On the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models

TL;DR

This paper explores the geometric duality between the hyperbolic Sutherland and rational Ruijsenaars-Schneider integrable models, revealing a gauge transformation as the core of their symplectic duality.

Contribution

It provides a geometric interpretation of the duality as a gauge transformation connecting cross sections in a symplectic reduction framework.

Findings

Duality between the models is interpreted as a gauge transformation.

Symplectic reduction links Hamiltonians to spectral invariants.

Geometric perspective clarifies the duality structure.

Abstract

We consider two families of commuting Hamiltonians on the cotangent bundle of the group GL(n,C), and show that upon an appropriate single symplectic reduction they descend to the spectral invariants of the hyperbolic Sutherland and of the rational Ruijsenaars-Schneider Lax matrices, respectively. The duality symplectomorphism between these two integrable models, that was constructed by Ruijsenaars using direct methods, can be then interpreted geometrically simply as a gauge transformation connecting two cross sections of the orbits of the reduction group.