Yangian Symmetry in Yang-Mills Theories S. G. Rajeev Seminar at Cornell University Dec 8 th 2004.

Yangian Symmetry in Yangian Symmetry in Yang-Mills TheoriesYang-Mills Theories

S. G. Rajeev S. G. Rajeev

Seminar at Cornell University Seminar at Cornell University

Dec 8Dec 8thth 2004 2004

Work in Collaboration with Work in Collaboration with Abhishek AgarwalAbhishek Agarwal• Also earlier work with Herbert Lee,Also earlier work with Herbert Lee,

Teoman Turgut and Govind Teoman Turgut and Govind KrishnaswamyKrishnaswamy

hep-th/0405116,hep-th/0409180

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?What is a Yangian?

It is a deformation of the associative algebra defined by the commutation relations

The most familiar deformation is the Kac—Moody algebra

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

D. Bernard and A. Leclair hep-th/9205064

[(Qm)ab;(Qn)c

d]= ±cb(Q

m+n)ad ¡ ±a

d(Qm+n)cb;

[(Qm)ab;(Qn)c

d]= ±cb(Q

m+n)ad ¡ ±a

d(Qm+n)cb + km±c

b±ad±(m+n)

Q0;Q1:

These satisfy the obvious relations

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:

These `Serre relations’ give a presentation of the algebra in terms of a finite number of generators and relations.

Finite Presentation

£(Q0)a

b;£(Q1)c

d;(Q1)ef

¤¤¡

£(Q1)a

b;£(Q0)c

d;(Q1)ef

¤¤= 0:

£(Q0)a

b;(Q0)cd

¤= ±c

b(Q0)a

d ¡ ±ad(Q0)c

b£(Q0)a

b;(Q1)cd

¤= ±c

b(Q1)a

d ¡ ±ad(Q1)c

b:

Equally important to representation theory is the co-product

The Co-product

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.

¢ (Q0)ab(u) = (Q0)a

b(u) I + I (Q0)ab(u)

¢ (Q1)ab(u) = (Q1)a

b(u) I + I (Q1)ab(u):

This is a homomorphism of theaboveproduct:

¢ ([(Qm)ab;(Qn)c

d]) = [¢ ((Qm)ab);¢ ((Qn)c

d)]

It is possible now to modify this structure so that the co-product is no longer co-commutative:

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 co-product is still a homomorphism? No change is needed in the relations

Lack of Co-Commutativity

¢ (Q0)ab = (Q0)a

b I + I (Q0)ab

¢ (Q1)ab = (Q1)a

b I + I (Q1)ab +(Q0)a

c (Q0)cb:

[Q0;Q0];[Q0;Q1]:

There is a modification of the Serre relations which are preserved by this new co-product:

The Terrific Relations

The Hopf algebra defined by these relations is the Yangian.

It is a true `quantum group’: neither commutative nor co-commutative. A new kind of symmetry that explains the integrability of many quantum systems: spin chains and matrix models.

£(Q0)a

b;£(Q1)c

d;(Q1)ef

¤¤¡

£(Q1)a

b;£(Q0)c

d;(Q1)ef

¤¤=

h(Q0)c

b;(Q0)c

f (Q0)ep(Q0)e

d ¡ (Q0)cp(Q0)p

f (Q0)ed

i

¡ ±ed

³(Q0)a

f (Q0)cp(Q0)p

b ¡ (Q0)ap(Q0)p

f (Q0)cb

´

+±cf

¡(Q0)a

d(Q0)ep(Q0)p

b ¡ (Q0)ap(Q0)p

d(Q0)eb

¢

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) would be

H =X

l

Pl ;l+1:

Here Pk;l is theoperator that interchanges the spins at sites 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

The stateof a spin chain is given by a sequence ji1;¢¢¢iL >.

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=£ i j

j i :

More complicated hamiltonians can be written as linear combinations of these operators.

It is useful to de ne theoperators

£ i1¢¢¢iaj 1¢¢¢j b

jk1 ¢¢¢kc >=c¡ bX

d=1

±kdj 1

¢¢¢±kd+ b¡ 1j b

jk1 ¢¢¢kd¡ 1i1 ¢¢¢iakd+b¢¢¢kc >

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 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 ;£ K

L ]= gI K MJ L P £ P

M

where the structure constants have a graphical interpretation.

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 = £ i

j

It is less obvious that there is another symmetry

the sum being over all possible sequences. This follows by expanding the transfer matrix around the point at infinity in the spectral parameter.

(Q1)ij =

X

I

£ i I kkI j ;

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 Matrix Models and Spin ChainsChains

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¯

i ®;ayj °± ] = ±j

i ±°®±¯

±;

H = hij

1N

ayj ai +hkli j

1N 2

ayi ayj akal

1N

n2

ayi1ayi2 ¢¢¢ayin j0>= ji1 ¢¢¢in >

The hamiltonian is a unitary invariant operator such as

The states that survive the large N limit are

Note that these states are in one-one correspondence with the states of a quantum spin chain.

£ IJ =

1p

N jI j+jJ j¡ 2Tr

¡ayi1 ¢¢¢ayi j I j aj j J j

¢¢¢aj 1

¢:

`Cut and Paste’ Operators in Large N Matrix Models

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

H =1N

Tr¡ayi ayj ai aj

¢

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 Yangian Symmetry in Matrix ModelsModels

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 = £ i

j = Trayi aj

(Q1)ij =

X

I

£ i I kkI j =

1

Nj I j + j J j + 1

2

Trai I kaj ¹I k:

Is there a further deformation of the Yangian which is also a symmetry of the finite N matrix model?

Matrix Approach to String Matrix Approach to String TheoryTheory

T. Banks, W. Fishler, S.H. Shenker and L. Susskind, Phys. Rev. D55, 5112(1995); R. Dijkgraaf, E. Verlinde, H. Verlinde hep-th/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:

This should represent string theory in a plane wave background:

¡12M 2Tr(X a)2 ¡

14M 2(X i )2 + ¢¢¢

ds2 = dx2 +¹ 2[(xa)2 + 14(xi )2](dx+)2

Maximal Super-Yang-Mills Maximal Super-Yang-Mills TheoryTheory

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 SYMThe 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 the kind of operators one can study are

tr©i1 ¢¢¢©ia

where ©i are the four scalars of the theory. At one loop they only mix witheach other. Remarkably the mixing matrix is the hamiltonian of theSU(4)Heisenberg spin chain in the 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 YMIntegrability 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 Will Integrability survive to Realistic Theories?Theories?

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.)

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.

Dilatation Dynamics of QCDDilatation 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 hep-th/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 hep-th/0412054

Beyond Perturbation TheoryBeyond 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!