In
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the
AdS/CFT correspondence (
anti de SitterIn mathematics and physics, n-dimensional anti de Sitter space, sometimes written AdS_n, is a maximally symmetric Lorentzian manifold with constant negative scalar curvature...
/conformal field theoryA conformal field theory is a quantum field theory that is invariant under conformal transformations...
correspondence), sometimes called the
Maldacena duality, is the conjectured equivalence between a
string theoryString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
and gravity defined on one space, and a
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
without gravity defined on the conformal
boundaryIn topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
of this space, whose
dimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
is lower by one or more. The name suggests that the first space is the product of
anti de Sitter spaceIn mathematics and physics, n-dimensional anti de Sitter space, sometimes written AdS_n, is a maximally symmetric Lorentzian manifold with constant negative scalar curvature...
(AdS) with some
closed manifoldIn mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
like
sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
,
orbifoldIn the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
, or noncommutative space, and that the quantum field theory is a
conformal field theoryA conformal field theory is a quantum field theory that is invariant under conformal transformations...
(CFT).
An example is the
dualityIn mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
between Type IIB string theory on AdS
5 ×
S5 space (a product of five dimensional AdS space with a five dimensional sphere) and a supersymmetric
N = 4
Yang–MillsYang–Mills theory is a gauge theory based on the SU group. Wolfgang Pauli formulated in 1953 the first consistent generalization of the five-dimensional theory of Kaluza, Klein, Fock and others to a higher dimensional internal space...
gauge theoryIn physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
(which is a
conformal field theoryA conformal field theory is a quantum field theory that is invariant under conformal transformations...
) on the 4-dimensional boundary of AdS
5. It is the most successful realization of the
holographic principleThe holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon...
, a speculative idea about
quantum gravityQuantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
originally proposed by Gerard 't Hooft and improved and promoted by
Leonard SusskindLeonard Susskind is the Felix Bloch Professor of Theoretical Physics at Stanford University. His research interests include string theory, quantum field theory, quantum statistical mechanics and quantum cosmology...
.
The AdS/CFT correspondence was originally proposed by Juan Maldacena in late 1997. Important aspects of the correspondence were given in articles by
Steven GubserSteven S. Gubser is a professor of physics at Princeton University. His research focuses on theoretical particle physics, especially string theory, and the AdS/CFT correspondence. He is a widely cited scholar in these and other related areas....
,
Igor KlebanovIgor R. Klebanov is a theoretical physicist whose research is centered on relations between string theory and quantum gauge field theory. Since 1989, he has been a Professor at Princeton University....
and Alexander Markovich Polyakov, and by
Edward WittenEdward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
. 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 gravityQuantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
and
quantum chromodynamicsIn theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
(QCD).
Maldacena's example
Maldacena's initial observation was 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
SUSY in 3+1D. The vector hypermultiplets form a gauge group
. 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
hypermultiplet decouples, but the
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
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
superconformal field theory. If we take the limit in which we are always in the near horizon region, the geometry becomes exactly
. 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
Conformal boundary
A suitable
Weyl transformation assures that AdS has a boundary. It turns out that this boundary is a conformal field theory having one less dimension. To make things more concrete, 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 Δ is the
conformal dimensionIn mathematics, the conformal dimension of a metric space X is the infimum of the Hausdorff dimension over the conformal gauge of X, that is, the class of all metric spaces quasisymmetric to X.-Formal definition:...
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 actionIn 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 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
, 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
.
Particles
A CFT
bound stateIn physics, a bound state describes a system where a particle is subject to a potential such that the particle has a tendency to remain localised in one or more regions of space...
of size
r is dual to a bulk
particleIn physics or chemistry, subatomic particles are the smaller particles composing nucleons and atoms. There are two types of subatomic particles: elementary particles, which are not made of other particles, and composite particles...
approximately localized at
z=
r.
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
real SUSY generators and the bosonic part consists of the conformal AdS group Spin(4,2) times an internal group
. See
superconformal algebraIn 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 .In two dimensions, the superconformal algebra is infinite-dimensional...
for more details.
For the case
, we have 32 real SUSY generators and an internal group
. Now,
and Spin(6) is the
isometry groupIn mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...
of S
5 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 AdS
5 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
dilatonIn particle physics, a dilaton is a hypothetical particle. It also appears in Kaluza-Klein theory's compactifications of extra dimensions when the volume of the compactified dimensions vary....
and
axionThe 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.
AdS4/CFT3
There are
real SUSY generators with
as the obligatory R-symmetry.
11D
supergravityIn 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, which preserves all 32 real generators. The bosonic isometry group is reduced to
. After a Kaluza–Klein decomposition over S
7, we get
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 system such as RHIC and
condensed matterCondensed matter may refer to several things*Condensed matter physics, the study of the physical properties of condensed phases of matter*European Physical Journal B: Condensed Matter and Complex Systems, a scientific journal published by EDP sciences...
systems.
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 holographyAlgebraic 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. It is sometimes described as an alternative formulation of the AdS/CFT...
or "Rehren duality"; although these are sometimes identified with AdS/CFT, string theorists agree that they are different things.
See also
- String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
- 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.Such an alternative...
- dS/CFT correspondence
In string theory, the dS/CFT correspondence is a de Sitter space analogue of the AdS/CFT correspondence, proposed originally by Andrew Strominger.- External links :* at the String Theory Wiki...
- Randall–Sundrum model
In physics, Randall–Sundrum models imagine that the real world is a higher-dimensional Universe described by warped geometry...
- 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 Poincaré manifold, or alternatively as the celestial sphere of a certain pseudo-Riemannian...
External links