Noncommutative Instantons via Dressing and Splitting Approaches

TL;DR

This paper introduces two alternative methods, dressing transformations and Ward's splitting, for constructing noncommutative instantons, expanding beyond the traditional ADHM approach and enabling potential multi-instanton solutions.

Contribution

It presents the first application of dressing and splitting methods to noncommutative instanton construction, generalizing classical techniques to the noncommutative setting.

Findings

Successfully constructed noncommutative U(2) BPST instanton solutions.

Established a connection between dressing transformations and twistor-based splitting methods.

Indicated the potential for generating nonsingular multi-instanton solutions.

Abstract

Almost all known instanton solutions in noncommutative Yang-Mills theory have been obtained in the modified ADHM scheme. In this paper we employ two alternative methods for the construction of the self-dual U(2) BPST instanton on a noncommutative Euclidean four-dimensional space with self-dual noncommutativity tensor. Firstly, we use the method of dressing transformations, an iterative procedure for generating solutions from a given seed solution, and thereby generalize Belavin's and Zakharov's work to the noncommutative setup. Secondly, we relate the dressing approach with Ward's splitting method based on the twistor construction and rederive the solution in this context. It seems feasible to produce nonsingular noncommutative multi-instantons with these techniques.