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

TL;DR

This paper develops an anomaly-preserving compactification framework for 4d gauge theories, enabling analytical study of nonperturbative phenomena like confinement and chiral symmetry breaking through semiclassical center vortex descriptions.

Contribution

It introduces a novel compactification method preserving anomalies, connecting weak and strong coupling regimes, and extends the analysis to QCD with fundamental quarks using 't Hooft flux.

Findings

Center vortex description reproduces 4d anomaly matching.

Weak-coupling vacuum structure connects to 4d strong-coupling regime.

Predictions align with chiral Lagrangian results for QCD.

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 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.