Topological Quantum Field Theory

拓扑量子场论

• 批准号: 2204297
• 负责人: Christopher Schommer-Pries
• 金额: $ 25.17万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204297&HistoricalAwards=false
• 关键词: Topological; Quantum; Field; Theory

In the 1980s, mathematics received an incredible influx of ideas coming from physics that had tremendous mathematical interest and applications. Topology is a field of mathematics that studies properties of shapes which are preserved under deformations such as stretching, crumpling, and bending, but not tearing or gluing. Quantum field theories that only depend on topological properties of space, and not geometric ones such as length and angle, became a field of mathematics, known as topological field theory (TFT). This project constitutes a multifaceted investigation of TFTs in dimensions three, four, and higher, using new techniques which have only been fully developed in the past decade. The problems considered are of a foundational nature, and answers to these are expected to lead to new techniques and interactions among different fields of mathematics and have potential applications in physics. Graduate students are an important part of this research program; graduate education, mentoring as well as dissemination of the results to the broader community of scientists via lectures and workshops is a key broader impact of the project.The PI's program will initially focus on recent advances relating TFT to the classification of smooth manifolds up to stable diffeomorphism. Special emphasis will be placed on 4-dimensional aspects and obtaining new examples of TFTs that distinguish homotopy equivalent manifolds. Another goal is to construct non-semisimple TFTs. Unlike the currently known examples, these have the potential to distinguish exotic smooth structures. A long-term goal of the program is to give a proof of the relative tangle hypothesis. This is the main missing part of Lurie’s proof of the cobordism hypothesis and fills a major gap in the proof of this important result.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.

在1980年代，数学从物理学产生了极大的数学兴趣和应用。拓扑是数学领域，研究形状的特性，这些特性保存在诸如伸展，巡航和弯曲之类的变形下，但不撕裂或粘合。仅取决于空间的拓扑特性而不是几何学（例如长度和角度）的量子场理论成为数学领域，称为拓扑场理论（TFT）。该项目构成了在过去十年中仅完全开发的新技术，构成了TFT在三，四和更高方面的多方面投资。所考虑的问题具有基本的性质，对这些问题的答案有望导致不同数学领域之间的新技术和相互作用，并在物理学中具有潜在的应用。研究生是该研究计划的重要组成部分；研究生教育，指导以及通过讲座和研讨会向更广泛的科学家社区传播结果是该项目的关键影响。特别重点将放在4维方面，并获得区分同等流形的TFT的新示例。另一个目标是构建非偏simimple tfts。与当前已知的示例不同，这些示例具有区分外来平滑结构的潜力。该计划的长期目标是提供相对纠缠假设的证明。这是Lurie证明COBORDISM假设的主要缺失部分，并填补了这一重要结果证明的主要空白。该奖项反映了NSF的法定任务，并通过评估该基金会的知识分子优点和更广泛的影响来审查标准。