Communications in Number Theory and Physics

Volume 12 (2018)

Number 2

Quantizing Weierstrass

Pages: 253 – 303

DOI: http://dx.doi.org/10.4310/CNTP.2018.v12.n2.a2

Authors

Vincent Bouchard (Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, AB, Canada)

Nitin K. Chidambaram (Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, AB, Canada)

Tyler Dauphinee (Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, AB, Canada)

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 differential operators that annihilate the perturbative and non-perturbative wave-functions. In particular, for the non-perturbative wave-function, we prove, up to order $\hslash^5$, that the differential operator is a quantum curve. As a side result, we obtain an infinite sequence of identities relating $A$-cycle integrals of elliptic functions and quasimodular forms.

Full Text (PDF format)

Received 5 October 2016

Accepted 24 February 2018