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

TL;DR

This paper reformulates eleven-dimensional supergravity using generalised geometry with an $E_{d(d)} imes \\mathbb{R}^+$ structure, unifying bosonic and fermionic fields and expressing equations of motion as generalized Ricci tensor conditions.

Contribution

It introduces a new reformulation of supergravity incorporating fermions within a generalised geometry framework with $E_{d(d)} imes \\mathbb{R}^+$ symmetry, extending previous approaches.

Findings

Bosonic fields unify into a generalised metric.

Equations of motion correspond to vanishing of the generalised Ricci tensor.

Fermionic equations and supersymmetry variations are expressed in terms of the generalised connection D.

Abstract

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 $D$. 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 $\tilde{H}_4=\mathit{Spin}(5)$ and $\tilde{H}_7=\mathit{SU}(8)$ representations for $d=4$ and $d=7$.