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

Möller, Sven; Scheithauer, Nils R.

doi:10.2140/ant.2024.18.1891

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

Sven Möller and Nils R. Scheithauer

Vol. 18 (2024), No. 10, 1891–1922

DOI: 10.2140/ant.2024.18.1891

Abstract

We associate with a generalised deep hole of the Leech lattice vertex operator algebra a generalised hole diagram. We show that this Dynkin diagram determines the generalised deep hole up to conjugacy and that there are exactly $70$ such diagrams. In an earlier work we proved a bijection between the generalised deep holes and the strongly rational, holomorphic vertex operator algebras of central charge $24$ with nontrivial weight- $1$ space. Hence, we obtain a new, geometric classification of these vertex operator algebras, generalising the classification of the Niemeier lattices by their hole diagrams.

Keywords

conformal field theory, vertex operator algebra, Leech lattice, Schellekens' list, deep hole, generalised deep hole, central charge 24

Mathematical Subject Classification

Primary: 17B69

Secondary: 17B22

Milestones

Received: 10 November 2022

Revised: 20 July 2023

Accepted: 12 October 2023

Published: 7 October 2024

Authors

Sven Möller
Fachbereich Mathematik Universität Hamburg Hamburg Germany

Nils R. Scheithauer
Fachbereich Mathematik Technische Universität Darmstadt Darmstadt Germany

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Vol. 18, No. 10, 2024