The gauge anomaly and the Seiberg–Witten map Original Research Article

The consistent form of the gauge anomaly is worked out at first order in θ 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 TrTaTbTc=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 renormalizability paradigm is relinquished and one admits noncommutative counterterms (of the gauge fields, its derivatives and θ) 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: TrTa{Tb,Tc}=0.