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

Abstract

We prove that given a pseudo-Riemannian conformal structure whose conformal holonomy representation fixes a totally isotropic subspace of arbitrary dimension, there is, w.r.t. a local metric in the conformal class defined off a singular set, a parallel, totally isotropic distribution on the tangent bundle which contains the image of the Ricci-tensor. This generalizes results obtained for invariant isotropic 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 described in terms of parallel spin tractors using conformal spin tractor calculus. As an example we obtain together with already known results about generic distributions in dimensions 5 and 6 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.