Can Seiberg-Witten Map Bypass Noncommutative Gauge Theory No-Go Theorem?

There are strong restrictions on the possible representations and in general on the matter content of gauge theories formulated on noncommutative Moyal spaces, termed as noncommutative gauge theory no-go theorem. According to the no-go theorem \cite{no-go}, matter fields in the noncommutative U(1) gauge theory can only have $\pm 1$ or zero charges and for a generic noncommutative $\prod_{i=1}^n U(N_i)$ gauge theory matter fields can be charged under at most two of the $U(N_i)$ gauge group factors. On the other hand, it has been argued in the literature that, since a noncommutative U(N) gauge theory can be mapped to an ordinary U(N) gauge theory via the Seiberg-Witten map, seemingly it can bypass the no-go theorem. In this note we show that the Seiberg-Witten map \cite{SW} can only be consistently defined and used for the gauge theories which respect the no-go theorem. We discuss the implications of these arguments for the particle physics model building on noncommutative space.
Comment: 16 pages; typos and formulas (2.29) and (2.30) corrected