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，该论文统一并概括了 Monstrous Moonshine 和 Modular Moonshine。我们的猜想断言，对于复数的任何子环 R、怪物的任何子群 G 以及从 RG 的表示环（绿环）到复数的任何环同态 f，“广义麦凯-汤普森级数”由下式给出将 f 应用于怪物顶点代数的分级部分是属零模函数的 q 展开。这在 Monstrous Moonshine 所涵盖的特殊情况下是已知的（其中 R 是复数数）和模月光（其中 R 与 p 进环同构，G 是 p 值为 1 的阶循环），但我们在一些其他情况下证明了这一点，并且我们已经证明所有广义 McKay-Thompson 级数满足我们称之为“准可复制性”的无限关系集合。这篇论文已被 IMRN 接受。我们还有其他结果尚未提交。首先，我们对所有 pq 阶群的格林环中的同态进行了分类，其中 p 和 q 是不同的素数，并且我们证明了对所有无平方阶的“完全弗里克”循环群的猜想。