Parafermionizing the Monster

TL;DR

This paper explores the parafermionization of the Monster conformal field theory, revealing its symmetry structure and computing associated McKay-Thompson series, with implications for understanding non-invertible gaugings.

Contribution

It demonstrates that the parafermionization corresponds to a non-invertible gauging of specific theories and identifies the resulting symmetry groups of the Monster CFT.

Findings

Parafermionization equals a non-invertible gauging of 
0(p) 
f
0(p)^\u2194.

The diagonal Monster CFT has (
0(3)_p) d7 (
0(3)_p)^ ext{op} symmetry.

Defect McKay-Thompson series are invariant under 
6(p+2) subgroup.

Abstract

We study the parafermionization of the Monster CFT with respect to its $\mathbb{Z}_{pA}$ subgroups, with $p$ an odd prime. Under certain assumptions, we show that the parafermionization is equal to a non-invertible gauging of $\mathcal{P}(p) \times \mathcal{P}(p)^\vee$, where $\mathcal{P}(p)$ is the theory of $\mathbb{Z}_p$-parafermions and $\mathcal{P}(p)^\vee$ is an appropriate dual theory, with global symmetry characterized by the centralizer of $\mathbb{Z}_{pA}$. By tracking the symmetries of $\mathcal{P}(p) \times \mathcal{P}(p)^\vee$ through the non-invertible gauging, we argue that the diagonal Monster CFT has $\mathrm{Rep}(\mathfrak{so}(3)_p) \boxtimes \mathrm{Rep}(\mathfrak{so}(3)_p)^\mathrm{op}$ symmetry, and hence that the holomorphic Monster theory has symmetry $\mathrm{Rep}(\mathfrak{so}(3)_p)$. We then compute the defect McKay-Thompson series associated to these symmetries, and prove that their invariance subgroups are $\Gamma_1(p+2)$.