Subtleties Concerning Conformal Tractor Bundles

TL;DR

This paper investigates the topological and geometric subtleties in the realization of conformal tractor bundles, revealing how different principal bundle choices affect the associated bundles and their implications for conformal holonomy.

Contribution

It demonstrates that different natural principal bundle choices can produce topologically distinct tractor bundles for the same conformal structure, impacting the understanding of conformal holonomy.

Findings

Different principal bundle choices lead to topologically distinct tractor bundles.

Implications for homogeneous models and conformal holonomy are discussed.

Provides detailed background on tractor bundles and parabolic geometries.

Abstract

The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing representation. Consequences for homogeneous models and conformal holonomy are described. A careful presentation is made of background material concerning standard tractor bundles and equivalence between parabolic geometries and underlying structures.