The gauge anomaly and the Seiberg-Witten map

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 Tr 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 renormalizability paradigm is relinquished and one admits noncommutative counterterms (of the gauge fields, its derivatives and 0) 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: Tr T-a {T-b, T-c} = 0. (C) 2003 Elsevier Science B.V. All rights reserved.