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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 Higgs branch 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 $SL(2, \mathbb{Z})$ and Langlands dual of the the algebra $L[\mathfrak{g}]$ is simply $\mathfrak{g}$) by Bourton et al. arXiv:1804.05396, 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. arXiv:1904.10969, who classified all $N\geq 3$ moduli space orbifold geometries at rank 2 and with the results of Bonetti et al. arXiv:1810.03612, who listed all possible orbifolding groups for the freely generated Coulomb branches of $N\geq 3$ 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.