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

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 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)

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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

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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

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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)

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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.
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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

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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

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

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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
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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
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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

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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.
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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.

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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.
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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

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Gauge theories with N2 SUSY versus integrable
systems
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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.

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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
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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

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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.
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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!

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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.)

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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)

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Integrability and the scattering amplitudes
Scattering with the mutireggeon exchanges
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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
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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

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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..