Editing AdS/CFT correspondence (section)

== Overview of the correspondence ==

[[Image:Uniform tiling 433-t0.png|thumb|A [[tritetragonal tiling|tessellation]] of the [[hyperbolic plane]] by triangles and squares.]]

=== Geometry of anti-de Sitter space ===

{{Further|topic=the mathematics described here|Anti-de Sitter space}}

In the AdS/CFT correspondence, one considers string theory or M-theory on an anti-de Sitter [[String background|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]].{{sfn|Klebanov|Maldacena|2009|p=28}}

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 [[Poincaré disk model|disk]] as illustrated on the right.{{sfn|Maldacena|2005|p=60}} This image shows a [[tessellation]] of a disk by triangles and squares. One can define the 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.{{sfn|ps=|Maldacena|2005|p=61}}

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.{{sfn|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 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.{{refn|The mathematical relationship between the interior and boundary of anti-de&nbsp;Sitter space is related to the [[ambient construction]] of [[Charles Fefferman]] and [[C. Robin Graham|Robin Graham]]. For details see {{harvnb|Fefferman|Graham|1985}}, {{harvnb|Fefferman|Graham|2011}}.}}
[[File:AdS3.svg|left|thumb|upright=1.25|Three-dimensional [[anti-de Sitter space]] is like a stack of [[Poincaré 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.{{sfn|Maldacena|2005|p=60}}

=== 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.{{sfn|ps=|Zwiebach|2009|p=552}}

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 the 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.{{sfn|ps=|Maldacena|2005|pp=61–62}}

[[File:Holomouse2.jpg|thumb|A [[hologram]] is a two-dimensional image that 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 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 to the relationship between a three-dimensional object and its image as a [[hologram]].{{sfn|ps=|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 that are related by the AdS/CFT correspondence are conjectured to be ''exactly'' equivalent, despite living in different numbers of dimensions. The conformal field theory is like a hologram that captures information about the higher-dimensional quantum gravity theory.{{sfn|ps=|Maldacena|2005|p=61}}

=== 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.{{refn|The known realizations of AdS/CFT typically involve unphysical numbers of spacetime dimensions and unphysical supersymmetries.}}

The most famous example of the AdS/CFT correspondence states that [[type IIB string theory]] on the [[product space]] {{nowrap|AdS<sub>5</sub> × ''S''<sup>5</sup>}} is equivalent to [[N = 4 supersymmetric Yang–Mills theory|''N'' = 4 supersymmetric Yang–Mills theory]] on the four-dimensional boundary.{{refn|This example is the main subject of the three pioneering articles on AdS/CFT: {{harvnb|Maldacena|1998}}; {{harvnb|Gubser|Klebanov|Polyakov|1998}}; and {{harvnb|Witten|1998}}.}} In this example, the spacetime on which the gravitational theory lives is effectively five-dimensional (hence the notation AdS<sub>5</sub>), and there are five additional [[compact dimension]]s (encoded by the ''S''<sup>5</sup> 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 [[fermion]]s.{{sfn|Maldacena|2005|p=62}} As a result, it has found applications in [[nuclear physics]], particularly in the study of the [[quark–gluon plasma]].{{sfn|ps=|Merali|2011|p=303}}{{sfn|ps=|Kovtun|Son|Starinets|2005}}

Another realization of the correspondence states that M-theory on {{nowrap|AdS<sub>7</sub> × ''S''<sup>4</sup>}} is equivalent to the so-called [[6D (2,0) superconformal field theory|(2,0)-theory]] in six dimensions.{{sfn|ps=|Maldacena|1998|loc=The pre-print was submitted in 1997 and published on January 1, 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 mechanical theory without a [[classical limit]].{{refn|For a review of the (2,0)-theory, see {{harvnb|Moore|2012}}.}} 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.{{refn|See {{harvnb|Moore|2012}} and {{harvnb|Alday|Gaiotto|Tachikawa|2010}}.}}

Yet another realization of the correspondence states that M-theory on {{nowrap|AdS<sub>4</sub> × ''S''<sup>7</sup>}} is equivalent to the [[ABJM superconformal field theory]] in three dimensions.{{sfn|ps=|Aharony|Bergman|Jafferis|Maldacena|2008}} Here the gravitational theory has four noncompact dimensions, so this version of the correspondence provides a somewhat more realistic description of gravity.{{sfn|ps=|Aharony|Bergman|Jafferis|Maldacena|2008|loc=sec. 1}}