Exceptional Field Theory II: E$_{7(7)}$

TL;DR

This work constructs a complete ${E}_{7(7)}$-covariant exceptional field theory on a $(4+56)$-dimensional spacetime with a covariant section constraint, organizing internal diffeomorphisms through an ${E}_{7(7)}$-covariant $E$-bracket and a tensor hierarchy that requires covariantly constrained compensator two-forms. The theory features non-abelian twisted self-duality relations for the 56 vector fields and a carefully designed action with kinetic terms for gravity, scalars, and vectors, plus a topological term and an ${E}_{7(7)}$-invariant potential; external diffeomorphisms fix all relative coefficients and ensure duality covariance. A key novelty is the inclusion of components of the eleven-dimensional dual graviton among the gauge fields, made consistent by the constrained compensator $B_{\mu u M}$ and the tensor hierarchy. The paper also demonstrates explicit embeddings of ${D=11}$ and ${ m IIB}$ supergravity by solving the section constraint and performing the required field redefinitions, illustrating how the EFT unifies higher-dimensional dynamics within a duality-covariant framework.

Abstract

We introduce exceptional field theory for the group E_{7(7)}, based on a (4+56)-dimensional spacetime subject to a covariant section condition. The `internal' generalized diffeomorphisms of the coordinates in the fundamental representation of E_{7(7)} are governed by a covariant `E-bracket', which is gauged by 56 vector fields. We construct the complete and unique set of field equations that is gauge invariant under generalized diffeomorphisms in the internal and external coordinates. Among them feature the non-abelian twisted self-duality equations for the 56 gauge vectors. We discuss the explicit solutions of the section condition describing the embedding of the full, untruncated 11-dimensional and type IIB supergravity, respectively. As a new feature compared to the previously constructed E_{6(6)} formulation, some components among the 56 gauge vectors descend from the 11-dimensional dual graviton but nevertheless allow for a consistent coupling by virtue of a covariantly constrained compensating 2-form gauge field.