On the algebraic structure of Killing superalgebras

TL;DR

This paper explores the algebraic structure of Killing superalgebras in 11-dimensional supergravity, showing they are filtered deformations of graded subalgebras of the Poincaré superalgebra, and links supersymmetry preservation to the bosonic field equations.

Contribution

It establishes a correspondence between highly supersymmetric backgrounds and filtered deformations of Poincaré superalgebra substructures, providing a new classification approach.

Findings

Killing superalgebra is a filtered deformation of a graded subalgebra of the Poincaré superalgebra.

Highly supersymmetric backgrounds can be reconstructed from their Killing superalgebras.

Preserving more than half the supersymmetry implies the bosonic field equations of supergravity.

Abstract

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 Poincar\'e 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 Poincar\'e 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.