The critical O($N$) $\sigma$-model at dimension $2<d<4$: Hardy-Ramanujan distribution of quasi-primary fields and a collective fusion approach

TL;DR

This paper analyzes the distribution of quasiprimary fields in an O(N) sigma model at large N, revealing a Hardy-Ramanujan law and introducing a collective fusion method to classify and resolve degeneracies of these fields.

Contribution

It introduces a novel collective fusion approach for classifying quasiprimary fields and demonstrates their distribution follows a Hardy-Ramanujan law at leading order in 1/N.

Findings

Distribution of quasiprimary fields follows Hardy-Ramanujan law

Developed a collective fusion method for field classification

Resolved degeneracies in quasiprimary field spectrum

Abstract

The distribution of quasiprimary fields of fixed classes characterized by their O$(N)$ representations $Y$ and the number $p$ of vector fields from which they are composed at $N=\infty$ in dependence on their normal dimension $[\delta]$ is shown to obey a Hardy-Ramanujan law at leading order in a $\frac{1}{N}$-expansion. We develop a method of collective fusion of the fundamental fields which yields arbitrary \qps and resolves any degeneracy.