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\({E_d}_{(d)}\times \mathbb{R}^{+}\) generalised geometry, connections and M theory. (English) Zbl 1333.83220
Summary: We show that generalised geometry gives a unified description of bosonic eleven-dimensional supergravity restricted to a \(d\)-dimensional manifold for all \(d\leq7\). The theory is based on an extended tangent space which admits a natural \(E_{d(d)}\times\mathbb R^+\) action. The bosonic degrees of freedom are unified as a ‘generalised metric’, as are the diffeomorphism and gauge symmetries, while the local \(O(d)\) symmetry is promoted to \(H_d\), the maximally compact subgroup of \(E_{d(d)}\). We introduce the analogue of the Levi-Civita connection and the Ricci tensor and show that the bosonic action and equations of motion are simply given by the generalised Ricci scalar and the vanishing of the generalised Ricci tensor respectively. The formalism also gives a unified description of the bosonic NSNS and RR sectors of type II supergravity in \(d-1\) dimensions. Locally the formulation also describes M-theory variants of double field theory and we derive the corresponding section condition in general dimension. We comment on the relation to other approaches to M theory with \(E_{d(d)}\) symmetry, as well as the connections to flux compactifications and the embedding tensor formalism.

MSC:
83E50 Supergravity
53C05 Connections (general theory)
53Z05 Applications of differential geometry to physics
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