We discuss the mathematical properties of six-dimensional non-Kähler manifolds which occur in the context of N = 1 supersymmetric heterotic and type IIA string compactifications with non-vanishing background H-field. The intrinsic torsion of the associated SU(3) structures falls into five different classes. For heterotic compactifications we present an explicit dictionary between the supersymmetry conditions and these five torsion classes. We show that the non-Ricci flat Iwasawa manifold solves the supersymmetry conditions with non-zero H-field, so that it is a consistent heterotic supersymmetric groundstate.

We examine gauge theories on Minkowski space-time times fuzzy coset spaces. This means that the extra space dimensions instead of being a continuous coset space S/R are a corresponding finite matrix approximation. The gauge theory defined on this non-commutative setup is reduced to four dimensions and the rules of the corresponding dimensional reduction are established. We investigate in particular the case of the fuzzy sphere including the dimensional reduction of fermion fields.

We discuss supersymmetric compactifications of heterotic strings in the presence of Hflux and general condensates using the formalism of G-structures and intrinsic torsion. We revisit the examples based on nearly-Kähler coset spaces and show that supersymmetric solutions, where the Bianchi identity is satisfied, can be obtained when both gaugino and dilatino condensates are present.1 Originally, gaugino condensation had been suggested as a supersymmetry breaking mechanism in a Calabi-Yau compactification [4,5]. Here, instead, we consider supersymmetric solutions on manifolds that are not Calabi-Yau.2 See however refs. [23,24] where condensation of fermions in the gravity sector is considered in a different context and ref.[25] which discusses a similar effect in a 5-brane background. Moreover, ref.[26] considered Minkowski vacua in 11-dimensional supergravity with gravitino condensates.

We study the dimensional reduction of a ten-dimensional supersymmetric E 8 gauge theory over six-dimensional coset spaces. We find that the coset space dimensional reduction over a symmetric coset space leaves the four dimensional gauge theory without any track of the original supersymmetry. On the contrary the dimensional reduction over a non symmetric coset space leads to a softly broken supersymmetric gauge theory in four dimensions. The SO 7 /SO 6 and G 2 /SU(3) are used as representative prototypes of symmetric and non symmetric coset spaces respectively. a e-mail address: pman@central.ntua.gr. Supported by ΓΓET grand 97EΛ/71. b e-mail address: George.Zoupanos@cern.ch. Partially supported by the EU project ERBFM-RXCT960090.

We formulate a model of noncommutative four-dimensional gravity on a covariant fuzzy space based on SO(1,4), that is the fuzzy version of the dS 4. The latter requires the employment of a wider symmetry group, the SO(1,5), for reasons of covariance. Addressing along the lines of formulating four-dimensional gravity as a gauge theory of the Poincaré group, spontaneously broken to the Lorentz, we attempt to construct a four-dimensional gravitational model on the fuzzy de Sitter spacetime. In turn, first we consider the SO(1,4) subgroup of the SO(1,5) algebra, in which we were led to, as we want to gauge the isometry part of the full symmetry. Then, the construction of a gauge theory on such a noncommutative space directs us to use an extension of the gauge group, the SO(1,5)×U(1), and fix its representation. Moreover, a 2-form dynamic gauge field is included in the theory for reasons of covariance of the transformation of the field strength tensor. Finally, the gauge theory is considered to be spontaneously broken to the Lorentz group with an extension of a U(1), i.e. SO(1,3)×U(1). The latter defines the four-dimensional noncommutative gravity action which can lead to equations of motion, whereas the breaking induces the imposition of constraints that will lead to expressions relating the gauge fields. It should be noted that we use the Euclidean signature for the formulation of the above programme.