Quantum dimensions and fusion rules of the VOA $V_{L_{C\times D}}^{\tau }$

Abstract

In this article, we determine quantum dimensions and fusion rules for the orbifold code VOA $V_{L_{C\times D}}^{\tau }$. As an application, we also construct certain $3$-local subgroups inside the automorphism group of the VOA $V^{♯}$, where $V^{♯}$ is a holomorphic VOA obtained by the ${\mathbb{Z}}_3$-orbifold construction on the Leech lattice VOA.