A-polynomial, B-model, and Quantization

TL;DR

This work establishes a universal framework to quantize classical algebraic curves via the A-polynomial, linking the Eynard–Orantin topological recursion to a Schrödinger-like quantum curve $\widehat{A}$ and a wave-function $Z$ satisfying $\widehat{A}Z=0$. By expressing $Z$ as $Z=\exp\big(\frac{1}{\hbar}\sum S_k \hbar^k\big)$ and developing a hierarchy of differential operators $\mathfrak{D}_r$, the authors derive the first quantum correction $\widehat{A}_1$ explicitly and show how higher corrections are determined from the leading data; they also formulate a K-theory criterion for when a curve is quantizable. The paper applies this machinery to knots, 3-manifolds, and toric Calabi–Yau geometries, including the Airy, $c=1$ models, the tetrahedron/${\mathbb C}^3$, and the conifold, demonstrating consistent quantum curves across multiple parametrizations and framings. The results illuminate the deep connections between quantum curves, knot invariants, and topological string theory, with practical computational schemes for obtaining $\widehat{A}$ from the classical curve and its TR data. Overall, the framework provides a systematic, largely universal method to derive quantum curves and their corrections from classical algebraic data in diverse physical settings.

Abstract

Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly in a suitable semi-classical limit, as $\hbar \to 0$, and becomes non-commutative or "quantum" away from this limit. For a classical curve defined by the zero locus of a polynomial $A(x,y)$, we provide a construction of its non-commutative counterpart $\hat{A} (\hat x, \hat y)$ using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing $\hat{A}$ that, surprisingly, turns out to be much simpler than any of the existent methods. In particular, as a bonus feature of our approach comes a curious observation that, for all curves that come from knots or topological strings, their non-commutative counterparts can be determined just from the first few steps of the topological recursion. We also propose a K-theory criterion for a curve to be "quantizable," and then apply our construction to many examples that come from applications to knots, strings, instantons, and random matrices.

Figures (4)

• Figure 1: A graphical representation of the Eynard-Orantin topological recursion.
• Figure 2: Mirror curve for ${\mathbb C}^3$ geometry.
• Figure 3: An example of mirror curve for a generalized conifold.
• Figure 4: Mirror curve for the conifold geometry.