Abstract: 
By now it is known that many interesting phenomena in geometry and representation theory can be understood as aspects of mirror symmetry of 3d N=4 SUSY QFTs. Such a QFT is associated to a hyper-Kähler manifold X equipped with a hyper-Hamiltonian action of a compact Lie group G and admits two topological twists. The first twist, which is known as the 3d B-model or Rozansky-Witten theory, is a TQFT of algebro-geometric flavor and has been studied extensively by Kapustin, Rozansky and Saulina. The second twist, which is known as the 3d A-model or 3d Seiberg-Witten theory, is a more mysterious TQFT of symplecto-topological flavor. The 2-categories of boundary conditions for these two TQFTs are expected to provide two distinct categorifications of category O for the hyperkahler quotient X///G and 3d mirror symmetry is expected to induce the the Koszul duality between categories O for mirror symplectic resolutions.


In this talk I will explain recently completed work with Ben Gammage realizing these expectations for abelian gauge theories. The key inputs are the theory of perverse schobers, which gives us a combinatorial description of the 3d A-model, and the compatibility of 2-categorical 3d mirror symmetry with the 3-categorical geometric Langlands program. The latter was formalized in my prior work with Gammage and Mazel-Gee. If time permits I will also discuss what is known about quiver gauge theories and the 3-Lie algebra action on 2-category O coming from Nakajima’s correspondences. These lift the well known 2-Lie algebra actions due to Webster and Kamnitzer-Webster-Weekes-Yacobi.