Finite Size Effects and Conformal Symmetry of $O(N)$ Nonlinear $\sigma$ Model in Three Dimensions

TL;DR

This paper investigates the finite size effects and conformal symmetry properties of the three-dimensional $O(N)$ nonlinear sigma model on a compact space, analyzing correlation length, Casimir energy, and modular invariance at criticality.

Contribution

It provides a large $N$ analysis of finite size effects and conformal properties, including the behavior of the partition function and thermodynamic quantities.

Findings

Correlation length and Casimir energy depend on space radii $L$ and $R$.

Partition function exhibits modular transformation properties.

Explores the extension of the $C$-theorem to three dimensions.

Abstract

We study the $O(N)$ nonlinear $\sigma$ model on a three-dimensional compact space $S^1 \times S^2$ (of radii $L$ and $R$ respectively) by means of large $N$ expansion, focusing on the finite size effects and conformal symmetries of this model at the critical point. We evaluate the correlation length and the Casimir energy of this model and study their dependence on $L$ and $R$. We examine the modular transformation properties of the partition function, and study the dependence of the specific heat on the mass gap in view of possible extension of the $C-$theorem to three dimensions.