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Momentum transport is anomalous in chiral $p+ip$ superfluids andsuperconductors in the presence of textures and superflow. Using the gradientexpansion of the semi-classical approximation, we show how gauge and Galileansymmetries induce an emergent curved spacetime with torsion and curvature forthe quasirelativistic low-energy Majorana-Weyl quasiparticles. We explicitlyshow the emergence of the spin-connection and curvature, in addition totorsion, using the superfluid hydrodynamics. The background constitutes anemergent quasirelativistic Riemann-Cartan spacetime for the Weyl quasiparticlesand they satisfy the conservation laws associated with local Lorentz symmetry,restricted to the plane of uniaxial anisotropy of the superfluid (or-conductor). Moreover, we show that the anomalous Galilean momentumconservation is a consequence of the gravitational Nieh-Yan (NY) chiral anomalythe Weyl fermions experience on the background geometry. Notably, the NYanomaly coefficient features a non-universal ultraviolet cut-off scale$\Lambda$, with canonical dimensions of momentum. Comparison of the anomalyequation and the hydrodynamic equations suggests that the value of the cut-offparameter $\Lambda$ is determined by the normal state Fermi liquid andnon-relativistic uniaxial symmetry of the $p$-wave superfluid orsuperconductor.

Emergent spacetime and gravitational Nieh-Yan anomaly in chiral $p+ip$ Weyl superfluids and superconductors