Index and duality of minimal $$ \mathcal{N} = 4 $$ Chern-Simons-matter theories

We perform a first step analysis toward generalization of the classification of $$ \mathcal{N} = 4 $$ N = 4 linear quiver gauge theories by Gaiotto and Witten including Chern-Simons interaction. For this we investigate minimal $$ \mathcal{N}=4\mathrm{U}{\left({N}_1\right)}_k\times \mathrm{U}{\left({N}_2\right)}_{-k} $$ N = 4 U N 1 k × U N 2 − k Chern-Simons theories and their superconformal indices. In the previous publication we analyzed the three-sphere partition function of the theories, which implies that the theory is good/ugly/bad if k − N 1 − N 2 is greater than/equal to/smaller than −1. In this paper we verify that this classification is consistent with the behavior of the superconformal index. We compare the superconformal indices for several pairs of non-bad theories connected by the Hanany-Witten transition and confirm their coincidence up to the contribution of one hypermultiplet.