$1/N^2$ correction to free energy in hermitian two-matrix model

TL;DR

This paper derives an explicit formula for the genus 1 correction to the free energy in the hermitian two-matrix model using loop equations, extending known results from the one-matrix case and exploring geometric connections.

Contribution

It provides a new explicit expression for the genus 1 correction in the two-matrix model and relates it to spectral curve geometry and known mathematical functions.

Findings

Derived explicit genus 1 correction formula using spectral curve data

Connected free energy correction to Bergmann tau-function and Frobenius manifold G-function

Extended known results from one-matrix to two-matrix models

Abstract

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, G-function of Frobenius manifolds and determinant of Laplacian over spectral curve.