About Twistor Spinors with Zero in Lorentzian Geometry

Abstract

We describe the local conformal geometry of a Lorentzian spin manifold $(M,g)$ admitting a twistor spinor $\varphi$ with zero. Moreover, we describe the shape of the zero set of $\varphi$. If $\varphi$ has isolated zeros then the metric $g$ is locally conformally equivalent to a static monopole. In the other case the zero set consists of null geodesic(s) and $g$ is locally conformally equivalent to a Brinkmann metric. Our arguments utilise tractor calculus in an essential way. The Dirac current of $\varphi$, which is a conformal Killing vector field, plays an important role for our discussion as well.

References in corpus (1)

• Normal Conformal Killing Forms

Cited by in corpus (6)

• Supersymmetry on Curved Spaces and Holography
• Supersymmetry in Lorentzian Curved Spaces and Holography
• Holonomy reductions of Cartan geometries and curved orbit decompositions
• Conformal superalgebras via tractor calculus
• Towards a Classification of pseudo-Riemannian Geometries Admitting Twistor Spinors
• Charged Conformal Killing Spinors