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Boundaries and Defects of N=4 SYM with 4 Supercharges, Part II: Brane Constructions and 3d N=2 Field Theories

We study the vacuum moduli spaces of 3d N=2 supersymmetric quantum field theories by applying the formalism developed in our previous paper arXiv:1404.5527. The 3d theories can be realized by branes in type IIB string theory, which in a decoupling limit reduce to 4d N=4 super-Yang-Mills theory on an interval with BPS defects inserted. The moduli space of a given 3d theory is obtained by solving a generalization of Nahm's equations with appropriate boundary/junction conditions, along with help from the S-duality of type IIB string theory. Our classical computations reproduce many known results about the quantum-corrected moduli spaces of 3d theories, e.g. U(N_c) theories with N_f flavors with mass and FI parameters turned on. In particular, our methods give first-principles derivations of several results in the literature, including the s-rule, quantum splitting of classical Coulomb branches, the lifting of the Coulomb branch by non-Abelian instantons, quantum merging of Coulomb and Higgs branches, and phase transitions from re-ordering 5-branes.