Reducible conformal holonomy in any metric signature and application to twistor spinors in low dimension

TL;DR

This paper establishes a link between reducible conformal holonomy and lightlike distributions in pseudo-Riemannian geometry, applying it to classify low-dimensional geometries with twistor spinors using conformal tractor calculus.

Contribution

It generalizes previous results on lightlike structures, providing a comprehensive geometric understanding of conformal holonomy and twistor spinors in various signatures.

Findings

Existence of parallel lightlike distributions containing Ricci tensor images.

Complete classification of geometries with twistor spinors in signatures (3,2) and (3,3).

Integrability of distributions in non-generic, zero-free cases.

Abstract

We prove that given a pseudo-Riemannian conformal structure whose conformal holonomy representation fixes a totally lightlike subspace of arbitrary dimension, there is, wrt. a local metric in the conformal class defined off a singular set, a parallel, totally lightlike distribution on the tangent bundle which contains the image of the Ricci-tensor. This generalizes results obtained for invariant lightlike lines and planes and closes a gap in the understanding of the geometric meaning of reducibly acting conformal holonomy groups. We show how this result naturally applies to the classification of geometries admitting twistor spinors in some low-dimensional split signatures when they are described using conformal spin tractor calculus. Together with already known results about generic distributions in dimensions 5 and 6 we obtain a complete geometric description of local geometries admitting real twistor spinors in signatures (3,2) and (3,3). In contrast to the generic case where generic geometric distributions play an important role, the underlying geometries in the non-generic case without zeroes turn out to admit integrable distributions.