Noncommutative Ward’s Conjecture and Integrable Systems

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-selfdual 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, NonLinear Schrödinger, Boussinesq, N-wave, (affine) Toda, sine-Gordon, Liouville, Tzitzéica, (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. The author visits Oxford from 16 August, 2005 to 15 August, 2006, supported by the Yamada Science Foundation. E-mail: [email protected], [email protected]