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 the first 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 and Sloane.

50 pages, LaTeX; some changes to the exposition, numbering in Section 6 changed; to appear in Ann. of Math

References in corpus (6)

• Schellekens' List and the Very Strange Formula
• Systematic Orbifold Constructions of Schellekens' Vertex Operator Algebras from Niemeier Lattices
• Integrability of C_2-cofinite vertex operator algebras
• On the Genus of the Moonshine Module
• On orbifold constructions associated with the Leech lattice vertex operator algebra
• On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras

Cited by in corpus (14)

• Classification of chiral fermionic CFTs of central charge $\le 16$
• Bosonic Rational Conformal Field Theories in Small Genera, Chiral Fermionization, and Symmetry/Subalgebra Duality
• Schellekens' List and the Very Strange Formula
• Classification of Unitary RCFTs with Two Primaries and Central Charge Less Than 25
• Systematic Orbifold Constructions of Schellekens' Vertex Operator Algebras from Niemeier Lattices
• Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list
• Two-dimensional conformal field theory, full vertex algebra and current-current deformation
• Holomorphic CFTs and topological modular forms
• On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras
• Orbifold construction and Lorentzian construction of Leech lattice vertex operator algebra
• A Geometric Classification of the Holomorphic Vertex Operator Algebras of Central Charge 24
• Unitarity and strong graded locality of holomorphic vertex operator superalgebras with central charge at most 24
• Automorphism groups of cyclic orbifold vertex operator algebras associated with the Leech lattice and some non-prime isometries
• Shift orbifolds, decompactification limits, and lattices