Classical Integrable Systems and Gauge Theory

TL;DR

This survey explores classical integrability through symmetry reduction of anti-self-dual Yang-Mills equations, discussing twistor methods, solutions via quasideterminants, and extensions to noncommutative and higher dimensions.

Contribution

It provides a comprehensive introduction to integrability, detailing twistor construction, explicit solutions, and extensions, with a focus on symmetry reduction and noncommutative generalizations.

Findings

Explicit local solutions using quasideterminants

Clarification of integrable aspects in lower dimensions

Discussion of noncommutative and higher-dimensional extensions

Abstract

This is an introductory survey written in Japanese on classical integrability from the viewpoint of symmetry reduction of 4-dimensional anti-self-dual Yang-Mills equations. Twistor construction methods are discussed in detail in 4-dimension. Exact local solutions are, in this article, presented in terms of quasideterminants in a compact form. Some integrable aspects in lower-dimension are clarified in the context of the reduction. Noncommutative extension and higher-dimensional extension are briefly discussed.