Integrable Chern-Simons Gauge Field Theory in 2+1 Dimensions

TL;DR

This paper develops an integrable 2+1-dimensional Chern-Simons gauge field theory for planar spin models, linking it to the Davey-Stewartson equations and exploring applications in anyon superfluidity and topological quantum field theory.

Contribution

It introduces a zero-curvature formulation of the Chern-Simons model, connecting it to integrable DS equations and deriving new reduction conditions.

Findings

Formulation of the model with gauge-invariant fields

Derivation of zero-curvature representation for DS equations

Discussion of applications in anyon superfluid and TQFT

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 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.