  We generalize S-duality to N=2 superconformal field theories (SCFTs) with
Coulomb branch operators of non-integer scaling dimension. As simple examples,
we find minimal generalizations of the S-dualities discovered in SU(2) gauge
theory with four fundamental flavors by Seiberg and Witten and in SU(3) gauge
theory with six fundamental flavors by Argyres and Seiberg. Our constructions
start by weakly gauging diagonal SU(2) and SU(3) flavor symmetry subgroups of
two copies of a particular rank-one Argyres-Douglas theory (along with
sufficient numbers of hypermultiplets to guarantee conformality of the
gauging). As we explore the resulting conformal manifold of the SU(2) SCFT, we
find an action of S-duality on the parameters of the theory that is reminiscent
of Spin(8) triality. On the other hand, as we explore the conformal manifold of
the SU(3) theory, we find that an exotic rank-two SCFT emerges in a dual SU(2)
description.
