Non-Abelian $A_4$ vortices in $SO(3)$ gauge theory and non-invertible symmetries

Abstract

We construct finite-tension non-Abelian vortex solutions in a renormalizable $(3+1)$-dimensional $SO(3)$ gauge theory Higgsed to the tetrahedral group $A_4$ by a Higgs field in the spin-3 representation. Since the vacuum manifold is $SO(3)/A_4$, the vortices are characterized by the non-Abelian fundamental group $π_1(SO(3)/A_4)\simeq \widetilde{A}_4$, the binary tetrahedral group. We obtain explicit axisymmetric vortex solutions carrying holonomies corresponding to the order-two and order-three conjugacy classes of $A_4$, determine their tensions numerically, and show that they exhibit type-I, type-II, and Bogomol'nyi--Prasad--Sommerfield-like behavior depending on the Higgs and gauge boson mass ratios. The vortices are classified by conjugacy classes of $\widetilde{A}_4$, while their infrared descriptions are labeled by conjugacy classes of $A_4$. We further demonstrate that the smooth finite-tension vortices reduce in the infrared to Gukov--Witten surface operators of the $A_4$ discrete gauge theory, thereby establishing a finite-energy ultraviolet completion of non-invertible defects in a renormalizable gauge-Higgs theory.