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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 certain quantum field theories, including theories similar to the Yang–Mills theories that describe elementary particles. On the other side of the correspondence are theories of quantum gravity, formulated in terms of string theory or M-theory.^[1]

The duality represents a major advance in our understanding of string theory and quantum gravity. 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^[2] and improved and promoted by Leonard Susskind.^[3]

In addition, it provides a powerful toolkit for studying strongly coupled quantum field theories.^[4] 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 and condensed matter physics by translating problems in those fields into more mathematically tractable problems in string theory.^[5]

The AdS/CFT correspondence was first proposed by Juan Maldacena in late 1997.^[6] Important aspects of the correspondence were elaborated in articles by Steven Gubser, Igor Klebanov and Alexander Markovich Polyakov,^[7] and by Edward Witten.^[8] In about five years, Maldacena's article had 3000 citations, becoming one of the most important conceptual breakthroughs in theoretical physics of the 1990s.^[9]

Overview

Intuition

M.C. Escher's *Circle Limit III*, 1959, is a representation of the hyperbolic plane. Three-dimensional anti-de Sitter space is a model of spacetime that has spacelike surfaces isometric to the hyperbolic plane.

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.^[10]

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 was famously depicted by M.C. Escher in a woodcut entitled Circle Limit III.^[11] This image shows a tesselation of a disk by fish shapes. One can define the distance between points of this disk in such a way that all the fish are the same size and the circular boundary is infinitely far from any point in the interior. In three dimensions, anti-de Sitter space can be viewed as the geometric object obtained by adding a third dimension to this picture (a "time" dimension). There is a similar description of anti-de Sitter space in any number of dimensions.^[12]

In anti-de Sitter space, it is possible to define a notion of "boundary" of spacetime at infinity. Such a boundary is called the conformal boundary. In the above description, the conformal boundary of three-dimensional anti-de Sitter space is the two-dimensional cylinder traced out by the boundary circle of the hyperbolic disk as one moves along in the time direction.^[13] The conformal boundary has an induced geometry, namely that of Minkowski space, the model of spacetime used in nongravitational physics.^[14]

The idea of the AdS/CFT correspondence is that the conformal boundary of anti-de Sitter space can be regarded as the "spacetime" for a special type of quantum field theory called a conformal field theory. The claim is that this quantum 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 identicle 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.^[15]

As mentioned above, the gravitational theories appearing in the correspondence are typically formulated in terms of string theory or M-theory. Both string theory and M-theory require extra dimensions of spacetime for their mathematical consistency: In string theory spacetime is ten-dimensional, while in M-theory it is eleven-dimensional.^[16] In the AdS/CFT correspondence, one considers theories 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.

A standard analogy for this is to consider multidimensional object such as a garden hose.^[17] 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.

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 properties which make them useful for solving problems in quantum field theory and quantum gravity.^[18]

The most famous example of the AdS/CFT correspondence states that type IIB string theory on the product $AdS_{5}\times S^{5}$ is equivalent to N=4 super Yang–Mills theory on the four-dimensional conformal boundary.^[19] In this example, the spacetime on which the gravitational theory lives is effectively five-dimensional (and there are five additional "compact" dimensions). 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. 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 $AdS_{7}\times S^{4}$ is equivalent to the so-called (2,0)-theory in six dimensions.^[20] 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.^[21] 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.^[22]

Yet another realization of the correspondence states that M-theory on $AdS_{4}\times S^{7}$ is equivalent to the ABJM superconformal field theory in three dimensions.^[23] Here the gravitational theory has four noncompact dimensions, so this version of the correspondence provides a somewhat more realistic description of gravity.

Applications to quantum gravity

A non-perturbative formulation of string theory

In quantum field theory, one computes the probabilities of various events using the techniques of perturbation theory. Developed by Richard Feynman in the first half of the twentieth century, perturbative quantum field theory uses special diagrams called Feynman diagram to organize computations. One imagines that these diagrams depict the paths of point-like particles and their interactions.^[24] Although this formalism is extremely useful for making predictions, these predictions are only possible in certain situations, and there are many problems for which one requires a non-perturbative formulation.^[25]

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 scattering 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.^[26]

The problem of developing a non-perturbative formulation of string theory was one of the original motivations for studying the AdS/CFT correspondence.^[6] 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.^[27]

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.^[28] This work extended previous results of Jacob Bekenstein who suggested that black holes have a well defined entropy.^[29] At first, Hawking's result posed a problem for theorists because it suggested that black holes destroy information. More precisely, Hawking's calculation appeared to show that matter in the presence of a black hole could evolve in a nonunitary fashion, contradicting one of the basic postulates of quantum mechanics. This apparent contradiction is now known as the black hole information paradox.

The AdS/CFT correspondence resolves the black hole information paradox to some extent because it shows how in certain contexts a black hole can evolve in a manner consistent with quantum mechanics. Indeed, one can consider black holes in the context of the AdS/CFT correspondence, and they have an alternate description in terms of a conformal field theory on the boundary of spacetime.^[30] Since the conformal field theory is a quantum mechanical theory, the black hole evolves in a manifestly unitary way, respecting the principles of quantum mechanics.^[31] In 2005, Hawking announced that the paradox had been settled in favor of information conservation by the AdS/CFT correspondence, and he wrote a paper in which he offered a concrete mechanism by which black holes might preserve information.^[32]

Applications to quantum field theory

Since it relates string theory to ordinary quantum field theory, the AdS/CFT correspondence can be used as a theoretical tool for doing calculations in quantum field theory using methods from string theory.^[4] As explained below, it has in fact been used to do approximate calculations in nuclear and condensed matter physics.

Early results

The idea that string theory might shed light on nuclear physics predates the discovery of the AdS/CFT correspondence. In fact, string theory was originally developed as a theory of nuclear physics before being abandoned in favor of quantum chromodynamics. It was subsequently realized that the very properties that made string theory unsuitable as a theory of nuclear physics made it an ideal candidate for a theory of quantum gravity unified with the other fundamental forces.

The connections between string theory and nuclear physics were also considered in a paper of Gerard 't Hooft from 1974.^[33] In this paper, 't Hooft considered theories similar to quantum chromodynamics, where the number of color charges is some arbitrary number $N$ , rather than three as in quantum chromodynamics. He then considered a certain limit where $N$ tends to infinity and argued that in this limit certain calculations in quantum field theory resemble calculations in string theory. These ideas became more specific with the discovery of the AdS/CFT correspondence, which identifies specific quantum field theories dual to string theory.^[34]

Quark-gluon plasma

One physical system which has been studied using these methods 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 and gluons that make up atomic nuclei to deconfine at temperatures of approximately two trillion degrees Kelvin, conditions similar to those present at around $10^{-11}$ seconds after the Big Bang.^[35]

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. 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.^[36] A calculation based on the AdS/CFT correspondence showed that the ratio of two quantities associated with the quark-gluon plasma, the shear viscosity $\eta$ and volume density of entropy $s$ , should be approximately equal to a certain universal constant:

{\frac {\eta }{s}}\approx {\frac {\hbar }{4\pi k}}

where $\hbar$ denotes Planck's constant and $k$ is Boltzmann's constant. In addition, the authors conjectured that this universal constant provides a lower bound for $\eta /s$ 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.^[37]

Condensed matter physics

Over the decades, experimental condensed matter physicists have discovered a number of exotic states of matter, including superconductors, superfluids and Bose-Einstein condensates. 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 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.^[38]

So far some success has been achieved in using string theory methods to describe the transition of a superfluid to an insulator.^[39] 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.^[39] These atoms initially behave as a superfluid, but eventually transition to an insulating state as experimentalists increase the intensity of the lasers. During the transition, the atoms behave in unusual ways which have recently been understood by considering a dual description where the fluid is viewed as a field theoretic analog of a black hole.^[39]

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 letter to Physics Today, Nobel laureate Philip W. Anderson wrote^[40]

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.
— x, x

Notes

^ Maldacena 2005
^ 't Hooft 1993
^ Susskind 1995
^ ^a ^b Klebanov and Maldacena 2009
^ Merali 2011
^ ^a ^b Maldacena 1998
^ Gubser, Klebanov, and Polyakov 1998
^ Witten 1998
^ "Top Cited Articles during 2010 in hep-th". Retrieved 25 July 2013.
^ Klebanov and Maldacena 2009, p.28
^ Maldacena 2005, p.60
^ Maldacena 2005, p.60
^ Maldacena 2005, p.61
^ Zwiebach 2009, p.552
^ Maldacena 2005, pp.61--62
^ Zwieback 2009, p.8
^ This analogy is used for example in Greene 2000, p.186
^ As explained below, the known realizations of AdS/CFT typically involve unphysical numbers of spacetime dimensions and unphysical supersymmetries.
^ This example is the main subject of the three pioneering articles on AdS/CFT: Maldacena 1998, Gubser, Klebanov, and Polyakov 1998, and Witten 1998.
^ Maldacena 1998
^ For a review of the (2,0)-theory, see Moore 2012.
^ See Moore 2012 and Alday, Gaiotto, and Tachikawa 2010.
^ Aharony et al. 2008
^ A standard textbook introducing the formalism of Feynman diagrams is Peskin and Schroeder 1995.
^ One cannot apply perturbation theory when the coupling constant in a theory is large.
^ Zwiebach 2009, p.12
^ Maldacena 2005, p.61
^ Hawking 1975
^ Bekenstein 1973
^ Zwiebach 2009, p.554
^ Maldacena 2005, p.63
^ Hawking 2005
^ 't Hooft 1974
^ Aharony et al. 2008
^ Zwiebach 2009, p.559
^ Kovtun, Son, and Starinets 2001
^ Luzum and Romatschke 2008
^ Merali 2011, p.303
^ ^a ^b ^c Sachdev 2013, p.51
^ Anderson, Philip. "Strange connections to strange metals". Physics Today. Retrieved 14 August 2013.
^ Klebanov and Polyakov 2002
^ Witten 2007
^ Rehren 2005
^ Witten 1989

References

Aharony, Ofer; Bergman, Oren; Jafferis, Daniel Louis; Maldacena, Juan (2008). "N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals". Journal of High Energy Physics. 2008 (10): 091. arXiv:0806.1218. doi:10.1088/1126-6708/2008/10/091.

Aharony, Ofer; Gubser, Steven; Maldacena, Juan; Ooguri, Hirosi; Oz, Yaron (2000). "Large N Field Theories, String Theory and Gravity". Phys. Rept. 323 (3–4): 183–386. arXiv:hep-th/9905111. doi:10.1016/S0370-1573(99)00083-6.

Alday, Luis; Gaiotto, Davide; Tachikawa, Yuji (2010). "Liouville correlation functions from four-dimensional gauge theories". Letters in Mathematical Physics. 91 (2): 167--197.

Bekenstein, Jacob (1973). "Black holes and entropy". Physical Review D. 7 (8).

Greene, Brian (2000). The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. Random House.

Gubser, Steven; Klebanov, Igor; Polyakov, Alexander (1998). "Gauge theory correlators from non-critical string theory". Physics Letters. B428: 105–114. arXiv:hep-th/9802109. Bibcode:1998PhLB..428..105G. doi:10.1016/S0370-2693(98)00377-3.

Hawking, Stephen (1975). "Particle creation by black holes". Communications in mathematical physics. 43 (3): 199--220.

Hawking, Stephen (2005). "Information loss in black holes". Physical Review D. 72 (8).

Klebanov, Igor; Maldacena, Juan (2009). "Solving Quantum Field Theories via Curved Spacetimes" (PDF). Physics Today. 62: 28. doi:10.1063/1.3074260. Retrieved May, 2013. {{cite journal}}: Check date values in: |accessdate= (help)

Klebanov, Igor; Polyakov, Alexander (2002). "The AdS dual of the critical O(N) vector model". Physics Letters B. 550: 213–219.

Kovtun, P. K.; Son, Dam T.; Starinets, A. O. (2001). "Viscosity in strongly interacting quantum field theories from black hole physics". Physical review letters. 94 (11).

Luzum, Matthew; Romatschke, Paul (2008). "Conformal relativistic viscous hydrodynamics: Applications to RHIC results at ${\sqrt {s_{NN}}}=200$ GeV". Physical Review C. 78 (3). arXiv:0804.4015. doi:10.1103/PhysRevC.78.034915.

Maldacena, Juan (1998). "The Large N limit of superconformal field theories and supergravity". Advances in Theoretical and Mathematical Physics. 2: 231–252. arXiv:hep-th/9711200. Bibcode:1998AdTMP...2..231M.

Maldacena, Juan (2005). "The Illusion of Gravity" (PDF). Scientific American. Retrieved July, 2013. {{cite journal}}: Check date values in: |accessdate= (help)

Merali, Zeeya (2011). "Collaborative physics: string theory finds a bench mate". Nature. 478 (7369): 302–304. doi:10.1038/478302a. PMID 22012369.

Moore, Gregory (2012). "Applications of the six-dimensional (2,0) theories to Physical Mathematics" (PDF). Retrieved 14 August 2013.

Peskin, Michael; Schroeder, Daniel (1995). An Introduction to Quantum Field Theory. Westview Press.

Rehren, Karl-Henning (2005). "QFT Lectures on AdS-CFT". In B. Dragovich; et al. (eds.). Proceedings of the 3rd Summer School in Modern Mathematical Physics. 2004 Zlatibor Summer School on Modern Mathematical Physics. Belgrade: Institute of Physics. pp. 95–118. arXiv:hep-th/0411086. {{cite conference}}: Explicit use of et al. in: |editor= (help); Unknown parameter |booktitle= ignored (|book-title= suggested) (help)

Sachdev, Subir (2013). "Strange and stringy". Scientific American. 308 (44): 44. doi:10.1038/scientificamerican0113-44.

Susskind, Leonard (1995). "The World as a Hologram". Journal of Mathematical Physics. 36 (11): 6377–6396. arXiv:hep-th/9409089. doi:10.1063/1.531249.

't Hooft, Gerard (1974). "A planar diagram theory for strong interactions". Nuclear Physics B. 72 (3): 461--473.

't Hooft, Gerard (1993). "Dimensional Reduction in Quantum Gravity". arXiv:gr-qc/9310026. {{cite journal}}: Cite journal requires |journal= (help)

Witten, Edward (1989). "Quantum Field Theory and the Jones Polynomial". Commun. Math. Phys. 121 (3): 351–399. Bibcode:1989CMaPh.121..351W. doi:10.1007/BF01217730. MR 0990772.

Witten, Edward (1998). "Anti-de Sitter space and holography". Advances in Theoretical and Mathematical Physics. 2: 253–291. arXiv:hep-th/9802150. Bibcode:1998hep.th....2150W.

Witten, Edward (2007). "Three-dimensional gravity revisited". arXiv:0706.3359. {{cite journal}}: Cite journal requires |journal= (help)

Zwiebach, Barton (2009). A First Course in String Theory. Cambridge University Press.

[1] Maldacena 2005

[2] 't Hooft 1993

[3] Susskind 1995

[solveqft-4] Klebanov and Maldacena 2009

[nature-5] Merali 2011

[largeN-6] Maldacena 1998

[noncrit-7] Gubser, Klebanov, and Polyakov 1998

[witten-8] Witten 1998

[9] "Top Cited Articles during 2010 in hep-th". Retrieved 25 July 2013.

[10] Klebanov and Maldacena 2009, p.28

[11] Maldacena 2005, p.60

[12] Maldacena 2005, p.60

[13] Maldacena 2005, p.61

[14] Zwiebach 2009, p.552

[15] Maldacena 2005, pp.61--62

[16] Zwieback 2009, p.8

[17] This analogy is used for example in Greene 2000, p.186

[18] As explained below, the known realizations of AdS/CFT typically involve unphysical numbers of spacetime dimensions and unphysical supersymmetries.

[19] This example is the main subject of the three pioneering articles on AdS/CFT: Maldacena 1998, Gubser, Klebanov, and Polyakov 1998, and Witten 1998.

[20] Maldacena 1998

[21] For a review of the (2,0)-theory, see Moore 2012.

[22] See Moore 2012 and Alday, Gaiotto, and Tachikawa 2010.

[23] Aharony et al. 2008

[24] A standard textbook introducing the formalism of Feynman diagrams is Peskin and Schroeder 1995.

[25] One cannot apply perturbation theory when the coupling constant in a theory is large.

[26] Zwiebach 2009, p.12

[27] Maldacena 2005, p.61

[28] Hawking 1975

[29] Bekenstein 1973

[30] Zwiebach 2009, p.554

[31] Maldacena 2005, p.63

[32] Hawking 2005

[33] 't Hooft 1974

[34] Aharony et al. 2008

[35] Zwiebach 2009, p.559

[36] Kovtun, Son, and Starinets 2001

[37] Luzum and Romatschke 2008

[38] Merali 2011, p.303

[strange-39] Sachdev 2013, p.51

[40] Anderson, Philip. "Strange connections to strange metals". Physics Today. Retrieved 14 August 2013.

[41] Klebanov and Polyakov 2002

[42] Witten 2007

[43] Rehren 2005

[44] Witten 1989

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@@ Line 21: / Line 21: @@
 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 was famously depicted by [[M.C. Escher]] in a woodcut entitled ''[[Circle Limit III]]''.<ref>Maldacena 2005, p.60</ref> This image shows a [[tesselation]] of a disk by fish shapes. One can define the distance between points of this disk in such a way that all the fish are the same size and the circular boundary is infinitely far from any point in the interior. In three dimensions, anti-de Sitter space can be viewed as the geometric object obtained by adding a third dimension to this picture (a "time" dimension). There is a similar description of anti-de Sitter space in any number of dimensions.<ref>Maldacena 2005, p.60</ref>
-In anti-de Sitter space, it is possible to define a notion of "boundary" of spacetime at infinity. Such a boundary is called the ''conformal boundary''. In the above description, the conformal boundary of three-dimensional anti-de Sitter space is the two-dimensional cylinder traced out by the boundary circle of the hyperbolic disk as one moves along in the time direction.<ref>Maldacena 2005, p.61</ref> The conformal boundary has an induced geometry, namely that of [[Minkowski space]], the model of spacetime used in nongravitational physics.<ref>Cf. the discussion in the next subsection.</ref>
+In anti-de Sitter space, it is possible to define a notion of "boundary" of spacetime at infinity. Such a boundary is called the ''conformal boundary''. In the above description, the conformal boundary of three-dimensional anti-de Sitter space is the two-dimensional cylinder traced out by the boundary circle of the hyperbolic disk as one moves along in the time direction.<ref>Maldacena 2005, p.61</ref> The conformal boundary has an induced geometry, namely that of [[Minkowski space]], the model of spacetime used in nongravitational physics.<ref>Zwiebach 2009, p.552</ref>
 The idea of the AdS/CFT correspondence is that the conformal boundary of anti-de Sitter space can be regarded as the "spacetime" for a special type of quantum field theory called a [[conformal field theory]]. The claim is that this quantum 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 identicle 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>
@@ Line 28: / Line 28: @@
 A standard analogy for this is to consider 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.
-===Some mathematical details===
-To make the above statements more precise, let us write down explicitly the metric on part of anti-de Sitter space. We will assume here that the reader is familiar with [[Riemannian geometry]] at the level assumed in most texts on [[quantum field theory]] and [[general relativity]].
-Rather than study the full <math>(n+1)</math>-dimensional anti-de Sitter space, we will consider a [[coordinate patch]] covering part of this space called the "half space" region. This patch is [[diffeomorphic]] to <math>\mathbb{R}\times\mathbb{R}^{n-1}\times\mathbb{R}_{>0}</math> and so admits a natural [[system of coordinates]] <math>t,x_1,\dots,x_{n-1}\in\mathbb{R}</math>, <math>y\in\mathbb{R}_{>0}</math>. The metric on this patch can be written<ref>Zwiebach 2009, p. 565</ref>
-:<math>ds^2=\frac{1}{y^2}\left(dy^2-dt^2+\sum_idx_i^2\right)</math>.
-The constant time slices of this coordinate patch carry an induced metric
-:<math>\frac{dy^2+\sum_idx_i^2}{y^2}</math>,
-which is precisely the metric for the [[Poincaré half-plane model]] of the [[hyperbolic plane]]. The boundary of the space is at <math>y=0</math>, where the induced metric is that of <math>n</math>-dimensional [[Minkowski space]]:
-:<math>dy^2=dt^2-\sum_idx_i^2</math>,
-As explained above, the AdS/CFT correspondence provides an equivalence between [[string theory]] and [[M-theory]] on anti-de Sitter space and [[conformal field theories]] on the conformal boundary. In [[Edward Witten]]'s formulation,<ref name="witten" /> the correspondence states that
-:<math>Z_{CFT}=\sum e^{-S(g)}</math>
-where <math>Z_{CFT}</math> is the [[partition function (quantum field theory)|partition function]] of a conformal field theory on the boundary and <math>S(g)</math> is the [[renormalized]] [[Einstein-Hilbert action]] of a metric <math>g</math> satisfying certain [[boundary condition]]s. The sum is over all [[manifold]]s and metrics in the bulk satisfying these boundary conditions.
 ===Examples of the correspondence===