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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，它是为了解释Monster简单有限群（最大的偶发有限群）与经典椭圆模函数之间的神秘关系（moonshine猜想）而构造的。它现在被理解为物理学中二维共形场论的一个严格的数学概念。也就是说，它为这样一个理论提供了公理。虽然24维亚纯共形场理论在物理学中具有特殊的重要价值，但目前我们所知道的例子只有Moonshine VOAs、由尼迈耶晶格构成的VOAs和它们的轨道VOAs。最近，Dong和Mason发现它们都包含48个Ising模的张量积。在这项研究的支持下，M.Miyamoto（本研究的负责人）一直在研究包含Ising模量积的VOAs，他发现了新的VOAs，称为“代码VOAs”，与1997年的其他VOAs相比，它更容易处理。我们还确定了它们的表示（所有模块），并引入了“诱导模块”的概念。1998年，我们发现了由扩展的[8,4,4]-Hamming码构造的Hamming码VOAs的一个特殊性质，并确定了其不可约模块间的融合规则。利用这一特殊性质，我们找到了著名的私酿VOA的新结构，然后找到了怪物简单群的新结构。它比原来的构造更容易，并且获得了很多怪物简单组的属性。2000年，我们将这种新的构造方法应用于已知的voa（晶格voa等），发现用这种方法可以构造所有已知的24维全纯voa。我们还成功地构造了代码voa的扭曲模块。对于扭曲模，我们已经从理论上证明了它的存在性，但是除了一个非常简单的例子外，我们没有其他的例子。因此，我们希望这种新结构有许多应用，特别是我们期望为月光voa构建扭曲模块，这是月光猜想的重要组成部分。少