Editing Conformal field theory (section)

== Two dimensions vs higher dimensions ==
The number of independent conformal transformations is infinite in two dimensions, and finite in higher dimensions. This makes conformal symmetry much more constraining in two dimensions.{{clarify|date=April 2023|reason=Counterintuitive}} All conformal field theories share the ideas and techniques of the [[conformal bootstrap]]. But the resulting equations are more powerful in two dimensions, where they are sometimes exactly solvable (for example in the case of [[Minimal model (physics)|minimal models]]), in contrast to higher dimensions, where numerical approaches dominate.

The development of conformal field theory has been earlier and deeper in the two-dimensional case, in particular after the 1983 article by Belavin, Polyakov and Zamolodchikov.<ref name="BPZ">{{Cite journal| doi = 10.1016/0550-3213(84)90052-X| issn = 0550-3213| volume = 241| issue = 2| pages = 333–380| last1 = Belavin| first1 = A.A.| last2 = Polyakov| first2 = A.M.| last3 = Zamolodchikov| first3 = A.B.| title = Infinite conformal symmetry in two-dimensional quantum field theory| journal = Nuclear Physics B| date = 1984| bibcode=1984NuPhB.241..333B| url = https://cds.cern.ch/record/152341/files/198407016.pdf}}</ref> 
The term ''conformal field theory'' has sometimes been used with the meaning of ''two-dimensional conformal field theory'', as in the title of a 1997 textbook.<ref name="BYB">P. Di Francesco, P. Mathieu, and D. Sénéchal, ''Conformal Field Theory'', 1997, {{ISBN|0-387-94785-X}}</ref>
Higher-dimensional conformal field theories have become more popular with the [[AdS/CFT correspondence]] in the late 1990s, and the development of numerical conformal bootstrap techniques in the 2000s.

=== Global vs local conformal symmetry in two dimensions ===

The global conformal group of the [[Riemann sphere]] is the group of [[Möbius transformation]]s {{tmath|1= \mathrm{PSL}_2(\mathbb{C}) }}, which is finite-dimensional. 
On the other hand, infinitesimal conformal transformations form the infinite-dimensional [[Witt algebra]]: the [[conformal Killing equation]]s in two dimensions, <math>\partial_\mu \xi_\nu + \partial_\nu \xi_\mu = \partial \cdot\xi \eta_{\mu \nu},~</math> reduce to just the Cauchy-Riemann equations, {{tmath|1= \partial_{\bar{z} } \xi(z) = 0 = \partial_z \xi (\bar{z}) }}, the infinity of modes of arbitrary analytic coordinate transformations <math>\xi(z)</math> yield the infinity of [[Killing vector field]]s {{tmath|1= z^n\partial_z }}.

Strictly speaking, it is possible for a two-dimensional conformal field theory to be local (in the sense of possessing a stress-tensor) while still only exhibiting invariance under the global {{tmath|1= \mathrm{PSL}_2(\mathbb{C}) }}. This turns out to be unique to non-unitary theories; an example is the biharmonic scalar.<ref name="raj11"/> This property should be viewed as even more special than scale without conformal invariance as it requires <math>T_\mu{}^\mu</math> to be a total second derivative.

Global conformal symmetry in two dimensions is a special case of conformal symmetry in higher dimensions, and is studied with the same techniques. This is done not only in theories that have global but not local conformal symmetry, but also in theories that do have local conformal symmetry, for the purpose of testing techniques or ideas from higher-dimensional CFT. In particular, numerical bootstrap techniques can be tested by applying them to [[Minimal model (physics)|minimal models]], and comparing the results with the known analytic results that follow from local conformal symmetry.

=== Conformal field theories with a Virasoro symmetry algebra ===
{{main|Two-dimensional conformal field theory}}

In a conformally invariant two-dimensional quantum theory, the Witt algebra of infinitesimal conformal transformations has to be [[Lie algebra extension#Virasoro algebra|centrally extended]]. The quantum symmetry algebra is therefore the [[Virasoro algebra]], which depends on a number called the '''central charge'''. This central extension can also be understood in terms of a [[conformal anomaly]].

It was shown by [[Alexander Zamolodchikov]] that there exists a function which decreases monotonically under the [[renormalization group]] flow of a two-dimensional quantum field theory, and is equal to the central charge for a two-dimensional conformal field theory. This is known as the Zamolodchikov [[C-theorem]], and tells us that [[renormalization group flow]] in two dimensions is irreversible.<ref name="zam86"/>

In addition to being centrally extended, the symmetry algebra of a conformally invariant quantum theory has to be complexified, resulting in two copies of the Virasoro algebra. 
In Euclidean CFT, these copies are called holomorphic and antiholomorphic. In Lorentzian CFT, they are called left-moving and right moving. Both copies have the same central charge.

The [[state space (physics)|space of states]] of a theory is a [[Lie algebra representation|representation]] of the product of the two Virasoro algebras. This space is a [[Hilbert space]] if the theory is unitary.
This space may contain a vacuum state, or in statistical mechanics, a thermal state. Unless the central charge vanishes, there cannot exist a state that leaves the entire infinite dimensional conformal symmetry unbroken. The best we can have is a state that is invariant under the generators <math>L_{n\geq -1}</math> of the Virasoro algebra, whose basis is {{tmath|1=(L_n)_{n\in\mathbb{Z} } }}. This contains the generators <math>L_{-1},L_0,L_1</math> of the global conformal transformations. The rest of the conformal group is spontaneously broken.