Deformations of calibrated D-branes in flux generalized complex manifolds

TL;DR

The paper identifies massless fluctuations of supersymmetric D-branes in flux backgrounds with the first Lie algebroid cohomology $H^1(L_{(\Sigma,{\cal F})})$ of the brane’s generalized tangent structure, tying D-brane moduli to generalized complex geometry. It develops a framework where deformations preserving generalized calibration are governed by holomorphic sections of the generalized normal bundle, with gauge equivalence captured by the complexified world-volume gauge group and the resulting cohomology $H^1(L_{(\Sigma,{\cal F})})$. The analysis uncovers how world-volume and background fluxes, as well as genuine type-changing generalized complex structures, can lift would-be massless moduli, and it provides a suite of explicit Calabi–Yau and SU(3) structure examples illustrating these effects. The work bridges generalized complex geometry, topological string BRST cohomology, and four-dimensional effective theory via a geometrical superpotential and a harmonic representative moduli space, offering a unified view of D-brane moduli in flux vacua and guiding extensions to more general backgrounds. Overall, it demonstrates that the moduli of calibrated D-branes in flux backgrounds are encoded in $H^1(L_{(\Sigma,{\cal F})})$, a structure inherited from the bulk integrable generalized complex geometry and robust under several physically relevant deformations.

Abstract

We study massless deformations of generalized calibrated cycles, which describe, in the language of generalized complex geometry, supersymmetric D-branes in N=1 supersymmetric compactifications with fluxes. We find that the deformations are classified by the first cohomology group of a Lie algebroid canonically associated to the generalized calibrated cycle, seen as a generalized complex submanifold with respect to the integrable generalized complex structure of the bulk. We provide examples in the SU(3) structure case and in a `genuine' generalized complex structure case. We discuss cases of lifting of massless modes due to world-volume fluxes, background fluxes and a generalized complex structure that changes type.