The twistor spinors of generic 2- and 3-distributions

TL;DR

This paper explores the relationship between generic distributions on 5- and 6-manifolds, conformal structures, and twistor spinors, providing explicit formulas and characterizations via holonomy and Fefferman-type constructions.

Contribution

It characterizes conformal spin structures arising from generic distributions using holonomy and twistor spinors, and derives explicit relations among conformal Killing fields, Einstein structures, and spinors.

Findings

Conformal structures are described as Fefferman-type constructions.

Conformal spin structures are characterized by their holonomy and twistor spinors.

Explicit formulas relate conformal Killing fields, Einstein structures, and twistor spinors.

Abstract

Generic distributions on 5- and 6-manifolds give rise to conformal structures that were discovered by P. Nurowski resp. R. Bryant. We describe both as Fefferman-type constructions and show that for orientable distributions one obtains conformal spin structures. The resulting conformal spin geometries are then characterized by their conformal holonomy and equivalently by the existence of a twistor spinor which satisfies a genericity condition. Moreover, we show that given such a twistor spinor we can decompose a conformal Killing field of the structure. We obtain explicit formulas relating conformal Killing fields, almost Einstein structures and twistor spinors.