Mirror Symmetry and Mixed Chern-Simons Levels for Abelian 3d $\mathcal{N} = 2$ theories

Abstract: We study the mirror symmetry of abelian 3d $\mathcal{N}=2$ theories with mixed Chern-Simons levels by turning them into $\mathcal{T}{A,N}$ theories that are defined as $N$ copies of $U(1)-[1]$ theory coupled together by mixed Chern-Simons levels $k{ij}$. We find that $\mathcal{T}{A,N}$ theories have many mirror dual theories with different mixed CS levels and FI parameters. As an example, we analyze $U(1)_k+ N_C \,\mathbf{C} + N{AC} \, \mathbf{AC}$ theories by transforming these theories into certain $\mathcal{T}_{A,N}$ theories and find many equivalent effective Chern-Simons levels. Finally, we analyze mirror symmetry for theories corresponding to knots. In this work we use sphere partition functions and vortex partition functions to derive dual theories.