Quark confinement due to unified magnetic monopoles and vortices reduced from symmetric instantons with holography

We develop a geometric framework to analyze quark confinement in four-dimensional Euclidean $SU(2)$ Yang–Mills theory in terms of finite-action topological defects. Starting from self-dual Yang–Mills configurations, we restrict to \emph{symmetric instantons} with spatial rotation symmetry so that dimensional reduction preserves conformal equivalence. This requirement maps ${\mathbb{R}}^4$ to curved backgrounds with compact directions and, in particular, identifies the reduced configurations with (i) hyperbolic magnetic monopoles of Atiyah type on $H^3\simeq {\mathrm{A}\mathrm{d}\mathrm{S}}_3$ (from an $SO(2)\simeq S^1$ symmetry) and (ii) hyperbolic vortices of Witten–Manton type on $H^2\simeq {\mathrm{A}\mathrm{d}\mathrm{S}}_2$ (from an $SO(3)\simeq SU(2)$ symmetry). We provide an explicit field map relating the monopole and vortex variables, enabling a unified treatment of these defects within the original four-dimensional setting. Moreover, the hyperbolic monopole on $H^3$ is completely determined by its holographic data on the conformal boundary $S_{\mathrm{\infty }}^2$, which reduces a non-Abelian Wilson loop placed on $\mathrm{\partial }{H}^3$ to an Abelian loop determined by the vortex $U(1)$ field (Abelian dominance and monopole dominance), without further dynamical assumptions beyond the symmetry reduction. In the semiclassical dilute-gas regime of these finite-action defects, the framework yields the Wilson area law, thereby providing analytic support for the dual-superconductor picture of confinement.