$\theta =\pi$ in $SU(N)/{\mathbb{Z}}_N$ gauge theories

Abstract

In $SU(N)$ gauge theory, it is argued recently that there exists a "mixed anomaly" between the CP symmetry and the 1-form ${\mathbb{Z}}_N$ symmetry at $\theta =\pi$, and the anomaly matching requires CP to be spontaneously broken at $\theta =\pi$ if the system is in the confining phase. In this paper, we elaborate on this discussion by examining the large volume behavior of the partition functions of the $SU(N)/{\mathbb{Z}}_N$ theory on $T^4$ a la 't Hooft. The periodicity of the partition function in $\theta$, which is not $2\pi$ due to fractional instanton numbers, suggests the presence of a phase transition at $\theta =\pi$. We propose lattice simulations to study the distribution of the instanton number in $SU(N)/{\mathbb{Z}}_N$ theories. A characteristic shape of the distribution is predicted when the system is in the confining phase. The measurements of the distribution may be useful in understanding the phase structure of the theory.

arXiv comment18 pages, 1 figure, reference added, typo fixed

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• Fractional instanton of the SU($3$) gauge theory in weak coupling regime
• Effective Field Theory of Force-Free Electrodynamics
• Domain Walls and the $CP$ Anomaly in Softly Broken Supersymmetric QCD
• Is $N=2$ Large?
• Violation of Eigenstate Thermalization Hypothesis in Quantum Field Theories with Higher-Form Symmetry
• Fractional topological charge in $SU(N)$ gauge theories without dynamical quarks
• Strongly coupled QFT dynamics via TQFT coupling
• $3d$ One-form Mixed Anomaly and Entanglement Entropy
• $\theta$ dependence of $T_c$ in 4d SU(3) Yang-Mills theory with histogram method and the Lee-Yang zeros in the large $N$ limit
• Phase structure of linear quiver gauge theories from anomaly matching
• Higher-form entanglement asymmetry and topological order