Singular ZN-curves and the Riemann-Hilbert problem

Abstract

We are solving the classical Riemann-Hilbert problem of rank N > 1 on the extended complex plane punctured in 2m+2 points, for N × N quasi-permutation monodromy matrices. The Riemann-Hilbert problem is solved in terms of the Szegökernel of a certain Riemann surface branched over the given 2m+2 points. The monodromy group of this Riemann surface is determined from the quasi-permutation monodromy matrices of the Riemann-Hilbert problem by setting all its nonzero entries equal to one. In our case, the monodromy group of the Riemann surface turns out to be the cyclic subgroup ZN of the symmetric group SN. This fact enables us to write the matrix entries of the solution of the N × N Riemann-Hilbert problem as a product of an algebraic function and θ-function quotients. The algebraic function is related to the Szegö kernel with zero characteristics. From the solution of the Riemann-Hilbert problem we automatically obtain a particular solution of the Schlesinger system. The τ-function of the Schlesinger system is computed explicitly in terms of θ-functions and the Bergmann projective connection of the Riemann surface. Finally, we study in detail the solution of the rank 3 problem with four singular points. In this case, the Riemann surface associated to the problem is a 2-sheeted cover of two elliptic curves which are 3-isogenous. As a result, the corresponding solution of the Riemann-Hilbert problem and the Schlesinger system is given in terms of Jacobi's ϑ-function.