FRG: Collaborative Research: Homotopy Renormalization of Topological Field Theories

FRG：协作研究：拓扑场论的同伦重正化

• 批准号: 1664454
• 负责人: Tudor Dan Dimofte
• 金额: $ 16.68万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1664454&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Homotopy; Renormalization

Mathematics inspired by physics has successfully provided the background and the language for most sophisticated areas of modern physics. In this project the principal investigators aim to create new algebraic and geometric tools helping to formulate ideas and methods of quantum physics in a precise mathematical way. Specifically, the team will combine its members' past experience and achievements to construct new Topological Quantum Field Theories for manifolds with additional structures. Inspired by Witten's Chern-Simons theory and invariants of 3-manifolds that are known now as Witten-Reshetikhin-Turaev invariants, the principal investigators will explore their new Field Theories with the objective to find physical definitions for the resulting topological invariants of manifolds. This work is stimulated by several fundamental examples and will open the door to new research avenues in algebra, topology, geometry, mathematical physics, and related areas of mathematics. The broader impacts of the project belong to two main categories: mentoring and outreach. The members of the research team are currently mentoring a total of twelve PhD students. They will advise graduate students on projects related to the main objectives of the grant. The outreach component is to organize several workshops and conferences aimed at developing communication and collaborative research between participants of the project, establishing scientific connections with other mathematicians, as well as fostering broader applications of this work.In 1999, Turaev introduced Homotopy Quantum Field Theories (HQFTs), which are generalizations of Topological Quantum Field Theories (TQFTs) studied by Schwartz, Witten, and Atiyah. HQFTs produce topological invariants of manifolds furnished with extra data that add supplementary topology/geometry to the context of TQFTs. Most of the theory of quantum invariants and HQFTs involves monoidal categories which have certain additional properties like being semi-simple. In various collaborations started in 2005, the team members developed a theory of re-normalized Quantum Invariants that derives non-trivial topological invariants from non-semi-simple categories. In this project the principal investigators will further use renormalization to develop non-trivial HQFTs, based on the examples coming from the theory of unrestricted quantum group. These new HQFTs should share the strength and the new features of the re-normalized Quantum Invariants. The principal investigators will further search for a physical interpretation of these new invariants.

受物理学启发的数学成功地为现代物理学最复杂的领域提供了背景和语言。 在这个项目中，主要研究人员的目标是创建新的代数和几何工具，帮助以精确的数学方式制定量子物理学的思想和方法。 具体来说，该团队将结合其成员过去的经验和成就，为具有附加结构的流形构建新的拓扑量子场论。受到 Witten 的 Chern-Simons 理论和 3 流形不变量（现在称为 Witten-Reshetikhin-Turaev 不变量）的启发，主要研究人员将探索他们的新场论，目标是找到由此产生的流形拓扑不变量的物理定义。 这项工作受到几个基本例子的启发，将为代数、拓扑、几何、数学物理和数学相关领域的新研究途径打开大门。该项目更广泛的影响分为两个主要类别：指导和推广。 研究团队成员目前正在指导总共十二名博士生。 他们将为研究生提供与补助金主要目标相关的项目建议。 外展部分是组织一些研讨会和会议，旨在发展项目参与者之间的交流和合作研究，与其他数学家建立科学联系，以及促进这项工作的更广泛应用。 1999 年，图拉耶夫引入了同伦量子场论 (HQFT)，它是 Schwartz、Witten 和 阿蒂亚。 HQFT 产生流形的拓扑不变量，并提供额外的数据，将补充拓扑/几何添加到 TQFT 的上下文中。 大多数量子不变量和 HQFT 理论都涉及幺半群范畴，这些范畴具有某些附加属性，例如半简单性。 在 2005 年开始的各种合作中，团队成员开发了一种重归一化量子不变量理论，该理论从非半简单类别中导出非平凡的拓扑不变量。 在这个项目中，主要研究人员将基于无限制量子群理论的例子，进一步使用重整化来开发非平凡的 HQFT。 这些新的 HQFT 应该具有重归一化量子不变量的优势和新特征。 主要研究人员将进一步寻找这些新不变量的物理解释。