15 May 2018 Monodromy dependence and connection formulae for isomonodromic tau functions
A. R. Its, O. Lisovyy, A. Prokhorov
Duke Math. J. 167(7): 1347-1432 (15 May 2018). DOI: 10.1215/00127094-2017-0055


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.


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A. R. Its. O. Lisovyy. A. Prokhorov. "Monodromy dependence and connection formulae for isomonodromic tau functions." Duke Math. J. 167 (7) 1347 - 1432, 15 May 2018. https://doi.org/10.1215/00127094-2017-0055


Received: 27 November 2016; Revised: 17 August 2017; Published: 15 May 2018
First available in Project Euclid: 9 March 2018

zbMATH: 06892361
MathSciNet: MR3799701
Digital Object Identifier: 10.1215/00127094-2017-0055

Primary: 33E17
Secondary: 34E05

Keywords: connection problem , isomonodromy , Painlevé equations , Riemann–Hilbert method , tau function

Rights: Copyright © 2018 Duke University Press


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Vol.167 • No. 7 • 15 May 2018
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