Tau-functions and monodromy symplectomorphisms

TL;DR

This paper introduces a new Hamiltonian framework for Schlesinger equations using dynamical r-matrices, revealing the symplectic structure of monodromy data and connecting the tau-function to generating functions, thus solving a recent conjecture.

Contribution

It develops a novel Hamiltonian formulation of Schlesinger equations with a dynamical r-matrix structure and interprets the tau-function as a generating function of the monodromy map.

Findings

Derived a Hamiltonian formulation using dynamical r-matrix

Identified Fock-Goncharov coordinates as log-canonical

Connected the tau-function to the generating function of the monodromy map

Abstract

We derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical $r$-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.