Weyl conformal geometry vs Riemannian geometry of Weyl gauge invariant (dressed) metric
Weyl conformal geometry is the natural underlying geometry of gauge theories of 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 ($g_{μν}^*$), at the "cost" of (gauge-induced) non-commutativity in the UV, due to 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 and dual description of the two geometries and actions in the symmetric phase. Unlike for the metric, the equation of motion of Weyl gauge field ($ω_μ$) does not commute with the dressing of the metric. Quantum non-locality, in particular entanglement, and non-commutativity are (gauge-invariant) physical artefacts of "translating" (local) Weyl geometry and Weyl gauge covariance into our real-world Riemannian geometry of Weyl gauge invariant observables, and they are evidence of Weyl gauge symmetry. At lower energies, $ω_μ$ becomes massive, can decouple and Einstein-Hilbert action and commutativity are recovered. The case of a light $ω_μ$ is also discussed.