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

合作研究：瞬时子、磁单极子及其不变量之间的关系

• 批准号: 1510063
• 负责人: Thomas Leness
• 金额: $ 13.66万
• 依托单位: Florida International University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1510063&HistoricalAwards=false
• 关键词: Collaborative; Research; Instantons; Monopoles; Relations

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.

流形是局部类似于欧几里得空间但可能具有复杂的全局结构的形状。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。理论物理学家在弦理论模型中使用其他维度的流形，例如二维或六维流形，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学、几何学和理论物理中都很有用。该项目所做的工作将为超对称量子场论的预言提供难得的数学证明。2012年在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发现超对称性。然而，首席调查者的研究表明，超对称性的一些结果至少可以从数学上得到验证。鉴于首席调查员的记录以及鼓励妇女和少数群体从事数学事业并提供指导和培训的愿望，该项目的活动还应促使少数群体，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，首席调查人员将根据这一项目撰写一本研究专著，并在罗格斯大学和佛罗里达国际大学组织会议，并在2015年至2018年每年举行的美国数学学会全国会议上举办特别会议。1994年，Edward Witten利用超对称量子场论，导出了闭合的、定向的、光滑的、具有允许拓扑和简单类型的四维流形的Donaldson和Seiberg-Witten不变量的公式。这个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的模空间的严格的数学方法；主要研究人员已经完成了这个程序的所有步骤，除了他们在非阿贝尔单极子的胶合定理上的工作。他们还将使用非阿贝尔单极子来定义四维流形的新不变量，研究高阶Donaldson不变量，并推导出闭三维流形的瞬子同调和Seiberg-Witten Floer同调之间的关系。