Reconstructing AdS/CFT

TL;DR

This work reframes AdS/CFT from a background‑independent quantum gravity viewpoint, showing that the radial Wheeler–DeWitt equation generically yields two boundary CFTs as asymptotic data and that asymptotically AdS spacetimes correspond to a single CFT by setting one branch to zero. It provides a constructive bulk reconstruction via a state‑independent kernel that maps boundary CFT data to the bulk wavefunctional, and proves a concrete 2+1 reconstruction formula relating a 2D CFT partition function to a 3D Wheeler–DeWitt wavefunctional. The analysis connects holographic renormalization with quantum gravitational dynamics, derives a Hamilton–Jacobi expansion that reproduces counterterm structure, and clarifies the first‑order (Cartan–Weyl) formulation's role in the bulk/boundary dictionary. The results illuminate the non‑uniqueness of the CFT–gravity map, the interpretation of radial states as quantum spacetimes, and offer a concrete path toward higher‑dimensional bulk reconstruction while highlighting open questions about Euclidean/Lorentzian relations and irreducibility of the boundary theories.

Abstract

In this note we clarify the dictionary between pure quantum gravity on the bulk in the presence of a cosmological constant and a CFT on the boundary. We show for instance that there is a general correspondence between quantum gravity ``radial states'' and a pair of CFT's. Restricting to one CFT is argued to correspond to states possessing an asymptotic infinity. This point of view allows us to address the problem of reconstructing the bulk from the boundary. And in the second part of this paper we present an explicit formula which gives, from the partition function of any 2 dimensional conformal field theory, a wave functional solution to the 3-dimensional Wheeler-DeWitt equation. This establishes at the quantum level a precise dictionary between 2d CFT and pure gravity.