Weyl gravity and Cartan geometry

TL;DR

This paper explores the geometric framework of conformal gravity via Cartan geometry, analyzing gauge theories, eliminating redundant degrees of freedom, and linking the Bach tensor to Yang-Mills currents in four dimensions.

Contribution

It demonstrates the use of Cartan geometry in conformal gravity, clarifies the role of the Weyl gauge potential, and connects the Bach tensor to Yang-Mills currents within a Lagrangian approach.

Findings

Weyl gauge potential is a spurious degree of freedom

Gauge field constrained to the normal conformal Cartan connection

Bach tensor identified with Yang-Mills current in 4D

Abstract

We point out that the Cartan geometry known as the second-order conformal structure provides a natural differential geometric framework underlying gauge theories of conformal gravity. We are concerned by two theories: the first one will be the associated Yang-Mills-like Lagrangian, while the second, inspired by~\cite{Wheeler2014}, will be a slightly more general one which will relax the conformal Cartan geometry. The corresponding gauge symmetry is treated within the BRST language. We show that the Weyl gauge potential is a spurious degree of freedom, analogous to a Stueckelberg field, that can be eliminated through the dressing field method. We derive sets of field equations for both the studied Lagrangians. For the second one, they constrain the gauge field to be the `normal conformal Cartan connection'. Finally, we provide in a Lagrangian framework a justification of the identification, in dimension $4$, of the Bach tensor with the Yang-Mills current of the normal conformal Cartan connection, as proved in \cite{Korz-Lewand-2003}.