A generalization of King's equation via noncommutative geometry (Q2055094)

In this nice paper, the authors give some applications of NCG to self-dual Yang-Mills equations and King's equation. There is a remarkable similarity between self-dual Yang-Mills equations and equations introduced by King for representations of quivers. The underlying reason is that both equations are obtained from appropriate moment maps. The authors introduce in this paper a common generalization based on noncommutative geometry. In this setup the moment map equation is governed by a cyclic 1-cochain. Examples of a generalized King's equation include ADHM equations, noncommutative instantons, vortex equations (in particular Hitchin and Vafa-Witten equations), as well as Bogomolny and Nahm equations for the gauge group \(U(k)\). Furthermore, the authors discuss Nekrasov's suggestion to reinterpret noncommutative instantons as infinite-dimensional versions of King's equation, also related to Quantum minimal. For the entire collection see [Zbl 1466.57001].