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

Abstract:
We solve an arbitrary matrix Riemann-Hilbert (inverse monodromy) problem with quasi-permutation monodromy representation outside of a divisor in the space of monodromy data. The 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 complex plane. The solution is given in terms of a generalization of Szego kernel on the branched covering. The links between corresponding
tau-function, determinants of Cauchy-Riemann operator, Liouville action and G-function of Frobenius manifolds are outlined.