Dynamical Implementation of the Constraints in Conformal Gravity

TL;DR

This work provides a first-order Cartan formulation of four-dimensional conformal gravity by introducing a geometric Lagrangian that dynamically enforces conformal constraints, notably via a 2-form Lagrange multiplier that imposes vanishing conformal torsion $oldsymbol{ m T}^a=0$. The theory is constructed to be invariant under the full conformal gauge group $ ext{H}_{ ext{C}}$ and reduces, after solving the auxiliary-field equations and imposing torsionlessness, to the standard Weyl gravity action with quadratic curvature invariants. A MacDowell–Mansouri–type structure emerges in the bulk, and the second-order reduction shows that Weyl symmetry becomes a global symmetry in the metric formulation. This framework clarifies conformal gravity as a gauge theory with dynamically generated constraints and opens avenues for higher-dimensional and supersymmetric extensions, where the geometric interpretation can be similarly dynamical and transparent.

Abstract

We propose a first-order geometric Lagrangian for four-dimensional conformal gravity within the Cartan formulation, which yields, dynamically, the standard constraints on the fields, expected for conformal gravity. Upon imposing the dynamical constraints, together with the request of conformal invariance of the off-shell Lagrangian, the theory reduces to the standard expression for conformal gravity, in terms of quadratic curvature invariants. Our results clarify the geometric status of conformal gravity as a gauge theory and open the way to a similar dynamical implementation of the constraints in higher dimensions and supersymmetric extensions.