Gauge theory and low-dimensional topology

规范理论和低维拓扑

• 批准号: 0125170
• 负责人: Paul Feehan
• 金额: $ 9.56万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-15 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0125170&HistoricalAwards=false
• 关键词: Gauge; theory; low; dimensional; topology

DMS-0125170Paul M. N. FeehanThe principal goal of our research is to prove the celebrated formula of Edward Witten (1994), relating the Donaldson and Seiberg-Witten series of smooth four-manifolds. Witten used quantum field theory methods to derive his celebrated formula, based on the concept of `duality', whereas the approach we have been pursuing jointly with Thomas Leness is a mathematical one, employing a moduli space of non-abelian monopoles as a cobordism between links of Donaldson and Seiberg-Witten moduli spaces. We have already partially verified Witten's formula using this approach, showing that in many cases the series agree up through terms of order depending only on the topology of the four-manifold. We aim to build on those results, by showing next that a relation between the two series of the overall expected shape exists and then applying auxiliary techniques to prove that his formula is the only one possible. A second goal of our research is todevelop a proof of Witten's formula, in the special case of symplectic four-manifolds equipped with Lefschetz fibrations, and shed light on the relationship between Seiberg-Witten invariants, Donaldson invariants, and the structure of Lefschetz fibrations. Ultimately, we also hope to explain the relationship between the mathematical and quantum field theory methods of deriving Witten's formula.Witten's formula, as understood by mathematicians, is part of a physical theory developed by Witten and Nathan Seiberg relating non-abelian Yang-Mills theory (a generalization of Maxwell's theory of electromagnetics) and the simpler abelian Seiberg-Witten gauge field theories. In the physical world, quantum Yang-Mills theories provide a theoretical basis for describing elementary particle interactions. In the mathematical world, beginning with the work of Simon Donaldson in 1983, classical Yang-Mills and Seiberg-Witten gauge theories have played a fundamental role in probing the geometry of four-dimensional spaces. Stringtheory is often cited by physicists as the most promising candidate for a unification of quantum field theory (itself is a unification of the electromagnetic, weak, and strong forces) and Einstein's theory of gravity. Such a `grand unified theory' would give a single theoretical framework for explaining both short-range, high-energy elementary particle interactions and the long-range gravitational force. However, discovery of such a unified framework has eluded physicists for nearly three-quarters of a century. Furthermore, while Einstein's theory of gravity is founded on rigorous mathematics, the goal of providing a mathematical foundation for quantum field theory has not yet been realized. Most physicists agree that particle accelerators of impossibly large size would be needed to experimentally verify that string theory is the `right' theory of nature. However, while independent, mathematical verifications of quantumfield theory predictions can never replace experimental tests, they may increase our confidence that quantum field theory predictions are correct when experimental checks are impossible with current technology. We hope that our project may serve as a small step in such directions.

我们研究的主要目的是证明Edward Witten(1994)的著名公式，该公式将Donaldson和Seiberg-Witten级数的光滑四维流形联系起来。Witten使用量子场论方法推导出他著名的公式，该公式基于“对偶性”的概念，而我们与Thomas Lness共同追求的方法是一种数学方法，使用非阿贝尔单极的模空间作为Donaldson模空间和Seiberg-Witten模空间的链接之间的协边。我们已经用这种方法部分验证了Witten的公式，表明在许多情况下，级数的一致只取决于四维流形的拓扑结构的阶数项。我们的目标是在这些结果的基础上，接下来证明总体预期形状的两个系列之间存在关系，然后应用辅助技术来证明他的公式是唯一可能的。我们研究的第二个目标是在配备了Lefschetz纤颤的辛四流形的特殊情况下证明Witten公式，并阐明Seiberg-Witten不变量、Donaldson不变量和Lefschetz纤颤的结构之间的关系。最后，我们还希望解释推导Witten公式的数学和量子场论方法之间的关系。数学家们理解的Witten公式是Witten和Nathan Seiberg发展的物理理论的一部分，该理论将非阿贝尔杨-米尔斯理论(麦克斯韦电磁学理论的推广)和更简单的阿贝尔Seiberg-Witten规范场理论联系在一起。在物理世界中，量子杨-米尔斯理论为描述基本粒子相互作用提供了理论基础。在数学界，从Simon Donaldson在1983年的工作开始，经典的Yang-Mills和Seiberg-Witten规范理论在探索四维空间的几何方面发挥了基础作用。弦理论经常被物理学家认为是量子场论(本身就是电磁力、弱力和强力的统一)和爱因斯坦引力理论统一的最有希望的候选者。这样的“大统一理论”将为解释短程、高能基本粒子相互作用和长程引力提供一个单一的理论框架。然而，近四分之三个世纪以来，物理学家一直未能发现这样一个统一的框架。此外，尽管爱因斯坦的引力理论建立在严格的数学基础上，但为量子场论提供数学基础的目标尚未实现。大多数物理学家都同意，需要有大得令人难以置信的粒子加速器来从实验上证实弦理论是自然界的“正确”理论。然而，尽管对量子场理论预测的独立的数学验证永远不能取代实验测试，但它们可能会增加我们的信心，即在目前的技术无法进行实验验证的情况下，量子场理论预测是正确的。我们希望我们的项目可以成为朝着这些方向迈出的一小步。