# Gauge Invariance and Anomalies in Condensed Matter Physics

Jürg Fröhlich

Jürg Fröhlich

**26/3/23**Published in : J. Math. Phys. 64, 031903 (2023)

This paper begins with a summary of a powerful formalism for the study of electronic states in condensed matter physics called "Gauge Theory of States/Phases of Matter." The chiral anomaly, which plays quite a prominent role in that formalism, is recalled. I then sketch an application of the chiral anomaly in 1+1 dimensions to quantum wires. Subsequently, some elements of the quantum Hall effect in two-dimensional (2D) gapped ("incompressible") electron liquids are reviewed. In particular, I discuss the role of anomalous chiral edge currents and of anomaly inflow in 2D gapped electron liquids with explicitly or spontaneously broken time reversal, i.e., in Hall- and Chern insulators. The topological Chern-Simons action yielding the transport equations valid in the bulk of such systems and the associated anomalous edge action are derived. The results of a general classification of "abelian" Hall insulators are outlined. After some remarks on induced Chern-Simons actions, I sketch results on certain 2D chiral photonic wave guides. I then continue with an analysis of chiral edge spin-currents and the bulk response equations in time-reversal invariant 2D topological insulators of electron gases with spin-orbit interactions. The "chiral magnetic effect" in 3D systems and axion-electrodynamics are reviewed next. This prepares the ground for an outline of a general theory of 3D topological insulators, including "axionic insulators". Some remarks on Weyl semi-metals, which exhibit the chiral magnetic effect, and on Mott transitions in 3D systems with dynamical axion-like degrees of freedom conclude this review.}