Higher S-dualities and Shephard-Todd groups

A bstract Seiberg and Witten have shown that in N = 2 SQCD with N f = 2 N c = 4 the S -duality group P S L 2 ℤ acts on the flavor charges, which are weights of Spin(8), by triality. There are other N = 2 SCFTs in which SU(2) SYM is coupled to strongly-interacting non-Lagrangian matter: their matter charges are weights of E 6 , E 7 and E 8 instead of Spin(8). The S -duality group P S L 2 ℤ acts on these weights: what replaces Spin(8) triality for the E 6 , E 7 , E 8 root lattices? In this paper we answer the question. The action on the matter charges of (a finite central extension of) P S L 2 ℤ factorizes trough the action of the exceptional Shephard-Todd groups G 4 and G 8 which should be seen as complex analogs of the usual triality group S 3 ≃ Weyl A 2 . 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.