Non-Commutative Geometry and the Spectral Flow Index Theorem

非交换几何和谱流指数定理

• 批准号: 12640086
• 负责人: MORIYOSHI Hitoshi
• 金额: $ 2.18万
• 依托单位: Keio University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640086/
• 关键词: NONCOMMUTATIVE GEOMETRY; K-TEHORY; THE INDEX THEOREM; SPECTRAL FLOW; CYCLIC COHOMOLORY; K-THEORY

In this research we proceeded to a generalization of the Atiyah-Singer index theorem on the bas1S of Low dimensional Topology. The objective of the project are the followings:1. Establish the elaborated Index Theorem in the framework of Non-commutative Geometry. Also study the relationship between the Index Theorem and the analytic secondary invariants like the eta invariants and the spectral flow;2. Study the relationship between the elaborated Index Theorem and secondary characteristic classes including the Maslov class.Here we state one of the results of the project, which is related to the Atiyah-Patodi-Singe Index Theorem in Non-commutative Geometry. In July 2001, we are invited to give a talk with the title 'Analytic K-theory and the index theorem' at Topology Symposium of Japan, Akita. We are also working on the project are presented with the title 'Eta invariants, the Godbillon-Vey classes and the index theorem' in March 2001 at Workshop on Non-commutative Geometry and String Theory, Keio University. We also developed the research on Non-commutative Hopf invariants, and presented some results at Nagoya Technology Institute with the title 'Spectral flow, the Teoplitz index and the Hopf invariant' in March 2001, and with the title 'A index theorem of vector field and the Hopf invariant' at Kagoshima University in February 2002.

在这项研究中，我们在低维拓扑的基础上对 Atiyah-Singer 指数定理进行了推广。该项目的目标如下： 1．在非交换几何框架中建立详细的指数定理。研究指数定理与eta不变量、谱流等解析二次不变量之间的关系； 2．研究阐述的指数定理与包括Maslov类在内的次级特征类之间的关系。这里我们陈述该项目的成果之一，该成果与非交换几何中的Atiyah-Patodi-Singe指数定理相关。 2001年7月，我们受邀在日本秋田拓扑学研讨会上作题为“解析K理论和指数定理”的报告。我们还致力于该项目，该项目于 2001 年 3 月在庆应义塾大学非交换几何和弦理论研讨会上发表，题为“Eta 不变量、Godbillon-Vey 类和索引定理”。我们还开展了非交换Hopf不变量的研究，并于2001年3月在名古屋技术研究所以“谱流、Teoplitz指数和Hopf不变量”为题发表了一些成果，并于2002年2月在鹿儿岛大学以“向量场的指数定理和Hopf不变量”为题发表了一些结果。

项目成果

M. Duboi-Violette, A. Krieg, Y. Maeda, P. Michor: "Smooth *- algebras"Progress of Theoretical Physics, Supplement. 144. (2001)

M. Duboi-Violette、A. Krieg、Y. Maeda、P. Michor：“平滑*-代数”理论物理进展，补充。

H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka: "Singular systems of exponential functions"Mathematical Physics Studies, Kluwer. 23. 169-186 (2001)

H. Omori、Y. Maeda、N. Miyazaki、A. Yoshioka：“指数函数的奇异系统”数学物理研究，Kluwer。

M.Duboi-Violette., et al.: "Smooth *-algebras"Progress of Theoretical Physics, Supplement. 144. (2001)

M.Duboi-Violette.，等人：“平滑*-代数”理论物理进展，补充。

H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka: "Convergent star products on Prechet linear Poisson algebras of Heizenberg type"Contemporary of Mathematics. 288. 391-395 (2001)

H. Omori、Y. Maeda、N. Miyazaki、A. Yoshioka：“Heizenberg 型 Prechet 线性泊松代数上的收敛星积”当代数学。

T. Natsume and H. Moriyoshi: "Operator algebras and Geometry"Suugaku memoire. Vol. 2, J.M.S.. (2001)

T. Natsume 和 H. Moriyoshi：《算子代数与几何》Suugaku 回忆录。