Nonabelian Monopoles and the Vortices that Confine Them

TL;DR

The paper develops a two-scale framework in softly broken ${\cal N}=2$ theories where nonabelian GNO monopoles of $G/H$ are confined by nonabelian vortices arising from breaking $H$. Using the explicit example $G=SU(N+1)$, $H={SU(N)\times U(1)}/{\mathbb Z}_N$, it demonstrates exact flux matching between minimal monopoles and minimal vortices, validating confinement and revealing that monopoles transform under the dual group ${\tilde H} = SU(N)\times U(1)$. The analysis shows that monopole flux, abelian and nonabelian components, agrees across the high-energy and low-energy effective theories, with vortex moduli described by $CP^{N-1}$ providing the nonabelian orientation degrees of freedom. The work highlights the role of flavor symmetry in stabilizing the nonabelian monopole/vortex system and suggests implications for confinement mechanisms and dualities in supersymmetric and QCD-like theories.

Abstract

Nonabelian magnetic monopoles of Goddard-Nuyts-Olive-Weinberg type have recently been shown to appear as the dominant infrared degrees of freedom in a class of softly broken ${\cal N}=2$ supersymmetric gauge theories in which the gauge group $G$ is broken to various nonabelian subgroups $H $ by an adjoint Higgs VEV. When the low-energy gauge group $H$ is further broken completely by e.g. squark VEVs, the monopoles representing $π_2(G/H)$ are confined by the nonabelian vortices arising from the breaking of $H$, discussed recently (hep-th/0307278). By considering the system with $G=SU(N+1)$, $ H = {SU(N) \times U(1) ø{\mathbb Z}_N}$, as an example, we show that the total magnetic flux of the minimal monopole agrees precisely with the total magnetic flux flowing along the single minimal vortex. The possibility for such an analysis reflects the presence of free parameters in the theory - the bare quark mass $m$ and the adjoint mass $μ$ - such that for $m \gg μ$ the topologically nontrivial solutions of vortices and of unconfined monopoles exist at distinct energy scales.

Figures (2)

• Figure 1: A single vortex ends on each monopole.
• Figure 2: The total flux around the monopole, integrated over a sphere of an arbitrary radius (hence with a radius much smaller than $1/v_2$ - where it lookes like an isotropic monopole), must match the total vortex flux integrated over a plane far enough from the monopole. On this plane the vortex flux is distributed over a region much larger than $1/v_2$.