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# Dimension Formulae and Generalised Deep Holes of the Leech Lattice Vertex Operator Algebra

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### Abstract

We prove a dimension formula for the weight-1 subspace of a vertex operator algebra $$V^{\operatorname{orb}(g)}$$ obtained by orbifolding a strongly rational, holomorphic vertex operator algebra $$V$$ of central charge 24 with a finite order automorphism $$g$$. Based on an upper bound derived from this formula we introduce the notion of a generalised deep hole in $$\operatorname{Aut}(V)$$. We then give a construction of all 70 strongly rational, holomorphic vertex operator algebras of central charge 24 with non-vanishing weight-1 space as orbifolds of the Leech lattice vertex operator algebra $$V_\Lambda$$ associated with generalised deep holes. This provides the first uniform construction of these vertex operator algebras and naturally generalises the construction of the 23 Niemeier lattices with non-vanishing root system from the deep holes of the Leech lattice $$\Lambda$$ by Conway, Parker and Sloane.

### Most cited references21

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### Vertex algebras, Kac-Moody algebras, and the Monster

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### Author and article information

###### Journal
10 October 2019
###### Article
1910.04947