The local structure of generalized complex branes

TL;DR

This paper characterizes the local structure of generalized complex branes, showing they are equivalent to holomorphic coisotropic submanifolds under certain conditions, and explores examples with nonstandard structures.

Contribution

It establishes a local equivalence between generalized complex branes and holomorphic coisotropic submanifolds, extending understanding of their geometric structure.

Findings

Generalized complex branes are locally equivalent to holomorphic coisotropic submanifolds.

Higher-rank branes correspond to holomorphic Poisson modules.

Examples of non-Lagrangian, non-complex branes are constructed on the Hopf surface.

Abstract

We show (modulo a parity condition) that, a generalized complex brane in a generalized complex manifold is locally equivalent to a holomorphic coisotropic submanifold of a holomorphic Poisson structure, with higher-rank branes corresponding to holomorphic Poisson modules. We describe (but do not prove here) the global version of this holomorphicity result. Finally, we use the "local holomorphic gauges" to give examples, in the Hopf surface with nonstandard generalized complex structure, of branes which are neither Lagrangian nor complex.