Communications in Number Theory and Physics
Volume 12 (2018)
Pages: 253 – 303
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.
Received 5 October 2016
Accepted 24 February 2018