Can Seiberg–Witten map bypass noncommutative gauge theory no-go theorem?

Abstract

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 (Chaichian, Prešnajder, Sheikh-Jabbari and Tureanu, 2002) [1], matter fields in the noncommutative U ( 1 ) gauge theory can only have ±1 or zero charges and for a generic noncommutative ∏ 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 (Seiberg and Witten, 1999) [2] 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.