## Duke Mathematical Journal

### Monodromy dependence and connection formulae for isomonodromic tau functions

#### 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 the generic Painlevé VI tau function. The result proves the conjectural formula for this constant proposed by Iorgov, Lisovyy, and Tykhyy. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of the Painlevé II tau function.

#### Article information

Source
Duke Math. J., Volume 167, Number 7 (2018), 1347-1432.

Dates
Revised: 17 August 2017
First available in Project Euclid: 9 March 2018

https://projecteuclid.org/euclid.dmj/1520586158

Digital Object Identifier
doi:10.1215/00127094-2017-0055

Mathematical Reviews number (MathSciNet)
MR3799701

Zentralblatt MATH identifier
06892361

Subjects
Primary: 33E17: Painlevé-type functions
Secondary: 34E05: Asymptotic expansions

#### Citation

Its, A. R.; Lisovyy, O.; Prokhorov, A. Monodromy dependence and connection formulae for isomonodromic tau functions. Duke Math. J. 167 (2018), no. 7, 1347--1432. doi:10.1215/00127094-2017-0055. https://projecteuclid.org/euclid.dmj/1520586158

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