The Ruijsenaars Self-Duality Map as a Mapping Class Symplectomorphism
This is a brief review of the main results of our paper [Nucl. Phys. B 860, 464–515 (2012)] 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.
KeywordsPoisson Bracket Symplectic Manifold Mapping Class Group Reduce Phase Space Flat Connection
We wish to thank the organizers, V. Dobrev in particular, for the pleasant atmosphere that we enjoyed at the LT-9 workshop in Varna. This work was supported in part by the Hungarian Scientific Research Fund (OTKA) under the grant K 77400.
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