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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, the AdS/CFT correspondence (anti-de-Sitter space/conformal field theory correspondence), sometimes called the Maldacena duality, is the conjectured equivalence between a string theory
String theory

String theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum gravity. The String s of string theory are one-dimensional oscillating lines, but they are no longer considered fundamental to the theory, which can be formulated in terms of points or surfaces too....
 defined on one space, and a quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
 without gravity defined on the conformal boundary
Boundary

Boundary may refer to:in mathematics,**A Boundary is the closure minus the interior of a subset of a topological space**The conditions of a boundary value problem in Mathematics...
 of this space, whose dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
 is lower by one or more. The name suggests that the first space is the product of anti de Sitter space
Anti de Sitter space

In mathematics and physics, n-dimensional anti de Sitter space, sometimes written , is a maximally symmetric Lorentzian manifold with constant negative scalar curvature....
 (AdS) with some closed manifold
Closed manifold

In mathematics, a closed manifold is a type of topological space, namely a compact space manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
 like sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
, orbifold
Orbifold

In the mathematical disciplines of topology and geometric group theory, an orbifold is a generalization of a manifold.It is a topological space with an orbifold structure ....
, or noncommutative space, and that the quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
 is a conformal field theory
Conformal field theory

A conformal field theory is a quantum field theory that is invariant under conformal symmetry. Conformal field theory is often studied in two-dimensional geometry dimensions where there is an infinite-dimensional group of local conformal transformations, described by the holomorphic functions....
 (CFT).

An example is the duality between Type IIB string theory on AdS5 × S5 space (a product of five dimensional AdS space with a five dimensional sphere
Hypersphere

In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real num...
) and a supersymmetric N=4 Yang-Mills gauge theory
Gauge theory

In physics, gauge theory is a quantum field theory where the Lagrangian is invariant under certain transformations.The transformations form a Lie group which is referred to as the symmetry group or the gauge group of the theory....
 (which is a conformal field theory
Conformal field theory

A conformal field theory is a quantum field theory that is invariant under conformal symmetry. Conformal field theory is often studied in two-dimensional geometry dimensions where there is an infinite-dimensional group of local conformal transformations, described by the holomorphic functions....
) on the 4-dimensional boundary of AdS5.






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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, the AdS/CFT correspondence (anti-de-Sitter space/conformal field theory correspondence), sometimes called the Maldacena duality, is the conjectured equivalence between a string theory
String theory

String theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum gravity. The String s of string theory are one-dimensional oscillating lines, but they are no longer considered fundamental to the theory, which can be formulated in terms of points or surfaces too....
 defined on one space, and a quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
 without gravity defined on the conformal boundary
Boundary

Boundary may refer to:in mathematics,**A Boundary is the closure minus the interior of a subset of a topological space**The conditions of a boundary value problem in Mathematics...
 of this space, whose dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
 is lower by one or more. The name suggests that the first space is the product of anti de Sitter space
Anti de Sitter space

In mathematics and physics, n-dimensional anti de Sitter space, sometimes written , is a maximally symmetric Lorentzian manifold with constant negative scalar curvature....
 (AdS) with some closed manifold
Closed manifold

In mathematics, a closed manifold is a type of topological space, namely a compact space manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
 like sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
, orbifold
Orbifold

In the mathematical disciplines of topology and geometric group theory, an orbifold is a generalization of a manifold.It is a topological space with an orbifold structure ....
, or noncommutative space, and that the quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
 is a conformal field theory
Conformal field theory

A conformal field theory is a quantum field theory that is invariant under conformal symmetry. Conformal field theory is often studied in two-dimensional geometry dimensions where there is an infinite-dimensional group of local conformal transformations, described by the holomorphic functions....
 (CFT).

An example is the duality between Type IIB string theory on AdS5 × S5 space (a product of five dimensional AdS space with a five dimensional sphere
Hypersphere

In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real num...
) and a supersymmetric N=4 Yang-Mills gauge theory
Gauge theory

In physics, gauge theory is a quantum field theory where the Lagrangian is invariant under certain transformations.The transformations form a Lie group which is referred to as the symmetry group or the gauge group of the theory....
 (which is a conformal field theory
Conformal field theory

A conformal field theory is a quantum field theory that is invariant under conformal symmetry. Conformal field theory is often studied in two-dimensional geometry dimensions where there is an infinite-dimensional group of local conformal transformations, described by the holomorphic functions....
) on the 4-dimensional boundary of AdS5. It is the most successful realization of the holographic principle
Holographic principle

The holographic principle is a property of quantum gravity theories which resolves the black hole information paradox within string theory. First proposed by Gerard 't Hooft, it was given a precise string-theory interpretation by Leonard Susskind....
, a speculative idea about quantum gravity
Quantum gravity

Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the Fundamental interaction , with general relativity, the theory of the fourth fundamental force: Gravitation....
 originally proposed by Gerard 't Hooft and improved and promoted by Leonard Susskind
Leonard Susskind

Leonard Susskind is the Felix Bloch professor of theoretical physics at Stanford University in the field of string theory and quantum field theory....
.

The AdS/CFT correspondence was originally proposed by Juan Maldacena in late 1997. Important aspects of the correspondence were given in articles by Gubser, Klebanov
Igor Klebanov

Igor R. Klebanov is a theoretical physicist whose research is centered on relations between string theory and quantum field theory. Born in Russia, he emigrated to the U.S....
 and Polyakov
Alexander Polyakov

Alexander M. Polyakov is a theoretical physicist, formerly at the Landau Institute for Theoretical Physics in Moscow, at Princeton University....
 and by Edward Witten
Edward Witten

Edward Witten is an United States theoretical physicist and professor at the Institute for Advanced Study. He is one of the world's leading researchers in superstring theory....
. The correspondence has also been generalized to many other (non-AdS) backgrounds and their dual (non-conformal) theories. In about five years, Maldacena's article had 3000 citations and became one of the most important conceptual breakthroughs in theoretical physics of the 1990s, providing many new lines of research into both quantum gravity and QCD
Quantum chromodynamics

Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....
.

The AdS/CFT correspondence should not be confused with algebraic holography
Algebraic holography

Algebraic holography, also sometimes called Rehren duality, is an attempt to understand the holographic principle of quantum gravity within the framework of algebraic quantum field theory, due to Karl-Henning Rehren....
 or "Rehren duality"; although these are sometimes identified with AdS/CFT, string theorists agree that they are different things.

Conformal boundary

A suitable Weyl transformation
Weyl transformation

In theoretical physics, the Weyl transformation is a local rescaling of the metric tensor:The invariance of a theory or an expression under this transformation is called the Weyl symmetry....
 assures that AdS has a boundary. If we do that, it turns out that its boundary is a conformal field theory having one less dimension. To make things more concrete, let's choose a particular coordinatization, the half-space coordinatization:

After a Weyl transformation ω = kz, we get

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

where ? is the conformal dimension of the local operator and k is the number of covariant indices of 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 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
Effective action

In quantum field theory, the effective action is a modified expression for the action , which takes into account quantum-mechanical corrections, in the following sense:...
 subject to the given boundary conditions.

Some 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 , 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 ? will be dual to a scalar bulk field with a bulk mass of .

Particles

A CFT bound state
Bound state

In physics, a bound state is a composite of two or more building blocks that behaves as a single object. In quantum mechanics , a bound state is a state in the Hilbert space that corresponds to two or more particles whose interaction energy is negative, and therefore these particles cannot be separated unless energy is spent....
 of size r is dual to a bulk particle
Subatomic particle

A subatomic particle is an elementary particle or composite particle particle smaller than an atom. Particle physics and nuclear physics are concerned with the study of these particles, their interactions, and non-atomic QCD matter....
 approximately localized at z=r.

Supersymmetry


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 real SUSY generators and the bosonic part consists of the conformal cum AdS group Spin(4,2) times an internal group . See superconformal algebra
Superconformal algebra

In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. It generates the superconformal group in some cases ....
 for more details.

For the case , we have 32 real SUSY generators and an internal group . Now, and Spin(6) is the isometry group
Isometry group

In mathematics, the isometry group of a metric space is the Set of all isometry from the metric space onto itself, with the function composition as group operation....
 of S5 with spinorial fields. The bosonic spatial isometry group of is .

In 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 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 with 30 generators. also has a symmetry and this is identified with .

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
Dilaton

Dilaton is a hypothetical particle that appears in Kaluza-Klein theory and string theory....
 and axion
Axion

The axion is a hypothetical elementary particle postulated by the Peccei-Quinn theory in 1977 to resolve the strong-CP problem in quantum chromodynamics ....
 field. They correspond to the gauge field coupling and theta angle of the dual superYang-Mills theory.

There are real SUSY generators with as the obligatory R-symmetry.

11D supergravity
Supergravity

In theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry ....
 contains 32 real SUSY generators. There is a particular compactification, , the Freund-Rubin compactification
Freund-Rubin compactification

11D supergravity contains a 3-form field C. It also contains an electric 2-brane and a magnetic 5-brane under C. These branes are BPS states and they are also black branes....
, which preserves all 32 real generators. The bosonic isometry group is reduced to . After a Kaluza-Klein decomposition over S7, we get SUSY. A 7-form magnetic flux is present over S7. Its integral over S7 has to be integral and nonzero.

See also


  • String theory
    String theory

    String theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum gravity. The String s of string theory are one-dimensional oscillating lines, but they are no longer considered fundamental to the theory, which can be formulated in terms of points or surfaces too....
  • Algebraic holography
    Algebraic holography

    Algebraic holography, also sometimes called Rehren duality, is an attempt to understand the holographic principle of quantum gravity within the framework of algebraic quantum field theory, due to Karl-Henning Rehren....
  • AdS/QCD
    AdS/QCD

    In theoretical physics, the AdS/QCD correspondence is a program to describe Quantum Chromodynamics in terms of a dual gravitational theory, following the principles of the AdS/CFT correspondence in a setup where the quantum field theory is not a conformal field theory....
  • Randall-Sundrum model
    Randall-Sundrum model

    In physics, Randall-Sundrum models imagine that the real world is a higher-dimensional Universe described by warped geometry. More concretely, our Universe is a five-dimensional anti de Sitter space and the elementary particles except for the graviton are localized on a -dimensional brane or branes....
  • Ambient construction
    Ambient construction

    In conformal geometry, the ambient construction refers to a construction of Charles Fefferman and Robin Graham for which a conformal manifold of dimension n is realized as the boundary of a certain hyperbolic manifold, or alternatively as the celestial sphere of a certain pseudo-Riemannian manifold....


External links


  • Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, .
  • Witten, Anti-de Sitter Space and Holography, .
  • Gubser, Klebanov and Polyakov, Gauge Theory Correlators from Non-Critical String Theory, .
  • Aharony et al, Large N Field Theories, String Theory and Gravity, . (261 pages of introductory text and review.)
  • - Juan Maldacena, Scientific American
    Scientific American

    Scientific American is a popular science science magazine, published since August 28, 1845, making it one of the oldest continuously published magazines in the United States....
    .
  • - Igor Klebanov
    Igor Klebanov

    Igor R. Klebanov is a theoretical physicist whose research is centered on relations between string theory and quantum field theory. Born in Russia, he emigrated to the U.S....
     and Juan Maldacena, Physics Today
    Physics Today

    Physics Today magazine, created in 1948, is the membership journal of The American Institute of Physics. It is provided to 130,000 members of twelve physics societies, including the American Physical Society....
    .
  • The state of the AdS/CFT conjecture on its 10th anniversary. Science News magazine, November 17, 2007