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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)
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Isaac Newton Institute, October 8
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.
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
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
Potential of the integrable Calogero many-body system
Ruijsenaars many-body system
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
Moduli of the complex structures of these Riemann surfaces are related to the integrals of motion. Summation over solutions=integration over the moduli
Theory has no dynamical field degrees of freedom. However there are N quantum mechanical degrees of freedom from the holonomy of the connection.
E=diag(p1,……,pn) + nondiag
Heavy fermion at rest
Standard YM Hamiltonian H=Tr E^2 yields the Calogero integrable system with trigonometric long-range interaction
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.
Vacuum expectation values of the complex scalars parameterize the moduli space of the Riemann surfaces.
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
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.
F(g) Log S (Beisert-Eden-Staudacher)
Scattering with the mutireggeon exchanges
Hk is the Hamiltonian of the spin chain with k sites