Different faces of integrability in the gauge theories or in hunting for the symmetries

Different faces of integrability Different faces of integrability in the gauge theories or in in the gauge theories or in

hunting for the symmetrieshunting for the symmetries

Isaac Newton Institute, October 8Isaac Newton Institute, October 8

Some history of the hidden integrabilitySome history of the hidden integrability

Matrix models for the quantum gravity –Douglas, Gross-Migdal, Brezin-Matrix models for the quantum gravity –Douglas, Gross-Migdal, Brezin-Kazakov (89-91)Kazakov (89-91)

Regge limit of scattering amplitudes in QCD- Lipatov,Korchemsky-Regge limit of scattering amplitudes in QCD- Lipatov,Korchemsky-Faddeev(93-94)Faddeev(93-94)

Topological gauge theories in D=2(YM) and D=3 (Chern-Simons)Topological gauge theories in D=2(YM) and D=3 (Chern-Simons) Nekrasov-A.G.(94-95)Nekrasov-A.G.(94-95) N=2 SUSY Yang-Mills theories- Krichever-Marshakov-Mironov-Morozov-N=2 SUSY Yang-Mills theories- Krichever-Marshakov-Mironov-Morozov-

A.G., Witten-Donagi(95) A.G., Witten-Donagi(95) Anomalous dimensions from integrability – Braun- Derkachev-Manashev-Anomalous dimensions from integrability – Braun- Derkachev-Manashev-

Belitsky-Korchemsky (in simplest one-loop cases in QCD-98-99)Belitsky-Korchemsky (in simplest one-loop cases in QCD-98-99) Anomalous dimensions in N=4 SYM Minahan- Zarembo, Beisert-Anomalous dimensions in N=4 SYM Minahan- Zarembo, Beisert-

Staudacher (02-03)Staudacher (02-03) Integrability of the dual sigma model for N=4 SYM- Bena-Polchinski-Roiban Integrability of the dual sigma model for N=4 SYM- Bena-Polchinski-Roiban

(04)(04) Matching of YM and stringy answers; Tseytlin-Frolov ;Minahan-Zarembo-Matching of YM and stringy answers; Tseytlin-Frolov ;Minahan-Zarembo-

Kazakov-Marshakov and many others (04-….)Kazakov-Marshakov and many others (04-….) Proposal for the all-loop result ; Beisert-Eden-Staudacher(06)Proposal for the all-loop result ; Beisert-Eden-Staudacher(06)

Integrability: what does it mean?Integrability: what does it mean?

Dynamical system with N degrees of Dynamical system with N degrees of freedom should have N conserved freedom should have N conserved integrals of motion {H,Iintegrals of motion {H,Inn}=0. They }=0. They commute that is one can consider the commute that is one can consider the different “time” directionsdifferent “time” directions

If number of the conserved integrals is If number of the conserved integrals is infinite - integrable field theories. Many infinite - integrable field theories. Many examples but mainly in (1+1) dimensionsexamples but mainly in (1+1) dimensions

Universality of the integrabilityUniversality of the integrability

Plasma, Hydrodynamics - KdV, KP Plasma, Hydrodynamics - KdV, KP equationsequations

2D Quantum gravity-matrix models – 2D Quantum gravity-matrix models – KdV,KP hierarchies KdV,KP hierarchies

Gauge theories in D=2,3,4 ; Quantum Hall Gauge theories in D=2,3,4 ; Quantum Hall effect in different geometry; Black holes -effect in different geometry; Black holes -Toda,Calogero and Ruijsenaars systemsToda,Calogero and Ruijsenaars systems

Evolution equations in D=4-spin chains Evolution equations in D=4-spin chains with the different groupswith the different groups

Integrability versus group theoryIntegrability versus group theory

Phase spaces of the integrable systems are closely Phase spaces of the integrable systems are closely related to the group -like manifolds which admit the related to the group -like manifolds which admit the Poisson structurePoisson structure

Examples of the finite dimensional “group” phase Examples of the finite dimensional “group” phase manifolds parameters: Coadjoint orbit ->T*gmanifolds parameters: Coadjoint orbit ->T*gT*G-T*G->Heisenberg Double>Heisenberg Double

More general integrable systems involves the phase More general integrable systems involves the phase spaces with additional parameters.spaces with additional parameters.

Finite dimensional examples :quantum groups(1 Finite dimensional examples :quantum groups(1 parameter), Sklyanin algebra(2 parameters),Mukai-parameter), Sklyanin algebra(2 parameters),Mukai-Odesskii algebra (many free parameters)Odesskii algebra (many free parameters)


Poisson structure is closely related to the geometric objects. Example – intersection of N quadrics Qk in CP(N+2) 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 Integrability versus groupgroup theory theory

Infinite dimensional examples; Kac-Moody algebra,Infinite dimensional examples; Kac-Moody algebra,

Virasoro algebra. Parameters: central charges and Virasoro algebra. Parameters: central charges and parameters of representationparameters of representation

Parameters of the “group” phase spaces are mapped into Parameters of the “group” phase spaces are mapped into the parameters of the integrable systemsthe parameters of the integrable systems

Generic situation: Integrable system follows Generic situation: Integrable system follows from the free motion on the group-like from the free motion on the group-like manifolds with possible constraintsmanifolds with possible constraints


Examples; KdV- free rotator on the coadjoint Examples; KdV- free rotator on the coadjoint Virasoro orbit Virasoro orbit uutt=uu=uuxx+u+uxxxxxx

Calogero and Toda systems - free motionCalogero and Toda systems - free motion on the T*(SU(N)) with the simple constrainton the T*(SU(N)) with the simple constraint

Relativistic Calogero system(Ruijsenaars)-Relativistic Calogero system(Ruijsenaars)-free motion on the Heisenberg Double with free motion on the Heisenberg Double with

constraintconstraint

Examples Examples

Potential of the integrable Calogero many-body system

Ruijsenaars many-body system

Integrability versus moduli spacesIntegrability versus moduli spaces

General comment:General comment:

Consider the solution to the equation of motion in some Consider the solution to the equation of motion in some gauge theorygauge theory

F=0, 3d Chern-Simons gauge theoryF=0, 3d Chern-Simons gauge theory

F=*F self-duality equation in 4d Yang-MillsF=*F self-duality equation in 4d Yang-Mills

F=*dZ BPS condition for the stable objects in SUSY YMF=*dZ BPS condition for the stable objects in SUSY YM

Solutions to these equations have nontrivial moduli spaces Solutions to these equations have nontrivial moduli spaces which enjoy the rich symmetry groups and provide the which enjoy the rich symmetry groups and provide the phase space for the integrable systemsphase space for the integrable systems

Integrability versus Riemann SurfacesIntegrability versus Riemann Surfaces

General comment: Solutions to the integrable systems General comment: Solutions to the integrable systems are parameterized by the Riemann surfaces (in general are parameterized by the Riemann surfaces (in general of infinite genus) which are related to the complex of infinite genus) which are related to the complex Liouville tori. In many interesting situations these Liouville tori. In many interesting situations these surfaces have finite genus. surfaces have finite genus.

Moduli of the complex structures of these Riemann surfaces are related to the integrals of motion. Summation over solutions=integration over the moduli

2D Yang-Mills on the cylinder2D Yang-Mills on the cylinder

CConsider SU(N) gauge theoryonsider SU(N) gauge theory

Heavy fermion at rest

Theory has no dynamical field degrees of freedom. However there are N quantum mechanical degrees of freedom from the holonomy of the connection.

A=diag(x1,…….,xn),

E=diag(p1,……,pn) + nondiag

Standard YM Hamiltonian H=Tr E^2 yields the Calogero integrable system with trigonometric long-range interaction

2

2D Yang-Mills theory and Calogero system2D Yang-Mills theory and Calogero system

What is the meaning of the time variables? What is the meaning of the time variables? The “first” time is the inverse coupling constantThe “first” time is the inverse coupling constant Higher “times” tHigher “times” tk k - - chemical potentials for the powers of chemical potentials for the powers of

the electric fieldthe electric field This is the generic situation – evolution parameters in This is the generic situation – evolution parameters in

the integrable systems relevant for the gauge theories the integrable systems relevant for the gauge theories are the couplings for the operators are the couplings for the operators

S=SS=S00 + t + tk k OOk k with some operators O with some operators Okk

In theories with running coupling tIn theories with running coupling t0 0 =log(scale) that is =log(scale) that is integrability is some property of RG evolutionintegrability is some property of RG evolution

Chern-Simons theory and Ruijsenaars Chern-Simons theory and Ruijsenaars systemsystem

Consider SU(N) Chern-Simons theory on the torus with Consider SU(N) Chern-Simons theory on the torus with marked point (Wilson line along the time direction)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 N=2 Supersymmetric gauge Integrability in N=2 Supersymmetric gauge theoriestheories

In N=2 theory there are physical variables protected by In N=2 theory there are physical variables protected by holomorphy; low-energy effective actions and spectrum holomorphy; low-energy effective actions and spectrum of stable particlesof stable particles

All these holomorphic data are fixed by finite-All these holomorphic data are fixed by finite-dimensional integrable system which captures the one-dimensional integrable system which captures the one-loop perturbative correction and contribution from the loop perturbative correction and contribution from the arbitrary number of instantons to the tree Lagrangian arbitrary number of instantons to the tree Lagrangian

Theory involves naturally two moduli spaces. Moduli Theory involves naturally two moduli spaces. Moduli space of vacua is parameterized by the vacuum space of vacua is parameterized by the vacuum condensates. Also moduli space of instantons. condensates. Also moduli space of instantons.

Integrability in N=2 SUSY theoriesIntegrability in N=2 SUSY theories

Seiberg and Witten found solution for the holomorphic data in terms 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 of the family of the Riemann surfaces of the genus (N-1) with some additional data (meromorphic differential) bundled over the moduli additional data (meromorphic differential) bundled over the moduli space of the vacua space of the vacua

Vacuum expectation values of the complex scalars parameterize the moduli space of the Riemann surfaces.

Mapping into the integrable systemMapping into the integrable system

Time variable in the integrable system t= log (IR scale)Time variable in the integrable system t= log (IR scale) Riemann surface = solution to the classical equations of motionRiemann surface = solution to the classical equations of motion Moduli space of vacua = half of the phase space of the integrable Moduli space of vacua = half of the phase space of the integrable

systemsystem Masses of the stable particles= “action” variablesMasses of the stable particles= “action” variables All N=2 gauge theories with the different matter content have the All N=2 gauge theories with the different matter content have the

corresponding integrable system under the carpetcorresponding integrable system under the carpet

Gauge theories with N=2 SUSY versus integrable systems

Integrability and N=2 gauge theoriesIntegrability and N=2 gauge theories

The very surface has even more “physical” interpretation The very surface has even more “physical” interpretation – this is the surface we would live on if we would enjoy – this is the surface we would live on if we would enjoy N=2 SUSY. Any “N=2 citizen” lives on the 5+1 N=2 SUSY. Any “N=2 citizen” lives on the 5+1 worldvolume of the soliton(M5 brane) in higher worldvolume of the soliton(M5 brane) in higher dimensions which looks as R(3,1)+(Riemann surface).dimensions which looks as R(3,1)+(Riemann surface).

Is it possible to derive integrable system Is it possible to derive integrable system “microscopically”? Yes, it follows from the consideration “microscopically”? Yes, it follows from the consideration of the instanton moduli space (Nekrasov 04).of the instanton moduli space (Nekrasov 04).

Hence we have situation when integrability related with Hence we have situation when integrability related with RG flows involves the summation over nonperturbative RG flows involves the summation over nonperturbative solutions. Symmetries behind moduli spaces. solutions. Symmetries behind moduli spaces.

Anomalous dimensions in the gauge Anomalous dimensions in the gauge theories and Integrabilitytheories and Integrability

Time variable T= log(RG scale), that is once again Time variable T= log(RG scale), that is once again integrability behind the RG evolutionintegrability 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 integrabilityAnomalous dimensions and integrability

Acting by the spin chain Hamiltonian on the set of Acting by the spin chain Hamiltonian on the set of operators one gets the spectrum of anomalous operators one gets the spectrum of anomalous dimensions upon the diagonalization of the mixing dimensions upon the diagonalization of the mixing matrix. The RG equation because of integrability has matrix. The RG equation because of integrability has hidden conserved quantum numbers – eigenvalues of hidden conserved quantum numbers – eigenvalues of the higher Hamiltonians commuting with dilatationthe higher Hamiltonians commuting with dilatation

In N=4 SuperYM spin chain responsible for one-loop In N=4 SuperYM spin chain responsible for one-loop evolution has the symmetry group SO(6)*SO(2,4) which evolution has the symmetry group SO(6)*SO(2,4) which is the global symmetry group of the N=4 SYMis the global symmetry group of the N=4 SYM

Higher loops integrable system involves the interaction Higher loops integrable system involves the interaction between nearest L neighbors at L loop order between nearest L neighbors at L loop order


Gauge-string duality ; N=4 SYM is dual to the Gauge-string duality ; N=4 SYM is dual to the superstring theory in superstring theory in

String tension is proportional to the square root of t’Hooft 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 Anomalous dimensions and integrabilityintegrability

Hamiltonian of the string = Dilatation operator in the Hamiltonian of the string = Dilatation operator in the gauge theorygauge theory

That is derivation of the spectrum of anomalous That is derivation of the spectrum of anomalous dimensions is equivalent to the derivation of the dimensions is equivalent to the derivation of the spectrum of the quantum string in the fixed backgroundspectrum of the quantum string in the fixed background

The main problem – there is no solution to the The main problem – there is no solution to the QUANTUM sigma model in this background yet. That is QUANTUM sigma model in this background yet. That is no exact quantum spectrum we look for.no exact quantum spectrum we look for.

The hint – consider the operators with large quantum The hint – consider the operators with large quantum numbers (R charge,Lorentz spin S e.t.c.). The numbers (R charge,Lorentz spin S e.t.c.). The corresponding string motion is quasiclassical!corresponding string motion is quasiclassical!


In this “forced” quasiclassical regime the comparison can In this “forced” quasiclassical regime the comparison can be made between perturbative YM calculations and be made between perturbative YM calculations and stringy answers. Complete agreement where possible.stringy answers. Complete agreement where possible.

First predictions from integrability for the all-loop First predictions from integrability for the all-loop answers for the simplest object – anomalous dimension answers for the simplest object – anomalous dimension of the operators with the large Lorentz spin S of the operators with the large Lorentz spin S

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

with dilatation. Their role is not completely clear yet.with dilatation. Their role is not completely clear yet. They imply the hidden symmetries behind the They imply the hidden symmetries behind the

perturbative YM ( Yangian symmetry,Dolan-Nappi-Witten perturbative YM ( Yangian symmetry,Dolan-Nappi-Witten e.t.c.)e.t.c.)

Integrability and the scattering amplitudesIntegrability and the scattering amplitudes

At the weak coupling the scattering amplitudes in the At the weak coupling the scattering amplitudes in the Regge limit are governed by the complex integrable Regge limit are governed by the complex integrable system SL(2,C) Heisenberg spin chain. Number of system SL(2,C) Heisenberg spin chain. Number of reggeons = number of sites in the spin chain. Pomeron-reggeons = number of sites in the spin chain. Pomeron-spin chain with 2 sites, Odderon- spin chain with 3 sitesspin chain with 2 sites, Odderon- spin chain with 3 sites

Time variable in the integrable evolution Time variable in the integrable evolution T= log (scale)=log s, where s-kinematical invariant of the T= log (scale)=log s, where s-kinematical invariant of the scattering problemscattering problem

There is holomorphic factorization of the HamiltonianThere is holomorphic factorization of the Hamiltonian

(Lipatov)(Lipatov)


Scattering with the mutireggeon exchanges


The integrability is the property of the evolution The integrability is the property of the evolution equations (BFKL) once againequations (BFKL) once again

Spectrum of the integrable system defines the Spectrum of the integrable system defines the asymptotic behavior of the scattering amplitudesasymptotic behavior of the scattering amplitudes

Hk is the Hamiltonian of the spin chain with k sites

Integrability and scattering amplitudesIntegrability and scattering amplitudes

Many questions; What happens with integrability (upon Many questions; What happens with integrability (upon the resummation of the gluons to reggeons) at higher the resummation of the gluons to reggeons) at higher loops. What is the meaning of higher conserved loops. What is the meaning of higher conserved charges? E.t.c.charges? E.t.c.

From the stringy side some progress as well. Attempts to From the stringy side some progress as well. Attempts to identify the stringy configurations responsible for the identify the stringy configurations responsible for the scattering amplitudes ( Alday-Maldacena). However no scattering amplitudes ( Alday-Maldacena). However no clear identification yet similar to the clear identification yet similar to the string energy=anomalous dimensionsstring energy=anomalous dimensions

ConclusionConclusion

Integrability is very general phenomenon behind the Integrability is very general phenomenon behind the evolution equations (T= log (scale)) and moduli spaces evolution equations (T= log (scale)) and moduli spaces in many different topological and nontopological gauge in many different topological and nontopological gauge theoriestheories

Perfect matching with gauge/string duality when possiblePerfect matching with gauge/string duality when possible First predictions for the all-loop answers in N=4 SYM First predictions for the all-loop answers in N=4 SYM

theorytheory Prediction for the hidden symmetries in YM gauge theory Prediction for the hidden symmetries in YM gauge theory

(Yangian e.t.c.) Meaning of higher charges in the RG (Yangian e.t.c.) Meaning of higher charges in the RG evolution not clear enoughevolution not clear enough

Just the very beginning of the story. A lot to be done…..Just the very beginning of the story. A lot to be done…..

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Different faces of integrability in the gauge theories or in hunting for the symmetries