Dimension Formulae and Generalised Deep Holes of the Leech Lattice Vertex Operator Algebra

We prove a dimension formula for the weight-1 subspace of a vertex 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 generalised deep hole in $\mathrm{Aut}(V)$. Then we show that the orbifold construction defines a bijection between the generalised deep holes of the Leech lattice vertex operator algebra $V_{\mathrm{\Lambda }}$ with non-trivial fixed-point Lie subalgebra and the strongly rational, holomorphic vertex operator algebras of central charge 24 with non-vanishing weight-1 space. This provides the first uniform construction of these vertex operator algebras and naturally generalises the correspondence between the deep holes of the Leech lattice $\mathrm{\Lambda }$ and the 23 Niemeier lattices with non-vanishing root system found by Conway, Parker and Sloane.