θ = π SU(N)/ZN gauge theories

In SU(N) gauge theory, it is argued recently that there exists a "mixed anomaly" between the CP symmetry and the 1-form 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 con fining phase. In this paper, we elaborate on this discussion by examining the large volume behavior of the partition functions of the SU(N) = 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) = Z(N) theories. A characteristic shape of the distribution is predicted when the system is in the con fi ning phase. The measurements of the distribution may be useful in understanding the phase structure of the theory.