Quantized Non-Abelian Monopoles on S^3

TL;DR

This paper investigates quantum fluctuations around non-Abelian monopoles on a three sphere, analyzing their stability and vacuum energy contributions, with implications for understanding confinement in gauge theories.

Contribution

It provides a detailed analysis of fluctuation modes and vacuum energy for non-Abelian monopoles on S^3, extending previous stability results and exploring quantum effects related to confinement.

Findings

Unstable modes found for monopoles with g ≥ 1.

Vacuum energy contribution behaves as -R^{-2/3}.

Quantum corrections decay slower than classical Coulomb potential.

Abstract

A possible electric-magnetic duality suggests that the confinement of non-Abelian electric charges manifests itself as a perturbative quantum effect for the dual magnetic charges. Motivated by this possibility, we study vacuum fluctuations around a non-Abelian monopole-antimonopole pair treated as point objects with charges g=\pm n/2 (n=1,2,...), and placed on the antipodes of a three sphere of radius R. We explicitly find all the fluctuation modes by linearizing and solving the Yang-Mills equations about this background field on a three sphere. We recover, generalize and extend earlier results, including those on the stability analysis of non-Abelian magnetic monopoles. We find that for g \ge 1 monopoles there is an unstable mode that tends to squeeze magnetic flux in the angular directions. We sum the vacuum energy contributions of the fluctuation modes for the g=1/2 case and find oscillatory dependence on the cutoff scale. Subject to certain assumptions, we find that the contribution of the fluctuation modes to the quantum zero point energy behaves as -R^{-2/3} and hence decays more slowly than the classical -R^{-1} Coulomb potential for large R. However, this correction to the zero point energy does not agree with the linear growth expected if the monopoles are confined.