1. Different faces of integrability in the gauge theories or in hunting for the symmetries Isaac Newton Institute, October 8

2. 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 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- 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 N=4 SYM Minahan- Zarembo, Beisert- Staudacher (02-03)  Integrability of the dual sigma model for N=4 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)

3. Integrability: what does it mean?  Dynamical system with N degrees of freedom should have N conserved integrals of motion {H,I n }=0. 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 (1+1) dimensions

4. Universality of the integrability  Plasma, Hydrodynamics - KdV, KP equations  2D Quantum gravity-matrix models – KdV,KP hierarchies  Gauge theories in D=2,3,4 ; Quantum Hall effect in different geometry; Black holes - Toda,Calogero and Ruijsenaars systems  Evolution equations in D=4-spin chains with the different groups

5. 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 ->T*g  T*G- >Heisenberg Double  More general integrable systems involves the phase spaces with additional parameters.  Finite dimensional examples :quantum groups(1 parameter), Sklyanin algebra(2 parameters),Mukai- Odesskii algebra (many free parameters)

6. Integrability versus group theory Poisson structure is closely related to the geometric objects. Example – intersection of N quadrics Q k in CP(N+2) with homogenious coordinates x k. Complicated polynomial algebras induced by geometry. The quadrics are Casimir operators of this algebra. A lot of Casimirs and free parameters.

7. 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

8. Integrability versus group theory  Examples; KdV- free rotator on the coadjoint Virasoro orbit u t =uu x +u xxx Calogero and Toda systems - free motion on the T*(SU(N)) with the simple constraint on the T*(SU(N)) with the simple constraint Relativistic Calogero system(Ruijsenaars)- free motion on the Heisenberg Double with constraint

9. Examples Potential of the integrable Calogero many-body system Ruijsenaars many-body system

10. Integrability versus moduli spaces  General comment: Consider the solution to the equation of motion in some gauge theory F=0, 3d Chern-Simons gauge theory F=*F self-duality equation in 4d Yang-Mills F=*dZ 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

11. 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 solutions=integration over the moduli

12. 2D Yang-Mills on the cylinder  C onsider 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(x 1,…….,x n ), E=diag(p 1,……,p n ) + nondiag Standard YM Hamiltonian H=Tr E^2 yields the Calogero integrable system with trigonometric long-range interaction 2

13. 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” t k - 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  S=S 0 + t k O k with some operators O k  In theories with running coupling t 0 =log(scale) that is integrability is some property of RG evolution

14. 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.

15. Integrability in N=2 Supersymmetric gauge theories  In N=2 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.

16. Integrability in N=2 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.

17. 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 N=2 gauge theories with the different matter content have the corresponding integrable system under the carpet

18. Gauge theories with N=2 SUSY versus integrable systems

19. Integrability and N=2 gauge theories  The very surface has even more “physical” interpretation – this is the surface we would live on if we would enjoy N=2 SUSY. Any “N=2 citizen” lives on the 5+1 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.

20. 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

21. 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 N=4 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 N=4 SYM  Higher loops integrable system involves the interaction between nearest L neighbors at L loop order

22. Anomalous dimensions and integrability  Gauge-string duality ; N=4 SYM is dual to the 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.

23. 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!

24. 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) 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-Witten e.t.c.)

25. 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) (Lipatov)

26. Integrability and the scattering amplitudes Scattering with the mutireggeon exchanges

27. 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 H k is the Hamiltonian of the spin chain with k sites

28. 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 energy=anomalous dimensions

29. 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 N=4 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…..