Integrating out geometry: Holographic Wilsonian RG and the membrane paradigm

TL;DR

The paper develops a holographic Wilsonian renormalization group by mapping boundary-scale integration to bulk geometry integration, formulating a flow equation for a boundary action S_B that encodes single- and multi-trace couplings in the large-N limit. By pushing the cut-off surface into infrared regions (e.g., toward black hole horizons or AdS2 throats), it isolates low-energy dynamics and yields a refined membrane-paradigm description, including a semi-holographic interface for emergent IR CFTs. The authors illustrate the framework with free scalar and vector fields, derive explicit beta functions for double-trace couplings, and analyze diffusion on the stretched horizon, connecting holographic RG to hydrodynamics and semi-holography. They also discuss how gapless UV modes influence IR physics and how to treat these modes explicitly within the effective action. The work provides a versatile, geometric approach to low-energy holographic dynamics and sets the stage for incorporating metric fluctuations and full hydrodynamic behavior.

Abstract

We formulate a holographic Wilsonian renormalization group flow for strongly coupled systems with a gravity dual, motivated by the need to extract efficiently low energy behavior of such systems. Starting with field theories defined on a cut-off surface in a bulk spacetime, we propose that integrating out high energy modes in the field theory should correspond to integrating out a part of the bulk geometry. We describe how to carry out this procedure in practice in the classical gravity approximation using examples of scalar and vector fields. By integrating out bulk degrees of freedom all the way to a black hole horizon, this formulation defines a refined version of the black hole membrane paradigm. Furthermore, it also provides a derivation of the semi-holographic description of low energy physics.

Figures (2)

• Figure 1: A schematic description of the effect of integrating out degrees of freedom in the field theory and a corresponding picture of the dual bulk spacetime, with boundary energy scales $\Lambda < \Lambda' < \Lambda_0$ and bulk radial coordinates $\epsilon > \epsilon' > \epsilon_0$. The boundary of the spacetime lies in the limit $z \to0$.
• Figure 2: A simple sketch of extremal black hole geometries with an infinite AdS$_2$ throat and the process of defining effective field theories via the holographic Wilsonian RG flow. To extract low energy physics, it is convenient to integrate out the bulk geometry all the way to the boundary of AdS$_2$, i.e. $z=\epsilon$ surface in the figure. The induced effective action on $z=\epsilon$ can be interpreted as multiple-trace deformations of the CFT$_1$ dual to AdS$_2$.