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

TL;DR

This paper develops a geometric framework to analyze quark confinement in 4D $SU(2)$ Yang–Mills theory by restricting to symmetric instantons and employing conformal equivalence to map the problem to finite-action hyperbolic defects. The self-dual equations reduce to hyperbolic magnetic monopoles on $\mathbb{H}^3$ and hyperbolic vortices on $\mathbb{H}^2$, which are related by an explicit field map and linked through holography to boundary data, enabling Abelian dominance of Wilson loops. In the holographic setup, the boundary Abelian fields encode the non-Abelian content, and the Wilson loop reduces to an Abelian integral, yielding an area law in the dilute-gas regime of monopoles and vortices. Collectively, these steps provide analytic support for the dual-superconductor picture of confinement and illustrate how lower-dimensional topological defects emerge naturally from the 4D Yang–Mills theory via symmetry, conformal equivalence, and holography.

Abstract

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 $H^3\simeq \mathrm{AdS}_3$ (from an $SO(2)\simeq S^1$ symmetry) and (ii) hyperbolic vortices of Witten--Manton type on $H^2\simeq \mathrm{AdS}_2$ (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 $S^2_\infty$, which reduces a non-Abelian Wilson loop placed on $\partial H^3$ 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.

Figures (4)

• Figure 1: Conformal equivalence and symmetry reduction: (Left) 4-dim. Euclidean space $\mathbb{R}^4({t},x^1,x^2,x^3)$ versus 3-dim. hyperbolic space $\mathbb{H}^3({t},x^3,\rho)$. (Right) 4-dim. Euclidean space $\mathbb{R}^4({t},x^1,x^2,x^3)$ versus 2-dim. hyperbolic space $\mathbb{H}^2({t},r)$.
• Figure 2: Symmetric instanton, conformal equivalence, and dimensional reduction.
• Figure 3: (Left) The Wilson loop $C$ located on the conformal boundary $\partial \mathbb{H}^3$, i.e., $t-x^3$ plane of the hyperbolic space $\mathbb{H}^3$. (Right) The Wilson loop $C$ located on $\partial \mathbb{H}^2$, i.e., $t-r$ plane of the hyperbolic plane $\mathbb{H}^2$. Hyperbolic vortices (black circles) and anti-vortices (white circles) located inside and outside of the Wilson loop $C$ on $\mathbb{H}^2$.
• Figure 4: The 1-vortex solution with the center at $(t,r)=(0,1)$ and the size $\lambda=1$. The distribution of gauge-invariant quantities . (Left) field strength $F_{tr}(t,r)$, (Right) $|\phi(t,r)|^2$.