AdS/CFT correspondence

For the relation of the AdS/CFT correspondence to the general context of string theory, see String theory § Gauge-gravity duality.

The conformal field theory, which may be a gauge theory lies on the conformal boundary of anti deSitter space with quantum gravity.

In theoretical physics, the AdS/CFT correspondence (anti de Sitter/conformal field theory correspondence), sometimes called the Maldacena duality or gauge/gravity duality, is the conjectured equivalence between a string theory with gravity defined on one space ${\mathcal {S}}$ (or more precisely: on a product space involving ${\mathcal {S}}$), and a quantum field theory without gravity defined on the conformal boundary of ${\mathcal {S}}$, whose dimension is lower by one or more. The name refers to the fact that the first space is typically the product of an anti de Sitter space (AdS) with some closed manifold, like spheres, orbifolds, or noncommutative spaces, while the dual theory is a conformal field theory (CFT).

The duality represents a major advance in our understanding of string theory and quantum gravity, and a powerful toolkit for studying strongly-coupled quantum field theories. In particular, it provides a non-perturbative formulation of string theory within the superselection sector of the given asymptotic structure of spacetime, and is also the most successful realization of the holographic principle, an idea in quantum gravity originally proposed by Gerard 't Hooft and improved and promoted by Leonard Susskind. Much of the AdS/CFT's usefulness derives from the fact that it is a weak-strong duality (S-duality?): one description is weakly-coupled and thus amenable to study within perturbation theory precisely when the other is not, and vice-versa. This has lead to a great number of insights flowing in both directions between the CFT and the gravity sides. It is also why proving the conjecture remains a difficult challenge, while allowing for many impressive checks of its consistency.

By far the most well-studied example is the duality between Type IIB string theory on AdS₅ × S⁵ space (a product of 5-dimensional AdS space with a 5-dimensional sphere) and a N=4 super Yang-Mills gauge theory (which is a conformal field theory) on the 4-dimensional boundary of AdS₅. This was also historically the first AdS/CFT correspondence proposed by Juan Maldacena in late 1997. Similar dualities have been used to study M-theory on Freund–Rubin compactifications, that is, on AdS₄ × S⁷ and AdS₇ × S⁴. Important aspects of the correspondence were given in articles by Steven Gubser, Igor Klebanov and Alexander Markovich Polyakov, and by Edward Witten. The correspondence has also been generalized to many other (non-AdS) backgrounds and their dual (non-conformal) theories. It has been used to study various systems in condensed matter physics and quantum chromodynamics (QCD), and opened up many new lines of research into quantum gravity. In about five years, Maldacena's article had 3000 citations, becoming one of the most important conceptual breakthroughs in theoretical physics of the 1990s.

Maldacena's example

Comparisons between a stack of D3-branes in type IIB string theory and the extremal charged 3-brane metric background were carried out by Klebanov and later sharpened by Maldacena. One starts with the observation that a stack of N D3-branes in type IIB string theory has massless brane fields residing on it. With respect to the brane, they form Yang–Mills supermultiplets transforming under N = 4 SUSY in 3+1D. The vector hypermultiplets form a gauge group U(N) ≅ SU(N) × U(1). This isn't quite a conformal field theory, even though it runs to one in the infrared once gravitation and string dynamics decouple. In the infrared, the U(1) hypermultiplet decouples, but the SU(N) hypermultiplets remain interacting as the beta function is zero. The metric background is given by an extremal 3-brane black hole. The event horizon is infinitely far away; the distance to it diverges logarithmically. The near horizon geometry is approximately AdS₅ × S⁵ with the approximation becoming more and more exact closer to the horizon. Now, take the scaling limit as the string scale goes to zero with the string coupling kept fixed. All the string and gravitational dynamics decouple, and the U(1) hypermultiplet too. We are left with a bona fide N = 4 superconformal field theory. If we take the limit in which we are always in the near horizon region, the geometry becomes exactly AdS₅ × S⁵. A D3-brane has a self-dual charge under the self-dual NS 5-form flux. A stack of N of them gives rise to an integral flux of N over S⁵

Conformal boundary

A suitable Weyl transformation assures that AdS has a conformal boundary. It turns out that this boundary supports a conformal field theory having one fewer dimension. To make things more concrete, choose a particular coordinatization, the half-space coordinatization:

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

where, e.g., η_μν are Minkowski's diag(+,-,-,-).

After "factorizing away" the denominator (by a Weyl transformation ω = kz), we get

$\ ds^{2}=dz^{2}+\eta _{\mu \nu }\,dx^{\mu }\,dx^{\nu },$

which has the Minkowski metric as the boundary at z = 0. This is called the conformal boundary.

Source fields

Basically, the correspondence runs as follows; if we deform the CFT by certain source fields by adding the source

$\int d^{d}xJ_{CFT}(x){\mathcal {O}}(x),$

this will be dual to an AdS theory with a bulk field J with the boundary condition

$\lim _{\text{boundary}}J\omega ^{\Delta -d+k}=J_{\text{CFT}}\,$

where Δ is the conformal dimension of the local operator ${\mathcal {O}}$ and k is the number of covariant indices of ${\mathcal {O}}$ minus the number of contravariant indices. Only gauge-invariant operators are allowed.

Here, we have a dual source field for every gauge-invariant local operator we have.

Using generating functionals, the relation is expressed as

$\left\langle {\mathcal {T}}\left\{\exp \left(\int d^{d}xJ_{4D}(x){\mathcal {O}}(x)\right)\right\}\right\rangle _{CFT}=Z_{AdS}\left[\lim _{\text{boundary}}J\omega ^{\Delta -d+k}=J_{4D}\right].$

The left hand side is the vacuum expectation value of the time-ordered exponential of the operators over the conformal field theory. The right hand side is the quantum gravity generating functional with the given conformal boundary condition. The right hand side is evaluated by finding the classical solutions to the effective action subject to the given boundary conditions.

Some AdS₅/CFT₄ examples

The stress-energy operator on the CFT side is dual to the transverse components of the metric on the AdS side. Since the stress-energy operator has a conformal weight of 4, the AdS metric ought to go as ω^-2}, which is true for AdS. Also, the graviton has to be massless, just as it should.

If there is a global internal symmetry G on the CFT side, its Noether current J will be dual to the transverse components of a gauge connection for a Yang–Mills gauge theory with G as the gauge group on the AdS side. Since J has a conformal weight of 3, the dual Yang–Mills gauge boson ought to have zero bulk mass, just as it should.

A scalar operator with conformal weight Δ will be dual to a scalar bulk field with a bulk mass of $k{\sqrt {\Delta (\Delta -4)}}$.

Particles

A CFT bound state of size r is dual to a bulk particle approximately localized at z=r.

Supersymmetry

AdS₅/CFT₄

We need to match up conformal supersymmetry in 4D with AdS supersymmetry in 5D. The symmetry supergroups in both cases happen to match up, as they should. There are 8N real SUSY generators and the bosonic part consists of the conformal AdS group Spin(4,2) times an internal group SU(N)_T × U(1)_A. See superconformal algebra for more details.

For the case N = 4, we have 32 real SUSY generators and an internal group SU(4)_T × U(1)_A. Now, SU(4) ≅ Spin(6) and Spin(6) is the isometry group of S⁵ with spinorial fields. The bosonic spatial isometry group of AdS₅ × S⁵ is Spin(4,2) × Spin (6).

In N = (2,0) 10D SUSY, we have 32 real SUSY generators. In a generic curved spacetime, some of the SUSY generators will be broken but in the special compactification of AdS₅ × S⁵ with both factors having the same radius, we are left 32 real unbroken generators. However, the bosonic spatial isometries with 55 generators in the flat case is now broken to Spin(4,2) × Spin(6) with 30 generators. N = (2,0) also has a U(1)_R symmetry and this is identified with U(1)_A.

The source of the curvature lies in the nonzero value of a self-dual 5-form flux belonging to the SUGRA multiplet. The integral of this 5-flux over S⁵ has to be a nonzero integer (if it's zero, we have no stress-energy tensor). Because the part of the 5-flux lying in AdS₅ contains a time component, it gives rise to negative curvature. The part of the 5-flux lying in S⁵ doesn't have a time component, and so, it gives rise to a positive curvature.

The SUGRA multiplet also contains a dilaton and axion field. They correspond to the gauge field coupling and theta angle of the dual superYang–Mills theory.

AdS₄/CFT₃

There are 4N real SUSY generators with Spin(N) as the obligatory R-symmetry.

11D N = 1 supergravity contains 32 real SUSY generators. There is a particular compactification, AdS₄ × S⁷, the Freund–Rubin compactification, which preserves all 32 real generators. The bosonic isometry group is reduced to Spin(3,2) × Spin(8). After a Kaluza–Klein decomposition over S⁷, we get N = 8 SUSY. A 7-form magnetic flux is present over S⁷. Its integral over S⁷ has to be integer and nonzero.

Applications

A plethora of papers is found in the literature which uses techniques of AdS/CFT to understand strongly coupled high-energy systems such as the RHIC and LHC experiments (Au-Au rsp. Pb-Pb collisions), and intricate condensed matter systems, or genuine problems of gravity physics, by relating, e.g., the entropy production problem in heavy-ion collisions to the entropy characterization of Black Hole physics.

Other topics

Certain "higher spin gauge theories" on AdS space appear to be holographically dual to a CFT with O(N) symmetry. This has been called the Klebanov–Polyakov correspondence.

The AdS/CFT correspondence should not be confused with algebraic holography or "Rehren duality"; although these are sometimes identified with AdS/CFT, string theorists agree that they are different things.

See also

• AdS/QCD
• dS/CFT correspondence
• Randall–Sundrum model
• Ambient construction

External links

• AdS/CFT correspondence on arxiv.org.
• [1] – Juan Maldacena, Scientific American.
• University of Michigan Lecture
• Professor Hirosi Ooguri's notes
• Meeting on the subject
• Database of articles; Supersymmetric Gauge Theories and the AdS/CFT Correspondence
• Shadow World. The state of the AdS/CFT conjecture on its 10th anniversary. Science News magazine, November 17, 2007.
• [2] A compilation of several papers deemed important