Coisotropic branes on tori and Homological mirror symmetry

TL;DR

This paper constructs an extended Fukaya category for symplectic tori that includes coisotropic branes, addressing gaps in the category needed for Homological Mirror Symmetry.

Contribution

It introduces a doubling procedure to incorporate coisotropic branes into the Fukaya category of tori, enhancing the framework for HMS.

Findings

Constructed a version of Fukaya category with coisotropic branes

Established relation between doubled and original Fukaya categories

Facilitated the verification of HMS for tori with coisotropic objects

Abstract

Homological mirror symmetry (HMS) asserts that the Fukaya category of a symplectic manifold is derived equivalent to the category of coherent sheaves on the mirror complex manifold. Without suitable enlargement (split closure) of the Fukaya category, certain objects of it are missing to prevent HMS from being true. One possible solution is to include coisotropic branes into the Fukaya category. This paper gives a construction for linear symplectic tori of a version of Fukaya category including coisotropic branes by using a doubling procedure, and discussing the relation between the Fukaya category of the doubling torus and the Fukaya category of the original torus.