Supergravity as Generalised Geometry II: $E_{d(d)} \times \mathbb{R}^+$ and M theory

Understand

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 $d\leq7$.

• The reformation has a $E_{d(d)} \times \mathbb{R}^+$ structure group and is has a local $\tilde{H}_d$ symmetry, where $\tilde{H}_d$ 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 generalised 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 $D$.