Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices

TL;DR

This paper presents a method to solve matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices using generalized Szeg"o kernels, leading to new solutions of the Schlesinger system and explicit tau-functions.

Contribution

It introduces a novel approach to solve inverse monodromy problems with quasi-permutation monodromy, expanding the class of solvable cases and providing explicit tau-function formulas.

Findings

Explicit solutions for Riemann-Hilbert problems with quasi-permutation monodromy.

Construction of new solutions to the Schlesinger system.

Calculation of the isomonodromy tau-function up to a constant factor.

Abstract

In this paper we solve an arbitrary matrix Riemann-Hilbert (inverse monodromy) problem with quasi-permutation monodromy representations outside of a divisor in the space of monodromy data. This divisor is characterized in terms of the theta-divisor on the Jacobi manifold of an auxiliary compact Riemann surface realized as an appropriate branched covering of $\CP1$ . The solution is given in terms of a generalization of Szeg\"o kernel on the Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. The isomonodromy tau-function of these solutions is computed up to a nowhere vanishing factor independent of the elements of monodromy matrices.