Periodic Reporting for period 3 - QuantGeomLangTFT (The Quantum Geometric Langlands Topological Field Theory)
Reporting period: 2018-06-01 to 2019-11-30
The proposal seeks to address an important question at the interface of all of these fields: over the past two decades, experts in the field of topological field theory -- the study of how topological properties of a space or space-time impact the kinds of information the space can carry, and how that information glues as we glue together the spacetimes coherently -- have understood that there should be a 4-dimensional topological field theory associated to the data of a quantum group. Here a quantum group is a fundamental deformation of an algebraic group of symmetries. Algebraically these quantum groups by now are quite well understood, and indeed many connections to topological field theories in dimension 3 have been rigourously developed. However, the constructions so far have had strong limitations on the nature of the quantum groups which are considered -- typically they are required to ""modular"" categories, which have found a number of applications for instance to statistical mechanics and quantum computing, but are ultimately too compact to yield examples related to algebraic geometry and representation theory."
Specific research outputs (papers authored, conferences organized, talks presented, etc) are given in Part B of this report.