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 ${\cal N}=4$ super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the ${\cal N}=4$ theory has a large $N$ phase transition related to the thermodynamics of $AdS$ black holes.

/pdf/anti-de-sitter-space-and-holography-58a7vi1qn6.pdf

Anti De Sitter Space And Holography

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.

Anti de sitter space and holography

http://inspirehep.net/record/499969/files/arXiv:hep-th_9905111.pdf

Large N Field Theories, String Theory and Gravity

The correspondence between supergravity (and string theory) on AdS space and boundary conformal eld theory relates the thermodynamics of N = 4 super Yang-Mills theory in four dimensions to the thermodynamics of Schwarzschild black holes in Anti-de Sitter space. In this description, quantum phenomena such as the spontaneous breaking of the center of the gauge group, magnetic connement, and the mass gap are coded in classical geometry. The correspondence makes it manifest that the entropy of a very large AdS Schwarzschild black hole must scale \holographically" with the volume of its horizon. By similar methods, one can also make a speculative proposal for the description of large N gauge theories in four dimensions without supersymmetry.

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/1998/0002/0003/ATMP-1998-0002-0003-a003.pdf

Anti-de Sitter Space, Thermal Phase Transition, And Confinement in Gauge Theories

Modified Gravity and Cosmology

The aim of this paper is to provide a general introduction to the problem of time in quantum gravity. This problem originates in the fundamental conflict between the way the concept of ‘time’ is used in quantum theory, and the role it plays in a diffeomorphism-invariant theory like general relativity. Schemes for resolving this problem can be sub-divided into three main categories: (I) approaches in which time is identified before quantising; (II) approaches in which time is identified after quantising; and (III) approaches in which time plays no fundamental role at all. Ten different specific schemes are discussed in this paper which also contain an introduction to the relevant parts of the canonical decomposition of general relativity.

Canonical quantum gravity and the problem of time

Recent developments in quantum theory have focused attention on fundamental questions, in particular on whether it might be necessary to modify quantum mechanics to reconcile quantum gravity and general relativity This book is based on a conference held in Oxford in the spring of 1984 to discuss quantum gravity It brings together contributors who examine different aspects of the problem, including the experimental support for quantum mechanics, its strange and apparently paradoxical features, its underlying philosophy, and possible modifications to the theory

Quantum Concepts in Space and Time

We consider the problem of quantizing scalar fields propagating in anti-de Sitter space-time. This space-time is static but not globally hyperbolic and hence the usual quantization procedures are inapplicable. Nevertheless, we show that a consistent quantization scheme can be devised by carefully controlling information entering and leaving the space-time through its timelike spatial infinity.

Quantum field theory in anti-de Sitter space-time

Non-local conformal anomalies

A Lagrangian theory is formulated describing the intrinsic mixing of the graviton with a massive ${2}^{+}$ $f$ meson which interacts universally with hadrons through the stress tensor. The theory is developed as an analog of the well-known $\ensuremath{\rho}\ensuremath{-}\ensuremath{\gamma}$ model of hadron electrodynamics, and in particular a field-current identity is exhibited which equates the massive ${2}^{+}$ meson with the hadronic energy-momentum tensor. An Einstein-type Lagrangian is used for both spin-2 particles, and general covariance is preserved throughout. The nonlinear coupling of the hadrons to the $f$ meson leads, within the framework of nonpolynomial field theories, to a universal cutoff for strong-interaction physics.

C. J. Isham

Papers

Canonical quantum gravity and the problem of time

Quantum Concepts in Space and Time

Quantum field theory in anti-de Sitter space-time

Non-local conformal anomalies

f -Dominance of Gravity