The profound, beautiful and, at times, rather mysterious symbiosis between mathematics and physics has always been a source of wonder, but, in the past twenty years, the intensity of the mutual interaction between these two has become nothing short of startling. Our objective here is to provide an introduction, in terms as elementary as possible, to one small aspect of this relationship. Toward this end we shall tell a story. Although we make no attempt to relate it chronologically, the story can be said to begin with the efforts of Yang and Mills to construct a nonabelian generalization of classical electromagnetic theory, and to culminate in a remarkable conjecture of Witten concerning the Donaldson invariants of a smooth four-manifold.

We show that the two orbits in the coadjoint representation of the group SO(4,2), the one, related with the Kepler's problem, and the other - modelling the phase space of a conformal particle with zero mass and spirality zero, coincide.