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<!---{{String theory|cTopic=Theory}}

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 of the correspondence are [[conformal field theory|conformal field theories]] (CFT) which are [[quantum field theory|quantum field theories]], including theories similar to the [[Yang–Mills theory|Yang–Mills theories]] that describe elementary particles. On the other side of the correspondence are [[anti-de Sitter space]]s (AdS) which are used in theories of [[quantum gravity]], formulated in terms of [[string theory]] or [[M-theory]].<ref>Maldacena 2005</ref>

The duality represents a major advance in our understanding of [[string theory]] and [[quantum gravity]].<ref name="de Haro et al. 2013, p.2">de Haro et al. 2013, p.2</ref> This is because it provides a [[non-perturbative]] formulation of string theory with certain [[boundary condition]]s and because it is the most successful realization of the [[holographic principle]], an idea in [[quantum gravity]] originally proposed by [[Gerard 't Hooft]]<ref name="'t Hooft 1993">'t Hooft 1993</ref> and improved and promoted by [[Leonard Susskind]].<ref name="Susskind 1995">Susskind 1995</ref>

In addition, it provides a powerful toolkit for studying [[coupling (physics)|strongly coupled]] [[quantum field theory|quantum field theories]].<ref>Klebanov and Maldacena 2009</ref> Much of the usefulness of the duality results from the fact that it is a weak-strong 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 features of [[nuclear physics|nuclear]] and [[condensed matter physics]] by translating problems in those fields into more mathematically tractable problems in string theory.<ref name="Merali 2011">Merali 2011</ref>-->

<!---The AdS/CFT correspondence was first proposed by [[Juan Maldacena]] in late 1997.<ref name="Maldacena 1998">Maldacena 1998</ref> Important aspects of the correspondence were elaborated in articles by [[Steven Gubser]], [[Igor Klebanov]] and [[Alexander Markovich Polyakov]],<ref name="ReferenceA">Gubser, Klebanov, and Polyakov 1998</ref> and by [[Edward Witten]].<ref name="Witten 1998">Witten 1998</ref> By 2010, Maldacena's article had over 7000 citations, becoming the most highly cited article in the field of [[high energy physics]].<ref name="slac.stanford">{{cite web |url=http://www.slac.stanford.edu/spires/topcites/2010/eprints/to_hep-th_annual.shtml |title=Top Cited Articles during 2010 in hep-th |author= |accessdate=25 July 2013}}</ref>-->

<!---==Background==

===Quantum gravity and strings===

{{main|Quantum gravity| String theory}}

Our current understanding of [[gravity]] is based on [[Albert Einstein]]'s [[general theory of relativity]].<ref>A standard textbook on general relativity is Wald 1984.</ref> Formulated in 1916, general relativity explains gravity in terms of the [[geometry]] of [[space]] and [[time]], or [[spacetime]]. It is formulated in the language of [[classical physics]] developed by physicists such as [[Isaac Newton]] and [[James Clerk Maxwell]].<ref>Maldacena 2005, p.58</ref> 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]].<ref>Griffiths 2004</ref>

[[Quantum gravity]] is the branch of physics that seeks to describe gravity using the principles of quantum mechanics. Currently, the most well studied approach to quantum gravity is [[string theory]],<ref name="Maldacena 2005, p.62">Maldacena 2005, p.62</ref> which models [[elementary particle]]s not as zero-dimensional points but as one-dimensional objects called [[string (physics)|strings]]. In the AdS/CFT correspondence, one typically considers theories of quantum gravity derived from string theory or its modern extension, [[M-theory]].<ref>See the subsection below entitled "Examples of the correspondence". For examples which do not involve string theory or M-theory, see the section entitled "Other topics".</ref>

In everyday life, we are familiar with three dimensions of [[space]] (up/down, left/right, and forward/backward), and one dimension of [[time]]. Thus, in the language of modern physics, one says that we live in a four-dimensional [[spacetime]]. 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.<ref>Zwieback 2009, p.8</ref> The quantum gravity theories appearing in the AdS/CFT correspondence are typically obtained from string and M-theory by a process known as [[compactification (physics)|compactification]]. This produces a theory in which spacetime has effectively a lower number of dimensions and the extra dimensions are "curled up" into circles.

A standard analogy for compactification is to consider a multidimensional object such as a [[garden hose]].<ref>This analogy is used for example in Greene 2000, p.186</ref> If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. However, 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.-->

<!---===Quantum field theory===

{{main|Quantum field theory|Conformal 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]].<ref>A standard text is Peskin and Schroeder 1995</ref> 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 [[quasiparticle]]s.<ref>For an introduction to the applications of quantum field theory to condensed matter physics, see Zee 2010.</ref>

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.<ref>Conformal field theories are characterized by their [[Invariant (physics)|invariance]] under [[conformal map|conformal transformations]].</ref> Such theories are often studied in the context of string theory, where they are associated with the [[worldsheet|surface]] swept out by a string propagating though spacetime, and in [[statistical mechanics]], where they model systems at a [[critical point (thermodynamics)|thermodynamic critical point]].<ref>For an introduction to conformal field theory emphasizing its applications to perturbative string theory, see Volume II of Deligne et al. 1999</ref>-->

<!---==Overview of the correspondence==

[[Image:Uniform tiling 433-t0.png|thumb|right|A [[teselation]] of the [[hyperbolic plane]] by triangles and quadrilaterals.]]

===The geometry of anti-de Sitter space===

{{details|Anti-de Sitter space|the mathematics described here}}

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]].<ref>Klebanov and Maldacena 2009, p.28</ref>

In very elementary terms, anti-de Sitter space is a mathematical model of [[spacetime]] in which the notion of distance between points (the [[metric tensor|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 [[Poincare disk model|disk]] as illustrated on the right.<ref name="Maldacena 2005, p.60">Maldacena 2005, p.60</ref> This image shows a [[tesselation]] of a disk by triangles and quadrilaterals. One can define the distance between points of this disk in such a way that all triangles and all quadrilaterals are the same size and the circular boundary is infinitely far from any point in the interior.

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.<ref name="Maldacena 2005, p.60"/> It looks like a solid [[cylinder (geometry)|cylinder]] in which any [[cross section (geometry)|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 we explain below. As with the hyperbolic plane, anti-de Sitter space is [[curvature|curved]] in such a way that any point in the interior is actually infinitely far from this boundary surface.

[[File:AdS3 (new).png|thumb|left|350px|Three-dimensional [[anti-de Sitter space]] is like a stack of [[Poincare disk model|hyperbolic disks]], each one representing the state of the [[universe]] at a given time. The resulting [[spacetime]] looks like a solid [[cylinder (geometry)|cylinder]].]]

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.<ref name="Maldacena 2005, p.60"/>-->

<!---===The idea of AdS/CFT===

We have now seen how anti-de Sitter space is related to hyperbolic space, and we have described the boundary (a cylinder in the case of three-dimensional anti-de Sitter space). An important property of this boundary is that, locally around any point, it looks just like [[Minkowski space]], the model of spacetime used in nongravitational physics.<ref>Zwiebach 2009, p.552</ref> 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.<ref name="Maldacena 2005, p.57">Maldacena 2005, p.57</ref> 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.<ref>Maldacena 2005, pp.61–62</ref>

[[File:Holomouse2.jpg|thumb|right|A [[hologram]] is a [[two-dimensional]] image which stores information about all three dimensions of the object it represents. The two images here are photographs of a single hologram taken from different angles.]]

Notice that the boundary of anti-de Sitter space has a lower number of dimensions than anti-de Sitter space itself. For instance, in the three-dimensional example we have been discussing, 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 to the relationship between a three-dimensional object and its image as a [[hologram]].<ref name="Maldacena 2005, p.57"/> Although a hologram is two-dimensional, it encodes information about all three dimensions of the object it represents. In the same way, theories which are related by the AdS/CFT correspondence are conjectured to be ''exactly'' equivalent, despite living in different numbers of dimensions.<ref name="Maldacena 2005, p.61">Maldacena 2005, p.61</ref> The conformal field theory is like a hologram which captures information about the higher-dimensional quantum gravity theory.-->

<!---===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 [[compactification (physics)|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 properties which make them useful for solving problems in [[quantum field theory]] and [[quantum gravity]].<ref>As explained below, the known realizations of AdS/CFT typically involve unphysical numbers of spacetime dimensions and unphysical [[supersymmetry|supersymmetries]].</ref>

The most famous example of the AdS/CFT correspondence states that [[type IIB string theory]] on the [[product space]] <math>AdS_5\times S^5</math> is equivalent to [[N=4 super Yang-Mills|N=4 super Yang–Mills theory]] on the four-dimensional boundary.<ref>This example is the main subject of the three pioneering articles on AdS/CFT: Maldacena 1998, Gubser, Klebanov, and Polyakov 1998, and Witten 1998.</ref> In this example, the spacetime on which the gravitational theory lives is effectively five-dimensional (hence the notation <math>AdS_5</math>), and there are five additional "[[compact space|compact]]" dimensions (encoded by the <math>S^5</math> 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 [[gluon]]s of quantum chromodynamics together with certain [[fermions]].<ref name="Maldacena 2005, p.62"/> As a result, it has found applications in [[nuclear physics]], particularly in the study of the [[quark-gluon plasma]].

Another realization of the correspondence states that M-theory on <math>AdS_7\times S^4</math> is equivalent to the so-called (2,0)-theory in six dimensions.<ref name="Maldacena 1998"/> 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 [[super conformal field theory|superconformal field theories]]. It is still poorly understood because it is a [[quantum mechanics|quantum mechanical]] theory without a [[classical limit]].<ref>For a review of the (2,0)-theory, see Moore 2012.</ref> 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.<ref>See Moore 2012 and Alday, Gaiotto, and Tachikawa 2010.</ref>

Yet another realization of the correspondence states that M-theory on <math>AdS_4\times S^7</math> is equivalent to the ABJM superconformal field theory in three dimensions.<ref name="Aharony et al. 2008">Aharony et al. 2008</ref> Here the gravitational theory has four noncompact dimensions, so this version of the correspondence provides a somewhat more realistic description of gravity.<ref>Aharony et al. 2008, sec.1</ref>-->

<!---==Applications to quantum gravity==

===A non-perturbative formulation of string theory===

[[Image:World lines and world sheet.svg|right|thumb|300px|Interaction in the quantum world: [[world line]]s of point-like [[particles]] or a [[world sheet]] swept up by closed [[string (physics)|strings]] in 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 diagram]]s to organize computations. One imagines that these diagrams depict the paths of point-like particles and their interactions.<ref>A standard textbook introducing the formalism of Feynman diagrams is Peskin and Schroeder 1995.</ref> Although this formalism is extremely useful for making predictions, these predictions are only possible when the strength of the interactions, the [[coupling constant]], is small enough to reliably describe the theory as being close to a theory [[free field|without interactions]].

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 [[string (physics)|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.<ref>Zwiebach 2009, p.12</ref>

The problem of developing a non-perturbative formulation of string theory was one of the original motivations for studying the AdS/CFT correspondence.<ref>Maldacena 1998, sec. 6</ref> 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.<ref name="Maldacena 2005, p.61"/>-->

<!---===Black hole information paradox===

{{Main|Black hole information paradox}}

In 1975, [[Stephen Hawking]] published a calculation which suggested that [[black hole]]s are not completely black but emit a dim radiation due to quantum effects near the [[event horizon]].<ref name="Hawking 1975">Hawking 1975</ref> 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 [[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]].<ref>For an accessible introduction to the black hole information paradox, and the related scientific dispute between Hawking and Leonard Susskind, see Susskind 2008.</ref>

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.<ref>Zwiebach 2009, p.554</ref> 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 fasion, respecting the principles of quantum mechanics.<ref name="Maldacena 2005, p.63">Maldacena 2005, p.63</ref> 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.<ref name="Hawking 2005">Hawking 2005</ref>-->

<!---==Applications to quantum field theory==

===Nuclear physics===

{{Main|AdS/QCD}}

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 accelerator]]s. This state of matter arises for brief instants when heavy [[ions]] such as [[gold]] or [[lead]] [[Atomic nucleus|nuclei]] are collided at high energies. Such collisions cause the [[quarks]] that make up [[atomic nuclei]] to [[deconfinement|deconfine]] at temperatures of approximately two [[1,000,000,000,000|trillion]] degrees [[Kelvin]], conditions similar to those present at around <math>10^{-11}</math> [[seconds]] after the [[Big Bang]].<ref>Zwiebach 2009, p.559</ref>

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.<ref>More precisely, one cannot apply the methods of perturbative quantum field theory.</ref> 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.<ref name="ReferenceB">Merali 2011, p.303; Kovtun, Son, and Starinets 2001</ref> 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]] <math>\eta</math> and volume density of [[entropy]] <math>s</math>, should be approximately equal to a certain universal [[constant (mathematics)|constant]]:

:<math>\frac{\eta}{s}\approx\frac{\hbar}{4\pi k}</math>

where <math>\hbar</math> denotes [[Planck's constant]] and <math>k</math> is [[Boltzmann's constant]].<ref>Zwiebach 2009, p.561; Kovtun, Son, and Starinets 2001</ref> In addition, the authors conjectured that this universal constant provides a [[lower bound]] for <math>\eta/s</math> 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]].<ref>Merali 2011, p.303; Luzum and Romatschke 2008</ref>

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 [[femtometer]]s. This phenomenon is characterized by a number <math>\widehat{q}</math> 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 <math>\widehat{q}</math>, 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.<ref>Zwiebach 2009, p.561</ref>-->

<!---===Condensed matter physics===

[[File:Meissner effect p1390048.jpg|thumb|A [[magnet]] [[Meissner effect|levitating]] above a [[high-temperature superconductor]]. Today some physicists are working to understand high-temperature superconductivity using the AdS/CFT correspondence.<ref name="Merali 2011"/>]]

{{Main|AdS/CMT}}

Over the decades, [[experimental physics|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.<ref name="Merali 2011, p.303">Merali 2011, p.303</ref>

So far some success has been achieved in using string theory methods to describe the transition of a [[superfluid]] to an [[insulator (electricity)|insulator]].<ref name="Sachdev 2013, p.51">Sachdev 2013, p.51</ref> A superfluid is a system of [[electrically neutral]] [[atoms]] that flows without any [[friction]]. Such systems are often produced in the laboratory using [[liquid helium]], but recently experimentalists have developed new ways of producing artificial superfluids by pouring trillions of cold atoms into a lattice of criss-crossing [[lasers]].<ref name="Sachdev 2013, p.51"/> 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 [[phase (matter)|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]].<ref name="Sachdev 2013, p.51"/>-->

<!---===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,<ref name="McLarren 2007">McLarren 2007</ref> 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,

{{Bquote|<math>N=4</math> 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 <math>N=4</math> supersymmetric Yang Mills results would reliably reflect QCD.<ref name="McLarren 2007" />|x|x|| }}

In a letter to [[Physics Today]], [[Nobel laureate]] [[Philip W. Anderson]] voiced similar concerns about applications of AdS/CFT to condensed matter physics, stating

{{Bquote|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.<ref>{{cite web |url=http://www.physicstoday.org/resource/1/phtoad/v66/i4/p9_s1?bypassSSO=1 |title=Strange connections to strange metals |last1=Anderson |first1=Philip |publisher=Physics Today |accessdate=14 August 2013}}</ref>|x|x|| }}-->

<!---==History and development==

[[File:Gerard 't Hooft.jpg|right|thumb|150px|[[Gerard 't Hooft]] obtained results related to the AdS/CFT correspondence in the 1970s by studying analogies between [[string theory]] and [[nuclear physics]].]]

===String theory and nuclear physics===

{{main|History of string theory|1/N expansion}}

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.<ref name="Zwiebach 2009, p.525">Zwiebach 2009, p.525</ref> In fact, string theory was originally developed during the late 1960s and early 1970s as a theory of [[hadron]]s, the [[subatomic particle]]s 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 (physics)|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 showed that this relationship emerges naturally from the physics of a rotating [[Principle of relativity|relativistic]] string.<ref name="ReferenceC">Aharony et al. 2008, sec. 1.1</ref>

On the other hand, attempts to model hadrons as strings faced serious problems. One problem was that string theory includes a [[mass|massless]] [[spin (physics)|spin-2]] particle whereas no such particle appears in the physics of hadrons.<ref name="Zwiebach 2009, p.525"/> Such a particle would mediate a force with the properties of [[gravity]]. In 1974, [[Joel Scherk]] and [[John Henry Schwarz|John 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.<ref>Scherk and Schwarz 1974</ref> 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]].<ref name="Zwiebach 2009, p.525"/>

In quantum chromodynamics, quarks have a kind of [[charge (physics)|charge]] that comes in three varieties called [[color charge|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 <math>N</math>, rather than three.<ref>'t Hooft 1974</ref> In this article, 't Hooft considered a certain limit where <math>N</math> tends to infinity and argued that in this limit certain calculations in quantum field theory resemble calculations in string theory.

[[Image:Stephen Hawking.StarChild.jpg|left|thumb|150px|[[Stephen Hawking]] predicted in 1975 that [[black hole]]s emit [[Hawking radiation|radiation]] due to quantum effects.]]-->

<!---===Black holes and holography===

{{main|Black hole information paradox|Thorne–Hawking–Preskill bet|Holographic principle}}

In 1975, [[Stephen Hawking]] published a calculation which suggested that [[black hole]]s are not completely black but emit a dim radiation due to quantum effects near the [[event horizon]].<ref name="Hawking 1975"/> This work extended previous results of [[Jacob Bekenstein]] who had suggested that black holes have a well defined [[entropy]].<ref>Bekenstein 1973</ref> 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]].<ref name="Susskind 2008">Susskind 2008</ref>

Hawking remained convinced that, according to his calculation, quantum information may be destroyed in the presence of black holes. Others disagreed with Hawking's conclusion.<ref name="Susskind 2008"/> In 1997, physicists [[John Preskill]] and [[Kip Thorne]] made a bet with Hawking that information was actually not lost in black holes. This bet came to be known as the [[Thorne-Hawking-Preskill bet]].

[[File:LeonardSusskindStanford2009 cropped.jpg|right|thumb|150px|[[Leonard Susskind]] made early contributions to the idea of [[holography]] in [[quantum gravity]].]]

Several years before the Thorne-Hawking-Preskill bet, Gerard 't Hooft had written a speculative paper on [[quantum gravity]] in which he revisited Hawking's work on [[black hole thermodynamics]], concluding that the total number of [[degree of freedom|degrees of freedom]] in a region of spacetime surrounding a black hole is proportional to the [[surface area]] of the horizon.<ref name="'t Hooft 1993"/> This idea was promoted by [[Leonard Susskind]] and is now known as the [[holographic principle]].<ref name="Susskind 1995"/> The holographic principle and its realization in string theory through the AdS/CFT correspondence have helped elucidate the mysteries of back holes suggested by Hawking's work and are believed to provide a resolution of the black hole information paradox.<ref name="Maldacena 2005, p.63"/> In 2004, Hawking conceded his bet with Thorne and Preskill,<ref>Susskind 2008, p.444</ref> and he suggested a concrete mechanism by which black holes might preserve information.<ref name="Hawking 2005"/>-->

<!---===Maldacena's paper===

In late 1997, [[Juan Maldacena]] published a landmark paper that initiated the study of AdS/CFT.<ref name="Maldacena 1998"/> According to [[Alexander Markovich Polyakov]], "[Maldacena's] work opened the flood gates."<ref name="Polyakov 2008, p.6">Polyakov 2008, p.6</ref> The conjecture immediately excited great interest in the string theory community<ref name="Maldacena 2005, p.63"/> and was considered in articles by [[Steven Gubser]], [[Igor Klebanov]] and Polyakov,<ref name="ReferenceA"/> and by [[Edward Witten]].<ref name="Witten 1998"/> 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.<ref name="Polyakov 2008, p.6"/>

[[Image:JuanMaldacena.jpg|left|thumb|175px|[[Juan Maldacena]] first proposed the AdS/CFT correspondence in late 1997.]]

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.<ref name="Aharony et al. 2008"/> 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.<ref name="ReferenceC"/> Maldacena's results also provided a concrete realization of the holographic principle with important implications for quantum gravity and black hole physics.<ref name="de Haro et al. 2013, p.2"/> By the year 2010, Maldacena's paper had become the most highly cited paper in [[high energy physics]] with over 7000 citations.<ref name="slac.stanford" />-->

<!---===AdS/CFT finds applications===

{{main|AdS/QCD|AdS/CMT}}

In 1999, after taking a job at [[Columbia University]], nuclear physicist [[Đàm Thanh Sơn]] paid a visit to Andrei Starinets, a friend from Sơn's undergraduate days who happened to be doing a Ph.D. in string theory at [[New York University]].<ref>Merali 2011, pp.302--3</ref> Although the two men had no intention of collaborating, Sơn soon realized that the AdS/CFT calculations Starinets was doing could shed light on some aspects of the [[quark-gluon plasma]], an exotic [[state of matter]] produced when heavy [[ion]]s are collided at high energies. In collaboration with Starinets and Pavel Kovtun, Sơn was able to use the AdS/CFT correspondence to calculate a key parameter of the plasma.<ref name="ReferenceB"/> As Sơn later recalled, "We turned the calculation on its head to give us a prediction for the value of the shear viscosity of a plasma... A friend of mine in nuclear physics joked that ours was the first useful paper to come out of string theory."<ref name="Merali 2011, p.303"/>

Today physicists continue to look for applications of the AdS/CFT correspondence in quantum field theory.<ref>Merali 2011; Sachdev 2013</ref> In addition to the applications to nuclear physics advocated by Đàm Thanh Sơn and his collaborators, condensed matter physicists such as [[Subir Sachdev]] have used string theory methods to understand some aspects of condensed matter physics. A notable result in this direction was the description, via the AdS/CFT correspondence, of the transition of a [[superfluid]] to an [[insulator (electricity)|insulator]].<ref name="Sachdev 2013, p.51"/> Another emerging subject is the fluid/gravity correspondence, which uses the AdS/CFT correspondence to translate problems in [[fluid dynamics]] into problems in [[general relativity]].<ref name="Rangamani 2009">Rangamani 2009</ref>-->

<!---==Generalizations==

===Three-dimensional gravity===

{{main|(2+1)-dimensional topological 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.<ref>For a review, see Carlip 2003.</ref> 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.<ref>According to the results of Witten 1988, three-dimensional quantum gravity can be understood by relating it to [[Chern-Simons theory]].</ref>

Beginning with the work of J.D. Brown and [[Marc Henneaux]] in 1986,<ref>Brown and Henneaux 1986</ref> 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]].<ref>Coussaert, Henneaux, and van Driel 1995</ref> Another conjecture due to [[Edward Witten]] states that three-dimensional gravity in anti-de Sitter space is equivalent to a conformal field theory with [[monster group]] symmetry.<ref>Witten 2007</ref> These conjectures provide examples of the AdS/CFT correspondence that do not require the full apparatus of string or M-theory.-->

<!---===dS/CFT correspondence===

{{main|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.<ref name ="Maldacena 2005, p.60"/> In more technical language, one says that anti-de Sitter space corresponds to a universe with negative [[cosmological constant]], whereas the real universe has a small positive cosmological constant.<ref>Perlmutter 2003</ref>

Although the properties of gravity at short distances should be somewhat independent of the value of the cosmological constant,<ref>Biquard 2005, p.33</ref> 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]].<ref>Strominger 2001</ref> 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.<ref name ="Maldacena 2005, p.60"/> Our universe may also resemble de Sitter space in the distant future.<ref name ="Maldacena 2005, p.60"/>-->

<!---===Kerr/CFT correspondence===

{{main|Kerr/CFT correspondence}}

Although the AdS/CFT correspondence is often useful for studying the properties of black holes,<ref>See the subsection entitled "Black hole information paradox".</ref> 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 black hole|extremal]] [[Kerr black hole|Kerr black holes]], which have the largest possible [[angular momentum]] compatible with a given [[mass]].<ref>Guica et al. 2009</ref> 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.<ref>Castro, Maloney, and Strominger 2010</ref>-->

<!---===Higher spin gauge theories===

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

<!---== See also ==

* [[Algebraic holography]]
* [[Ambient construction]]
* [[Randall–Sundrum model]]

==Notes==

{{reflist|2}}-->

<!---==References==

{{refbegin|2}}

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{{refend}}-->


{{デフォルトソート:AdS/CFTたいおう}}
{{デフォルトソート:AdS/CFTたいおう}}
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[[Category:重力理論]]
[[Category:重力理論]]
[[Category:場の量子論]]
[[Category:場の量子論]]

[[de:Holografisches Prinzip#AdS/CFT-Korrespondenz]]

2013年10月22日 (火) 06:16時点における版

AdS/CFT対応（AdS/CFTたいおう、英: AdS/CFT correspondence）とは、1997年にフアン・マルダセナによって提唱された理論で、AdS (Anti de Sitter) 時空上の重力の弱結合領域と共形場理論 (Conformal Field Theory) の強結合領域との双対性のことである。

具体的には、10次元時空として5次元AdS時空（AdS5）と5次元球面（S5）の直積空間を考える。AdS5時空の境界は4次元ミンコフスキー時空（M4）であり、上記の共形場理論とはAdS5×S5時空の等長変換群SO(4,2)×SO(6)を対称性として持つ境界M4上の超対称ゲージ理論のことである。また、AdS/CFT対応は、4次元超対称ゲージ理論の強結合領域での相関関数の計算が、AdS5を背景時空とする5次元超重力理論の弱結合領域で計算できることを示したという面で画期的であった。

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