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We review (non-supersymmetric) gauge theories of four-dimensional space-timesymmetries and their quadratic action. The only true gauge theory of such asymmetry (with a physical gauge boson) that has an exact geometricinterpretation, generates Einstein gravity in its spontaneously broken phaseand is anomaly-free, is that of Weyl gauge symmetry (of dilatations). Gaugingthe full conformal group does not generate a true gauge theory of physical(dynamical) associated gauge bosons. Regarding the Weyl gauge symmetry, it isnaturally realised in Weyl conformal geometry, where it admits two differentbut equivalent geometric formulations, of same quadratic action: one non-metricbut torsion-free, the other Weyl gauge-covariant and metric (with respect to anew differential operator). To clarify the origin of this intriguing result, athird equivalent formulation of this gauge symmetry is constructed in thestandard way on the tangent space (uplifted to space-time by the vielbein),which is metric but has vectorial torsion. This shows an interesting dualityvectorial non-metricity vs vectorial torsion, related to a projectivetransformation. We comment on the physical meaning of these results.

Weyl quadratic gravity as a gauge theory and non-metricity vs torsion duality