SQCD and pairs of pants

TL;DR

This paper connects 4d ${ m N}=1$ SQCD theories with 6d SCFT compactifications on punctured spheres, revealing new dualities and symmetry structures through geometric decompositions.

Contribution

It identifies specific 4d SQCD models as compactifications of 6d SCFTs on punctured spheres, providing a geometric framework for understanding their dualities and symmetries.

Findings

Constructed Lagrangians flow to E-string compactifications.

Identified dualities as pair-of-pants decompositions.

Revealed symmetry decompositions matching puncture structures.

Abstract

We show that the $4d$ ${\cal N}=1$ $SU(3)$ $N_f=6$ 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 $SU(6)\times SU(6)\times U(1)$ global symmetry of the theory, when decomposed into the $SU(2)^3\times U(1)^3\times SU(6)$ subgroup, corresponds to the three $SU(2)$ symmetries associated to the three punctures and the $U(1)^3 \times SU(6)$ 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 ${\cal N}=1$ $SU(N+2)$ SQCD in the middle of the conformal window, $N_f=2N+4$, is the theory obtained by compactifying the $6d$ minimal $(D_{N+3},D_{N+3})$ 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 $SU(2N+4)\times SU(2N+4)\times U(1)$ symmetry of the UV Lagrangian decomposes into $SU(N+1)^2\times SU(2)$ puncture symmetries and the $U(1)^3\times SU(2N+4)$ 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.