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 with all the matrix elements being the same 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.