Editing AdS/CFT correspondence (section)

=== 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}}