A geometric classification of the holomorphic vertex operator algebras of central charge 24 (Q6621742)

One of the most important results in the theory of vertex operator algebras (VOAs), due to its connections with the theory of unimodular lattices and the Monster VOA, is the classification of the strongly rational holomorphic VOAs of central charge 24. It is due to the work of numerous authors over many years and was proved with a case-by-case approach. The present paper provides a new, unified proof based on the classification of generalised deep holes of the Leech lattice.\N\NLet \(V\) be a (strongly rational holomorphic) VOA of central charge \(24\). Its subspace of degree \(1\) is a reductive Lie algebra. In [\textit{A. N. Schellekens}, Commun. Math. Phys. 153, No. 1, 159--185 (1993; Zbl 0782.17014)], Schellekens proved that \(V_1\) can only have one of \(70\) nonzero Lie algebra structures. He conjectured, and it was later proved, that every Lie algebra on Schellekens' list can be realized as the \(V_1\) of some VOA of central charge \(24\), and that these VOAs are determined up to isomorphisms by their \(V_1\). As said, the proof is case-by-case and uses a variety of methods.\N\NIn the present paper, the authors employ a new approach based on the classification of the generalised deep holes of the Leech lattice. These are certain automorphisms of the Leech lattice VOA \(V_\Lambda\). The authors associate to every generalised deep hole a generalised hole diagrams, which is a certain affine Dynkin diagram. The authors prove that there are exactly 70 generalised hole diagrams and that they parametrise all the generalised deep holes up to conjugacy. In a previous paper [\textit{S. Möller} and \textit{N. Scheithauer}, Ann. Math. (2) 197, No. 1, 221--288 (2023; Zbl 1529.17040)], the authors showed that to any generalised deep hole can be associated a VOA with central charge \(24\) via cyclic orbifold construction. This defines a bijection between conjugacy classes of generalised deep holes and isomorphism classes of VOAs of central charge \(24\). Together with the results of the present paper, this completes the proof. The new geometric approach can be seen as a generalisation of the Niemeier classification of unimodular lattices of rank \(24\), which can be deduced from Conway, Parker and Sloane's classification of deep holes of the Leech lattice [\textit{J. H. Conway} et al., Proc. R. Soc. Lond., Ser. A 380, 261--290 (1982; Zbl 0496.10020)].