Comment on: Topological invariants, instantons, and the chiral anomaly on spaces with torsion

Quantum anomalies both in the Riemannian and in the Riemann-Cartan spacetimes were calculated previously using different methods, see e.g. [11,25]. However, recently [3] the completeness of these earlier calculations have been questioned which all demonstrated that the Nieh–Yan four-form [18] is irrelevant to the axial anomaly. For the axial anomaly, we have a couple of distinguished features. Most prominent is its relation with the Atiyah–Singer index theorem. But also from the viewpoint of perturbative quantum field theory (QFT), the chiral anomaly has some features which signal its conceptual importance. For all topological field theories like BF-theories, Chern–Simons, and for all topological effects like the anomaly, the remarkable fact holds that the relevant invariants do not renormalize — higher order loop corrections do not alter the one-loop value of the anomaly, for example. The fact that the anomaly is stable against radiative corrections guarantees that it can be given a topological interpretation. For the anomaly, this is the Adler–Bardeen theorem, while other topological field theories are carefully designed to have, amongst other properties, vanishing beta functions. Another feature is finiteness: in any approach, the chiral anomaly as a topological invariant is a finite quantity. In a spacetime with torsion, Chandia and Zanelli [3] argue that the Nieh–Yan (NY) four-form d ∗A will add to