Monodromy dependence and connection formulae for isomonodromic tau functions


Alexander Its

10:00:00 - 10:50:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

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 equation 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 generic isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works by calculating the connection constants for generic Painlevé VI and Painlevé III tau functions. The result proves the conjectural formulae for these constants earlier proposed by N. Iorgov, O. Lisovyy, and Yu. Tykhyy (PVI) and by O. Lisovyy, Y. Tykhyy, and the speaker (PIII) with the help of the recently discovered connection of the Painlevé tau-functions with the Virasoro conformal blocks. The conformal block approach will be also outlined. The talk is based on the joint works with O. Lisovyy, Y. Tykhyy and A. Prokhorov.