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

TL;DR

The paper develops an $E_{6(6)}$-covariant exceptional field theory (EFT) formulated on a $5+27$ dimensional spacetime with internal coordinates in the $\bar{\mathbf{27}}$ of $E_{6(6)}$ and subject to a covariant section constraint. It constructs a unique gauge-invariant action including a tensor hierarchy and a topological term, and demonstrates two inequivalent section choices that reproduce the full $D=11$ supergravity and the full Type IIB supergravity, respectively, yielding an off-shell action for IIB. The EFT framework generalizes double field theory to the exceptional series, unifying M-theory and Type IIB at the level of a higher-dimensional, duality-covariant formulation and clarifying how higher-dimensional diffeomorphisms and gauge symmetries organize into an $E_{6(6)}$ structure. External diffeomorphisms fix the relative coefficients and ensure the non-manifest covariance aligns with standard supergravity reductions, while the section constraint selects the physical subspace of the extended coordinates.

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 $\bar{\bf 27}$ 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.

Figures (1)

• Figure 1: ${\rm E}_{6(6)}$ EFT embedding of $D=11$ supergravity, IIB supergravity, and $D=5$ supergravity.