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

TL;DR

This paper clarifies the behavior of dyonic loops in 4d $SU(N)$ gauge theories on $R^3 imes S^1$, resolving previous inconsistencies by introducing the concept of screening via twist vortices, and confirms the Wilson-'t Hooft classification predictions.

Contribution

It introduces the notion of screening by twist vortices for dyonic loops, enabling consistent computation within the 3d monopole semiclassics framework.

Findings

Dyonic loops are screened by twist vortices, resolving the area law contradiction.

Properly dressed dyonic loops match Wilson-'t Hooft classification predictions.

The framework applies to the thermal phase, revealing the topological nature of $Z_N$ domain walls.

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 $\theta=\pi$ in the thermal deconfined phase despite the fact that the bulk state is smooth there.