Anomaly-corrected supersymmetry algebra and supersymmetric holographic renormalization

TL;DR

The paper constructs a comprehensive holographic renormalization program for 5D ${\cal N}=2$ gauged supergravity with fermions, enabling a holographic derivation of the 4D ${\cal N}=1$ superconformal Ward identities and their Weyl and super-Weyl anomalies. By solving the Hamilton–Jacobi equations for the radial evolution, it identifies the full set of divergent and finite boundary counterterms and demonstrates that the resulting anomalies satisfy Wess–Zumino consistency. It shows that the supercurrent and SUSY algebra receive anomalous contributions on curved backgrounds with conformal Killing spinors, leading to an anomaly-corrected SUSY algebra and modified SUSY transformation rules for operators. The work also discusses boundary-condition consistency (Dirichlet/Neumann choices) and illustrates the framework with a toy model, highlighting implications for holography, SUSY on curved spaces, and potential insights for SUSY localization and higher-derivative SUGRA constructions.

Abstract

We present a systematic approach to supersymmetric holographic renormalization for a generic 5D $\mathcal{N}=2$ gauged supergravity theory with matter multiplets, including its fermionic sector, with all gauge fields consistently set to zero. We determine the complete set of supersymmetric local boundary counterterms, including the finite counterterms that parameterize the choice of supersymmetric renormalization scheme. This allows us to derive holographically the superconformal Ward identities of a 4D superconformal field theory on a generic background, including the Weyl and super-Weyl anomalies. Moreover, we show that these anomalies satisfy the Wess-Zumino consistency condition. The super-Weyl anomaly implies that the fermionic operators of the dual field theory, such as the supercurrent, do not transform as tensors under rigid supersymmetry on backgrounds that admit a conformal Killing spinor, and their anticommutator with the conserved supercharge contains anomalous terms. This property is explicitly checked for a toy model. Finally, using the anomalous transformation of the supercurrent, we obtain the anomaly-corrected supersymmetry algebra on curved backgrounds admitting a conformal Killing spinor.