Unifying Monopole and Center Vortex as the Semiclassical Confinement Mechanism

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.