On gauge invariance of noncommutative Chern-Simons theories

Motivated by possible applications to condensed matter systems, in this paper we construct U(N) noncommutative Chern-Simons (NCCS) action for a disc and for a double-layer geometry, respectively. In both cases, gauge invariance severely constrains the form of the NCCS action. In the first case, it is necessary to introduce a group-valued boson field with a non-local chiral boundary action, whose gauge variation cancels that of the bulk action. In the second case, the coefficient matrix K in the double U(N) NCCS action is restricted to be of the form K = k (1 1) ( 1 1) with integer k. We suggest that this double NCCS theory with U(1) gauge group describes the so-called Halperin (kkk) state in a double-layer quantum Hall system. Possible physical consequences are addressed.