International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization International Centre for Theoretical Physics Integrable Chern-simons Gauge Field Theory in 2+1 Dimensions

The classical spin model in planar condensed media is represented as the U (1) Chern-Simons gauge field theory. When the vorticity of the continuous flow of the media coincides with the statistical magnetic field, which is necessary for the model's integrability, the theory admits zero curvature connection. This allows me to formulate the model in terms of gauge-invariant fields whose evolution is described by the Davey-Stewartson (DS) equations. The Self-dual Chern-Simons solitons described by the Liouville equation are subjected to corresponding integrable dynamics. As a by-product the 2+1-dimensional zero-curvature representation for the DS equation is obtained as well as the new reduction conditions related to the DS-I case. Some possible applications for the statistical transmutation in the anyon superfluid and TQFT are briefly discussed. 1 0 Introduction Recently, the Chern-Simons (CS) solitons are derived as solutions of the self-dual CS system [1]. These solutions have an important quantum meaning producing excitations known as anyons possessing arbitrary statistics [2]. They are relevant to the Quantum Hall Effect (QHE) and the High Temperature Superconductivity since they are still being intensively studied [3,4]. In fact, the effective Ginzburg-Landau theory of the QHE turns out to be described by the Nonlinear Schrödinger Equation (NLSE) in 2+1 dimensions for a complex matter field which couples with the Abelian Chern-Simons gauge field. When the field's self-interaction coupling strength is related to the Chern-Simons coupling constant the model reduces to the static self-dual CS equations [1]. These equations, being conformal invariant, are completely integrable and admit the Lax pair representation [5]. The time evolution of the model, in general, can be arranged as a conformal mapping of the plane to itself at different times. But analysis of the time dependent solutions with integrable dynamics remains an open problem. Jackiw and Pi [6] mentioned the tantalizing similarity of the problem with the Davey-Stewartson equation, being a well known integrable extension of the NLSE in 2+1 dimensions [13]. In the present paper I reduce the self-dual CS model from an integrable model in 2+1 dimensions. This model was first derived by Ishimori and has integrable dynamics of the magnetic vortices [7]. Its formulation in terms of the vorticity for continuous flow tends to clarify the physical meaning of the model and allows an extension for higher dimensions, admitting the bilinear Hirota representation [8,10]. The physical applications of the model are connected with the resolution of the anomalous behavior of …