The S[duality group S[double-struck](F) of a 4d N = 2 supersymmetric theory F is identified with the group of triangle equivalences of its cluster category C (F) modulo the subgroup acting trivially on the physical quantities. S[double-struck](F) is a discrete group commensurable to a subgroup of the Siegel modular group Sp(2g,) (g being the dimension of the Coulomb branch). This identification reduces the determination of the S-duality group of a given N = 2 theory to a problem in homological algebra. In this paper we describe the techniques which make the computation straightforward for a large class of N = 2 QFTs. The group S[double-struck](F) is naturally presented as a generalized braid group. The S-duality groups are often larger than expected. In some models the enhancement of S-duality is quite spectacular. For instance, a QFT with a huge S-duality group is the Lagrangian SCFT with gauge group SO(8) × SO(5)3 × SO(3)6 and half-hypermultiplets in the bi- and tri-spinor representations. We focus on four families of examples: the N = 2 SCFTs of the form (G,G"), Dp(G), and E r(1,1) (G), as well as the asymptoticallyfree theories (G, Ĥ) (which contain N = 2 SQCD as a special case). For the E r(1,1) (G) models we confirm the presence of the PSL(2,) S-duality group predicted by Del Zotto, Vafa and Xie, but for most models in this class S-duality gets enhanced to a larger group.
