Noncommutative SU(N) theories, the axial anomaly, Fujikawa's method and the Atiyah-Singer index

Fujikawa's method is employed to compute at first order in the noncommutative parameter the U(1)A anomaly for noncommutative SU(N). We consider the most general Seiberg-Witten map which commutes with hermiticity and complex conjugation and a noncommutative matrix parameter, θμν, which is of “magnetic” type. Our results for SU(N) can be readily generalized to cover the case of general nonsemisimple gauge groups when the symmetric Seiberg-Witten map is used. Connection with the Atiyah-Singer index theorem is also made.