The Index Theorem and Analytic Secondary Invariants in Symplectic Geometry

辛几何中的指数定理和解析二次不变量

基本信息

批准号：
09640081
负责人：
MORIYOSHI Hitoshi
金额：
$ 1.98万
依托单位：
Keio University (1998-1999)Hokkaido University (1997)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640081/
关键词：
NONCOMMUTATIVE GEOMETRY K-TEHORY CYCLIC COHOMOLORY THE INDEX THEOREM SYMPLECTIC GEOMETRY 非可換微分幾何学 K-理論 Maslov類 spectral flow

项目摘要

The objective of the project are the followings:1. Establish the elaborated Index Theorem in the framework of Noncommutative Geometry. Also study the relationship between the Index Theorem and the analytic secondary invariants like the eta invarinats and the spectral flow;2. Apply the elaborated Index Theorem to Symplectic Geometry and study the Maslov class from the viewpoint of secondary classes.Here we state one of the results of the project, which is related to the Atiyah-Patodi-Singe Index Theorem in Noncommutative Geometry. Let X be a compact even-dimensional manifold with boundary Y. We equip X with a Riemaniann metric and assume that X is isometric to the product space Y×(-1,0) in a neighborhood of Y. We then denote by X the complete manifold obtained by attaching the half cylinder Y×[0,+∞] to X. To understand the Atiyah-Patodi-Singer Index Theorem in a framework of Noncommutative Geometry, we first introduce a notion of group quasi-action and understand X as the quotient with … More respect to a quasi-action of R. Next we construct a short exact sequence of CィイD1*ィエD1-algebras involved with kernel functions on X. We then define the index of operators on X as elements in a relative K-group. The short exact sequence constructed above is also interesting itself since it yields the Wiener-Hopf extension for CィイD1*ィエD1R even in the simplest case. Given the K-theoretic definition of index, we construct a relative cyclic cocycle that is related to the eta invariant of Y. This description makes clear the role of the integral on the L-polynomial and the eta invariant appeared in the Atiyah-Patodi-Singer Index Theorem, which are a priori depending on the choice of Riemannian metric on X. In short, the eta invariant appears as the transgression form connecting the local invariant with the index of an R-invariant operator on the cylinder Y×R. We also developed the research toi obtain the result that clearify the relation between the eta invarinats and the spectral flow for type II von Neumann algebras. Less

该项目的目的是：1。在非交通性几何形状框架中建立精心设计的索引定理。还要研究索引定理与分析次级不变性之间的关系，例如ETA不变性和光谱流； 2。将详细的索引定理应用于相似的几何形状，并从中等类的角度研究Maslov类。在这里，我们陈述了该项目的结果之一，该项目与非交互性几何形状中的Atiyah-Patodi-Singe指数定理有关。令X成为边界Y的紧凑型均匀歧管。我们为X装备了X型公制，并假设X在Y附近的产品空间y×（-1,0）等等速线。然后，我们用X表示x通过将半圆柱体y×[0，+∞]与x. x. acter noncoter in nonc andie andii incort andie in acter nonc所获得的完整歧管表示，以供x。几何形状，我们首先引入了群体表演的通知，并将x理解为……更尊重R。上面构建的简单序列也很有趣，因为它即使在最简单的情况下也会产生CII D1*II D1R的Wiener-HOPF扩展。鉴于索引的K理论定义，我们构建了与Y的ETA不变性相关的相对环状共生。此描述清楚地表明了积分对L-多种状态和ETA不变的作用，并且ETA不变性出现在Atiyah-Patodi-Singer-Singer-Singer-Singer Index Theorem中，这是根据Riemann Metrric Metrric Metrorrir的X. Interiant的先验性，是X. In Intriant的先验。在气缸y×r上具有R-In-Invariant运算符索引的本地不变。我们还开发了研究，从而获得了消除ETA不变性与II型Von Neumann代数的光谱流之间关系的结果。较少的