Chern-simons Vortices

In (2 + 1) dimensions, the Maxwell term − 1 4 F αβ F αβ can be replaced by the Chern-Simons three-form (κ/4)ǫ αβγ A α F βγ , yielding a novel type of 'electromagnetism'. This has been proposed for studying the Quantum Hall Effect as well as High-Temperature Superconductivity. The gauge field can be coupled to a scalar field either relativistically or non-relativistically. In both cases, one admits finite-energy, vortex solutions. 1. Introduction The phenomenological description of 'ordinary' superconductivity is provided by Landau-Ginzburg theory [1]: the Cooper pairs formed by the electrons are represented by a scalar field, whose charge is twice that of the electron. The scalar fields interact through their electromagnetic fields, governed by the Maxwell equations. The theory admits static, vortex-like solutions [2]. Landau-Ginzburg theory is non-relativistic; a relativistic version is provided by the Abelian Higgs model [3], which admits again static and purely magnetic vortex-type solutions. In the eighties, however, new phenomena were discovered, namely the Quantum Hall Effect [4] and High-Temperature Superconductivity [5]. None of these is described by Landau-Ginzburg theory. Hence the need for a new theory. Both phenomena mentioned above are essentially planar. The mathematical room for a modification is provided by the three-form