Fields and gravitational theories: on the role of the Weyl symmetry and the trace anomaly

In this thesis, we explore both the classical and quantum consequences of Weyl symmetry in the contexts of Riemannian and non-Riemannian geometries. In particular, we delve into classical Weyl-invariant theories in a metric-affine setting and examine the arguably most significant quantum phenomenon associated with this symmetry: the conformal anomaly. Indeed, in the context of quantum fields on curved backgrounds, the conformal anomaly plays a crucial role in studying the renormalized generating functional for one-particle irreducible correlation functions. This work presents some new results related to these topics. We complete the classification of conformal actions for all the tensors species within a general metric-affine framework, and investigate the physical implications of relaxing the requirement of full Weyl invariance by considering some substructures of the Weyl group in a Riemannian setting. Among other results, these substructures reveal an intriguing connection between conformal and higher-derivative gravity, where the latter can be viewed as a partially gauge-fixed version of the former. On the quantum side, we systematize the use of the ambient space formalism to construct and integrate the conformal anomaly in each even dimension considering scenarios where the metric is the only background field. Additionally, we propose an extension of the concept of anomaly to the aforementioned Weyl subgroups, and employ cohomological methods to derive the most general trace anomalies and anomalous actions in the presence of torsion in $d=2,4$ in a completely model-independent way.