CR-tractors and the Fefferman space

Abstract

We develop the natural tractor calculi associated to conformal and CR structures as a fundamental tool for the study of Fefferman's construction of a canonical conformal class on the total space of a circle bundle over a non-degenerate CR manifold of hypersurface type. In particular, we construct and treat the basic objects that relate the natural bundles and natural operators on the two spaces. This is illustrated with several applications: We prove that a number of conformally invariant overdetermined systems, including Killing form equations and the equations for twistor spinors, admit non-trivial solutions on any Fefferman space. We show that, on a Fefferman space, the space of infinitesimal conformal isometries naturally decomposes into a direct sum of subspaces, which admit an interpretation as solutions of certain CR invariant PDE's. Finally we explicitly analyse the relation between tractor calculus on a CR manifold and the complexified conformal tractor calculus on its Fefferman space, thus obtaining a powerful computational tool for working with the Fefferman construction.