More on N $$ \mathcal{N} $$ =2 S-folds

We carry out a systematic study of 4d $$ \mathcal{N} $$ = 2 preserving S-folds of F-theory 7-branes and the worldvolume theories on D3-branes probing them. They consist of two infinite series of theories, which we denote following [1, 2] by $$ {\mathcal{S}}_{G,\mathrm{\ell}}^{(r)} $$ for l = 2, 3, 4 and $$ {\mathcal{T}}_{G,\mathrm{\ell}}^{(r)} $$ for l = 2, 3, 4, 5, 6. Their distinction lies in the discrete torsion carried by the S-fold and in the difference in the asymptotic holonomy of the gauge bundle on the 7-brane. We study various properties of these theories, using diverse field theoretical and string theoretical methods.