Collaborative Research: Instantons, Monopoles, and Relations among their invariants

流形是局部类似于欧几里得空间但可能具有复杂全局结构的形状。四维流形（具有三个空间方向和一个时间方向）在广义相对论中被用作宇宙的模型。其他维度的流形，如二维或六维，被理论物理学家用于弦理论模型，这可能导致量子场论和引力的统一。由于理论物理和应用数学中出现的许多方程的解集都是流形，因此区分流形的能力在整个数学科学以及拓扑、几何和理论物理中都很有用。在这个项目中进行的工作将导致超对称量子场论预测的罕见数学证明。2012年欧洲核子研究中心对希格斯玻色子的发现证实了标准模型，但大型强子对撞机实验尚未发现超对称。然而，首席研究人员的研究表明，至少超对称的一些结果可以在数学上得到验证。鉴于首席研究人员的记录和鼓励妇女和少数民族从事数学事业并提供指导和培训的持续愿望，本项目中的活动还应导致少数民族，特别是非洲裔和西班牙裔学生，以及妇女更多地参与数学研究。为了将这项工作传达给更广泛的受众，主要研究者将根据该项目撰写研究专著，并从2015年到2018年每年在罗格斯大学和佛罗里达国际大学组织会议，并在美国数学学会会议上组织特别会议。1994年，Edward Witten利用超对称量子场论，导出了具有可容许拓扑和简单型的封闭、定向、光滑四维流形的Donaldson和Seiberg-Witten不变量的著名公式。本项目的第一个目标是完成Witten公式的证明，采用基于非阿贝尔单极子模空间的数学严格方法；主要研究人员已经完成了这个项目的所有步骤，除了他们对非阿贝尔单极子胶合定理的研究。他们还将使用非阿贝耳单极子定义四维流形的新不变量，研究高阶Donaldson不变量，并推导封闭三维流形的瞬子和Seiberg-Witten Floer同调之间的关系。

Manifolds are shapes which locally resemble Euclidean space but may have complicated global structure. Four-dimensional manifolds (with three spatial and one temporal direction) are used in General Relativity as models for the Universe. Manifolds of other dimensions, such as two or six, are used by theoretical physicists in String Theory models which may lead to a unification of Quantum Field Theory and Gravity. Because the solution sets of many equations arising in theoretical physics and applied mathematics are manifolds, the ability to distinguish between manifolds is useful throughout the mathematical sciences, as well as in Topology, Geometry, and Theoretical Physics. The work undertaken in this project will lead to a rare mathematical proof of a prediction from supersymmetric quantum field theory. The discovery in 2012 at CERN of the Higgs boson confirms the Standard Model, but supersymmetry has not yet been detected by the Large Hadron Collider experiments. However, the Principal Investigators' research shows that at least some consequences of supersymmetry can be mathematically verified. The activities in this project should also lead to greater involvement of minorities, especially African-American and Hispanic students, and women in mathematics research, given the Principal Investigators' record and continuing desire to encourage women and minorities to pursue careers in mathematics and to provide mentorship and training. To communicate this work to a wider audience, the Principal Investigators will write a research monograph based on this project, and organize conferences at Rutgers University and Florida International University and special sessions at national American Mathematical Society meetings each year from 2015 through 2018. In 1994, using supersymmetric quantum field theory, Edward Witten derived his celebrated formula relating Donaldson and Seiberg-Witten invariants of a closed, oriented, smooth four-dimensional manifold with admissible topology and simple type. The first goal of this project is to complete the proof of Witten's formula, employing a mathematically rigorous method based on moduli spaces of non-Abelian monopoles; the Principal Investigators have completed all steps of this program except their work on the gluing theorem for non-Abelian monopoles. They will also use non-Abelian monopoles to define new invariants of four-dimensional manifolds, investigate higher- rank Donaldson invariants, and derive relations between the instanton and Seiberg-Witten Floer homologies of closed three-dimensional manifolds.