Towards 2-Categorical 3d Mirror Symmetry

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. In this talk I will discuss what is known about the 2-categories of boundary conditions for these two TQFTs. They are expected to provide two distinct categorifications of category O for the hyperkahler quotient X///G and 3d mirror symmetry is expected to induce a categorification of the Koszul duality between categories O for mirror symplectic resolutions. For abelian gauge theories this picture is work in progress with Ben Gammage and Aaron Mazel-Gee. This generalizes the works of Kapustin-Vyas-Setter and Teleman on pure gauge theory.

algebraic geometrysymplectic geometry

Audience: researchers in the topic

M-seminar