11D supergravity as a gauge theory for the M-algebra

The eleven-dimensional gravitational action invariant under local Poincaré transformations is given by the dimensional continuation of the Euler class of ten dimensions. Here we show that the supersymmetric extension of this action leads, through the Noether procedure, to a theory with the local symmetry group given by the M-algebra. The fields of the theory are the vielbein e^a_µ, the Lorentz (spin) connection ω^ab_µ , one gravitino (ψ_µ), and two 1-forms, b^ab_µ and b^abcde_µ , which transform as antisymmetric Lorentz tensors. These fields are components of a single connection for the M-algebra and the supersymmetric Lagrangian can be seen to be a Chern-Simons form. The dynamics has a multiplicity of degenerate vacua without propagating degrees of freedom. The theory is shown to admit solutions of the form S^10−d × X_d+₁, where X_d+₁ is a warped product of R with a d-dimensional spacetime. Among this class, the gravitational effective action describes a propagating graviton only if d = 4 and the spacetime has positive cosmological constant. The perturbations around this solution reproduce linearized General Relativity around four-dimensional de Sitter spacetime.