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\({\mathfrak{h}=\mathfrak{h}_{-2} \oplus\mathfrak{h}_{-1} \oplus\mathfrak{h}_{0}}\) 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 \({\mathfrak{h}_\mathfrak{j}=\mathfrak{g}_\mathfrak{j}}\) 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.
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Communicated by X. Yin
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Figueroa-O’Farrill, J., Santi, A. Spencer Cohomology and 11-Dimensional Supergravity. Commun. Math. Phys. 349, 627–660 (2017). https://doi.org/10.1007/s00220-016-2700-1
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DOI: https://doi.org/10.1007/s00220-016-2700-1