Duality in integrable systems and gauge theories

Abstract

We discuss various dualities, relating integrable systems and show that these dualities are explained in the framework of Hamiltonian and Poisson reductions. The dualities we study shed some light on the known integrable systems as well as allow to construct new ones, double elliptic among them. We also discuss applications to the (supersymmetric) gauge theories in various dimensions. 06/99

Dualities in Integrable Systems: Geometrical Aspects

“…The first type of duality we would like to discuss concerns the dualities between the pair of dynamical systems [1,2]. It has been formulated by Ruijsenaars [1] in the context of the transition to the action-angle variables.…”

“…It has been actually forgotten for a decade and was revived recently after the breakthrough in the string theory. In the recent works [2,3] it was reformulated in more geometric terms utilizing the fact that the phase spaces of the corresponding systems are the manifolds coinciding with some moduli spaces. These moduli spaces or their close relatives enjoy a large symmetry group due to their origin and we shall demonstrate that part of the symmetries can be precisely formulated in terms of duality.…”

“…In the separation of variables procedure one maps the initial phase space into the symmetric power of the symplectic manifold which serves as the phase space for the one-body problem. Finally one more duality which is analogue of the S duality in the string theory can be formulated [2] as the change of the basis in the base of the Liouville tori fibration over the space of the integrals of motion.…”

“…We shall unify two seemingly different approaches to double elliptic two-body system [2,12] utilizing the Mukai-Odesskii algebra associated with this manifold. This paper is an extended version of the V.R.…”

“…talk given during the Kiev NATO Advanced Research Workshop "Dynamical Symmetries of Integrable Quantum Field Theories and Lattice models" in September 2000. We are based heavily on our papers [2,3] but we also present some new results which are part of our joint works with H. Braden and A. Odesski in progress.…”