  Topology enters in quantum field theory (qft) in multiple forms: one of the
most important, in non-abelian gauge theories, being in the identification of
the $\theta$ vacuum in QCD. A very relevant aspect of this connection is
through the phenomenon of chiral and conformal qft anomalies. It has been
realized that a class of materials, comprising topological insulators and Weyl
semimetals, also exhibit the phenomenon of anomalies, which are responsible for
several exotic phenomena, such as the presence of edge currents, resilient
under perturbations and scattering by impurities. Another example comes from
the response functions of these materials under thermal and mechanical
stresses, that may be performed using correlation functions of stress energy
tensors in General Relativity. In this case the conformal anomaly plays an
important role. We briefly illustrate some salient features of this
correspondence, and the effective action describing the long-range interactions
that may account for such topological effects.
