  The topology of closed manifolds forces interacting charges to appear in
pairs. We take advantage of this property in the setting of the conformal
boundary of $\mathrm{AdS}_5$ spacetime, topologically equivalent to the closed
manifold $S^1\times S^3$, by considering the coupling of two massless opposite
charges on it. Taking the interaction potential as the analog of Coulomb
interaction (derived from a fundamental solution of the $S^3$ Laplace-Beltrami
operator), a conformal $S^1\times S^3$ metric deformation is proposed, such
that free motion on the deformed metric is equivalent to motion on the round
metric in the presence of the interaction potential. We give explicit
expressions for the generators of the conformal algebra in the representation
induced by the metric deformation.
  By identifying the charge as the color degree of freedom in QCD, and the two
charges system as a quark--anti-quark system, we argue that the associated
conformal wave operator equation could provide a realistic quantum mechanical
description of the simplest QCD system, the mesons.
  Finally, we discuss the possibility of employing the compactification radius,
$R$, as another scale along $\Lambda_{QCD}$, by means of which, upon
reparametrizing $Q^2c^2$ as $\left( Q^2c^2 +\hbar^2 c^2/R^2\right)$, a
pertubative treatment of processes in the infrared could be approached.
