We consider flux compactification of type IIB string theory as the orientifold limit of an F-theory on a Calabi–Yau 4-fold. We show that when supersymmetry is dominantly broken by the axion–dilaton and the contributions of the F-terms associated with complex structure moduli are small, the Hessian of the flux potential always has tachyonic modes for de Sitter vacua. This implies that there exist no meta-stable de Sitter vacua in this limit. Moreover, we find that the stability requirement imposes a relation between the values of cosmological constant and the scale of supersymmetry-breaking for non-supersymmetric anti-de Sitter vacua in this limit. The proof is general and does not rely on the details of the geometry of the compact Calabi–Yau internal space. We finally analyze the consequences of these constraints on the statistics of metastable de Sitter vacua and address some other related issues.
"Constraints on meta-stable de Sitter flux vacua." Adv. Theor. Math. Phys. 13 (1) 187 - 215, January 2009.