Department of Mathematics and Statistics,

Concordia University, 7141 Sherbrooke West,

Montreal H4B1R6 Quebec, CANADA

E-mail: korotkin@mathstat.concordia.ca

**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.