Matrix Riemann-Hilbert problems related to branched coverings of $\CP1$

Details Matrix Riemann-Hilbert problems related to branched coverings of $\CP1$ Matrix Riemann-Hilbert problems related to branched coverings of $\CP1$

2001-06-13

arxiv.org/abs/math-ph/0106009v2

Physics

Mathematical Physics

Minor misprints are corrected. To appear in "Operator Theory: Advances and Application", Proceedings of the Summer School on F

Scientific paper

In these notes we solve a class of Riemann-Hilbert (inverse monodromy) problems with quasi-permutation monodromy groups which correspond to non-singular branched coverings of $\CP1$. The solution is given in terms of Szeg\"o kernel on the underlying Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. We present some results on explicit calculation of the corresponding tau-function, and describe divisor of zeros of the tau-function (so-called Malgrange divisor) in terms of the theta-divisor on the Jacobi manifold of the Riemann surface. We discuss the relationship of the tau-function to determinant of Laplacian operator on the Riemann surface.

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