Homological 3d Mirror Symmetry

Three-dimensional N=4 supersymmetric gauge theories admit two topological “A” and “B” twists. They are analogous to the more familiar A and B twists in two dimensions, and they are exchanged by a physical duality known as “3d mirror symmetry.” Many mathematical consequences of 3d mirror symmetry, especially at the homological/categorical level, are just beginning to be explored. I’ll discuss a proposed identification of categories of line operators in the A and B twists, some partial structure of the 2-category of boundary conditions, and an application to HOMFLY-PT knot homology.