3d N=4 gauge theories admit two topological twists, “A” and “B”, which are expected to give rise to extended TQFT’s. Aspects of these TQFT’s are starting to be defined mathematically. My goal in this talk is to discuss recent developments in understanding line operators (the value of the TQFT’s on a circle), which are expected to form a dg braided tensor category in the A and B twists — analogous to familiar structure of Wilson lines in 3d Chern-Simons theory. I will motivate two definitions of the categories of line operators, one intrinsically 3-dimensional (following work of Hiilburn-Yoo and Hilburn-Raskin, and work with Garner-Geracie-Hilburn) and one via boundary VOA’s (following work of Costello-Creutzig-Gaiotto and Ballin-Niu, and work in progress with Ballin-Creutzig-Niu). I will then explain how continuous global symmetries, which are prevalent in 3d N=4 theories, lead to deformations of the categories of line operators, enhancing them to certain sheaves of categories. Finally, I will touch on a recent application to non-semisimple TQFT’s related to quantum groups at roots of unity, based on work with Creutzig-Garner-Geer.
 
virtual (zoom): Virtual: http://berkeley.zoom.us/j/93328405860?pwd=Um1GbHBCSUJMdUlWWnd0ZVMxQmwwdz09