### 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)×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.

Original language | English |
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Pages (from-to) | 464-515 |

Number of pages | 52 |

Journal | Nuclear Physics B |

Volume | 860 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jul 21 2012 |

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

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## Cite this

*Nuclear Physics B*,

*860*(3), 464-515. https://doi.org/10.1016/j.nuclphysb.2012.03.005