Explicit derivation of Yang-Mills self-dual solutions on non-commutative harmonic space

We develop the non-commutative harmonic space (NHS) analysis to study the problem of solving the nonlinear constraint equations of non-commutative 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)) parametrizing, respectively, two non-commutative harmonic subspaces NHS(eta, 0) and NHS(0, theta) used to study the self-dual and anti self-dual noncommutative Yang-Mills solutions. We reformulate the Yang-Mills self-dual constraint equations on NHS(eta, 0) by extending the idea of harmonic analyticity to linearize them. We then give a perturbative self-dual solution recovering the ordinary one. Finally, we present the explicit computation of an exact self-dual solution.