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arxiv: 2606.08080 · v2 · pith:TED2MMTJnew · submitted 2026-06-06 · ✦ hep-th · gr-qc· hep-ph· math-ph· math.MP

Weyl conformal geometry vs Riemannian geometry of Weyl gauge invariant dressed metric

D. M. Ghilencea , V.-M. Mandric This is my paper

Pith reviewed 2026-06-27 19:38 UTC · model grok-4.3

classification ✦ hep-th gr-qchep-phmath-phmath.MP
keywords Weyl conformal geometrydressed metricWilson lineWeyl quadratic gravityWDBI actiongauge invariancenon-local mapRiemannian equivalence
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The pith

Weyl geometry is equivalent to Riemannian geometry of a non-local dressed metric via Wilson lines.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that Weyl conformal geometry, the natural setting for gauge theories of dilatations and Poincaré symmetry, can be rewritten from a Riemannian viewpoint once fields are dressed by the Wilson line of dilatations. This produces a non-local dressed metric whose Riemannian curvature scalars reproduce the original Weyl-invariant actions exactly. The equivalence applies to both Weyl quadratic gravity and the Weyl-Dirac-Born-Infeld action in any dimension d. Non-commutativity appears in the ultraviolet because the equation of motion for the Weyl gauge field does not commute with the dressing operation. At lower energies the Weyl vector becomes massive, decouples, and the standard commutative Einstein-Hilbert theory is recovered.

Core claim

Weyl geometry can be seen as Riemannian geometry of the non-local dressed metric g^*_{\mu\nu}, at the cost of non-commutativity in the UV also due to 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^*_{\mu\nu}. This is a non-local map and a dual description of the two geometries and actions in the symmetric phase. Unlike for the metric, the equation of motion of the Weyl gauge field does not commute with the dressing. Quantum non-locality and non-commutativity are artefacts of translating Weyl geometry and Weyl gauge covariance into Riemannian geometry o

What carries the argument

The non-local dressed metric g^*_{\mu\nu} constructed by multiplying the original metric by the Wilson line of dilatations, which converts Weyl gauge invariance into a Riemannian description.

If this is right

  • The Weyl quadratic gravity and WDBI actions remain Weyl gauge invariant in any dimension when rewritten in the Riemannian geometry of the dressed metric.
  • Non-commutativity and non-locality in the UV are direct consequences of the translation between the two geometries.
  • The map is non-local and applies in the symmetric phase where the Weyl field is massless.
  • At low energies the Weyl vector decouples, commutativity is restored, and the Einstein-Hilbert action emerges unchanged.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same dressing procedure might convert other conformal or scale-invariant gravity theories into effective Riemannian descriptions with controlled non-local corrections.
  • Quantization of Weyl gravity could be attempted with standard Riemannian methods once the non-local dressed metric is used as the fundamental variable.
  • The recovered commutativity at low energies suggests Weyl symmetry functions as a UV completion that becomes hidden below the mass scale of the Weyl vector.
  • Non-commutativity induced by the Wilson line could appear in high-energy scattering amplitudes or early-universe observables as a signature of the underlying gauge symmetry.

Load-bearing premise

The Wilson line dressing produces an exact equivalence of the actions and equations without extra terms or loss of physical content even though the Weyl gauge field equation does not commute with the dressing.

What would settle it

An explicit expansion of the dressed equations of motion that produces terms absent from the original Weyl theory after the Wilson line is inserted.

read the original abstract

Weyl conformal geometry is the natural underlying geometry of gauge theories of the Weyl group (of dilatations and Poincar\'e symmetry), such as Weyl quadratic gravity and its generalisation, Weyl-Dirac-Born-Infeld action (WDBI). These are local, Weyl-anomaly free (quantum) gauge theories of 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 ($g_{\mu\nu}^*$), at the "cost" of non-commutativity in the UV, also due to 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 $g^*_{\mu\nu}$. This is a {\it non-local} map and a dual description of the two geometries and actions in the symmetric phase. Unlike for the metric, the equation of motion of the Weyl gauge field ($\omega_\mu$) does not commute with the dressing of the metric. Quantum non-locality and non-commutativity are then artefacts of "translating" Weyl geometry and Weyl gauge covariance into our Riemannian geometry of Weyl gauge invariant observables and are indirect evidence of Weyl gauge symmetry. At lower energies, $\omega_\mu$ becomes massive, decouples and commutativity and Einstein-Hilbert action are recovered.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper claims that Weyl conformal geometry can be equivalently reformulated as the Riemannian geometry of a non-local dressed metric g^*_{\mu\nu} obtained via the Wilson line of dilatations. Under this map, the Weyl-quadratic-gravity and WDBI actions (Weyl-gauge-invariant in d dimensions) take identical form in the Riemannian description; non-commutativity in the UV is presented as an artefact of the translation, while the equation of motion for the Weyl gauge field \omega_\mu does not commute with the dressing. At lower energies \omega_\mu becomes massive, commutativity is restored, and the Einstein-Hilbert action is recovered.

Significance. If the claimed exact equivalence of actions and equations holds, the work supplies a concrete dual description that recasts Weyl-gauge theories of gravity in more familiar Riemannian language while preserving gauge invariance. This could clarify the role of non-locality and non-commutativity as indirect signatures of Weyl symmetry and might aid the study of the symmetric phase of gravity. The low-energy recovery of standard GR is a physically reassuring feature.

major comments (1)
  1. [Abstract (paragraph on non-commutativity)] Abstract (paragraph beginning 'Weyl geometry can then be seen...'): The central claim is that the Wilson-line dressing yields identical actions and equations in the Riemannian description. The manuscript explicitly states that the EOM for \omega_\mu does not commute with the dressing. Because the variation that produces the equations of motion involves \delta/\delta\omega_\mu, this non-commutativity risks generating extra terms or constraints in the dressed formulation that are absent in the original Weyl theory. An explicit demonstration that these contributions either vanish or are identically zero (e.g., by direct variation of the dressed action) is required to establish that the equivalence is exact rather than approximate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important subtlety regarding the non-commutativity of the dressing operation with the variation that yields the equations of motion. We address the comment below and will revise the manuscript to include the requested explicit demonstration.

read point-by-point responses

  1. Referee: Abstract (paragraph beginning 'Weyl geometry can then be seen...'): The central claim is that the Wilson-line dressing yields identical actions and equations in the Riemannian description. The manuscript explicitly states that the EOM for ω_μ does not commute with the dressing. Because the variation that produces the equations of motion involves δ/δω_μ, this non-commutativity risks generating extra terms or constraints in the dressed formulation that are absent in the original Weyl theory. An explicit demonstration that these contributions either vanish or are identically zero (e.g., by direct variation of the dressed action) is required to establish that the equivalence is exact rather than approximate.

    Authors: We agree that the potential for additional terms arising from the ω_μ-dependence of the Wilson line in the dressing map requires explicit verification to confirm that the equivalence of the actions extends to their equations of motion. The statement in the abstract that the EOM for ω_μ does not commute with the dressing refers to the operator ordering in the non-local map itself, not to a mismatch in the physical content. Because the dressed action is constructed to be identically equal to the original Weyl action (as shown in Sections 3 and 4), its functional variation with respect to ω_μ must reproduce the same EOM once the dependence of g^*_{ u ho} on the Wilson line is properly accounted for. In the revised manuscript we will add a dedicated calculation (new subsection in Section 4) that performs the variation of the dressed action explicitly, demonstrates that all extra contributions from δg^*/δω_μ cancel identically against the variation of the Wilson line, and thereby confirms that no additional constraints appear. This will establish the exact equivalence at the level of both actions and equations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; equivalence follows from explicit Wilson-line construction with noted caveats

full rationale

The central claim is a geometric re-description: the Wilson line of dilatations defines a non-local dressed metric g^*_{\mu\nu} such that Weyl-geometry actions (quadratic gravity, WDBI) take the same functional form when rewritten in the Riemannian geometry of g^*. This is presented as a derived consequence of the dressing map rather than an input. The abstract explicitly flags that the EOM for \omega_\mu does not commute with the dressing and treats the resulting non-locality/non-commutativity as an artefact of the translation, not as a hidden assumption. No fitted parameters are relabelled as predictions, no uniqueness theorem is imported from the authors' prior work to force the result, and no ansatz is smuggled via self-citation. The derivation chain is therefore self-contained against the stated construction; any load-bearing self-citations (if present in the full text) would concern background results on the original Weyl actions, not the equivalence map itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the definition of the dressed metric via Wilson lines and the assumption that this map preserves the form of the actions exactly; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math Standard differential geometry and gauge transformation rules for Weyl conformal geometry hold.
    The paper assumes the usual properties of Weyl geometry and Wilson lines as background.

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discussion (0)

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