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.
Highlights
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 prove that the conformal symmetry in d dimensions converts to the AdS isometry in d +1 dimensions after d-dimensional quantum averaging
The anti-de Sitter/conformal field theory (AdS/CFT) correspondence [1] is a promising tool to crack a hard problem in strongly coupled gauge theories, but is still mysterious even after many pieces of evidence and application appeared after the first proposal
Summary
The anti-de Sitter/conformal field theory (AdS/CFT) (or gravity/gauge theory) correspondence [1] is a promising tool to crack a hard problem in strongly coupled gauge theories (see Refs. [2,3,4] for some reviews), but is still mysterious even after many pieces of evidence and application appeared after the first proposal. 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 verify that the induced metric computed explicitly with the free flow equation always becomes the AdS metric when the theory is conformal.
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