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| Paper IPM / Physic / 17713 |
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In this note we present a solution to the question of whether or not, in the presence of torsion, the topological NiehâYan term contributes to chiral anomaly. The integral of NiehâYan term is non-zero if topology is non-trivial; the manifold has a boundary or vierbeins have singularities. Noting that singular NiehâYan 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 NiehâYan 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 NiehâYan 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 NiehâYan 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 NiehâYan term can be smoothly rotated away, does not contribute to topological structure and consequently does not contribute to chiral anomaly.
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