Quantization of Anosov foliations and noncommutative geometry

该项目的目的是获得“哥德亿-维循环环和纵向狄rac算子（与研究者Hitoshi Moriyoshi）”和“量子曲面的拓扑方法（与哥本哈根大学的Ryszard Nest）”中结果的量子版本，更准确地说，在非对易黎曼曲面上的“单位切线束”上构造非对易的Anosov叶。这种非交换的Anosov叶理被认为是单位切线束上与测地线流动相关的（交换的）Anosov叶理的量化。该项目的最终目标是证明A. cones的叶化指数定理，对于非交换的Anosov叶化。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，作为格大于1的封闭黎曼曲面的单位圆束的严格量化。这些非交换的3流形是这样构造的，黎曼曲面和它的单位切线束之间的关系通过一个合适的群作用保持完整。在此基础上，借鉴a . cones的非交换几何思想，构造了非交换3流形上的一个叶形，作为C^*-代数。我们正在准备一篇论文“非交换的Anosov叶化（暂定标题）”。我们正在做细节方面的工作。正如人们所期望的那样，在交换情况下，表示非交换叶的一个“叶”的C^*代数是一个覆盖空间。对于非交换的Anosov叶理，我们提出了如何将量子化Riemann曲面上的Dirac算子提升为纵向椭圆算子的一些想法。不幸的是，我们没能完成这个项目。然而，我们肯定会继续在这个项目上工作，因为我们现在对如何实现目标有了一个清晰的想法。

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prove the foliation index theorem of A. Connes, for the noncommutative Anosov foliations.In a joint project with Nest (unpublished) we constructed noncommutative 3-manifolds as strict quantizations of unit circle bundles of closed Riemann surfaces of genus greater than 1.These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.