Wilson-'t Hooft classification and the perimeter law for dyonic loops in 3d monopole semiclassics

Abstract

We investigate the long-distance behavior of dyonic loop operators in 4d $SU(N)$ gauge theories on $\mathbb{R}^3 \times S^1$ using the 3d monopole semiclassics. If we employ the naive definition of the 't Hooft loop in the Abelianized regime, the dyonic loop operators do not admit the well-defined computations within the effective field theory. Moreover, if one forcibly proceeds with the computations of their expectation values, all the dyonic loops turn out to show the area law, which contradicts the prediction of the Wilson-'t Hooft classification. In this paper, we resolve this puzzle by employing the notion of screening for line operators, and we argue that the dyonic loops are screened by a defect known as the twist vortex, which is non-dynamical in the infrared effective theory but is dynamical in the original ultraviolet theory. The dyonic loops properly dressed by twist vortices admit the well-defined computations within the effective field theory, and we reproduce the kinematic prediction of the Wilson-'t~Hooft classification using the $3$d monopole semiclassics. Furthermore, we apply our framework to the thermal deconfined phase to evaluate the dual string tension, elucidating the topological nature of $\mathbb{Z}_N$ domain walls. We confirm that the domain-wall state has the phase transition at $θ=π$ in the thermal deconfined phase despite the fact that the bulk state is smooth there.

Figures (3)

• Figure 1: Contour plots of the potential $V(\varphi, \sigma)/V_0$ with $\gamma = 10$ for representative values of $\theta$: $\theta = 0, ~ \pi/2$, and $3 \pi/2$. The minima are indicated by cross symbols. For visual clarity, the potential minimum is set to zero in these plots. At this mass deformation parameter $\gamma = 10$, the $\mathbb{Z}_2$ center symmetry is spontaneously broken. Kink configurations interpolating between two vacua correspond to domain walls. The insertion of a spatial 't Hooft loop $H(C;\Sigma)$ induces a kink on the surface $\Sigma$ connecting $(\sigma, \varphi) = (\sigma_*, \varphi_*)$ and $(-\sigma_*, -\varphi_*)$. For $(-\pi<)\theta < \pi$, this configuration represents the kink with the minimum tension. However, as illustrated in the plot for $\theta = 3 \pi/2$, in the range $\pi < \theta (< 3\pi)$, a different kink connecting $(\sigma, \varphi) = (\sigma_*, \varphi_*)$ and $(2\pi-\sigma_*, -\varphi_*)$ has the minimum tension. Since the $2\pi$ shift of $\sigma$ corresponds to attaching a Wilson loop to the boundary, the lightest kink in this regime is generated by the composite operator $H(C;\Sigma) W(C)$.
• Figure 2: An illustration of the spatial 't Hooft loop $H(C;\Sigma)$ (in the 2d cross section) . The $\mathbb{Z}_N$ twist is inserted in the hopping term which crosses the dual-lattice surface $\Sigma$.
• Figure 3: A configuration which gives the perimeter law of the spatial 't Hooft loop $H(C;\Sigma)$ (in the 2d cross section). If we take $U_4 = C$, the $S_N$ gauge configuration $U_\ell = S^{-1}$ for $\ell \in (\Sigma)^*$ can minimize both the $U_4$-hopping term and the deformation potential. The flatness condition of the spatial plaquette is broken on the boundary $C$, which only gives the perimeter law.