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.

The moduli in a 4D N=1 heterotic compactification on an elliptic CY, as well as in the dual F-theoretic compactification, break into "base" parameters which are even (under the natural involution of the elliptic curves), and "fiber" or twisting parameters; the latter include a continuous part which is odd, as well as a discrete part. We interpret all the heterotic moduli in terms of cohomology groups of the spectral covers, and identify them with the corresponding F-theoretic moduli in a certain stable degeneration. The argument is based on the comparison of three geometric objects: the spectral and cameral covers and the ADE del Pezzo fibrations. For the continuous part of the twisting moduli, this amounts to an isomorphism between certain abelian varieties: the connected component of the heterotic Prym variety (a modified Jacobian) and the F-theoretic intermediate Jacobian. The comparison of the discrete part generalizes the matching of heterotic 5brane / F-theoretic 3brane impurities. 1 1 60 Λ, and therefore D ′ Z[W/W 0 ] = D ′ 1 60 Λ = 1 60 D ′ Λ = 1 60 60Λ = Λ.) As we promised in the previous section, this provides the justification for the matching of brane impurities.

We discuss the vacuum structure of type IIA/B Calabi-Yau string compactifications to four dimensions in the presence of n-form H-fluxes. These will lift the vacuum degeneracy in the Calabi-Yau moduli space, and for generic points in the moduli space, N = 2 supersymmetry will be broken. However, for certain 'aligned' choices of the H-flux vector, supersymmetric ground states are possible at the degeneration points of the Calabi-Yau geometry. We will investigate in detail the H-flux induced superpotential and the corresponding scalar potential at several degeneration points, such as the Calabi-Yau large volume limit, the conifold loci, the Seiberg-Witten points, the strong coupling point and the conformal points. Some emphasis is given to the question whether partial supersymmetry breaking can be realized at those points. We also relate the H-flux induced superpotential to the formalism of gauged N = 2 supergravity. Finally we point out the analogies between the Calabi-Yau vacuum structure due to H-fluxes and the attractor formalism of N = 2 black holes.

We consider N = 1 compactifications to four dimensions of heterotic string theory in the presence of fluxes. We show that up to order O (α ′2 ) the associated action can be written as a sum of squares of BPS-like quantities. In this way we prove that the equations of motion are solved by backgrounds which fulfill the supersymmetry conditions and the Bianchi identities. We also argue for the expression of the related superpotential and discuss the radial modulus stabilization for a class of examples.

In the framework of N = 1 supersymmetric string models given by the heterotic string on an elliptic Calabi-Yau π : Z → B together with a SU(n) bundle we compute the chiral matter content of the massless spectrum. For this purpose the net generation number, i.e. half the third Chern class, is computed from data related to the heterotic vector bundle in the spectral cover description; a non-technical introduction to that method is supplied. This invariant is, in the class of bundles considered, shown to be related to a discrete modulus which is the heterotic analogue of the F -theory four-flux. We consider also the relevant matter which is supported along certain curves in the base B and derive the net generation number again from the independent matter-related computation. We then illustrate these considerations with two applications. First we show that the construction leads to numerous 3 generation models of unbroken gauge group SU(5), SO(10) or E 6 . Secondly we discuss the closely related issue of the heterotic 5-brane/instanton transition resp. the F-theoretic 3-brane/instanton transition. The extra chiral matter in these transitions is related to the Hecke transform of the direct sum of the original bundle and the dissolved 5-brane along the intersection of their spectral covers. Finally we point to the corresponding F -theory interpretation of chiral matter from the intersection of 7-branes where the influence of four-flux on the twisting along the intersection curve plays a crucial role.