Z N Curves , Riemann - Hilbert Problem and Modular Solutions of the Schlesinger Equations

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 with 2m(N − 1) parameters. Our approach is based on the finite gap integration method applied to study the Riemann-Hilbert by Deift, Its, Kapaev and Zhou [1] and Kitaev and Korotkin [2, 3]. This permits us to solve the Riemann-Hilbert problem in terms of the Szegö 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 N × N Riemann-Hilbert problem as a product of an algebraic function and θ-function quotients. The algebraic function turns out to be related to the Szegö kernel with zero characteristics. The 2N (m − 1) monodromy parameters are in one to one correspondence with the 2N (m − 1) characteristics of the θ-functions. The symmetry of the problem enables us to show that if two monodromy representations are equivalent up to multiplication by N-th root of unity, then the corresponding θ-characteristics differ only at rational numbers k/N , k = 1,. .. , N − 1. 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 holomorphic projective connection of the Riemann surface. In the course of the computation we also derive Thomae-type formulae for a class of non-singular 1/N-periods. Finally we study in detail the solution of the rank 3 problem with four singular points (0, t, 1, ∞). The corresponding Riemann surface C 3,1 is of genus two branched at the above four points and admits the dihedral group D 3 of automorphisms. This implies that C 3,1 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 with modulus T = T (t), Im T > 0. The function …