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 $V_{{\Lambda }_g}^{\hat{g}}$.

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

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• Systematic Orbifold Constructions of Schellekens' Vertex Operator Algebras from Niemeier Lattices
• Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list
• Two-dimensional conformal field theory, full vertex algebra and current-current deformation
• Automorphism groups and uniqueness of holomorphic vertex operator algebras of central charge $24$
• Unitary forms for holomorphic vertex operator algebras of central charge $24$
• Natural Construction of Ten Borcherds-Kac-Moody Algebras Associated with Elements in $M_{23}$
• A Geometric Classification of the Holomorphic Vertex Operator Algebras of Central Charge 24
• Unitarity and strong graded locality of holomorphic vertex operator superalgebras with central charge at most 24
• Automorphism groups of cyclic orbifold vertex operator algebras associated with the Leech lattice and some non-prime isometries
• Automorphism groups of certain orbifold vertex operator algebras arising from coinvariant lattices associated with the Leech lattice