Minkowski flux vacua of type II supergravities

TL;DR

This work analyzes Minkowski flux vacua of 10d type II supergravities with parallel, space-time filling O_p/D_p sources for 3 ≤ p ≤ 8, showing that the 4d Ricci scalar $ ilde{{ m R}}_4$ can be written as a negative sum of squares of BPS-like conditions. Setting these squares to zero yields solutions to the full 10d equations of motion, thereby characterizing a broad class of Minkowski flux vacua without relying on supersymmetry, and extending the construction from O3 in IIB to general O_p configurations. The allowed flux content is constrained to $H$, $F_{6-p}$, and $F_{8-p}$ with a specific parallel/transverse decomposition, and the Bianchi identities ensure tadpole cancellation; a complete verification of the e.o.m. is provided under the stated geometric restrictions. The authors also propose a non-geometric flux extension, conjecturing a T-duality invariant framework that includes NSNS non-geometric fluxes $Q$ and $R$, with potential connections to 4d gauged supergravity or β-supergravity. Overall, the results offer a constructive path toward a classification of Minkowski flux vacua in type II theories with parallel sources, with implications for moduli stabilization and the landscape of consistent backgrounds.

Abstract

We study flux compactifications of 10d type II supergravities to 4d Minkowski space-time, supported by parallel orientifold Op-planes with 3 $\leq$ p $\leq$ 8. With some restrictions, the 4d Ricci scalar can be written as a negative sum of squares involving BPS-like conditions. Setting all squares to zero provides automatically a solution to 10d equations of motion. This way, we characterize a broad class, if not the complete set, of Minkowski flux vacua. We also conjecture an extension to include non-geometric fluxes. None of our results rely on supersymmetry.