B\"acklund Transformations and the Atiyah-Ward ansatz for Noncommutative Anti-Self-Dual Yang-Mills Equations

TL;DR

This paper develops Backlund transformations for noncommutative anti-self-dual Yang-Mills equations, generating exact solutions expressed via quasideterminants, and links these solutions to noncommutative twistor theory and the Atiyah-Ward ansatz.

Contribution

It introduces Backlund transformations for noncommutative gauge theories and connects them to noncommutative twistor geometry, providing a method to generate explicit solutions.

Findings

Generated exact solutions using quasideterminants.

Linked solutions to noncommutative Atiyah-Ward ansatz.

Established a framework within noncommutative twistor theory.

Abstract

We present Backlund transformations for the noncommutative anti-self-dual Yang-Mills equation where the gauge group is G=GL(2) and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasideterminants. We also explain the origins of all of the ingredients of the Backlund transformations within the framework of noncommutative twistor theory. In particular we show that the generated solutions belong to a noncommutative version of the Atiyah-Ward ansatz.