Noncommutative Monopoles and Riemann-Hilbert Problems

TL;DR

This paper extends the Riemann-Hilbert problem approach to construct and analyze noncommutative monopole solutions in Yang-Mills-Higgs theory, providing a unified framework for various monopole types.

Contribution

It introduces a method to solve noncommutative monopoles using Riemann-Hilbert problems, generalizing previous techniques to noncommutative spaces.

Findings

Constructed noncommutative Dirac, Wu-Yang, and BPS monopoles.

Solved Riemann-Hilbert problems for charge one monopoles.

Unified approach for multiple monopole configurations.

Abstract

The Bogomolny equations for Yang-Mills-Higgs monopoles follow from a system of linear equations which may be solved through a parametric Riemann-Hilbert problem. We extend this approach to noncommutative R^3 and use it to (re)construct noncommutative Dirac, Wu-Yang, and BPS monopole configurations in a unified manner. In all cases we write down the underlying matrix-valued functions for multi-monopoles and solve the corresponding Riemann-Hilbert problems for charge one.