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Different faces of integrability in the gauge

theories or in hunting for the symmetries

- Isaac Newton Institute, October 8

Some history of the hidden integrability

- Matrix models for the quantum gravity Douglas,

Gross-Migdal, Brezin-Kazakov (89-91) - Regge limit of scattering amplitudes in QCD-

Lipatov,Korchemsky-Faddeev(93-94) - Topological gauge theories in D2(YM) and D3

(Chern-Simons) - Nekrasov-A.G.(94-95)
- N2 SUSY Yang-Mills theories- Krichever-Marshakov-

Mironov-Morozov-A.G., Witten-Donagi(95) - Anomalous dimensions from integrability Braun-

Derkachev-Manashev-Belitsky-Korchemsky (in

simplest one-loop cases in QCD-98-99) - Anomalous dimensions in N4 SYM Minahan-

Zarembo, Beisert-Staudacher (02-03) - Integrability of the dual sigma model for N4

SYM- Bena-Polchinski-Roiban (04) - Matching of YM and stringy answers

Tseytlin-Frolov Minahan-Zarembo-Kazakov-Marshakov

and many others (04-.) - Proposal for the all-loop result

Beisert-Eden-Staudacher(06)

Integrability what does it mean?

- Dynamical system with N degrees of freedom should

have N conserved integrals of motion H,In0.

They commute that is one can consider the

different time directions - If number of the conserved integrals is infinite

- integrable field theories. Many examples but

mainly in (11) dimensions

Universality of the integrability

- Plasma, Hydrodynamics - KdV, KP equations
- 2D Quantum gravity-matrix models KdV,KP

hierarchies - Gauge theories in D2,3,4 Quantum Hall effect

in different geometry Black holes -Toda,Calogero

and Ruijsenaars systems - Evolution equations in D4-spin chains with the

different groups

Integrability versus group theory

- Phase spaces of the integrable systems are

closely related to the group -like manifolds

which admit the Poisson structure - Examples of the finite dimensional group phase

manifolds parameters Coadjoint orbit

-gtTg?TG-gtHeisenberg Double - More general integrable systems involves the

phase spaces with additional parameters. - Finite dimensional examples quantum groups(1

parameter), Sklyanin algebra(2 parameters),Mukai-O

desskii algebra (many free parameters)

Integrability versus group theory

Poisson structure is closely related to the

geometric objects. Example intersection of N

quadrics Qk in CP(N2) with homogenious

coordinates xk.

Complicated polynomial algebras induced by

geometry. The quadrics are Casimir operators of

this algebra. A lot of Casimirs and free

parameters.

Integrability versus group theory

- Infinite dimensional examples Kac-Moody algebra,
- Virasoro algebra. Parameters central charges and

parameters of representation - Parameters of the group phase spaces are mapped

into the parameters of the integrable systems - Generic situation Integrable system follows from

the free motion on the group-like manifolds with

possible constraints

Integrability versus group theory

- Examples KdV- free rotator on the coadjoint

Virasoro orbit

utuuxuxxx - Calogero and Toda systems - free motion
- on the T(SU(N)) with the simple constraint
- Relativistic Calogero system(Ruijsenaars)-
- free motion on the Heisenberg Double with

constraint

Examples

Potential of the integrable Calogero many-body

system

Ruijsenaars many-body system

Integrability versus moduli spaces

- General comment
- Consider the solution to the equation of motion

in some gauge theory - F0, 3d Chern-Simons gauge theory
- FF self-duality equation in 4d Yang-Mills
- FdZ BPS condition for the stable objects in

SUSY YM - Solutions to these equations have nontrivial

moduli spaces which enjoy the rich symmetry

groups and provide the phase space for the

integrable systems

Integrability versus Riemann Surfaces

- General comment Solutions to the integrable

systems are parameterized by the Riemann surfaces

(in general of infinite genus) which are related

to the complex Liouville tori. In many

interesting situations these surfaces have finite

genus.

Moduli of the complex structures of these Riemann

surfaces are related to the integrals of motion.

Summation over solutionsintegration over the

moduli

2D Yang-Mills on the cylinder

- Consider SU(N) gauge theory

Theory has no dynamical field degrees of freedom.

However there are N quantum mechanical degrees of

freedom from the holonomy of the

connection. Adiag(x1,.,xn), Ediag(p1,,pn)

nondiag

Heavy fermion at rest

2

Standard YM Hamiltonian HTr E2 yields the

Calogero integrable system with trigonometric

long-range interaction

2D Yang-Mills theory and Calogero system

- What is the meaning of the time variables?
- The first time is the inverse coupling constant
- Higher times tk - chemical potentials for the

powers of the electric field - This is the generic situation evolution

parameters in the integrable systems relevant for

the gauge theories are the couplings for the

operators - SS0 tk Ok with some operators Ok
- In theories with running coupling t0 log(scale)

that is integrability is some property of RG

evolution

Chern-Simons theory and Ruijsenaars system

- Consider SU(N) Chern-Simons theory on the torus

with marked point (Wilson line along the time

direction)

The phase space is related to the moduli space

of flat connections on the torus. Coordinates

follows from the holonomy along A-cycle and

momenta from holonomy along B-cycle. The emerging

dynamical system on the moduli space

relativistic generalization of the Calogero

system with N degrees of freedom. When one of

the radii degenerates Ruijsenaars system

degenerates to the Calogero model. These are

examples of integrability in the perturbed

topological theory.

Integrability in N2 Supersymmetric gauge theories

- In N2 theory there are physical variables

protected by holomorphy low-energy effective

actions and spectrum of stable particles - All these holomorphic data are fixed by

finite-dimensional integrable system which

captures the one-loop perturbative correction and

contribution from the arbitrary number of

instantons to the tree Lagrangian - Theory involves naturally two moduli spaces.

Moduli space of vacua is parameterized by the

vacuum condensates. Also moduli space of

instantons.

Integrability in N2 SUSY theories

- Seiberg and Witten found solution for the

holomorphic data in terms of the family of the

Riemann surfaces of the genus (N-1) with some

additional data (meromorphic differential)

bundled over the moduli space of the vacua

Vacuum expectation values of the complex scalars

parameterize the moduli space of the Riemann

surfaces.

Mapping into the integrable system

- Time variable in the integrable system t log

(IR scale) - Riemann surface solution to the classical

equations of motion - Moduli space of vacua half of the phase space

of the integrable system - Masses of the stable particles action

variables - All N2 gauge theories with the different matter

content have the corresponding integrable system

under the carpet

Gauge theories with N2 SUSY versus integrable

systems

Integrability and N2 gauge theories

- The very surface has even more physical

interpretation this is the surface we would

live on if we would enjoy N2 SUSY. Any N2

citizen lives on the 51 worldvolume of the

soliton(M5 brane) in higher dimensions which

looks as R(3,1)(Riemann surface). - Is it possible to derive integrable system

microscopically? Yes, it follows from the

consideration of the instanton moduli space

(Nekrasov 04). - Hence we have situation when integrability

related with RG flows involves the summation over

nonperturbative solutions. Symmetries behind

moduli spaces.

Anomalous dimensions in the gauge theories and

Integrability

- Time variable T log(RG scale), that is once

again integrability behind the RG evolution

One loop renormalization of the composite

operators in YM theory is governed by the

integrable Heisenberg spin chains

Example of the operator TrXXXZXZZZXXX, the number

of sites in the chain coincides with the number

of fields involved in the composite operator

Anomalous dimensions and integrability

- Acting by the spin chain Hamiltonian on the set

of operators one gets the spectrum of anomalous

dimensions upon the diagonalization of the mixing

matrix. The RG equation because of integrability

has hidden conserved quantum numbers

eigenvalues of the higher Hamiltonians commuting

with dilatation - In N4 SuperYM spin chain responsible for

one-loop evolution has the symmetry group

SO(6)SO(2,4) which is the global symmetry group

of the N4 SYM - Higher loops integrable system involves the

interaction between nearest L neighbors at L

loop order

Anomalous dimensions and integrability

- Gauge-string duality N4 SYM is dual to the

superstring theory in

String tension is proportional to the square root

of tHooft coupling

That is weak coupling in the gauge theory

correspond to the deep quantum regime in the

string sigma model while strong coupling

corresponds to the quasiclassical

string(Maldacena 97). Could gauge/string duality

explain the origin of integrability? The answer

is partially positive. Stringy sigma model on

this background is CLASSICALLY integrable.

Anomalous dimensions and integrability

- Hamiltonian of the string Dilatation operator

in the gauge theory - That is derivation of the spectrum of anomalous

dimensions is equivalent to the derivation of

the spectrum of the quantum string in the fixed

background - The main problem there is no solution to the

QUANTUM sigma model in this background yet. That

is no exact quantum spectrum we look for. - The hint consider the operators with large

quantum numbers (R charge,Lorentz spin S e.t.c.).

The corresponding string motion is quasiclassical!

Anomalous dimensions and integrability

- In this forced quasiclassical regime the

comparison can be made between perturbative YM

calculations and stringy answers. Complete

agreement where possible. - First predictions from integrability for the

all-loop answers for the simplest object

anomalous dimension of the operators with the

large Lorentz spin S - F(g) Log S (Beisert-Eden-Staudacher)
- There are a lot of higher conserved charges

commuting with dilatation. Their role is not

completely clear yet. - They imply the hidden symmetries behind the

perturbative YM ( Yangian symmetry,Dolan-Nappi-Wit

ten e.t.c.)

Integrability and the scattering amplitudes

- At the weak coupling the scattering amplitudes in

the Regge limit are governed by the complex

integrable system SL(2,C) Heisenberg spin chain.

Number of reggeons number of sites in the spin

chain. Pomeron-spin chain with 2 sites, Odderon-

spin chain with 3 sites - Time variable in the integrable evolution

T log (scale)log s, where

s-kinematical invariant of the scattering problem - There is holomorphic factorization of the

Hamiltonian - (Lipatov)

Integrability and the scattering amplitudes

Scattering with the mutireggeon exchanges

Integrability and the scattering amplitudes

- The integrability is the property of the

evolution equations (BFKL) once again - Spectrum of the integrable system defines the

asymptotic behavior of the scattering amplitudes

Hk is the Hamiltonian of the spin chain with k

sites

Integrability and scattering amplitudes

- Many questions What happens with integrability

(upon the resummation of the gluons to reggeons)

at higher loops. What is the meaning of higher

conserved charges? E.t.c. - From the stringy side some progress as well.

Attempts to identify the stringy configurations

responsible for the scattering amplitudes (

Alday-Maldacena). However no clear identification

yet similar to the

string energyanomalous dimensions

Conclusion

- Integrability is very general phenomenon behind

the evolution equations (T log (scale)) and

moduli spaces in many different topological and

nontopological gauge theories - Perfect matching with gauge/string duality when

possible - First predictions for the all-loop answers in N4

SYM theory - Prediction for the hidden symmetries in YM gauge

theory (Yangian e.t.c.) Meaning of higher charges

in the RG evolution not clear enough - Just the very beginning of the story. A lot to be

done..