On Classical de Sitter Vacua in String Theory

We review the prospect of obtaining tree-level de Sitter (dS) vacua and slow-roll inflation models in string compactifications. Restricting ourselves to the closed string sector and assuming the absence of NSNS-sources, we classify the minimal classical ingredients that evade the simplest no-go theorems against dS vacua and inflation. Spaces with negative integrated curvature together with certain combinations of low-dimensional orientifold planes and low-rank RR-fluxes emerge as the most promising setups of this analysis. We focus on two well-controlled classes that lead to an effective 4D, N=1 supergravity description: Type IIA theory on group or coset manifolds with SU(3)-structure and O6-planes, as well as type IIB compactifications on SU(2)-structure manifolds with O5- and O7-planes. While fully stabilized AdS vacua are generically possible, a number of problems encountered in the search for dS vacua are discussed.