Spencer Cohomology and 11-Dimensional Supergravity

Abstract

We recover the classification of the maximally supersymmetric bosonic backgrounds of 11-dimensional supergravity by Lie algebraic means. We classify all filtered deformations of the {{mathbb{Z}}}-graded subalgebras{mathfrak{h}=mathfrak{h}_{-2}oplusmathfrak{h}_{-1}oplusmathfrak{h}_{0}} of the Poincaré superalgebra {mathfrak{g}=mathfrak{g}_{-2}oplusmathfrak{g}_{-1}oplusmathfrak{g}_{0}=VoplusSoplus mathfrak{so}(V)} which differ only in zero degree, that is {mathfrak{h}_0subsetmathfrak{g}_0} and {mathfrak{h}_mathfrak{j}=mathfrak{g}_mathfrak{j}} for {mathfrak{j}<0}. Aside from the Poincaré superalgebra itself and its {{mathbb{Z}}}-graded subalgebras, there are only three other Lie superalgebras, which are the symmetry superalgebras of the non-flat maximally supersymmetric backgrounds. In passing we identify the gravitino variation with (a component of) a Spencer cocycle.