Categorification at Roots of Unity

拓扑量子场论是量子物理学的基础数学，它是理解三维和四维几何空间的一些基本定量特征的一种方便的组织原则，我们的宇宙和时空就是基本的例子。自20世纪90年代中期以来，许多数学家推测，某些四维拓扑场论（4d tqft）的明确结构应该存在，它的发现将增强我们对四维量子物理的数学理解。该项目的目标是开发一种明确实现这种四维tqft的可访问方法。四维tqft将允许组合描述并且在实际应用中是可计算的。它们与早期的三维结构相关，但比前者更微妙。这个项目很自然地融入了一个更大的项目，通过一个叫做分类的过程将某些三维场论和四维场论联系起来，在这个过程中，三维的“阴影”概念被四维的更丰富的结构所取代。得到的较高的结构可以用来解决较低维的问题。此外，代数结构将沿着这条道路发展，它将为经典代数研究中的主题提供新的亮点。更详细地说，这是一个研究量子群在统一素根上的范畴及其范畴表征理论的项目。这是对Witten-Reshetikhin-Turaev三维拓扑量子场论进行分类的长期计划的一部分，从而产生一个四维理论。1994年，Crane和Frenkel推测了拓扑量子场论通过分类提升的存在性。本计划的一个关键组成部分是在单位素根上找到某些链接同源理论的范畴增强，例如Khovanov同源。该项目将进一步发展最近引入的广义同调代数理论，称为“hopfological algebra”。将这一新发展的hopfological理论与早期关于量子群在一元素根上的分类表示理论相结合，将允许人们明确地构建链接和3流形的hopfological不变量，从而通过分类明确地提升Witten-Reshetikhin-Turaev的三维拓扑量子场论。在此过程中，将研究量子群的一素根的范畴表示理论及其新的形态学特征。

Topological quantum field theory, the mathematics underlying quantum physics, serves as a convenient organizational principle in understanding some basic quantitative features of three- and four-dimensional geometric spaces, of which our universe and space-time are fundamental examples. Since mid-1990s, various mathematicians have conjectured that an explicit construction of certain four-dimensional topological field theories (4d TQFTs) should exist, the discovery of which will enhance our mathematical understanding of four-dimensional quantum physics. The objective of this project is to develop an accessible approach towards explicitly realizing such 4d TQFTs. The 4d TQFTs will admit combinatorial descriptions and be computable in practice for applications. They are related to, yet more subtle than, an earlier construction in dimension three. The project naturally fits into a larger program of relating certain three-dimensional field theories and four-dimensional ones by a process called categorification, where "shadow" concepts in three dimensions are replaced by richer structures in four dimensions. The higher structures obtained can be utilized to resolve problems in lower dimensions. In addition, algebraic structures will be developed along the the way that it will shed new light on topics in classical algebraic studies.In more detail, this is a project to investigate categorifications of quantum groups at prime roots of unity, and their categorical representation theory. This constitutes part of a long term program to categorify the Witten-Reshetikhin-Turaev three-dimensional topological quantum field theory, producing a four-dimensional theory. The existence of such liftings of topological quantum field theories by categorification was conjectured by Crane and Frenkel in 1994. A key ingredient in this program is to find a categorical enhancement of certain link homology theories, such as Khovanov homology, at prime roots of unity. The project will be carried out by further developing a recently introduced generalized homological algebraic theory called "hopfological algebra." Incorporating this newly developed hopfological theory into earlier works on categorified representation theory of quantum group at a prime root of unity will allow one to explicitly construct hopfological invariant for links and 3-manifolds, thereby explicitly lifting Witten-Reshetikhin-Turaev's three-dimensional topological quantum field theorys via categorification. In the process, categorified representation theory of quantum groups a prime root of unity will be investigated, together with their new hopofological features.