We derive necessary and sufficient conditions for N = 1 compactifications of (massive) IIA supergravity to AdS 4 in the language of SU (3) structures. We find new solutions characterized by constant dilaton and nonzero fluxes for all form fields. All fluxes are given in terms of the geometrical data of the internal compact space. The latter is constrained to belong to a special class of half-flat manifolds.

We consider string theory compactifications of the form AdS 4 ×M 6 with orientifold six-planes, where M 6 is a six-dimensional compact space that is either a nilmanifold or a coset. For all known solutions of this type we obtain the four-dimensional N = 1 low energy effective theory by computing the superpotential, the Kähler potential and the mass spectrum for the light moduli. For the nilmanifold examples we perform a cross-check on the result for the mass spectrum by calculating it alternatively from a direct Kaluza-Klein reduction and find perfect agreement. We show that in all but one of the coset models all moduli are stabilized at the classical level. As an application we show that all but one of the coset models can potentially be used to bypass a recent no-go theorem against inflation in type IIA theory.

In the context of supersymmetric compactifications of type II supergravity to four dimensions, we show that orientifold sources can be compatible with a generalized SU (3) × SU (3)-structure that is neither strictly SU (3) nor static SU (2). We illustrate this with explicit examples, obtained by suitably T-dualizing known solutions on the sixtorus. In addition we prove the following integrability statements, valid under certain mild assumptions: (a) for general type II supergravity backgrounds with orientifold and/or Dbrane generalized-calibrated sources, the source-corrected Einstein and dilaton equations of motion follow automatically from the supersymmetry equations once the likewise sourcecorrected form equations of motion and Bianchi identities are imposed; (b) in the special case of supersymmetric compactifications to four-dimensional Minkowski space, the equations of motion of all fields, including the NSNS three-form, follow automatically once the supersymmetry and the Bianchi identities of the forms are imposed. Both (a) and (b) are equally valid whether the sources are smeared or localized. As a byproduct we obtain the calibration form for a space-filling NS5-brane.

We discuss a novel strategy to construct 4D N = 0 stable flux vacua of type II string theory, based on the existence of BPS bounds for probe D-branes in some of these backgrounds. In particular, we consider compactifications where D-branes filling the 4D space-time obey the same BPS bound as they would in an N = 1 compactification, while other D-branes, like those appearing as domain walls from the 4D perspective, can no longer be BPS. We construct a subfamily of such backgrounds giving rise to 4D N = 0 Minkowski no-scale vacua, generalizing the well-known case of type IIB on a warped Calabi-Yau. We provide several explicit examples of these constructions, and compute quantities of phenomenological interest like flux-induced soft terms on D-branes. Our results have a natural, simple description in the language of Generalized Complex Geometry, and in particular in terms of D-brane generalized calibrations. Finally, we extend the integrability theorems for 10D supersymmetric type II backgrounds to the N = 0 case and use the results to construct a new class of N = 0 AdS 4 compactifications. DWSB AdS4 vacua 61 11. Integrability of N = 0 vacua 63 11.1 Spinorial factorization of sourceless equations of motion 64 11.2 Adding calibrated sources 66 11.3 Integrability conditions for flux compactifications 69 11.4 Non-supersymmetric AdS 4 vacua from integrability 69 12. Conclusions and outlook 72 A. Supergravity conventions 75 A.1 Bosonic sector 75 A.2 Fermionic sector 76 A.3 Splitting to 4+6 dimensions and pure spinors 77 B. SUSY-breaking and pure spinors 79 B.1 Pure DWSB Minkowski backgrounds 81 C. The scalar curvature from pure spinors 82 D. Comments on non-geometric backgrounds 83 E. 10d integrability 84 F. Integrability of GKP vacua 87Indeed, from [6] we know that the supersymmetry conditions for a general flux background are equivalent to requiring that certain kinds of probe D-branes obey a BPS bound. 2 As we will show, this is only partially true in N = 0 GKP vacua, where a particular set of D-branes, namely some of those that look like domain-walls from the 4D viewpoint, no longer obey this BPS bound, and are thus intrinsically unstable in this background. On the other hand, D-branes that fill the 4D spacetime directions or look like strings in 4D still maintain their BPS properties unchanged with respect to the N = 1 case. This observation suggests an immediate way to generalize the GKP construction to other settings. Indeed, instead of considering the whole set of N = 0 supergravity compactifications to 4D Minkowski, we may restrict to those where 4D space-filling and string-like D-branes develop a BPS bound, while 4D domain walls will be lacking such a 'BPSness' property. The analysis of these backgrounds, which we dub 'Domain Wall SUSY-breaking' (DWSB) backgrounds, will be organized as follows:In Section 3 we translate the DWSB pattern in terms of the usual 10D gravitino and dilatino variations, in order to parameterize the space of DWSB backgrounds. Within this parameter space we single out a particular one-pa...

We present a classification of a large class of type IIA N = 1 supersymmetric compactifications to AdS 4 , based on left-invariant SU(3)-structures on coset spaces. In the absence of sources the parameter spaces of all cosets leading to a solution contain regions corresponding to nearly-Kähler structure. I.e. all these cosets can be viewed as deformations of nearly-Kähler manifolds. Allowing for (smeared) six-brane/orientifold sources we obtain more possibilities. In the second part of the paper, we use a simple ansatz, which can be applied to all six-dimensional coset manifolds considered here, to construct explicit thick domain wall solutions separating two AdS 4 vacua of different radii. We also consider smooth interpolations between AdS 4 × M 6 and R 1,2 × M 7 , where M 6 is a nearly-Kähler manifold and M 7 is the G 2 -holonomy cone over M 6 .