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.

• Received 7 September 2014

DOI:https://doi.org/10.1103/PhysRevD.90.105004