AdS CFT

This article is or was once a featured article on the Mathematics and Physics Wiki, which generally indicates a relatively well-written, or interesting article.

String Theory
All Roads Lead to String Theory (Polchinski)
Prior to the First Superstring Revolution
Early History	S-Matrix Theory Regge Trajectory
Bosonic String Theory	Worldsheet String Bosonic String Theory String Perturbation Theory Tachyon Condensation
Supersymmetric Revolution	Supersymmetry RNS Formalism GS Formalism BPS
Superstring Revolutions
First Superstring Revolution	GSO Projection Type II String Theory Type IIB String Theory Type IIA String Theory Type I String Theory Type H String Theory Type HO String Theory Type HE String Theory
Second Superstring Revolution	T-Duality D-Brane S-Duality Horava-Witten String Theory M-Theory Holographic Principle N=4 Super-Yang-Mills Theory AdS CFT BFSS Matrix Theory Matrix String Theory (2,0) Theory Twistor String Theory F-Theory String Field Theory Pure Spinor Formalism
After the Revolutions
Phenomenology	String Theory Landscape Minimal Supersymmetric Standard Model String Phenomenology

The AdS/CFT correspondence is an implementation of the Holographic Principle relating String Theory in Anti-de Sitter Space to a Conformal Field Theory on the conformal boundary of that space.

Intuition

Consider the Holographic Principle. The principle tells us that all information stored within a region is encoded on it's boundary. See Holographic Principle for the intuition behind this principle. Now, this applies to any region, including an Anti-de Sitter Space. The Anti-de Sitter Space has a conformal boundary, on which a Conformal Field Theory,or CFT, is defined. Therefore, a Quantum Gravitational Theory in Anti-de Sitter Space, or AdS (space) for short, is equivalent to a Conformal Field Theory on it's Conformal Boundary..

Examples

Maldacena's Original Example

Maldacena's paper discusses three examples of AdS/CFT duality. First and best-known is the duality between Type IIB String Theory on $AdS_5 \times S^5$, and N=4 Super-Yang-Mills Theory. Also considered were M-Theory on $AdS_4 \times S^7$, which is dual to ABJM Theory (but the equations for ABJM Theory were not figured out until 2008), and M-Theory on $AdS_7 \times S^4$, which is dual to (2,0) Theory.

Consider a stack of $N$ D3 Branes. They form Yang-Mills Supermultiplets with 4-Dimensional $\mathcal{N} = 4$ Supersymmetry. The vector hypermultiplets are transforming under a gauge group $U(N)\cong SU(N)\times U(1)$. In the infrared, the graviational dynamics and strings decouple, and the $U(1)$ hypermultiplet decouples, while the $SU(N)$ hypermultiplets remain interacting as the beta function is zero. Notice that we know have a Conformal Field Theory with $\mathcal{N} = 4$ Supersymmetry. Note that this metric background is that of an extremal 3-Brane black hole. The distance to the event horizon is logarithmically divergent, so it is infinitely far away. As one approaches the event horizon, the geometry approaches $AdS_5 \times S^5$. This is the original motivation for AdS/CFT .

AdS₅/CFT₄ example

The Stress-Energy Operator on the Conformal Field Theory is dual to the transverse components of the metric on the Anti-de Sitter.

The Conformal Boundary

Parameterise the Anti-de Sitter Space as follows:

$ds^2 = (kz)^{-2}\left( dz^2 + \eta_{\mu\nu} \, dx^\mu \, dx^\nu \right)$

Apply a Conformal Transformation , we get

$\mathrm{d}s^2 = \mathrm{d}z^2 + \eta_{\mu\nu}\mathrm{d}x^\mu \mathrm{d}x^\nu$

Which becomes the Minkowski Metric when $z=0$, which is the boundary. This is called the Conformal Boundary.

Mathematical Formulation

We deform the Conformal Field Theory by adding the Source Fields

$\int \mbox{d}x^D j_{CFT}(x)\mathcal{O}(x)$

This is now dual to a theory on Anti-de Sitter Space with a bulk field $j$. This space would have the boundary condition

$\lim\limits_{\mathrm{boundary} } j \omega^{\Delta-d+n} = J_{\text{CFT}}$

Here, $\Delta$ is the Conformal Dimension of the Gauge-Invariant Operator $\mathcal{O}$ and $n$ is the number of covariant indices of $\mathcal{O}$ minus the number of contravariant indices.

$\left\langle \mathcal{T}\left\{ \exp\left(\int \mathrm{d}x^D J_{4D}(x)\mathcal{O}(x)\right) \right\} \right\rangle_{\mathrm{CFT}} = Z_{\mathrm{AdS}}\left[\lim_{\mathrm{boundary}} J \omega^{\Delta-D+n } = J_{4D}\right]$

The LHS is the Vacuum Expectation Value of the time-ordered exponential of the operators over the Conformal Field Theory. The right hand side is the generating functional for the Quantum Gravity with the given Conformal Boundary Condition. The RHS is evaluated by finding the classical solutions to the effective action subject to the given boundary conditions.

Black Hole Firewalls

AdS CFT strongly suggests that the formation and evaporation of a black hole are unitary. It has however been argued , , , that this does not allow for a smooth event horizon to a black hole. The argument for this goes that the CFT lacks certain operators that describe the interior.

However, in , such operators were constructed by Suvrat Raju and Kyriakos Papadodimias.

Related dualities

AdS/CMT

AdS/CMT is an application of AdS/CFT to apply String Theory to Condensed Matter Physics. See .

AdS/QCD

AdS/QCD is a generalisation of AdS/CFT in which the "CFT" is not a real CFT, but instead Quantum Chromodynamics, which is not a Conformal Field Theory, but still a Quantum Field Theory. See and

Kerr/CFT

This section is empty. You can help by adding to it.

Fluid/Gravity

This section is empty. You can help by adding to it.