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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 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.

Submission history

From: D. Ghilencea Dr [view email]
[v1] Sat, 6 Jun 2026 10:03:39 UTC (25 KB)
[v2] Tue, 23 Jun 2026 15:52:07 UTC (27 KB)