Aconformal field theory (CFT) is aquantum field theory that isinvariant underconformal transformations. Intwodimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified.

Conformal field theory has important applications^[1] tocondensed matter physics,statistical mechanics,quantum statistical mechanics, andstring theory. Statistical and condensed matter systems are indeed often conformally invariant at theirthermodynamic orquantum critical points.

Inquantum field theory,scale invariance is a common and natural symmetry, because any fixed point of therenormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions^[2] to argue that it should appear in nature. The basic idea behind its plausibility is thatlocal scale invariant theories have their currents given by${\displaystyle T_{\mu \nu }{\xi }^{\nu }}$ where${\displaystyle {\xi }^{\nu }}$ is aKilling vector and${\displaystyle T_{\mu \nu }}$ is a conserved operator (the stress-tensor) of dimension exactly⁠${\displaystyle d}$⁠. For the associated symmetries to include scale but not conformal transformations, the trace${\displaystyle T_{\mu }{}^{\mu }}$ has to be a non-zero total derivative implying that there is a non-conserved operator of dimension exactly⁠${\displaystyle d-1}$⁠.

Under some assumptions it is possible to completely rule out this type of non-renormalization and hence prove that scale invariance implies conformal invariance in a quantum field theory, for example inunitary compact conformal field theories in two dimensions.

While it is possible for aquantum field theory to bescale invariant but not conformally invariant, examples are rare.^[3] For this reason, the terms are often used interchangeably in the context of quantum field theory.

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.^{[clarification needed]} All conformal field theories share the ideas and techniques of theconformal bootstrap. But the resulting equations are more powerful in two dimensions, where they are sometimes exactly solvable (for example in the case ofminimal 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.^[4] The termconformal field theory has sometimes been used with the meaning oftwo-dimensional conformal field theory, as in the title of a 1997 textbook.^[5]Higher-dimensional conformal field theories have become more popular with theAdS/CFT correspondence in the late 1990s, and the development of numerical conformal bootstrap techniques in the 2000s.

The global conformal group of theRiemann sphere is the group ofMöbius transformations⁠${\displaystyle {\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})}$⁠, which is finite-dimensional. On the other hand, infinitesimal conformal transformations form the infinite-dimensionalWitt algebra: theconformal Killing equations in two dimensions,${\displaystyle {\mathrm{\partial }}_{\mu }{\xi }_{\nu }+{\mathrm{\partial }}_{\nu }{\xi }_{\mu }=\mathrm{\partial }\cdot \xi {\eta }_{\mu \nu },}$ reduce to just the Cauchy-Riemann equations,⁠${\displaystyle {\mathrm{\partial }}_{\overline{z}}\xi (z)=0={\mathrm{\partial }}_z\xi (\overline{z})}$⁠, the infinity of modes of arbitrary analytic coordinate transformations${\displaystyle \xi (z)}$ yield the infinity ofKilling vector fields⁠${\displaystyle z^n{\mathrm{\partial }}_z}$⁠.

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${\displaystyle {\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})}$⁠. This turns out to be unique to non-unitary theories; an example is the biharmonic scalar.^[6] This property should be viewed as even more special than scale without conformal invariance as it requires${\displaystyle T_{\mu }{}^{\mu }}$ 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 tominimal models, and comparing the results with the known analytic results that follow from local conformal symmetry.

In a conformally invariant two-dimensional quantum theory, the Witt algebra of infinitesimal conformal transformations has to becentrally extended. The quantum symmetry algebra is therefore theVirasoro algebra, which depends on a number called thecentral charge. This central extension can also be understood in terms of aconformal anomaly.

It was shown byAlexander Zamolodchikov that there exists a function which decreases monotonically under therenormalization 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 ZamolodchikovC-theorem, and tells us thatrenormalization group flow in two dimensions is irreversible.^[7]

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.

Thespace of states of a theory is arepresentation of the product of the two Virasoro algebras. This space is aHilbert 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${\displaystyle L_{n\ge -1}}$ of the Virasoro algebra, whose basis is⁠${\displaystyle (L_n{)}_{n\in \mathbb{Z}}}$⁠. This contains the generators${\displaystyle L_{-1},L_0,L_1}$ of the global conformal transformations. The rest of the conformal group is spontaneously broken.

For a given spacetime and metric, a conformal transformation is a transformation that preserves angles. We will focus on conformal transformations of the flat${\displaystyle d}$-dimensional Euclidean space${\displaystyle {\mathbb{R}}^d}$ or of the Minkowski space⁠${\displaystyle {\mathbb{R}}^{1,d-1}}$⁠.

If${\displaystyle x\to f(x)}$ is a conformal transformation, the Jacobian${\displaystyle J_{\nu }^{\mu }(x)=\frac{\mathrm{\partial }{f}^{\mu }(x)}{\mathrm{\partial }{x}^{\nu }}}$ is of the form

${\displaystyle J_{\nu }^{\mu }(x)=\mathrm{\Omega }(x)R_{\nu }^{\mu }(x),}$

where${\displaystyle \mathrm{\Omega }(x)}$ is the scale factor, and${\displaystyle R_{\nu }^{\mu }(x)}$ is a rotation (i.e. an orthogonal matrix) or Lorentz transformation.

Theconformal group of Euclidean space is locally isomorphic to⁠${\displaystyle \mathrm{S}\mathrm{O}(1,d+1)}$⁠, and of Minkowski space is⁠${\displaystyle \mathrm{S}\mathrm{O}(2,d)}$⁠. This includes translations, rotations (Euclidean) or Lorentz transformations (Minkowski), and dilations i.e. scale transformations

${\displaystyle x^{\mu }\to \lambda x^{\mu }.}$

This also includes special conformal transformations. For any translation⁠${\displaystyle T_a(x)=x+a}$⁠, there is aspecial conformal transformation

${\displaystyle S_a=I\circ T_a\circ I,}$

where${\displaystyle I}$ is theinversion such that

${\displaystyle I\left(x^{\mu }\right)=\frac{x^{\mu }}{x^2}.}$

In the sphere⁠${\displaystyle S^d={\mathbb{R}}^d\cup \{\mathrm{\infty }\}}$⁠, the inversion exchanges${\displaystyle 0}$ with⁠${\displaystyle \mathrm{\infty }}$⁠. Translations leave${\displaystyle \mathrm{\infty }}$ fixed, while special conformal transformations leave${\displaystyle 0}$ fixed.

The commutation relations of the corresponding Lie algebra are

where${\displaystyle P}$ generatetranslations,${\displaystyle D}$ generates dilations,${\displaystyle K_{\mu }}$ generate special conformal transformations, and${\displaystyle M_{\mu \nu }}$ generate rotations or Lorentz transformations. The tensor${\displaystyle {\eta }_{\mu \nu }}$ is the flat metric.

In Minkowski space, the conformal group does not preservecausality. Observables such as correlation functions are invariant under the conformal algebra, but not under the conformal group. As shown by Lüscher and Mack, it is possible to restore the invariance under the conformal group by extending the flat Minkowski space into a Lorentzian cylinder.^[8] The original Minkowski space is conformally equivalent to a region of the cylinder called a Poincaré patch. In the cylinder, global conformal transformations do not violate causality: instead, they can move points outside the Poincaré patch.

In theconformal bootstrap approach, a conformal field theory is a set of correlation functions that obey a number of axioms.

The${\displaystyle n}$-point correlation function${\displaystyle ⟨O_1(x_1)\cdots O_n(x_n)⟩}$ is a function of the positions${\displaystyle x_i}$ and other parameters of the fields⁠${\displaystyle O_1,\dots ,O_n}$⁠. In the bootstrap approach, the fields themselves make sense only in the context of correlation functions, and may be viewed as efficient notations for writing axioms for correlation functions. Correlation functions depend linearly on fields, in particular⁠${\displaystyle {\mathrm{\partial }}_{x_1}⟨O_1(x_1)\cdots ⟩=⟨{\mathrm{\partial }}_{x_1}{O}_1(x_1)\cdots ⟩}$⁠.

We focus on CFT on the Euclidean space⁠${\displaystyle {\mathbb{R}}^d}$⁠. In this case, correlation functions areSchwinger functions. They are defined for⁠${\displaystyle x_i\ne x_j}$⁠, and do not depend on the order of the fields. In Minkowski space, correlation functions areWightman functions. They can depend on the order of the fields, as fields commute only if they are spacelike separated. A Euclidean CFT can be related to a Minkowskian CFT byWick rotation, for example thanks to theOsterwalder-Schrader theorem. In such cases, Minkowskian correlation functions are obtained from Euclidean correlation functions by an analytic continuation that depends on the order of the fields.

Any conformal transformation${\displaystyle x\to f(x)}$ acts linearly on fields⁠${\displaystyle O(x)\to {\pi }_f(O)(x)}$⁠, such that${\displaystyle f\to {\pi }_f}$ is a representation of the conformal group, and correlation functions are invariant:

${\displaystyle ⟨{\pi }_f(O_1)(x_1)\cdots {\pi }_f(O_n)(x_n)⟩=⟨O_1(x_1)\cdots O_n(x_n)⟩.}$

Primary fields are fields that transform into themselves via⁠${\displaystyle {\pi }_f}$⁠. The behaviour of a primary field is characterized by a number${\displaystyle \mathrm{\Delta }}$ called itsconformal dimension, and a representation${\displaystyle \rho }$ of the rotation or Lorentz group. For a primary field, we then have

${\displaystyle {\pi }_f(O)(x)=\mathrm{\Omega }(x^{\prime }{)}^{-\mathrm{\Delta }}\rho (R(x^{\prime }))O(x^{\prime }),\text{where}{x}^{\prime }=f^{-1}(x).}$

Here${\displaystyle \mathrm{\Omega }(x)}$ and${\displaystyle R(x)}$ are the scale factor and rotation that are associated to the conformal transformation⁠${\displaystyle f}$⁠. The representation${\displaystyle \rho }$ is trivial in the case of scalar fields, which transform as⁠${\displaystyle {\pi }_f(O)(x)=\mathrm{\Omega }(x^{\prime }{)}^{-\mathrm{\Delta }}O(x^{\prime })}$⁠. For vector fields, the representation${\displaystyle \rho }$ is the fundamental representation, and we would have⁠${\displaystyle {\pi }_f(O_{\mu })(x)=\mathrm{\Omega }(x^{\prime }{)}^{-\mathrm{\Delta }}{R}_{\mu }^{\nu }(x^{\prime })O_{\nu }(x^{\prime })}$⁠.

A primary field that is characterized by the conformal dimension${\displaystyle \mathrm{\Delta }}$ and representation${\displaystyle \rho }$ behaves as a highest-weight vector in aninduced representation of the conformal group from the subgroup generated by dilations and rotations. In particular, the conformal dimension${\displaystyle \mathrm{\Delta }}$ characterizes a representation of the subgroup of dilations. In two dimensions, the fact that this induced representation is aVerma module appears throughout the literature. For higher-dimensional CFTs (in which the maximally compact subalgebra is larger than theCartan subalgebra), it has recently been appreciated that this representation is a parabolic orgeneralized Verma module.^[9]

Derivatives (of any order) of primary fields are calleddescendant fields. Their behaviour under conformal transformations is more complicated. For example, if${\displaystyle O}$ is a primary field, then${\displaystyle {\pi }_f({\mathrm{\partial }}_{\mu }O)(x)={\mathrm{\partial }}_{\mu }\left({\pi }_f(O)(x)\right)}$ is a linear combination of${\displaystyle {\mathrm{\partial }}_{\mu }O}$ and⁠${\displaystyle O}$⁠. Correlation functions of descendant fields can be deduced from correlation functions of primary fields. However, even in the common case where all fields are either primaries or descendants thereof, descendant fields play an important role, because conformal blocks and operator product expansions involve sums over all descendant fields.

The collection of all primary fields⁠${\displaystyle O_p}$⁠, characterized by their scaling dimensions${\displaystyle {\mathrm{\Delta }}_p}$ and the representations⁠${\displaystyle {\rho }_p}$⁠, is called thespectrum of the theory.

The invariance of correlation functions under conformal transformations severely constrain their dependence on field positions. In the case of two- and three-point functions, that dependence is determined up to finitely many constant coefficients. Higher-point functions have more freedom, and are only determined up to functions of conformally invariant combinations of the positions.

The two-point function of two primary fields vanishes if their conformal dimensions differ.

${\displaystyle {\mathrm{\Delta }}_1\ne {\mathrm{\Delta }}_2⟹⟨O_1(x_1)O_2(x_2)⟩=0.}$

If the dilation operator is diagonalizable (i.e. if the theory is not logarithmic), there exists a basis of primary fields such that two-point functions are diagonal, i.e.⁠${\displaystyle i\ne j⟹⟨O_i{O}_j⟩=0}$⁠. In this case, the two-point function of a scalar primary field is^[10]

${\displaystyle ⟨O(x_1)O(x_2)⟩=\frac{1}{|x_1-x_2{|}^{2\mathrm{\Delta }}},}$

where we choose the normalization of the field such that the constant coefficient, which is not determined by conformal symmetry, is one. Similarly, two-point functions of non-scalar primary fields are determined up to a coefficient, which can be set to one. In the case of a symmetric traceless tensor of rank⁠${\displaystyle \ell }$⁠, the two-point function is

${\displaystyle ⟨O_{{\mu }_1,\dots ,{\mu }_{\ell }}(x_1)O_{{\nu }_1,\dots ,{\nu }_{\ell }}(x_2)⟩=\frac{\prod _{i=1}^{\ell }{I}_{{\mu }_i,{\nu }_i}(x_1-x_2)-\text{traces}}{|x_1-x_2{|}^{2\mathrm{\Delta }}},}$

where the tensor${\displaystyle I_{\mu ,\nu }(x)}$ is defined as

${\displaystyle I_{\mu ,\nu }(x)={\eta }_{\mu \nu }-\frac{2x_{\mu }{x}_{\nu }}{x^2}.}$

The three-point function of three scalar primary fields is

${\displaystyle ⟨O_1(x_1)O_2(x_2)O_3(x_3)⟩=\frac{C_{123}}{|x_{12}{|}^{{\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2-{\mathrm{\Delta }}_3}|x_{13}{|}^{{\mathrm{\Delta }}_1+{\mathrm{\Delta }}_3-{\mathrm{\Delta }}_2}|x_{23}{|}^{{\mathrm{\Delta }}_2+{\mathrm{\Delta }}_3-{\mathrm{\Delta }}_1}},}$

where⁠${\displaystyle x_{ij}=x_i-x_j}$⁠, and${\displaystyle C_{123}}$ is athree-point structure constant. With primary fields that are not necessarily scalars, conformal symmetry allows a finite number of tensor structures, and there is a structure constant for each tensor structure. In the case of two scalar fields and a symmetric traceless tensor of rank⁠${\displaystyle \ell }$⁠, there is only one tensor structure, and the three-point function is

${\displaystyle ⟨O_1(x_1)O_2(x_2)O_{{\mu }_1,\dots ,{\mu }_{\ell }}(x_3)⟩=\frac{C_{123}\left(\prod _{i=1}^{\ell }{V}_{{\mu }_i}-\text{traces}\right)}{|x_{12}{|}^{{\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2-{\mathrm{\Delta }}_3}|x_{13}{|}^{{\mathrm{\Delta }}_1+{\mathrm{\Delta }}_3-{\mathrm{\Delta }}_2}|x_{23}{|}^{{\mathrm{\Delta }}_2+{\mathrm{\Delta }}_3-{\mathrm{\Delta }}_1}},}$

where we introduce the vector

${\displaystyle V_{\mu }=\frac{x_{13}^{\mu }{x}_{23}^2-x_{23}^{\mu }{x}_{13}^2}{|x_{12}||x_{13}||x_{23}|}.}$

Four-point functions of scalar primary fields are determined up to arbitrary functions${\displaystyle g(u,v)}$ of the two cross-ratios

${\displaystyle u=\frac{x_{12}^2{x}_{34}^2}{x_{13}^2{x}_{24}^2},v=\frac{x_{14}^2{x}_{23}^2}{x_{13}^2{x}_{24}^2}.}$

The four-point function is then^[11]

${\displaystyle ⟨\prod _{i=1}^4{O}_i(x_i)⟩=\frac{{\left(\frac{|x_{24}|}{|x_{14}|}\right)}^{{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2}{\left(\frac{|x_{14}|}{|x_{13}|}\right)}^{{\mathrm{\Delta }}_3-{\mathrm{\Delta }}_4}}{|x_{12}{|}^{{\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2}|x_{34}{|}^{{\mathrm{\Delta }}_3+{\mathrm{\Delta }}_4}}g(u,v).}$

Theoperator product expansion (OPE) is more powerful in conformal field theory than in more general quantum field theories. This is because in conformal field theory, the operator product expansion's radius of convergence is finite (i.e. it is not zero).^[12] Provided the positions${\displaystyle x_1,x_2}$ of two fields are close enough, the operator product expansion rewrites the product of these two fields as a linear combination of fields at a given point, which can be chosen as${\displaystyle x_2}$ for technical convenience.

The operator product expansion of two fields takes the form

${\displaystyle O_1(x_1)O_2(x_2)=\sum _k{c}_{12k}(x_1-x_2)O_k(x_2),}$

where${\displaystyle c_{12k}(x)}$ is some coefficient function, and the sum in principle runs over all fields in the theory. (Equivalently, by the state-field correspondence, the sum runs over all states in the space of states.) Some fields may actually be absent, in particular due to constraints from symmetry: conformal symmetry, or extra symmetries.

If all fields are primary or descendant, the sum over fields can be reduced to a sum over primaries, by rewriting the contributions of any descendant in terms of the contribution of the corresponding primary:

${\displaystyle O_1(x_1)O_2(x_2)=\sum _p{C}_{12p}{P}_p(x_1-x_2,{\mathrm{\partial }}_{x_2})O_p(x_2),}$

where the fields${\displaystyle O_p}$ are all primary, and${\displaystyle C_{12p}}$ is the three-point structure constant (which for this reason is also calledOPE coefficient). The differential operator${\displaystyle P_p(x_1-x_2,{\mathrm{\partial }}_{x_2})}$ is an infinite series in derivatives, which is determined by conformal symmetry and therefore in principle known.

Viewing the OPE as a relation between correlation functions shows that the OPE must be associative. Furthermore, if the space is Euclidean, the OPE must be commutative, because correlation functions do not depend on the order of the fields, i.e.⁠${\displaystyle O_1(x_1)O_2(x_2)=O_2(x_2)O_1(x_1)}$⁠.

The existence of the operator product expansion is a fundamental axiom of the conformal bootstrap. However, it is generally not necessary to compute operator product expansions and in particular the differential operators⁠${\displaystyle P_p(x_1-x_2,{\mathrm{\partial }}_{x_2})}$⁠. Rather, it is the decomposition of correlation functions into structure constants and conformal blocks that is needed. The OPE can in principle be used for computing conformal blocks, but in practice there are more efficient methods.

Using the OPE⁠${\displaystyle O_1(x_1)O_2(x_2)}$⁠, a four-point function can be written as a combination of three-point structure constants ands-channel conformal blocks,

${\displaystyle ⟨\prod _{i=1}^4{O}_i(x_i)⟩=\sum _p{C}_{12p}{C}_{p34}{G}_p^{(s)}(x_i).}$

The conformal block${\displaystyle G_p^{(s)}(x_i)}$ is the sum of the contributions of the primary field${\displaystyle O_p}$ and its descendants. It depends on the fields${\displaystyle O_i}$ and their positions. If the three-point functions${\displaystyle ⟨O_1{O}_2{O}_p⟩}$ or${\displaystyle ⟨O_3{O}_4{O}_p⟩}$ involve several independent tensor structures, the structure constants and conformal blocks depend on these tensor structures, and the primary field${\displaystyle O_p}$ contributes several independent blocks. Conformal blocks are determined by conformal symmetry, and known in principle. To compute them, there are recursion relations^[9] and integrable techniques.^[13]

Using the OPE${\displaystyle O_1(x_1)O_4(x_4)}$ or⁠${\displaystyle O_1(x_1)O_3(x_3)}$⁠, the same four-point function is written in terms oft-channel conformal blocks oru-channel conformal blocks,

${\displaystyle ⟨\prod _{i=1}^4{O}_i(x_i)⟩=\sum _p{C}_{14p}{C}_{p23}{G}_p^{(t)}(x_i)=\sum _p{C}_{13p}{C}_{p24}{G}_p^{(u)}(x_i).}$

The equality of the s-, t- and u-channel decompositions is calledcrossing symmetry: a constraint on the spectrum of primary fields, and on the three-point structure constants.

Conformal blocks obey the same conformal symmetry constraints as four-point functions. In particular, s-channel conformal blocks can be written in terms of functions${\displaystyle g_p^{(s)}(u,v)}$ of the cross-ratios. While the OPE${\displaystyle O_1(x_1)O_2(x_2)}$ only converges if⁠⁠, conformal blocks can be analytically continued to all (non pairwise coinciding) values of the positions. In Euclidean space, conformal blocks are single-valued real-analytic functions of the positions except when the four points${\displaystyle x_i}$ lie on a circle but in a singly-transposedcyclic order [1324], and only in these exceptional cases does the decomposition into conformal blocks not converge.

A conformal field theory in flat Euclidean space${\displaystyle {\mathbb{R}}^d}$ is thus defined by its spectrum${\displaystyle \{({\mathrm{\Delta }}_p,{\rho }_p)\}}$ and OPE coefficients (or three-point structure constants)⁠${\displaystyle \{C_{p{p}^{\prime }{p}^{″}}\}}$⁠, satisfying the constraint that all four-point functions are crossing-symmetric. From the spectrum and OPE coefficients (collectively referred to as theCFT data), correlation functions of arbitrary order can be computed.

A conformal field theory isunitary if its space of states has a positive definitescalar product such that the dilation operator is self-adjoint. Then the scalar product endows the space of states with the structure of aHilbert space.

In Euclidean conformal field theories, unitarity is equivalent toreflection positivity of correlation functions: one of theOsterwalder-Schrader axioms.^[11]

Unitarity implies that the conformal dimensions of primary fields are real and bounded from below. The lower bound depends on the spacetime dimension⁠${\displaystyle d}$⁠, and on the representation of the rotation or Lorentz group in which the primary field transforms. For scalar fields, the unitarity bound is^[11]

${\displaystyle \mathrm{\Delta }\ge \frac{1}{2}(d-2).}$

In a unitary theory, three-point structure constants must be real, which in turn implies that four-point functions obey certain inequalities. Powerful numerical bootstrap methods are based on exploiting these inequalities.

A conformal field theory iscompact if it obeys three conditions:^[14]

• All conformal dimensions are real.
• For any${\displaystyle \mathrm{\Delta }\in \mathbb{R}}$ there are finitely many states whose dimensions are less than⁠${\displaystyle \mathrm{\Delta }}$⁠.
• There is a unique state with the dimension⁠${\displaystyle \mathrm{\Delta }=0}$⁠, and it is thevacuum state, i.e. the corresponding field is theidentity field.

(The identity field is the field whose insertion into correlation functions does not modify them, i.e.⁠${\displaystyle ⟨I(x)\cdots ⟩=⟨\cdots ⟩}$⁠.) The name comes from the fact that if a 2D conformal field theory is also asigma model, it will satisfy these conditions if and only if its target space is compact.

It is believed that all unitary conformal field theories are compact in dimension⁠${\displaystyle d>2}$⁠. Without unitarity, on the other hand, it is possible to find CFTs in dimension four^[15] and in dimension${\displaystyle 4-ϵ}$^[16] that have a continuous spectrum. And in dimension two,Liouville theory is unitary but not compact.

A conformal field theory may have extra symmetries in addition to conformal symmetry. For example, the Ising model has a${\displaystyle {\mathbb{Z}}_2}$ symmetry, and superconformal field theories have supersymmetry.

Ageneralized free field is a field whose correlation functions are deduced from its two-point function byWick's theorem. For instance, if${\displaystyle \varphi }$ is a scalar primary field of dimension⁠${\displaystyle \mathrm{\Delta }}$⁠, its four-point function reads^[17]

${\displaystyle ⟨\prod _{i=1}^4\varphi (x_i)⟩=\frac{1}{|x_{12}{|}^{2\mathrm{\Delta }}|x_{34}{|}^{2\mathrm{\Delta }}}+\frac{1}{|x_{13}{|}^{2\mathrm{\Delta }}|x_{24}{|}^{2\mathrm{\Delta }}}+\frac{1}{|x_{14}{|}^{2\mathrm{\Delta }}|x_{23}{|}^{2\mathrm{\Delta }}}.}$

For instance, if${\displaystyle {\varphi }_1,{\varphi }_2}$ are two scalar primary fields such that${\displaystyle ⟨{\varphi }_1{\varphi }_2⟩=0}$ (which is the case in particular if${\displaystyle {\mathrm{\Delta }}_1\ne {\mathrm{\Delta }}_2}$), we have the four-point function

${\displaystyle ⟨{\varphi }_1(x_1){\varphi }_1(x_2){\varphi }_2(x_3){\varphi }_2(x_4)⟩=\frac{1}{|x_{12}{|}^{2{\mathrm{\Delta }}_1}|x_{34}{|}^{2{\mathrm{\Delta }}_2}}.}$

Mean field theory is a generic name for conformal field theories that are built from generalized free fields. For example, a mean field theory can be built from one scalar primary field⁠${\displaystyle \varphi }$⁠. Then this theory contains⁠${\displaystyle \varphi }$⁠, its descendant fields, and the fields that appear in the OPE\phi \phi. The primary fields that appear in${\displaystyle \varphi \varphi }$ can be determined by decomposing the four-point function${\displaystyle ⟨\varphi \varphi \varphi \varphi ⟩}$ in conformal blocks:^[17] their conformal dimensions belong to${\displaystyle 2\mathrm{\Delta }+2\mathbb{N}}$: in mean field theory, the conformal dimension is conserved modulo integers. Structure constants can be computed exactly in terms of theGamma function.^[18]

Similarly, it is possible to construct mean field theories starting from a field with non-trivial Lorentz spin. For example, the 4dMaxwell theory (in the absence of charged matter fields) is a mean field theory built out of an antisymmetric tensor field${\displaystyle F_{\mu \nu }}$ with scaling dimension⁠${\displaystyle \mathrm{\Delta }=2}$⁠.

Mean field theories have a Lagrangian description in terms of a quadratic action involving Laplacian raised to an arbitrary real power (which determines the scaling dimension of the field). For a generic scaling dimension, the power of the Laplacian is non-integer. The corresponding mean field theory is then non-local (e.g. it does not have a conserved stress tensor operator).^{[citation needed]}

Thecritical Ising model is the critical point of theIsing model on a hypercubic lattice in two or three dimensions. It has a${\displaystyle {\mathbb{Z}}_2}$ global symmetry, corresponding to flipping all spins. Thetwo-dimensional critical Ising model includes the${\displaystyle \mathcal{M}(4,3)}$Virasoro minimal model, which can be solved exactly. There is no Ising CFT in${\displaystyle d\ge 4}$ dimensions.

Thecritical Potts model with${\displaystyle q=2,3,4,\cdots }$ colors is a unitary CFT that is invariant under thepermutation group⁠${\displaystyle S_q}$⁠. It is a generalization of the critical Ising model, which corresponds to⁠${\displaystyle q=2}$⁠. The critical Potts model exists in a range of dimensions depending on⁠${\displaystyle q}$⁠.

The critical Potts model may be constructed as thecontinuum limit of thePotts model ond-dimensional hypercubic lattice. In the Fortuin-Kasteleyn reformulation in terms of clusters, the Potts model can be defined for⁠${\displaystyle q\in \mathbb{C}}$⁠, but it is not unitary if${\displaystyle q}$ is not integer.

Thecritical O(N) model is a CFT invariant under theorthogonal group. For any integer⁠${\displaystyle N}$⁠, it exists as an interacting, unitary and compact CFT in${\displaystyle d=3}$ dimensions (and for${\displaystyle N=1}$ also in two dimensions). It is a generalization of the critical Ising model, which corresponds to the O(N) CFT at⁠${\displaystyle N=1}$⁠.

The O(N) CFT can be constructed as thecontinuum limit of a lattice model with spins that areN-vectors, called then-vector model.

Alternatively, the critical${\displaystyle O(N)}$ model can be constructed as the${\displaystyle \epsilon \to 1}$ limit ofWilson–Fisher fixed point in${\displaystyle d=4-\epsilon }$ dimensions. At⁠${\displaystyle \epsilon =0}$⁠, the Wilson–Fisher fixed point becomes the tensor product of${\displaystyle N}$ free scalars with dimension⁠${\displaystyle \mathrm{\Delta }=1}$⁠. For the model in question is non-unitary.^[19]

WhenN is large, the O(N) model can be solved perturbatively in a 1/N expansion by means of theHubbard–Stratonovich transformation. In particular, the${\displaystyle N\to \mathrm{\infty }}$ limit of the critical O(N) model is well-understood.

The conformal data of the critical O(N) model are functions ofN and of the dimension, on which many results are known.^[20]

Some conformal field theories in three and four dimensions admit a Lagrangian description in the form of agauge theory, either abelian or non-abelian. Examples of such CFTs areconformal QED with sufficiently many charged fields in${\displaystyle d=3}$ or theBanks-Zaks fixed point in⁠${\displaystyle d=4}$⁠.

Continuous phase transitions (critical points) of classical statistical physics systems withD 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.

Continuousquantum phase transitions in condensed matter systems withD spatial dimensions may be described by LorentzianD+1 dimensional conformal field theories (related byWick rotation to Euclidean CFTs inD + 1 dimensions). Apart from translation and rotation invariance, an additional necessary condition for this to happen is that the dynamical critical exponentz should be equal to 1. CFTs describing such quantum phase transitions (in absence of quenched disorder) are always unitary.

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 bec = 26 in bosonic string theory andc = 10 in superstring theory. Coordinates of the spacetime in which string theory lives correspond to bosonic fields of this CFT.

Conformal field theories play a prominent role in theAdS/CFT correspondence, in which a gravitational theory inanti-de Sitter space (AdS) is equivalent to a conformal field theory on the AdS boundary. Notable examples ared = 4,N = 4 supersymmetric Yang–Mills theory, which is dual toType IIB string theory on AdS₅ × S⁵, andd = 3,N = 6 super-Chern–Simons theory, which is dual toM-theory on AdS₄ × S⁷. (The prefix "super" denotessupersymmetry,N denotes the degree ofextended supersymmetry possessed by the theory, andd the number of space-time dimensions on the boundary.)

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 calledconformal 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.^[21]

• Rychkov, Slava (2016).EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions. SpringerBriefs in Physics.arXiv:1601.05000.doi:10.1007/978-3-319-43626-5.ISBN 978-3-319-43625-8.S2CID 119192484.
• Martin Schottenloher,A Mathematical Introduction to Conformal Field Theory, Springer-Verlag, Berlin, Heidelberg, 1997.ISBN 3-540-61753-1, 2nd edition 2008,ISBN 978-3-540-68625-5.

• CFT Zoo
• Media related toConformal field theory at Wikimedia Commons

• Conformal field theory
• Symmetry
• Scaling symmetries
• Mathematical physics

• Articles with short description
• Short description matches Wikidata
• Wikipedia articles needing clarification from April 2023
• All articles with unsourced statements
• Articles with unsourced statements from May 2021
• Commons category link from Wikidata