The recent mathematical literature introduces generalised geometries which are defined by a reduction from the structure group SO(d, d) of the vector bundle T d ⊕ T d * to a special subgroup. In this article we show that compactification of IIB superstring vacua on 7manifolds with two covariantly constant spinors leads to a generalised G 2 -structure associated with a reduction from SO(7, 7) to G 2 ×G 2 . We also consider compactifications on 6-manifolds where analogously we obtain a generalised SU (3)-structure associated with SU (3) × SU (3), and show how these relate to generalised G 2 -structures.
We consider compactifications of M-theory on 7-manifolds in the presence of 4-form fluxes, which leave at least four supercharges unbroken. We focus especially on the case, where the 7-manifold supports two spinors which are SU(3) singlets and the fluxes appear as specific SU(3) structures. We derive the constraints on the fluxes imposed by supersymmetry and calculate the resulting 4-dimensional superpotential.
We consider topological sigma models with generalized Kähler target spaces. The mirror map is constructed explicitly for a special class of target spaces and the topological A and B model are shown to be mirror pairs in the sense that the observables, the instantons and the anomalies are mapped to each other. We also apply the construction to open topological models and show that A branes are mapped to B branes. Furthermore, we demonstrate a relation between the field strength on the brane and a two-vector on the mirror manifold.
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