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

TL;DR

This paper develops a geometric framework linking symmetric instantons, magnetic monopoles, and vortices via holography to analytically support the Wilson area law and the dual-superconductor model of quark confinement.

Contribution

It introduces a unified geometric and holographic approach connecting instantons, monopoles, and vortices in Yang--Mills theory, providing analytic evidence for confinement.

Findings

Hyperbolic monopoles are determined by boundary data on $S^2_ ext{infty}$.

Wilson loops reduce to Abelian loops via monopole dominance.

The framework yields the Wilson area law in the dilute-gas regime.

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.