SQCD and pairs of pants

We show that the 4d$$ \mathcal{N} $$ N = 1 SU(3) Nf = 6 SQCD is the model obtained when compactifying the rank one E-string theory on a three punctured sphere (a trinion) with a particular value of flux. The SU(6) × SU(6) × U(1) global symmetry of the theory, when decomposed into the SU(2)3× U(1)3× SU(6) subgroup, corresponds to the three SU(2) symmetries associated to the three punctures and the U(1)3× SU(6) subgroup of the E8 symmetry of the E-string theory. All the puncture symmetries are manifest in the UV and thus we can construct ordinary Lagrangians flowing in the IR to any compactification of the E-string theory. We generalize this claim and argue that the $$ \mathcal{N} $$ N = 1 SU(N + 2) SQCD in the middle of the conformal window, Nf = 2N + 4, is the theory obtained by compactifying the 6d minimal (DN +3, DN +3) conformal matter SCFT on a sphere with two maximal SU(N + 1) punctures, one minimal SU(2) puncture, and with a particular value of flux. The SU(2N + 4) × SU(2N + 4) × U(1) symmetry of the UV Lagrangian decomposes into SU(N + 1)2× SU(2) puncture symmetries and the U(1)3× SU(2N + 4) subgroup of the SO(12 + 4N ) symmetry group of the 6d SCFT. The models constructed from the trinions exhibit a variety of interesting strong coupling effects. For example, one of the dualities arising geometrically from different pair-of-pants decompositions of a four punctured sphere is an SU(N + 2) generalization of the Intriligator-Pouliot duality of SU(2) SQCD with Nf = 4, which is a degenerate, N = 0, instance of our discussion.