FRG: Collaborative Research: Homotopy Renormalization of Topological Field Theories

受物理学启发的数学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。 具体而言，该团队将结合联合收割机成员过去的经验和成就，为具有额外结构的流形构建新的拓扑量子场论。受维滕的陈-西蒙斯理论和现在被称为维滕-雷舍季欣-图拉耶夫不变量的3-流形不变量的启发，主要研究人员将探索他们的新场论，目的是找到流形的拓扑不变量的物理定义。 这项工作是刺激了几个基本的例子，并将打开大门，以新的研究途径，代数，拓扑，几何，数学物理和相关领域的数学。该项目的更广泛影响属于两大类：辅导和外联。 研究团队成员目前正在指导总共12名博士生。 他们将就与赠款主要目标有关的项目向研究生提供建议。 推广部分是组织几次讲习班和会议，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，Turaev介绍了同伦量子场论（HQFTs），这是Schwartz，维滕，阿蒂亚 HQFT产生的拓扑不变量的流形提供额外的数据，添加补充拓扑/几何的上下文中的TQFT。 大多数量子不变量和HQFT的理论都涉及到具有某些额外性质的monoidal范畴，例如半单。 在2005年开始的各种合作中， 团队成员开发了一种重新规范化的量子不变量理论，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，主要研究人员将进一步使用重整化来开发非平凡的HQFT，基于来自非限制量子群理论的例子。 这些新的HQFT应该共享重新归一化的量子不变量的强度和新特征。 主要研究人员将进一步寻找这些新不变量的物理解释。

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.