Monodromy dependence and connection formulae for isomonodromic tau functions

TL;DR

This paper extends the Jimbo-Miwa-Ueno form to the full monodromy data space, enabling explicit computation of connection constants for isomonodromic tau functions, including Painlevé VI and II cases, confirming conjectures.

Contribution

It introduces a new method based on Bertola's generalization to compute connection constants explicitly for isomonodromic tau functions.

Findings

Computed the connection constant for Painlevé VI tau function, confirming a conjecture.

Evaluated constant factors in Painlevé II tau function asymptotics.

Extended the framework to non-Fuchsian systems for connection constant evaluation.

Abstract

We discuss an extension of the Jimbo-Miwa-Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for generic Painlev\'e VI tau function. The result proves the conjectural formula for this constant proposed in \cite{ILT13}. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of Painlev\'e II tau function.