CHERN-SIMONS TERMS IN NONCOMMUTATIVE GEOMETRY AND ITS APPLICATION TO BILAYER QUANTUM HALL SYSTEMS

Considering bilayer systems as extensions of the planar ones by an int ernal space of two discrete points, we use the ideas of Noncommutative Geometry to construct the gauge theories for these systems. After int egrating over the discrete space we find an effective 2 + 1 action inv olving an extra complex scalar field, which can be interpreted as aris ing from the tunneling between the layers. The gauge fields are found in different phases corresponding to the different correlations due to the Coulomb interaction between the layers. In a particular phase, wh en the radial part of the complex scalar field is a constant, we recov er the Wen-Zee model [X.G. Wen and A, Zee, Phys. Rev. Lett. 69 (1992) 1811; Phys. Rev. B 47 (1993) 2265] of Bilayer Quantum Hall systems. Th ere are some circumstances, where this radial part may become dynamica l and cause dissipation in the oscillating supercurrent between the la yers.