Deformations of generalized complex branes

TL;DR

This paper develops a formal deformation theory for rank 1 branes on generalized complex manifolds, constructing a groupoid that encodes deformations and proving an unobstructedness result for these deformations.

Contribution

It introduces a new formal deformation groupoid for generalized complex branes and constructs a governing differential graded Lie algebra, extending deformation theory in this geometric context.

Findings

Constructed a formal deformation groupoid for GC branes

Developed a DGLA that governs local trivializations of deformations

Proved an unobstructedness theorem for deformations

Abstract

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.