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.

受物理启发的数学已经成功地为现代物理学的大多数复杂领域提供了背景和语言。 在这个项目中，主要研究人员旨在创建新的代数和几何工具，以精确的数学方式帮助制定量子物理学的思想和方法。 具体而言，团队将结合其成员的过去经验和成就，以构建具有其他结构的歧管的新拓扑量子场理论。受Witten的Chern-Simons理论和3个manifolds的不变式的启发，这些理论现已被称为Witten-Reshetikhin-Turaev不变式，主要研究人员将探索其新的领域理论，目的是为产生的股票拓扑拓扑室找到物理定义。 这项工作受到了几个基本例子的刺激，将为代数，拓扑，几何，数学物理学以及数学相关领域的新研究途径打开大门。该项目的更广泛影响属于两个主要类别：指导和宣传。 研究小组的成员目前正在指导十二名博士学位学生。 他们将向研究生提供与赠款主要目标有关的项目。 该外展部分是组织几个旨在在项目参与者之间建立沟通和协作研究的研讨会和会议，建立了与其他数学家建立科学联系，并促进了这项工作的更广泛的应用。在1999年，Turaev引入了同质量子量子量化的量子量化理论（HQFTS），这是对TOPOLICE量子量化的综合性（TQFTS）（TQFTS）（TQFTS）（TQFTS）（TQ）（TQ），TQ综合了TQ，TQ）和TQ综合综合综合综合综合症（TQ），综合综合症，综合综合症，综合综合症，综合综合综合型，综合了众多的领域，综合了众多的综合量。阿蒂亚。 HQFTS产生的流形的拓扑不变性，并配备了额外的数据，可在TQFT的背景下添加补充拓扑/几何形状。 大多数量子不变性和HQFT的理论都涉及单素类别，这些类别具有某些其他特性，例如半简单。 在2005年始于2005年的各种合作中，团队成员开发了一种重新归一化的量子不变性理论，该理论从非避免类别的类别中得出了非平凡的拓扑不变性。 在该项目中，主要研究人员将根据不受限制的量子组理论的示例进一步使用重新归化来开发非平凡的HQFT。 这些新的HQFT应该共享重新归一化量子不变的力量和新功能。 主要研究人员将进一步寻找对这些新不变的物理解释。