范畴化与低维拓扑中的高维范畴

拓扑场论是低维拓扑中重要的研究工具。一个自然的推广是延伸的拓扑场论，其中自然地需要引入高维范畴。范畴化是把一个n维拓扑场论升级成n+1维拓扑场论的过程。一个很重要的例子是三维球面里扭结的Khovanov同调作为Jones多项式的范畴化。把扭结的Jones多项式推广到一般的三维流形，就是Witten-Reshetikhin-Turaev三维拓扑场论，其与量子群sl(2)的表示密切相关。本次申请的项目主要研究与Witten-Reshetikhin-Turaev三维拓扑场论的范畴化相关的问题。特别地，希望从代数和几何的角度，对sl(2)的范畴化的表示构成的辫子2维范畴有更好的理解。此外，我们还希望计算更多辫子2维范畴的例子。

Topological field theory is an important tool in low dimensional topology. A natural generalization is called extended topological field theory that builds on higher category theory. Categorification can be viewed as upgrading an n dimensional topological field theory to n+1 dimensional. A celebrated example is the Khovanov homology of links in the three sphere as a categorification of the Jones polynomial. The corresponding topological field theory for three manifolds is Witten-Reshetikhin-Turaev theory that is related to the representation of the quantum group sl(2). The goal of our project is to study categorification of Witten-Reshetikhin-Turaev theory. In particular, we want to understand the braided 2-category of categorified representations of sl(2) from both algebraic and geometric perspectives. More examples of braided 2-categories will be studied.