The metamorphosis of semi-classical mechanisms of confinement: From monopoles on ${\mathbb R}^3 \times S^1$ to center-vortices on ${\mathbb R}^2 \times T^2$

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 $S^1 \times {\mathbb R}^3$ with a small circle and a double-trace deformation, and Yang-Mills theory on $T^2 \times {\mathbb R}^2$ where the torus $T^2$ 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. On ${\mathbb R \times T^2 \times S^1}$, we are able to extrapolate both monopole regime and vortex regime to a quantum mechanical domain, where a cross-over takes place. Both sides of the cross-over are described by a deformed $\mathbb Z_N$ TQFT. On ${\mathbb R^2 \times S^1 \times S^1}$, we derive the effective field theory of vortices from the effective theory of monopoles in the presence of a 't Hooft flux. This results from a two-stage adjoint Higgs mechanism, to $U(1)^{N-1}$ in 3d first and a $\mathbb Z_N$ EFT in 2d second. This proves adiabatic continuity of the two confinement mechanisms across dimensions and shows how monopoles and their magnetic flux transmute into center-vortices. 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.