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We study the response of a class of topological systems to electromagneticand gravitational sources, including torsion and curvature. By using thetechnology of anomaly polynomials, we derive the parity-odd response of amassive Dirac fermion in $d=2+1$ and $d=4+1$, which provides a simple model fora topological insulator. We discuss the covariant anomalies of thecorresponding edge states, from a Callan-Harvey anomaly-inflow, as well as aHamiltonian spectral flow point of view. We also discuss the applicability ofour results to other systems such as Weyl semi-metals. Finally, usingdimensional reduction from $d=4+1$, we derive the effective action for a$d=3+1$ time-reversal invariant topological insulator in the presence oftorsion and curvature, and discuss its various physical consequences.

Torsion, Parity-odd Response and Anomalies in Topological States