Boundaries and defects of N = 4 $$ \mathcal{N}=4 $$ SYM with 4 supercharges. Part II: Brane constructions and 3d N = 2 $$ \mathcal{N}=2 $$ field theories

Abstract

We study the vacuum moduli spaces of 3d $$ \mathcal{N}=2 $$ supersymmetric quantum field theories by applying the formalism developed in our previous paper [1]. The 3d theories can be realized by branes in type IIB string theory, which in a decoupling limit reduce to 4d $$ \mathcal{N}=4 $$ 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.