General · EdgepediaSeptember 17, 20268 min read

AdS/CFT correspondence

The AdS/CFT correspondence is a conjectured equivalence between string or M-theory formulated on spaces with anti-de Sitter (AdS) asymptotics and conformal field theories (CFTs) living in fewer dimensions on the boundary of those spaces. It was proposed by Juan Maldacena in November 1997 and is the sharpest known realization of the holographic principle, the idea that the physics of a gravitational bulk can be encoded in a nongravitational theory on its boundary.¹^•²

Key fact	Detail
Original proposal	Maldacena, "The Large N Limit of Superconformal Field Theories and Supergravity", submitted 27 November 1997 (hep-th/9711200)¹
Canonical example	Four-dimensional $\mathcal{N}=4$ super-Yang-Mills theory with gauge group SU(N) is dual to type IIB string theory on $AdS_{5} \times S^{5}$ ¹^•³
Derivation route	Take branes in M/string theory and a low-energy limit in which the field theory on the brane decouples from the bulk; the near-horizon geometry remains trustworthy at large N¹
Correlator prescription	Gauge-theory correlation functions are related to the dependence of the supergravity action on boundary conditions at the AdS boundary (Gubser, Klebanov, Polyakov 1998)³
Viscosity result	For a large class of theories with AdS black-hole duals, $\eta/s = \hbar/4\pi k_{B} \approx 6.08 \times 10^{-13} \ \mathrm{K \cdot s}$ , proposed as a lower bound (Kovtun, Son, Starinets 2004)⁴
Entanglement formula	Entanglement entropy of a boundary region equals the area of a minimal bulk surface divided by $4G$ (Ryu-Takayanagi, 2006)⁵
Status	A conjecture with a great deal of supporting evidence but no general proof⁶

Origin in brane physics

Maldacena's argument began with branes, extended objects in M/string theory on which open strings can end. Taking a low-energy limit in which the field theory living on the branes decouples from excitations in the surrounding bulk, he showed that the large N limit of certain conformal field theories contains in its Hilbert space a sector describing supergravity on the product of Anti-de Sitter spacetimes, spheres, and other compact manifolds. In this limit the near-horizon geometry of the branes can still be trusted, and the enhanced supersymmetries of that geometry match the extra supersymmetry generators of the superconformal group, as opposed to the super-Poincaré group alone.¹

The central case is the 't Hooft limit of four-dimensional $\mathcal{N}=4$ super-Yang-Mills theory at the conformal point, which Maldacena showed contains strings, specifically type IIB strings. He conjectured that compactifications of M/string theory on various Anti-de Sitter spacetimes are dual to various conformal field theories, and noted that this gives a new proposal for a definition of M-theory.¹

The two early companion papers

Two 1998 papers turned the conjecture into a usable dictionary. Edward Witten, in "Anti De Sitter Space And Holography", developed the proposal that large N limits of certain conformal field theories in d dimensions are described by supergravity and string theory on the product of (d+1)-dimensional AdS space with a compact manifold.⁷ Gubser, Klebanov, and Polyakov showed how to obtain Green's functions in 3+1-dimensional $\mathcal{N}=4$ super-Yang-Mills theory with a large number of colors: correlation functions of gauge-theory operators are related to the dependence of the supergravity action on boundary conditions at the AdS boundary. From the quadratic terms in supergravity they read off anomalous dimensions, and they predicted that operators coupling to massive string states at level n acquire anomalous dimensions growing as $2(n g_{YM} \sqrt{2N})^{1/2}$ at large 't Hooft coupling, a new prediction about strong-coupling behavior of large N SYM theory.³

The comprehensive 1999 review by Ofer Aharony, Steven S. Gubser, Juan Maldacena, Hirosi Ooguri, and Yaron Oz (a Rutgers-Harvard-IAS-Berkeley-CERN collaboration) organized the field, describing the evidence for the correspondence, the results derived in the regime where the field theory is approximated by classical or semiclassical gravity, and extensions to non-conformal theories and to QCD.²

What the duality claims

In its strong form, the correspondence states that the dynamics of string theory, or quantum gravity, in asymptotically AdS spacetimes is encoded in a field theory without gravity.⁸ When the two sides form a precisely defined dual pair, the statement is an exact duality between complete theories, not merely a relation between opposite corners of perturbation theory; the boundary CFT supplies a nonperturbative definition of the quantum gravity theory in the bulk.⁹ The practical power comes from the coupling regimes: one theory's strong-coupling regime, where perturbation theory fails, maps to the other's weak-coupling regime where calculations are tractable, and vice versa.⁶ The often-quoted slogan that AdS/CFT computes strongly coupled field theory using classical gravity refers to the special corner where both the rank N and the 't Hooft coupling $\lambda$ are large.¹⁰

Despite the accumulated evidence, the correspondence remains a conjecture; no general proof exists.⁶

Black holes and finite temperature

AdS spacetime admits black-hole solutions whose thermodynamics translates directly into field-theory language. The Hawking-Page transition, at which the free energy of an AdS black hole crosses that of thermal AdS, is interpreted through the correspondence as a phase transition in the dual field theory, associated with confinement.¹¹ At finite temperature the dual of a uniform, strongly coupled plasma is a planar black brane: the Hawking temperature and Bekenstein-Hawking entropy of the brane are identified with the gauge-theory temperature and entropy, and the energy-momentum tensor of the plasma can be read from the expansion of the bulk metric near the boundary.¹²

Entanglement and the emergence of geometry

In 2006 Shinsei Ryu and Tadashi Takayanagi, then at the Kavli Institute for Theoretical Physics, proposed that the entanglement entropy of a region in a (d+1)-dimensional conformal field theory can be obtained from the area of d-dimensional minimal surfaces in $AdS_{d+2}$ , in direct analogy to the Bekenstein-Hawking formula for black hole entropy. The proposal agrees with the known entanglement entropy of two-dimensional CFTs when applied to $AdS_{3}$ .⁵ The formula reads

$S_{EE}(A) = \frac{\mathcal{A}(\gamma_{A})}{4 G_{d+2}}$

where $\gamma_{A}$ is the minimal surface whose boundary is homologous to the entangling surface. The proposal was subsequently proved and extended to include quantum corrections and to CFTs dual to higher-derivative theories of gravity.¹³ When the minimal surface is a black-hole horizon, holographic entanglement entropy reproduces the Bekenstein-Hawking entropy, an observation that supports the idea that spacetime is built from entanglement.¹⁴

Viscosity and the quark-gluon plasma

A striking quantitative result came from Kovtun, Son, and Starinets in 2004. Applying techniques from string theory, they demonstrated that in a broad family of strongly coupled quantum field theories with a dual formulation in terms of black holes in anti-de Sitter space, the shear viscosity divided by the entropy density assumes a universal value

$\frac{\eta}{s} = \frac{\hbar}{4\pi k_{B}} \approx 6.08 \times 10^{-13} \ \mathrm{K \cdot s}$

and they provided evidence that this value may serve as a lower bound for a wide class of systems, suggesting that black hole horizons are dual to the most ideal fluids.⁴ The bound is not universal: Buchel, Myers, and Sinha argued in 2008 that it is generically violated in superconformal gauge theories with non-equal central charges $c \neq a$ , in controllable settings.¹⁵

These results made AdS/CFT a tool for the quark-gluon plasma, the strongly coupled state of matter formed in relativistic heavy-ion collisions. Lattice QCD handles static thermodynamic quantities but works in Euclidean signature and cannot easily address real-time properties such as the plasma's expansion and approach to equilibrium; holographic methods capture nonperturbative plasma physics with comparatively simple analytical tools.¹² The comparison has limits. $\mathcal{N}=4$ super-Yang-Mills has no running coupling, an exactly conformal equation of state, and no confinement-deconfinement transition, so an expanding $\mathcal{N}=4$ plasma cools indefinitely; how well the simplest version of the duality models real QCD depends on the questions asked.¹²

Scope and open directions

The correspondence applies to spacetimes with AdS asymptotics, where a timelike conformal boundary hosts the dual theory. Our universe, with its positive cosmological constant, is better modeled by de Sitter space, and a naive translation of the AdS construction to de Sitter does not work; extending holography to de Sitter and to flat space remains an active research direction.¹⁶^•¹⁷ Within its domain, however, AdS/CFT provides what the field regards as a nonperturbative definition of quantum gravity in AdS spacetimes and a practical tool for computing strongly coupled field-theory observables using classical gravity.⁹

References

Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › Quantum gravity and unification › String-theoretic gravity and holography › AdS/CFT correspondence

Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026

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