Non-commutative Ward's conjecture and integrable systems

Abstract

Non-commutative Ward's conjecture is a non-commutative 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 non-commutative integrable equations in both ( 2 + 1 ) - and ( 1 + 1 ) -dimensions are actually reductions of non-commutative anti-self-dual Yang–Mills equations with finite gauge groups, which include non-commutative versions of Calogero–Bogoyavlenskii–Schiff equation, Zakharov system, Ward's chiral and topological chiral models, (modified) Korteweg–de Vries, non-linear Schrödinger, Boussinesq, N-wave, (affine) Toda, sine-Gordon, Liouville, Tzitzéica, (Ward's) harmonic map equations, 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 non-commutative integrable systems and string theories. Some integrable aspects of them are also discussed.