Seiberg–Witten map and the axial anomaly in non-commutative field theory

Using the point-splitting regularisation, we calculate the axial anomaly in an arbitrary even dimensional non-commutative (NC) field theory. Our result is (star) gauge invariant in its unintegrated form, to the leading order in the NC parameter. Exploiting the Seiberg–Witten map, this result gets transformed to the familiar Adler–Bell–Jackiw anomaly in ordinary spacetime. Furthermore, using this map, we derive an expression for the unintegrated axil anomaly for constant fields in NC spacetime, that is valid to all finite orders of the NC parameter.