On Torsion Contribution to Chiral Anomaly via Nieh-Yan Term

TL;DR

This paper investigates whether the Nieh-Yan term contributes to chiral anomaly in the presence of torsion, revealing conditions under which it affects the anomaly and how it depends on spacetime and IR cut-offs.

Contribution

It demonstrates that the Nieh-Yan term's contribution to chiral anomaly depends on topology, singularities, and IR cut-offs, and introduces a quantization condition to remove ill-behaved dependencies.

Findings

Nieh-Yan term contributes to chiral anomaly with a dimensionful coefficient in certain conditions.

Singularities in the Nieh-Yan term lead to sinusoidal dependence on UV cut-off.

Contribution can be nullified if singularities are absent, making the term smoothly rotatable away.

Abstract

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