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


Different faces of integrability in the gauge theories or in hunting for the symmetries Isaac Newton Institute, October 8 – PowerPoint PPT presentation

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

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-
  • Topological gauge theories in D2(YM) and D3
  • 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

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
  • 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
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
  • 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

Potential of the integrable Calogero many-body
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
  • 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

Moduli of the complex structures of these Riemann
surfaces are related to the integrals of motion.
Summation over solutionsintegration over the
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)
Heavy fermion at rest
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
  • SS0 tk Ok with some operators Ok
  • In theories with running coupling t0 log(scale)
    that is integrability is some property of RG

Chern-Simons theory and Ruijsenaars system
  • Consider SU(N) Chern-Simons theory on the torus
    with marked point (Wilson line along the time

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

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
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
  • All N2 gauge theories with the different matter
    content have the corresponding integrable system
    under the carpet

Gauge theories with N2 SUSY versus integrable
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
  • 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
  • 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
  • (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
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

  • 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
  • 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
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