We present a brief overview of attempts to construct de Sitter vacua in string theory and explain how the results of this 20-year endeavor could point to the fact that string theory harbours no de Sitter vacua at all. Making such a statement is often considered controversial and "bad news for string theory". We discuss how perhaps the opposite can be true.2 An interesting light-version of a review on dark energy in string theory can be found in [27] 3 See also [34].

We elaborate on the construction of de Sitter solutions from IIA orientifolds of SU(3)-structure manifolds that solve the 10-dimensional equations of motion at tree-level in the approximation of smeared sources. First we classify geometries that are orbifolds of a group manifold covering space which, upon the proper inclusion of O6 planes, can be described within the framework of N=1 supergravity in 4D. Then we scan systematically for de Sitter solutions, obtained as critical points of an effective 4D potential. Apart from finding many new solutions we emphasize the challenges in constructing explicit classical de Sitter vacua, which have sofar not been met. These challenges are interesting avenues for further research and include finding solutions that are perturbatively stable, satisfy charge and flux quantization, and have genuine localized (versus smeared) orientifold sources. This paper intends to be self-contained and pedagogical, and thus can serve as a guide to the necessary technical tools required for this line of research. In an appendix we explain how to study flux and charge quantization in the presence of a non-trivial H-field using twisted homology.Comment: 50 pages, 3 figure

We investigate the type II string effective potential at tree-level and derive necessary ingredients for having de Sitter solutions in orientifold models with fluxes. Furthermore, we examine some explicit O6 compactifications in IIA supergravity on manifolds with SU(3)-structure in the limit where the orientifold sources are smeared. In particular, we use a simple ten-dimensional Ansatz for four-dimensional de Sitter solutions and find the explicit criteria in terms of the torsion classes such that these de Sitter solutions solve the equations of motion. We have verified these torsion conditions for the cosets and the Iwasawa manifold and it turns out that the conditions cannot be fulfilled for these spaces. However this investigation allows us to find new non-supersymmetric AdS solutions for some cosets. It remains an open question whether there exist SU(3)-structure manifolds that satisfy the conditions on the torsion classes for the simple de Sitter solutions to exist. 5 Discussion 20 A Form conventions and useful formulae 22 B 10D Einstein and dilaton equation 23 C IIA SUGRA 25 D SU(3)-structure equations 25

We investigate whether vacuum solutions in flux compactifications that are obtained with smeared sources (orientifolds or D-branes) still survive when the sources are localised. This seems to rely on whether the solutions are BPS or not. First we consider two sets of BPS solutions that both relate to the GKP solution through T-dualities: (p + 1)-dimensional solutions from spacetime-filling Op-planes with a conformally Ricci-flat internal space, and p-dimensional solutions with Op-planes that wrap a 1-cycle inside an everywhere negatively curved twisted torus. The relation between the solution with smeared orientifolds and the localised version is worked out in detail. We then demonstrate that a class of non-BPS AdS 4 solutions that exist for IASD fluxes and with smeared D3-branes (or analogously for ISD fluxes with anti-D3-branes) does not survive the localisation of the (anti) D3-branes. This casts doubts on the stringy consistency of non-BPS solutions that are obtained in the limit of smeared sources.

In this paper we investigate the localisation of SUSY-breaking branes which, in the smeared approximation, support specific non-BPS vacua. We show, for a wide class of boundary conditions, that there is no flux vacuum when the branes are described by a genuine delta-function. Even more, we find that the smeared solution is the unique solution with a regular brane profile. Our setup consists of a non-BPS AdS_7 solution in massive IIA supergravity with smeared anti-D6-branes and fluxes T-dual to ISD fluxes in IIB supergravity.Comment: 27 pages, Latex2e, 5 figure