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

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.