E7(7) invariant Lagrangian of d=4 N=8 supergravity

TL;DR

This paper constructs an off-shell $E_{7(7)}$-invariant Lagrangian for $d=4$ $\mathcal{N}=8$ supergravity using 56 vector fields and an ADM-type split, avoiding Lagrange multipliers. It demonstrates superinvariance and SUSY-closure for the modified bosonic terms, derives a manifest $E_{7(7)}$ Noether current and charge, and develops a Hamiltonian framework that establishes general covariance via the Dirac algebra and a first-class diffeomorphism action on the vector sector. The formalism reconciles the $E_{7(7)}$ symmetry with standard supergravity by showing equivalence to the Cremmer–Julia theory on-shell and provides a robust path to quantization in phase space through Dirac brackets and a fully covariant Noether structure. These results offer new insights into duality symmetries, gauge dynamics of the 56-vector system, and potential implications for UV properties and extended symmetry structures in M-theory contexts.

Abstract

We present an E7(7) invariant Lagrangian that leads to the equations of motion of d=4 N=8 supergravity without using Lagrange multipliers. The superinvariance of this new action and the closure of the supersymmetry algebra are proved explicitly for the terms that differ from the Cremmer--Julia formulation. Since the diffeomorphism symmetry is not realized in the standard way on the vector fields, we switch to the Hamiltonian formulation in order to prove the invariance of the E7(7) invariant action under general coordinate transformations. We also construct the conserved E7(7)-Noether current of maximal supergravity and we conclude with comments on the implications of this manifest off-shell E7(7)-symmetry for quantizing d=4 N=8 supergravity, in particular on the E7(7)-action on phase space.