A remark about mirror symmetry of elliptic curves and generalized complex geometry

TL;DR

This paper demonstrates that for elliptic curves, the isomorphism of generalized complex structures under T-duality aligns with the mirror map described by Polishchuk and Zaslow, linking two perspectives in complex geometry.

Contribution

It establishes the equivalence between the T-duality isomorphism of generalized complex structures and the mirror map for elliptic curves.

Findings

T-duality and mirror symmetry coincide for elliptic curves.

Generalized complex structures are preserved under this isomorphism.

The result bridges two different approaches to mirror symmetry in elliptic curves.

Abstract

In this short note we prove that in the case of elliptic curves, the isomorphism of generalized complex structure between $T$-dual manifolds described by Cavalcanti-Gualtieri coincides with the mirror map for elliptic curves described by Polishchuk and Zaslow.