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

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

46 pages

References in corpus (6)

• Double Field Theory
• Generalised Geometry for M-Theory
• Differential geometry with a projection: Application to double field theory
• Generalized E(7(7)) coset dynamics and D=11 supergravity
• E(7(7)) and d=11 supergravity
• E11 and Supersymmetry

Cited by in corpus (31)

• Extended geometry and gauged maximal supergravity
• Covariant action for a string in doubled yet gauged spacetime
• Generalised geometry for string corrections
• M-theory and Type IIB from a Duality Manifest Action
• The Odd story of ${\alpha }^{\prime }$-corrections
• Gauged supergravities in 5 and 6 dimensions from generalised Scherk-Schwarz reductions
• U-geometry : SL(5)
• An Action for F-theory: $\mathrm{SL}(2)\times {\mathbb{R}}^{+}$ Exceptional Field Theory
• Fermions and Supersymmetry in ${\mathrm{E}}_{6(6)}$ Exceptional Field Theory
• Effective actions of non-geometric fivebranes
• Generalized gravity from modified DFT
• Duality Symmetric String and M-Theory
• The gauge structure of Exceptional Field Theories and the tensor hierarchy
• Generalised diffeomorphisms for E${}_9$
• Exactly marginal deformations from exceptional generalised geometry
• On the moduli space of spontaneously broken N = 8 supergravity
• Type IIB flux vacua from G-theory II
• Generalised Space-time and Gauge Transformations
• On the exceptional generalised Lie derivative for $d\ge 7$
• Gaugings from E7 Extended Geometries
• New gaugings and non-geometry
• U-folds as K3 fibrations
• Extended generalized geometry and a DBI-type effective action for branes ending on branes
• Gauge Transformation of Double Field Theory for Open String
• Supersymmetric AdS backgrounds and weak generalised holonomy
• Dimensional Reduction of the Generalized DBI
• Generalised Calibrations in AdS backgrounds from Exceptional Sasaki-Einstein Structures
• C-spaces, generalized geometry and double field theory
• Geometry of Membrane Sigma Models
• Algebraic structures in exceptional geometry
• F-theory amplitudes