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Scattering of instantons, monopoles and vortices in higher dimensions

We consider Yang-Mills theory on manifolds ${\mathbb R}\times X$ with a $d$-dimensional Riemannian manifold $X$ of special holonomy admitting gauge instanton equations. Instantons are considered as particle-like solutions in $d+1$ dimensions whose static configurations are concentrated on $X$. We study how they evolve in time when considered as solutions of the Yang-Millsequations on ${\mathbb R}\times X$ with moduli depending on time $t\in{\mathbb R}$. It is shown that in the adiabatic limit, when the metric in the $X$ direction is scaled down, the classical dynamics of slowly moving instantons corresponds to a geodesic motion in the moduli space $\cal M$ of gauge instantons on $X$. Similar results about geodesic motion in the moduli space of monopoles and vortices in higher dimensions are briefly discussed.