On classical stability with broken supersymmetry

We study the perturbative stability of four settings that arise in String Theory, when dilaton potentials accompany the breaking of Supersymmetry, in the tachyon-free USp(32) and U(32) orientifold models, and also in the heterotic SO(16) × SO(16) model. The first two settings are a family of AdS3 ×S7 vacua of the orientifold models and a family of AdS7 × S3 vacua of the heterotic model, supported by form fluxes, with small world-sheet and string-loop corrections within wide ranges of parameters. In both cases we find some unstable scalar perturbations, as a result of mixings induced by fluxes, confirming for the first class of vacua a previous result. However, in the second class of vacua they only affect the ℓ = 1 modes, so that a ℤ2 projection induced by an overall parity in the internal space suffices to eliminate them, leading to perturbative stability. Moreover, the constant dilaton profiles of these vacua allow one to extend the analysis to generic potentials, thus exploring the possible effects of higher-order corrections, and we exhibit wide nearby regions of perturbative stability. The solutions in the third setting have nine-dimensional Poincaré symmetry. They include regions with large world-sheet or string-loop corrections, but we show that these vacua have no perturbative instabilities. Finally, the last setting concerns cosmological solutions in ten dimensions where the “climbing” phenomenon takes place: they have bounded string-loop corrections but large world-sheet ones close to the initial singularity. In this case we find that perturbations generally decay, but homogeneous tensor modes exhibit an interesting logarithmic growth that signals a breakdown of isotropy. If the Universe then proceeds to lower dimensions, milder potentials from other branes force all perturbations to remain bounded.