The Line operators in the G2HDM model
We investigate the global structure of the Gauged Two-Higgs-Doublet Model (G2HDM), a framework that extends the Standard Model by introducing a dark sector governed by the gauge symmetry $U(1)_X \times SU(2)_H$. The full gauge symmetry of the theory, including the visible sector, is given by the universal covering group $\tilde{G} = U(1)_Y \times SU(2)_L \times SU(3)_C \times U(1)_X \times SU(2)_H$. However, the true gauge group may instead be a quotient $G = \tilde{G} / \Gamma$, where $\Gamma$ is the center of $\tilde{G}$ or a subgroup thereof, leading to different global structures that cannot be distinguished by local experiments.We explore the physical implications of these global structures, analyzing their effects on Wilson, 't Hooft, and dyonic line operators, as well as the periodicity of the CP-violating $\theta$-angles associated with each group factor. Furthermore, we determine the minimal electric and magnetic charges that arise after electroweak symmetry breaking, highlighting their dependence on the choice of $\Gamma$. These findings provide a systematic characterization of the G2HDM's global properties and their potential phenomenological consequences.