Quantum gravity from Weyl conformal geometry

TL;DR

This paper reviews Weyl conformal geometry's role in formulating a gauge theory of gravity, highlighting its spontaneous symmetry breaking, geometric origin of fields, and embedding of the Standard Model within this framework.

Contribution

It introduces a gauge-invariant Weyl quadratic gravity theory with spontaneous symmetry breaking, unifying gravity and Standard Model interactions through geometric principles.

Findings

Weyl gauge symmetry is spontaneously broken, recovering Einstein-Hilbert gravity.

The theory is Weyl-anomaly free and provides a geometric regularisation in any dimension.

Standard Model naturally embeds in conformal geometry without additional degrees of freedom.

Abstract

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.