Isotropic A-branes and the stability condition

TL;DR

This paper introduces isotropic A-branes as a new class of branes in the topological A-model, based on generalized complex geometry, and explores their stability and implications for mirror symmetry.

Contribution

It proposes a new definition of generalized complex submanifolds leading to isotropic A-branes, expanding the Fukaya category and impacting homological mirror symmetry.

Findings

Existence of isotropic A-branes in symplectic manifolds.

Generalized complex submanifolds can be isotropic, coisotropic, or Lagrangian.

Stability conditions for isotropic A-branes are analyzed using worldsheet methods.

Abstract

The existence of a new kind of branes for the open topological A-model is argued by using the generalized complex geometry of Hitchin and the SYZ picture of mirror symmetry. Mirror symmetry suggests to consider a bi-vector in the normal direction of the brane and a new definition of generalized complex submanifold. Using this definition, it is shown that there exists generalized complex submanifolds which are isotropic in a symplectic manifold. For certain target space manifolds this leads to isotropic A-branes, which should be considered in addition to Lagrangian and coisotropic A-branes. The Fukaya category should be enlarged with such branes, which might have interesting consequences for the homological mirror symmetry of Kontsevich. The stability condition for isotropic A-branes is studied using the worldsheet approach.