Isomonodromic Tau-Function of Hurwitz Frobenius Manifolds and Its Applications

In this work we find the isomonodromic (Jimbo-Miwa) tau-function corresponding to Frobenius manifold structures on Hurwitz spaces. We discuss several applications of this result. First, we get an explicit expression for the G-function (solution of Getzler’s equation) of the Hurwitz Frobenius manifolds. Second, in terms of this tau-function we compute the genus one correction to the free energy of hermitian two-matrix model. Third, we find the Jimbo-Miwa tau-function of an arbitrary Riemann-Hilbert problem with quasi-permutation monodromy matrices. Finally, we get a new expression (analog of genus one Ray-Singer formula) for the determinant of Laplace operator in the Poincaré metric on Riemann surfaces of an arbitrary genus. MSC 1991: 53D45, 34M55 Short title: “Tau-function of Hurwitz Frobenius manifolds”