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.

20 世纪 80 年代，数学从物理学中获得了令人难以置信的大量思想，这些思想具有巨大的数学兴趣和应用。拓扑学是一个数学领域，研究在拉伸、皱缩和弯曲等变形（但不包括撕裂或粘合）下保留的形状的属性。量子场论仅依赖于空间的拓扑性质，而不依赖于长度和角度等几何性质，成为一个数学领域，称为拓扑场论（TFT）。该项目利用过去十年才完全开发出来的新技术，对第三、第四和更高维度的 TFT 进行了多方面的研究。所考虑的问题具有基础性，这些问题的答案预计将带来新技术和不同数学领域之间的相互作用，并在物理学中具有潜在的应用。研究生是该研究项目的重要组成部分；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群体传播研究结果是该项目更广泛的关键影响。PI 的项目最初将重点关注 TFT 与光滑流形分类直至稳定微分同胚相关的最新进展。将特别强调 4 维方面并获得区分同伦等效流形的 TFT 的新示例。另一个目标是构建非半简单 TFT。与目前已知的例子不同，它们有可能区分奇异的光滑结构。该计划的长期目标是证明相对缠结假设。这是卢里证明共配假设的主要缺失部分，填补了证明这一重要结果的一个重大空白。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。