Remarks on A-branes, Mirror Symmetry, and the Fukaya category

TL;DR

This paper explores the nature of A-branes in topological string theory, proposing that they include more general coisotropic branes beyond Lagrangian submanifolds, which impacts the structure of the Fukaya category and mirror symmetry.

Contribution

It introduces the concept of coisotropic A-branes with non-flat line bundles and argues for their inclusion in the Fukaya category to support Homological Mirror Symmetry.

Findings

A-branes can be coisotropic, not just Lagrangian.

Coisotropic A-branes have a foliated manifold structure.

Enlarging the Fukaya category with coisotropic branes supports mirror symmetry.

Abstract

We discuss D-branes of the topological A-model (A-branes), which are believed to be closely related to the Fukaya category. We give string theory arguments which show that A-branes are not necessarily Lagrangian submanifolds in the Calabi-Yau: more general coisotropic branes are also allowed, if the line bundle on the brane is not flat. We show that a coisotropic A-brane has a natural structure of a foliated manifold with a transverse holomorphic structure. We argue that the Fukaya category must be enlarged with such objects for the Homological Mirror Symmetry conjecture to be true.