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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 $2pi$ 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.
We review results concerning the theta dependence of 4D SU(N) gauge theories and QCD, where theta is the coefficient of the CP-violating topological term in the Lagrangian. In particular, we discuss theta dependence in the large-N limit. Most resul
We review recent results on the theta dependence of the ground-state energy and spectrum of four-dimensional SU(N) gauge theories, where theta is the coefficient of the CP-violating topological term F-Fdual in the Lagrangian. In particular, we discus
We discuss the existence of a conformal phase in SU(N) gauge theories in four dimensions. In this lattice study we explore the model in the bare parameter space, varying the lattice coupling and bare mass. Simulations are carried out with three color
Generalizations of the AGT correspondence between 4D $mathcal{N}=2$ $SU(2)$ supersymmetric gauge theory on ${mathbb {C}}^2$ with $Omega$-deformation and 2D Liouville conformal field theory include a correspondence between 4D $mathcal{N}=2$ $SU(N)$ su