Gauge Symmetry and Integrable Models

TL;DR

This paper demonstrates a novel connection between nonlinear sigma models and abelian gauge theories on curved backgrounds, enabling derivation of integrable models and Lax pairs through gauge theory methods, with spectral parameters linked to spacetime symmetries.

Contribution

It introduces a gauge-theoretic framework to derive integrable models and their Lax representations, relating spectral parameters to spacetime symmetries and compactified dimensions.

Findings

Established isomorphism between sigma models and gauge theories on curved backgrounds.

Derived integrable models and Lax pairs from gauge theoretical perspective.

Connected Bäcklund transformations to Chern-Simons theory with spectral parameters.

Abstract

We establish the isomorphism between a nonlinear $\sigma$-model and the abelian gauge theory on an arbitrary curved background, which allows us to derive integrable models and the corresponding Lax representations from gauge theoretical point of view. In our approach the spectral parameter is related to the global degree of freedom associated with the conformal or Galileo transformations of the spacetime. The B$\ddot{\rm a}$cklund transformations are derived from Chern-Simons theory where the spectral parameter is defined in terms of the extract compactified space dimension coordinate.