arXiv subjects

Andrew D. K. Beckett

Publications and source records attributed to Andrew D. K. Beckett.

4 recordsLinked to original sources

arXiv2026

Killing (super)algebras for generalised spin manifolds

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 Poincaré 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.

math.DG↗

arXiv2026

Notes on generalised spin structures

We review some definitions and basic notions relating to generalised spin structures and introduce the notion of reducibility. We discuss connections on these structures, define a covariant Lie derivative for associated bundles and develop a covariant Cartan calculus. We introduce an extension of the Lie algebra of Killing vectors, the symmetry algebra, and show that it has a representation on sections of associated bundles. We discuss homogeneous generalised spin structures and provide a characterisation of them in terms of lifts of the isotropy representation.

math.DG↗

arXiv2025

Killing (Super)Algebras Associated to Connections on Spinors

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 Poincaré 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.

math.DG↗

arXiv2025

Killing Superalgebras in 2 Dimensions

We provide some examples of Killing superalgebras on 2-dimensional pseudo-Riemannian manifolds within the theoretical framework established in [SIGMA 21 (2025), 081, 61 pages, arXiv:2409.11306]. We compute the Spencer cohomology group $\mathsf{H}^{2,2}(\mathfrak{s}_-;\mathfrak{s})$ and filtered deformations of the non-chiral flat model (Euclidean and Poincaré) superalgebra $\mathfrak{s}$ for various Dirac currents and show these arise as Killing superalgebras for (imaginary) geometric Killing and skew-Killing spinors in both Riemannian and Lorentzian signature.

math.DG↗