  For two-dimensional topological insulators, the integer and intrinsic
(without external magnetic field) quantum Hall effect is described by the gauge
anomalous (2+1)-dimensional [2+1d] Chern-Simons (CS) response for the
background gauge potential of the electromagnetic U(1) field. The Hall
conductance is given by the quantized prefactor of the CS term, which is a
momentum-space topological invariant. Here, we show that three-dimensional
crystalline topological insulators with no other symmetries are described by a
topological (3+1)-dimensional [3+1d] mixed CS term. In addition to the
electromagnetic U(1) gauge field, this term contains elasticity tetrad fields
$E^{\ a}_{\mu}({\bf r},t) = \partial_{\mu}X^a(\mathbf{r},t)$ which are
gradients of crystalline U(1) phase fields $X^a(\mathbf{r},t)$ and describe the
deformations of the crystal. For a crystal in three spatial dimensions
$a=1,2,3$ and the mixed axial-gravitational response contains three parameters
protected by crystalline symmetries: the weak momentum-space topological
invariants. The response of the Hall conductance to the deformations of the
crystal is quantized in terms of these invariants. In the presence of
dislocations, the anomalous 3+1d CS term describes the Callan-Harvey anomaly
inflow mechanism. The response can be extended to all odd spatial dimensions.
The elasticity tetrads, being the gradients of the lattice U(1) fields, have
canonical dimension of inverse length. Similarly, if such tetrad fields enter
general relativity, the metric becomes dimensionful, but the physical
parameters, such as Newton's constant, the cosmological constant, and masses of
particles, become dimensionless.
