Schellekens' List and the Very Strange Formula

TL;DR

This paper provides a new, simpler proof of Schellekens' classification of 71 possible Lie algebras as weight-one spaces of certain vertex operator algebras at central charge 24, using orbifold constructions and lattice theory.

Contribution

It introduces a novel proof approach leveraging the dimension formula and Kac's very strange formula, connecting all such vertex operator algebras to the Leech lattice VOA.

Findings

Confirmed all 71 Lie algebras can be realized via orbifolds from the Leech lattice VOA.

Restricted possible Lie algebras to Schellekens' list using the new proof method.

Partially classified 43 of the 70 non-zero Lie algebra cases.

Abstract

In 1993 Schellekens proved that the weight-one space $V_1$ of a strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 must be one of 71 Lie algebras. During the following three decades, in a combined effort by many authors, it was proved that each of these Lie algebras is realised by such a vertex operator algebra and that, except for $V_1=\{0\}$, this vertex operator algebra is uniquely determined by $V_1$. In this paper we give a fundamentally different, simpler proof of Schellekens' list of 71 Lie algebras. Using the dimension formula in arXiv:1910.04947 and Kac's "very strange formula" we show that every strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with $V_1\neq\{0\}$ can be obtained by an orbifold construction from the Leech lattice vertex operator algebra $V_\Lambda$. This suffices to restrict the possible Lie algebras that can occur as weight-one space of $V$ to the 71 of Schellekens. Moreover, the fact that each strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 comes from the Leech lattice $\Lambda$ can be used to classify these vertex operator algebras by studying properties of the Leech lattice. We demonstrate this for 43 of the 70 non-zero Lie algebras on Schellekens' list, omitting those cases that are too computationally expensive.