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

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

2011-01-10

arxiv.org/abs/1101.1759v4

Physics

Mathematical Physics

Final version to appear in Nuclear Physics B, with simplified proof of Theorem 1, 56 pages

Scientific paper

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.

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