We compute an the genus 1 correction to free energy of Hermitian two-matrix model in terms of theta-functions associated to spectral curve arising in large N limit. We discuss the relationship of this expression to isomonodromic tau-function, Bergmann tau-function on Hurwitz spaces, G-function of Frobenius manifolds and determinant of Laplacian in a… (More)
The semisimple Frobenius manifolds related to the Hurwitz spaces H g,N (k 1 ,. .. , k l) are considered. We show that the corresponding isomonodromic tau-function τ I coincides with (−1/2)-power of the Bergmann tau-function which was introduced in a recent work by the authors . This enables us to calculate explicitly the G-function of Frobenius manifolds… (More)
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… (More)
Contents 1 Introduction 2 2 Tau-function on spaces of Abelian differentials over Riemann surfaces 7 2. 1 4 Variational formulas for determinants of Laplacians in Strebel metrics of finite volume 34 4.
Using the loop equations we find an explicit expression for genus 1 correction in hermitian two-matrix model in terms of holomorphic objects associated to spectral curve arising in large N limit. Our result generalises known expression for F 1 in hermitian one-matrix model. We discuss the relationship between F 1 , Bergmann tau-function on Hurwitz spaces,… (More)
The main result of this work is a computation of the Bergmann tau-function on Hurwitz spaces in any genus. This allows to get an explicit formula for the G-function of Frobenius manifolds associated to arbitrary Hurwitz spaces, get a new expression for determinant of Laplace operator in Poincaré metric on Riemann surfaces of arbitrary genus, and compute… (More)
Let w be an Abelian differential on compact Riemann surface of genus g ≥ 1. We obtain an explicit holomorphic factorization formula for ζ-regularized determinant of the Laplacian in flat conical metrics with trivial holonomy |w| 2 , generalizing the classical Ray-Singer result in g = 1.
In this paper, we introduce a new class of integrable systems, naturally associated to Hurwitz spaces (spaces of meromorphic functions over Riemann surfaces). The critical values of the meromorphic functions play the role of "times". Our systems give a natural generalization of the Ernst equation; in genus zero, they realize the scheme of deformation of… (More)
The isomonodromic tau-function for the Hurwitz spaces of branched coverings of genus zero and one are constructed explicitly. Such spaces may be equipped with the structure of a Frobenius manifold and this introduces a flat coordinate system on the manifold. The isomonodromic tau-function, and in particular the associated G-function, are rewritten in these… (More)
We study extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of compact genus two Riemann surfaces. By a combination of analytical and numerical methods we identify four non-degenerate critical points of this function and compute the signature of the Hessian at these points. The curve with the maximal number of… (More)