Free Energy of the Two-Matrix Model/dToda Tau-Function

TL;DR

This paper derives a comprehensive integral formula for the free energy of the two-matrix model with polynomial potentials, linking it to the tau-function of the dispersionless Toda hierarchy and extending previous results to higher genus spectral curves.

Contribution

It introduces a generalized integral formula for the free energy of the two-matrix model applicable to arbitrary genus spectral curves, expanding prior work limited to genus zero.

Findings

Integral formula for free energy of two-matrix model

Connection to tau-function of dispersionless Toda hierarchy

Extension to spectral curves of arbitrary genus

Abstract

We provide an integral formula for the free energy of the two-matrix model with polynomial potentials of arbitrary degree (or formal power series). This is known to coincide with the tau-function of the dispersionless two--dimensional Toda hierarchy. The formula generalizes the case studied by Kostov, Krichever, Mineev-Weinstein, Wiegmann, Zabrodin and separately Takhtajan in the case of conformal maps of Jordan curves. Finally we generalize the formula found in genus zero to the case of spectral curves of arbitrary genus with certain fixed data.