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

TL;DR

The paper argues that the Homological Mirror Symmetry conjecture requires extending the Fukaya category to include coisotropic A-branes, not just Lagrangian ones. Through world-sheet analysis and bihamiltonian geometry, it shows that A-branes come with a foliation and a transverse holomorphic structure, with endomorphisms described by $H^*(\mathcal{O}_{\mathcal{F}}(Y))$ and the curvature $F$ constrained by $(\omega^{-1}F)^2 = -id$. It also uses mirror symmetry for abelian varieties to illustrate that mirrors of B-branes can be non-Lagrangian A-branes, motivating the category extension. The work connects A-brane geometry to transverse holomorphic structures and deformation-quantization, highlighting the need for a more general HMS framework applicable to all weak Calabi–Yau manifolds.

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.