Systematic Orbifold Constructions of Schellekens' Vertex Operator Algebras from Niemeier Lattices
Gerald Höhn, Sven Möller
TL;DR
This work delivers a unified, systematic realization of all strongly rational, holomorphic VOAs of central charge $24$ with nonzero weight-one space, by realizing each such VOA as a cyclic orbifold $V_N^{\mathrm{orb}(g)}$ of a Niemeier lattice VOA $V_N$ along one of $226$ short automorphisms $g$, which are shown to be generalised deep holes. The construction pairs with a Leech-lattice viewpoint, via inverse orbifolds, to prove that the weight-one Lie algebra $V_1$ uniquely determines the VOA, thereby completing a uniform proof of Schellekens’ list. Central to the approach are the dimension and rank formulas for orbifolds at $c=24$, a rank criterion ensuring orbifold-rank equality, and a precise classification of short automorphisms by Frame shapes corresponding to the $11$ conjugacy classes in $\operatorname{O}(\Lambda)$. Together these results provide a concrete, fully uniform path from Niemeier lattices to all $V$ with $V_1\neq\{0\}$ and establish the strong form of uniqueness of holomorphic $c=24$ VOAs from the Lie algebra structure of $V_1$.
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 $V_1$.