Center vortex and confinement in Yang-Mills theory and QCD with anomaly-preserving compactifications

We construct an anomaly-preserving compactification of 4d gauge theories, including $SU(N)$ Yang-Mills theory, $\mathcal{N}=1$ supersymmetric Yang-Mills theory, and QCD, down to 2d by turning on 't Hooft flux through $T^2$. It provides a new framework to analytically calculate nonperturbative properties such as confinement, chiral symmetry breaking, and multi-branch structure of vacua. We give the semiclassical description of these phenomena based on the center vortex and show that it enjoys the same anomaly matching condition with the original $4$d gauge theory. We conjecture that the weak-coupling vacuum structure on small $T^2 \times \mathbb{R}^2$ is adiabatically connected to the strong-coupling regime on $\mathbb{R}^4$ without any phase transitions. In QCD with fundamental quarks as well, we can turn on 't Hooft flux either by activating $SU(N_f)_{\mathrm{V}}$ symmetry twist for $N_f=N$ flavors or by introducing a magnetic flux of baryon number $U(1)_{\mathrm{B}}$ for arbitrary $N_f$ flavors. In both cases, the weak-coupling center-vortex theory gives the prediction consistent with chiral Lagrangian of $4$d QCD.