# Refined Counting of Necklaces in One-loop $\mathcal{N}=4$ SYM

## TL;DR

This paper calculates the grand partition function of one-loop $ ext{SU}(2)$ sector in $ ext{N}=4$ SYM using finite group theory, revealing that only planar diagrams contribute and relating it to XXX$_{1/2}$ spin chains, with analysis of Hagedorn temperature.

## Contribution

It extends Pólya's theorem to compute the grand partition function with chemical potentials and shows the dominance of planar terms, linking gauge theory to spin chain models.

## Key findings

- Grand partition function computed at one-loop with chemical potentials.
- Only planar diagrams contribute to the grand partition function.
- Hagedorn temperature behavior analyzed on the complex chemical potential plane.

## Abstract

We compute the grand partition function of $\mathcal{N}=4$ SYM at one-loop in the $SU(2)$ sector with general chemical potentials, extending the results of P\'olya's theorem. We make use of finite group theory, applicable to all orders of $1/N_c$ expansion. We show that only the planar terms contribute to the grand partition function, which is therefore equal to the grand partition function of an ensemble of XXX$_\frac12$ spin chains. We discuss how Hagedorn temperature changes on the complex plane of chemical potentials.