Exceptional Field Theory I: $E_{6(6)}$ covariant Form of M-Theory and Type IIB

Abstract

We present the details of the recently constructed $E_{6(6)}$ covariant extension of 11-dimensional supergravity. This theory requires a 5+27 dimensional spacetime in which the `internal' coordinates transform in the $\overline{27}$ of $E_{6(6)}$. All fields are $E_{6(6)}$ tensors and transform under (gauged) internal generalized diffeomorphisms. The `Kaluza-Klein' vector field acts as a gauge field for the $E_{6(6)}$ covariant `E-bracket' rather than a Lie bracket, requiring the presence of two-forms akin to the tensor hierarchy of gauged supergravity. We construct the complete and unique action that is gauge invariant under generalized diffeomorphisms in the internal and external coordinates. The theory is subject to covariant section constraints on the derivatives, implying that only a subset of the extra 27 coordinates is physical. We give two solutions of the section constraints: the first preserves GL(6) and embeds the action of the complete (i.e. untruncated) 11-dimensional supergravity; the second preserves GL(5) x SL(2) and embeds complete type IIB supergravity. As a by-product, we thus obtain an off-shell action for type IIB supergravity.

52 pages, 1 figure, v2: version published in PRD, v3: typos in eqs. (5.22), (5.24) corrected

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• Branes are Waves and Monopoles
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• Duality Symmetric String and M-Theory
• Systematics of consistent truncations from generalised geometry
• Kaluza-Klein Spectrometry from Exceptional Field Theory
• The Geometry, Branes and Applications of Exceptional Field Theory
• Homogeneous Yang-Baxter deformations as generalized diffeomorphisms
• U-duality extension of Drinfel'd double
• A non-relativistic limit of M-theory and 11-dimensional membrane Newton-Cartan geometry
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• Generalised diffeomorphisms for E${}_9$
• E${}_{6(6)}$ Exceptional Drinfel'd Algebras
• Beyond E11
• Extended geometries
• E${}_9$ exceptional field theory I. The potential
• Exotic branes and non-geometric fluxes
• IIB Supergravity and the E6(6) covariant vector-tensor hierarchy
• Supersymmetric Backgrounds and Generalised Special Holonomy
• T-duality and non-geometric solutions from double geometry
• Supersymmetric AdS${}_7$ and AdS${}_6$ vacua and their consistent truncations with vector multiplets
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• Dualising consistent IIA / IIB truncations
• E${}_{6(6)}$ Exceptional Field Theory: Review and Embedding of Type IIB
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• Non-abelian U-duality at work
• Higher Gauge Structures in Double and Exceptional Field Theory
• Lobotomy of Flux Compactifications
• $L_{\infty }$ algebras and Tensor Hierarchies in Exceptional Field Theory and Gauged Supergravity
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• Exceptional M-brane sigma models and $\eta$-symbols
• Gaugings from E7 Extended Geometries
• The Coulomb branch of N=4 SYM and dilatonic scions in supergravity
• U-gravity : $\mathbf{SL}(N)$
• Critical Super F-theories
• Locally non-geometric fluxes and missing momenta in M-theory
• Tensor hierarchy algebra extensions of over-extended Kac--Moody algebras
• Super-Chern-Simons spectra from Exceptional Field Theory
• New gaugings and non-geometry
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• Unified non-metric (1,0) tensor-Einstein supergravity theories and (4,0) supergravity in six dimensions
• Born sigma model for branes in exceptional geometry
• Fluxes in Exceptional Field Theory and Threebrane Sigma-Models
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• E11 and exceptional field theory
• Particle actions and brane tensions from double and exceptional geometry
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• Type II strings are Exceptional
• Infinity-enhancing of Leibniz algebras
• Consistent truncations in higher derivative supergravity
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• One-Loop $\beta$ Function of the Double Sigma Model with Constant Background
• Towards Weakly Constrained Double Field Theory
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• Finite Transformations in Doubled and Exceptional Space
• A Kaluza-Klein Approach to Double and Exceptional Field Theory
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• Extended geometry of magical supergravities
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• Uplifts of maximal supergravities and transitions to non-geometric vacua
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• Generalized Cartan Calculus in general dimension
• Strings, Branes and the Self-dual Solutions of Exceptional Field Theory
• The teleparallel complex
• Geometry of ${\mathbb{R}}^{+}\times E_{3(3)}$ Exceptional Field Theory and F-theory
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• Exceptional Field Theory for $E_{6(6)}$ supergravity
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• Classical worldvolumes as generalised geodesics
• Tri-vector deformations with external fluxes
• Exotic potentials and Bianchi identities in SL(5) exceptional field theory
• $L_{\infty }$ algebras for extended geometry
• Solution-generating transformations in duality-invariant theories and the fluid/gravity correspondence
• Exotic branes and mixed-symmetry potentials II: Duality rules and exceptional $p$-form gauge fields
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• Canonical analysis of E${}_{6(6)}(\mathbb{R})$ invariant five dimensional (super-)gravity
• M Theory from F Theory
• Open F-branes
• Exotic Branes from del Pezzo Surfaces
• C-spaces, generalized geometry and double field theory
• Non-isometric U-dualities
• Reductions of Exceptional Field Theories
• N-Extended Lorentzian Kac-Moody algebras
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• Duality Covariant Solutions in Extended Field Theories
• Double Field Theory and Pseudo-Supersymmetry
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• Algebraic structures in exceptional geometry
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• Half-maximal Extended Drinfel'd Algebras
• Localized Kaluza-Klein 6-brane
• Extended Field Theories as higher Kaluza-Klein theories
• Exotic Aspects of Extended Field Theories