Spheres, generalised parallelisability and consistent truncations

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 math formula geometries admit such generalised parallelisations with an math formula frame algebra. Thus we show that the remarkable consistent truncations on S3, S4, S5 and S7 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 S5

Ectoplasm with an Edge

The construction of supersymmetric invariant actions on a spacetime manifold with a boundary is carried out using the "ectoplasm" formalism for the construction of closed forms in superspace. Non-trivial actions are obtained from the pull-backs to the bosonic bodies of closed but non-exact forms in superspace; finding supersymmetric invariants thus becomes a cohomology problem. For a spacetime with a boundary, the appropriate mathematical language changes to relative cohomology, which we use to give a general formulation of off-shell supersymmetric invariants in the presence of boundaries. We also relate this construction to the superembedding formalism for the construction of brane actions, and we give examples with bulk spacetimes of dimension 3, 4 and 5. The closed superform in the 5D example needs to be constructed as a Chern-Simons type of invariant, obtained from a closed 6-form displaying Weil triviality.Comment: 25 page

Supergravity as generalised geometry II: Ed(d) x R+ and M theory

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 Ed(d) × R + structure group and it has a local H˜ d symmetry, where H˜ d is the double cover of the maximally compact subgroup of Ed(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 H˜ 4 = Spin(5) and H˜ 7 = SU(8) representations for d = 4 and d = 7

Exceptional generalised geometry for massive IIA and consistent reductions

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

New Gaugings and Non-Geometry

We discuss the possible realisation in string/M theory of the recently discovered family of four-dimensional maximal math formula gauged supergravities, and of an analogous family of seven-dimensional half-maximal math formula gauged supergravities. We first prove a no-go theorem that neither class of gaugings can be realised via a compactification that is locally described by ten- or eleven-dimensional supergravity. In the language of Double Field Theory and its M theory analogue, this implies that the section condition must be violated. Introducing the minimal number of additional coordinates possible, we then show that the standard S3 and S7 compactifications of ten- and eleven-dimensional supergravity admit a new class of section-violating generalised frames with a generalised Lie derivative algebra that reproduces the embedding tensor of the math formula and math formula gaugings respectively. The physical meaning, if any, of these constructions is unclear. They highlight a number of the issues that arise when attempting to apply the formalism of Double Field Theory to non-toroidal backgrounds. Using a naive brane charge quantisation to determine the periodicities of the additional coordinates restricts the math formula gaugings to an infinite discrete set and excludes all the math formula gaugings other than the standard one

Ed(d) × R+ generalised geometry, connections and M theory

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 Ed(d)×R+Ed(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 Hd, the maximally compact subgroup of Ed(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 Ed(d) symmetry, as well as the connections to flux compactifications and the embedding tensor formalism

Supergravity Fluxes and Generalised Geometry : LMS/EPSRC Durham Symposium on Higher Structures in M theory

11 pages, Contribution to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 2018. © 2019 WILEY‐VCH Verlag GmbH & Co. KGaA, WeinheimWe briefly review the description of the internal sector of supergravity theories in the language of generalised geometry and how this gives rise to a description of supersymmetric backgrounds as integrable geometric structures. We then review recent work, featuring holomorphic Courant algebroids, on the description of $\mathcal N=1$ heterotic flux vacua. This work studied the finite deformation problem of the Hull-Strominger system, guided by consideration of the superpotential functional on the relevant space of geometries. It rewrote the system in terms of the Maurer-Cartan set of a particular $L_\infty$-algebra associated to a holomorphic Courant algebroid, with the superpotential itself becoming an analogue of a holomorphic Chern-Simons functional.Peer reviewedFinal Accepted Versio