Mathieu Moonshine in the elliptic genus of K3
Mathieu Moonshine in the elliptic genus of K3
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K3 椭圆属中的 Mathieu Moonshine
DOI：
10.1007/jhep10(2010)062
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发表时间：
2010
影响因子：
5.4
通讯作者：
R. Volpato
中科院分区：
文献类型：
--
作者：
M. Gaberdiel;S. Hohenegger;R. Volpato
最近有人推测 K3 的椭圆亏格可以用 Mathieu 群 $$ {\mathbb{M}_{24}} $$ 表示的维数来写。随后，通过研究从椭圆属中获得的缠绕属，在用其特征替换马蒂厄群表示的维数后，发现了这一想法的一些进一步的证据。在本文中，我们通过对所有（剩余的）孪生属的一般模属性进行有根据的猜测，找到了它们的显式公式。这使我们能够根据 $$ {\mathbb{M}_{24}} $$ 表示的维度来识别所有展开系数的分解。对于前 500 个系数，我们验证这些表示出现的重数确实都是非负整数。这代表了支持该猜想的非常令人信服的证据。
It has recently been conjectured that the elliptic genus of K3 can be written in terms of dimensions of Mathieu group $$ {\mathbb{M}_{24}} $$ representations. Some further evidence for this idea was subsequently found by studying the twining genera that are obtained from the elliptic genus upon replacing dimensions of Mathieu group representations by their characters. In this paper we find explicit formulae for all (remaining) twining genera by making an educated guess for their general modular properties. This allows us to identify the decomposition of all expansion coefficients in terms of dimensions of $$ {\mathbb{M}_{24}} $$-representations. For the first 500 coefficients we verify that the multiplicities with which these representations appear are indeed all non-negative integers. This represents very compelling evidence in favour of the conjecture.