Applications of quantum groups to 4-dimensional topological field theory

最近，Ben-Zvi，Brochier和Jordan [BZBJ]引入了一个4维拓扑场论-称为Betti几何朗兰兹（BGL）理论-与N=4的超级杨-米尔斯理论密切相关，并与曲面和3-流形上的Betti局部系统的模空间密切相关。BGL理论产生除其他事项外，量子化的泊松结构空间的G$-局部系统的黎曼surface.It是一个有趣的一个重要问题，单独的辛叶这些模空间，对应于本地系统的monodromy周围的每个穿刺在于在一个规定的共轭类，并利用输出产生新的拓扑不变量。具体项目包括：1）在约化群中构造共轭类的量子化，作为共理想子代数，并为量子群发展由此产生的辫子模范畴2）研究轨道上的拓扑场论，并将其与量子对称对联系起来。

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.