Categorial mirror symmetry for K3 surfaces

TL;DR

This paper explores the mirror symmetry between K3 surfaces by analyzing a modified Fukaya category and establishing an equivalence with a subcategory of the derived category of coherent sheaves on the mirror surface, specifically for elliptic K3 surfaces with a section.

Contribution

It demonstrates a categorical mirror symmetry for elliptic K3 surfaces with a section by relating the Fukaya category to the derived category of the mirror.

Findings

Derived category of the modified Fukaya category is equivalent to a subcategory of the mirror's coherent sheaves.

Establishes categorical mirror symmetry for elliptic K3 surfaces with a section.

Provides new insights into the structure of K3 surfaces in the context of mirror symmetry.

Abstract

We study the structure of a modified Fukaya category ${\frak F}(X)$ associated with a K3 surface $X$, and prove that whenever $X$ is an elliptic K3 surface with a section, the derived category of $\fF(X)$ is equivalent to a subcategory of the derived category ${\bold D}(\hat X)$ of coherent sheaves on the mirror K3 surface $\hat X$.