Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra
Sven Möller, Nils R. Scheithauer
Abstract
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 $V_\Lambda$ 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 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.