A-branes and Noncommutative Geometry

TL;DR

This paper proposes that for holomorphic symplectic manifolds, the category of A-branes is equivalent to a noncommutative deformation of the B-brane category on the same space, realized via the Seiberg-Witten transform in the direction of Ω^{-1}. It extends the idea to generalized complex branes, showing that GC branes similarly correspond to noncommutative B-branes on an appropriate complex manifold, with the deformation parameter tied to holomorphic Poisson data. The authors provide a concrete torus test and a construction of noncommutative B-branes on complex tori using a Dolbeault DG-algebra and Fedosov-type deformation. The work offers an algebraic framework to understand A-branes beyond the Fukaya category, linking them to noncommutative geometry without invoking T-duality, and suggests robust consistency with mirror-symmetric expectations in concrete torus settings.

Abstract

We argue that for a certain class of symplectic manifolds the category of A-branes (which includes the Fukaya category as a full subcategory) is equivalent to a noncommutative deformation of the category of B-branes (which is equivalent to the derived category of coherent sheaves) on the same manifold. This equivalence is different from Mirror Symmetry and arises from the Seiberg-Witten transform which relates gauge theories on commutative and noncommutative spaces. More generally, we argue that for certain generalized complex manifolds the category of generalized complex branes is equivalent to a noncommutative deformation of the derived category of coherent sheaves on the same manifold. We perform a simple test of our proposal in the case when the manifold in question is a symplectic torus.