Schellekens' list and the very strange formula

In [45] (see also [20]) Schellekens proved that the weight-one space V₁ 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 V₁={0}, this vertex operator algebra is uniquely determined by V₁. Uniform proofs of these statements were given in [42,26]. In this paper we give a fundamentally different, simpler proof of Schellekens' list of 71 Lie algebras. Using the dimension formula in [42] and Kac's “very strange formula” [28] we show that every strongly rational, holomorphic vertex operator algebra V of central charge 24 with V₁≠{0} can be obtained by an orbifold construction from the Leech lattice vertex operator algebra V_Λ. 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 Λ 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.