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.
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Hashimoto, A., Ouyang, P. & Yamazaki, M. Boundaries and defects of \( \mathcal{N}=4 \) SYM with 4 supercharges. Part II: Brane constructions and 3d \( \mathcal{N}=2 \) field theories. J. High Energ. Phys. 2014, 108 (2014). https://doi.org/10.1007/JHEP10(2014)108
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DOI: https://doi.org/10.1007/JHEP10(2014)108