Tau-functions and monodromy symplectomorphisms
Abstract
We derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical -matrix structure. The corresponding symplectic form is shown to be the pullback, under the monodromy map, of a natural symplectic form on the extended monodromy manifold. We show that Fock-Goncharov coordinates are log-canonical for the symplectic form on the extended monodromy manifold. Using these coordinates we define the symplectic potential on the monodromy manifold and interpret the isomonodromic tau-function as the generating function of the monodromy map. This, in particular, solves a recent conjecture by A.Its, O.Lisovyy and A.Prokhorov.
Cite
@article{arxiv.1910.03370,
title = {Tau-functions and monodromy symplectomorphisms},
author = {Marco Bertola and Dmitry Korotkin},
journal= {arXiv preprint arXiv:1910.03370},
year = {2022}
}
Comments
37 pages , 6 figures. This is a significantly extended version of arXiv:1903.09197, and it supersedes it. V2: Major revision V3: minor corrections. V4: further minor corrections. V5: typo corrected