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Theory of elasticity in topological insulators has many common features withrelativistic quantum fields interacting with gravitational field in the tetradform. Here we discuss several issues in the effective topological(pseudo)electromagnetic response in three-dimensional weak crystallinetopological insulators with no time-reversal symmetry that feature elasticitytetrads, including a mixed "axial-gravitational" anomaly. This response hassome resemblance to "quasitopological" terms proposed for massless Weylquasiparticles with separate, emergent fermion tetrads. As an example, wediscuss the chiral/axial anomaly in superfluid 3He-A. We demonstrate theprincipal difference between the elasticity tetrads and the Weyl fermiontetrads in the construction of the topological terms in the action. Inparticular, the topological action expressed in terms of the elasticitytetrads, cannot be expressed in terms of the Weyl fermion tetrads since in thiscase the gauge invariance is lost.

Tetrads in solids: from elasticity theory to topological quantum Hall systems and Weyl fermions