SQCD and pairs of pants

We show that the 4d N = 1 SU(3) N-f = 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) x SU(6) x U(1) global symmetry of the theory, when decomposed into the SU(2)(3)x U(1)(3)x SU(6) subgroup, corresponds to the three SU(2) symmetries associated to the three punctures and the U(1)(3)x SU(6) subgroup of the E-8 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 N = 1 SU(N + 2) SQCD in the middle of the conformal window, N-f = 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) x SU(2N + 4) x U(1) symmetry of the UV Lagrangian decomposes into SU(N + 1)(2)x SU(2) puncture symmetries and the U(1)(3)x 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 N-f = 4, which is a degenerate, N = 0, instance of our discussion.