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

A simple cosmological model with only six parameters (matter density, Omega_m h^2, baryon density, Omega_b h^2, Hubble Constant, H_0, amplitude of fluctuations, sigma_8, optical depth, tau, and a slope for the scalar perturbation spectrum, n_s) fits not only the three year WMAP temperature and polarization data, but also small scale CMB data, light element abundances, large-scale structure observations, and the supernova luminosity/distance relationship. Using WMAP data only, the best fit values for cosmological parameters for the power-law flat LCDM model are (Omega_m h^2, Omega_b h^2, h, n_s, tau, sigma_8) = 0.1277+0.0080-0.0079, 0.02229+-0.00073, 0.732+0.031-0.032, 0.958+-0.016, 0.089+-0.030, 0.761+0.049-0.048). The three year data dramatically shrink the allowed volume in this six dimensional parameter space. Assuming that the primordial fluctuations are adiabatic with a power law spectrum, the WMAP data_alone_ require dark matter, and favor a spectral index that is significantly less than the Harrison-Zel'dovich-Peebles scale-invariant spectrum (n_s=1, r=0). Models that suppress large-scale power through a running spectral index or a large-scale cut-off in the power spectrum are a better fit to the WMAP and small scale CMB data than the power-law LCDM model: however, the improvement in the fit to the WMAP data is only Delta chi^2 = 3 for 1 extra degree of freedom. The combination of WMAP and other astronomical data yields significant constraints on the geometry of the universe, the equation of state of the dark energy, the gravitational wave energy density, and neutrino properties. Consistent with the predictions of simple inflationary theories, we detect no significant deviations from Gaussianity in the CMB maps.

/pdf/wilkinson-microwave-anisotropy-probe-wmap-three-year-results-2l0o1lq78p.pdf

Wilkinson Microwave Anisotropy Probe (WMAP) three year results: implications for cosmology

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

Large N Field Theories, String Theory and Gravity

We extend earlier ideas about the appearance of noncommutative geometry in string theory with a nonzero B-field. We identify a limit in which the entire string dynamics is described by a minimally coupled (supersymmetric) gauge theory on a noncommutative space, and discuss the corrections away from this limit. Our analysis leads us to an equivalence between ordinary gauge fields and noncommutative gauge fields, which is realized by a change of variables that can be described explicitly. This change of variables is checked by comparing the ordinary Dirac-Born-Infeld theory with its noncommutative counterpart. We obtain a new perspective on noncommutative gauge theory on a torus, its T-duality, and Morita equivalence. We also discuss the D0/D4 system, the relation to M-theory in DLCQ, and a possible noncommutative version of the six-dimensional (2,0) theory.

https://iopscience.iop.org/article/10.1088/1126-6708/1999/09/032/pdf

String theory and noncommutative geometry

We suggest and motivate a precise equivalence between uncompactified 11-dimensional $M$ theory and the $N=\ensuremath{\infty}$ limit of the supersymmetric matrix quantum mechanics describing $D0$ branes. The evidence for the conjecture consists of several correspondences between the two theories. As a consequence of supersymmetry the simple matrix model is rich enough to describe the properties of the entire Fock space of massless well separated particles of the supergravity theory. In one particular kinematic situation the leading large distance interaction of these particles is exactly described by supergravity. The model appears to be a nonperturbative realization of the holographic principle. The membrane states required by $M$ theory are contained as excitations of the matrix model. The membrane world volume is a noncommutative geometry embedded in a noncommutative spacetime.

/pdf/m-theory-as-a-matrix-model-a-conjecture-1zi5yivqwj.pdf

M theory as a matrix model: A conjecture

We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λ
 L
 ≤ 2πk
 B
 T/ℏ. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.

A bound on chaos

We use holography to study sensitive dependence on initial conditions in strongly coupled field theories. Specifically, we mildly perturb a thermofield double state by adding a small number of quanta on one side. If these quanta are released a scrambling time in the past, they destroy the local two-sided correlations present in the unperturbed state. The corresponding bulk geometry is a two-sided AdS black hole, and the key effect is the blueshift of the early infalling quanta relative to the t = 0 slice, creating a shock wave. We comment on string- and Planck-scale corrections to this setup, and discuss points that may be relevant to the firewall controversy.

/pdf/black-holes-and-the-butterfly-effect-4v604bigor.pdf

Black holes and the butterfly effect

https://www.physics.rutgers.edu/~friedan/papers/Nucl_Phys_B271_93_1986.pdf

Conformal invariance, supersymmetry and string theory

https://web.phys.ntu.edu.tw/string/matrix/NuclPhysB335(1990)635_Douglas_Shenker_StringsInLessThanOneDimension.pdf

Stephen H. Shenker

Papers

M theory as a matrix model: A conjecture

A bound on chaos

Black holes and the butterfly effect

Conformal invariance, supersymmetry and string theory

Strings in less than one dimension