A Mathematical Analysis of the Axial Anomaly

TL;DR

This paper provides a new mathematical derivation of the axial anomaly in quantum field theory using BV formalism and equivariant quantization, linking anomalies to topological invariants of the Dirac operator.

Contribution

It formalizes the axial anomaly within the BV and equivariant quantization frameworks, connecting it to cohomology classes and the index theorem.

Findings

The relevant cohomology complex is quasi-isomorphic to de Rham forms.

The anomaly corresponds to a top-degree cohomology class.

The anomaly is trivial if and only if the Dirac operator's index is zero.

Abstract

As is well known to physicists, the axial anomaly of the massless free fermion in Euclidean signature is given by the index of the corresponding Dirac operator. We use the Batalin-Vilkovisky (BV) formalism and the methods of equivariant quantization of Costello and Gwilliam to produce a new, mathematical derivation of this result. Using these methods, we formalize two conventional interpretations of the axial anomaly, the first as a violation of current conservation at the quantum level and the second as the obstruction to the existence of a well-defined fermionic partition function. Moreover, in the formalism of Costello and Gwilliam, anomalies are measured by cohomology classes in a certain obstruction-deformation complex. Our main result shows that---in the case of the axial symmetry---the relevant complex is quasi-isomorphic to the complex of de Rham forms of the spacetime manifold and that the anomaly corresponds to a top-degree cohomology class which is trivial if and only if the index of the corresponding Dirac operator is zero.