Self-duality of the compactified Ruijsenaars–Schneider system from quasi-Hamiltonian reduction Original Research Article

The Delzant theorem of symplectic topology is used to derive the completely integrable compactified Ruijsenaars–Schneider image system from a quasi-Hamiltonian reduction of the internally fused double image. In particular, the reduced spectral functions depending respectively on the first and second image factor of the double engender two toric moment maps on the image phase space image that play the roles of action-variables and particle-positions. A suitable central extension of the image 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 image system in terms of flat image connections on the one-holed torus.