Gap Equations of O(N) Non-linear Sigma Model in Three Dimensions

TL;DR

This paper investigates the $O(N)$ non-linear sigma model in three dimensions using large $N$ expansion, revealing instability of solutions on certain curved manifolds and analyzing saddle point equations under specific boundary conditions.

Contribution

It provides a detailed analysis of the gap equations for the $O(N)$ non-linear sigma model on curved three-dimensional manifolds at the critical point, highlighting stability issues under anti-periodic boundary conditions.

Findings

Solution on $S^1 imes S^2$ is unstable.

Saddle point equations are affected by boundary conditions.

Brief discussion on $S^1 imes S^1 imes S^1$ case.

Abstract

We study the $O(N)$ non-linear $\sigma$ model on three-dimensional manifolds of constant curvature by means of the large $N$ expansion at the critical point. We examine saddle point equations imposing anti-periodic boundary condition in time direction. In the case $S^1 \times S^2$ we find that a solution is inevitably unstable. We briefly refer to the case $S^1 \times S^1 \times S^1$.