Abstract
Abstract We write the loop equations for the β 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.
Highlights
Random matrix models have played a very important role in physics and mathematics [9, 26, 36,37,38, 50, 62, 66]
We write the loop equations for the β 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
Summary
Random matrix models have played a very important role in physics and mathematics [9, 26, 36,37,38, 50, 62, 66]. Some progress was made [6] in those angular integrals, and we don’t know yet how to compute angular integrals for arbitrary β, we know already that those angular integrals have to satisfy some differential equations, and this is sufficient to derive loop equations. This is what we do in this article, we derive the loop equations, and solve them perturbatively by expanding in some small parameter
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