The Lax Pair by Dimensional Reduction of Chern-Simons Gauge Theory

TL;DR

This paper demonstrates how the Nonlinear Schrödinger Equation and its Lax pair in 1+1 dimensions can be derived from a 2+1 dimensional Chern-Simons gauge theory through dimensional reduction, revealing new insights into integrable systems.

Contribution

It introduces a novel derivation of the 1+1 dimensional NLS equation from 2+1D Chern-Simons theory, connecting gauge theory and integrable systems via dimensional reduction.

Findings

Spectral parameter arises as a condensate of the gauge field

Derivation of the Lax pair from topological gauge theory

Interpretation of Chern-Simons Gauss law in soliton context

Abstract

We show that the Nonlinear Schr\"odinger Equation and the related Lax pair in 1+1 dimensions can be derived from 2+1 dimensional Chern-Simons Topological Gauge Theory. The spectral parameter, a main object for the Loop algebra structure and the Inverse Spectral Transform, has appear as a homogeneous part (condensate) of the statistical gauge field, connected with the compactified extra space coordinate. In terms of solitons, a natural interpretation for the one-dimensional analog of Chern-Simons Gauss law is given.