Deformations of generalized complex branes

We investigate the formal deformation theory of (rank 1) branes on generalized complex (GC) manifolds. This generalizes, for example, the deformation theory of a complex submanifold in a fixed complex manifold. For each GC brane $\mathcal{B}$ on a GC manifold $(X,\mathbb{J})$, we construct a formal (pointed) groupoid $\textbf{Def}^{\mathcal{B}}(X,\mathbb{J})$ (defined over a certain category of real Artin algebras) that encodes the formal deformations of $\mathcal{B}$. We study the geometric content of this groupoid in a number of different situations. Using the theory of (bi)semicosimplicial differential graded Lie algebras (DGLAs), we construct for each brane $\mathcal{B}$ a DGLA $L_{\mathcal{B}}$ that governs the "locally trivializable" deformations of $\mathcal{B}$. As a concrete application of this construction, we prove an unobstructedness result.