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世界的最佳实验验证模型。拓扑量子场论（TQFT）是一种相关函数与时空几何无关的量子场论。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计算的量称为量子不变量。它们提供了理解弯曲时空的有力工具——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。本课题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。项目更广泛的影响包括指导和拓展。PI将继续指导学生和博士后研究与项目相关的课题。PI还将组织会议，作为促进合作和扩大研究参与的平台。本课题的目的是研究三流形和四流形的非半简单量子不变量。此前，PI发起了利用三截面图和Hopf三元组构造4流形的kuperberg型不变量的程序。Hopf三元组，作为构造的代数输入，最初被假设为半简单的，最近被推广到非半简单的情况，但没有发现这种三元组的具体例子。该项目将继续发展该计划。首先，PI计划系统地搜索产生4流形不变量的非半简单Hopf三元组。除了对其性质进行理论研究外，还将对奇异流形进行大量计算以评估不变量的强度。其次，利用由Hopf超代数组成的Hopf三元组，改进了推导4流形复自旋结构的不变量的过程。第三，通过引入三角代数的概念，将该构造扩展到其完全的普遍性。在三维空间，该项目旨在基于弱Hopf代数扩展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利用几何拓扑技术从3流形构造模张量范畴的新程序。具体来说，PI将开发方法来产生目前无法实现的范畴的F-和r -矩阵。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。

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.