Genus one contribution to free energy in hermitian two-matrix model

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 singular metric over spectral curve. 1 Two-matrix models: introduction In this paper we study the partition function of multi-cut two-matrix model: ZN ≡ e 2F := ∫ dM1dM2e −Ntr{V1(M1)+V2(M2)−M1M2} (1.1) where the integral is taken over all independent entries of two hermitian matrices M1 and M2 such that the eigenvalues of M1 are concentrated over a finite set of intervals (cuts) with given filling fractions. This integral is to be understood as a formal asymptotic series in N and in the coefficients of the two potentials V1 and V2. As a formal series, the questions of convergence of the matrix integral is irrelevant, and the model can be extended to matrices with eigenvalues constrained on contours in the complex plane. Such asymptotic series play an important role in physics, as generating functions of statistical physics on random discretized polygonal surfaces, i.e. a simplified model of euclidean 2D quantum gravity [3, 6]. The large N expansion F = ∑∞ G=0 N −2GFG (N is the matrix size), called topological expansion, is one of the cornerstones of the theory, since FG has the meaning of generating function for random discretized polygonal surfaces of genus G. Double scaling limits of these models correspond to statistical physics models on continuous surfaces, with conformal invariance properties. Matrix models thus provide realizations of minimal (p, q) conformal models. The 1-matrix model was shown to correspond to pure gravity (i.e. q = 2), and the 2-matrix model was introduced as it produces all (p, q) minimal models. Recently, the interest in large N matrix models was renewed as it was understood [27], that the large N free energy of matrix models is the low energy effective action for some string theories. The computation of 1/N2 expansion for both one-matrix and two-matrix models is based on the loop equations, which was first derived for 1-matrix 1-cut in [8], then for 1-matrix 2-cuts in [9, 10], and 1 recently derived in [12, 13, 11] for 1-matrix model multicut, and in [14, 15] for two-matrix model 1 and 2-cuts. Here, we will extend the results of [14, 15] for an arbitrary number of cuts i.e. for an arbitrary (up to maximal) genus of the spectral curve. Writing down polynomials V1 and V2 in the form V1(x) = d1+1 ∑ k=1 uk k x , V2(y) = d2+1 ∑