Editing Conformal field theory (section)

== Applications ==

=== Continuous phase transitions ===
{{main|Phase transition}}
Continuous phase transitions (critical points) of classical statistical physics systems with ''D'' spatial dimensions are often described by Euclidean conformal field theories. A necessary condition for this to happen is that the critical point should be invariant under spatial rotations and translations. However this condition is not sufficient: some exceptional critical points are described by scale invariant but not conformally invariant theories. If the classical statistical physics system is reflection positive, the corresponding Euclidean CFT describing its critical point will be unitary.

Continuous [[quantum phase transition]]s in condensed matter systems with ''D'' spatial dimensions may be described by Lorentzian ''D+1'' dimensional conformal field theories (related by [[Wick rotation]] to Euclidean CFTs in {{nowrap|''D'' + 1}} dimensions). Apart from translation and rotation invariance, an additional necessary condition for this to happen is that the dynamical critical exponent ''z'' should be equal to 1. CFTs describing such quantum phase transitions (in absence of quenched disorder) are always unitary.

=== String theory ===
{{main|String theory}}
World-sheet description of string theory involves a two-dimensional CFT coupled to dynamical two-dimensional quantum gravity (or supergravity, in case of superstring theory). Consistency of string theory models imposes constraints on the central charge of this CFT, which should be {{nowrap|1=''c'' = 26}} in bosonic string theory and {{nowrap|1=''c'' = 10}} in superstring theory. Coordinates of the spacetime in which string theory lives correspond to bosonic fields of this CFT.

=== AdS/CFT correspondence ===
{{main|AdS/CFT correspondence}}
Conformal field theories play a prominent role in the [[AdS/CFT correspondence]], in which a gravitational theory in [[anti-de Sitter space]] (AdS) is equivalent to a conformal field theory on the AdS boundary. Notable examples are ''d''&nbsp;=&nbsp;4, [[N = 4 supersymmetric Yang–Mills theory|''N'' = 4 supersymmetric Yang–Mills theory]], which is dual to [[Type IIB string theory]] on AdS<sub>5</sub>&nbsp;×&nbsp;''S''<sup>5</sup>, and ''d''&nbsp;=&nbsp;3, ''N''&nbsp;=&nbsp;6 super-[[Chern–Simons theory]], which is dual to [[M-theory]] on AdS<sub>4</sub>&nbsp;×&nbsp;''S''<sup>7</sup>. (The prefix "super" denotes [[supersymmetry]], ''N'' denotes the degree of [[extended supersymmetry]] possessed by the theory, and ''d'' the number of space-time dimensions on the boundary.)

=== Conformal perturbation theory ===
By perturbing a conformal field theory, it is possible to construct other field theories, conformal or not. Their correlation functions can be computed perturbatively from the correlation functions of the original CFT, by a technique called '''conformal perturbation theory'''. 

For example, a type of perturbation consists in discretizing a conformal field theory by studying it on a discrete spacetime. The resulting finite-size effects can be computed using conformal perturbation theory.<ref name="lhwr25"/>