AdS geometry from CFT on a general conformally flat manifold

TL;DR

This paper demonstrates how to derive an AdS geometry from a CFT on a conformally flat manifold using a primary flow equation, revealing the deep connection between boundary conformal symmetry and bulk AdS structure.

Contribution

It introduces a primary flow equation approach to construct AdS geometry from CFT on general conformally flat manifolds, extending previous flat boundary results.

Findings

AdS geometry can be obtained from CFT on conformally flat manifolds.

The induced AdS metric is related to Poincare AdS via a bulk diffeomorphism.

Conformal symmetry at the boundary guarantees the emergence of AdS space.

Abstract

We construct an anti-de-Sitter (AdS) geometry from a conformal field theory (CFT) defined on a general conformally flat manifold via a flow equation associated with the curved manifold, which we refer to as the primary flow equation. We explicitly show that the induced metric associated with the primary flow equation becomes AdS whose boundary is the curved manifold. Interestingly, it turns out that such an AdS metric with conformally flat boundary is obtained from the usual Poincare AdS by a simple bulk diffeomorphism transformation. We also demonstrate that the emergence of such an AdS space is guaranteed only by the conformal symmetry at boundary, which converts to the AdS isometry after quantum averaging, as in the case of the flat boundary.