Flow equation, conformal symmetry and AdS geometry

TL;DR

This paper demonstrates that AdS geometry naturally arises from conformal field theories through the free flow equation, linking conformal symmetry to AdS isometry without explicit metric calculations.

Contribution

It shows that the AdS metric emerges from conformal field theories via the free flow equation and connects conformal symmetry to AdS isometry.

Findings

Induced metric from flowed fields matches the quantum information metric.

Explicit calculations confirm the emergence of AdS geometry.

Conformal symmetry transforms into AdS isometry after quantum averaging.

Abstract

We argue that the Anti-de-Sitter (AdS) geometry in d+1 dimensions naturally emerges from an arbitrary conformal field theory in d dimensions using the free flow equation. We first show that an induced metric defined from the flowed field generally corresponds to the quantum information metric, called the Bures or Helstrom metric, if the flowed field is normalized appropriately. We next verify that the induced metric computed explicitly with the free flow equation always becomes the AdS metric when the theory is conformal. We finally prove that the conformal symmetry in d dimensions converts to the AdS isometry in d+1 dimensions after d dimensional quantum averaging. This guarantees the emergence of AdS geometry without explicit calculation.