Gauge theory and low-dimensional topology

我们研究的主要目标是证明Edward Witten（1994）的著名公式，该公式涉及光滑四流形的Donaldson和Seiberg-Witten系列。Witten使用量子场论方法推导出他著名的公式，基于“对偶性”的概念，而我们与Thomas philippe共同追求的方法是一种数学方法，使用非阿贝尔单极子的模空间作为Donaldson和Seiberg-Witten模空间之间的协点。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，级数通过仅依赖于四流形拓扑的顺序项而一致。我们的目标是在这些结果的基础上，通过展示两个整体期望形状系列之间存在的关系，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具有Lefschetz纤维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不变量与Lefschetz纤维结构之间的关系。最后，我们还希望解释推导威腾公式的数学和量子场论方法之间的关系。数学家们所理解的威腾公式是威腾和内森·塞伯格（Nathan Seiberg）发展的物理理论的一部分，该理论与非阿贝尔杨-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-威腾规范场理论有关。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。在数学世界，从1983年西蒙·唐纳森的工作开始，经典的杨-米尔斯和塞伯格-威顿规范理论在探索四维空间的几何方面发挥了重要作用。弦理论经常被物理学家引用为量子场论（本身是电磁、弱和强力的统一）和爱因斯坦引力理论的最有希望的统一候选者。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远距离引力提供一个单一的理论框架。然而，物理学家们在近四分之三个世纪的时间里都没有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。大多数物理学家都认为，要在实验上验证弦理论是“正确的”自然理论，就需要超大尺寸的粒子加速器。然而，虽然量子场论预测的独立数学验证永远不能取代实验测试，但它们可能会增加我们对量子场论预测是正确的信心，因为目前的技术无法进行实验验证。我们希望我们的项目可以成为朝着这个方向迈出的一小步。

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.