On the stability of open-string orbifold models with broken supersymmetry

We consider an open-string realisation of $\mathcal{N}=2\to \mathcal{N}=0$ spontaneous breaking of supersymmetry in four-dimensional Minkowski spacetime. It is based on type IIB orientifold theory compactified on $T^2\times T^4/\mathbb{Z}_2$, with Scherk--Schwarz supersymmetry breaking implemented along $T^2$. We show that in the regions of moduli space where the supersymmetry breaking scale is lower than the other scales, there exist configurations with minima that have massless Bose-Fermi degeneracy and hence vanishing one-loop effective potential, up to exponentially suppressed corrections. These backgrounds describe non-Abelian gauge theories, with all open-string moduli and blowing up modes of $T^4/\mathbb{Z}_2$ stabilized, while all untwisted closed-string moduli remain flat directions. Other backgrounds with strictly positive effective potentials exist, where the only instabilities arising at one loop are associated with the supersymmetry breaking scale, which runs away. All of these backgrounds are consistent non-perturbatively.