Spacetime and the Holographic Renormalization Group

TL;DR

The authors introduce a Wilsonian holographic RG framework for interior AdS volumes, foliating spacetime by surfaces ${\partial{\cal M}}_{\rho}$ and deriving an RG equation that governs the flow of inner correlators as a function of the foliation parameter $\rho$. By integrating out the exterior region, they obtain a nonlocal boundary action $Z_\rho[{\Phi}_{\rho}]$ that encodes the physics of the interior and show how inner correlators relate to outer CFT correlators via a convolution with a kernel $G_{\epsilon\rho}$. In the Poincaré patch they derive explicit RG flow equations in momentum space and discuss how, in the semiclassical limit, bulk field equations may be mapped to RG constraints in the CFT, hinting that bulk dynamics could emerge from boundary RG structure. Overall, the work connects interior holography to RG flow geometry, proposing that spacetime itself may arise from the geometry of RG flows between coarsened boundary theories.

Abstract

Anti-de Sitter (AdS) space can be foliated by a family of nested surfaces homeomorphic to the boundary of the space. We propose a holographic correspondence between theories living on each surface in the foliation and quantum gravity in the enclosed volume. The flow of observables between our ``interior'' theories is described by a renormalization group equation. The dependence of these flows on the foliation of space encodes bulk geometry.