The dependence on the monodromy data of the isomonodromic tau function

TL;DR

This paper explores how the isomonodromic tau function depends on monodromy data, providing variational formulas and generalizations applicable to Riemann-Hilbert problems, with examples including Painleve II and Toeplitz determinants.

Contribution

It offers a simplified, more general description of the tau function's dependence on monodromy data, including variational formulas and the concept of discrete Schlesinger transformations.

Findings

Derived variational formulas for the tau function with respect to monodromy data

Introduced discrete Schlesinger transformations and their impact on RHP solutions

Provided examples with Painleve II and Toeplitz determinants

Abstract

[Note: important Corrigendum now available at arXiv:1601.04790] The isomonodromic tau function defined by Jimbo-Miwa-Ueno vanishes on the Malgrange's divisor of generalized monodromy data for which a vector bundle is nontrivial, or, which is the same, a certain Riemann-Hilbert problem has no solution. In their original work, Jimbo, Miwa, Ueno did not derive the dependence on the (generalized) monodromy data (i.e. monodromy representation and Stokes' parameters). We fill the gap by providing a (simpler and more general) description in which all the parameters of the problem (monodromy-changing and monodromy-preserving) are dealt with at the same level. We thus provide variational formulae for the isomonodromic tau function with respect to the (generalized) monodromy data. The construction applies more generally: given any (sufficiently well-behaved) family of Riemann-Hilbert problems (RHP) where the jump matrices depend arbitrarily on deformation parameters, we can construct a one-form Omega (not necessarily closed) on the deformation space (Malgrange's differential), defined off Malgrange's divisor. We then introduce the notion of discrete Schlesinger transformation: it means that we allow the solution of the RHP to have poles (or zeros) at prescribed point(s). Even if Omega is not closed, its difference evaluated along the original solution and the transformed one, is shown to be the logarithmic differential (on the deformation space) of a function. As a function of the position of the points of the Schlesinger transformation, yields a natural generalization of Sato formula for the Baker-Akhiezer vector even in the absence of a tau function, and it realizes the solution of the RHP as such BA vector. Some exemples (Painleve' II and finite Toplitz/Hankel determinants) are provided.