
There were indications of integrability much earlier in the
related study of structure functions; e.g., MULTI-COLOR QCD AT
HIGH ENERGIES AND ONE-DIMENSIONAL HEISENBERG MAGNET L.D.
Faddeev , G.P. Korchemsky (1994). There is much more work,
see for references A.V. Belitsky, G.P. Korchemsky, D. Muller hepth/0412054
Beyond Perturbation Theory
When an infinite number of operators mix even one loop anomalous
dimensions can lead to sophisticated dynamical problems. If some
kind of gauge-string duality holds we can translate the problem of
solving the full Callan-Symanzik equation (`dilatation dynamics’) to a
matrix model or a sigma model at least in the large N limit ( classical
limit of string theory). While realistic systems like QCD are unlikely to
be integrable, there might be supersymmetric variants which are. Then
we can study the formation of hadronic bound states as a problem in
this dynamics: at short distances we have the boundary conditions of
a free theory and at long distances we get out the hadronic states.
Although only a dream in QCD, I have done exactly this in a quantum
mechanical toy model with asymptotic freedom. It is possible to determine
explicitly the operator that represents the violation of scale invariance due
to renormalization.
Quantum Field Theory is back!