Einstein-Cartan Calculus for Exceptional Geometry

Abstract

In this paper we establish and clarify the link between the recently found E7 generalised geometric structures, which are based on the SU(8) invariant reformulation of D=11 supergravity proposed long ago, and newer results obtained in the framework of recent approaches to generalised geometry, where E7 duality is built in and manifest from the outset. In making this connection, the so-called generalised vielbein postulate plays a key role. We explicitly show how this postulate can be used to define an E7 valued affine connection and an associated covariant derivative, which yields a generalised curvature tensor for the E7 based exceptional geometry. The analysis of the generalised vielbein postulate also provides a natural explanation for the emergence of the embedding tensor from higher dimensions.

18 pages; v2: we have added some minor calculational details

References in corpus (5)

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

Cited by in corpus (15)

• Supersymmetric E${}_{7(7)}$ Exceptional Field Theory
• FRW in quadratic form of $f(T)$ gravitational theories
• Reconstruction, Thermodynamics and Stability of $\Lambda$CDM Model in $f(T,\mathcal{T})$ Gravity
• Beyond E11
• On supersymmetric E11 exceptional field theory
• Seeking the balance: Patching double and exceptional field theories
• An SO(3)$\times$SO(3) invariant solution of $D=11$ supergravity
• E${}_{7(7)}$ Exceptional Field Theory in Superspace
• Gaugings from E7 Extended Geometries
• Exceptional versus superPoincaré algebra as the defining symmetry of maximal supergravity
• Non-linear uplift Ansätze for the internal metric and the four-form field-strength of maximal supergravity
• A geometric formulation of exceptional field theory
• The E10 Wheeler-DeWitt operator at low levels
• About the (in)equivalence between holonomic versus non-holonomic theories of gravity
• Canonical analysis of E${}_{6(6)}(\mathbb{R})$ invariant five dimensional (super-)gravity