Quantizing Weierstrass

TL;DR

This paper explores the relationship between topological recursion and quantum curves for elliptic spectral curves, constructing quantizations that connect to matrix models and revealing new identities involving elliptic functions and modular forms.

Contribution

It introduces a method to quantize genus one spectral curves associated with Weierstrass equations, linking quantum curves to matrix model properties and elliptic function identities.

Findings

Quantum curves annihilate wave-functions up to order hbar^5.

Established identities between A-cycle integrals of elliptic functions and quasi-modular forms.

Connected topological recursion with quantum curve properties for elliptic spectral curves.

Abstract

We study the connection between the Eynard-Orantin topological recursion and quantum curves for the family of genus one spectral curves given by the Weierstrass equation. We construct quantizations of the spectral curve that annihilate the perturbative and non-perturbative wave-functions. In particular, for the non-perturbative wave-function, we prove, up to order hbar^5, that the quantum curve satisfies the properties expected from matrix models. As a side result, we obtain an infinite sequence of identities relating A-cycle integrals of elliptic functions and quasi-modular forms.