Metastable flux configurations and de Sitter spaces

We derive stability conditions for the critical points of the no-scale scalar potential governing the dynamics of the complex structure moduli and the axio-dilaton in compactifications of type IIB string theory on Calabi-Yau three-folds. We discuss a concrete example of a T6 orientifold. We then consider the four-dimensional theory obtained from compactifications of type IIB string theory on non-geometric backgrounds which are mirror to rigid Calabi-Yau manifolds and show that the complex structure moduli fields can be stabilized in terms of HRR only, i.e. with no need of orientifold projection. The stabilization of all the fields at weak coupling, including the axio-dilaton, may require to break supersymmetry in the presence of HNS flux or corrections to the scalar potential.