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
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Kazakov (89-91)
Regge limit of scattering amplitudes in QCD- Lipatov,KorchemskyFaddeev(93-94)
Topological gauge theories in D=2(YM) and D=3 (Chern-Simons)
Nekrasov-A.G.(94-95)
N=2 SUSY Yang-Mills theories- Krichever-Marshakov-Mironov-MorozovA.G., Witten-Donagi(95)
Anomalous dimensions from integrability – Braun- Derkachev-ManashevBelitsky-Korchemsky (in simplest one-loop cases in QCD-98-99)
Anomalous dimensions in N=4 SYM Minahan- Zarembo, BeisertStaudacher (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-ZaremboKazakov-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,In}=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
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
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),MukaiOdesskii 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(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 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
ut=uux+uxxx
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
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
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
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.
Heavy
fermion at rest
A=diag(x1,…….,xn),
E=diag(p1,……,pn) + nondiag
2
Standard YM Hamiltonian H=Tr E^2 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” t - chemical potentials for the powers of
k
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=S0 + 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 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 finitedimensional integrable system which captures the oneloop 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 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.
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
Gauge theories with N=2 SUSY versus integrable systems
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.
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 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
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.
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-Witten
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. Pomeronspin 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 energy=anomalous dimensions
Conclusion
 Integrability is very general phenomenon behind the
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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…..