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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.

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          Most cited references 21

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          The Magma Algebra System I: The User Language

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            Vertex algebras, Kac-Moody algebras, and the Monster

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              Vertex Algebras Associated with Even Lattices

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                Author and article information

                Journal
                10 October 2019
                Article
                1910.04947

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                Custom metadata
                17B69, 11F11, 11F27
                48 pages, LaTeX
                math.QA math.NT math.RT

                Number theory, Algebra

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