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

TL;DR

This paper systematically constructs all 70 specific holomorphic vertex operator algebras of central charge 24 with non-zero weight-one space using orbifold methods from Niemeier lattice VOAs and automorphisms, establishing their uniqueness via Lie algebra structure.

Contribution

It provides a comprehensive, uniform construction of these VOAs through orbifold techniques and clarifies their automorphism structures and classification.

Findings

All 70 VOAs constructed from Niemeier lattices and automorphisms.

Automorphisms correspond to generalized deep holes with specific order properties.

Each VOA uniquely determined by the Lie algebra of its weight-one space.

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 $\operatorname{Aut}(V_N)$. 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 $\operatorname{Co}_0$. 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 $V_1$.