Classification of subfactors in theory of operator algebras and its applications

算子代数理论子因子的分类及其应用

基本信息

批准号：
13640204
负责人：
KAWAHIGASHI Asuyuki
金额：
$ 2.43万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640204/
关键词：
Operator algebra Quantum field theory Subfactor Conformal field theory Modular invariant Tensor cateeor モジュラー不変 subfactor conformal field theory alpha-induction Virasoro algebra central charge topological invariant quantum field

项目摘要

I have obtained a classification result in operator algebraic approach to conformal field theory. A one-dimensional conformal field theory in the operator algebraic formulation is called a conformal net, and I have shown, with Longo, that if the symmetry group is a diffeomorphism group, then the central charge of a conformal net is defined and if furthermore it is less than 1, then it is completely classified with an invariant labeled with pairs of A-D-E Dynkin diagrams. This is a complete solution to a well known problem. I fully use theory of alpha-induction and modular invariants I have studied before. Next, I have studied classification theory of two-dimensional conformal nets and obtained a complete classification result with Longo for the case with central charge less than 1. In the one-dimensional classification, the diagrams D 2n+1 and E_7 did not appear, but now they. do appear. Strictly speaking, in the two-dimensional classification results, I have assumptions that a net has parity symmetry and is maximal, and these two are equivalent to the condition of the mu-index being 1, but these two conditions can be easily dropped, and we would have merely more combinatorial complecity. We use the above-mentioned one-dimensional classification for this, and the new point is theory of 2苗ohomology group of a tensor category. A key step is a proof of 2-chomology vanishing for tensor categories related to the Virasoro algebra

我已经获得了连续性代数方法的分类结果。操作员代数公式中的一维综合场理论称为保形网络，我已经表明，如果对称组是一个差异群体，则定义了对称组的核心范围，那么将其定义的核心负责人，如果它是一个完全不小于1的，那么它是与dy-a-a-a-a-pairs a-n-a-n-a-n-a-dy-a-n-a-de-a-n-a-n-a-de-a-n-a-pairs-a-n-pairs-a-n-a-pairs-a-n-a-n a-pair a-n-paire n a-n-n-n-and-a-d-a。这是解决一个众所周知的问题的完整解决方案。我完全使用了以前研究的α诱导和模块化不变性理论。接下来，我研究了二维形成共形网的分类理论，并获得了longo的完整分类结果，其中中央电荷小于1。在一维分类中，该图D 2N+1和E_7没有出现，但是现在它们。确实出现。严格来说，在二维分类结果中，我假设网络具有均衡对称性并且是最大的，并且这两个等同于Mu-Index的条件为1，但是这两个条件很容易掉落，我们只会更加组合复杂性。为此，我们使用上述一维分类，新观点是张量类别的OHomology组理论。一个关键步骤是证明与Virasoro代数相关的张量类别的2个知识论消失