34 research outputs found
Deformations of circle-valued Morse functions on surfaces
Let be a smooth connected orientable compact surface. Denote by
the space of all Morse functions having no critical
points on the boundary of and such that for every boundary component of
the restriction is either a constant map or a covering
map. Endow with the -topology. In this note the
connected components of are classified. This result extends the
results of S. V. Matveev, V. V. Sharko, and the author for the case of Morse
functions being locally constant on the boundary of .Comment: 8 pages, 4 figure
On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system
We suggest a Hamiltonian formulation for the spin Ruijsenaars–Schneider system in the trigonometric case. Within this interpretation, the phase space is obtained by a quasi-Hamiltonian reduction performed on (the cotangent bundle to) a representation space of a framed Jordan quiver. For arbitrary quivers, analogous varieties were introduced by Crawley-Boevey and Shaw, and their interpretation as quasi-Hamiltonian quotients was given by Van den Bergh. Using Van den Bergh’s formalism, we construct commuting Hamiltonian functions on the phase space and identify one of the flows with the spin Ruijsenaars–Schneider system. We then calculate all the Poisson brackets between local coordinates, thus answering an old question of Arutyunov and Frolov. We also construct a complete set of commuting Hamiltonians and integrate all the flows explicitly