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FRG: Collaborative Research: Categorifying Quantum Three-Manifold Invariants

FRG：合作研究：量子三流形不变量的分类

基本信息

批准号：
1664240
负责人：
Aaron Lauda
金额：
$ 14.74万
依托单位：
University of Southern California
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1664240&HistoricalAwards=false
关键词：
FRG Collaborative Research Categorifying Quantum

项目摘要

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.

量子拓扑是数学的一个分支，为量子引力理论所需的结构提供了试验场。这个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，并且在 3 维方面得到了充分的研究。然而，由于我们生活在 4 维（包括时间）中，完整的量子引力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、更稳健的理论。该项目将使用分类的角度将 3 维中最成功的理论之一增强为完整的 4 维理论。这种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。通过这种协调一致的努力，PI 将在 3 流形不变量的分类问题上取得实质性进展。 PI 将利用理论物理和更高表示理论的最新突破，为解决这一问题创造重大进展。所采用的技术包括：五膜紧化提供对 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更高表示的张量积，以及用于单位根分类的 Hopfological 代数理论，以及最近关于奇链接同调理论和 Habiro 普遍不变量分类的工作。