Gauge theory and low-dimensional topology

DMS-0125170 Paul M. N.本文的主要目的是证明Edward维滕（1994）关于光滑四维流形的唐纳森和Seiberg-维滕级数的著名公式。维滕使用量子场论的方法来推导出他的著名公式，其基础是“对偶性”的概念，而我们一直在与托马斯·勒内斯共同追求的方法是一种数学方法，采用非阿贝尔单极的模空间作为唐纳森和塞伯格-维滕模空间之间的协边。我们已经用这种方法部分地验证了维滕的公式，表明在许多情况下级数通过仅依赖于四维流形拓扑的阶项而一致。我们的目标是建立在这些结果的基础上，接下来通过展示整体预期形状的两个系列之间的关系存在，然后应用辅助技术来证明他的公式是唯一可能的。 我们的第二个研究目标是证明维滕公式，在特殊的情况下，辛四流形配备Lefschetz纤维，并阐明之间的关系Seiberg-维滕不变量，唐纳森不变量，和结构的Lefschetz纤维。最后，我们还希望解释推导维滕公式的数学方法和量子场论方法之间的关系。维滕公式，正如数学家所理解的那样，是由维滕和内森·塞伯格发展的物理理论的一部分，涉及非阿贝尔杨-米尔斯理论（麦克斯韦电磁学理论的推广）和更简单的阿贝尔塞伯格-维滕规范场理论。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。在数学领域，从1983年西蒙·唐纳森的工作开始，经典的杨-米尔斯规范理论和塞伯格-威滕规范理论在探索四维空间的几何学中发挥了重要作用。弦论经常被物理学家引用为统一量子场论（本身是电磁力、弱力和强力的统一）和爱因斯坦引力理论的最有希望的候选者。这样一个“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，这样一个统一的框架的发现已经困扰了物理学家将近四分之三的世纪。此外，虽然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，要用实验来证实弦论是“正确的”自然理论，就需要有大得不可能的粒子加速器。然而，虽然量子场论预测的独立数学验证永远无法取代实验测试，但它们可能会增加我们的信心，即量子场论预测是正确的，而实验检查是不可能的。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。

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.