INTEGRABLE CHERN-SIMONS GAUGE FIELD THEORY IN 2+1 DIMENSIONS

Abstract

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 us 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 the 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. The physical interpretation of the Bäcklund transformations in terms of the Chern-Simons gauge theory is given. Some possible applications for the statistical transmutation in the anyon superfluid and TQFT are briefly discussed.