Zeta-Regularization of the O(N) Non-Linear Sigma Model in D dimensions

TL;DR

This paper applies zeta regularization to analyze the O(N) non-linear sigma model in various D-dimensional geometries, deriving partition functions, gap equations, and exploring large D asymptotics with numerical solutions.

Contribution

It introduces a novel application of zeta regularization to the O(N) sigma model in diverse geometries, providing new analytical and numerical insights.

Findings

Partition functions and gap equations derived for various geometries.

Numerical solutions of gap equations at critical couplings.

Asymptotic analysis of partition functions for large D.

Abstract

The O(N) non-linear sigma model in a $D$-dimensional space of the form ${\bf R}^{D-M} \times {\bf T}^M$, ${\bf R}^{D-M} \times {\bf S}^M$, or ${\bf T}^M \times {\bf S}^P$ is studied, where ${\bf R}^M$, ${\bf T}^M$ and ${\bf S}^M$ correspond to flat space, a torus and a sphere, respectively. Using zeta regularization and the $1/N$ expansion, the corresponding partition functions and the gap equations are obtained. Numerical solutions of the gap equations at the critical coupling constants are given, for several values of $D$. The properties of the partition function and its asymptotic behaviour for large $D$ are discussed. In a similar way, a higher-derivative non-linear sigma model is investigated too. The physical relevance of our results is discussed.