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Based on the supersymmetric quantum mechanical approach, we havesystematically studied both the $U\left(1\right)$ gauge anomaly and thediffeomorphism anomaly in Weyl systems with torsion, curvature and externalelectromagnetic fields. These anomalies relate to the chiral current (orcurrent) non-conservation and chiral energy-momentum (or energy-momentum)non-conservation, respectively, which can be applied to the $^{3}\text{He-A}$phase, the chiral superconductors and the Weyl semimetals with dislocations anddisclinations. In sharp difference with other anomalies, there exist torsionalanomalies depending on the position of Weyl nodes in the energy-momentum space.These anomalies originate from particles pumped up through the Weyl nodes andthey are thus insensitive to the ultra-violet physics, while the Nieh-Yananomaly is from the particle inflow through the ultra-violet cut-off. Thecurrent non-conservation as well as the energy-momentum non-conservation arefound, which stem from the zero modes trapped in the dislocations and they canbe understood from the Callan-Harvey mechanism. Finally, by comparing ourresults with the well-established momentum anomaly in the $^{3}\text{He-A}$phase, the Nieh-Yan term as well as other cut-off dependent terms are shown tobe negligible, because the ratio between the Lorentz symmetry breaking scaleand the chemical potential is of order $10^{-5}$.

Torsional Anomalies and Bulk-Dislocation Correspondence in Weyl Systems