FRG: Collaborative Research: Categorifying Quantum Three-Manifold Invariants

量子拓扑学是数学的一个分支，它为量子引力理论中所需的结构提供了试验场。这一领域带来了数学与理论物理之间前所未有的互动。它已经非常成功，并在3个维度上得到了很好的研究。然而，由于我们生活在4维(包括时间)中，量子引力的完整理论需要将这些工具扩展到4维。一种被称为“分类”的新兴数学哲学提供了一条揭示数学结构中隐藏的层面的途径，揭示了能够描述更复杂现象的更丰富、更可靠的理论。这个项目将使用分类的观点，将三维中最成功的理论之一提升为完整的四维理论。这一合作将利用低维几何、表示理论和高维规范理论之间的相互作用。通过这种协调努力，PI将在对3-流形不变量进行分类的问题上取得实质性进展。PI将利用最近在理论物理和更高表示理论方面的突破，这些突破为在这个问题上取得重大进展创造了新的可能性。其中将使用的技术包括：提供对3-流形的各种新旧同调不变量的通用描述的五层紧凑化，使用无限范畴来定义量子群的更高表示的张量积，用于单位根分类的Hopfology代数理论，以及最近关于奇链同调理论和Habiro万能不变量的分类的工作。

Quantum topology is a branch of mathematics that provides a testing ground for the structures needed in a quantum theory of gravity. This field has brought about unprecedented interaction between mathematics and theoretical physics. It has been extremely successful and well studied in 3-dimensions. However, since we live in 4-dimensions (including time), a full theory of quantum gravity requires an extension of these tools to 4-dimensions. An emerging mathematical philosophy known as "categorification" provides an avenue to uncover a hidden layer in mathematical structures, revealing a richer and more robust theory capable of describing more complex phenomenon. This project will use the perspective of categorification to enhance one of the most successful theories in 3-dimensions to a full 4-dimensional theory.This collaboration will harness the interplay between low-dimensional geometry, representation theory, and higher-dimensional gauge theory. Through this coordinated effort the PIs will make substantial progress on the problem of categorifying 3-manifold invariants. The PIs will capitalize on recent breakthroughs in theoretical physics and higher representation theory that have created new possibilities for significant progress on this problem. Among the techniques to be employed include: fivebrane compactifications to provide a universal description of various old and new homological invariants of 3-manifolds, the use of infinity categories for defining tensor products of higher representations of quantum groups, and the theory of Hopfological algebra for categorifications at roots of unity, as well as recent work on odd link homology theory and categorifications of Habiro's universal invariant.