Quantizing Weierstrass

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.

40 pages. v2: published version (improved presentation)

References in corpus (5)

• Open string amplitudes and large order behavior in topological string theory
• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Hurwitz numbers, matrix models and enumerative geometry
• Universal scaling limits of matrix models, and (p,q) Liouville gravity
• The SYZ mirror symmetry and the BKMP remodeling conjecture

Cited by in corpus (3)

• 2-parameter $\tau$-function for the first Painlevé equation -Topological recursion and direct monodromy problem via exact WKB analysis-
• Witten-Reshetikhin-Turaev function for a knot in Seifert manifolds
• From CFT to Ramond super-quantum curves