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One of the hallmarks of 6D superconformal field theories (SCFTs) is that on apartial tensor branch, all known theories resemble quiver gauge theories withlinks comprised of 6D conformal matter, a generalization of weakly coupledhypermultiplets. In this paper we construct 4D quiverlike gauge theories inwhich the links are obtained from compactifications of 6D conformal matter onRiemann surfaces with flavor symmetry fluxes. This includes generalizations ofsuper QCD with exceptional gauge groups and quarks replaced by 4D conformalmatter. Just as in super QCD, we find evidence for a conformal window as wellas confining gauge group factors depending on the total amount of matter. Wealso present F-theory realizations of these field theories via ellipticallyfibered Calabi-Yau fourfolds. Gauge groups (and flavor symmetries) come from7-branes wrapped on surfaces, conformal matter localizes at the intersection ofpairs of 7-branes, and Yukawas between 4D conformal matter localize at pointscoming from triple intersections of 7-branes. Quantum corrections can alsomodify the classical moduli space of the F-theory model, matching expectationsfrom effective field theory.

4D Gauge Theories with Conformal Matter