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Killing (super)algebras for generalised spin manifolds

Andrew D. K. Beckett

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 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.

Killing (super)algebras for generalised spin manifolds

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 Poincaré superalgebra extended by the -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.

Paper Structure

This paper contains 43 sections, 37 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Flat model (super)algebras and $R$-symmetry
  4. Flat model (super)algebras
  5. Schur algebra and $R$-symmetry group
  6. Extending the algebra
  7. The spin-$R$ group and spin-$R$ structures
  8. The spin-$R$ group
  9. Spin-$R$ structures
  10. Representations of the spin-$R$ group and associated bundles
  11. Connections and curvatures
  12. Covariant Lie derivative and Cartan calculus
  13. Time orientability and bundle Dirac current
  14. Killing spinors and Killing (super)algebras on generalised spin manifolds
  15. The symmetry algebra of a spin-$R$ structure
  16. ...and 28 more sections

Key Result

Proposition 5

The covariant Lie derivative described above satisfies the following properties: for $X\in\mathfrak{iso}(M,g)$ and $Y\in\mathfrak{X}(M)$ (equivalently, $\comm{\widehat{\mathscr{L}}_X}{\widehat{\nabla}} = \imath_X F$ for all $X\in\mathfrak{iso}(M,g)$) and for $X,Y\in\mathfrak{iso}(M,g)$.Note that the $F(X,Y)$ terms in these equations are to be understood as operators acting on sections of an asso

Theorems & Definitions (77)

  • Definition 1: Dirac current, flat model (super)algebra Beckett2024_ksa
  • Definition 2
  • Definition 3: Spin-$R$ structure
  • Definition 4
  • Proposition 5
  • proof
  • Proposition 6: Covariant Cartan calculus Beckett2025_gen_spin
  • Lemma 7: Beckett2025_gen_spin
  • Lemma 8: Existence of bundle Dirac current
  • proof
  • ...and 67 more