Parafermionizing the Monster

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 $\mathrm{Rep}(\mathfrak{so}(3{)}_p)⊠\mathrm{Rep}(\mathfrak{so}(3{)}_p{)}^{\mathrm{op}}$ 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 $\hat{I}{“}_1(p+2)$.

38 pages; v2: publication version