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

Keith-Hynes, Patrick; Thacker, H. B.

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

Authors

Description

The behavior of Wilson loops with fractional charge is used to study the $θ$-dependence of the free energy density $ε(θ)$ for the $CP^1$, $CP^5$, and $CP^9$ sigma models in two spacetime dimensions. The function $ε(θ)$ is extracted from the area law for a Wilson loop of charge $q=θ/2π$. For $CP^1$, $ε(θ)$ is smooth in the region $θ\approxπ$ and well-described by a dilute instanton gas throughout the range $0<θ<2π$. For $CP^5$ and $CP^9$ the energy exhibits a clear cusp and evidence for discrete, degenerate vacua at $θ= π$, as expected from large $N$ arguments. For $CP^9$ the $θ$-dependence is in good quantitative agreement with the leading order large $N$ prediction $ε(θ)={1/2}χ_tθ^2$ throughout the range $0<θ<π$.
16 pages, 10 figures