Open String BRST Cohomology for Generalized Complex Branes

TL;DR

This work identifies open-string endomorphisms for generalized complex (GC) branes with the Lie algebroid cohomology of the brane’s associated $E_Y$, the $-i$ eigenbundle of the GC structure restricted to the brane. In the B-brane limit, it provides a canonical, non-spectral description of endomorphisms as the cohomology of a differential $Q_Y$ on a graded bundle over $Y$, with a first differential matching the product of the Atiyah class of the brane and the Dolbeault obstruction $\beta_Y$. For coisotropic A-branes, the results corroborate a construction by Orlov and one of the authors, showing the BRST cohomology aligns with a foliation-holomorphic structure. Collectively, the paper bridges GC geometry with open-string BRST theory, offering concrete computational tools for endomorphisms in GC brane categories and explicit formulas in key brane examples.

Abstract

It has been shown recently that the geometry of D-branes in general topologically twisted (2,2) sigma-models can be described in the language of generalized complex structures. On general grounds such D-branes (called generalized complex (GC) branes) must form a category. We compute the BRST cohomology of open strings with both ends on the same GC brane. In mathematical terms, we determine spaces of endomorphisms in the category of GC branes. We find that the BRST cohomology can be expressed as the cohomology of a Lie algebroid canonically associated to any GC brane. In the special case of B-branes, this leads to an apparently new way to compute Ext groups of holomorphic line bundles supported on complex submanifolds: while the usual method leads to a spectral sequence converging to the Ext, our approach expresses the Ext group as the cohomology of a certain differential acting on the space of smooth sections of a graded vector bundle on the submanifold. In the case of coisotropic A-branes, our computation confirms a proposal of D. Orlov and one of the authors (A.K.).