The metamorphosis of semi-classical mechanisms of confinement: From monopoles on R³ S¹ to center-vortices on R² T²

There are two distinct regimes of Yang-Mills theory where we can demonstrate confinement, the existence of a mass gap, and fractional theta angle dependence using a reliable semi-classical calculation. The two regimes are Yang-Mills theory on S¹ R³ with a small circle and a double-trace deformation, and Yang-Mills theory on T² R² where the torus T² is small and threaded by a 't Hooft flux. In the first case the confinement mechanism is related to self-dual monopoles, whereas in the second case self-dual center-vortices play a crucial role. These two topological objects are distinct. In particular, they have different mutual statistics with Wilson loops. On the other hand, they carry the same topological charge and action. We show that one can derive the effective field theory of vortices, a deformed ZN TQFT, from the effective theory of the magnetic Coulomb gas in the presence of a 't Hooft flux. We describe this result as a two-stage adjoint Higgs mechanism, where the first stage yields U (1) ^N-1 EFT in 3d and second stage yields a ZN EFT in 2d and 1d. This basic mechanism is flux fractionalization: The magnetic flux of the monopoles fractionalizes and collimates in such a way that 2d Wilson loops detect it as a center vortex.