Categorification at Roots of Unity

拓扑量子场论是量子物理的数学基础，它是理解三维和四维几何空间的一些基本数量特征的方便的组织原则，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，许多数学家猜测，应该存在某些四维拓扑场论(4D TQFT)的显式构造，它的发现将加深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可获得的方法，以明确实现这种4d全面质量转移。4D TQFT将接受组合描述，并可在实际应用中进行计算。它们与三维空间中的早期结构有关，但比之更为微妙。该项目自然适合于一个更大的计划，通过一个称为归类的过程将某些三维场理论与四维场理论联系起来，在这个过程中，三维中的“影子”概念被四维中更丰富的结构所取代。得到的较高结构可以用来解决较低维度的问题。此外，代数结构将沿着这条路发展，它将为经典代数研究中的主题提供新的线索。更详细地说，这是一个研究单位素根量子群的范畴化及其范畴表示理论的项目。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，产生了一个四维理论。1994年，Crane和Frenkel猜想，拓扑型量子场论的范畴化提升是存在的。这个程序的一个关键成分是找到某些链接同调理论的绝对增强，例如Khovanov同调，在统一的素根上。该项目将通过进一步发展最近引入的广义同调代数理论来实施，该理论被称为“Hopfology代数”。将这一新发展的Hopfology理论与早期关于单位素根量子群范畴化表示理论的工作相结合，可以显式地构造链环和3-流形的Hopfology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。在这一过程中，将研究量子群的范畴化表示理论，以及它们的新的霍普夫学特征。

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.