Anomalies are the breaking of classical symmetries by quantum effects, and their non-renormalization properties play a crucial role in a wide range of phenomena. I present some rigorous theorems on the (non-perturbative) anomaly non-renormalization in QFT models, based on Renormalization Group, cluster or tree expansion and determinant bounds, proving the exact cancellation of the terms coming by the lattice cut-offs. I discuss in particular a lattice fermion-vector model in d=3+1, the Sommerfield model in d=1+1 and the anomaly cancellation in a chiral lattice d=3+1 model. Analogous results on universality in transport coefficients in Graphene, Hall insulators and Weyl semimetals in presence of a many body interactions will be also briefly presented.
Irene Giardina