Applications of quantum groups to 4-dimensional topological field theory

最近，Ben-Zvi，Brochier和Jordan[BZBJ]引入了一个与N=4的超杨-Mills理论以及曲面和三维流形上的Betti局部系统的模空间密切相关的四维拓扑场理论，称为Betti几何朗兰兹理论(BGL)。其中，BGL理论产生了$G-局部黎曼曲面系空间上Poisson结构的量子化。一个有趣的重要问题是，分别量子化这些模空间的辛叶，这些辛叶对应于每个穿孔的单行位于指定的共轭类中的局部系统，并利用输出产生新的拓扑不变量。具体项目包括：1)构造约化群中共轭类的量子化，作为上理想子代数，并发展所得到的量子群的辫子模范类；2)研究奥尔布洛德上的拓扑场论，并将其与量子对称对联系起来；3)研究量子参数为单位根时这些结构及相关结构的特殊性质。

Recently, Ben-Zvi, Brochier, and Jordan [BZBJ] have introduced a 4-dimensional topological field theory - dubbed Betti Geometric Langlands (BGL) theory - intimately related to N=4 super Yang-Mills theory, and to the moduli spaces of Betti local systems on surfaces and 3-manifolds. The BGL theory produces among other things, quantizations of the Poisson structures on spaces of $G$-local systems of Riemann surfaces.It is an interesting an important problem to quantize individually the symplectic leaves of these moduli spaces, which correspond to local systems whose monodromy around each puncture lies in a prescribed conjugacy class, and to leverage the output to produce new topological invariants. Specific projects include:1) Constructing quantizations of conjugacy classes in reductive groups, as co-ideal sub-algebras, and developing the resulting braided module categories for the quantum group2) Studying topological field theories on orbifolds, and relating these to quantum symmetric pairs.3) Studying special features of these and related constructions when the quantum parameter is a root of unity.