Editing M-theory (section)

==AdS/CFT correspondence==

===Overview===

{{main article|AdS/CFT correspondence}}

[[Image:Uniform tiling 433-t0 (formatted).svg|thumb|left|alt=A disk tiled by triangles and quadrilaterals which become smaller and smaller near the boundary circle.|A [[alternated octagonal tiling|tessellation]] of the [[hyperbolic plane]] by triangles and squares]]

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]].{{efn|A standard text is Peskin and Schroeder 1995.}} 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.{{efn|For an introduction to the applications of quantum field theory to condensed matter physics, see Zee 2010.}}

One approach to formulating M-theory and studying its properties is provided by the [[AdS/CFT correspondence|anti-de Sitter/conformal field theory (AdS/CFT) correspondence]]. Proposed by [[Juan Maldacena]] in late 1997, the AdS/CFT correspondence is a theoretical result which implies that M-theory is in some cases equivalent to a quantum field theory.<ref name="Maldacena_a"/> In addition to providing insights into the mathematical structure of string and M-theory, the AdS/CFT correspondence has shed light on many aspects of quantum field theory in regimes where traditional calculational techniques are ineffective.<ref>Klebanov and Maldacena 2009</ref>

In the AdS/CFT correspondence, 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 [[Poincaré disk model|disk]] as illustrated on the left.<ref name="Maldacena 2005, p. 60">Maldacena 2005, p. 60</ref> 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.<ref name="Maldacena 2005, p. 61">Maldacena 2005, p. 61</ref>

[[File:AdS3.svg|thumb|upright=1.4|alt=A cylinder formed by stacking copies of the disk illustrated in the previous figure.|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. One can study theories of [[quantum gravity]] such as M-theory in the resulting [[spacetime]].]]

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 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.<ref name="Maldacena 2005, p. 61"/>

This construction describes a hypothetical universe with only two space dimensions 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"/>

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, within a small region on the surface around any given 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 quantum 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 entities and calculations in one theory into their counterparts 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>

===6D (2,0) superconformal field theory===

{{main article|6D (2,0) superconformal field theory}}

[[Image:Knot table-blank unknot.svg|left|thumb|upright=1.6|alt=A collection of knot diagrams in the plane.|The six-dimensional [[6D (2,0) superconformal field theory|(2,0)-theory]] has been used to understand results from the [[knot theory|mathematical theory of knots]].]]

One particular realization of the AdS/CFT correspondence states that M-theory on the [[product space]] {{math|''AdS''<sub>7</sub>×''S''<sup>4</sup>}} is equivalent to the so-called [[6D (2,0) superconformal field theory|(2,0)-theory]] on the six-dimensional boundary.<ref name="Maldacena_a">Maldacena 1998</ref> Here "(2,0)" refers to the particular type of supersymmetry that appears in the theory. In this example, the spacetime of the gravitational theory is effectively seven-dimensional (hence the notation {{math|''AdS''<sub>7</sub>}}), and there are four additional "[[compact space|compact]]" dimensions (encoded by the {{math|[[n-sphere|''S''<sup>4</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 since it describes a world with six spacetime dimensions.{{efn|For a review of the (2,0)-theory, see Moore 2012.}}

Nevertheless, the (2,0)-theory has proven to be important for studying the general properties of quantum field theories. Indeed, this theory subsumes many mathematically interesting [[effective field theory|effective quantum field theories]] and points to new dualities relating these theories. For example, Luis Alday, Davide Gaiotto, and Yuji Tachikawa showed that by compactifying this theory on a [[Surface (topology)|surface]], one obtains a four-dimensional quantum field theory, and there is a duality known as the [[AGT correspondence]] which relates the physics of this theory to certain physical concepts associated with the surface itself.<ref>Alday, Gaiotto, and Tachikawa 2010</ref> More recently, theorists have extended these ideas to study the theories obtained by compactifying down to three dimensions.<ref>Dimofte, Gaiotto, and Gukov 2010</ref>

In addition to its applications in quantum field theory, the (2,0)-theory has spawned important results in [[pure mathematics]]. For example, the existence of the (2,0)-theory was used by Witten to give a "physical" explanation for a conjectural relationship in mathematics called the [[geometric Langlands correspondence]].<ref>Witten 2009</ref> In subsequent work, Witten showed that the (2,0)-theory could be used to understand a concept in mathematics called [[Khovanov homology]].<ref>Witten 2012</ref> Developed by [[Mikhail Khovanov]] around 2000, Khovanov homology provides a tool in [[knot theory]], the branch of mathematics that studies and classifies the different shapes of knots.<ref>Khovanov 2000</ref> Another application of the (2,0)-theory in mathematics is the work of [[Davide Gaiotto]], [[Greg Moore (physicist)|Greg Moore]], and [[Andrew Neitzke]], which used physical ideas to derive new results in [[hyperkähler manifold|hyperkähler geometry]].<ref>Gaiotto, Moore, and Neitzke 2013</ref>

===ABJM superconformal field theory===

{{main article|ABJM superconformal field theory}}

Another realization of the AdS/CFT correspondence states that M-theory on {{math|''AdS''<sub>4</sub>×''S''<sup>7</sup>}} is equivalent to a quantum field theory called the [[ABJM superconformal field theory|ABJM theory]] in three dimensions. In this version of the correspondence, seven of the dimensions of M-theory are curled up, leaving four non-compact dimensions. Since the spacetime of our universe is four-dimensional, this version of the correspondence provides a somewhat more realistic description of gravity.<ref name="Aharony et al. 2008">Aharony et al. 2008</ref>

The ABJM theory appearing in this version of the correspondence is also interesting for a variety of reasons. Introduced by Aharony, Bergman, Jafferis, and Maldacena, it is closely related to another quantum field theory called [[Chern–Simons theory]]. The latter theory was popularized by Witten in the late 1980s because of its applications to knot theory.<ref>Witten 1989</ref> In addition, the ABJM theory serves as a semi-realistic simplified model for solving problems that arise in condensed matter physics.<ref name="Aharony et al. 2008"/>