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

L Fehér (1) , C Klimcik (1)

1 I2M - Institut de Mathématiques de Marseille

The Delzant theorem of symplectic topology is used to derive the completely integrable compacti-fied Ruijsenaars–Schneider III b system from a quasi-Hamiltonian reduction of the internally fused double SU(n) × 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.

Domaines: Physique mathématique [math-ph]

L Fehér, C Klimcik. Self-duality of the compactified Ruijsenaars–Schneider system from quasi-Hamiltonian reduction. Nuclear Physics B, 2012, B 860, pp.464. ⟨10.1016/j.nuclphysb.2012.03.005⟩. ⟨hal-01267532⟩