3d $\mathcal{N}=2$ Brane Webs and Quivers

We discuss the effective Chern-Simons levels for 3d $\mathcal{N}=2$ gauge theories and their relations to the relative angles between NS5-brane and NS5'-brane. We find that turning on real masses for chiral multiplets leads to various equivalent brane webs that are related by flipping the sign of mass parameters. This flip can be interpreted as 3d mirror symmetry for abelian theories. Each of these webs has a corresponding mathematical quiver structure. We check the equivalence of vortex partition functions for these brane webs by implementing topological vertex method. In addition, we compute the vortex partition functions of nonabelian theories with gauge group $U(N)$ and find the associated quiver structures and brane webs. We find that on Higgs branch nonabelian brane webs are broken to abelian brane webs with gauge group $U(1)^{\otimes N}$. We also discuss the Ooguri-Vafa invariants for nonabelian theories and the movement of flavor D5-branes that leads to equivalent brane webs.