Higher S-dualities and Shephard-Todd groups

Abstract: Seiberg and Witten have shown that in N=2$$ \mathcal{N}=2 $$ SQCD with Nf = 2Nc = 4 the S-duality group PSL2ℤ$$ \mathrm{P}\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right) $$ acts on the flavor charges, which are weights of Spin(8), by triality. There are other N=2$$ \mathcal{N}=2 $$ SCFTs in which SU(2) SYM is coupled to strongly-interacting non-Lagrangian matter: their matter charges are weights of E6, E7 and E8 instead of Spin(8). The S-duality group PSL2ℤ$$ \mathrm{P}\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right) $$ acts on these weights: what replaces Spin(8) triality for the E6, E7, E8root lattices? In this paper we answer the question. The action on the matter charges of (a finite central extension of) PSL2ℤ$$ \mathrm{P}\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right) $$ factorizes trough the action of the exceptional Shephard-Todd groups G4 and G8 which should be seen as complex analogs of the usual triality group S3≃WeylA2$$ {\mathfrak{S}}_3\simeq \mathrm{Weyl}\left({A}_2\right) $$. Our analysis is based on the identification of S-duality for SU(2) gauge SCFTs with the group of automorphisms of the cluster category of weighted projective lines of tubular type. © 2015, The Author(s)