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

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.