Gauge Symmetry, T-Duality and Doubled Geometry

String compactifications with T-duality twists are revisited and the gauge algebra of the dimensionally reduced theories calculated. These reductions can be viewed as string theory on T-fold backgrounds, and can be formulated in a `doubled space' in which each circle is supplemented by a T-dual circle to construct a geometry which is a doubled torus bundle over a circle. We discuss a conjectured extension to include T-duality on the base circle, and propose the introduction of a dual base coordinate, to give a doubled space which is locally the group manifold of the gauge group. Special cases include those in which the doubled group is a Drinfel'd double. This gives a framework to discuss backgrounds that are not even locally geometric.Comment: 16 page

Scaling Cosmologies from Duality Twisted Compactifications

Oscillating moduli fields can support a cosmological scaling solution in the presence of a perfect fluid when the scalar field potential satisfies appropriate conditions. We examine when such conditions arise in higher-dimensional, non-linear sigma-models that are reduced to four dimensions under a generalized Scherk-Schwarz compactification. We show explicitly that scaling behaviour is possible when the higher-dimensional action exhibits a global SL(n,R) or O(2,2) symmetry. These underlying symmetries can be exploited to generate non-trivial scaling solutions when the moduli fields have non-canonical kinetic energy. We also consider the compactification of eleven-dimensional vacuum Einstein gravity on an elliptic twisted torus.Comment: 21 pages, 3 figure

Non-geometric backgrounds, doubled geometry and generalised T-duality

String backgrounds with a local torus fibration such as T-folds are naturally formulated in a doubled formalism in which the torus fibres are doubled to include dual coordinates conjugate to winding number. Here we formulate and explore a generalisation of this construction in which all coordinates are doubled, so that the doubled space is a twisted torus, i.e. a compact space constructed from identifying a group manifold under a discrete subgroup. This incorporates reductions with duality twists, T-folds and a class of flux compactifications, together with the non-geometric backgrounds expected to arise from these through T-duality. It also incorporates backgrounds that are not even locally geometric, and suggests a generalisation of T-duality to a more general context. We discuss the effective field theory arising from such an internal sector, give a world-sheet sigma model formulation of string theory on such backgrounds and illustrate our discussion with detailed examples.Comment: 81 page

Radiation-dominated area metric cosmology

We provide further crucial support for a refined, area metric structure of spacetime. Based on the solution of conceptual issues, such as the consistent coupling of fermions and the covariant identification of radiation fields on area metric backgrounds, we show that the radiation-dominated epoch of area metric cosmology is equivalent to that epoch in standard Einstein cosmology. This ensures, in particular, successful nucleosynthesis. This surprising result complements the previously derived prediction of a small late-time acceleration of an area metric universe.Comment: 23 pages, no figures; references adde

A Massive S-duality in 4 dimensions

We reduce the Type IIA supergravity theory with a generalized Scherk-Schwarz ansatz that exploits the scaling symmetry of the dilaton, the metric and the NS 2-form field. The resulting theory is a new massive, gauged supergravity theory in four dimensions with a massive 2-form field and a massive 1-form field. We show that this theory is S-dual to a theory with a massive vector field and a massive 2-form field, which are dual to the massive 2-form and 1-form fields in the original theory, respectively. The S-dual theory is shown to arise from a Scherk-Schwarz reduction of the heterotic theory. Hence we establish a massive, S-duality type relation between the IIA theory and the heterotic theory in four dimensions. We also show that the Lagrangian for the new four dimensional theory can be put in the most general form of a D=4, N=4 gauged Lagrangian found by Schon and Weidner, in which (part of) the SL(2) group has been gauged.Comment: 20 pages, references adde

Lectures on Gauged Supergravity and Flux Compactifications

The low-energy effective theories describing string compactifications in the presence of fluxes are so-called gauged supergravities: deformations of the standard abelian supergravity theories. The deformation parameters can be identified with the various possible (geometric and non-geometric) flux components. In these lecture notes we review the construction of gauged supergravities in a manifestly duality covariant way and illustrate the construction in several examples.Comment: 48 pages, lectures given at the RTN Winter School on Strings, Supergravity and Gauge Theories, CERN, January 200

Flux moduli stabilisation, Supergravity algebras and no-go theorems

We perform a complete classification of the flux-induced 12d algebras compatible with the set of N=1 type II orientifold models that are T-duality invariant, and allowed by the symmetries of the T^6/(Z_2 x Z_2) isotropic orbifold. The classification is performed in a type IIB frame, where only H_3 and Q fluxes are present. We then study no-go theorems, formulated in a type IIA frame, on the existence of Minkowski/de Sitter (Mkw/dS) vacua. By deriving a dictionary between the sources of potential energy for the three moduli (S, T and U) in types IIA and IIB, we are able to combine algebra results and no-go theorems. The outcome is a systematic procedure for identifying phenomenologically viable models where Mkw/dS vacua may exist. We present a complete table of the allowed algebras and the viability of their resulting scalar potential, and we point at the models which stand any chance of producing a fully stable vacuum.Comment: Version published in JHE

Type II compactifications on manifolds with SU(2) x SU(2) structure

We study compactifications of type II theories on SU(2) x SU(2) structure manifolds to six, five and four spacetime dimensions. We use the framework of generalized geometry to describe the NS-NS sector of such compactifications and derive the structure of their moduli spaces. We show that in contrast to SU(3) x SU(3) structure compactifications, there is no dynamical SU(2) x SU(2) structure interpolating between an SU(2) structure and an identity structure. Furthermore, we formulate type II compactifications on SU(2) x SU(2) structures in the context of exceptional generalized geometry which makes the U-duality group manifest and naturally incorporates the scalar degrees of freedom arising in the Ramond-Ramond sector. Via this formalism we derive the structure of the moduli spaces as it is expected from N=4 supergravity.Comment: 69 pages, v2 published versio

Lectures on Nongeometric Flux Compactifications

These notes present a pedagogical review of nongeometric flux compactifications. We begin by reviewing well-known geometric flux compactifications in Type II string theory, and argue that one must include nongeometric "fluxes" in order to have a superpotential which is invariant under T-duality. Additionally, we discuss some elementary aspects of the worldsheet description of nongeometric backgrounds. This review is based on lectures given at the 2007 RTN Winter School at CERN.Comment: 31 pages, JHEP

A ten-dimensional action for non-geometric fluxes

The NSNS Lagrangian of ten-dimensional supergravity is rewritten via a change of field variables inspired by Generalized Complex Geometry. We obtain a new metric and dilaton, together with an antisymmetric bivector field which leads to a ten-dimensional version of the non-geometric Q-flux. Given the involved global aspects of non-geometric situations, we prescribe to use this new Lagrangian, whose associated action is well-defined in some examples investigated here. This allows us to perform a standard dimensional reduction and to recover the usual contribution of the Q-flux to the four-dimensional scalar potential. An extension of this work to include the R-flux is discussed. The paper also contains a brief review on non-geometry.Comment: 47 pages; v2: minor modifications, references added, version to be published in JHE