NC Branes and Hierarchies in Quantum Hall Fluids

We develop the non commutative Chern-Simons gauge model analysis modeling the description of the hierarchical states of fractional quantum Hall fluids. For a generic level \(n\) of the hierarchy, we show that the order parameter matrix \(K_{ab}\) is given by \( \theta^{-2}Tr(\tau_a \tau_b)\), where \(\{\tau_a, 1\leq a \leq n \}\) define a specific set of \(n\times n\) matrices depending on the parameter \(\theta\) and the levels \(l_a\) of the CS effective field theory. Our analysis predicts the existence of a third order tensor of order parameters \(C_{abc}\) induced by the external magnetic field. It is shown that the \(C_{abc}\)'s are not new order parameters and are given by \(\theta d_{abc}\), where \(d_{abc}\) are numbers depending on the \(l_a\)'s. We also give the generalized quantum Hall soliton extending the one obtained in the case of the Laughlin state.