Chiral Anomaly, Topological Field Theory, and Novel States of Matter

Starting with a description of the motivation underlying the analysis presented in this paper and a brief survey of the chiral anomaly, I proceed to review some basic elements of the theory of the quantum Hall effect in 2D incompressible electron gases in an external magnetic field, ("Hall insulators"). I discuss the origin and role of anomalous chiral edge currents and of anomaly inflow in 2D insulators with explicitly or spontaneously broken time reversal, i.e., in Hall insulators and "Chern insulators". The topological Chern-Simons action yielding the large-scale response equations for the 2D bulk of such states of matter is displayed. A classification of Hall insulators featuring quasi-particles with abelian braid statistics is sketched. Subsequently, the chiral edge spin currents encountered in some time-reversal invariant 2D topological insulators with spin-orbit interactions and the bulk response equations of such materials are described. A short digression into the theory of 3D topological insulators, including "axionic insulators", follows next. To conclude, some open problems are described and a problem in cosmology related to axionic insulators is mentioned. As far as the quantum Hall effect and the spin currents in time-reversal invariant 2D topological insulators are concerned this review is based on extensive work my collaborators and I carried out in the early 1990's.