AdS-CFT correspondence

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Originally conjectured by Maldacena in 1997^[1], the duality known as the anti de Sitter/Conformal Field Theory correspondence, AdS-CFT for short, is the statement that type IIB string theory on $\mathrm{AdS}_5 \times \mathbb{S}^5\,$ may be dual to a conformal field theory on the boundary of that space, namely $\mathcal{N}=4\,$ super Yang-Mills theory with gauge group $U(N)\,$ .

To develop the correspondence, one considers a configuration of $N\,$ coincident D3-branes in type IIB string theory. The lowest energy (massless) open string modes are described by 4-dimensional $\mathcal{N}=4\,$ super Yang-Mills theory on the branes. If we take the large $N\,$ limit while simultaneously taking the string coupling $g_s\,$ to be small (such that $g_s N\,$ is kept fixed), then the configuration can be described as a solution to type IIB supergravity, namely an extremal black p-brane. It is found that a number of thermodynamic quantities on the string theory side agree with of those on the gravity side, in the limit that $\alpha' \to 0\,$ in both theories. This limit is realized as the near-horizon region of the black p-brane, which has the geometry of $\mathrm{AdS}_5 \times \mathbb{S}^5\,$ , and as the level of massless excitations on the brane. This leads to the conjecture that type IIB supergravity on the 10-dimensional space $\mathrm{AdS}_5 \times \mathbb{S}^5\,$ may be dual to $\mathcal{N}=4\,$ SYM in 4 dimensions. The $\mathrm{AdS}\,$ geometry is insensitive to rescalings of $\alpha'\,$ , and therefore all mass levels are expected to participate on the supergravity side. Therefore, it is believed that the full type IIB string theory on $\mathrm{AdS}_5 \times \mathbb{S}^5\,$ is equivalent to $\mathcal{N}=4\,$ SYM on the 4-dimensional boundary of that space.

1 Background
2 Correspondence
3 Finite Temperature
4 Integrability
5 See also
6 References

Background

Prerequisites

Large-N

It was realized by 't Hooft that Feynman diagrams in $SU(N)\,$ gauge theories organize themselves into groups that resemble the topological expansion of the string ^[2]. Specifically, each diagram depends on the gauge coupling $g_{YM}\,$ and number of colours $N\,$ according to

$(g_{YM}^2)^{E-V} N^F = \lambda^{E-V} N^{2-2g} = (g_{YM}^2)^{2-2g} \lambda^F\,$ ,

where $\lambda = g_{YM}^2 N\,$ is the 't Hooft coupling and $V\,$ , $E\,$ and $F\,$ are the number of vertices, edges and loops in the diagram, respectively. We have made use of the identity obeyed by the Euler characteristic, $\chi = V-E+I = 2 - 2g\,$ for a surface of genus $g\,$ . If we take $N\,$ to be large, holding $\lambda\,$ fixed, then the perturbation series of Feynman diagrams resembles the topological expansion of the string partition function, where the string coupling is $g_s \sim g_{YM}^2 \sim 1/N\,$ .

Holography

Correspondence

Each D-brane couples to the metric with strength $g_s\,$ , so that a stack of $N\,$ coincident D-branes source the metric with strength $g_s N\,$ . In the limit that $g_s N \ll 1\,$ , the branes gravitate weakly, whereas if $g_s N \gg 1\,$ , spacetime becomes highly curved in the vicinity of the branes.

Geometry

The background $\mathrm{AdS}\,$ geometry arises from expanding the string theory around the configuration of $N\,$ parallel D3-branes. The $10\,$ -dimensional metric is ^[3]

$ds^2 = \left(1 + \frac{L^4}{y^4} \right)^{-\frac{1}{2}} \eta_{ij}dx^i dx^j + \left(1 + \frac{L^4}{y^4} \right)^{\frac{1}{2}} \left( dy^2 + y^2 d\Omega_5^2 \right)\,$ ,

where $L^4 = 4\pi g_s N (\alpha')^2\,$ is the radius of the D3-brane, while $\eta_{ij} = \operatorname{diag}(- +++)\,$ is the metric on $4\,$ -dimensional Minkowski space. The limit $y \gg L\,$ corresponds to flat $10\,$ -dimensional Minkowski space. To analyze the decoupling limit $\alpha' \to 0\,$ , let us perform a rescaling so that $\tilde{y} = \tfrac{y}{{\alpha'}}\,$ . Then

$ds^2 =\left(1 + \frac{4\pi g_s N}{ {\alpha'}^2 \tilde y^4} \right)^{-\frac{1}{2}} \eta_{ij}dx^i dx^j + \left( 1 + \frac{4\pi g_s N}{ {\alpha'}^2 \tilde y^4} \right)^{\frac{1}{2}} {\alpha'}^2 \left( d\tilde y^2 + \tilde y^2 d\Omega_5^2 \right)\,$ .

Taking the limit $\alpha' \to 0\,$ while holding $\tilde{y}\,$ fixed corresponds to "dropping the 1", and is equivalent to the limit $y \ll L\,$ , which is the so-called throat. This region can be analyzed by change of coordinate to $u = \tfrac{L^2}{y}\,$ and taking $u\,$ to be large, for which

$ds^2 = L^2 \left[ \frac{ \eta_{ij}dx^i dx^j + du^2 }{u^2} + d\Omega_5^2 \right]\,$ .

The throat geometry, close to the branes, is therefore that of $\mathrm{AdS}_5 \times \mathbb{S}^5\,$ (anti de Sitter space times the 5-sphere), and the two spaces have identical radii $L\,$ .

Symmetry

Note that the isometry group of $\mathrm{AdS}_{d+1}\,$ is $SO(2,d)\,$ while the conformal group on $d\,$ -dimensional Minkowski space is isomorphic to $SO(2,d)\,$ also.

Field-operator correspondence

The correspondence may be visualized as follows: The string theory on $\mathrm{AdS}_{d+1} \times \mathcal{M}\,$ can be thought of as a collection of infinitely many quantum field theories. Suppose $\phi\,$ is one of these fields, e.g. a massless scalar field on the bulk, and $\phi_0\,$ is its boundary data (i.e. $\phi\,$ approaches $\phi_0\,$ at the boundary). The bulk field $\phi\,$ is related to a gauge invariant boundary operator $\mathcal{O}\,$ . Specifically, $\phi_0\,$ couples to a conformal field $\mathcal{O}\,$ on the boundary via $\int_{\mathbb{S}^d} \phi_0 \mathcal{O}\,$ . The correspondence is then that

$\left\langle e^{ \int_{\mathbb{S}^d} \phi_{i,0} \mathcal{O}^i } \right\rangle_{CFT} = Z_S[\phi_i]\,$ ,

where $Z_S\,$ is the string theory partition function, so that $\phi_0\,$ may be considered a source for $\mathcal{O}\,$ . This is the precise form of the correspondence, but in practice string theory on $\mathrm{AdS}\,$ is too difficult to quantize. We therefore usually use a semiclassical approximation in which we write

$Z_S[\phi_i]\approx e^{-S_{eff}[\phi_i]}\,$

where $S_{eff}\,$ is the low-energy effective action and $\phi_i\,$ are the fields corresponding to the low-lying excitations of the full type IIB theory. A weaker form of the correspondence is then the approximation

$\left\langle e^{ \int_{\mathbb{S}^d} \phi_{i,0} \mathcal{O}^i } \right\rangle_{CFT} \approx e^{- S_{eff}[\phi_i]}\,$ .

correlation functions of operators $\mathcal{O}^i\,$ in the CFT can be found by functionally differentiating with respect to $\phi_i\,$ .

To refine the relationship between $\phi\,$ and $\phi_0\,$ , we need to make use of the bulk-boundary propagator which is the Green's function which solves the equations of motion for $\phi(u, \mathbf{x})\,$ with the inclusion of a source $\phi_0(\mathbf{x})\,$ on the boundary. Generally, $\phi\,$ behaves $\phi(u) \sim \phi_0 u^{2h_-} + \tilde{\phi} u^{2h_+}\,$ as $u \to 0\,$ . The leading term $\phi_0\,$ is dual to the source term $\phi_0 \mathcal{O}\,$ on the CFT, while the sub-leading term $\tilde{\phi}\,$ is dual to the VEV, $\langle\mathcal{O}\rangle\,$ , calculated in the presence of the source term $\phi_0 \mathcal{O}\,$ ^[4].

Correlation functions

For 3-point functions, see ^[5] ^[6] ^[7]

Parameters

There are $N\,$ units of Ramond-Ramond self-dual 5-form flux through $\mathbb{S}^5\,$ , i.e.,

$\int_{\mathbb{S}^5} F_5 = N\,$ .

The SYM coupling $g_{YM}\,$ is related to the string coupling

$g_s = \frac{g_{YM}^2}{2\pi}\,$ ,

while the 't Hooft coupling $\lambda = g_{YM}^2 N\,$ is related to the AdS scale $L\,$ and string length $\ell_s\,$ through

$\lambda = \frac{1}{2} \frac{L^4}{\ell_s^4}\,$ .

To have perturbative control over the string theory side we demand a weak curvature and weak coupling. In other words, large $\lambda\,$ and consequently large $N\,$ .

Wilson loops

The expectation value of some Wilson loop operator in the gauge theory is given by the string partition function evaluated over strings that have the loop as their boundary.^[8] ^[9] That is,

$\left\langle W[C] \right\rangle = \int_{\partial X = C}\! \mathcal{D}X \mathcal{D}\Theta \mathcal{D}g \, e^{-S[X,\Theta,g]}\,$ ,

where $X\,$ and $\Theta\,$ are the Bosonic and Fermionic coordinates of the string.

Since N=4 SYM does not contain quarks in the fundamental representation, the Wilson loop in question is motivated by considering first quantized action of a massive W-boson obtained by first breaking the $SU(N+1)\,$ symmetry into $SU(N)\times U(1)\,$ . The relevant operator is then^[10]

$W[C] = \mathrm{Tr}\, \mathcal{P} \exp {i \int_0^1 \!ds\, A_\mu \dot{x}^\mu + \left|\dot{x}\right|\Phi_i \theta^i}$ , (Lorentzian signature),

where $\theta^i\,$ may be considered a variables over $\mathbb{S}^5\,$ , and $\Phi_i\,$ are six scalars in the adjoint representation. This expression may also be seen as the dimensional reduction of the Wilson loop of the 10-dimensional N=1 super Yang-Mills theory.

Finite Temperature

At finite temperature, a black hole appears in the supergravity background^[11] with temperature $T=\frac{\kappa}{2\pi}\,$ where $\kappa\,$ is the surface gravity at the horizon. The Hawking-Bekenstein entropy

$\frac{S}{V_x} = \frac{A}{4G V_x} = \frac{\pi^6}{4G}\frac{L^8}{\beta^3} = \frac{\pi^2}{2} N^2 T^3\,$ ,

using $16 \pi G = (2\pi)^7 \ell_s^8 g_s^2\,$ , gives a prediction of the entropy density of the gauge theory at strong coupling, which differs from the weak coupling result^[12] by a factor of $\tfrac{3}{4}\,$ . Here $V_x\,$ is the (formally infinite) volume of Minkowski space.