New directions in vertex algebras and moonshine

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怪异月光和模块化月光。我们的猜想断言，对于复数的任何子群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，将f应用于怪物顶点代数的梯度块所得到的“广义McKay-Thompson级数”是一个属零模函数的q展开式。这在怪异的Moonshine（其中R是复数）和模Moonshine（其中R同构于p进环，G是p值为1的阶循环）所涵盖的特殊情况下是已知的，但我们在一些额外的情况下证明了这一点，并且我们已经证明了所有广义McKay-Thompson级数满足无限的关系集合，我们称之为“准可复制性”。这篇论文已被IMRN接受。我们还有更多的结果还没有提交。首先，我们对所有pq阶群的Green环的同态进行了分类，其中p和q是不同素数，并证明了我们对所有无平方阶的“完全Fricke”循环群的猜想。

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.