Pluriharmonic solutions to Yang-Mills equations: a $C^*$-algebras approach

TL;DR

This paper explores Yang-Mills equations using complex analysis and $C^*$-algebras, introducing a new class of instanton solutions via operator-valued pluriharmonic forms.

Contribution

It develops a novel approach to Yang-Mills equations employing complex variables and operator theory, leading to new instanton solutions and a complex variable formulation of the Lagrangian.

Findings

Constructed a new class of instanton solutions

Provided a complex variable version of the Yang-Mills Lagrangian

Linked operator-valued pluriharmonic forms to gauge theory

Abstract

This partially expository paper provides a view of Yang-Mills equations from the perspective of complex variables, operator theory, and $C^{*}$-algebras. Through operator-valued pluriharmonic and skew-Hermitian differential forms, it constructs a new class of instanton solutions. Furthermore, it provides a complex variable version of the Yang-Mills Lagrangian and the Belavin-Polyakov-Schwartz-Tyupkin instanton.