Killing (Super)Algebras Associated to Connections on Spinors

TL;DR

This paper generalizes Killing superalgebras to various dimensions and signatures, linking geometric structures on spinor bundles with algebraic deformations, and classifies their realizations on Lorentzian manifolds.

Contribution

It introduces a broad framework for Killing (super)algebras associated with connections on spinors, extending previous supergravity models and analyzing their algebraic deformations.

Findings

Killing superalgebras are filtered deformations of graded subalgebras of the Poincaré superalgebra.

Provides conditions for the existence of Killing (super)algebras based on spinor bundle connections.

Classifies deformations and constructs examples on Lorentzian spin manifolds.

Abstract

We generalise the notion of a Killing superalgebra, which arises in the physics literature on supergravity, to general dimension, signature and choice of spinor module and Dirac current. We also allow for Lie algebras as well as superalgebras, capturing a set of examples previously defined using geometric Killing spinors on higher-dimensional spheres. Our definition requires a connection on a spinor bundle - provided by supersymmetry transformations in the supergravity examples and by the Killing spinor equation on the spheres - and we obtain a set of sufficient conditions on such a connection for the Killing (super)algebra to exist. We show that these Lie (super)algebras are filtered deformations of graded subalgebras of (a generalisation of) the Poincar\'e superalgebra and then study such deformations abstractly using Spencer cohomology. In the highly supersymmetric Lorentzian case, we describe the filtered subdeformations which are of the appropriate form to arise as Killing superalgebras, lay out a classification scheme for their odd-generated subalgebras and prove that, under certain technical conditions, there exist homogeneous Lorentzian spin manifolds on which these deformations are realised as Killing superalgebras. Our results generalise previous work in the 11-dimensional supergravity literature.