Singular Z_N curves, Riemann-Hilbert problem and modular solutions of the Schlesinger equation

TL;DR

This paper solves a classical Riemann-Hilbert problem with quasi-permutation monodromy matrices using Riemann surfaces with Z_N symmetry, leading to explicit solutions of the Schlesinger system and related modular functions.

Contribution

It introduces a novel method to explicitly solve the Riemann-Hilbert problem for rank N>1 with Z_N symmetry, connecting it to theta functions and modular solutions of the Schlesinger equation.

Findings

Explicit solutions for the Riemann-Hilbert problem using theta functions.

Derivation of the tau-function of the Schlesinger system.

Detailed analysis of the rank 3 case with four singular points.

Abstract

We are solving the classical Riemann-Hilbert problem of rank N>1 on the extended complex plane punctured in 2m+2 points, for NxN quasi-permutation monodromy matrices. Following Korotkin we solve the Riemann-Hilbert problem in terms of the Szego kernel of certain Riemann surfaces branched over the given 2m+2 points. These Riemann surfaces are constructed from a permutation representation of the symmetric group S_N to which the quasi-permutation monodromy representation has been reduced. The permutation representation of our problem generates the cyclic subgroup Z_N. For this reason the corresponding Riemann surfaces of genus N(m-1) have Z_N symmetry. This fact enables us to write the matrix entries of the solution of the NxN Riemann-Hilbert problem as a product of an algebraic function and theta-function quotients. The algebraic function turns out to be related to the Szego kernel with zero characteristics. From the solution of the Riemann- Hilbert problem we automatically obtain a particular solution of the Schlesinger system. The tau-function of the Schlesinger system is computed explicitly. The rank 3 problem with four singular points (0,t,1,\infty) is studied in detail. The corresponding solution of the Riemann-Hilbert problem and the Schlesinger system is given in terms of Jacobi's theta-function with modulus T=T(t), Im(T)>0. The function T=T(t) is invertible if it belongs to the Siegel upper half space modulo the subgroup \Gamma_0(3) of the modular group. The inverse function t=t(T) generates a solution of a general Halphen system.