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Advances in Mathematics

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.

Schellekens' list and the very strange formula