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 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.