ar X iv : h ep - t h / 06 06 28 6 v 1 3 0 Ju n 20 06 On the NCCS model of the quantum Hall fluid March 5 , 2008

Area non-preserving transformations in the non-commutative plane are introduced with the aim to map the ν = 1 IQHE state on the ν < 1 FQHE states. Using the hydrodynamical description of the quantum Hall fluid it is shown, that these transformations are generated by the vector fields satisfying the Gauss law in the non-commutative Chern-Simons gauge theory and the corresponding field-theory Lagrangian is reconstructed. It is demonstrated that the geometric transformations induce quantum-mechanical similarity transformations , which establish the interplay between integral and fractional QHEs Apparent similarity between integral and fractional quantum Hall effects challenges the search of the theoretical schemes which bridge the gap between them. One of the physically appealing models is the Jain's composite fermion (CF) picture [1], which links FQHE of interacting electrons to IQHE of composite fermions. CF picture gains mathematical content in the Chern-Simons (CS) gauge theory (see e.g. [2]): the principal role of the CS gauge field is to attach an even number of the elementary magnetic flux quanta to each electron, converting it to the composite particle.