Anti De Sitter Space And Holography

TL;DR

This paper discusses the AdS/CFT correspondence, linking conformal field theories to supergravity in Anti-de Sitter space, and confirms this with matching spectra and phase transition insights.

Contribution

It elaborates on the precise correspondence between CFT observables and supergravity, including correlation functions and operator dimensions, and confirms the duality with spectral matches.

Findings

Kaluza-Klein modes match with chiral operators in N=4 super Yang-Mills.

Correlation functions in CFT relate to supergravity action dependence.

Large N phase transition linked to AdS black hole thermodynamics.

Abstract

Recently, it has been proposed by Maldacena that large $N$ limits of certain conformal field theories in $d$ dimensions can be described in terms of supergravity (and string theory) on the product of $d+1$-dimensional $AdS$ space with a compact manifold. Here we elaborate on this idea and propose a precise correspondence between conformal field theory observables and those of supergravity: correlation functions in conformal field theory are given by the dependence of the supergravity action on the asymptotic behavior at infinity. In particular, dimensions of operators in conformal field theory are given by masses of particles in supergravity. As quantitative confirmation of this correspondence, we note that the Kaluza-Klein modes of Type IIB supergravity on $AdS_5\times {\bf S}^5$ match with the chiral operators of $\N=4$ super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the $\N=4$ theory has a large $N$ phase transition related to the thermodynamics of $AdS$ black holes.