D-terms and F-terms from D7-brane fluxes

TL;DR

The paper presents a thorough derivation of D- and F-terms in the $\mathcal{N}=1$ four-dimensional effective action for type IIB Calabi–Yau orientifolds with space-time filling D7-branes and background worldvolume fluxes. By performing a Kaluza-Klein reduction of the D7-brane fermionic sector and analyzing gravitino couplings, it yields explicit expressions for the D-term $D=\frac{12\kappa_4^2\mu_7\ell}{\mathcal{K}}\int_{S^P} J\wedge\mathcal{F}$ and the flux-induced superpotential $W(\zeta)=\kappa_4^2\mu_7\, Q_A\, \zeta^A$ with $Q_A=\ell\int_{S^\Lambda}\tilde{s}_A\wedge\tilde{f}$. Fluxes decompose into $^Y f$ and $\tilde f$, shifting chiral coordinates or contributing directly to F-terms, and the authors connect these results to a holomorphic Chern–Simons action reduction. Two explicit moduli scenarios demonstrate flux-driven scalar potentials and their cosmological relevance, including potential D-term uplifting alongside non-perturbative stabilization. The work clarifies how D7-brane fluxes influence moduli stabilization and SUSY-breaking mechanisms in string compactifications, with implications for de Sitter vacua and F-theory generalizations.

Abstract

Using a Kaluza-Klein reduction of the fermionic part of the D-brane action we compute D- and F-terms of the N=1 effective action for generic Calabi-Yau orientifold compactifications in the presence of a space-time filling D7-brane. We include non-trivial background fluxes for the D7-brane U(1) field strength on the internal four-cycle wrapped by the brane. First the four-dimensional fermionic spectrum arising from the D7-brane is derived and then the D- and F-terms are obtained by computing appropriate couplings of these fermionic fields. For specific examples we examine the resulting flux-induced scalar potentials and comment on their relevance in string cosmology.