论文标题
Schellekens的列表和非常奇怪的公式
Schellekens' List and the Very Strange Formula
论文作者
论文摘要
1993年，Schellekens证明了一个强大的，霍明型顶点操作员代数$ V $中央电荷24必须是71个谎言代数之一，这是一个强大的，全态顶点操作员代数。在接下来的三十年中，在许多作者的共同努力中，证明这些谎言代数都由这样的顶点操作员代数实现，除了$ v_1 = \ {0 \} $之外，此顶点操作符代数是由$ v_1 $ niquy by $ v_1 $确定的。 在本文中，我们给出了Schellekens的71个代数列表的根本不同，更简单的证明。使用Arxiv：1910.04947和KAC的“非常奇怪的公式”中的尺寸公式，我们表明，每一个强烈理性的，全态的顶点操作员代数$ V $ central Charge 24，$ v_1 \ neq \ neq \ neq \ neq \ {0 \} $都可以通过$ v_1 \ neq \} $从Leech lattice Verte $ $ verte $ $ verte $ verte $ $ verte contra $ v_1 \ neq \ {0 \} $获得。这足以限制可能以$ v $的重量空间为代数的代数。 此外，每个强烈理性的，全态顶点操作员代数$ v $ of Central Charge 24来自Leech Lattice $λ$，可用于通过研究Leech Lattice的属性来对这些顶点操作员代数进行分类。我们为Schellekens列表中的70个非零谎言代数中的43个证明了这一点，从而省略了那些在计算上太昂贵的案例。
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 $V_Λ$. 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.
