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In this note we present a solution to the question of whether or not, in thepresence of torsion, the topological Nieh-Yan term contributes to chiralanomaly. The integral of Nieh-Yan term is non-zero if topology is non-trivial;the manifold has a boundary or vierbeins have singularities. Noting thatsingular Nieh-Yan term could be written as a sum of delta functions, we arguethat the heat kernel expansion cannot end at finite steps. This leads to asinusoidal dependence on the Nieh-Yan term and the UV cut-off of the theory (oralternatively the minimum length of spacetime). We show this ill-behaveddependence can be removed if a quantization condition on length scales isapplied. It is expected as the Nieh-Yan term can be derived as the differenceof two Chern class integrals (i.e. Pontryagin terms). On the other hand, in thepresence of a cosmological constant, we find that indeed the Nieh-Yan termcontributes to the index with a dimensionful anomaly coefficient that dependson the de Sitter length or equivalently inverse Hubble rate. We find similarresult in thermal field theory where the anomaly coefficient depends ontemperature. In both examples, the anomaly coefficient depends on IR cut-off ofthe theory. Without singularities, the Nieh-Yan term can be smoothly rotatedaway, does not contribute to topological structure and consequently does notcontribute to chiral anomaly.

On Torsion Contribution to Chiral Anomaly via Nieh-Yan Term