A lattice theoretical interpretation of generalized deep holes of the Leech lattice vertex operator algebra

Abstract

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. H\"ohn, 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.