An Introduction to Homological Mirror Symmetry and the Case of Elliptic Curves

TL;DR

This paper introduces homological mirror symmetry through the example of elliptic curves, establishing an equivalence between algebraic and symplectic categories, and provides background on the broader Calabi-Yau case.

Contribution

It constructs an explicit equivalence for elliptic curves, making homological mirror symmetry more accessible and offering foundational background for the general Calabi-Yau case.

Findings

Established an equivalence between derived categories and Fukaya categories for elliptic curves

Provided accessible exposition on homological mirror symmetry concepts

Laid groundwork for understanding the conjecture in Calabi-Yau contexts

Abstract

Here we carefully construct an equivalence between the derived category of coherent sheaves on an elliptic curve and a version of the Fukaya category on its mirror. This is the most accessible case of homological mirror symmetry. We also provide introductory background on the general Calabi-Yau case of The Homological Mirror Symmetry Conjecture.