General self-dual Yang-Mills solutions

Abstract

The recent work of Atiyah, Hitchin, Drinfeld, and Manin is used to discuss self-dual Yang-Mills solutions for the compact gauge groups $O\left(n\right)\mathrm{}$, $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$, and $\mathrm{Sp}\mathrm{}\left(n\right)\mathrm{}$. It is shown that the resulting solutions contain the correct number of parameters for all values of the topological charge. Although explicit construction of a general self-dual field requires the solution of a finite-dimensional, nonlinear matrix equation, we show that for widely separated instantons this equation can be solved perturbatively, providing a systematic expansion about the dilute-gas limit and a physical interpretation of the independent parameters in this limit. Further, closed-form expressions can be obtained for the general SU(2) solutions with topological charge 2 or 3. Finally, explicit isospin-1/2 and isospin-1 propagators are derived for a massless scalar field in the presence of the general self-dual SU(2) solution.