A step to the perfect classification of 24dimensional meromorphic vertex operator algebras

迈向24维亚纯顶点算子代数完美分类的一步

基本信息

批准号：
09440004
负责人：
MIYAMOTO Masahiko
金额：
$ 6.21万
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 (B).
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-09440004/
关键词：
Ising models Lattices Meromorphic Virasoro algebra Automorphism groups Monster simple group Vertex operator algebra Moonshine conjecture ムーンシャン予想

项目摘要

A concept of vertex operator algebras (VOA shortly) has originated from the moonshine vertex operator algebra, which was constructed in order to explain a mysterious relation (the moonshine conjecture) between Monster simple finite group (the largest sporadic finite simple group) and the classical elliptic modular function. It is now understand to be a rigorous and mathematical concept of 2 dimensional conformal field theory in physics. Namely, it offers axioms on such a theory. Although 24 dimensional meromorphic conformal field theory have a special important value in physics, the examples we know at present are only Moonshine VOA, VOAs constructed from the Niemeier lattices and their orbifold VOAs. Recently, Dong and Mason found that all of them contain a tensor product of 48 Ising modules.With the support of this Grant, M.Miyamoto (Head of this research) has been studying VOAs containing a tensor product of Ising modules, he found new VOAs called "code VOAs", which are easy to hand … More le compared with the other VOAs in 1997. We also determined their representations (all modules) and introduced a concept of "induced modules." In 1998, we found a special property of Hamming code VOAs (constructed from an extended [8, 4, 4]-Hamming code) and determined its fusion rules among its irreducible modules. Using this special property, we found a new construction of the famous moonshine VOA and then a new construction of Monster simple group. It is very easier than the original construction and obtained a lot of properties of Monster simple group. In 2000, we applied the new method of construction to the known VOAs (lattice VOAs, etc.) and found that it is possible to construct all known 24 dimensional holomorphic VOAs by this way. We also succeed to construct twisted modules of code VOAs. For twisted modules, the existence was proved theoretically, but we don't know examples except very easy one. So we hope that this new construction have many applications, especially we expect to construct twisted modules for the moonshine VOAs, which are the essential parts of the moonshine conjectures. Less

顶点操作员代数（VOA）的概念源自Moonshine顶点操作员代数，该代数是为了解释Monster Simple Firite Group（最大的零星有限有限的群体）与经典椭圆形模块化功能之间的神秘关系（Moonshine Concept）。现在，它是物理学中2维综合场理论的严格和数学概念。也就是说，它提供了这种公理，尽管有24维的仿产子形态结构理论在物理学方面具有特殊的重要价值，目前我们知道的示例仅是月光voa，是由Niemeier Lattices及其Orbifold Voas构建的VOA。最近，东和梅森发现，所有这些都包含48个Ising模块的张量产品。在这笔赠款的支持下，M.Miyamoto（这项研究的负责人）一直在研究包含Ising模块的张量产品的VOA，他发现了新的称为“代码VOAS”的新Voas，与其他voas相比，它与其他voas相比，我们在199年的概述中都更加确定。 “诱导的模块。”在1998年，我们发现了锤子代码VOA的特殊属性（由扩展[8，4，4] - 戴锤法构建），并确定其在其不可约模块中的融合规则。使用此特殊物业，我们发现了著名的Moonshine VOA的新结构，然后是Monster Simple Group的新结构。它比原始结构要容易得多，并且获得了Monster Simple组的许多属性。在2000年，我们将新的构建方法应用于已知的VOA（晶格VOA等），并发现可以通过这种方式构造所有已知的24维霍明型VOA。我们还成功地构建了代码VOA的扭曲模块。对于扭曲的模块，存在被证明是理论上的，但是我们不知道例子，除了非常容易的例子。因此，我们希望这种新结构有许多应用，尤其是我们期望为Moonshine Voas构建扭曲的模块，这是Moonshine猜想的重要部分。较少的