Mirror symmetry of 3d N=4 abelian gauge theories

Abstract: We study A and B twists of 3d N=4 abelian gauge theories, as well as their mirror symmetries. Such a theory is defined by an n by r charge matrix q, and we refer to it as T_q. Our goal is twofold: firstly, the definition and mirror symmetry of boundary VOAs and secondly, the definition and mirror symmetry of the category of line operators. For the first goal, following the construction of Costello-Gaiotto and Costello-Creutzig-Gaiotto, we define the VOA of boundary local operators for specific boundary conditions of T_q: for the A twist, the Neumann boundary condition and for the B twist, the Dirichlet boundary condition. Using free field realizations, we show that for a pair of mirror dual theories T_q and T_{p}, the boundary VOA for the A twist of T_q is isomorphic to the boundary VOA for the B twist of T_p. For the second goal, we define the category of line operators as a category of modules of the boundary VOA. We show that for a pair of mirror dual theories T_q and T_{p}, the category of line operators for the A twist of T_q and for the B twist of T_p are equivalent as braided tensor categories. This is joint with A. Ballin, T. Creutzig and T. Dimofte.