Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction

TL;DR

This paper derives the self-duality of the compactified Ruijsenaars-Schneider system using quasi-Hamiltonian reduction, revealing new symplectic and mapping class group symmetries and interpretations in terms of flat SU(n) connections.

Contribution

It provides a rigorous derivation of the Ruijsenaars self-duality from quasi-Hamiltonian reduction and introduces new presentations of the duality map via symplectomorphisms.

Findings

The self-duality map exchanges action-variables and particle-positions.

Mapping class group actions induce automorphisms on the system.

The results connect the system to flat SU(n) connections on a one-holed torus.

Abstract

The Delzant theorem of symplectic topology is used to derive the completely integrable compactified Ruijsenaars-Schneider III(b) system from a quasi-Hamiltonian reduction of the internally fused double SU(n) x SU(n). In particular, the reduced spectral functions depending respectively on the first and second SU(n) factor of the double engender two toric moment maps on the III(b) phase space CP(n-1) that play the roles of action-variables and particle-positions. A suitable central extension of the SL(2,Z) mapping class group of the torus with one boundary component is shown to act on the quasi-Hamiltonian double by automorphisms and, upon reduction, the standard generator S of the mapping class group is proved to descend to the Ruijsenaars self-duality symplectomorphism that exchanges the toric moment maps. We give also two new presentations of this duality map: one as the composition of two Delzant symplectomorphisms and the other as the composition of three Dehn twist symplectomorphisms realized by Goldman twist flows. Through the well-known relation between quasi-Hamiltonian manifolds and moduli spaces, our results rigorously establish the validity of the interpretation [going back to Gorsky and Nekrasov] of the III(b) system in terms of flat SU(n) connections on the one-holed torus.