Noncommutative Ward's Conjecture and Integrable Systems

TL;DR

This paper proves that many noncommutative integrable equations are reductions of noncommutative anti-self-dual Yang-Mills equations, establishing a unifying framework with implications for twistor theory and string theory.

Contribution

It demonstrates that a wide class of noncommutative integrable equations originate from noncommutative anti-self-dual Yang-Mills equations, extending Ward's conjecture to the noncommutative setting.

Findings

Many noncommutative integrable equations are reductions of noncommutative anti-self-dual Yang-Mills equations.

Establishes the existence of twistor descriptions for these equations.

Implications for noncommutative string theories and integrable systems.

Abstract

Noncommutative Ward's conjecture is a noncommutative version of the original Ward's conjecture which says that almost all integrable equations can be obtained from anti-self-dual Yang-Mills equations by reduction. In this paper, we prove that wide class of noncommutative integrable equations in both (2+1)- and (1+1)-dimensions are actually reductions of noncommutative anti-self-dual Yang-Mills equations with finite gauge groups, which include noncommutative versions of Calogero-Bogoyavlenskii-Schiff eq., Zakharov system, Ward's chiral and topological chiral models, (modified) Korteweg-de Vries, Non-Linear Schroedinger, Boussinesq, N-wave, (affine) Toda, sine-Gordon, Liouville, Tzitzeica, (Ward's) harmonic map eqs., and so on. This would guarantee existence of twistor description of them and the corresponding physical situations in N=2 string theory, and lead to fruitful applications to noncommutative integrable systems and string theories. Some integrable aspects of them are also discussed.