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

Fractionally charged Wilson loops as a probe of $θ$-dependence in $CP^{N-1}$ sigma models: Instantons vs. large N - High Energy Physics - Lattice - Download this document for free, or read online. Document in PDF available to download.

Abstract: The behavior of Wilson loops with fractional charge is used to study the$\theta$-dependence of the free energy density $\epsilon\theta$ for the$CP^1$, $CP^5$, and $CP^9$ sigma models in two spacetime dimensions. Thefunction $\epsilon\theta$ is extracted from the area law for a Wilson loop ofcharge $q=\theta-2\pi$. For $CP^1$, $\epsilon\theta$ is smooth in the region$\theta\approx\pi$ and well-described by a dilute instanton gas throughout therange $0<\theta<2\pi$. For $CP^5$ and $CP^9$ the energy exhibits a clear cuspand evidence for discrete, degenerate vacua at $\theta = \pi$, as expected fromlarge $N$ arguments. For $CP^9$ the $\theta$-dependence is in good quantitativeagreement with the leading order large $N$ prediction$\epsilon\theta={1-2}\chi t\theta^2$ throughout the range $0<\theta<\pi$.

Author: ** Patrick Keith-Hynes, H. B. Thacker**

Source: https://arxiv.org/