Comparison of mirror functors of elliptic curves via LG/CY correspondence

TL;DR

This paper compares two approaches to homological mirror symmetry for elliptic curves, linking the Fukaya category and matrix factorizations through explicit relations, advancing understanding of mirror functors.

Contribution

It establishes an explicit connection between the Lagrangian torus fibration approach and the localized mirror functor method for elliptic curves.

Findings

Explicit relation between two mirror symmetry approaches

Bridging Fukaya categories and matrix factorizations

Enhanced understanding of mirror functors for elliptic curves

Abstract

Polishchuk-Zaslow explained the homological mirror symmetry between Fukaya category of symplectic torus and the derived category of coherent sheaves of elliptic curves via Lagrangian torus fibration. Recently, Cho-Hong-Lau found another proof of homological mirror symmetry using localized mirror functor, whose target category is given by graded matrix factorizations. We find an explicit relation between these two approaches.