The Ruijsenaars Self-Duality Map as a Mapping Class Symplectomorphism

L Fehér , C Klimcik (1)

1 I2M - Institut de Mathématiques de Marseille

This is a brief review 1 of the main results of our paper arXiv:1101.1759 that contains a complete global treatment of the compactified trigonometric Ruijsenaars-Schneider system by quasi-Hamiltonian reduction. Confirming previous conjectures of Gorsky and collaborators, we have rigorously established the interpretation of the system in terms of flat SU (n) connections on the one-holed torus and demonstrated that its self-duality symplectomorphism represents the natural action of the standard mapping class generator S on the phase space. The pertinent quasi-Hamiltonian reduced phase space turned out to be symplectomorphic to the complex projective space equipped with a multiple of the Fubini-Study symplectic form and two toric moment maps playing the roles of particle-positions and action-variables that are exchanged by the duality map. Open problems and possible directions for future work are also discussed.

Domaines: Physique mathématique [math-ph]

L Fehér, C Klimcik. The Ruijsenaars Self-Duality Map as a Mapping Class Symplectomorphism. Lie theory and its applications in physics, Jun 2011, Varna, Bulgaria. pp.423-437, ⟨10.1007/978-4-431-54270-4_30⟩. ⟨hal-01267692⟩