Loop equations and topological recursion for the arbitrary-$\beta$ two-matrix model

TL;DR

This paper develops loop equations and a topological recursion method for the arbitrary-beta two-matrix model, revealing a quantum spectral curve that satisfies a Bethe ansatz and Baxter TQ relation, advancing understanding of matrix models.

Contribution

It introduces a novel topological recursion framework for the beta two-matrix model, extending classical methods to a quantum spectral curve setting.

Findings

Spectral curve is a differential operator (quantum spectral curve).

Quantum spectral curve satisfies a Bethe ansatz.

Spectral curve relates to Baxter TQ relation.

Abstract

We write the loop equations for the $\beta$ two-matrix model, and we propose a topological recursion algorithm to solve them, order by order in a small parameter. We find that to leading order, the spectral curve is a "quantum" spectral curve, i.e. it is given by a differential operator (instead of an algebraic equation for the hermitian case). Here, we study the case where that quantum spectral curve is completely degenerate, it satisfies a Bethe ansatz, and the spectral curve is the Baxter TQ relation.