Curves , Riemann - Hilbert Problem and 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. Our approach is based on the finite gap integration method applied to study the Riemann-Hilbert by Kitaev and Korotkin [1], Deift, Its, Kapaev and Zhou [2] and Korotkin, [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. The monodromy group of these Riemann surfaces is determined from the quasi-permutation monodromy matrices of the Riemann-Hilbert problem by setting all their non-zero entries equal to one. In our case, the monodromy group of the Riemann surfaces turns out to be the cyclic subgroup Z N of the symmetric group S N and for this reason these 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 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 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 (λ 1 , λ 2 , λ 3 , ∞). 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), t = λ 2 −λ 1 λ 3 −λ 1 and Im T > 0. The inverse function t = t(T) is automorphic under the action of the subgroup Γ 0 (3) of the modular group and generates a solution of a general Halphen system. The analytic counterpart of this picture is given by Goursat's …