Color confinement at the boundary of the conformally compactified $\mathrm{AdS}_5$

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.