Isomonodromic deformations and Hurwitz spaces

TL;DR

This paper explores the connection between isomonodromic deformations, Riemann-Hilbert problems, and Hurwitz spaces, revealing new relationships involving tau-functions, Szeg"o kernels, and divisors on Riemann surfaces.

Contribution

It introduces solutions to Riemann-Hilbert problems with quasi-permutation monodromy matrices using Szeg"o kernels and links tau-functions to determinants of Cauchy-Riemann operators.

Findings

Explicit solutions to specific Riemann-Hilbert problems.

Established relationship between tau-function and Cauchy-Riemann operator determinant.

Connected theta-divisor with Malgrange's divisor in Schlesinger equations.

Abstract

A class of Riemann-Hilbert problems corresponding to quasi-permutation monodromy matrices is solved in terms of Szeg\"o kernel on auxiliary Riemann surfaces. The tau-function of Schlesinger system turns out to be closely related to determinant of Cauchy-Riemann operator. The link between theta-divisor and Malgrange's divisor in the theory of Schlesinger equations is established.