Matrix Riemann-Hilbert problems related to branched coverings of $\CP1$
Abstract
Description
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 Szegö 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.
Minor misprints are corrected. To appear in "Operator Theory: Advances and Application", Proceedings of the Summer School on Factorization and Integrable Systems, Algarve, September 6-9, 2000. Ed by I.Gohberg, A. F. dos Santos and N.Manojlovic, Birkhauser, Boston, 2002
Minor misprints are corrected. To appear in "Operator Theory: Advances and Application", Proceedings of the Summer School on Factorization and Integrable Systems, Algarve, September 6-9, 2000. Ed by I.Gohberg, A. F. dos Santos and N.Manojlovic, Birkhauser, Boston, 2002