Four-Dimensional Superconformal Theories with Interacting Boundaries or Defects

TL;DR

The work develops a controlled framework for four-dimensional superconformal field theories coupled to three-dimensional boundary or defect degrees of freedom by exploiting an $\mathcal{N}=2$, d=3 superspace formulation embedded in $\mathcal{N}=2$, d=4 language. It presents abelian and non-abelian constructions, shows that bulk-boundary couplings preserve the relevant superconformal symmetries, and proves perturbative finiteness via Callan-Symanzik analysis and non-renormalization arguments. A central achievement is the demonstration that beta functions vanish to all orders in both abelian and (assumed unbroken) $\mathcal{N}=4$, d=3 cases, ensuring conformality in the presence of a boundary or defect; the boundary fields can acquire anomalous dimensions without breaking the overall conformal structure. The results have potential implications for holographic duals with interfaces or defects, and point toward rich avenues for boundary/defect conformal field theories in higher-dimensional holography and beyond.

Abstract

We study four-dimensional superconformal field theories coupled to three-dimensional superconformal boundary or defect degrees of freedom. Starting with bulk N=2, d=4 theories, we construct abelian models preserving N=2, d=3 supersymmetry and the conformal symmetries under which the boundary/defect is invariant. We write the action, including the bulk terms, in N=2, d=3 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 N=4 SU(N) Yang-Mills theory in the bulk coupled to a N=4, d=3 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 N=2, d=3 superspace, which has the distinct advantage that non-renormalization theorems become transparent. Using N=4, d=3 supersymmetry, we argue that the model is conformal.

Figures (4)

• Figure 1: A $z$ dependent contribution to a bulk-bulk propagator.
• Figure 2: One loop contribution to the bulk propagator.
• Figure 3: One loop contribution to the boundary propagator.
• Figure 4: One loop contribution to the boundary propagator.