  We study the transport properties of topological insulators, encoding them in
a generating functional of gauge and gravitational sources. Much of our focus
is on the simple example of a free massive Dirac fermion, the so-called Chern
insulator, especially in 2+1 dimensions. In such cases, when parity and
time-reversal symmetry are broken, it is necessary to consider the
gravitational sources to include a frame and an independent spin connection
with torsion. In 2+1 dimensions, the simplest parity-odd response is the Hall
viscosity. We compute the Hall viscosity of the Chern insulator using a careful
regularization scheme, and find that although the Hall viscosity is generally
divergent, the difference in Hall viscosities of distinct topological phases is
well-defined and determined by the mass gap. Furthermore, on a 1+1-dimensional
edge between topological phases, the jump in the Hall viscosity across the
interface is encoded, through familiar anomaly inflow mechanisms, in the
structure of anomalies. In particular, we find new torsional contributions to
the covariant diffeomorphism anomaly in 1+1 dimensions. Including parity-even
contributions, we find that the renormalized generating functionals of the two
topological phases differ by a chiral gravity action with a negative
cosmological constant. This (non-dynamical) chiral gravity action and the
corresponding physics of the interface theory is reminiscent of well-known
properties of dynamical holographic gravitational systems. Finally, we consider
some properties of spectral flow of the edge theory driven by torsional
dislocations.
