Categorification at Roots of Unity

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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.