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We construct an anomaly-preserving compactification of 4d gauge theories,including $SU(N)$ Yang-Mills theory, $\mathcal{N}=1$ supersymmetric Yang-Millstheory, and QCD, down to 2d by turning on 't Hooft flux through $T^2$. Itprovides a new framework to analytically calculate nonperturbative propertiessuch as confinement, chiral symmetry breaking, and multi-branch structure ofvacua. We give the semiclassical description of these phenomena based on thecenter vortex and show that it enjoys the same anomaly matching condition withthe original $4$d gauge theory. We conjecture that the weak-coupling vacuumstructure on small $T^2 \times \mathbb{R}^2$ is adiabatically connected to thestrong-coupling regime on $\mathbb{R}^4$ without any phase transitions. In QCDwith fundamental quarks as well, we can turn on 't Hooft flux either byactivating $SU(N_f)_{\mathrm{V}}$ symmetry twist for $N_f=N$ flavors or byintroducing 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 theprediction consistent with chiral Lagrangian of $4$d QCD.

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