TY - JOUR

T1 - Schellekens' list and the very strange formula

AU - van Ekeren, Jethro

AU - Lam, Ching Hung

AU - Möller, Sven

AU - Shimakura, Hiroki

N1 - Funding Information:
The authors would like to thank Gerald Höhn and Nils Scheithauer for helpful discussions. The third and fourth authors are grateful for the hospitality of Academia Sinica in Taipei, where parts of this research were conducted. The authors thank the organisers of the 2018 RIMS workshop “VOAs and Symmetries” in Kyoto, where also part of this work was done. The first author was supported by CNPq grants 409598/2016-0 and 303806/2017-6 , the second by grant AS-IA-107-M02 of Academia Sinica and MOST grant 107-2115-M-001-003-MY3 of Taiwan, the third by an AMS-Simons Travel Grant and the fourth by JSPS KAKENHI Grant Numbers JP17K05154 , JP19KK0065 and JP20K03505 .
Funding Information:
The authors would like to thank Gerald H?hn and Nils Scheithauer for helpful discussions. The third and fourth authors are grateful for the hospitality of Academia Sinica in Taipei, where parts of this research were conducted. The authors thank the organisers of the 2018 RIMS workshop ?VOAs and Symmetries? in Kyoto, where also part of this work was done. The first author was supported by CNPq grants 409598/2016-0 and 303806/2017-6, the second by grant AS-IA-107-M02 of Academia Sinica and MOST grant 107-2115-M-001-003-MY3 of Taiwan, the third by an AMS-Simons Travel Grant and the fourth by JSPS KAKENHI Grant Numbers JP17K05154, JP19KK0065 and JP20K03505.
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/3/26

Y1 - 2021/3/26

N2 - 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 VΛ. 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.

AB - 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 VΛ. 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.

KW - Classification

KW - Conformal field theory

KW - Leech lattice

KW - Schellekens' list

KW - Vertex operator algebra

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U2 - 10.1016/j.aim.2021.107567

DO - 10.1016/j.aim.2021.107567

M3 - Article

AN - SCOPUS:85099339649

SN - 0001-8708

VL - 380

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 107567

ER -