New supersymmetric vacua on solvmanifolds

We obtain new supersymmetric flux vacua of type II supergravities on four-dimensional Minkowski times six-dimensional solvmanifolds. The orientifold O4, O5, O6, O7, or O8-planes and D-branes are localized. All vacua are in addition not T-dual to a vacuum on the torus. The corresponding solvmanifolds are proven to be Calabi-Yau, with explicit metrics. Other Ricci flat solvmanifolds are shown to be only K\"ahler.Comment: v2: few additions and minor modifications, published versio

Non-geometric fluxes versus (non)-geometry

Non-geometry has been introduced when considering a new type of string backgrounds, for which stringy symmetries serve as transition functions between patches of the target space. Then, some terms in the potential of four-dimensional gauged supergravities, generated by so-called non-geometric fluxes, have been argued to find a higher-dimensional origin in these backgrounds, even if a standard compactification on those cannot be made. We present here recent results clarifying the relation between these two settings. Thanks to a field redefinition, we reformulate the NSNS Lagrangian in such a way that the non-geometric fluxes appear in ten dimensions. In addition, if an NSNS field configuration is non-geometric, its reformulation in terms of the new fields can restore a standard geometry. A dimensional reduction is then possible, and leads to the non-geometric terms in the four-dimensional potential. Reformulating similarly doubled field theory, we get a better understanding of the role of the non-geometric fluxes, and rewrite the Lagrangian in a manifestly diffeomorphism-covariant manner. We finally discuss the relevance of the field redefinition and the non-geometric fluxes when studying the non-commutativity of string coordinates.Comment: 12 pages; this paper is based on a talk given at String-Math 2012 in Bonn, Germany, and contributes to the proceedings of this conference; v2: references added; v3: published versio

\beta-supergravity: a ten-dimensional theory with non-geometric fluxes, and its geometric framework

We present a ten-dimensional theory, named \beta-supergravity, that contains non-geometric fluxes and could uplift some four-dimensional gauged supergravities. Building on earlier work, we study here its NSNS sector, where Q- and R-fluxes are precisely identified. Interestingly, the Q-flux is captured in an analogue of the Levi-Civita spin connection, giving rise to a second curvature scalar. We reproduce the ten-dimensional Lagrangian using the Generalized Geometry formalism; this provides us with enlightening interpretations of the new structures. Then, we derive the equations of motion of our theory, and finally discuss further aspects: the dimensional reduction to four dimensions and comparison to gauged supergravities, the obtention of ten-dimensional purely NSNS solutions, the extensions to other sectors and new objects, the supergravity limit, and eventually the symmetries, in particular the \beta gauge transformation. We also introduce the related notion of a generalized cotangent bundle.Comment: v3: typos corrected, published versio

NS-branes, source corrected Bianchi identities, and more on backgrounds with non-geometric fluxes

In the first half of the paper, we study in details NS-branes, including the NS5-brane, the Kaluza-Klein monopole and the exotic $5_2^2$- or Q-brane, together with Bianchi identities for NSNS (non)-geometric fluxes. Four-dimensional Bianchi identities are generalized to ten dimensions with non-constant fluxes, and get corrected by a source term in presence of an NS-brane. The latter allows them to reduce to the expected Poisson equation. Without sources, our Bianchi identities are also recovered by squaring a nilpotent $Spin(D,D) \times \mathbb{R}^+$ Dirac operator. Generalized Geometry allows us in addition to express the equations of motion explicitly in terms of fluxes. In the second half, we perform a general analysis of ten-dimensional geometric backgrounds with non-geometric fluxes, in the context of $\beta$-supergravity. We determine a well-defined class of such vacua, that are non-geometric in standard supergravity: they involve $\beta$-transforms, a manifest symmetry of $\beta$-supergravity with isometries. We show as well that these vacua belong to a geometric T-duality orbit.Comment: v2: minor changes and additions, few references added, published versio

Refining the boundaries of the classical de Sitter landscape

We derive highly constraining no-go theorems for classical de Sitter backgrounds of string theory, with parallel sources; this should impact the embedding of cosmological models. We study ten-dimensional vacua of type II supergravities with parallel and backreacted orientifold Op-planes and Dp-branes, on four-dimensional de Sitter space-time times a compact manifold. Vacua for p=3, 7 or 8 are completely excluded, and we obtain tight constraints for p=4, 5, 6. This is achieved through the derivation of an enlightening expression for the four-dimensional Ricci scalar. Further interesting expressions and no-go theorems are obtained. The paper is self-contained so technical aspects, including conventions, might be of more general interest.Comment: 15 pages + appendices and references; v2: few additions; v3: requirements on the sources and internal geometry clarified, version accepted for publication; v4: erratum added, minor impact on the result

Signatures of extra dimensions in gravitational waves

Considering gravitational waves propagating on the most general 4+N-dimensional space-time, we investigate the effects due to the N extra dimensions on the four-dimensional waves. All wave equations are derived in general and discussed. On Minkowski4 times an arbitrary Ricci-flat compact manifold, we find: a massless wave with an additional polarization, the breathing mode, and extra waves with high frequencies fixed by Kaluza-Klein masses. We discuss whether these two effects could be observed.Comment: v1: 21 pages + appendices, comments welcome! v2: few minor additions; v3: minor mistake corrected in the warped case, no impact on the main results, appendix A.2 modifie

A geometric action for non-geometric fluxes

We give a geometrical interpretation of the non-geometric Q and R fluxes. To this end we consider double field theory in a formulation that is related to the conventional one by a field redefinition taking the form of a T-duality inversion. The R flux is a tensor under diffeomorphisms and satisfies a non-trivial Bianchi identity. The Q flux can be viewed as part of a connection that covariantizes the winding derivatives with respect to diffeomorphisms. We give a higher-dimensional action with a kinetic term for the R flux and a 'dual' Einstein-Hilbert term containing the connection Q.Comment: 4 pages; v2: discussion extended, version to be published in PR

Towards Kaluza-Klein Dark Matter on Nilmanifolds

We present a first study of the field spectrum on a class of negatively-curved compact spaces: nilmanifolds or twisted tori. This is a case where analytical results can be obtained, allowing to check numerical methods. We focus on the Kaluza-Klein expansion of a scalar field. The results are then applied to a toy model where a natural Dark Matter candidate arises as a stable massive state of the bulk scalar.Comment: 32 pages, 8 figures; v2: few additions, published versio