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

TL;DR

This paper investigates the phase structure of $SU(N)/ ext{Z}_N$ gauge theories at $ heta=\pi$, analyzing partition functions, proposing lattice simulations, and exploring the implications of mixed anomalies and fractional instanton effects.

Contribution

It provides a detailed analysis of the large volume behavior of partition functions and proposes lattice methods to study instanton distributions in $SU(N)/ ext{Z}_N$ theories.

Findings

Partition function periodicity is affected by fractional instantons.

A phase transition is suggested at $ heta=\pi$ due to non-$2\pi$ periodicity.

Predicted characteristic instanton distribution shape in the confining phase.

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.