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

TL;DR

This paper investigates the $ heta$-dependence of free energy in $CP^{N-1}$ sigma models using fractional Wilson loops, revealing instanton effects at low N and large N vacua degeneracy, with results matching large N predictions.

Contribution

It introduces a novel method using fractional Wilson loops to probe $ heta$-dependence in $CP^{N-1}$ models, providing detailed comparisons between instanton gas and large N behaviors.

Findings

$ heta$-dependence is smooth for $CP^1$, consistent with instanton gas.

$CP^5$ and $CP^9$ show cusps at $ heta=\pi$, indicating degenerate vacua.

Quantitative agreement with large N predictions for $CP^9$.

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. The function $\epsilon(\theta)$ is extracted from the area law for a Wilson loop of charge $q=\theta/2\pi$. For $CP^1$, $\epsilon(\theta)$ is smooth in the region $\theta\approx\pi$ and well-described by a dilute instanton gas throughout the range $0<\theta<2\pi$. For $CP^5$ and $CP^9$ the energy exhibits a clear cusp and evidence for discrete, degenerate vacua at $\theta = \pi$, as expected from large $N$ arguments. For $CP^9$ the $\theta$-dependence is in good quantitative agreement with the leading order large $N$ prediction $\epsilon(\theta)={1/2}\chi_t\theta^2$ throughout the range $0<\theta<\pi$.