Low-dimensional tori in Calogero-Moser-Sutherland systems

TL;DR

This paper provides an explicit stratification of the phase space of Calogero-Moser-Sutherland systems for SU(n), describing symplectic structures and action-angle coordinates on each stratum, revealing a detailed geometric structure.

Contribution

It offers a detailed description of the phase space stratification and explicit construction of action-angle coordinates for CMS systems related to SU(n).

Findings

Phase space decomposes into symplectic strata of dimensions 2s.

Explicit action-angle coordinates constructed on each positive-dimensional stratum.

Zero-dimensional stratum corresponds to equilibrium points.

Abstract

The main result of this paper is an explicit description of the stratification of the phase space of Calogero--Moser--Sutherland (CMS) integrable systems corresponding to Lie groups $SU(n)$. The phase space decomposes into symplectic strata of dimensions $2s$, where $s = 0, 1, \ldots, n - 1$. On each stratum of the positive dimension, we construct natural action-angle coordinates and compute the symplectic form explicitly, showing that every stratum is symplectomorphic to $\mathbb{R}_{> 0}^s \times \mathbb{T}^s$. The zero-dimensional stratum corresponds to the equilibrium point of the multi-time CMS dynamics.