# Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra

### Abstract

We prove a dimension formula for the weight-$1$ subspace of a vertical operator algebra $V^{\mathrm{orb}(g)}$ 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 generalized deep hole in $\mathrm{Aut}(V)$.

Then we show that the orbifold construction defines a bijection between the generalized deep holes of the Leech lattice vertex operator algebra $V_\Lambda$ with non-trivial fixed-point Lie subalgebra and the strongly rational holomorphic vertex algebras of central charge $24$ with non-vanishing weight-$1$ space. This provides a uniform construction of these vertex operator algebras and naturally generalises the correspondence between the deep holes of the Leech lattice $\Lambda$ and the $23$ Niemeier lattices with non-vanishing root system found by Conway, Parker, Sloane and Borcherds.

## Authors

Sven Möller

Department of Mathematics, Rutgers University, Piscataway, NJ and Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

Nils R. Scheithauer