Non local parity transformations and anomalies

TL;DR

This paper offers a new derivation of the parity anomaly in 2+1 dimensions, showing how a non local symmetry leads to the Chern-Simons action via a well-defined Jacobian.

Contribution

It introduces an alternative approach using non local symmetry and Jacobian analysis to derive the parity anomaly for massless Dirac fields.

Findings

Derivation of the parity anomaly from a non local symmetry

Explicit connection between Jacobian and Chern-Simons action

Finite, well-defined Jacobian reproduces standard anomaly in the limit

Abstract

We present an alternative derivation of the parity anomaly for a massless Dirac field in 2+1 dimensions coupled to a gauge field. The anomaly functional, a Chern-Simons action for the gauge field, is obtained from the non-trivial Jacobian corresponding to a non local symmetry of the Pauli-Villars regularized action. That Jacobian is well-defined, finite, and yields the standard Chern-Simons term when the cutoff tends to infinity.