Isomonodromic tau-function of Hurwitz Frobenius manifolds and its applications

TL;DR

This paper derives the isomonodromic tau-function for Hurwitz Frobenius manifolds and applies it to compute G-functions, matrix model corrections, Riemann-Hilbert tau-functions, and determinants of Laplace operators on Riemann surfaces.

Contribution

It provides an explicit expression for the tau-function associated with Hurwitz Frobenius manifolds and demonstrates its applications in various mathematical physics contexts.

Findings

Explicit formula for the G-function of Hurwitz Frobenius manifolds.

Computed genus one correction to hermitian two-matrix model free energy.

Derived a new expression for the Laplace operator determinant on Riemann surfaces.

Abstract

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 Poincar\'e metric on Riemann surfaces of an arbitrary genus.