A Unifying conformal field theory approach to the quantum Hall effect

We review the main results of the effective description of the Quantum Hall fluid for the Jain fillings, nu=m/2pm+1, and the non-standard ones nu=m/pm+2 by a conformal field theory (CFT) in two dimensions. It is stressed the unifying character of the m-reduction procedure to construct appropriate twisted CFT models, called Twisted Models (TM), which by construction reproduce the Quantum Hall topological properties at those fillings. Indeed for the Jain plateaux we find that the different descriptions given in the literature fall into different sectors of the TM for the torus topology. Other interesting aspects are explicitly seen for the m=2 non standard filling nu=1/p+1 (the pairing case) as the merging of non-Abelian statistics or the instability of the TM model (c=2) versus the Moore-Read one (c=3/2). Furthermore by using Boundary CFT techniques the presence of localized impurities and/or dissipation is shown to be closely connected with the twisted sector of the TM, whose presence assures the consistency of the construction and whose role in describing non trivial global properties of 2D quantum condensed matter systems is still under study.