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In $SU(N)$ gauge theory, it is argued recently that there exists a "mixedanomaly" between the CP symmetry and the 1-form $\mathbb{Z}_N$ symmetry at$\theta=\pi$, and the anomaly matching requires CP to be spontaneously brokenat $\theta=\pi$ if the system is in the confining phase. In this paper, weelaborate on this discussion by examining the large volume behavior of thepartition 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 $2\pi$ dueto fractional instanton numbers, suggests the presence of a phase transition at$\theta=\pi$. We propose lattice simulations to study the distribution of theinstanton number in $SU(N)/\mathbb{Z}_N$ theories. A characteristic shape ofthe distribution is predicted when the system is in the confining phase. Themeasurements of the distribution may be useful in understanding the phasestructure of the theory.

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