Conformal and projective tractors from 2-frame bundles by the dressing field method

TL;DR

This paper explores conformal and projective structures using 2-frame bundles and the dressing field method to derive gauge-invariant connections, also constructing tractor bundles for both geometries.

Contribution

It introduces a novel application of the dressing field method to derive local gauge-invariant connections for conformal and projective structures from 2-frames.

Findings

Derived local gauge-invariant connections for conformal and projective structures.

Constructed tractor bundles for both conformal and projective geometries.

Showed the equivalence of tractor bundle constructions in different geometric contexts.

Abstract

This paper is devoted to the study of conformal and projective structures, and especially their connections, in the language of 2-frames, or $G$-structures of 2nd-order. While their normal Cartan connections are well-known, we use the dressing field method to obtain their local, mostly gauge invariant, connections. Lastly, we recall how to obtain the tractor bundle from conformal structures, and then show that we can define a projective tractor bundle using the same kind of construction.