Superconformal Index for $\mathcal{N}=3$ $\widehat{ADE}$ Chern-Simons Quiver Gauge Theories

We compute superconformal indices for $\mathcal{N}=3$ $\widehat{ADE}$ Chern-Simons quiver gauge theories with a product gauge group ${\prod }_{i}U(N{)}_i$, using the method of supersymmetric localization. We also perform a large $N$ analysis of the index. This index includes contribution from non zero magnetic flux sector. The fact that these theories have a weakly coupled UV completion in terms of $\mathcal{N}=3$ supersymmetric Chern-Simons Yang-Mills theories enables us to apply the localization technique. Such theories have dual M-theory description on ${\mathrm{A}\mathrm{d}\mathrm{S}}_4\times M_7$, where $M_7$ is a tri-Sasaki Einstein manifold.