Description
The compactified Ruijsenaars-Schneider systems are completely integrable Hamiltonian systems on smooth moduli spaces of flat SU(n) connections on the one-holed torus. The monodromy around the hole is constrained to the smallest non-central conjugacy classes of SU(n), which can be labelled by a parameter x that is between 0 and 1. Depending on the value of x, the systems arise in two drastically different forms: in type (i) these are toric systems, while in the type (ii) cases they possess globally continuous action variables that generate a Hamiltonian torus action (only) on a dense open subset of the phase space of dimension 2(n-1). After reviewing background material, we report our recent study of the momentum polytopes and the fibers of the action map (alias the momentum map) that are contained in the complement of the domain of the densely defined torus action in the type (ii) cases. We have shown that these fibers are smooth connected isotropic submanifolds, diffeomorphic to the 3-dimensional sphere in the simplest cases, similarly to the fibers occurring, e.g., in the classical Gelfand-Cetlin systems.
The talk is based on joint work with Holger Dullin (https://arxiv.org/abs/2604.18023) and earlier publications.
The talk will also be broadcast on Zoom.