Isotropic A-branes and the stability condition

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.