Higher S-dualities and Shephard-Todd groups

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Seiberg and Witten have shown that in N=2 SQCD with Nf=2Nc=4 the S-duality group PSL(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 E₆, E₇ and E₈ instead of Spin(8). The S-duality group PSL(2,Z) acts on these weights: what replaces Spin(8) triality for the E₆,E₇,E₈ root lattices? In this paper we answer the question. The action on the matter charges of (a finite central extension of) PSL(2,Z) 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≃ Weyl(A₂). 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.