Anomaly Cancellation in 2+1 dimensions in the presence of a domainwall mass

TL;DR

This paper explicitly calculates the anomaly cancellation mechanism in a 2+1 dimensional fermion model with a domain wall mass, confirming the cancellation of gauge variation contributions from Chern-Simons and chiral fermions.

Contribution

It provides a complete one-loop calculation of anomaly cancellation near the domain wall, including the effects of massive modes and high-energy behavior.

Findings

Chern-Simons term is generated by integrating out massive modes

The effective chiral theory's high-energy behavior is softened

An explicit cancellation of gauge anomalies is demonstrated

Abstract

A Fermion in 2+1 dimensions, with a mass function which depends on one spatial coordinate and passes through a zero ( a domain wall mass), is considered. In this model, originally proposed by Callan and Harvey, the gauge variation of the effective gauge action mainly consists of two terms. One comes from the induced Chern-Simons term and the other from the chiral fermions, bound to the 1+1 dimensional wall, and they are expected to cancel each other. Though there exist arguments in favour of this, based on the possible forms of the effective action valid far from the wall and some facts about theories of chiral fermions in 1+1 dimensions, a complete calculation is lacking. In this paper we present an explicit calculation of this cancellation at one loop valid even close to the wall. We show that, integrating out the ``massive'' modes of the theory does produce the Chern-Simons term, as appreciated previously. In addition we show that it generates a term that softens the high energy behaviour of the 1+1 dimensional effective chiral theory thereby resolving an ambiguity present in a general 1+1 dimensional theory.