Killing (super)algebras for generalised spin manifolds

TL;DR

This paper introduces a new concept of Killing superalgebras for generalised spin structures on pseudo-Riemannian manifolds, extending classical symmetries to include gauge transformations and analyzing their algebraic deformations.

Contribution

It defines Killing superalgebras for generalised spin structures, incorporating gauge transformations, and studies their algebraic deformations using Spencer cohomology.

Findings

Killing superalgebras extend classical symmetries with gauge transformations.

Deformations of these algebras relate to Poincaré superalgebra extensions.

Reconstruction of supersymmetric backgrounds from algebraic deformations.

Abstract

We define the notion of a Killing (super)algebra for a connection on a spinor bundle associated to a generalised spin structure on a pseudo-Riemannian manifold of any signature. We are led naturally to include in the even subspace not only Killing vectors but also certain infinitesimal gauge transformations, and we show that the definition of the (super)algebra requires, in addition to the spinor connection and a Dirac current, a map to pair spinor fields into infinitesimal gauge transformations. We show that these (super)algebras are filtered subdeformations of (an analogue of) the Poincar\'e superalgebra extended by the \(R\)-symmetry algebra. By employing Spencer cohomology, we study such deformations from a purely algebraic point of view and, at least in the case of Lorentzian signature and high supersymmetry, identify the subclass of deformations to which the Killing superalgebras belong. Finally, we show that, with some caveats, one can reconstruct a supersymmetric background geometry from such a deformation as a homogeneous space on which the deformation is realised as a subalgebra of the Killing superalgebra.