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

TL;DR

The paper investigates IR dualities between 3d ${\cal N}=8$ super Yang–Mills theories with ABCD gauge groups and ABJ(M) models at Chern–Simons levels $k=1$ or $k=2$, using the superconformal index as a precise diagnostic. By constructing and comparing indices for the ${U(N)}$ case (via its ${\cal N}=4$ mirror) and for BCD families (including ${\mathbb Z}_2$ orbifolds), and by analyzing large-$N$ limits with twisted sectors, the authors establish matches with ABJM/ABJ(M) duals and identify the corresponding gravitational backgrounds $AdS_4\times S^7/\mathbb{Z}_m$. Extending the analysis to ${\cal N}=4$ mirror-symmetric CS theories, they map FI and mass parameters across dual pairs and demonstrate index-level consistency of the dualities, Higgsing patterns, and orbifold/orientifold constructions. The results provide nontrivial evidence for the proposed IR equivalences and offer a framework for connecting YM dynamics to CS-matter theories and their gravity duals in diverse gauge groups and topologies.

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.

Figures (4)

• Figure 1: (a) A configuration of $N$ D3s, 2 NS5s, and one D5. (b) T-dual transformation of (a). (c) S-dual transformation of (a).
• Figure 2: Two possible configurations of 2 NS5s and 2 D5s
• Figure 3: D3-brane creation due to Hanany-Witten effect
• Figure 4: (a) A generic Hanany-Witten set up. (b) T-dual transformation from (a)