  Magnetic excitations play a crucial role in understanding the color
confinement of $4$d Yang-Mills theory, and we have the monopole and the center
vortex as plausible candidates to explain its mechanism. Under suitable
compactified setups of $4$d Yang-Mills theory, we can achieve different
weakly-coupled descriptions of confinement phenomena: The monopole mechanism
takes place on $\mathbb{R}^3\times S^1$ with the double-trace deformation, and
the center-vortex mechanism is effective on $\mathbb{R}^2\times T^2$ with the
't Hooft flux. We unify these two semiclassical descriptions by showing the
explicit relation between the monopole and center vortex.
