The hidden on-shell E 7(7) symmetry of maximal supergravity is usually discussed in a truncation from D = 11 to four dimensions. In this article, we reverse the logic and start from a theory with manifest off-shell E 7(7) symmetry inspired by West's coset construction. Following de Wit's and Nicolai's idea that a 4 + 56 dimensional "exceptional geometry" underlies maximal supergravity, we construct the corresponding Lagrangian and the supersymmetry variations for the 56 dimensional subsector. We prove that both the dynamics and the supersymmetry coincide with D = 11 supergravity in a truncation to d = 7 in the expected way. ‡

Abstract:We study the perturbative quantisation of N = 8 supergravity in a formulation where its E 7(7) symmetry is realised off-shell. Relying on the cancellation of SU(8) current anomalies we show that there are no anomalies for the non-linearly realised E 7(7) either; this result extends to all orders in perturbation theory. As a consequence, the e 7(7) Ward identities can be consistently implemented and imposed at all orders in perturbation theory, and therefore potential divergent counterterms must in particular respect the full non-linear E 7(7) symmetry.

We study the "fermionic billiards", i.e. the chaotic dynamics of the gravitino, that arise in the near-spacelike-singularity limit of elevendimensional supergravity and of its dimensional truncations (notably fourdimensional simple supergravity). By exploiting the gravity-coset correspondence, we show that the billiard dynamics of the gravitino is described by a 'spin extension' of the Weyl group of the hyperbolic Kac-Moody algebra E 10 . This spin extension is a discrete subgroup of (a spin covering of) the maximal compact subgroup K(E 10 ) of E 10 that is generated by ten (simple-root-related) idempotent elements of order 8. The 'super-billiard' that combines the bosonic and fermionic billiards is found to have a remarkably simple structure, which exhibits a striking analogy with a polarized photon propagating in the tendimensional Lorentzian Weyl chamber of E P 10 a=1 β a collapses at each spatial point). This limit shall also be referred to as the 'BKL-limit' in the following. Let us first recall that the 1 See Appendix A for our conventions in the Cremmer-Julia-Scherk action [20]. 2 The proper time T is related to t by dT = N (t, x)dt at each spatial point x [14].

We present an E 7(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 E 7(7) invariant action under general coordinate transformations. We also construct the conserved E 7(7) -Noether current of maximal supergravity and we conclude with comments on the implications of this manifest off-shell E 7(7) -symmetry for quantizing d = 4 N = 8 supergravity, in particular on the E 7(7) -action on phase space.

We establish a dynamical equivalence between the bosonic part of pure type I supergravity in D = 10 and a D = 1 nonlinear σ-model on the Kac-Moody coset space DE 10 /K(DE 10 ) if both theories are suitably truncated. To this end we make use of a decomposition of DE 10 under its regular SO(9, 9) subgroup. Our analysis also deals partly with the fermionic fields of the supergravity theory and we define corresponding representations of the generalized spatial Lorentz group K(DE 10 ).