The origin of Weyl gauging in metric-affine theories

In the first part, we discuss the interplay between local scale invariance and metric-affine degrees of freedom from few distinct points of view. We argue, rather generally, that the gauging of Weyl symmetry is a natural byproduct of requiring that scale invariance is a symmetry of a gravitational theory that is based on a metric and on an independent affine structure degrees of freedom. In the second part, we compute the N\"other identities associated with all the gauge symmetries, including Weyl, Lorentz and diffeomorphisms invariances, for general actions with matter degrees of freedom, exploiting a gauge covariant generalization of the Lie derivative. We find two equivalent ways to approach the problem, based on how we regard the spin-connection degrees of freedom, either as an independent object or as the sum of two Weyl invariant terms. The latter approach, which rests upon the use of a new connection, denoted $\hat{\nabla}$, is particularly convenient and constitutes one of our main results.