3+1d QHE, elasticity tetrads and mixed axial-gravitational anomalies
Elasticity tetrads with torsion describe the hydrodynamic elasticity theory of crystals with dislocations. These tetrads have the canonical dimensions of inverse length and are the proper variables allowing the description of the integer quantum Hall effect - as well as the intrinsic, anomalous quantum Hall effect in topological insulators - in \emph{odd} spatial dimensions in terms of "quasi-topological" Chern-Simons -like terms in the response. The CS terms describe the universal mixed current response to background electromagnetic field and to the deformation of crystal in terms of the elasticity tetrads, and represent the analogue of a mixed axial-gravitational anomaly. In particular, the current is conserved for crystals without dislocations, and the variation of the Hall conductivity with respect to deformations is quantized in terms of topological momentum-space invariants. In the presence of dislocations, the mixed response satisfies the Callan-Harvey anomaly inflow with zero modes along dislocations.