On the algebraic structure of Killing superalgebras

We study the algebraic structure of the Killing superalgebra of a supersymmetric background of $11$-dimensional supergravity and show that it is isomorphic to a filtered deformation of a $\mathbb{Z}$-graded subalgebra of the Poincaré superalgebra. We are able to map the classification problem for highly supersymmetric backgrounds (i.e., those which preserve more than half the supersymmetry) to the classification problem of a certain class of filtered deformations of graded subalgebras of the Poincaré superalgebra. We show that one can reconstruct a highly supersymmetric background from its Killing superalgebra; in so doing, we relate the bosonic field equations of $11$-dimensional supergravity to the Jacobi identity of the Killing superalgebra and show in this way that preserving more than half the supersymmetry implies the bosonic field equations.

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We are grateful to the anonymous referee for comments on a previous versionof the paper. We believe that the paper has improved as a result of the peer-reviewprocess. The present version of the paper was finished while we wereparticipating in the workshop “Geometry, Gravity and Supersymmetry” atthe Mainz ITP. It is our pleasure to thank them for their hospitality and forproviding a pleasant collaborative environment.

The research of JMF is supported in part by the grant ST/L000458/1 “Particle Theory at the Higgs Centre” from the UK Science and Technology Facilities Council. The research of AS is fully supported by a Marie-Curie research fellowship of the ”Istituto Nazionale di Alta Matematica” (Italy). We are grateful to these funding agencies for their support.

Published 8 March 2018