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

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.

83 pages; JHEP 2012

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• Open string amplitudes and large order behavior in topological string theory
• Remodeling the B-model
• Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach
• Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry
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Cited by in corpus (7)

• Moments of the Gaussian $\beta$ Ensembles and the large-$N$ expansion of the densities
• Moments of the eigenvalue densities and of the secular coefficients of $\beta$-ensembles
• $b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral
• Loop Equation Analysis of the Circular $\beta$ Ensembles
• Two $\beta$-ensemble realization of $\beta$-deformed WLZZ models
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• $\beta$-WLZZ models from $\beta$-ensemble integrals directly