Quantum gravity from Weyl conformal geometry

We review recent developments in physical implications of Weyl conformal geometry. The associated Weyl quadratic gravity action is a gauge theory of the Weyl group of dilatations and Poincar\'e symmetry. Weyl conformal geometry is defined by equivalence classes of the metric and Weyl gauge field ($\omega_\mu$), related by Weyl gauge transformations. Weyl geometry can be seen as a covariantised version of Riemannian geometry with respect to Weyl gauge symmetry (of dilatations). This Weyl gauge-covariant formulation of Weyl geometry is metric, which avoids century-old criticisms on the physical relevance of this geometry, that ignored its gauge symmetry. Weyl quadratic gravity and its geometry have interesting properties: a) Weyl gauge symmetry is spontaneously broken and Einstein-Hilbert gravity and Riemannian geometry are recovered, with $\Lambda>0$; b) this is the only true gauge theory of a space-time symmetry i.e. with a physical (Weyl) gauge boson ($\omega_\mu$); c) all fields and masses have geometric origin (with no added scalar fields); d) the theory has a Weyl gauge invariant geometric regularisation (by $\hat R$) in $d$ dimensions and it is Weyl-anomaly free; this anomaly is recovered in the broken phase after massive $\omega_\mu$ decouples; e) the theory is the leading order of the general Weyl gauge invariant Dirac-Born-Infeld (WDBI) action of Weyl conformal geometry in $d$ dimensions; f) in the limit of vanishing Weyl gauge current, one obtains conformal gravity; g) finally, Standard Model (SM) has a natural embedding in conformal geometry with no new degrees of freedom, with successful Starobinsky-Higgs inflation. Briefly, Weyl conformal geometry generates a (quantum) gauge theory of gravity, given by Weyl quadratic gravity, and leads to a unified description, by the gauge principle, of Einstein-Hilbert gravity and SM interactions.