N = 3 SCFTs in 4 dimensions and non-simply laced groups

In this paper we discuss various N = 3 SCFTs in 4 dimensions and in particular those which can be obtained as a discrete gauging of an N = 4 SYM theories with non- simply laced groups. The main goal of the project was to compute the Coulomb branch superconformal index and moduli space Hilbert series for the N = 3 SCFTs that are obtained from gauging a discrete subgroup of the global symmetry group of N = 4 Super Yang-Mills theory. The discrete subgroup contains elements of both SU(4) R-symmetry group and the S-duality group of N = 4 SYM. This computation was done for the simply laced groups (where the S-duality groups is SL(2 , ℤ) and Langlands dual of the algebra ^L𝔤 is simply 𝔤 ) by Bourton et al. [ 1 ], and we extended it to the non-simply laced groups. We also considered the orbifolding groups of the Coulomb branch for the cases when Coulomb branch is relatively simple; in particular, we compared them with the results of Argyres et al. [ 2 ], who classified all N ≥ 3 moduli space orbifold geometries at rank 2 and with the results of Bonetti et al. [ 3 ], who listed all possible orbifolding groups for the freely generated Coulomb branches of N ≥ 3 SCFTs. Finally, we have considered sporadic complex crystallographic reflection groups with rank greater than 2 and analyzed, which of them can correspond to an N = 3 SCFT with a principal Dirac pairing.