Mapping an Instanton to Spacetime

A mapping from the Lie algebra of the complexified Lorentz group to the $\mathfrak{s}\mathfrak{u}(2)\times \mathfrak{s}\mathfrak{u}(2)\sim \mathfrak{s}\mathfrak{p}(1)\times \mathfrak{s}\mathfrak{p}(1)$ part of the algebra the coset space $Sp(2)\mathrm{/}[Sp(1)\times Sp(1)]$ is presented. The coset space is shown to be home to the instanton, the curvature form that optimizes the Yang-Mills functional. Arguments are presented to support the generalization to $Sp(n)\mathrm{/}Sp(1{)}^n$ to yield a self-consistent many-body theory for $n$ particles interacting with one another via fields that reside in the coset space.