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

顶点算子代数（简称VOA）的概念起源于Moonshine顶点算子代数，它的构造是为了解释Monster简单有限群（最大的零星有限简单群）与经典群之间的神秘关系（Moonshine猜想）。椭圆模函数现在被理解为物理学中二维共形场论的严格数学概念，即它提供了这种理论的公理。维度亚纯共形场理论在物理学中具有特殊的重要价值，目前我们知道的例子只有Moonshine VOA、由尼迈尔晶格构造的VOA及其轨道折叠VOA，最近Dong和Mason发现它们都包含张量积。 48 个 Ising 模块。在这项资助的支持下，M.Miyamoto（这项研究的负责人）一直在研究包含 Ising 模块张量积的 VOA，他发现新的 VOA 称为“代码 VOA”，与 1997 年的其他 VOA 相比，它更容易掌握。我们还确定了它们的表示形式（所有模块）并引入了“诱导模块”的概念。1998 年，我们发现了一个特殊的 VOA。我们研究了汉明码 VOA（由扩展的 [8, 4, 4]-汉明码构成）的性质，并确定了其不可约模块之间的融合规则。利用这一特殊性质，我们发现了一种新的构造。著名的moonshine VOA，然后是Monster简单群的新构造等），发现可以通过这种方式构造所有已知的24维全纯VOA，我们也成功构造了代码VOA的扭曲模块。扭曲模块，理论上证明了存在性，但我们不知道除了非常简单的例子之外，所以我们希望这种新结构有很多应用，特别是我们期望为 Moonshine VOA 构建扭曲模块，这是 Moonshine 猜想的重要组成部分。