Systematic Orbifold Constructions of Schellekens' Vertex Operator Algebras from Niemeier Lattices

Abstract

We present a systematic, rigorous construction of all 70 strongly rational, holomorphic vertex operator algebras $V$ of central charge 24 with non-zero weight-one space $V_1$ as cyclic orbifold constructions associated with the 24 Niemeier lattice vertex operator algebras $V_N$ and certain 226 short automorphisms in $\mathrm{Aut}(V_N)$. We show that up to algebraic conjugacy these automorphisms are exactly the generalised deep holes, as introduced in arXiv:1910.04947, of the Niemeier lattice vertex operator algebras with the additional property that their orders are equal to those of the corresponding outer automorphisms. Together with the constructions in arXiv:1708.05990 and arXiv:1910.04947 this gives three different uniform constructions of these vertex operator algebras, which are related through 11 algebraic conjugacy classes in ${\mathrm{Co}}_0$. Finally, by considering the inverse orbifold constructions associated with the 226 short automorphisms, we give the first systematic proof of the result that each strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with non-zero weight-one space $V_1$ is uniquely determined by the Lie algebra structure of $V_1$.

42 pages, LaTeX; added a missing assumption in Proposition 2.2 and fixed a typo in Table 1

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• Modular invariance of trace functions in orbifold theory
• Meromorphic c=24 Conformal Field Theories
• C_2-cofiniteness of cyclic-orbifold models
• Construction and Classification of Holomorphic Vertex Operator Algebras
• Conformal Field Theories, Representations and Lattice Constructions
• Schur-Weyl Duality for Heisenberg Cosets
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Cited by in corpus (12)

• 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
• Dimension Formulae and Generalised Deep Holes of the Leech Lattice Vertex Operator Algebra
• Classification of Unitary RCFTs with Two Primaries and Central Charge Less Than 25
• Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list
• Holomorphic CFTs and topological modular forms
• The operator algebra of cyclic orbifolds
• 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