Schellekens' List and the Very Strange Formula

[2005.12248] Schellekens' List and the Very Strange Formula Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Quantum Algebra arXiv:2005.12248 (math) [Submitted on 25 May 2020 ( v1 ), last revised 18 Jan 2021 (this version, v2)] Title: Schellekens' List and the Very Strange Formula Authors: Jethro van Ekeren , Ching Hung Lam , Sven Möller , Hiroki Shimakura View a PDF of the paper titled Schellekens' List and the Very Strange Formula, by Jethro van Ekeren and 3 other authors View PDF HTML (experimental) 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. Comments: 28 pages, LaTeX; minor changes, Sections 4 and 5 swapped; to appear in Adv. Math Subjects: Quantum Algebra (math.QA) ; Representation Theory (math.RT) MSC classes: 17B69 Cite as: arXiv:2005.12248 [math.QA] (or arXiv:2005.12248v2 [math.QA] for this version) https://doi.org/10.48550/arXiv.2005.12248 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Adv. Math., 380:107567, 2021 Related DOI : https://doi.org/10.1016/j.aim.2021.107567 Focus to learn more DOI(s) linking to related resources Submission history From: Sven Möller [ view email ] [v1] Mon, 25 May 2020 17:48:55 UTC (30 KB) [v2] Mon, 18 Jan 2021 07:34:36 UTC (28 KB) Full-text links: Access Paper: View a PDF of the paper titled Schellekens' List and the Very Strange Formula, by Jethro van Ekeren and 3 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.QA < prev | next > new | recent | 2020-05 Change to browse by: math math.RT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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