Gauge theory, homology cobordisms, and Rohlin's invariant

规范场论、同调配边主义和罗林不变量

• 批准号: 0505605
• 负责人: Daniel Ruberman
• 金额: $ 17.77万
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505605&HistoricalAwards=false
• 关键词: Gauge; theory; homology; cobordisms; Rohlin

Daniel Ruberman will carry out research in geometric topology, using Seiberg-Witten and Yang-Mills gauge theory. The first part of the project, to be carried out in collaboration with Nikolai Saveliev and Tomasz Mrowka, is to understand the smooth topology of 4-manifolds that homologically resemble a product of a 3-dimensional manifold with a circle. The central questions center around the interpretation of the classical Rohlin invariant in terms of gauge theory; an important facet is the analysis of end-periodic Dirac operators. Solutions of the main problems posed will decide the existence of some manifolds predicted by high-dimensional topology (surgery theory) and settle a fundamental question about the homology cobordism group. This latter question is key in understanding the triangulability of manifolds of dimension 5 or more. The second part of the project proposes joint work with Saso Strle. We plan to investigate the degree to which invariants of a 3-manifold derived from the Seiberg-Witten equations constrain the topology of those 4-manifolds which it bounds. The results will be used to investigate a classical problem about group actions on the K3 surface and other algebraic varieties. Many of the most difficult problems in topology are concerned with the structure of three and four-dimensional manifolds. Questions about phenomena in these dimensions have particular interest because those are the dimension of space and time that make up our world. The research to be carried out under this grant makes use of powerful geometric and analytical techniques that have been introduced into the field over the last twenty years, under the general rubric of gauge theory. Much of the application of four-dimensional gauge theory has been to manifolds that are simply-connected, in which any loop can be shrunk to a point. The research proposed will use Yang-Mills and Seiberg-Witten gauge theories to shed light on manifolds which are not simply-connected, where new phenomena are expected. By studying the Dirac operator (originally introduced to give a relativistic quantum theory of the electron) on manifolds with a certain type of periodicity, we seek to get restrictions on an invariant of non-simply connected manifolds, called the Rohlin invariant. This subtle invariant is connected to three-dimensional topology, and its behavior determines whether high-dimensional manifolds may be triangulated.

丹尼尔·鲁贝曼（Daniel Ruberman）将使用塞伯格·韦滕（Seiberg-Witten）和杨麦格斯（Yang-Mills）仪表理论进行几何拓扑研究。该项目的第一部分与尼古拉·萨维尔夫（Nikolai Saveliev）和托马斯·莫罗卡（Tomasz Mrowka）合作进行，是要了解4个manifolds的平滑拓扑结构，这些拓扑在同源上类似于带有圆圈的三维流形的产物。 中央问题围绕着对仪表理论的古典罗林不变的解释。一个重要的方面是对最终周期性狄拉克操作员的分析。提出的主要问题的解决方案将决定存在高维拓扑（手术理论）预测的一些流形，并解决有关同源性共同体群体的基本问题。 后一个问题是理解维度5或更多歧管的三角测量性的关键。 该项目的第二部分提出了与Saso Strle的联合合作。我们计划研究从塞伯格（Seiberg）的方程式得出的3个manifold的不变性限制了其界限的4个manifolds的拓扑。 结果将用于研究有关K3表面和其他代数品种的小组作用的经典问题。 拓扑中的许多最困难的问题都与三维和四维流形的结构有关。 这些维度中有关现象的问题具有特别的兴趣，因为这些问题是构成我们世界的时空的维度。 根据仪表理论的一般规则，在过去二十年中，使用了强大的几何和分析技术，使用了强大的几何和分析技术。 四维规定理论的许多应用都是在简单连接的流形中，其中任何循环都可以缩小到某个点。 拟议的研究将使用杨利尔和塞伯格（Seiberg-witten）理论的理论来阐明不简单连接的歧管，而在这些歧管上，预期的是新现象。 通过研究具有一定类型的周期性的歧管上的狄拉克运算符（最初是为了给出电子的相对论量子理论），我们试图对不变连接的流形的不变构成限制，称为Rohlin不变。 这种微妙的不变剂与三维拓扑相关，其行为决定了是否可以将高维流形进行三角测量。