2006
DOI: 10.1016/j.nuclphysb.2006.02.014
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Non-commutative Ward's conjecture 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-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., Zak… Show more

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Cited by 31 publications
(32 citation statements)
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“…We note that the noncommutative ASDYM equation gives rise by reduction to various noncommutative lower-dimensional integrable equations including all equations above except for noncommutative Burgers equation. More examples are summarized in [25,26], which would be evidence for noncommutative version [29] of the Ward conjecture [58]. (See also [1,40].…”
Section: Integrable Equations In Noncommutative Spacesmentioning
confidence: 99%
“…We note that the noncommutative ASDYM equation gives rise by reduction to various noncommutative lower-dimensional integrable equations including all equations above except for noncommutative Burgers equation. More examples are summarized in [25,26], which would be evidence for noncommutative version [29] of the Ward conjecture [58]. (See also [1,40].…”
Section: Integrable Equations In Noncommutative Spacesmentioning
confidence: 99%
“…. ., and if the Wronski matrix W is invertible, then φ = ∂(W )W −1 (25) solves the pKP hierarchy in the algebra of N × N matrices with entries in R and product A · B = AQB with Q defined in (20). Furthermore, ϕ defined in (21) then solves the pKP hierarchy in R.…”
Section: Propositionmentioning
confidence: 99%
“…Among them, the noncommutative anti-self-dual Yang-Mills (ASDYM) equation in 4dimensions is important because in the Euclidean signature (++++), the ADHM construction can be used to find all exact instanton solutions and gives rise to new physical objects such as U(1) instantons (Nekrasov and Schwarz (1998)). In the split signature (++−−), many noncommutative integrable equations can be derived from the noncommutative ASDYM equation by a reduction process (See Hamanaka (2005b); Hamanaka (2006) and references therein). Integrable aspects of the noncommutative ASDYM equation can be understood in the geometrical framework of noncommutative twistor theory (Brain (2005); Brain and Majid (2008); Hannabuss (2001) ;Horváth, Lechtenfeld and Wolf (2002); Ihl and Uhlmann (2003); Kapustin, Kuznetsov and Orlov (2001); Lechtenfeld and Popov (2002); Takasaki (2001)).…”
Section: Introductionmentioning
confidence: 99%