It is shown that the effective five-dimensional theory of the strongly coupled heterotic string is a gauged version of Nϭ1 five-dimensional supergravity with four-dimensional boundaries. For the universal supermultiplets, this theory is explicitly constructed by a generalized dimensional reduction procedure on a Calabi-Yau manifold. A crucial ingredient in the reduction is the retention of a ''non-zero mode'' of the four-form field strength, leading to the gauging of the universal hypermultiplet by the graviphoton. We show that this theory has an exact three-brane domain wall solution which reduces to Witten's ''deformed'' Calabi-Yau background upon linearization. This solution consists of two parallel three-branes with sources provided by the fourdimensional boundary theories and constitutes the appropriate background for a reduction to four dimensions. Four-dimensional space-time is then identified with the three-brane world volume. ͓S0556-2821͑98͒02122-5͔

We present a countably infinite number of new explicit co-homogeneity one Sasaki-Einstein metrics on S 2 × S 3 of both quasi-regular and irregular type. These give rise to new solutions of type IIB supergravity which are expected to be dual to N = 1 superconformal field theories in four dimensions with compact or non-compact R-symmetry and rational or irrational central charges, respectively.

We discuss mirror symmetry in generalized Calabi-Yau compactifications of type II string theories with background NS fluxes. Starting from type IIB compactified on CalabiYau threefolds with NS three-form flux we show that the mirror type IIA theory arises from a purely geometrical compactification on a different class of six-manifolds. These mirror manifolds have SU(3) structure and are termed half-flat; they are neither complex nor Ricci-flat and their holonomy group is no longer SU (3). We show that type IIA appropriately compactified on such manifolds gives the correct mirror-symmetric lowenergy effective action.

We systematically analyse the necessary and sufficient conditions for the preservation of supersymmetry for bosonic geometries of the form Ê 1,9−d × M d , in the common NS-NS sector of type II string theory and also type I/heterotic string theory. The results are phrased in terms of the intrinsic torsion of G-structures and provide a comprehensive classification of static supersymmetric backgrounds in these theories. Generalised calibrations naturally appear since the geometries always admit NS or type I/heterotic fivebranes wrapping calibrated cycles.Some new solutions are presented. In particular we find d = 6 examples with a fibred structure which preserve N = 1, 2, 3 supersymmetry in type II and include compact type I/heterotic geometries.

We consider type II string theory in space-time backgrounds which admit eight supercharges. Such backgrounds are characterized by the existence of a (generically nonintegrable) generalized SU (3)×SU (3) structure. We demonstrate how the corresponding ten-dimensional supergravity theories can in part be rewritten using generalised O(6, 6)covariant fields, in a form that strongly resembles that of four-dimensional N = 2 supergravity, and precisely coincides with such after an appropriate Kaluza-Klein reduction. Specifically we demonstrate that the NS sector admits a special Kähler geometry with Kähler potentials given by the Hitchin functionals. Furthermore we explicitly compute the N = 2 version of the superpotential from the transformation law of the gravitinos, and find its N = 1 counterpart.May 2005 Furthermore, the extra deformations which modify the SU (3) structure but leave g invariant are also projected out. Thus none of the subtleties discussed in section 2.4.6 is of any concern here. Instead the moduli space of metric deformations is simply M finite = M finite J × M finite ρ (3.26) where M finite J and M finite ρ are the spaces U finite J and U finite ρ modulo rescalings of c in Φ + and the magnitude and phase of Ω in Φ − , that is M finite J = U finite J /C * , M finite ρ = U finite ρ /C * . (3.27)These spaces can be parametrized by the local "special" coordinates t a = X a /X 0 and z k = Z K /Z 0 . In the latter case we isolate one (labeled α 0 ) of the α K , and assume that we can consistently scale its coefficient to unity when expanding Ω. The corresponding Kähler potentials are(3.28)Note the natural scalar Hitchin functionals (3.19), allow us to define scalar Kähler potentials, as should appear in a four-dimensional theory, rather than the six-forms e −K J and e −Kρ that appear in the general ten-dimensional theory of sec. 2.4.

Inflationary solutions are constructed in a specific five-dimensional model with boundaries motivated by heterotic M theory. We concentrate on the case where the vacuum energy is provided by potentials on those boundaries. It is pointed out that the presence of such potentials necessarily excites bulk fields. We distinguish a linear and a non-linear regime for those modes. In the linear regime, inflation can be discussed in an effective four-dimensional theory in the conventional way. This effective action is derived by integrating out the bulk modes. Therefore, these modes do not give rise to excited Kaluza-Klein modes from a four-dimensional perspective. We lift a four-dimensional inflating solution up to five dimensions where it represents an inflating domain wall pair. This shows explicitly the inhomogeneity in the fifth dimension. We also demonstrate the existence of inflating solutions with unconventional properties in the non-linear regime. Specifically, we find solutions with and without an horizon between the two boundaries. These solutions have certain problems associated with the stability of the additional dimension and the persistence of initial excitations of the Kaluza-Klein modes.

We derive the five-dimensional effective action of strongly coupled heterotic string theory for the complete (1, 1) sector of the theory by performing a reduction, on a Calabi-Yau three-fold, of M-theory on S 1 /Z 2 . A crucial ingredient for a consistent truncation is a non-zero mode of the antisymmetric tensor field strength which arises due to magnetic sources on the orbifold planes.The correct effective theory is a gauged version of five-dimensional N = 1 supergravity coupled to Abelian vector multiplets, the universal hypermultiplet and four-dimensional boundary theories with gauge and gauge matter fields. The gauging is such that the dual of the four-form field strength in the universal multiplet is charged under a particular linear combination of the Abelian vector fields. In addition, the theory has potential terms for the moduli in the bulk as well as on the boundary. Because of these potential terms, the supersymmetric ground state of the theory is a multi-charged BPS three-brane domain wall, which we construct in general. We show that the five-dimensional theory together with this solution provides the correct starting point for particle phenomenology as well as early universe cosmology. As an application, we compute the four-dimensional N = 1 supergravity theory for the complete (1, 1) sector to leading nontrivial order by a reduction on the domain wall background. We find a correction to the matter field Kähler potential and threshold corrections to the gauge kinetic functions.

We show that generalised geometry gives a unified description of bosonic eleven-dimensional supergravity restricted to a d-dimensional manifold for all d ≤ 7. The theory is based on an extended tangent space which admits a natural E d(d) × R + action. The bosonic degrees of freedom are unified as a "generalised metric", as are the diffeomorphism and gauge symmetries, while the local O(d) symmetry is promoted to H d , the maximally compact subgroup of E d(d) . We introduce the analogue of the Levi-Civita connection and the Ricci tensor and show that the bosonic action and equations of motion are simply given by the generalised Ricci scalar and the vanishing of the generalised Ricci tensor respectively. The formalism also gives a unified description of the bosonic NSNS and RR sectors of type II supergravity in d − 1 dimensions. Locally the formulation also describes M-theory variants of double field theory and we derive the corresponding section condition in general dimension. We comment on the relation to other approaches to M theory with E d(d) symmetry, as well as the connections to flux compactifications and the embedding tensor formalism.