Gauge Theory and Invariants of Symplectic Manifolds

这个项目的第一个目标是研究一类被称为广义Seiberg-Witten方程的方程。这些鲜为人知的方程属于被称为规范理论的数学领域，它起源于物理学，描述了规范场的动力学，如电磁场和其他携带基本自然力的场。规范场在纯数学中的重要性源于它们与流形几何的关系。这就引出了这个项目的第二个目标，即应用广义Seiberg-Witten方程的理论来寻找区分不同流形的新方法。主要研究员（PI）特别感兴趣的应用程序躺在不同的研究领域的交叉点，如代数几何（研究方程描述的流形），辛几何（研究与经典力学相关的流形），拓扑学（研究变形下保持不变的流形的性质）和弦理论（现代理论物理学的分支）。此外，该项目的目标是向广大受众，包括研究生和本科生，以及其他领域的研究人员，介绍规范理论和几何的一些想法和技术。作为该项目的一部分，PI将为学生组织研讨会和小型课程，并针对非专业人员编写规范理论的简要说明。该项目将发展三维和四维流形上广义Seiberg-Witten方程研究的分析基础。这类方程的例子包括Vafa-Witten和Kapustin-Witten方程，它们有望导致低维流形和纽结的新拓扑不变量，以及ADHM Seiberg-Witten方程，它们在定义具有特殊完整性的高维黎曼流形的拓扑不变量方面发挥重要作用。近年来，在Taubes的开创性工作之后，这些方程的紧性问题有了很大的进展。该项目的第一个目标是解决描述边界附近解的模空间的匡威问题。这将涉及到理解奇异的解决方案，富特方程，一个非线性推广的狄拉克方程，这似乎是极限的重名广义塞伯格-威滕方程。特别是，目前知之甚少的变形理论和胶合结构等奇异部分，并在这一方向取得进展将涉及开发新的分析工具，研究奇异的椭圆型微分方程的解。PI将应用这个一般理论来定义卡-丘三重的Pandharipande-Thomas不变量的辛模拟。不变量计数嵌入的伪全纯曲线，其权重使用ADHM Seiberg-Witten方程的模空间定义。Pandharipande-Thomas不变量的辛解释可能会为Maulik-Nekrasov-Okounkov-Pandharipande猜想提供新的线索，该猜想仍然是Calabi-Yau三重枚举几何的主要开放问题之一。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

The first objective of this project is to study a class of equations known as generalized Seiberg-Witten equations. These little understood equations belong to the area of mathematics known as gauge theory, which originates from physics and describes the dynamics of gauge fields, such as the electromagnetic field and other fields carrying fundamental forces of nature. The importance of gauge fields in pure mathematics stems from their relationship to the geometry of shapes known as manifolds. This leads to the second objective of this project, which is to apply the theory of generalized Seiberg-Witten equations to find new ways of distinguishing different manifolds from one another. The Principal Investigator (PI) is specifically interested in applications lying at the intersection of different areas of research, such as algebraic geometry (which studies manifolds described by equations), symplectic geometry (which studies manifolds related to classical mechanics), topology (which studies properties of manifolds which remain unchanged under deformations), and string theory (a branch of modern theoretical physics). In addition, the project's goal is to expose a broad audience, including graduate and undergraduate students, as well as researchers in other areas, to some of the ideas and techniques of gauge theory and geometry. As part of the project, the PI will organize seminars and minicourses for students and write expository notes on gauge theory aimed at non-specialists.This project will develop analytic foundations in the study of generalized Seiberg-Witten equations on three- and four-dimensional manifolds. Examples of such equations include the Vafa–Witten and Kapustin–Witten equations, which are expected to lead to new topological invariants of low-dimensional manifolds and knots, and the ADHM Seiberg–Witten equations which play an important role in defining conjectural invariants of higher-dimensional Riemannian manifolds with special holonomy. In recent years, there has been a lot of progress on the compactness problem for these equations, following groundbreaking work of Taubes. The first goal of the project is to solve the converse problem of describing the moduli spaces of solutions near the boundary. This will involve understanding singular solutions to the Fueter equation, a nonlinear generalization of the Dirac equation, which appears as the limit of rescallings of generalized Seiberg–Witten equations. In particular, little is at present known about deformation theory and gluing constructions for such singular sections, and making progress in this direction will involve developing new analytical tools for studying singular solutions of elliptic differential equations. The PI will apply this general theory to define a symplectic analog of the Pandharipande–Thomas invariants of Calabi–Yau threefolds. The invariant counts embedded pseudo-holomorphic curves with weights defined using moduli spaces of the ADHM Seiberg–Witten equations. A symplectic interpretation of the Pandharipande–Thomas invariant is likely to shed a new light on the Maulik–Nekrasov–Okounkov–Pandharipande conjecture, which remains one of the major open problems of enumerative geometry of Calabi–Yau threefolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.