A vertex operator algebra for the S-fold realisation of a non-unitary Haagerup--Izumi modular fusion category

Abstract

We show that there exists a strongly rational vertex operator algebra $V_{11}$, proposed by Flohr, whose representation theory $\mathrm{Rep}\,V_{11}\simeq\mathrm{Sem}\otimes\mathcal Z(\sigma_2\mathcal F)$ as braided categories. Here $\mathrm{Sem}$ is the semion category (with the spherical structure in which its generator has dimension $-1$), $\mathcal F$ is Ostrik's twisted Haagerup category and $\sigma_2$ is the Galois automorphism $\zeta_{117}\mapsto\zeta_{117}^2$.