Killing superalgebras for Lorentzian four-manifolds

TL;DR

This paper classifies the Killing superalgebras of Lorentzian four-manifolds with rigid supersymmetry by analyzing filtered deformations of the N=1 Poincaré superalgebra, linking them to supersymmetric backgrounds in supergravity.

Contribution

It introduces a novel approach to classify Killing superalgebras via filtered deformations and Spencer cohomology, providing a systematic classification of maximally supersymmetric backgrounds.

Findings

Classification of maximally supersymmetric backgrounds.

Identification of Killing superalgebras as filtered deformations.

Connection between Killing spinors and Lie superalgebras.

Abstract

We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian four-manifolds by re-interpreting them as filtered deformations of $\mathbb{Z}$-graded subalgebras with maximum odd dimension of the $N{=}1$ Poincar\'e superalgebra in four dimensions. Part of this calculation involves computing a Spencer cohomology group which, by analogy with a similar result in eleven dimensions, prescribes a notion of Killing spinor, which we identify with the defining condition for bosonic supersymmetric backgrounds of minimal off-shell supergravity in four dimensions. We prove that such Killing spinors always generate a Lie superalgebra, and that this Lie superalgebra is a filtered deformation of a subalgebra of the $N{=}1$ Poincar\'e superalgebra in four dimensions. Demanding the flatness of the connection defining the Killing spinors, we obtain equations satisfied by the maximally supersymmetric backgrounds. We solve these equations, arriving at the classification of maximally supersymmetric backgrounds whose associated Killing superalgebras are precisely the filtered deformations we classify in this paper.