Categorification at Roots of Unity

拓扑量子场论是量子物理学的数学基础，它是理解三维和四维几何空间的一些基本定量特征的一个方便的组织原则，我们的宇宙和时空就是其中的基本例子。自20世纪90年代中期以来，许多数学家都提出了四维拓扑场论（4d TQFTs）的显式构造，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易于使用的方法来明确实现此类4d TQFT。四维TQFT将允许组合描述，并且在实际应用中是可计算的。它们与第三维度的早期构造有关，但更为微妙。这个项目很自然地适合于一个更大的程序，这个程序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，三维中的“阴影”概念被四维中更丰富的结构所取代。得到的更高的结构可以用来解决低维的问题。此外，代数结构将沿着它将在经典代数研究中揭示新的主题的方式发展。更详细地说，这是一个研究量子群在单位素根上的代数化及其范畴表示理论的项目。这构成了一个长期计划的一部分，以分类维滕-列舍季欣-图拉耶夫三维拓扑量子场论，产生一个四维理论。起重机和Frenkel在1994年证明了拓扑量子场论通过分类的这种提升的存在性。在这个计划中的一个关键因素是找到一个明确的增强某些链接同源理论，如Khovanov同源，在素根的单位。该项目将通过进一步发展最近引入的广义同调代数理论（称为“hopfological代数”）来进行。将这个新发展的hopfological理论扩展到早期的关于量子群在一个素单位根上的范畴化表示理论的工作中，将允许人们显式地为链和3-流形构造hopfological不变量，从而通过范畴化显式地提升Witten-Reshetikhin-Turaev的三维拓扑量子场论。在这个过程中，将研究量子群的范畴化表示理论，以及它们的新的hopofological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.