Spencer cohomology and eleven-dimensional supergravity

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.