Torsion, parity-odd response, and anomalies in topological states

We study the response of a class of topological systems to electromagnetic and gravitational sources, including torsion and curvature. By using the technology of anomaly polynomials, we derive the parity-odd response of a massive Dirac fermion in d = 2 + 1 and d = 4 + 1, which provides a simple model for a topological insulator. We discuss the covariant anomalies of the corresponding edge states, from a Callan-Harvey anomaly inflow, as well as a Hamiltonian spectral flow point of view. We also discuss the applicability of our results to other systems such as Weyl semimetals. Finally, using dimensional reduction from d = 4 + 1, we derive the effective action for a d = 3 + 1 time-reversal invariant topological insulator in the presence of torsion and curvature, and discuss its various physical consequences.