Non-semisimple quantum invariants of three and four manifolds

量子拓扑学是经典拓扑学与量子物理学相互作用的产物。作为现代物理学的基础，量子场论仍然是真实的世界的最佳实验验证模型。拓扑量子场论（英语：Topological quantum field theory，TQFT）是一种关联函数不依赖于时空几何的量子场论。因此，它计算的量对时空的形状不敏感;这就是拓扑学的本质。从TQFT计算的量被称为量子不变量。它们为理解弯曲时空提供了强有力的工具--流形。在数学上，TQFT连接了几个学科，包括张量范畴，低维拓扑和量子计算。这个项目旨在研究三维和四维流形的量子不变量。PI将使用创新的方法来构建有趣的量子不变量。该项目更广泛的影响包括辅导和外联。PI将继续指导学生和博士后研究与该项目相关的主题。PI还将组织会议作为促进合作和扩大研究参与的平台。该项目的目标是研究三个和四个流形的非半单量子不变量。在此之前，PI发起了利用三分图和Hopf三元组构建4-流形的Kuperberg型不变量的计划。作为代数输入的霍普夫三联体最初被假设为半单的，最近被推广到非半单的情况，但还没有发现这种三联体的具体例子。该项目将继续开发该程序。首先，PI计划系统地搜索产生4-流形不变量的非半单Hopf三元组。除了对它的性质进行理论研究外，还将对奇异流形进行广泛的计算，以评估不变量的强度。其次，PI将改进程序，通过使用由Hopf超代数组成的Hopf三元组导出4-流形的复自旋结构的不变量。第三，通过引入三代数的概念，将该构造推广到其全部的一般性。在三维空间中，该项目的目标是在弱Hopf代数的基础上扩展Kuperberg不变量，从而提供各种三维量子不变量的统一。此外，PI将在数学上探索一个新的程序，使用几何拓扑学的技术从3-流形构建模张量类别。具体来说，PI将开发方法，以产生目前无法实现的类别的F-和R-矩阵。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Quantum topology emerged from the interaction of classical topology and quantum physics. As a foundation in modern physics, quantum field theory remains the best experimentally-verified model of the real world. A topological quantum field theory (TQFT) is a quantum field theory whose correlation functions do not depend on the geometries of the spacetime. Hence it computes quantities which are not sensitive to the shape of the spacetime; this is essentially what topology is about. The quantities computed from a TQFT are called quantum invariants. They provide powerful tools to understand curved spacetimes -- manifolds. Mathematically, TQFTs bridge several subjects including tensor categories, low dimensional topology, and quantum computing. This project aims to study quantum invariants of three- and four-dimensional manifolds. The PI will use innovative methods to construct interesting quantum invariants. The broader impacts of the project contain mentoring and outreach. The PI will continue mentoring students and postdocs to work on topics related to the project. The PI will also organize conferences as a platform to foster collaboration and broaden participation of research.The objective of the project is to investigate non-semisimple quantum invariants of three and four manifolds. Previously, the PI initiated the program of constructing Kuperberg-type invariants of 4-manifolds utilizing trisection diagrams and Hopf triplets. The Hopf triplets, serving as the algebraic input to the construction, were initially assumed to be semisimple and this was recently generalized to the non-semisimple case, but specific examples of such triplets have not been found. This project will continue the development of the program. Firstly, the PI plans to systematically search for non-semisimple Hopf triplets that produce invariants of 4-manifolds. Besides theoretical studies of its properties, extensive computations will be conducted on exotic manifolds to assess the strength of the invariant. Secondly, the PI will refine the procedure to derive an invariant of complex spin structures of 4-manifolds by using Hopf triplets consisting of Hopf superalgebras. Thirdly, the construction will be extended to its full generality by introducing the notion of trialgebras. In dimension three, the project aims to extend the Kuperberg invariant on the basis of weak Hopf algebras, and thus provides a unification of various 3-dimensional quantum invariants. Additionally, the PI will mathematically explore a novel program of constructing modular tensor categories from 3-manifolds using techniques from geometric topology. Specifically, the PI will develop methods to produce the F- and R-matrices of the category which are currently not achievable.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.