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

Abstract

Abstract We construct an anomaly-preserving compactification of 4D gauge theories, including SU(N) Yang–Mills theory, $\mathcal {N}=1$ supersymmetric Yang–Mills theory, and quantum chromodynamics (QCD), down to 2D by turning on the ’t Hooft flux through T2. This provides a new framework to analytically calculate nonperturbative properties such as confinement, chiral symmetry breaking, and the multi-branch structure of vacua. We give a semiclassical description of these phenomena based on the center vortex and show that it enjoys the same anomaly-matching condition as the original 4D 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 the ’t Hooft flux either by activating the SU(Nf)V symmetry twist for Nf = N flavors or by introducing a magnetic flux of baryon number U(1)B for arbitrary Nf flavors. In both cases, the weak-coupling center-vortex theory gives a prediction consistent with the chiral Lagrangian of 4D QCD.