FRG: Collaborative Research: Categorifying Quantum Three-Manifold Invariants

FRG：合作研究：量子三流形不变量的分类

• 批准号: 1664227
• 负责人: Sergei Gukov
• 金额: $ 21万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1664227&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Categorifying; Quantum

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，它为重力理论中所需的结构提供了测试地面。该领域带来了数学与理论物理学之间空前的相互作用。它非常成功，并且在三维方面进行了很好的研究。但是，由于我们生活在四维（包括时间）中，因此量子重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”的新兴数学哲学为揭示了数学结构中隐藏层的途径，揭示了一种更丰富，更坚固的理论，能够描述更复杂的现象。 该项目将利用分类的观点来增强三维中最成功的理论之一，以实现完整的4维理论。此协作将利用低维几何形状，表示理论和较高维度计的理论之间的相互作用。通过这项协调的努力，PI将在分类3个manifold不变性的问题上取得重大进展。 PI将利用理论物理学的最新突破和更高的代表理论，这些理论为这一问题带来了重大进展的新可能性。 在要采用的技术中包括：五脑压缩，以提供三个三个manifolds的各种新旧和新的同源性不变的普遍描述，使用无限类别来定义量子群的较高表示的张量产品，以及Hopfological代数理论在Unity的依据和奇数链接中的跨性别链接和分类的HABRIANT和ODD TROCERIANT的分类。

项目成果

VOA[ M 4 ]

美国之音[ M 4 ]

• DOI: 10.1063/1.5100059
• 发表时间: 2020
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Feigin, Boris;Gukov, Sergei
• 通讯作者: Gukov, Sergei

3d modularity

3D模块化

• DOI: 10.1007/jhep10(2019)010
• 发表时间: 2019
• 期刊: Journal of High Energy Physics
• 影响因子: 5.4
• 作者: Cheng, Miranda C.N.;Chun, Sungbong;Ferrari, Francesca;Gukov, Sergei;Harrison, Sarah M.
• 通讯作者: Harrison, Sarah M.

Equivariant Verlinde Algebra from Superconformal Index and Argyres–Seiberg Duality

• DOI: 10.1007/s00220-017-3074-8
• 发表时间: 2016-05
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: S. Gukov;Du Pei;Wenbin Yan;Ke Ye

IR duality in 2D N=(0,2) gauge theory with noncompact dynamics

二维 N=(0,2) 非紧动力学规范理论中的红外对偶性

• DOI: 10.1103/physrevd.99.066005
• 发表时间: 2019
• 期刊: Physical Review D
• 影响因子: 5
• 作者: Dedushenko, Mykola;Gukov, Sergei
• 通讯作者: Gukov, Sergei

Trisecting non-Lagrangian theories

三等分非拉格朗日理论

• DOI: 10.1007/jhep11(2017)178
• 发表时间: 2017
• 期刊: Journal of High Energy Physics
• 影响因子: 5.4
• 作者: Gukov, Sergei