Tractors and Twistors from conformal Cartan geometry: a gauge theoretic approach I. Tractors
Résumé
Tractors and Twistors bundles both provide natural conformally covariant calculi on $4D$-Riemannian manifolds. They have different origins but are closely related, and usually constructed bottom-up from prolongation of defining differential equations. We propose alternative top-down gauge theoretic constructions starting from the conformal Cartan bundle $\mathcal{P}$ and its vectorial $E$ and spinorial $\mathsf{E}$ associated bundles. Our key ingredient is the dressing field method of gauge symmetry reduction, which allows to exhibit tractors and twistors and their associated connections as gauge fields of a nonstandard kind as far as Weyl rescaling symmetry is concerned. By which we mean that they implement the gauge principle but are of a different geometric nature than the well known differential geometric objects usually underlying gauge theories. We provide the corresponding BRST treatment. The present paper deals with the case of tractors while a companion paper deals with twistors.