Topological expansion of the Bethe ansatz, and quantum algebraic geometry

TL;DR

This work develops a quantum extension of topological recursion for β-ensembles by promoting the classical spectral curve to a quantum curve defined by the Schrödinger equation ħ^2 ψ'' = U(x). It builds a quantum Riemann surface with generalized genus, A-/B-cycles, first/second/third kind differentials, and a quantum Bergman kernel, all consistent with a Bethe-ansatz vanishing-monodromy condition. The authors formulate a β-topological recursion that yields quantum invariants F_g and correlators W_n^{(g)}, and apply the formalism to matrix models and non-oriented ribbon graphs, including an integration-theoretic treatment via Stokes sectors and WKB limits. This framework unifies loop equations, quantum geometry, and enumerative combinatorics, and points to extensions to higher-order differential systems and deeper links with integrable systems and τ-functions.

Abstract

In this article, we solve the loop equations of the β-random matrix model, in a way similar to what was found for the case of hermitian matrices β=1. For β=1, the solution was expressed in terms of algebraic geometry properties of an algebraic spectral curve of equation y^2=U(x). For arbitrary β, the spectral curve is no longer algebraic, it is a Schroedinger equation ((\hbar\partial)^2-U(x)).ψ(x)=0 where \hbar\propto (\sqrtβ-1/\sqrtβ). In this article, we find a solution of loop equations, which takes the same form as the topological recursion found for β=1. This allows to define natural generalizations of all algebraic geometry properties, like the notions of genus, cycles, forms of 1st, 2nd and 3rd kind, Riemann bilinear identities, and spectral invariants F_g, for a quantum spectral curve, i.e. a D-module of the form y^2-U(x), where [y,x]=\hbar. Also, our method allows to enumerate non-oriented discrete surfaces.

Figures (8)

• Figure 1: Example of sectors for a potential of degree $\deg V=3$, i.e. $d=2$. If $\deg V=d+1$ there are $2d+2$ sectors.
• Figure 2: The zeroes of $\psi$ accumulate near $\infty$ along the half-lines bordering sectors where $\psi_0$ is exponentially large on both sides. In particular, there is no accumulation of zeroes along $L_0$ and $L_{2d+1}$.
• Figure 3: Representation in two sheets of a Riemann surface of genus $1$.
• Figure 4: Picture of the path of integration used for the definition of the kernel $K(x,z)$.
• Figure 5: Classical case $\hbar=0$ of a two sheeted Riemann surface. The branchpoints are paired (in an arbitrary way) to form cuts, and the two sheets are glued along the cuts. Another possibility, is to draw a cut from each branchpoint to $\infty$. The ${\cal A}$-cycles surround pairs of branchpoints in the physical sheet. There are also some degenerate branchpoints, which correspond to cuts of vanishing length.

Theorems & Definitions (42)

• Remark 2.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.1
• Definition 3.1
• Definition 3.2
• Theorem 3.1
• Remark 3.1
• Theorem 3.2