Editing M-theory (section)

===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>