3d $\mathcal{N}=2$ $\widehat{ADE}$ Chern-Simons Quivers

We study 3d $\mathcal{N}=2$ Chern-Simons (CS) quiver theories on $S^3$ and ${\mathrm{\Sigma }}_{\mathfrak{g}}\times S^1$. Using localization results, we examine their partition functions in the large rank limit and requiring the resulting matrix models to be local, find a large class of quiver theories that include quivers in one-to-one correspondence with the $\widehat{ADE}$ Dynkin diagrams. We compute explicitly the partition function on $S^3$ for $\widehat{D}$ quivers and that on ${\mathrm{\Sigma }}_{\mathfrak{g}}\times S^1$ for $\widehat{AD}$ quivers, which lead to certain predictions for their holographic duals. We also provide a new and simple proof of the “index theorem”, extending its applicability to a larger class of theories than considered before in the literature.