Anomaly cancellation in 2+1 dimensions in the presence of a domain wall mass

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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), in the background of an Abelian gauge field is considered. In this model, originally proposed in a non-Abelian version 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 favor 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 which is 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 behavior of the (1+1)-dimensional effective chiral theory thereby resolving an ambiguity present in a general (1+1)-dimensional theory.
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