Fractionally charged Wilson loops as a probe of $\theta$-dependence in $C{P}^{N-1}$ sigma models: Instantons vs. large N

The behavior of Wilson loops with fractional charge is used to study the $\theta$-dependence of the free energy density $ϵ(\theta )$ for the $C{P}^1$, $C{P}^5$, and $C{P}^9$ sigma models in two spacetime dimensions. The function $ϵ(\theta )$ is extracted from the area law for a Wilson loop of charge $q=\theta \mathrm{/}2\pi$. For $C{P}^1$, $ϵ(\theta )$ is smooth in the region $\theta \approx \pi$ and well-described by a dilute instanton gas throughout the range $0<\theta <2\pi$. For $C{P}^5$ and $C{P}^9$ the energy exhibits a clear cusp and evidence for discrete, degenerate vacua at $\theta =\pi$, as expected from large $N$ arguments. For $C{P}^9$ the $\theta$-dependence is in good quantitative agreement with the leading order large $N$ prediction $ϵ(\theta )=1\mathrm{/}2{\chi }_t{\theta }^2$ throughout the range $0<\theta <\pi$.