Matrix Riemann-Hilbert problems related to branched coverings of $\CP1$

TL;DR

This paper solves a class of Riemann-Hilbert problems linked to branched coverings of the complex projective line, providing explicit solutions via Szeg"o kernels and exploring their relation to tau-functions and Laplacian determinants.

Contribution

It introduces a new approach to solving Riemann-Hilbert problems with quasi-permutation monodromy groups using Szeg"o kernels, and connects these solutions to tau-functions and geometric divisors.

Findings

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

Calculation of tau-functions and description of Malgrange divisor.

Relationship between tau-function and Laplacian determinant on Riemann surfaces.

Abstract

In these notes we solve a class of Riemann-Hilbert (inverse monodromy) problems with quasi-permutation monodromy groups which correspond to non-singular branched coverings of $\CP1$. The solution is given in terms of Szeg\"o kernel on the underlying Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. We present some results on explicit calculation of the corresponding tau-function, and describe divisor of zeros of the tau-function (so-called Malgrange divisor) in terms of the theta-divisor on the Jacobi manifold of the Riemann surface. We discuss the relationship of the tau-function to determinant of Laplacian operator on the Riemann surface.