FRG: Collaborative Research: Categorifying Quantum Three-Manifold Invariants

量子拓扑学是数学的一个分支，它为量子引力理论所需的结构提供了一个试验场。这一领域带来了数学与理论物理之间前所未有的相互作用。它已经非常成功，并且在三维空间中得到了很好的研究。然而，由于我们生活在四维空间（包括时间），一个完整的量子引力理论需要将这些工具扩展到四维空间。一种被称为“分类”的新兴数学哲学为揭示数学结构中的隐藏层提供了一条途径，揭示了能够描述更复杂现象的更丰富、更健壮的理论。本项目将运用分类的视角，将最成功的三维理论之一提升为完整的四维理论。这次合作将利用低维几何、表征理论和高维规范理论之间的相互作用。通过这种协调的努力，pi将在三流形不变量的分类问题上取得实质性进展。pi将利用理论物理学和高级表示理论的最新突破，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。其中所采用的技术包括：五膜紧化提供了3流形的各种新旧同调不变量的通用描述，使用无穷范畴来定义量子群的高级表示的张量积，以及在单位根处的分类的Hopfological代数理论，以及最近关于奇链同调理论和Habiro的通用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.