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

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

2003-06-25

arxiv.org/abs/math-ph/0306061v1

Physics

Mathematical Physics

submitted to Math.Annalen 08.2002

Scientific paper

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.

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