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New directions in vertex algebras and moonshine

顶点代数和 Moonshine 的新方向

基本信息

批准号：
22K03264
负责人：
CARNAHAN Scott
金额：
$ 2.66万
依托单位：
University of Tsukuba
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2022
资助国家：
日本
起止时间：
2022-04-01 至 2026-03-31
项目状态：
未结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-22K03264/
关键词：
moonshine vertex algebra weak Hopf algebra vertex operator algebra

项目摘要

Together with my student Satoru Urano, I have submitted a paper about our conjecture that unifies and generalizes Monstrous Moonshine and Modular Moonshine. Our conjecture asserts that for any subring R of the complex numbers, any subgroup G of the monster, and any ring homomorphism f from the representation ring (Green ring) of RG to the complex numbers, the "generalized McKay-Thompson series" given by applying f to the graded pieces of the monster vertex algebra is the q-expansion of a genus zero modular function.This is known in the special cases covered by Monstrous Moonshine (where R is the complex numbers) and Modular Moonshine (where R is isomorphic to a p-adic ring and G is cyclic of order with p-valuation 1), but we proved it in some additional cases, and we have shown that all generalized McKay-Thompson series satisfy an infinite collection of relations that we call "quasi-replicability".This paper has been accepted at IMRN.We have additional results that have not been submitted yet. First, we have classified homomorphisms from the Green rings of all groups of order pq, where p and q are distinct primes, and we have proved our conjecture for all "totally Fricke" cyclic groups of square-free order.

我和我的学生Satoru Urano一起提交了一篇关于我们的猜想的论文，该猜想统一并推广了Monte-Moonshine和Modular Moonshine。我们的猜想断言，对于复数的任何子环R，Monster的任何子群G，以及表示环的任何环同态f，（绿色环）RG的复数，通过将f应用于怪物顶点代数的分级部分而给出的“广义麦凯-汤普森级数”是亏格零模函数的q展开。这在Monstrous Moonshine所涵盖的特殊情况中是已知的（其中R是复数）和Modular Moonshine（其中R同构于p-adic环，G是p-赋值为1的阶循环环），但我们在一些额外的情况下证明了它，我们证明了所有的广义McKay-Thompson级数都满足一个我们称之为“拟可复制性”的关系的无限集合.这篇论文已经被IMRN接受.我们还有其他的结果尚未提交. 首先，我们对pq阶群的绿色环上的同态进行了分类，其中p和q是不同的素数，并证明了我们对所有无平方阶的“全环”循环群的猜想。