Abstract
We show that generalised geometry gives a unified description of bosonic eleven-dimensional supergravity restricted to a d-dimensional manifold for all d ≤ 7. 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.
Highlights
The idea that eleven-dimensional supergravity, or for that matter M theory, might have a more unified description incorporating a larger symmetry group is a long-standing one
We show that generalised geometry gives a unified description of bosonic eleven-dimensional supergravity restricted to a d-dimensional manifold for all d ≤ 7
Even though here we focus our attention on the bosonic sector, we will find that, the supersymmetry variations of the fermions are already encoded by the geometry
Summary
The idea that eleven-dimensional supergravity, or for that matter M theory, might have a more unified description incorporating a larger symmetry group is a long-standing one. We find a generic Ed(d) covariant form of the “section condition” [17] that encodes the restriction of the M theory version of double field theory to d coordinates At their core, generalised geometries1 [19, 20] rely on the idea of extending the tangent space of a manifold M , such that it can accommodate a larger symmetry group that includes diffeomorphisms and the gauge transformations of supergravity. The original version of generalised geometry was extended by Hull [21] and Pacheco and Waldram [22] to include the symmetries appearing in M theory This gives a generalised tangent space E ≃ T M ⊕ Λ2T ∗M ⊕ Λ5T ∗M ⊕ T ∗M ⊗ Λ7T ∗M , relevant to eleven-dimensional supergravity restricted to d ≤ 7 dimensions and admitting a natural Ed(d) structure. The need for this extra factor in the context of E7(7) geometries has already been identified in [12, 13, 18, 59]
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