Generalized parity transformations in the regularized Chern-Simons theory

TL;DR

This paper investigates how regularization affects parity symmetry in Abelian Chern-Simons theory, revealing an anomaly and proposing generalized parity transformations that restore symmetry in the regularized model.

Contribution

It introduces a framework for defining nonlocal, generalized parity transformations that preserve the odd nature of the regularized Chern-Simons action.

Findings

Regularization induces a parity anomaly in the Abelian Chern-Simons theory.

Generalized, nonlocal parity transformations can restore the odd symmetry of the regularized action.

These transformations approach the usual parity as the cutoff is removed.

Abstract

We study renormalization effects in the Abelian Chern-Simons (CS) action. These effects can be non-trivial when the gauge field is coupled to dynamical matter, since the regularization of the UV divergences in the model forces the introduction of a parity even piece in the gauge field action. This changes the classical (odd) transformation properties of the pure CS action. This effect, already discussed for the case of a lattice regularization by F. Berruto, M.C. Diamantini and P. Sodano in hep-th/0004203, is also present when the theory is defined in the continuum and, indeed, it is a manifestation of a more general `anomalous' effect, since it happens for every regularization scheme. We explore the physical consequences of this anomaly. We also show that generalized, nonlocal parity transformations can be defined in such a way that the regularized theory is odd, and that those transformations tend to the usual ones when the cutoff is removed. These generalized transformations play a role that is tantamount to the deformed symmetry corresponding to Ginsparg-Wilson fermions [2] (in an even number of spacetime dimensions).