Holographic Renormalization

TL;DR

The paper develops and applies the holographic renormalization framework to holographic RG flows described by asymptotically AdS domain walls, showing how a finite S_ren and covariant counterterms yield well-defined, anomaly-aware correlators. It provides a systematic near-boundary analysis, derives Ward identities, and demonstrates how bulk fluctuations—including gravity, scalars, and vectors—inform the full set of 1- and 2-point functions. Explicit results are obtained for two known D=5 supergravity flows, CB and GPPZ, including new vector-current correlators and detailed stress-tensor sector data, with careful treatment of Goldstone poles and beta-function relations. The work clarifies how nonlocal information in correlators arises from bulk dynamics beyond local near-boundary data and validates holographic renormalization as a complete, background-independent method for computing CFT data from gravity duals. Its findings deepen the connection between bulk renormalization, boundary anomalies, and the structure of correlation functions in deformed CFTs.

Abstract

We systematically develop the procedure of holographic renormalization for RG flows dual to asymptotically AdS domain walls. All divergences of the on-shell bulk action can be cancelled by adding covariant local boundary counterterms determined by the near-boundary behavior of bulk fields. This procedure defines a renormalized action from which correlation functions are obtained by functional differentiation. The correlators are finite and well behaved at coincident points. Ward identities, corrected for anomalies, are satisfied. The correlators depend on parts of the solution of the bulk field equations which are not determined by near-boundary analysis. In principle a full nonlinear solution is required, but one can solve linearized fluctuation equations to define a bulk-to-boundary propagator from which 2-point correlation functions are easily obtained. We carry out the procedure explicitly for two known RG flows obtained from the maximal gauged D=5 supergravity theory, obtaining new results on correlators of vector currents and related scalar operators and giving further details on a recent analysis of the stress tensor sector.