Large $\theta$ angle in two-dimensional large $N$ ${\mathbb{CP}}^{N-1}$ model

Abstract

In confining large $N$ theories with a $\theta$ angle such as four-dimensional $\mathrm{SU}(N)$ pure Yang-Mills theory, there are multiple metastable vacua and it makes sense to consider the parameter region of ``large $\theta$ of order $N$'' despite the fact that $\theta$ is a $2\pi$-periodic parameter. We investigate this parameter region in the two-dimensional ${\mathbb{CP}}^{N-1}$ model by computing the partition function on $T^2$. When $\theta /N$ is of order $\mathcal{O}(0.1)$ or less, we get perfectly sensible results for the vacuum energies and decay rates of metastable vacua. However, when $\theta /N$ is of order $\mathcal{O}(1)$, we encounter a problem about saddle points that would give larger contributions to the partition function than the true vacuum. We discuss why it might not be straightforward to resolve this problem.

35 pages, 4 figures; v2: references and explanations added, published version

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