# Generating functions of planar polygons from homological mirror symmetry of elliptic curves

## TL;DR

This paper explores the generating functions of planar polygons linked to elliptic curves' mirror symmetry, expressing them through special functions and analyzing their mathematical properties and geometric significance.

## Contribution

It introduces explicit formulas for these generating functions using theta and mock theta functions, and investigates their Jacobi properties and singularities.

## Key findings

- Generating functions expressed via Jacobi theta and mock theta functions.
- Identification of (mock) Jacobi properties of the generating functions.
- Analysis of special values and singularities with geometric implications.

## Abstract

We study generating functions of certain shapes of planar polygons arising from homological mirror symmetry of elliptic curves. We express these generating functions in terms of rational functions of the Jacobi theta function and Zwegers' mock theta function and determine their (mock) Jacobi properties. We also analyze their special values and singularities, which are of geometric interest as well.