Four-dimensional superconformal theories with interacting boundaries or defects

We study four-dimensional superconformal field theories coupled to three-dimensional superconformal boundary or defect degrees of freedom. Starting with bulk $\mathcal{N}=2,$ $d=4\mathrm{}$ theories, we construct Abelian models preserving $\mathcal{N}=2,$ $d=3\mathrm{}$ supersymmetry and the conformal symmetries under which the boundary/defect is invariant. We write the action, including the bulk terms, in $\mathcal{N}=2,$ $d=3\mathrm{}$ superspace. Moreover we derive Callan-Symanzik equations for these models using their superconformal transformation properties and show that the beta functions vanish to all orders in perturbation theory, such that the models remain superconformal upon quantization. Furthermore, we study a model with $\mathcal{N}=4\mathrm{}\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ Yang-Mills theory in the bulk coupled to an $\mathcal{N}=4,$ $d=3\mathrm{}$ hypermultiplet on a defect. This model was constructed by DeWolfe, Freedman, and Ooguri, and conjectured to be conformal based on its relation to an AdS configuration studied by Karch and Randall. We write this model in $\mathcal{N}=2,$ $d=3\mathrm{}$ superspace, which has the distinct advantage that nonrenormalization theorems become transparent. Using $\mathcal{N}=4,$ $d=3\mathrm{}$ supersymmetry, we argue that the model is conformal.