Geodesic
A lattice theoretical interpretation of generalized deep holes of the Leech lattice vertex operator algebra
Forum of Mathematics, Sigma, Tome 11 (2023)

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We give a lattice theoretical interpretation of generalized deep holes of the Leech lattice VOA $V_\Lambda $. We show that a generalized deep hole defines a ‘true’ automorphism invariant deep hole of the Leech lattice. We also show that there is a correspondence between the set of isomorphism classes of holomorphic VOA V of central charge $24$ having non-abelian $V_1$ and the set of equivalence classes of pairs $(\tau , \tilde {\beta })$ satisfying certain conditions, where $\tau \in Co.0$ and $\tilde {\beta }$ is a $\tau $-invariant deep hole of squared length $2$. It provides a new combinatorial approach towards the classification of holomorphic VOAs of central charge $24$. In particular, we give an explanation for an observation of G. Höhn, which relates the weight one Lie algebras of holomorphic VOAs of central charge $24$ to certain codewords associated with the glue codes of Niemeier lattices. 
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     author = {Ching Hung Lam and Masahiko Miyamoto},
     title = {A lattice theoretical interpretation of generalized deep holes of the {Leech} lattice vertex operator algebra},
     journal = {Forum of Mathematics, Sigma},
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     year = {2023},
     doi = {10.1017/fms.2023.86},
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Ching Hung Lam; Masahiko Miyamoto. A lattice theoretical interpretation of generalized deep holes of the Leech lattice vertex operator algebra. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.86

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