The Gauge Anomaly and the Seiberg-Witten Map

TL;DR

This paper analyzes gauge anomalies in noncommutative chiral gauge theories using the Seiberg-Witten map, deriving conditions for anomaly cancellation under different renormalizability assumptions.

Contribution

It provides a detailed first-order analysis of gauge anomalies in noncommutative gauge theories and clarifies how anomaly cancellation conditions depend on renormalizability assumptions.

Findings

Anomaly cancellation condition is tra T^a T^b T^c = 0 for power-counting renormalizable counterterms.

If non-power-counting renormalizable counterterms are allowed, the condition simplifies to tra T^a race T^b, T^c  = 0.

The analysis is performed within the framework of the Seiberg-Witten map for noncommutative gauge theories.

Abstract

The consistent form of the gauge anomaly is worked out at first order in $\theta$ for the noncommutative three-point function of the ordinary gauge field of certain noncommutative chiral gauge theories defined by means of the Seiberg-Witten map. We obtain that for any compact simple Lie group the anomaly cancellation condition of this three-point function reads $\traza \T^a \T^b \T^c = 0$, if one restricts the type of noncommutative counterterms that can be added to the classical action to restore the gauge symmetry to those which are renormalizable by power-counting. On the other hand, if the power-counting remormalizability paradigm is relinquished and one admits noncommutative counterterms (of the gauge fields, its derivatives and $\theta$) which are not power-counting renormalizable, then, the anomaly cancellation condition for the noncommutative three-point function of the ordinary gauge field becomes the ordinary one: $\traza \T^a \{\T^b,\T^c\} = 0$.