Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus

TL;DR

This paper analyzes the monodromy dependence of isomonodromic tau functions on a torus with multiple singularities, revealing their symplectic structure and providing explicit formulas using Fredholm determinants.

Contribution

It introduces a method to compute the monodromy dependence of tau functions on a torus, connecting it with symplectic geometry and extending genus zero results.

Findings

Explicit Fredholm determinant representation of tau functions.

Identification of the tau function's exterior logarithmic derivative as a generating function.

Recovery of genus zero results using the new techniques.

Abstract

We compute the monodromy dependence of the isomonodromic tau function on a torus with $n$ Fuchsian singularities and $SL(N)$ residue matrices by using its explicit Fredholm determinant representation. We show that the exterior logarithmic derivative of the tau function defines a closed one-form on the space of monodromies and times, and identify it with the generating function of the monodromy symplectomorphism. As an illustrative example, we discuss the simplest case of the one-punctured torus in detail. Finally, we show that previous results obtained in the genus zero case can be recovered in a straightforward manner using the techniques presented here.