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

We discuss 3d brane webs and effective Chern-Simons levels for 3d $\mathcal{N}=2$ gauge theories. We find that turning on real masses for chiral multiplets leads to various equivalent brane webs that are related by flipping positions of D5-branes. We interpret flips as $ST$-transformations for chiral multiplets. $ST$-transformations could turn abelian theories into dual theories with mixed Chern-Simons levels that are interpreted as quiver matrices $C_{ij}$ encoding DT-invariants. We notice that each brane web corresponds to a quiver matrix. $ST$-transformations of holomorphic blocks are discussed to verify results. We also discuss the movement of flavor D5-branes, which leads to double-layer brane webs and manifests fiber-base duality. In the second part, we compute refined vortex partition functions of nonabelian theories with the gauge group $U\left(N\right)$ and find corresponding quiver matrices. The computation shows that on Higgs branch nonabelian groups are broken to abelian groups.