Non Commutative Field Theories and Integrable Models in 2d

TL;DR

This paper investigates noncommutative extensions of integrable 2D field theories, showing that some deformations lose integrability while others can be consistently defined with additional constraints and conserved charges.

Contribution

It demonstrates how to construct integrable noncommutative versions of certain 2D models using a generalized zero-curvature approach, highlighting which models remain integrable.

Findings

Moyal deformations of sG and shG are not integrable due to particle production.

Noncommutative sG and shG models can be made integrable with extra constraints.

Noncommutative PCM remains integrable with non-trivial conserved charges.

Abstract

We study the noncommutative extensions of certain integrable field theories, namely the sine- and sinh-Gordon (sG and shG) models, and the U(N) principal chiral model (pcm). We argue that the Moyal deformations of the sG and shG models are not integrable, by looking at tree-level amplitudes where there is particle production. By considering the noncommutative generalization of the zero-curvature method, it is possible to define integrable versions of the noncommutative sG and shG models, which introduce extra constraints. The noncommutative pcm is shown to be integrable and we discuss the existence of non-trivial non-local conserved charges, and the associated noncommutative zero-curvature condition.