Parafermionizing the monster

A bstract We study the parafermionization of the Monster CFT with respect to its ℤ pA subgroups, with p an odd prime. Under certain assumptions, we show that the parafermionization is equal to a non-invertible gauging of $$ \mathcal{P}(p)\times \mathcal{P}{(p)}^{\vee } $$ P p × P p ∨ , where $$ \mathcal{P}(p) $$ P p is the theory of ℤ p -parafermions and $$ \mathcal{P}{(p)}^{\vee } $$ P p ∨ is an appropriate dual theory, with global symmetry characterized by the centralizer of ℤ pA . By tracking the symmetries of $$ \mathcal{P}(p)\times \mathcal{P}{(p)}^{\vee } $$ P p × P p ∨ through the non-invertible gauging, we argue that the diagonal Monster CFT has $$ \mathrm{Rep}\left(\mathfrak{so}{(3)}_p\right)\boxtimes \mathrm{Rep}{\left(\mathfrak{so}{(3)}_p\right)}^{\mathrm{op}} $$ Rep so 3 p ⊠ Rep so 3 p op symmetry, and hence that the holomorphic Monster theory has symmetry $$ \mathrm{Rep}\left(\mathfrak{so}{(3)}_p\right) $$ Rep so 3 p . We then compute the defect McKay-Thompson series associated to these symmetries, and prove that their invariance subgroups are Γ 1 ( p + 2).