A Chern-Simons action written with Christoffel Symbols has a natural gauge symmetry of diffeomorphisms. This Chern-Simons action will induce a WessZumio-Witten model on the boundary of the manifold. If we restrict the diffeomorphisms to chiral diffeomorphism, the Wess-Zumio-Witten model is equivalent to a quantum Liouville action. email: [email protected]

We discuss a model for the integer quantum Hall effect which is based on a Schroedinger-Chern-Simons-action functional for a non-interacting system of electrons in an electromagnetic field on a mutiply connected manifold. In this model the integer values of the Hall conductivity arises in view of the quanti-zation of the Chern-Simons-action functional for electromagnetic potential.

An influence of the quantum potential on the Chern–Simons solitons leads to quantization of the statistical parameter κ = me/g, and the quantum potential strenght s = 1 − m. A new type of exponentially localized Chern–Simons solitons for the Bloch electrons near the hyperbolic energy band boundary are found.

We examine a criterion for the anyonic superconductivity at zero temperature in Abelian matter-coupled Chern-Simons gauge field theories in three dimensions. By solving the Dyson-Schwinger equations, we obtain a critical value of the statistical charge for the superconducting phase in a massless fermion-Chern-Simons model.

We provide a unified proof of the asymptotics of the self-dual MaxwellChern-Simons vortices, as the Maxwell term is neglected, in both the U(1) and CP (1) case. This result is achieved by identifying and analyzing a suitable class of nonlinear elliptic systems with exponential type nonlinearities. key words: nonlinear elliptic system, Chern-Simons vortex theory MCS 2000 subject classification: ...

The three dimensional Chern-Simons theory on the noncommutative plane is considered. We show that the action is invariant under arbitrary (finite) gauge transformations and hence, we can relax the level quantization condition which we have in the usual (commutative) Chern-Simons theories. We also briefly discuss the case of the noncommutative torus.