Generalised Geometries, Constrained Critical Points and Ramond–Ramond Fields

We describe the geometry of type II string compactifications to 6– and 7–dimensional backgrounds with Ramond–Ramond–fields in terms of principal fibre bundle as generalised SU(3)– and G2– structures. We endow these structures with an integrability condition which can be characterised as a constrained critical point for the generalised Hitchin functional and provides a solution to the supersymmetry variations. Moreover, we discuss topological and geometrical properties of generalised SU(3)–structures which we show to coincide with the notion of a generalised Calabi-Yau metric in dimension 6. Subj. Class.: special Riemannian manifolds, generalised geometries, supergravity, string theory 2000 MSC: 53C10, 53C25, 58E15, 83E30, 83E50