On solutions of the Schlesinger equations in terms of theta-functions

Abstract

The Schlesinger equations (see [18]) arise in the context of the following Riemann-Hilbert (inverse monodromy) problem: For an arbitrary g ∈ N and distinct 2g + 2 points λj ∈ C, construct a function Ψ(λ): CP1 {λ1, . . . , λ2g+2} → SL(2,C) which has the following properties: (1) Ψ(∞) = I; (2) Ψ(λ) is holomorphic for all λ ∈ CP1 {λ1, . . . , λ2g+2}; (3) Ψ(λ) has regular singular points at λ = λj, j = 1, . . . ,2g + 2, with given monodromy matrices, Mj ∈ SL(2,C). In the case where the monodromy matrices are independent of the parameters λ1, . . . , λ2g+2, the function Ψ ≡ Ψ(λ) solves the matrix differential equation