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

• 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

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

在共形场理论的算符代数方法中，我得到了一个分类结果。算符代数形式中的一维共形场理论称为共形网，我用Longo证明了，如果对称群是微分同胚群，则定义了共形网的中心电荷，如果中心电荷小于1，则用A-D-E动态图对标记的不变量完全分类。这是一个众所周知的问题的完整解决方案。我充分利用了我以前学习过的阿尔法归纳法和模不变量理论。其次，研究了二维共形网络的分类理论，对于中心电荷小于1的情况，用Longo得到了一个完整的分类结果。在一维分类中，没有出现D2n+1和E7图，但现在它们出现了。确实出现了。严格地说，在二维分类结果中，我假设一个网具有宇称对称性并且是极大的，并且这两个条件等价于u-指数为1的条件，但这两个条件很容易被丢弃，我们只会有更多的组合完备性。我们使用了上述一维分类，新的观点是张量范畴的2苗同调群理论。一个关键的步骤是证明与Virasoro代数相关的张量范畴的2-Chomology零化

项目成果

Y.Kawahigashi: "Generalizediongo-Rehren subfactors and alpha-induction"Comm.Math.Phys.. 226. 269-287 (2002)

Y.Kawahigashi：“广义iongo-Rehren 子因子和α 感应”Comm.Math.Phys.. 226. 269-287 (2002)

Y.Kawahigashi, R.Longo: "Classification of local conformal nets : Case c<1"Ann.Math.. (印刷中). (2004)

Y.Kawahigashi、R.Longo：“局部共形网络的分类：案例 c<1”Ann.Math..（出版中）。

Y.Kawahigashi, R.Longo: "Classification of local conformal nets Case c<1"Ann.Math.. (in press).

Y.Kawahigashi、R.Longo：“局部共形网络的分类案例 c<1”Ann.Math..（正在印刷中）。

Y.Kawahigashi, R.Longo: "Classification of two-dimensional local conformal nets with c < 1 and 2-cohomology vanishing for tensor categories"Commun.Math.Phys.. 244. 63-97 (2004)

Y.Kawahigashi、R.Longo：“张量类别中 c < 1 和 2-上同调消失的二维局部共形网络的分类”Commun.Math.Phys.. 244. 63-97 (2004)

Y.Kawahigashi: "Conformal quantum field theory and subfactors"Acta Math.Sci.. 19. 557-566 (2003)

Y.Kawahigashi：“共形量子场论和子因子”Acta Math.Sci.. 19. 557-566 (2003)