4d $\mathcal{N}=3$ indices via discrete gauging

TL;DR

<3-5 sentence high-level summary> This work provides a concrete procedure to obtain 4d $\mathcal{N}=3$ SCFTs by discrete gauging of $\mathcal{N}=4$ SYM, encoding the gauging in a refined superconformal index and a Higgs branch Hilbert series. It demonstrates that for rank-1 theories the index and HBHS reproduce known results, while for higher rank theories the Coulomb branches are generically not freely generated, with explicit examples across $U(N)$, $SU(N+1)$, $SO(2N)$, and $E_N$ families. The analysis yields detailed CB/HB structures, clarifies when CBs are complete intersections, and confirms large-$N$ AdS/CFT expectations via a KK graviton index comparison, while outlining future directions for Schur limits and lens-space indices to probe non-local operator data.

Abstract

A class of 4d $\mathcal{N}=3$ SCFTs can be obtained from gauging a discrete subgroup of the global symmetry group of $\mathcal{N}=4$ Super Yang-Mills theory. This discrete subgroup contains elements of both the $SU(4)$ R-symmetry group and the $SL(2,\mathbb{Z})$ S-duality group of $\mathcal{N}=4$ SYM. We give a prescription for how to perform the discrete gauging at the level of the superconformal index and Higgs branch Hilbert series. We interpret and match the information encoded in these indices to known results for rank one $\mathcal{N}=3$ theories. Our prescription is easily generalised for the Coloumb branch and the Higgs branch indices of higher rank theories, allowing us to make new predictions for these theories. Most strikingly we find that the Coulomb branches of higher rank theories are generically not-freely generated.