6 research outputs found
On the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models
We consider two families of commuting Hamiltonians on the cotangent bundle of
the group GL(n,C), and show that upon an appropriate single symplectic
reduction they descend to the spectral invariants of the hyperbolic Sutherland
and of the rational Ruijsenaars-Schneider Lax matrices, respectively. The
duality symplectomorphism between these two integrable models, that was
constructed by Ruijsenaars using direct methods, can be then interpreted
geometrically simply as a gauge transformation connecting two cross sections of
the orbits of the reduction group.Comment: 16 pages, v2: comments and references added at the end of the tex
On the spectra of the quantized action-variables of the compactified Ruijsenaars-Schneider system
A simple derivation of the spectra of the action-variables of the quantized
compactified Ruijsenaars-Schneider system is presented. The spectra are
obtained by combining Kahler quantization with the identification of the
classical action-variables as a standard toric moment map on the complex
projective space. The result is consistent with the Schrodinger quantization of
the system worked out previously by van Diejen and Vinet.Comment: Based on talk at the workshop CQIS-2011 (Protvino, Russia, January
2011), 12 page
Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction
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) x 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.Comment: Final version to appear in Nuclear Physics B, with simplified proof
of Theorem 1, 56 page