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The $S$-duality group $\mathbb{S}(\mathcal{F})$ of a 4d $\mathcal{N}=2$supersymmetric theory $\mathcal{F}$ is identified with the group of triangleauto-equivalences of its cluster category $\mathscr{C}(\mathcal{F})$ modulo thesubgroup acting trivially on the physical quantities. $\mathbb{S}(\mathcal{F})$is a discrete group commensurable to a subgroup of the Siegel modular group$Sp(2g,\mathbb{Z})$ ($g$ being the dimension of the Coulomb branch). Thisidentification reduces the determination of the $S$-duality group of a given$\mathcal{N}=2$ theory to a problem in homological algebra. In this paper wedescribe the techniques which make the computation straightforward for a largeclass of $\mathcal{N}=2$ QFTs. The group $\mathbb{S}(\mathcal{F})$ is naturallypresented as a generalized braid group.  The $S$-duality groups are often larger than expected. In some models theenhancement of $S$-duality is quite spectacular. For instance, a QFT with ahuge $S$-duality group is the Lagrangian SCFT with gauge group $SO(8)\timesSO(5)^3\times SO(3)^6$ and half-hypermultiplets in the bi- and tri-spinorrepresentations.  We focus on four families of examples: the $\mathcal{N}=2$ SCFTs of the form$(G,G^\prime)$, $D_p(G)$, and $E_r^{(1,1)}(G)$, as well as theasymptotically-free theories $(G,\widehat{H})$ (which contain $\mathcal{N}=2$SQCD as a special case). For the $E_r^{(1,1)}(G)$ models we confirm thepresence of the $PSL(2,\mathbb{Z})$ $S$-duality group predicted by Del Zotto,Vafa and Xie, but for most models in this class $S$-duality gets enhanced to amuch larger group.

Homological S-Duality in 4d N=2 QFTs