Supergravity as generalised geometry II: E d(d) × ℝ+ and M theory

We reformulate eleven-dimensional supergravity, including fermions, in terms of generalised geometry, for spacetimes that are warped products of Minkowski space with a d-dimensional manifold M with d ≤ 7. The reformulation has an E d (d) × ℝ+ structure group and it has a local {Hd} symmetry, where {Hd} is the double cover of the maximally compact subgroup of E d (d). The bosonic degrees for freedom unify into a generalised metric, and, defining the generalisked analogue D of the Levi-Civita connection, one finds that the corresponding equations of motion are the vanishing of the generalised Ricci tensor. To leading order, we show that the fermionic equations of motion, action and supersymmetry variations can all be written in terms of D. Although we will not give the detailed decompositions, this reformulation is equally applicable to type IIA or IIB supergravity restricted to a (d − 1) -dimensional manifold. For completeness we give explicit expressions in terms of {H₄} = Spin (5) and {H₇} = SU (8) representations for d = 4 and d = 7.