Relating U(N) x U(N) to SU(N) x SU(N) Chern-Simons membrane theories

By integrating out the U(1) B gauge field, we show that the U(n) x U(n) ABJM theory at level k is equivalent to a Z(k) identification of the (SU(n) x SU(n))/Z(n) Chern-Simons theory, but only when n and k are coprime. As a consequence, the k = 1 ABJM model for two M2-branes in R-8 can be identified with the N = 8 (SU(2)x SU(2))/Z(2) theory. We also conjecture that the U(2) x U(2) ABJM model at k = 2 is equivalent to the N = 8 SU(2) x SU(2)-theory.