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

流形是局部类似于欧几里得空间的形状，但可能具有复杂的全局结构。四维流形（具有三个空间方向和一个时间方向）在广义相对论中用作宇宙模型。理论物理学家在弦理论模型中使用其他维度的流形，例如二维或六维，这可能会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学、几何学和理论物理中都很有用。 该项目中进行的工作将为超对称量子场论的预测提供罕见的数学证明。 2012 年欧洲核子研究中心发现的希格斯玻色子证实了标准模型，但大型强子对撞机实验尚未检测到超对称性。 然而，首席研究员的研究表明，至少超对称性的一些后果可以在数学上得到验证。 鉴于首席研究员的记录以及鼓励妇女和少数族裔从事数学职业并提供指导和培训的持续愿望，该项目的活动还应导致少数族裔，特别是非裔美国人和西班牙裔学生以及妇女更多地参与数学研究。 为了向更广泛的受众传播这项工作，主要研究人员将根据该项目撰写研究专着，并在 2015 年至 2018 年每年在罗格斯大学和佛罗里达国际大学组织会议，并在美国数学会全国会议上组织特别会议。 1994 年，Edward Witten 使用超对称量子场论推导了他著名的公式，该公式涉及具有可接受拓扑的闭合、有向、光滑四维流形的 Donaldson 和 Seiberg-Witten 不变量和简单类型。该项目的首要目标是采用基于非阿贝尔单极子模空间的严格数学方法来完成维滕公式的证明；主要研究人员已经完成了该计划的所有步骤，除了非阿贝尔单极子粘合定理的工作。他们还将使用非阿贝尔单极子定义四维流形的新不变量，研究高阶唐纳森不变量，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.