The 't Hooft-Polyakov Monopole in the Presence of a 't Hooft Operator

TL;DR

The paper constructs exact BPS configurations of a single nonabelian monopole in the presence of a minimal 't Hooft operator for $G=SU(2)$ and $SO(3)$ using the Nahm transform. In the large-separation limit the fields reduce to the embedded 't Hooft-Polyakov monopole, while at collision ($d\to 0$) the two gauge theories exhibit distinct screening behavior: complete screening in $SU(2)$ with $F=0$ and constant $\Phi$, versus persistent nonabelian structure in $SO(3)$ where the monopole spreads and an $\mathcal{O}$-shaped region of vanishing Higgs forms at $z=\frac{1}{2\lambda}$. A topological interpretation via the 't Hooft charge shows $SO(3)$ supports a charge-one, gauge-protected configuration ($\mathbb{Z}/2\mathbb{Z}$), while the $SU(2)$ case corresponds to a charge-two, screenable configuration—interpreted as two minimal operators coalescing. The results illuminate how duality concepts connect line operators and monopoles in nonabelian gauge theories and provide exact, checkable solutions across gauge groups.

Abstract

We present explicit BPS field configurations representing one nonabelian monopole with one minimal weight 't Hooft operator insertion. We explore the SO(3) and SU(2) gauge groups. In the case of SU(2) gauge group the minimal 't Hooft operator can be completely screened by the monopole. If the gauge group is SO(3), however, such screening is impossible. In the latter case we observe a different effect of the gauge symmetry enhancement in the vicinity of the 't Hooft operator.

Figures (5)

• Figure 1: Higgs field profiles for $\lambda=1.$ Dashed lines correspond to $|\Phi|^2=\frac{1}{2},$ the shaded area $|\Phi|^2\leq\frac{1}{2},$ and the dark region indicates the position of the monopole core.
• Figure 2: Energy density contour plots for $\lambda=1.$
• Figure 3: Positions of the monopole $\vec{T}_{'tHP}$, the 't Hooft operator $\vec{T}_{D}$, and the observation point $\vec{x}.$
• Figure 4: Brane diagram signifying the Nahm data on an interval $(-\lambda, \lambda)$ and a semi-infinite interval $(\lambda, \infty).$ This diagram depicts the Nahm data defining an $SU(2)$ monopole in the presence of one minimal charge 't Hooft operator.
• Figure 5: Brane diagram for the Nahm data corresponding to an $SO(3)$ monopole at $\vec{T}_{'tHP}$ and one minimal charge 't Hooft operator at $\vec{T}_D.$ The data is defined on ${\mathbb R}$ and is continuous outside the points $s=-\lambda,\ \lambda.$