Umbral moonshine and the Niemeier lattices
Umbral moonshine and the Niemeier lattices
复制标题
本影月光和尼迈尔晶格
DOI：
10.1186/2197-9847-1-3
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发表时间：
2013
影响因子：
1.2
通讯作者：
Jeffrey A. Harvey
中科院分区：
文献类型：
--
作者：
Miranda C. N. Cheng;John FR Duncan;Jeffrey A. Harvey
在本文中，我们将阴影月光的尼迈耶格-23甚至unimodular正定格秩24与非平凡的根系统。对于每个尼迈耶格，我们附加一个有限群，考虑一个自然定义的商的格自同构群，并为每个共轭类的每个这些群体，我们确定了一个向量值模拟模块化的形式，其组件符合模拟θ函数的拉马努金在许多情况下。这导致了暗月光猜想，指出一个无限维的模块被分配给每个尼迈耶格的方式，相关的分级跟踪功能是一个独特的性质模拟模块的形式。这些结构和构造扩展了我们早期的论文，特别是包括江口，大栗和立川观察到的马蒂厄月光作为一个特殊情况。我们的分析还突出了亏格零群和尼迈耶格之间的对应关系。作为这个关系的一部分，我们将具有A型分量的Niemeier根系的Coxeter数精确地识别为对应的经典模曲线亏格为零的那些水平。AMS主题分类11 F22; 11 F37; 11 F46; 11 F50; 20 C34; 20 C35
In this paper, we relate umbral moonshine to the Niemeier lattices - the 23 even unimodular positive-definite lattices of rank 24 with non-trivial root systems. To each Niemeier lattice, we attach a finite group by considering a naturally defined quotient of the lattice automorphism group, and for each conjugacy class of each of these groups, we identify a vector-valued mock modular form whose components coincide with mock theta functions of Ramanujan in many cases. This leads to the umbral moonshine conjecture, stating that an infinite-dimensional module is assigned to each of the Niemeier lattices in such a way that the associated graded trace functions are mock modular forms of a distinguished nature. These constructions and conjectures extend those of our earlier paper and in particular include the Mathieu moonshine observed by Eguchi, Ooguri and Tachikawa as a special case. Our analysis also highlights a correspondence between genus zero groups and Niemeier lattices. As a part of this relation, we recognise the Coxeter numbers of Niemeier root systems with a type A component as exactly those levels for which the corresponding classical modular curve has genus zero.AMS subject classification11F22; 11F37; 11F46; 11F50; 20C34; 20C35