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A lattice theoretical interpretation of generalized deep holes

By Ching Hung Lam

Appears in collection : Vertex Algebras and Representation Theory / Algèbres vertex et théorie des représentations

We will 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 will also discuss 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 $24$. Finally, we will discuss an observation of G.Höhn, which relates the weight one Lie algebra of holomorphic VOAs of central charge $24$ to certain codewords associated with the glue codes of Niemeier lattices.

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Citation data

  • DOI 10.24350/CIRM.V.19932703
  • Cite this video Lam Ching Hung (6/10/22). A lattice theoretical interpretation of generalized deep holes. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19932703
  • URL https://dx.doi.org/10.24350/CIRM.V.19932703

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Bibliography

  • LAM, Ching Hung et MIYAMOTO, Masahiko. A lattice theoretical interpretation of generalized deep holes of the Leech lattice vertex operator algebra. arXiv preprint arXiv:2205.04681, 2022. - https://arxiv.org/pdf/2205.04681.pdf

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