Cyclic orbifolds of lattice vertex operator algebras having group like fusions

Abstract

Let $L$ be an even (positive definite) lattice and $g\in O(L)$. In this article, we prove that the orbifold vertex operator algebra $V_L^{\hat{g}}$ has group-like fusion if and only if $g$ acts trivially on the discriminant group $\mathcal{D}(L)=L^{\ast }/L$ (or equivalently $(1-g)L^{\ast }<L$). We also determine their fusion rings and the corresponding quadratic space structures when $g$ is fixed point free on $L$. By applying our method to some coinvariant sublattices of the Leech lattice $\Lambda$, we prove a conjecture proposed by G. Höhn. In addition, we also discuss a construction of certain holomorphic vertex operator algebras of central charge $24$ using the the orbifold vertex operator algebra $V_{{\Lambda }_g}^{\hat{g}}$.

The main theorem was proved in a slightly more general setting and the title of the article has been changed