Conformal field theory in momentum space and anomaly actions in gravity:The analysis of three- and four-point functions

After a brief outline of general aspects of conformal field theories in coordinate space, in a first part we review the solution of the conformal constraints of three- and four-point functions in momentum space in dimensions $d\geq 2$, in the form of conformal Ward identities (CWI's). We center our discussion on the analysis of correlators containing stress-energy tensors $(T)$, conserved currents $(J)$, and scalar operators $(O)$. For scalar four-point functions, we briefly discuss our method for determining the dual conformal solutions of such equations, identified only by the CWI's, and related to the conformal Yangian symmetry, introduced by us in previous work. In correlation functions with $T$ tensors, evaluated around a flat spacetime, the conformal anomaly is characterized by the (non-local) exchange of massless poles in specific form factors, a signature that has been investigated both in free field theory and non-perturbatively, by solving the conformal constraints. We discuss the anomaly effective action, and illustrate the derivation of the CWI's directly from its path integral definition and its Weyl symmetry, which is alternative to the standard operatorial approach used in conformal field theories in flat space. For two- and three-point functions, we elaborate on the matching of these types of correlators to free-field theories. Perturbative realizations of CFTs at one-loop provide the simplest expressions of the general solutions identified by the CWI's, for generic operators $T$, $J$, and scalars of specific scaling dimensions, by an appropriate choice of their field content. In a technical appendix we offer details on the reconstruction of the $TTO$ and $TTT$ correlators in the approach of Bzowski, McFadden and Skenderis, and specifically on the secondary Ward identities of the method, in order to establish a complete match with the perturbative description.