Noncommutative Integrable Field Theories in 2d

TL;DR

This paper investigates noncommutative versions of 2D integrable models, revealing that naive generalizations are not integrable, but with added constraints, models like sinh-Gordon and U(N) principal chiral become integrable, with explicit conserved charges constructed.

Contribution

It demonstrates how to restore integrability in noncommutative 2D models by adding constraints and constructs explicit conserved charges for the U(N) principal chiral model.

Findings

Naive noncommutative sinh-Gordon model is not integrable.

Adding constraints restores integrability.

Explicit conserved charges are constructed for the U(N) model.

Abstract

We study the noncommutative generalization of (euclidean) integrable models in two-dimensions, specifically the sine- and sinh-Gordon and the U(N) principal chiral models. By looking at tree-level amplitudes for the sinh-Gordon model we show that its na\"\i ve noncommutative generalization is {\em not} integrable. On the other hand, the addition of extra constraints, obtained through the generalization of the zero-curvature method, renders the model integrable. We construct explicit non-local non-trivial conserved charges for the U(N) principal chiral model using the Brezin-Itzykson-Zinn-Justin-Zuber method.