ABCD of 3d ${\cal N}=8$ and 4 Superconformal Field Theories

Abstract

We argue the equivalence between the infrared conformal field theory of the 3d $\mathcal{N}=8$ supersymmetric Yang-Mills theories of ABCD ($U(N), SO(2N+1), Sp(2N), O(2N)$) gauge groups and the ABJ(M) theories of $U(N)_k\times U(\tilde N)_{-k}$ for $k=1,2$. We support this duality by comparing the superconformal index of the IR limit of these super Yang-Mills theories and that of those ABJ(M) models. Especially we find the match between two indices of (mirror dual of) the $\mathcal{N}=8$ U(N) SYM and of $U(N)_1\times U(N)_{-1}$ ABJM model. Also we take large $N$ limit of ABCD super Yang-Mills theories with additional fundamental hyper-multiplets and infer the large N limit of $\mathcal{N}=8$ ABCD theories themselves, finding the expected gravitational duals. With the additional input on finite N, we argue the equivalence of Yang-Mills and ABJ(M) theories for all N. We further explore similar dualities to Chern-Simons matter theories for $\mathcal{N}=4$ Yang-Mills theories related by mirror symmetry.