权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Quantization of Anosov foliations and noncommutative geometry

Anosov 叶状结构和非交换几何的量子化

基本信息

批准号：
15540203
负责人：
NATSUME Toshikazu
金额：
$ 2.18万
依托单位：
Nagoya Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540203/
关键词：
Anosov foliations C^*-algebra noncommutative geometry K-theory cyclic cohomology quantization C*-環非可換幾何

项目摘要

The purpose of this project is to obtain a quantum version of the results in "The Godbillon-Vey cyclic cocycle and longitudinal Dirac operators (with the investigator Hitoshi Moriyoshi)" and "Topological approach to quantum surfaces( with Ryszard Nest of the University of Copenhagen)", more precisely to construct noncommutative Anosov foliations on "the unit tangent bundles" over noncommutative Riemann surfaces. This noncommutative Anosov foliations are regarded as quantizations of the (commutative) Anosov foliations associated with geodesic flows on the unit tangent bundles. The ultimate goal of the project is to prove the foliation index theorem of A. Connes, for the noncommutative Anosov foliations.In a joint project with Nest (unpublished) we constructed noncommutative 3-manifolds as strict quantizations of unit circle bundles of closed Riemann surfaces of genus greater than 1.These noncommutaive 3-manifolds were constructed in such a way that the relationship between the Riemann surface and its unit tangent bundle is kept intact through a suitable group action. Moreover we constructed a foliation on the noncommutaive 3-manifold as a certain C^*-algebra in the spirit of A. Connes's noncommutative geometry. We are preparing a paper "Noncommutaive Anosov foliations (tentative title)". We are currently working on detail. As one expects, on view of commutative case, the C^*-algebra representing a "leaf of the noncommutaive foliation is a covering space. We developed some idea how to lift the Dirac operator on the quantized Riemann surface to a longitudinal elliptic operator for the noncommutative Anosov foliation.Unfortunately we were unable to complete the project. However, we certainly continue to work on the project, as we now have a clear idea how to achieve the goal.

这个项目的目的是得到“Godillon-Vey循环上循环和纵向Dirac算子(与研究员Hitoshi Moriyoshi)”和“量子曲面的拓扑方法(与哥本哈根大学的Ryszard Nest)”中的结果的量子版本，更准确地说，在非对易Riemann曲面上的“单位切丛”上构造非对易Anosov叶。这种非对易的Anosov叶被认为是与单位切丛上的测地线流有关的(对易)Anosov叶的量化。在与Nest(未发表)的一个联合项目中，我们构造了非交换3-流形，作为亏格大于1的闭黎曼曲面的单位圆丛的严格量化。这些非交换3-流形的构造方法是这样的：通过适当的群作用保持黎曼曲面与其单位切丛之间的关系不变。此外，在A.Connes的非对易几何的精神下，我们在非对易3-流形上构造了一个叶层作为某种C^*-代数。我们正在准备一篇题为《非交换的阿诺索夫叶(暂定标题)》的论文。我们目前正在处理细节问题。不出所料，在交换的情况下，表示非对易叶的C^*-代数是覆盖空间。我们发展了一些想法，如何将量子化的Riemann曲面上的Dirac算子提升为非对易Anosov叶的纵向椭圆算子。不幸的是，我们无法完成这个项目。然而，我们肯定会继续这个项目，因为我们现在有了一个明确的目标。