Matrix RiemannHilbert problems related to branched coverings of $\CP1$
Abstract
In these notes we solve a class of RiemannHilbert (inverse monodromy) problems with quasipermutation monodromy groups which correspond to nonsingular 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 taufunction, and describe divisor of zeros of the taufunction (socalled Malgrange divisor) in terms of the thetadivisor on the Jacobi manifold of the Riemann surface. We discuss the relationship of the taufunction to determinant of Laplacian operator on the Riemann surface.
 Publication:

arXiv eprints
 Pub Date:
 June 2001
 arXiv:
 arXiv:mathph/0106009
 Bibcode:
 2001math.ph...6009K
 Keywords:

 Mathematical Physics;
 Mathematics  Mathematical Physics;
 Primary 35Q15;
 Secondary 30F60;
 32G81
 EPrint:
 Minor misprints are corrected. To appear in "Operator Theory: Advances and Application", Proceedings of the Summer School on Factorization and Integrable Systems, Algarve, September 69, 2000. Ed by I.Gohberg, A. F. dos Santos and N.Manojlovic, Birkhauser, Boston, 2002