Quantum Topology beyond Semi-Simplicity

受物理学的启发，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。该项目旨在创造新的代数和几何工具，以精确的数学方式制定量子物理的思想和方法。具体来说，与李代数相关的量子群理论在低维拓扑，特别是量子不变量的创建中得到了广泛而富有成效的应用。在这种背景下，首席研究员（PI）和他的合作者提供了新的系统策略来定义由非半简单类别产生的再标准化量子不变量。这项拨款的重点是为来自半简单和非半简单类别的许多量子不变量开发一个框架。这种框架的独特性质为代数、拓扑、几何、数学物理和相关数学领域的新研究途径打开了大门。这项奖助金更广泛的影响涉及两大类：辅导和外展。在整个项目中，PI将为研究生和博士后提供与基金主要目标相关的项目建议。PI已经共同组织了许多会议和研讨会，并将继续这样的扩展，以发展与其他数学家的交流和合作研究，以及促进本基金工作的更广泛应用。这项拨款考虑了非半简单类别产生的四种不变量：Hennings， Kuperberg， Reshetikhin-Turaev再标准化量子不变量（RQIs）和Turaev-Viro RQIs。所有这些不变量都具有独特的特性和优势。特别是，Hennings和Kuperberg不变量是使用Hopf代数的结构定义的（因此有许多例子），但有某些消失，不允许扩展到完整的tqft；rqi非常强大，它们的定义是基于表示理论的，这使得它们的存在对于一些重要的例子来说是难以捉摸的。该项目将推广和重新规范化Hennings和Kuperberg不变量，并表明新的不变量导致具有附加数据的tqft（克服消失障碍）。此外，PI将构建一个将Hennings和Kuperberg型不变量与先前定义的Reshetikhin-Turaev和Turaev-Viro rqi联系起来的一般框架，允许所有四个不变量的强度统一。PI将使用这个框架来推进和绘制与复杂（超）群和Levin-Wen模型的chen - simons理论的联系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Inspired by physics, mathematics has successfully provided the background and language for most of the sophisticated areas of modern physics. This project aims to create new algebraic and geometric tools to formulate ideas and methods of quantum physics in a precise mathematical way. Specifically, the theory of quantum groups associated with Lie algebras has been widely and productively used in low-dimensional topology, and in particular, with the creation of quantum invariants. Within this context, the Principal Investigator (PI) and his collaborators have offered new systematic strategies to define re-normalized quantum invariants arising from non-semi-simple categories. The focus of this grant is to develop a framework for many of the quantum invariants coming from both semi-simple and non-semi-simple categories. The unique properties of such a framework opens the door to new research avenues in algebra, topology, geometry, mathematical physics, and related areas of mathematics. The broader impacts of this grant address two main categories: mentoring and outreach. Throughout the project, the PI will advise graduate students and postdocs on projects related to the main objectives of the grant. The PI has co-organized many conferences and a workshop and will continue such outreach with the aim of developing communication and collaborative research with other mathematicians, as well as fostering broader applications of the work of this grant.This grant considers four types of invariants arising from non-semi-simple categories: Hennings, Kuperberg, Reshetikhin-Turaev re-normalized Quantum Invariants (RQIs) and Turaev-Viro RQIs. All of these invariants have unique features and strengths. In particular, Hennings and Kuperberg invariants are defined using the structure of Hopf algebras (and thus have many examples) but have certain vanishings which do not allow extensions to full TQFTs; the RQIs are quite powerful and their definitions are based on representation theory making their existence for some significant examples elusive. This project will generalize and re-normalize both Hennings and Kuperberg invariants and show that the new invariants lead to TQFTs with additional data (overcoming the vanishing obstruction). Furthermore, the PI will construct a general framework which relates the Hennings and Kuperberg type invariants to the previously defined Reshetikhin-Turaev and Turaev-Viro RQIs, allowing a unification of the strengths of all four invariants. The PI will use this framework to advance and draw connections with Chern-Simons theory with complex (super) groups and Levin-Wen models.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.