Dynamical Implementation of the Constraints in Conformal Gravity

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.

💡 Research Summary

The paper presents a novel first‑order geometric formulation of four‑dimensional conformal gravity (CG) within the Cartan framework, aiming to derive the usual algebraic constraints (vanishing conformal torsion and dilatation curvature) dynamically rather than imposing them by hand. The authors start by recalling that standard CG is usually expressed as the square of the Weyl tensor, ∫ C_{μνρσ}C^{μνρσ} √−g d⁴x, and that this formulation hides the gauge structure of the full conformal group SO(2,4).

In Section 2 they introduce the conformal algebra generated by Lorentz rotations J_{ab}, translations P_a, special conformal transformations K_a, and dilatations D. By performing an Inönü‑Wigner contraction they isolate a non‑semisimple solvable subgroup H_C = (SO(1,3)×SO(1,1))⋉ℝ^{1,3}, which will serve as the gauge group of the Cartan connection. The Maurer‑Cartan equations are written for the curvature two‑forms: the Lorentz curvature R^{ab}, the conformal torsion T^a, the special‑conformal curvature F^a, and the dilatation curvature G.

Section 3 constructs the first‑order Lagrangian. The field content consists of the vierbein V^a, the Lorentz spin connection ω^{ab}, the gauge fields associated with K_a (denoted b^a) and D (denoted f^a), together with auxiliary 1.5‑order fields χ^{ab}, χ and two Lagrange multipliers λ_{ab}, λ. The Lagrangian is a linear combination of the curvature two‑forms and their Hodge duals, plus the multiplier terms λ_{ab} T^{ab} + λ G, which enforce the constraints T^a=0 and G=0.

Varying with respect to the multipliers yields precisely the conformal‑torsion and dilatation constraints. Variation with respect to the auxiliary fields gives algebraic relations that express χ^{ab} and χ in terms of the curvatures and the vierbein. Crucially, the equations of motion for ω^{ab}, b^a, f^a become non‑dynamical once the constraints are imposed, allowing these gauge fields to be solved algebraically in terms of the vierbein and its first derivatives. Substituting these solutions back into the Lagrangian eliminates all first‑order fields and produces a second‑order action that is exactly the Weyl‑squared action, confirming the equivalence with standard CG.

Section 4 analyses the symmetries of the constructed Lagrangian. Two perspectives are considered: (i) invariance under the full conformal group SO(2,4) as a Cartan connection, which automatically guarantees conformal invariance; (ii) a Yang‑Mills‑type invariance under the subgroup H_C alone. The latter requires the conformal torsion to vanish; otherwise the Yang‑Mills structure is broken. Hence the torsion constraint is not merely a simplifying assumption but a necessary condition for preserving the H_C‑Yang‑Mills symmetry.

Section 5 rewrites the final Lagrangian in a compact Cartan notation and displays its second‑order form explicitly, showing the precise match with the Weyl‑squared term. The authors also discuss how the vierbein provides the soldering map between the tangent space of the base manifold and the quotient algebra 𝔤/𝔥, thereby reproducing the metric structure without ever introducing a metric a priori.

In the conclusions (Section 6) the authors emphasize that their approach clarifies the geometric status of CG as a genuine gauge theory, with constraints emerging from dynamics rather than being imposed externally. They outline several promising extensions: applying the same method to higher‑dimensional conformal gravities (especially six dimensions, where the algebraic structure is richer), constructing supersymmetric versions by adding fermionic partners to the Cartan connection, and investigating the role of the Lagrange multipliers in the quantum theory (e.g., BRST treatment of the constraints).

Appendix A collects the detailed conformal algebra, the explicit form of the Cartan connection, and the associated Bianchi identities. Appendix B lists the standard constraints used in the literature on CG, facilitating comparison with the dynamically derived constraints of the present work.

Overall, the paper provides a coherent and mathematically rigorous framework that unifies the gauge‑theoretic and geometric viewpoints of conformal gravity, and it opens a clear pathway toward higher‑dimensional and supersymmetric generalizations.