AdS/CFT correspondence

In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called
Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical
theories. On one side are anti-de Sitter spaces (AdS) which are used in theories of quantum gravity, formulated
in terms of string theory or M-theory. On the other side of the correspondence are conformal field theories
(CFT) which are quantum field theories, including theories similar to the Yang–Mills theories that describe
elementary particles.

The duality represents a major advance in the understanding of string theory and quantum gravity.[1] This is
because it provides a non-perturbative formulation of string theory with certain boundary conditions and
because it is the most successful realization of the holographic principle, an idea in quantum gravity originally
proposed by Gerard 't Hooft and promoted by Leonard Susskind.

It also provides a powerful toolkit for studying strongly coupled quantum field theories.[2] Much of the
usefulness of the duality results from the fact that it is a strong-weak duality: when the fields of the quantum
field theory are strongly interacting, the ones in the gravitational theory are weakly interacting and thus more
mathematically tractable. This fact has been used to study many aspects of nuclear and condensed matter
physics by translating problems in those subjects into more mathematically tractable problems in string theory.

The AdS/CFT correspondence was first proposed by Juan Maldacena in late 1997. Important aspects of the
correspondence were elaborated in articles by Steven Gubser, Igor Klebanov, and Alexander Polyakov, and
by Edward Witten. By 2015, Maldacena's article had over 10,000 citations, becoming the most highly cited
article in the field of high energy physics,[3] reaching almost 20,000 citations by 2020.

Contents
Background
Quantum gravity and strings
Quantum field theory
Overview of the correspondence
The geometry of anti-de Sitter space
The idea of AdS/CFT
Examples of the correspondence
Applications to quantum gravity
A non-perturbative formulation of string theory
Black hole information paradox
Applications to quantum field theory
Nuclear physics
Condensed matter physics
Criticism
History and development
String theory and nuclear physics
Black holes and holography
 Maldacena's paper
Generalizations
Three-dimensional gravity
dS/CFT correspondence
Kerr/CFT correspondence
Higher spin gauge theories
See also
Notes
References

Background

Quantum gravity and strings

Current understanding of gravity is based on Albert Einstein's general theory of relativity.[4] Formulated in
1915, general relativity explains gravity in terms of the geometry of space and time, or spacetime. It is
formulated in the language of classical physics[5] developed by physicists such as Isaac Newton and James
Clerk Maxwell. The other nongravitational forces are explained in the framework of quantum mechanics.
Developed in the first half of the twentieth century by a number of different physicists, quantum mechanics
provides a radically different way of describing physical phenomena based on probability.[6]

Quantum gravity is the branch of physics that seeks to describe gravity using the principles of quantum
mechanics. Currently, a popular approach to quantum gravity is string theory,[7] which models elementary
particles not as zero-dimensional points but as one-dimensional objects called strings. In the AdS/CFT
correspondence, one typically considers theories of quantum gravity derived from string theory or its modern
extension, M-theory.[8]

In everyday life, there are three familiar dimensions of space (up/down, left/right, and forward/backward), and
there is one dimension of time. Thus, in the language of modern physics, one says that spacetime is four-
dimensional.[9] One peculiar feature of string theory and M-theory is that these theories require extra
dimensions of spacetime for their mathematical consistency: in string theory spacetime is ten-dimensional,
while in M-theory it is eleven-dimensional.[10] The quantum gravity theories appearing in the AdS/CFT
correspondence are typically obtained from string and M-theory by a process known as compactification. This
produces a theory in which spacetime has effectively a lower number of dimensions and the extra dimensions
are "curled up" into circles.[11]

A standard analogy for compactification is to consider a multidimensional object such as a garden hose. If the
hose is viewed from a sufficient distance, it appears to have only one dimension, its length, but as one
approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant
crawling inside it would move in two dimensions.[12]

Quantum field theory

The application of quantum mechanics to physical objects such as the electromagnetic field, which are
extended in space and time, is known as quantum field theory.[13] In particle physics, quantum field theories
form the basis for our understanding of elementary particles, which are modeled as excitations in the
fundamental fields. Quantum field theories are also used throughout condensed matter physics to model
particle-like objects called quasiparticles.[14]

In the AdS/CFT correspondence, one considers, in addition to a theory of quantum gravity, a certain kind of
quantum field theory called a conformal field theory. This is a particularly symmetric and mathematically well
behaved type of quantum field theory.[15] Such theories are often studied in the context of string theory, where
they are associated with the surface swept out by a string propagating through spacetime, and in statistical
mechanics, where they model systems at a thermodynamic critical point.[16]

Overview of the correspondence

The geometry of anti-de Sitter space

In the AdS/CFT correspondence, one considers string theory or M-

theory on an anti-de Sitter background. This means that the geometry
of spacetime is described in terms of a certain vacuum solution of
Einstein's equation called anti-de Sitter space.[17]

In very elementary terms, anti-de Sitter space is a mathematical model

of spacetime in which the notion of distance between points (the
metric) is different from the notion of distance in ordinary Euclidean
geometry. It is closely related to hyperbolic space, which can be
viewed as a disk as illustrated on the right.[18] This image shows a A tessellation of the hyperbolic plane
tessellation of a disk by triangles and squares. One can define the by triangles and squares.
distance between points of this disk in such a way that all the triangles
and squares are the same size and the circular outer boundary is
infinitely far from any point in the interior.[19]

Now imagine a stack of hyperbolic disks where each disk represents the state of the universe at a given time.
The resulting geometric object is three-dimensional anti-de Sitter space.[18] It looks like a solid cylinder in
which any cross section is a copy of the hyperbolic disk. Time runs along the vertical direction in this picture.
The surface of this cylinder plays an important role in the AdS/CFT correspondence. As with the hyperbolic
plane, anti-de Sitter space is curved in such a way that any point in the interior is actually infinitely far from
this boundary surface.[20]

This construction describes a hypothetical universe with only two space and one time dimension, but it can be
generalized to any number of dimensions. Indeed, hyperbolic space can have more than two dimensions and
one can "stack up" copies of hyperbolic space to get higher-dimensional models of anti-de Sitter space.[18]

The idea of AdS/CFT

An important feature of anti-de Sitter space is its boundary (which looks like a cylinder in the case of three-
dimensional anti-de Sitter space). One property of this boundary is that, locally around any point, it looks just
like Minkowski space, the model of spacetime used in nongravitational physics.[21]

One can therefore consider an auxiliary theory in which "spacetime" is given by the boundary of anti-de Sitter
space. This observation is the starting point for AdS/CFT correspondence, which states that the boundary of
anti-de Sitter space can be regarded as the "spacetime" for a conformal field theory. The claim is that this
conformal field theory is equivalent to the gravitational theory on the bulk anti-de Sitter space in the sense that
there is a "dictionary" for translating calculations in one theory into calculations in the other. Every entity in
 one theory has a counterpart in the other theory.
For example, a single particle in the gravitational
theory might correspond to some collection of
particles in the boundary theory. In addition, the
predictions in the two theories are quantitatively
identical so that if two particles have a 40 percent
chance of colliding in the gravitational theory,
then the corresponding collections in the
boundary theory would also have a 40 percent
chance of colliding.[22]

Three-dimensional anti-de Sitter space is like a stack of

hyperbolic disks, each one representing the state of the
universe at a given time. The resulting spacetime looks like
a solid cylinder.

Notice that the boundary of anti-de Sitter space has fewer dimensions
than anti-de Sitter space itself. For instance, in the three-dimensional
example illustrated above, the boundary is a two-dimensional surface.
The AdS/CFT correspondence is often described as a "holographic
duality" because this relationship between the two theories is similar A hologram is a two-dimensional
to the relationship between a three-dimensional object and its image as image which stores information about
a hologram.[23] Although a hologram is two-dimensional, it encodes all three dimensions of the object it
represents. The two images here are
information about all three dimensions of the object it represents. In
photographs of a single hologram
the same way, theories which are related by the AdS/CFT
taken from different angles.
correspondence are conjectured to be exactly equivalent, despite
living in different numbers of dimensions. The conformal field theory
is like a hologram which captures information about the higher-
dimensional quantum gravity theory.[19]

Examples of the correspondence

Following Maldacena's insight in 1997, theorists have discovered many different realizations of the AdS/CFT
correspondence. These relate various conformal field theories to compactifications of string theory and M-
theory in various numbers of dimensions. The theories involved are generally not viable models of the real
world, but they have certain features, such as their particle content or high degree of symmetry, which make
them useful for solving problems in quantum field theory and quantum gravity.[24]

The most famous example of the AdS/CFT correspondence states that type IIB string theory on the product
space is equivalent to N = 4 supersymmetric Yang–Mills theory on the four-dimensional
boundary. [25] In this example, the spacetime on which the gravitational theory lives is effectively five-
dimensional (hence the notation ), and there are five additional compact dimensions (encoded by the
factor). In the real world, spacetime is four-dimensional, at least macroscopically, so this version of the
correspondence does not provide a realistic model of gravity. Likewise, the dual theory is not a viable model of
any real-world system as it assumes a large amount of supersymmetry. Nevertheless, as explained below, this
boundary theory shares some features in common with quantum chromodynamics, the fundamental theory of
the strong force. It describes particles similar to the gluons of quantum chromodynamics together with certain
fermions.[7] As a result, it has found applications in nuclear physics, particularly in the study of the quark–
gluon plasma.[26]

Another realization of the correspondence states that M-theory on is equivalent to the so-called
(2,0)-theory in six dimensions. [27] In this example, the spacetime of the gravitational theory is effectively
seven-dimensional. The existence of the (2,0)-theory that appears on one side of the duality is predicted by the
classification of superconformal field theories. It is still poorly understood because it is a quantum mechanical
theory without a classical limit.[28] Despite the inherent difficulty in studying this theory, it is considered to be
an interesting object for a variety of reasons, both physical and mathematical.[29]

Yet another realization of the correspondence states that M-theory on is equivalent to the ABJM
superconformal field theory in three dimensions.[30] Here the gravitational theory has four noncompact
dimensions, so this version of the correspondence provides a somewhat more realistic description of
gravity.[31]

Applications to quantum gravity

A non-perturbative formulation of string theory

In quantum field theory, one typically computes the

probabilities of various physical events using the
techniques of perturbation theory. Developed by Richard
Feynman and others in the first half of the twentieth
century, perturbative quantum field theory uses special
diagrams called Feynman diagrams to organize
computations. One imagines that these diagrams depict
the paths of point-like particles and their interactions.[32]
Although this formalism is extremely useful for making
predictions, these predictions are only possible when the
strength of the interactions, the coupling constant, is Interaction in the quantum world: world lines of
small enough to reliably describe the theory as being point-like particles or a world sheet swept up by
closed strings in string theory.
close to a theory without interactions.[33]

The starting point for string theory is the idea that the
point-like particles of quantum field theory can also be modeled as one-dimensional objects called strings. The
interaction of strings is most straightforwardly defined by generalizing the perturbation theory used in ordinary
quantum field theory. At the level of Feynman diagrams, this means replacing the one-dimensional diagram
representing the path of a point particle by a two-dimensional surface representing the motion of a string.
Unlike in quantum field theory, string theory does not yet have a full non-perturbative definition, so many of
the theoretical questions that physicists would like to answer remain out of reach.[34]

The problem of developing a non-perturbative formulation of string theory was one of the original motivations
for studying the AdS/CFT correspondence.[35] As explained above, the correspondence provides several
examples of quantum field theories which are equivalent to string theory on anti-de Sitter space. One can
alternatively view this correspondence as providing a definition of string theory in the special case where the
gravitational field is asymptotically anti-de Sitter (that is, when the gravitational field resembles that of anti-de
Sitter space at spatial infinity). Physically interesting quantities in string theory are defined in terms of
quantities in the dual quantum field theory.[19]

Black hole information paradox

In 1975, Stephen Hawking published a calculation which suggested that black holes are not completely black
but emit a dim radiation due to quantum effects near the event horizon.[36] At first, Hawking's result posed a
problem for theorists because it suggested that black holes destroy information. More precisely, Hawking's
calculation seemed to conflict with one of the basic postulates of quantum mechanics, which states that
physical systems evolve in time according to the Schrödinger equation. This property is usually referred to as
unitarity of time evolution. The apparent contradiction between Hawking's calculation and the unitarity
postulate of quantum mechanics came to be known as the black hole information paradox.[37]

The AdS/CFT correspondence resolves the black hole information paradox, at least to some extent, because it
shows how a black hole can evolve in a manner consistent with quantum mechanics in some contexts. Indeed,
one can consider black holes in the context of the AdS/CFT correspondence, and any such black hole
corresponds to a configuration of particles on the boundary of anti-de Sitter space.[38] These particles obey the
usual rules of quantum mechanics and in particular evolve in a unitary fashion, so the black hole must also
evolve in a unitary fashion, respecting the principles of quantum mechanics.[39] In 2005, Hawking announced
that the paradox had been settled in favor of information conservation by the AdS/CFT correspondence, and
he suggested a concrete mechanism by which black holes might preserve information.[40]

Applications to quantum field theory

Nuclear physics

One physical system which has been studied using the AdS/CFT correspondence is the quark–gluon plasma,
an exotic state of matter produced in particle accelerators. This state of matter arises for brief instants when
heavy ions such as gold or lead nuclei are collided at high energies. Such collisions cause the quarks that make
up atomic nuclei to deconfine at temperatures of approximately two trillion kelvins, conditions similar to those
present at around seconds after the Big Bang.[41]

The physics of the quark–gluon plasma is governed by quantum chromodynamics, but this theory is
mathematically intractable in problems involving the quark–gluon plasma.[42] In an article appearing in 2005,
Đàm Thanh Sơn and his collaborators showed that the AdS/CFT correspondence could be used to understand
some aspects of the quark–gluon plasma by describing it in the language of string theory.[26] By applying the
AdS/CFT correspondence, Sơn and his collaborators were able to describe the quark gluon plasma in terms of
black holes in five-dimensional spacetime. The calculation showed that the ratio of two quantities associated
with the quark–gluon plasma, the shear viscosity and volume density of entropy , should be approximately
equal to a certain universal constant:

where denotes the reduced Planck's constant and is Boltzmann's constant.[43] In addition, the authors
conjectured that this universal constant provides a lower bound for in a large class of systems. In 2008,
the predicted value of this ratio for the quark–gluon plasma was confirmed at the Relativistic Heavy Ion
Collider at Brookhaven National Laboratory.[44]
Another important property of the quark–gluon plasma is that very high energy quarks moving through the
plasma are stopped or "quenched" after traveling only a few femtometers. This phenomenon is characterized
by a number called the jet quenching parameter, which relates the energy loss of such a quark to the squared
distance traveled through the plasma. Calculations based on the AdS/CFT correspondence have allowed
theorists to estimate , and the results agree roughly with the measured value of this parameter, suggesting that
the AdS/CFT correspondence will be useful for developing a deeper understanding of this phenomenon.[45]

Condensed matter physics

Over the decades, experimental condensed matter physicists have

discovered a number of exotic states of matter, including
superconductors and superfluids. These states are described using the
formalism of quantum field theory, but some phenomena are difficult
to explain using standard field theoretic techniques. Some condensed
matter theorists including Subir Sachdev hope that the AdS/CFT
correspondence will make it possible to describe these systems in the
language of string theory and learn more about their behavior.[47]
A magnet levitating above a high-
So far some success has been achieved in using string theory methods temperature superconductor. Today
to describe the transition of a superfluid to an insulator. A superfluid is some physicists are working to
a system of electrically neutral atoms that flows without any friction. understand high-temperature
Such systems are often produced in the laboratory using liquid superconductivity using the
helium, but recently experimentalists have developed new ways of AdS/CFT correspondence.[46]
producing artificial superfluids by pouring trillions of cold atoms into
a lattice of criss-crossing lasers. These atoms initially behave as a
superfluid, but as experimentalists increase the intensity of the lasers, they become less mobile and then
suddenly transition to an insulating state. During the transition, the atoms behave in an unusual way. For
example, the atoms slow to a halt at a rate that depends on the temperature and on Planck's constant, the
fundamental parameter of quantum mechanics, which does not enter into the description of the other phases.
This behavior has recently been understood by considering a dual description where properties of the fluid are
described in terms of a higher dimensional black hole.[48]

Criticism

With many physicists turning towards string-based methods to attack problems in nuclear and condensed
matter physics, some theorists working in these areas have expressed doubts about whether the AdS/CFT
correspondence can provide the tools needed to realistically model real-world systems. In a talk at the Quark
Matter conference in 2006,[49] an American physicist, Larry McLerran pointed out that the N=4 super Yang–
Mills theory that appears in the AdS/CFT correspondence differs significantly from quantum
chromodynamics, making it difficult to apply these methods to nuclear physics. According to McLerran,

supersymmetric Yang–Mills is not QCD ... It has no mass scale and is conformally
invariant. It has no confinement and no running coupling constant. It is supersymmetric. It has no
chiral symmetry breaking or mass generation. It has six scalar and fermions in the adjoint
representation ... It may be possible to correct some or all of the above problems, or, for various
physical problems, some of the objections may not be relevant. As yet there is not consensus nor
compelling arguments for the conjectured fixes or phenomena which would insure that the
supersymmetric Yang Mills results would reliably reflect QCD.[49]
In a letter to Physics Today, Nobel laureate Philip W. Anderson voiced similar concerns about applications of
AdS/CFT to condensed matter physics, stating

As a very general problem with the AdS/CFT approach in condensed-matter theory, we can point
to those telltale initials "CFT"—conformal field theory. Condensed-matter problems are, in
general, neither relativistic nor conformal. Near a quantum critical point, both time and space may
be scaling, but even there we still have a preferred coordinate system and, usually, a lattice. There
is some evidence of other linear-T phases to the left of the strange metal about which they are
welcome to speculate, but again in this case the condensed-matter problem is overdetermined by
experimental facts.[50]

History and development

String theory and nuclear physics

The discovery of the AdS/CFT correspondence in late 1997 was the culmination
of a long history of efforts to relate string theory to nuclear physics.[51] In fact,
string theory was originally developed during the late 1960s and early 1970s as a
theory of hadrons, the subatomic particles like the proton and neutron that are
held together by the strong nuclear force. The idea was that each of these
particles could be viewed as a different oscillation mode of a string. In the late
1960s, experimentalists had found that hadrons fall into families called Regge
trajectories with squared energy proportional to angular momentum, and theorists
Gerard 't Hooft obtained
showed that this relationship emerges naturally from the physics of a rotating
results related to the
relativistic string.[52]
AdS/CFT
correspondence in the
On the other hand, attempts to model hadrons as strings faced serious problems.
1970s by studying
One problem was that string theory includes a massless spin-2 particle whereas
analogies between string
no such particle appears in the physics of hadrons.[51] Such a particle would theory and nuclear
mediate a force with the properties of gravity. In 1974, Joël Scherk and John physics.
Schwarz suggested that string theory was therefore not a theory of nuclear
physics as many theorists had thought but instead a theory of quantum
gravity.[53] At the same time, it was realized that hadrons are actually made of quarks, and the string theory
approach was abandoned in favor of quantum chromodynamics.[51]

In quantum chromodynamics, quarks have a kind of charge that comes in three varieties called colors. In a
paper from 1974, Gerard 't Hooft studied the relationship between string theory and nuclear physics from
another point of view by considering theories similar to quantum chromodynamics, where the number of
colors is some arbitrary number , rather than three. In this article, 't Hooft considered a certain limit where
tends to infinity and argued that in this limit certain calculations in quantum field theory resemble calculations
in string theory.[54]

Black holes and holography

In 1975, Stephen Hawking published a calculation which suggested that black holes are not completely black
but emit a dim radiation due to quantum effects near the event horizon.[36] This work extended previous
results of Jacob Bekenstein who had suggested that black holes have a well defined entropy.[55] At first,
Hawking's result appeared to contradict one of the main postulates of quantum mechanics, namely the unitarity
 of time evolution. Intuitively, the unitarity postulate says that quantum
mechanical systems do not destroy information as they evolve from one state to
another. For this reason, the apparent contradiction came to be known as the
black hole information paradox.[56]

Later, in 1993, Gerard 't Hooft wrote a speculative

paper on quantum gravity in which he revisited
Hawking's work on black hole thermodynamics,
concluding that the total number of degrees of
freedom in a region of spacetime surrounding a
black hole is proportional to the surface area of the
horizon.[57] This idea was promoted by Leonard
Stephen Hawking Susskind and is now known as the holographic
predicted in 1975 that principle.[58] The holographic principle and its
black holes emit realization in string theory through the AdS/CFT
radiation due to quantum correspondence have helped elucidate the mysteries
effects. of black holes suggested by Hawking's work and
are believed to provide a resolution of the black Leonard Susskind made
hole information paradox.[39] In 2004, Hawking early contributions to the
conceded that black holes do not violate quantum mechanics,[59] and he idea of holography in
suggested a concrete mechanism by which they might preserve information.[40] quantum gravity.

Maldacena's paper

In late 1997, Juan Maldacena published a landmark paper that initiated the study of AdS/CFT.[27] According
to Alexander Markovich Polyakov, "[Maldacena's] work opened the flood gates."[60] The conjecture
immediately excited great interest in the string theory community[39] and was considered in articles by Steven
Gubser, Igor Klebanov and Polyakov,[61] and by Edward Witten.[62] These papers made Maldacena's
conjecture more precise and showed that the conformal field theory appearing in the correspondence lives on
the boundary of anti-de Sitter space.[60]

One special case of Maldacena's proposal says that N=4 super Yang–Mills
theory, a gauge theory similar in some ways to quantum chromodynamics, is
equivalent to string theory in five-dimensional anti-de Sitter space.[30] This
result helped clarify the earlier work of 't Hooft on the relationship between
string theory and quantum chromodynamics, taking string theory back to its
roots as a theory of nuclear physics.[52] Maldacena's results also provided a
concrete realization of the holographic principle with important implications
for quantum gravity and black hole physics.[1] By the year 2015,
Maldacena's paper had become the most highly cited paper in high energy
physics with over 10,000 citations.[3] These subsequent articles have
provided considerable evidence that the correspondence is correct, although
so far it has not been rigorously proved.[63]
Juan Maldacena first
proposed the AdS/CFT
correspondence in late 1997.
Generalizations

Three-dimensional gravity
In order to better understand the quantum aspects of gravity in our four-dimensional universe, some physicists
have considered a lower-dimensional mathematical model in which spacetime has only two spatial dimensions
and one time dimension.[64] In this setting, the mathematics describing the gravitational field simplifies
drastically, and one can study quantum gravity using familiar methods from quantum field theory, eliminating
the need for string theory or other more radical approaches to quantum gravity in four dimensions.[65]

Beginning with the work of J. D. Brown and Marc Henneaux in 1986,[66] physicists have noticed that
quantum gravity in a three-dimensional spacetime is closely related to two-dimensional conformal field theory.
In 1995, Henneaux and his coworkers explored this relationship in more detail, suggesting that three-
dimensional gravity in anti-de Sitter space is equivalent to the conformal field theory known as Liouville field
theory.[67] Another conjecture formulated by Edward Witten states that three-dimensional gravity in anti-de
Sitter space is equivalent to a conformal field theory with monster group symmetry.[68] These conjectures
provide examples of the AdS/CFT correspondence that do not require the full apparatus of string or M-
theory.[69]

dS/CFT correspondence

Unlike our universe, which is now known to be expanding at an accelerating rate, anti-de Sitter space is
neither expanding nor contracting. Instead it looks the same at all times.[18] In more technical language, one
says that anti-de Sitter space corresponds to a universe with a negative cosmological constant, whereas the real
universe has a small positive cosmological constant.[70]

Although the properties of gravity at short distances should be somewhat independent of the value of the
cosmological constant,[71] it is desirable to have a version of the AdS/CFT correspondence for positive
cosmological constant. In 2001, Andrew Strominger introduced a version of the duality called the dS/CFT
correspondence.[72] This duality involves a model of spacetime called de Sitter space with a positive
cosmological constant. Such a duality is interesting from the point of view of cosmology since many
cosmologists believe that the very early universe was close to being de Sitter space.[18] Our universe may also
resemble de Sitter space in the distant future.[18]

Kerr/CFT correspondence

Although the AdS/CFT correspondence is often useful for studying the properties of black holes,[73] most of
the black holes considered in the context of AdS/CFT are physically unrealistic. Indeed, as explained above,
most versions of the AdS/CFT correspondence involve higher-dimensional models of spacetime with
unphysical supersymmetry.

In 2009, Monica Guica, Thomas Hartman, Wei Song, and Andrew Strominger showed that the ideas of
AdS/CFT could nevertheless be used to understand certain astrophysical black holes. More precisely, their
results apply to black holes that are approximated by extremal Kerr black holes, which have the largest
possible angular momentum compatible with a given mass.[74] They showed that such black holes have an
equivalent description in terms of conformal field theory. The Kerr/CFT correspondence was later extended to
black holes with lower angular momentum.[75]

Higher spin gauge theories

The AdS/CFT correspondence is closely related to another duality conjectured by Igor Klebanov and
Alexander Markovich Polyakov in 2002.[76] This duality states that certain "higher spin gauge theories" on
anti-de Sitter space are equivalent to conformal field theories with O(N) symmetry. Here the theory in the bulk
is a type of gauge theory describing particles of arbitrarily high spin. It is similar to string theory, where the
excited modes of vibrating strings correspond to particles with higher spin, and it may help to better understand
the string theoretic versions of AdS/CFT and possibly even prove the correspondence.[77] In 2010, Simone
Giombi and Xi Yin obtained further evidence for this duality by computing quantities called three-point
functions.[78]

See also
Algebraic holography
Ambient construction
Randall–Sundrum model

Notes
1. de Haro et al. 2013, p. 2
2. Klebanov and Maldacena 2009
3. "Top Cited Articles of All Time (2014 edition)" (https://inspirehep.net/info/hep/stats/topcites/201
4/alltime.html). INSPIRE-HEP. Retrieved 26 December 2015.
4. A standard textbook on general relativity is Wald 1984.
5. Maldacena 2005, p. 58
6. Griffiths 2004
7. Maldacena 2005, p. 62
8. See the subsection entitled "Examples of the correspondence". For examples which do not
involve string theory or M-theory, see the section entitled "Generalizations".
9. Wald 1984, p. 4
10. Zwiebach 2009, p. 8
11. Zwiebach 2009, pp. 7–8
12. This analogy is used for example in Greene 2000, p. 186.
13. A standard text is Peskin and Schroeder 1995.
14. For an introduction to the applications of quantum field theory to condensed matter physics, see
Zee 2010.
15. Conformal field theories are characterized by their invariance under conformal transformations.
16. For an introduction to conformal field theory emphasizing its applications to perturbative string
theory, see Volume II of Deligne et al. 1999.
17. Klebanov and Maldacena 2009, p. 28
18. Maldacena 2005, p. 60
19. Maldacena 2005, p. 61
20. The mathematical relationship between the interior and boundary of anti-de Sitter space is
related to the ambient construction of Charles Fefferman and Robin Graham. For details see
Fefferman and Graham 1985, Fefferman and Graham 2011.
21. Zwiebach 2009, p. 552
22. Maldacena 2005, pp. 61–62
23. Maldacena 2005, p. 57
24. The known realizations of AdS/CFT typically involve unphysical numbers of spacetime
dimensions and unphysical supersymmetries.
25. This example is the main subject of the three pioneering articles on AdS/CFT: Maldacena
1998; Gubser, Klebanov, and Polyakov 1998; and Witten 1998.
26. Merali 2011, p. 303; Kovtun, Son, and Starinets 2001
27. Maldacena 1998
28. For a review of the (2,0)-theory, see Moore 2012.
29. See Moore 2012 and Alday, Gaiotto, and Tachikawa 2010.
30. Aharony et al. 2008
31. Aharony et al. 2008, sec. 1
32. A standard textbook introducing the formalism of Feynman diagrams is Peskin and Schroeder
1995.
33. Zee 2010, p. 43
34. Zwiebach 2009, p. 12
35. Maldacena 1998, sec. 6
36. Hawking 1975
37. For an accessible introduction to the black hole information paradox, and the related scientific
dispute between Hawking and Leonard Susskind, see Susskind 2008.
38. Zwiebach 2009, p. 554
39. Maldacena 2005, p. 63
40. Hawking 2005
41. Zwiebach 2009, p. 559
42. More precisely, one cannot apply the methods of perturbative quantum field theory.
43. Zwiebach 2009, p. 561; Kovtun, Son, and Starinets 2001
44. Merali 2011, p. 303; Luzum and Romatschke 2008
45. Zwiebach 2009, p. 561
46. Merali 2011
47. Merali 2011, p. 303
48. Sachdev 2013, p. 51
49. McLerran 2007
50. Anderson 2013
51. Zwiebach 2009, p. 525
52. Aharony et al. 2008, sec. 1.1
53. Scherk and Schwarz 1974
54. 't Hooft 1974
55. Bekenstein 1973
56. Susskind 2008
57. 't Hooft 1993
58. Susskind 1995
59. Susskind 2008, p. 444
60. Polyakov 2008, p. 6
61. Gubser, Klebanov, and Polyakov 1998
62. Witten 1998
63. Maldacena 2005, p. 63; Cowen 2013
64. For a review, see Carlip 2003.
65. According to the results of Witten 1988, three-dimensional quantum gravity can be understood
by relating it to Chern–Simons theory.
66. Brown and Henneaux 1986
67. Coussaert, Henneaux, and van Driel 1995
68. Witten 2007
69. Guica et al. 2009, p. 1
70. Perlmutter 2003
71. Biquard 2005, p. 33
72. Strominger 2001
73. See the subsection entitled "Black hole information paradox".
74. Guica et al. 2009
75. Castro, Maloney, and Strominger 2010
76. Klebanov and Polyakov 2002
77. See the Introduction in Klebanov and Polyakov 2002.
78. Giombi and Yin 2010

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