Weyl conformal geometry vs Riemannian geometry of Weyl invariant dressed metric

Abstract: Weyl conformal geometry is the natural underlying geometry of gauge theories of the Weyl group (of dilatations and Poincaré symmetry), such as Weyl quadratic gravity and its generalisation, Weyl-Dirac-Born-Infeld action (WDBI). These gauge theories are Weyl-anomaly-free candidates for quantum gravity. We describe Weyl gauge symmetry from a more familiar Riemannian view of Weyl gauge invariant dressed fields by the Wilson line of dilatations. Weyl geometry can then be seen as Riemannian geometry of non-local dressed metric ($g_{μν}^*$), at the cost of non-commutativity in the UV, also induced by the Wilson line. Then Weyl quadratic gravity and WDBI actions of Weyl geometry, which are Weyl gauge invariant in $d$ dimensions, have the same expression in Riemannian geometry defined by $g^*_{μν}$. This is a non-local map between the two geometries and actions in the symmetric phase. Unlike for the metric, the equation of motion of the Weyl gauge field ($ωμ$) does not commute with the dressing of the metric. When $ωμ$ becomes massive and decouples (broken phase), commutativity and Einstein-Hilbert action are recovered.

• The paper establishes a formal mapping between Weyl conformal geometry and a Riemannian framework through non-local Wilson line integrals.
• It demonstrates that the dressed metric encapsulates gauge invariance, non-commutativity, and a minimal length scale intrinsic to quantum gravity.
• The study confirms that both Weyl quadratic and WDBI actions remain anomaly-free and regularized when expressed via the invariant dressed metrics.

Weyl Conformal Geometry and the Riemannian Geometry of Weyl Invariant Dressed Metrics

Introduction

The study "Weyl conformal geometry vs Riemannian geometry of Weyl invariant dressed metric" (2606.08080) conducts a rigorous examination of the relationship between Weyl conformal geometry—central to Weyl gauge theories of gravity—and Riemannian geometry, by establishing a formal, non-local mapping mediated by Weyl gauge invariant “dressed” metrics obtained through Wilson line integrals of the Weyl gauge field. The work situates itself within the context of anomaly-free quantum gravity models, including Weyl quadratic gravity and its Weyl-Dirac-Born-Infeld (WDBI) generalization, and addresses both the structural and dynamical distinctions and connectivities between the two geometric frameworks in both the symmetric (unbroken) and broken phases of Weyl gauge symmetry.

Weyl Conformal Geometry and Gauge Theory Framework

Weyl conformal geometry is framed as a gauge theory of the Weyl group (dilatations and Poincaré transformations), wherein both the metric $g_{\mu\nu}$ and the Weyl gauge boson $\omega_\mu$ possess independent transformation properties. A central point is the existence of a Stueckelberg mechanism: spontaneous breaking of Weyl gauge symmetry results in a massive $\omega_\mu$ (mass scale near $M_p$), decoupling non-metricity effects at sub-Planckian scales, and yielding the Einstein-Hilbert term as an infrared limit.

Within this setup, the quadratic Weyl gauge gravity action and its generalizations are manifestly gauge invariant, anomaly-free, and sourced entirely by geometric fields without additional degrees of freedom. The Standard Model is naturally embedded within this geometric structure, yielding compelling inflationary scenarios and accommodating astrophysical observations (e.g., SPARC galactic rotation curves).

Dressed Fields and the Wilson Line Construction

The paper’s primary technical advancement is the introduction of Weyl gauge invariant “dressed” fields, most notably a dressed metric $g_{\mu\nu}^*$ constructed by a non-local Wilson line integral of $\omega_\mu$:

$$g^*_{\mu\nu}(x) = \exp\left\{2\int_{-\infty}^x \omega_\mu(y) dy^\mu\right\} g_{\mu\nu}(x)$$

This construction ensures that $g^*_{\mu\nu}$ (and other similarly dressed fields) are strictly gauge invariant, absorbing the local rescaling induced by Weyl transformations via a global, path-dependent phase. The method follows earlier work by Dirac and others on gauge-invariant observables, generalizing the concept to the non-abelian context of gravity with local scale invariance.

Crucially, this mapping transports the non-metricity of Weyl geometry into a Riemannian geometry of a non-local metric, at the explicit cost of introducing non-commutativity in the ultraviolet regime due to the path dependence and the curvature $$F_{\mu\nu} = \partial_\mu \omega_\nu - \partial_\nu \omega_\mu$$. The commutator of partial derivatives acting on dressed fields is no longer zero, proportional instead to $F_{\mu\nu}$, which encodes quantum-like non-locality and encapsulates information regarding scale holonomy.

Non-Local Map Between Weyl and Riemannian Geometries

By using the mapping $\omega_\mu$0, the action functionals of both Weyl quadratic gravity and its WDBI generalization are shown to have exactly the same form in terms of the dressed Riemannian metric as in the original Weyl geometric formulation—aside from the induced non-commutativity. Explicitly, all curvature tensors, the Euler-Gauss-Bonnet term, and the Weyl tensor within the action can be equivalently re-expressed in terms of $\omega_\mu$1, while the field strength $\omega_\mu$2 continues to control the non-local and non-commutative structure of the theory.

This mapping is non-local; the dressed metric encodes information from the history of the Weyl gauge field along a chosen path (Weyl geodesic), not just its value at a single spacetime point. In the symmetric Weyl gauge invariant phase, this Riemannian description is strictly valid, but as the Weyl symmetry is broken and massive $\omega_\mu$3 decouples ($\omega_\mu$4), the non-commutativity vanishes and standard Riemannian geometry is restored.

Implications for Actions and Dynamics

The duality between the Weyl and dressed Riemannian formulations persists at the level of equations of motion for the metric and dilaton-like fields, provided the non-commutativity is properly handled. A subtlety arises with respect to the equation of motion for $\omega_\mu$5, where the process of "dressing" does not generally commute with functional variation—the path-dependent nature of the dressed metric introduces terms (Weyl gauge current contributions) missed in a naive Riemannian treatment, underscoring the genuinely distinct physical content encoded in the two formulations at the quantum effective level.

Furthermore, the analysis extends to $\omega_\mu$6 dimensions, demonstrating that both Weyl and Riemannian actions—when expressed in terms of dressed metrics—are manifestly regularized (no need for an additional UV regulator) and Weyl gauge invariant. In particular, the WDBI action represents, to date, one of the few structures where regularization is entirely geometrical, with physical couplings rendered dimensionless and all mass scales of geometric origin.

Theoretical and Physical Consequences

Several theoretical implications are highlighted:

• Minimal Length Scale: The presence of a non-zero commutator $\omega_\mu$7 on dressed metrics, proportional to $\omega_\mu$8, enforces a minimal area scale determined by the Weyl gauge boson mass $\omega_\mu$9, consistent with the existence of a minimal length in quantum gravity phenomenology.
• Non-Commutative Geometry: The non-commutativity arising in dressed fields is interpreted as a geometric quantum effect, with the Wilson line integrating quantum fluctuations of $\omega_\mu$0 and non-commutativity vanishing upon symmetry breaking.
• Anomaly Freedom and UV Completion: Both the Weyl quadratic and WDBI actions, formulated via dressed metrics, remain anomaly-free and regularized at the quantum level; this asserts the viability of Weyl-conformal approaches to quantum gravity.
• Geometrically Regularized Standard Model Couplings: The inclusion of Standard Model fields under the determinant in the WDBI action, and the existence of a geometric regularization for all couplings, position this framework as a promising pathway to unification scenarios.

Conclusion

This work rigorously formalizes an equivalence between Weyl conformal geometry and a Riemannian geometry built from Weyl invariant dressed metrics, unified by a non-local, gauge invariant mapping via Wilson lines of the Weyl gauge field. The equivalence is exact at the level of action functionals in both the quadratic and WDBI gauge theories of gravity and extends (with some limitations) to the dynamical equations, underpinning the physical meaningfulness of the dressed, non-local operators.

Non-commutativity in the space of dressed fields constitutes a robust ultraviolet geometric effect and embodies the quantum aspects of Weyl gauge symmetry. The existence of a minimal scale and the manifest regularization of the action highlight the deep interplay between geometry and quantum consistency in this approach. These results clarify the correspondence between the physically realized (broken symmetry) Riemannian geometry and its underlying Weyl symmetric phase, providing a more transparent interpretation of gauge symmetry, gravitational dynamics, and possibly opening pathways to new quantum gravity models.