FRG: Collaborative Research: Categorifying Quantum Three-Manifold Invariants

量子拓扑学是数学的一个分支，它为引力的量子理论所需的结构提供了一个试验场。这一领域带来了数学和理论物理之间前所未有的互动。它非常成功，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（包括时间）中，一个完整的量子引力理论需要将这些工具扩展到四维。一种被称为“分类”的新兴数学哲学提供了一种揭示数学结构中隐藏层的途径，揭示了能够描述更复杂现象的更丰富，更强大的理论。 这个项目将使用范畴化的观点来将三维中最成功的理论之一提升为一个完整的四维理论。这个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。通过这种协调的努力，PI将取得实质性进展的问题上的分类3流形不变量。PI将利用理论物理学和更高表示理论的最新突破，为这个问题的重大进展创造新的可能性。 将采用的技术包括：五膜紧化，以提供一个通用的描述各种旧的和新的同调不变量的3流形，使用无穷大类别定义张量积的更高表示的量子群，和理论的Hopfological代数的消隐在根的单位，以及奇环同调理论和哈比罗普适不变量的简化的最新工作。

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.