  We discuss fundamental aspects of chiral anomaly-driven interactions in
conformal field theory (CFT) in four spacetime dimensions. They find
application in very general contexts, from early universe plasma to topological
condensed matter. We outline the key shared characteristics of these
interactions, specifically addressing the case of chiral anomalies, both for
vector currents and gravitons. In the case of topological materials, the
gravitational chiral anomaly is generated by thermal gradients via the
(Tolman-Ehrenfest) Luttinger relation. In the CFT framework, a nonlocal
effective action, derived through perturbation theory, indicates that the
interaction is mediated by an excitation in the form of an anomaly pole, which
appears in the conformal limit of the vertex. To illustrate this, we
demonstrate how conformal Ward identities (CWIs) in momentum space allow us to
reconstruct the entire chiral anomaly interaction in its longitudinal and
transverse sectors just by inclusion of a pole in the longitudinal sector. Both
sectors are coupled in amplitudes with an intermediate chiral fermion or a
bilinear Chern-Simons current with intermediate photons. In the presence of
fermion mass corrections, the pole transforms into a cut, but the absorption
amplitude in the axial-vector channel satisfies mass-independent sum rules
related to the anomaly in any chiral interaction. The detection of an
axion-like/quasiparticle in these materials may rely on a combined
investigation of these sum rules, along with the measurement of the angle of
rotation of the plane of polarization of incident light when subjected to a
chiral perturbation. This phenomenon serves as an analogue of a similar one in
ordinary axion physics, in the presence of an axion-like condensate, that we
rederive using axion electrodynamics.
