Schellekens' list and the very strange formula
Schellekens' list and the very strange formula
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DOI：
10.1016/j.aim.2021.107567
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发表时间：
2020-05
影响因子：
1.7
通讯作者：
J. Ekeren;C. Lam;S. Møller;Hiroki Shimakura
中科院分区：
文献类型：
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作者：
J. Ekeren;C. Lam;S. Møller;Hiroki Shimakura
摘要 在[45]（另见[20]）中，Schellekens证明了中心电荷24的强有理全纯顶点算子代数V的权一空间V 1 必定是71个李代数之一。在接下来的三十年里，在许多作者的共同努力下，证明了这些李代数中的每一个都是由这样一个顶点算子代数实现的，并且除了V 1 = {0}之外，这个顶点算子代数是由V 1 唯一确定的。这些陈述的统一证明在[42]，[26]中给出。在本文中，我们对 Schellekens 的 71 个李代数列表给出了一个根本不同的、更简单的证明。利用[42]中的维数公式和Kac的“非常奇怪的公式”[28]，我们证明了中心电荷24的每个强有理全纯顶点算子代数V，其中V 1≠{0} 都可以通过Leech点阵顶点算子代数VΛ的环折构造来获得。这足以将可能出现的李代数限制为 V 的权重一空间到 Schellekens 的 71。此外，中心电荷24的每个强有理全纯顶点算子代数V来自Leech晶格Λ的事实可用于通过研究Leech晶格的性质来对这些顶点算子代数进行分类。我们针对 Schellekens 列表中 70 个非零李代数中的 43 个证明了这一点，忽略了那些计算成本太高的情况。
Abstract In [45](see also [20]) 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. 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 V 1≠{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.