Explicit Derivation of Yang-Mills Self-Dual Solutions on non-Commutative Harmonic Space

TL;DR

This paper develops a noncommutative harmonic space framework to analyze and explicitly solve the self-dual Yang-Mills equations in four dimensions, extending harmonic analyticity methods to noncommutative settings.

Contribution

It introduces a new noncommutative harmonic space approach for solving Yang-Mills self-duality equations and provides explicit solutions including an exact one.

Findings

Formulated noncommutative Yang-Mills self-duality constraints on harmonic space.

Extended harmonic analyticity to linearize noncommutative equations.

Derived explicit solutions, including an exact self-dual solution.

Abstract

We develop the noncommutative harmonic space (NHS) analysis to study the problem of solving the non-linear constraint eqs of noncommutative Yang-Mills self-duality in four-dimensions. We show that this space, denoted also as NHS($\eta,\theta$), has two SU(2) isovector deformations $\eta^{(ij)}$ and $\theta^{(ij)}$ parametrising respectively two noncommutative harmonic subspaces NHS($\eta,0$) and NHS($0,\theta$) used to study the self-dual and anti self-dual noncommutative Yang-Mills solutions. We formulate the Yang-Mills self-dual constraint eqs on NHS($\eta,0$) by extending the idea of harmonic analyticity to linearize them. Then we give a perturbative self-dual solution recovering the ordinary one. Finally we present the explicit computation of an exact self-dual solution.