%0 Journal Article
%T Self-duality of the compactified Ruijsenaars–Schneider system from quasi-Hamiltonian reduction
%+ Institut de Mathématiques de Marseille (I2M)
%A Fehér, L
%A Klimcik, C
%< avec comité de lecture
%@ 0550-3213
%J Nuclear Physics B
%I Elsevier
%V B 860
%P 464
%8 2012
%D 2012
%R 10.1016/j.nuclphysb.2012.03.005
%Z Mathematics [math]/Mathematical Physics [math-ph]Journal articles
%X 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.
%G English
%2 https://hal.science/hal-01267532/document
%2 https://hal.science/hal-01267532/file/FKNPB.pdf
%L hal-01267532
%U https://hal.science/hal-01267532
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ INSMI
%~ I2M
%~ I2M-2014-
%~ TDS-MACS