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

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• 作者：L. Fehé ; r^a ; ^b ; ^{lfeher@rmki.kfki.hu} ; C. Klimč ; í ; k^c ; ^{klimcik@univmed.fr}
• 刊名：Nuclear Physics B
• 出版年：2012
• 期刊代码：45_05503213
• 类别：ph
• 出版时间：21 July, 2012
• 卷：860
• 期：3
• 页码：464-515
• 文件大小：536 K

摘要

The Delzant theorem of symplectic topology is used to derive the completely integrable compactified Ruijsenaars-Schneider system from a quasi-Hamiltonian reduction of the internally fused double . In particular, the reduced spectral functions depending respectively on the first and second factor of the double engender two toric moment maps on the phase space that play the roles of action-variables and particle-positions. A suitable central extension of the 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 system in terms of flat connections on the one-holed torus.