FRG: Collaborative Research: Homotopy Renormalization of Topological Field Theories

受物理学启发的数学已经成功地为现代物理学中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造新的代数和几何工具，帮助以精确的数学方式阐述量子物理的思想和方法。具体地说，该团队将结合其成员过去的经验和成果，为具有额外结构的流形构建新的拓扑量子场理论。受Witten的Chern-Simons理论和现在称为Witten-Reshetikhin-Turaev不变量的三维流形的不变量的启发，主要研究人员将探索他们的新的场论，目的是为由此产生的流形的拓扑不变量找到物理定义。这项工作受到几个基本例子的启发，并将为代数、拓扑、几何、数学物理和相关数学领域的新研究途径打开大门。该项目的更广泛影响主要分为两大类：辅导和外联。研究团队的成员目前总共指导了12名博士生。他们将就与助学金主要目标有关的项目向研究生提供建议。外展部分是组织几个研讨会和会议，旨在促进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)，这是Schwartz，Witten和Atiyah研究的拓扑型量子场论(TQFT)的推广。HQFT生成具有额外数据的流形的拓扑不变量，这些额外数据将补充的拓扑/几何添加到TQFT的上下文中。量子不变量和HQFT的大多数理论都涉及到具有诸如半单的某些附加性质的么半范畴。在2005年开始的各种合作中，团队成员发展了一种重新归一化量子不变量理论，该理论从非半简单范畴派生出非平凡的拓扑不变量。在这个项目中，主要研究人员将在无限制量子群理论的基础上，进一步利用重整化来发展非平凡的HQFT。这些新的HQFT应该具有重新归一化的量子不变量的强度和新的特征。主要研究人员将进一步寻找这些新不变量的物理解释。

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.