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) × 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.

We reformulate ten-dimensional type II supergravity as a generalised geometrical analogue of Einstein gravity, defined by an O(9, 1)×O(1, 9) ⊂ O(10, 10)×R + structure on the generalised tangent space. Using the notion of generalised connection and torsion, we introduce the analogue of the Levi-Civita connection, and derive the corresponding tensorial measures of generalised curvature. We show how, to leading order in the fermion fields, these structures allow one to rewrite the action, equations of motion and supersymmetry variations in a simple, manifestly Spin(9, 1) ×Spin(1, 9)-covariant form. The same formalism also describes d-dimensional compactifications to flat space. 1 a.coimbra08@imperial.ac.uk

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 localH d symmetry, whereH 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 ofH 4 = Spin(5) andH 7 = SU(8) representations for d = 4 and d = 7.

We show that generalised geometry gives a unified description of maximally supersymmetric consistent truncations of ten-and eleven-dimensional supergravity. In all cases the reduction manifold admits a "generalised parallelisation" with a frame algebra with constant coefficients. The consistent truncation then arises as a generalised version of a conventional Scherk-Schwarz reduction with the frame algebra encoding the embedding tensor of the reduced theory. The key new result is that all round-sphere S d geometries admit such generalised parallelisations with an SO(d + 1) frame algebra. Thus we show that the remarkable consistent truncations on S 3 , S 4 , S 5 and S 7 are in fact simply generalised Scherk-Schwarz reductions. This description leads directly to the standard non-linear scalar-field ansatze and as an application we give the full scalar-field ansatz for the type IIB truncation on S 5 .

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We define intrinsic torsion in generalised geometry and use it to introduce a new notion of generalised special holonomy. We then consider generic warped supersymmetric flux compactifications of M theory and Type II of the form R D−1,1 × M . Using the language of E d(d) ×R + generalised geometry, we show that, for D ≥ 4, preserving minimal supersymmetry is equivalent to the manifold M having generalised special holonomy and list the relevant holonomy groups. We conjecture that this result extends to backgrounds preserving any number of supersymmetries. As a prime example, we consider N = 1 in D = 4. The corresponding generalised special holonomy group is SU (7), giving the natural M theory extension to the notion of a G 2 manifold, and, for Type II backgrounds, reformulating the pure spinor SU (3) × SU (3) conditions as an integrable structure.

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Abstract:We develop an exceptional generalised geometry formalism for massive type IIA supergravity. In particular, we construct a deformation of the generalised Lie derivative, which generates the type IIA gauge transformations as modified by the Romans mass. We apply this new framework to consistent Kaluza-Klein reductions preserving maximal supersymmetry. We find a generalised parallelisation of the exceptional tangent bundle on S 6 , and from this reproduce the consistent truncation ansatz and embedding tensor leading to dyonically gauged ISO(7) supergravity in four dimensions. We also discuss closely related hyperboloid reductions, yielding a dyonic ISO(p, 7 − p) gauging. Finally, while for vanishing Romans mass we find a generalised parallelisation on S d , d = 4, 3, 2, leading to a maximally supersymmetric reduction with gauge group SO(d + 1) (or larger), we provide evidence that an analogous reduction does not exist in the massive theory.