An Orientifold with Fluxes and Branes via T-duality

TL;DR

The paper develops a concrete, string-theoretic framework for flux compactifications with D-branes by deriving the bosonic action of a type I' orientifold through six T-dualities on a six-torus, including the coupling to D3-brane gauge fields and background 3-form fluxes. It demonstrates warped Minkowski vacua and a no-scale 4D potential that stabilizes many moduli, and shows precise agreement with ${\cal N}=4$ gauged supergravity upon identifying fields and flux parameters. The authors further discuss a generalized superpotential incorporating both $G_3$ and $\omega_3$ fluxes, and analyze the truncation to ${\cal N}=1$ in toroidal settings, highlighting potential dilaton stabilization and the role of the warp factor. They also extend the discussion to higher-dimensional open-string sectors (e.g., D9-branes) with calibration and κ-symmetry considerations, illustrating how brane tensions and fluxes interplay with moduli stabilization in warped backgrounds. Overall, the work provides a detailed bridge between explicit string constructions and gauged supergravity descriptions, with implications for moduli stabilization and model-building in flux compactifications.

Abstract

String compactifications with non-abelian gauge fields localized on D-branes, with background NSNS and RR 3-form fluxes, and with non-trivial warp factors, can naturally exist within T-dual versions of type I string theory. We develop a systematic procedure to construct the effective bosonic Lagrangian of type I T-dualized along a six-torus, including the coupling to gauge multiplets on D3-branes and the modifications due to 3-form fluxes. Looking for solutions to the ten-dimensional equations of motion, we find warped products of Minkowski space and Ricci-flat internal manifolds. Once the warp factor is neglected, the resulting no-scale scalar potential of the effective four-dimensional theory combines those known for 3-form fluxes and for internal Yang-Mills fields and stabilizes many of the moduli. We perform an explicit comparison of our expressions to those obtained from N=4 gauged supergravity and find agreement. We also comment on the possibility to include D9-branes with world-volume gauge fluxes in the background with 3-form fluxes.