| Summary: | We reformulate type II supergravity and dimensional restrictions of eleven- dimensional supergravity as generalised geometrical analogues of Einstein gravity. The bosonic symmetries are generated by generalised vectors, while the bosonic fields are unified into a generalised metric. The generalised tangent space features a natural action of the relevant (continuous) duality group. Also, the analogues of orthonormal frames for the generalised metric are related by the well-known enhanced local symmetry groups, which provide the analogue of the local Lorentz symmetry in general relativity. Generalised connections and torsion feature prominently in the construction, and we show that the analogue of the Levi-Civita connection is not uniquely determined by metric compatibility and vanishing torsion. However, connections of this type can be used to extract the derivative operators which appear in the supergravity equations, and the undetermined pieces of the connection cancel out from these, leaving the required unique expressions. We find that the bosonic action and equations of motion can be interpreted as generalised curvatures, while the derivative operators appearing in the supersymmetry variations and equations of motion for the fermions become very simple expressions in terms of the generalised connection. In the final chapter, the construction is used to reformulate supersymmetric flux backgrounds as torsion-free generalised G-structures. This is the direct analogue of the special holonomy condition which arises for supersymmetric backgrounds without flux in ordinary Riemannian geometry.
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