Mirror symmetry of elliptic curves and Ising model

TL;DR

This paper explores the differential equations of mirror symmetry for elliptic curves, linking them to theta functions, Landau-Ginzburg potentials, and the Ising model, providing explicit formulas and characterizations.

Contribution

It characterizes the differential equations for mirror symmetry of elliptic curves and connects the Landau-Ginzburg potential with the Ising model's spectral curve using theta functions.

Findings

Characterization of ODEs for mirror maps.

Expression of mirror correspondence via theta constants.

Identification of Landau-Ginzburg potential with Ising model spectral curve.

Abstract

We study the differential equations governing mirror symmetry of elliptic curves, and obtain a characterization of the ODEs which give rise to the integral ${\bf q}$-expansion of mirror maps. Through theta function representation of the defining equation, we express the mirror correspondence in terms of theta constants. By investigating the elliptic curves in $X_9$-family, the identification of the Landau-Ginzburg potential with the spectral curve of Ising model is obtained. Through the Jacobi elliptic function parametrization of Boltzmann weights in the statistical model, an exact Jacobi form-like formula of mirror map is described .