Applications of quantum groups to 4-dimensional topological field theory

量子群在 4 维拓扑场论中的应用

• 批准号: 1974971
• 金额: --
• 依托单位: University of Edinburgh
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2015
• 资助国家: 英国
• 起止时间: 2015 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1974971
• 关键词: Applications; quantum; groups; dimensional; topological

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.

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。除其他外，BGL理论产生了$G$-Riemann曲面局部系统空间上的Poisson结构的量化。单独量化这些模空间的辛叶是一个有趣的重要问题，这些模空间对应于局部系统，其每个穿孔周围的单值位于指定的共轭类中，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。