The Ruijsenaars self-duality map as a mapping class symplectomorphism

Details The Ruijsenaars self-duality map as a mapping class symplectomorphism The Ruijsenaars self-duality map as a mapping class symplectomorphism

2012-03-15

arxiv.org/abs/1203.3300v1

Physics

Mathematical Physics

Contribution to the proceedings of the workshop `Lie Theory and its Applications in Physics IX' (Varna, June 2011), 13 pages

Scientific paper

This is a brief review 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.

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