Abstract
We show that the 4d mathcal{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} = 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.
Highlights
SCFTs and 4d QFTs, which was obtained in a rather indirect way, has an intriguing structure [17]
In this paper we have suggested a geometric interpretation of SU(M ) SQCD in the middle of the conformal window, that is Nf = 2M
These models can be obtained as compactifications of DM+1 minimal conformal matter in 6d on three punctured spheres with two
Summary
The second generalization of the above statement is that we will argue that trinions with some flux, one maximal, one minimal, and one generic puncture are related to N = 1 SU(N + 2) SQCD with Nf ≤ 2N + 4. We give a geometric interpretation of Seiberg duality [29] of SQCD in the middle of conformal window and generalize the Intriligator-Pouliot duality [30] . In appendix A we detail how the trinion of the current paper is related to the one of [1]
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