E7(7) invariant Lagrangian of d = 4 $$ \mathcal{N} $$ = 8 supergravity

Abstract

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.

$ {E_d}_{(d)}\times {{\mathbb{R}}^{+}} $ generalised geometry, connections and M theory

“…Physically, it is known to be related to the "warp factor" of warped supergravity reductions. The need for this extra factor in the context of E 7(7) geometries has already been identified in [12,13,18,59]. …”

“…Following the original observation that the dimensionally reduced supergravity has a hidden E d(d) global symmetry [1][2][3], formulations using exceptional groups, as well as their infinitedimensional extensions, have appeared in various guises [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].…”

“…The corresponding E 7(7) non-linear realisation, following the construction of West and using the finite extended spacetime originally conjectured in [9], has been discussed in considerable detail by Hillmann [12,13]. Truncating to conventional spacetime, he was able to show an equivalence with [4,5], and, again, the current paper can be viewed as the corresponding geometrical formulation, analogous to the relation between gravity as Riemannian geometry and as a non-linear realisation of GL(4) ⋉ R 4 introduced by Borisov and Ogievetsky [33].…”

Generalised space-time and gauge transformations

“…In 2007 all gauged maximal supergravities in five dimensions we constructed, for the first time, by taking the fields to depend on some of the generalised coordinates [9]. In 2009 the papers [22,23] computed the E 11 ⊗ s l 1 non-linear realisation in four dimension keeping the 56 Lorentz scalar coordinates in addition to the usual four coordinates of space-time, but only the fields at level zero, that is, the metric and the scalar fields. More recently a more systematic construction of the E 11 ⊗ s l 1 has been undertaken in eleven dimensions [24] and four dimensions [25].…”

“…Unfortunately there has not so far been found a clear way to extend this method to the full E 11 algebra. A different approach was taken in references [22,23] mentioned above; the undetermined coefficients that arise when constructing the dynamics of the E 11 ⊗ s l 1 non-linear realisation in four dimensions at low levels were fixed by demanding that the theory be diffeomorphism and gauge invariant.…”