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

Technische Universität Darmstadt, Darmstadt, Germany