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 Poincaré 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 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.

32 pages (v2: minor comments, reference added, final version to be published in CMP)

References in corpus (3)

• (M-theory-)Killing spinors on symmetric spaces
• Deformations of M-theory Killing superalgebras
• Homogeneous M2 duals

Cited by in corpus (11)

• Gauge enhancement of super M-branes via parametrized stable homotopy theory
• Real ADE-equivariant (co)homotopy and Super M-branes
• Killing superalgebras for Lorentzian four-manifolds
• Eleven-dimensional supergravity from filtered subdeformations of the Poincaré superalgebra
• Killing superalgebras for Lorentzian six-manifolds
• Non-Relativistic Supersymmetry on Curved Three-Manifolds
• Generalized symmetry superalgebras
• Components of Eleven-dimensional Supergravity with Four Off-shell Supersymmetries
• Supergravities and Branes from Hilbert-Poincaré Series
• Killing superalgebras for lorentzian five-manifolds
• Decomposable $(4,7)$ solutions in eleven-dimensional supergravity