D-branes on general N=1 backgrounds: superpotentials and D-terms

TL;DR

The paper develops a unified description of space-time filling D-branes on general Type II ${\cal N}=1$ flux backgrounds using generalized calibrations and pure spinors. It derives a universal four-dimensional superpotential $W$ and D-term ${\cal D}$ from the calibration conditions, relates them to BPS domain walls and cosmic strings, and confirms a canonical normalization through domain-wall arguments. Holomorphy of $W$ is established with an almost complex structure on the brane configuration space, which reduces to the integrable generalized complex structure on the holomorphic subspace; D-flatness is interpreted as a moment-map condition, yielding a moduli space ${\cal M}$ of SUSY configurations. The paper also works out explicit D3, D5, D6, and D7-brane examples in $SU(3)$-structure backgrounds, showing flux-induced lifting of moduli and reproducing known results in a broader framework.

Abstract

We study the dynamics governing space-time filling D-branes on Type II flux backgrounds preserving four-dimensional N=1 supersymmetry. The four-dimensional superpotentials and D-terms are derived. The analysis is kept on completely general grounds thanks to the use of recently proposed generalized calibrations, which also allow one to show the direct link of the superpotentials and D-terms with BPS domain walls and cosmic strings respectively. In particular, our D-brane setting reproduces the tension of D-term strings found from purely four-dimensional analysis. The holomorphicity of the superpotentials is also studied and a moment map associated to the D-terms is proposed. Among different examples, we discuss an application to the study of D7-branes on SU(3)-structure backgrounds, which reproduces and generalizes some previous results.