Weyl gravity and Cartan geometry

Jérémy Attard (1, 2) , Jordan François (3) , Serge Lazzarini (1, 2)

1 CPT - Centre de Physique Théorique - UMR 7332
2 CPT - E2 Géométrie, Physique et Symétries
3 Leibniz Universität Hannover=Leibniz University Hannover

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 J.T. Wheeler in Phys. Rev. D90 (2014), 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 Class. Quant. Grav. 20 (2003).

Mots clés: Gauge Theories; Dressing; Cartan connection; Gravitation and relativity

Domaines: Physique mathématique [math-ph]

Jérémy Attard, Jordan François, Serge Lazzarini. Weyl gravity and Cartan geometry. Physical Review D, 2016, 93 (8), pp.085032. ⟨10.1103/PhysRevD.93.085032⟩. ⟨hal-01248843⟩