Marginal operators and supersymmetry enhancement in 3d $S$-fold SCFTs

The study of exactly marginal deformations of superconformal field theories is a topic that has received considerable attention due to their rich properties. We investigate the $\mathcal{N}=2$ preserving exactly marginal operators of 3d $S$-fold SCFTs. Two families of such theories are considered: one is constructed by gauging the diagonal flavour symmetry of the $T(U(2))$ and $T(U(3))$ theories, and the other by gauging the diagonal flavour symmetry of the $T^{[2,1^2]}_{[2,1^2]}(SU(4))$ theory. In both families, it is possible to turn on a Chern--Simons level for each gauge group and to couple to each theory various numbers of hypermultiplets. The detailed analysis of the exactly marginal operators, along with the superconformal indices, allows us to determine whether supersymmetry gets enhanced in the infrared and to deduce the amount of supersymmetry of the corresponding SCFT.