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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 $SU(2)^3times U(1)^3times SU(6)$ 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 $SU(N+1)^2times SU(2)$ puncture symmetries and the $U(1)^3times SU(2N+4)$ 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.
We proceed to study Yang-Baxter deformations of 4D Minkowski spacetime based on a conformal embedding. We first revisit a Melvin background and argue a Lax pair by adopting a simple replacement law invented in 1509.00173. This argument enables us to
We explicitly derive Lax pairs for string theories on Yang-Baxter deformed backgrounds, 1) gravity duals for noncommutative gauge theories, 2) $gamma$-deformations of S$^5$, 3) Schrodinger spacetimes and 4) abelian twists of the global AdS$_5$,. Then
We study deformations of N=1 supersymmetric QCD that exhibit a rich landscape of supersymmetric and non-supersymmetric vacua.
There exists a large class of groups of operators acting on Hilbert spaces, where commutativity of group elements can be expressed in the geometric language of symplectic polar spaces embedded in the projective spaces PG($n, p$), $n$ being odd and $p
We study gauge theories with N=1 supersymmetry in 2+1 dimensions. We start by calculating the 1-loop effective superpotential for matter in an arbitrary representation. We then restrict ourselves to gauge theories with fundamental matter. Using the 1