Trigonometric Sutherland systems and their Ruijsenaars duals from symplectic reduction

TL;DR

This paper explores the duality relations between trigonometric Sutherland systems and rational Ruijsenaars-Schneider systems through symplectic reduction, providing a geometric interpretation that clarifies and simplifies their understanding.

Contribution

It offers a new geometric perspective on the dualities using symplectic reduction of cotangent bundles of specific groups, clarifying the origin of these dualities.

Findings

Duality relations arise from symplectic reductions of cotangent bundles.

Provides a geometric interpretation that simplifies previous proofs.

Enhances understanding of the duality between particle systems.

Abstract

Besides its usual interpretation as a system of $n$ indistinguishable particles moving on the circle, the trigonometric Sutherland system can be viewed alternatively as a system of distinguishable particles on the circle or on the line, and these 3 physically distinct systems are in duality with corresponding variants of the rational Ruijsenaars-Schneider system. We explain that the 3 duality relations, first obtained by Ruijsenaars in 1995, arise naturally from the Kazhdan-Kostant-Sternberg symplectic reductions of the cotangent bundles of the group U(n) and its covering groups $U(1) \times SU(n)$ and ${\mathbb R}\times SU(n)$, respectively. This geometric interpretation enhances our understanding of the duality relations and simplifies Ruijsenaars' original direct arguments that led to their discovery.