Spencer cohomology and eleven-dimensional supergravity

TL;DR

This paper classifies maximally supersymmetric backgrounds in eleven-dimensional supergravity using Lie algebra techniques, identifying all relevant deformations and their associated symmetry superalgebras.

Contribution

It introduces a Lie algebraic approach to classify supersymmetric backgrounds and identifies all relevant deformations of the Poincaré superalgebra.

Findings

Classification of all filtered deformations of certain subalgebras

Identification of three non-flat maximally supersymmetric backgrounds

Connection between gravitino variation and Spencer cocycle

Abstract

We recover the classification of the maximally supersymmetric bosonic backgrounds of eleven-dimensional supergravity by Lie algebraic means. We classify all filtered deformations of the $\mathbb Z$-graded subalgebras $\mathfrak{h}=\mathfrak{h}_{-2}\oplus\mathfrak{h}_{-1}\oplus\mathfrak{h}_{0}$ of the Poincar\'e superalgebra $\mathfrak{g}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}=V\oplus S\oplus \mathfrak{so}(V)$ which differ only in zero degree, that is $\mathfrak{h}_0\subset\mathfrak{g}_0$ and $\mathfrak{h}_j=\mathfrak{g}_j$ for $j<0$. Aside from the Poincar\'e 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.