The Holographic Renormalization Group

TL;DR

The holographic renormalization group establishes a local, spacetime-dependent RG framework by applying the Hamilton-Jacobi formalism to five-dimensional gravity with scalars. It derives holographic beta-functions $\beta^I(\phi)$ from a decomposed action $S=S_{\rm loc}+\Gamma$, obtains Callan-Symanzik equations for boundary correlators, and reproduces the conformal anomaly with holographic coefficients $c$ and $d$ tied to the bulk potential and couplings. The approach unifies domain-wall/domain-wall-like flows, brane-world dynamics, and warped compactifications, linking radial AdS evolution to boundary RG data and illuminating issues such as the cosmological constant problem and IR boundary conditions. It also sets the stage for extensions to higher-form fields and fermions, and connects with other RG formalisms in string theory, including Polchinski’s exact RG. Overall, it provides a robust, geometrical route to compute RG quantities from bulk gravity and to study a broad class of holographic applications.

Abstract

In this lecture, we review the derivation of the holographic renormalization group given in hep-th/9912012. Some extra background material is included, and various applications are discussed.