Global geometric properties of AdS space and the AdS/CFT correspondence

TL;DR

This paper explores the global geometric structure of Anti-de Sitter (AdS) space, revealing its isomorphism to a product space and demonstrating a bulk-boundary correspondence akin to AdS/CFT using these geometric insights.

Contribution

It establishes the isomorphism of AdS space to a product space and derives the bulk-boundary propagator based on these geometric properties.

Findings

AdS space is isomorphic to RP^1×B^n.

Boundary of AdS is isomorphic to RP^1×S^{n-1}.

Bulk-boundary propagator demonstrates an AdS/CFT-like correspondence.

Abstract

The Poisson kernels and relations between them for a massive scalar field in a unit ball $B^n$ with Hua's metric and conformal flat metric are obtained by describing the $B^n$ as a submanifold of an $(n+1)$-dimensional embedding space. Global geometric properties of the AdS space are discussed. We show that the $(n+1)$-dimensional AdS space AdS$_{n+1}$ is isomorphic to $RP^1\times B^n$ and boundary of the AdS is isomorphic to $RP^1\times S^{n-1}$. Bulk-boundary propagator and the AdS/CFT like correspondence are demonstrated based on these global geometric properties of the $RP^1\times B^n$.