Reconstructing AdS/CFT

TL;DR

This paper clarifies the AdS/CFT correspondence by establishing a detailed dictionary between quantum gravity in the bulk and boundary CFTs, including a formula linking 2D CFT partition functions to 3D quantum gravity wave functionals.

Contribution

It provides a precise quantum-level dictionary between 2D CFTs and pure 3D gravity, including an explicit formula for reconstructing bulk states from boundary data.

Findings

Established a correspondence between quantum gravity radial states and pairs of CFTs.

Presented an explicit formula linking 2D CFT partition functions to 3D Wheeler-DeWitt wave functionals.

Clarified the boundary conditions corresponding to asymptotic infinity in the holographic setup.

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.