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Construction of Holomorphic Vertex Operator Algebras of Central Charge 24 Using the Leech Lattice and Level p Lattices
by Ching Hung Lam   Hiroki Shimakura

Vol. 12 No. 1 (2017) P.39~P.70
 DOI: https://doi.org/10.21915/BIMAS.2017102 10.21915/BIMAS.2017102

ABSTRACT

In this article, we discuss a more uniform construction of all $71$ holomorphic vertex operator algebras in Schellekens' list using an idea proposed by G. H\"ohn. The main idea is to try to construct holomorphic vertex operator algebras of central charge $24$ using some sublattices of the Leech lattice $\Lambda$ and level $p$ lattices. We study his approach and try to elucidate his ideas. As our main result, we prove that for an even unimodular lattice $L$ and a prime order isometry $g$, the orbifold vertex operator algebra $V_{L_g}^{\hat{g}}$ has group-like fusion. We also realize the construction proposed by H\"ohn for some special isometry of the Leech lattice of prime order.

KEYWORDS
holomorphic vertex operator algebras; orbifold vertex operator algebras; Leech lattice; level p lattices.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 17B69.

MILESTONES