Weyl gravity and Cartan geometry

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}

The Dressing Field Method of Gauge Symmetry Reduction, a Review with Examples

“…For the complex case, we have thenḠ = SU (2, 2) ≃ Spin (2,4), which is the group preserving the metricΣ = 0 12 12 0 , andH is given in matrix notation bȳ…”

Constructing an explicit AdS/CFT correspondence with Cartan geometry

“…In addition, the ω a have been shown to be auxiliary, in the sense that no matter which Lagrangian is chosen the ω a can always be written in terms of the Riemann curvature and its traces [24], see also [4].…”

“…The two theories, based on AdS or conformal symmetry, have a natural correspondence in the context of their Lie algebras alone where neither SUSY, nor holography, is necessary. 2 We define a Yang-Mills theory as having symmetry generators with a Euclidean inner product, distinct from the interesting use in [4], where the definition is more general.…”

Foundations of Mathematics and Physics One Century After Hilbert

“…Such an object appears in the representation theory of crossed products of C * -algebras and is known as a 1-α-cocycle (see [50; 69]). 4 Then, defining C pj (j ) := α j [C p (j )], one has an example of (7), and the above result applies to the 1-α-cocycle C. As a particular case, consider the following…”