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

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.