Dimensional reduction, Seiberg-Witten map, and supersymmetry

It is argued that dimensional reduction of the Seiberg-Witten map for a gauge field induces Seiberg-Witten maps for the other noncommutative fields of a gauge invariant theory. We demonstrate this observation by dimensionally reducing the noncommutative N = 1 super Yang-Mills (SYM) theory in 6 dimensions to obtain noncommutative N = 2 SYM in 4 dimensions. We explicitly derive Seiberg-Witten maps of the component fields in 6 and 4 dimensions. Moreover, we give a general method to define the deformed supersymmetry transformations that leave the actions invariant after performing Seiberg-Witten maps.