Conformal field theories
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A conformal field theory (CFT) is a
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
that is invariant under conformal transformations. In two
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
s, 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 to
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...
,
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
,
quantum statistical mechanics Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is ...
, and
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
. Statistical and condensed matter systems are indeed often conformally invariant at their
thermodynamic Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of ...
or quantum critical points.


Scale invariance vs conformal invariance

In
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
,
scale invariance In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term ...
is a common and natural symmetry, because any fixed point of the
renormalization group In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in t ...
is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions to argue that it should appear in nature. The basic idea behind its plausibility is that ''local'' scale invariant theories have their currents given by T_ \xi^\nu where \xi^\nu is a
Killing vector In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal g ...
and T_ is a conserved operator (the stress-tensor) of dimension exactly d. For the associated symmetries to include scale but not conformal transformations, the trace T_\mu^\mu has to be a non-zero total derivative implying that there is a non-conserved operator of dimension exactly 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 in
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation In mathematics, a unitary representation of a grou ...
compact conformal field theories in two dimensions. While it is possible for a
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
to be
scale invariant In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical ter ...
but not conformally invariant, examples are rare. For this reason, the terms are often used interchangeably in the context of quantum field theory.


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. 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 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. 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.P. Di Francesco, P. Mathieu, and D. Sénéchal, ''Conformal Field Theory'', 1997, Higher-dimensional conformal field theories have become more popular with the
AdS/CFT correspondence In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter ...
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 In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
is the group of
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad' ...
s PSL_2(\mathbb) , which is finite-dimensional. On the other hand, infinitesimal conformal transformations form the infinite-dimensional
Witt algebra In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algeb ...
: the
conformal Killing equation In conformal geometry, a conformal Killing vector field on a manifold of dimension ''n'' with (pseudo) Riemannian metric g (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field X whose (locally defined) flo ...
s in two dimensions, \partial_\mu \xi_\nu + \partial_\nu \xi_\mu = \partial \cdot\xi \eta_,~ reduce to just the Cauchy-Riemann equations, \partial_ \xi(z) = 0 = \partial_z \xi (\bar) , the infinity of modes of arbitrary analytic coordinate transformations \xi(z) yield the infinity of
Killing vector field In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric tensor, metric. Killing fields are the Lie g ...
s 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 PSL_2(\mathbb) . This turns out to be unique to non-unitary theories; an example is the biharmonic scalar. This property should be viewed as even more special than scale without conformal invariance as it requires 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 to 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

In a conformally invariant two-dimensional quantum theory, the Witt algebra of infinitesimal conformal transformations has to be 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 A conformal anomaly, scale anomaly, trace anomaly or Weyl anomaly is an anomaly, i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory. A classically conformal theory is a theory which, when placed on a surface ...
. It was shown by
Alexander Zamolodchikov Alexander Borisovich Zamolodchikov (russian: Алекса́ндр Бори́сович Замоло́дчиков; born September 18, 1952) is a Russian physicist, known for his contributions to condensed matter physics, two-dimensional conformal ...
that there exists a function which decreases monotonically under the
renormalization group In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in t ...
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 In quantum field theory the ''C''-theorem states that there exists a positive real function, C(g^_i,\mu), depending on the coupling constants of the quantum field theory considered, g^_i, and on the energy scale, \mu^_, which has the following pr ...
, and tells us that
renormalization group flow In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in th ...
in two dimensions is irreversible. 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
space of states Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually conside ...
of a theory is a representation of the product of the two Virasoro algebras. This space is a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
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 L_ of the Virasoro algebra, whose basis is (L_n)_. This contains the generators L_,L_0,L_1 of the global conformal transformations. The rest of the conformal group is spontaneously broken.


Conformal symmetry


Definition and Jacobian

For a given spacetime and metric, a conformal transformation is a transformation that preserves angles. We will focus on conformal transformations of the flat d-dimensional Euclidean space \mathbb^d or of the Minkowski space \mathbb^. If x\to f(x) is a conformal transformation, the Jacobian J^\mu_\nu(x) = \frac is of the form : J^\mu_\nu(x) = \Omega(x) R^\mu_\nu(x), where \Omega(x) is the scale factor, and R^\mu_\nu(x) is a rotation (i.e. an orthogonal matrix) or Lorentz transformation.


Conformal group

The
conformal group In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space. Se ...
is locally isomorphic to SO(1, d + 1) (Euclidean) or SO(2,d) (Minkowski). This includes translations, rotations (Euclidean) or Lorentz transformations (Minkowski), and dilations i.e. scale transformations : x^\mu \to \lambda x^\mu. This also includes special conformal transformations. For any translation T_a(x) = x + a, there is a special conformal transformation : S_a = I \circ T_a \circ I, where I is the inversion such that : I\left(x^\mu\right) = \frac. In the sphere S^d = \mathbb^d \cup \, the inversion exchanges 0 with \infty. Translations leave \infty fixed, while special conformal transformations leave 0 fixed.


Conformal algebra

The commutation relations of the corresponding Lie algebra are : \begin[] [P_\mu, P_\nu] &= 0, \\[] [D, K_\mu] &= -K_\mu, \\[] [D, P_\mu] &= P_\mu, \\[] [K_\mu, K_\nu] &= 0, \\[] [K_\mu, P_\nu] &= \eta_D - iM_, \end where P generate
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
s, D generates dilations, K_\mu generate special conformal transformations, and M_ generate rotations or Lorentz transformations. The tensor \eta_ is the flat metric.


Global issues in Minkowski space

In Minkowski space, the conformal group does not preserve
causality Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
. 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. 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.


Correlation functions and conformal bootstrap

In the conformal bootstrap approach, a conformal field theory is a set of correlation functions that obey a number of axioms. The n-point correlation function \left\langle O_1(x_1)\cdots O_n(x_n)\right\rangle is a function of the positions x_i and other parameters of the fields 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 \partial_ \left\langle O_1(x_1)\cdots \right\rangle = \left\langle \partial_O_1(x_1)\cdots \right\rangle . We focus on CFT on the Euclidean space \mathbb^d. In this case, correlation functions are Schwinger functions. They are defined for x_i\neq x_j, and do not depend on the order of the fields. In Minkowski space, correlation functions are Wightman 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 by
Wick rotation In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that s ...
, for example thanks to the Osterwalder-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.


Behaviour under conformal transformations

Any conformal transformation x\to f(x) acts linearly on fields O(x) \to \pi_f(O)(x), such that f\to \pi_f is a representation of the conformal group, and correlation functions are invariant: : \left\langle\pi_f(O_1)(x_1)\cdots \pi_f(O_n)(x_n) \right\rangle = \left\langle O_1(x_1)\cdots O_n(x_n)\right\rangle. Primary fields are fields that transform into themselves via \pi_f. The behaviour of a primary field is characterized by a number \Delta called its conformal dimension, and a representation \rho of the rotation or Lorentz group. For a primary field, we then have : \pi_f(O)(x) = \Omega(x')^ \rho(R(x')) O(x'), \quad \text\ x'=f^(x). Here \Omega(x) and R(x) are the scale factor and rotation that are associated to the conformal transformation f. The representation \rho is trivial in the case of scalar fields, which transform as \pi_f(O)(x) = \Omega(x')^ O(x') . For vector fields, the representation \rho is the fundamental representation, and we would have \pi_f(O_\mu)(x) = \Omega(x')^ R_\mu^\nu(x') O_\nu(x') . A primary field that is characterized by the conformal dimension \Delta and representation \rho behaves as a highest-weight vector in an
induced representation In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" represe ...
of the conformal group from the subgroup generated by dilations and rotations. In particular, the conformal dimension \Delta characterizes a representation of the subgroup of dilations. In two dimensions, the fact that this induced representation is a Verma module appears throughout the literature. For higher-dimensional CFTs (in which the maximally compact subalgebra is larger than the
Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
), it has recently been appreciated that this representation is a parabolic or generalized Verma module. Derivatives (of any order) of primary fields are called descendant fields. Their behaviour under conformal transformations is more complicated. For example, if O is a primary field, then \pi_f(\partial_\mu O)(x) = \partial_\mu\left(\pi_f(O)(x)\right) is a linear combination of \partial_\mu O and 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 O_p, characterized by their scaling dimensions \Delta_p and the representations \rho_p, is called the spectrum of the theory.


Dependence on field positions

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. : \Delta_1\neq \Delta_2 \implies \left\langle O_(x_1)O_(x_2)\right\rangle= 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. i\neq j\implies \left\langle O_i O_j\right\rangle = 0. In this case, the two-point function of a scalar primary field is : \left\langle O(x_1)O(x_2) \right\rangle = \frac, 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 \ell, the two-point function is : \left\langle O_(x_1) O_(x_2)\right\rangle = \frac, where the tensor I_(x) is defined as : I_(x) = \eta_ - \frac. The three-point function of three scalar primary fields is : \left\langle O_(x_1)O_(x_2)O_(x_3)\right\rangle = \frac, where x_=x_i-x_j, and C_ is a three-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 \ell, there is only one tensor structure, and the three-point function is : \left\langle O_(x_1)O_(x_2)O_(x_3)\right\rangle = \frac, where we introduce the vector : V_\mu = \frac. Four-point functions of scalar primary fields are determined up to arbitrary functions g(u,v) of the two cross-ratios : u = \frac \ , \ v = \frac. The four-point function is then : \left\langle \prod_^4O_i(x_i)\right\rangle = \fracg(u,v).


Operator product expansion

The
operator product expansion In quantum field theory, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. As an axiom, it offers a non-perturbative approach to quantum field theory. One example is the ver ...
(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). Provided the positions 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 x_2 for technical convenience. The operator product expansion of two fields takes the form : O_1(x_1)O_2(x_2) = \sum_k c_(x_1-x_2) O_k(x_2), where c_(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: : O_1(x_1)O_2(x_2) = \sum_p C_P_p(x_1-x_2,\partial_) O_p(x_2), where the fields O_p are all primary, and C_ is the three-point structure constant (which for this reason is also called OPE coefficient). The differential operator P_p(x_1-x_2,\partial_) 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. 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 P_p(x_1-x_2,\partial_). 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.


Conformal blocks and crossing symmetry

Using the OPE O_1(x_1)O_2(x_2), a four-point function can be written as a combination of three-point structure constants and s-channel conformal blocks, : \left\langle \prod_^4 O_i(x_i) \right\rangle = \sum_p C_C_ G_p^(x_i). The conformal block G_p^(x_i) is the sum of the contributions of the primary field O_p and its descendants. It depends on the fields O_i and their positions. If the three-point functions \left\langle O_1O_2O_p\right\rangle or \left\langle O_3O_4O_p\right\rangle involve several independent tensor structures, the structure constants and conformal blocks depend on these tensor structures, and the primary field O_p contributes several independent blocks. Conformal blocks are determined by conformal symmetry, and known in principle. To compute them, there are recursion relations and integrable techniques. Using the OPE O_1(x_1)O_4(x_4) or O_1(x_1)O_3(x_3), the same four-point function is written in terms of t-channel conformal blocks or u-channel conformal blocks, : \left\langle \prod_^4 O_i(x_i) \right\rangle = \sum_p C_C_ G_p^(x_i) =\sum_p C_C_ G_p^(x_i). The equality of the s-, t- and u-channel decompositions is called
crossing symmetry In quantum field theory, a branch of theoretical physics, crossing is the property of scattering amplitudes that allows antiparticles to be interpreted as particles going backwards in time. Crossing states that the same formula that determines ...
: 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 g_p^(u,v) of the cross-ratios. While the OPE O_1(x_1)O_2(x_2) only converges if , x_, <\min(, x_, ,, x_, ), 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 x_i lie on a circle but in a singly-transposed
cyclic order In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. Ins ...
324 __NOTOC__ Year 324 ( CCCXXIV) was a leap year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Crispus and Constantinus (or, less frequently, year ...
and only in these exceptional cases does the decomposition into conformal blocks not converge. A conformal field theory in flat Euclidean space \mathbb^d is thus defined by its spectrum \ and OPE coefficients (or three-point structure constants) \, satisfying the constraint that all four-point functions are crossing-symmetric. From the spectrum and OPE coefficients (collectively referred to as the CFT data), correlation functions of arbitrary order can be computed.


Features of conformal field theories


Unitarity

A conformal field theory is unitary if its space of states has a positive definite
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
such that the dilation operator is self-adjoint. Then the scalar product endows the space of states with the structure of a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. In Euclidean conformal field theories, unitarity is equivalent to reflection positivity of correlation functions: one of the Osterwalder-Schrader axioms. Unitarity implies that the conformal dimensions of primary fields are real and bounded from below. The lower bound depends on the spacetime dimension d, and on the representation of the rotation or Lorentz group in which the primary field transforms. For scalar fields, the unitarity bound is : \Delta \geq \frac12(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.


Compactness

A conformal field theory is compact if it obeys three conditions: * All conformal dimensions are real. * For any \Delta\in\mathbb there are finitely many states whose dimensions are less than \Delta. * There is a unique state with the dimension \Delta =0, and it is the vacuum state, i.e. the corresponding field is the identity field. (The identity field is the field whose insertion into correlation functions does not modify them, i.e. \left\langle I(x)\cdots \right\rangle = \left\langle \cdots \right\rangle .) The name comes from the fact that if a 2D conformal field theory is also a
sigma model In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold, although it is most commonly taken to be either a Lie group or ...
, 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 d>2. Without unitarity, on the other hand, it is possible to find CFTs in dimension four and in dimension 4 - \epsilon that have a continuous spectrum. And in dimension two, Liouville theory is unitary but not compact.


Extra symmetries

A conformal field theory may have extra symmetries in addition to conformal symmetry. For example, the Ising model has a \mathbb_2 symmetry, and superconformal field theories have supersymmetry.


Examples


Mean field theory

A generalized free field is a field whose correlation functions are deduced from its two-point function by
Wick's theorem Wick's theorem is a method of reducing high-order derivatives to a combinatorics problem. It is named after Italian physicist Gian-Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihila ...
. For instance, if \phi is a scalar primary field of dimension \Delta, its four-point function reads : \left\langle \prod_^4\phi(x_i) \right\rangle = \frac + \frac + \frac. For instance, if \phi_1,\phi_2 are two scalar primary fields such that \langle \phi_1\phi_2\rangle=0 (which is the case in particular if \Delta_1\neq\Delta_2), we have the four-point function : \Big\langle \phi_1(x_1)\phi_1(x_2)\phi_2(x_3)\phi_2(x_4)\Big\rangle = \frac. 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 \phi. Then this theory contains \phi, its descendant fields, and the fields that appear in the OPE \phi \phi. The primary fields that appear in \phi \phi can be determined by decomposing the four-point function \langle\phi\phi\phi\phi\rangle in conformal blocks: their conformal dimensions belong to 2\Delta+2\mathbb: in mean field theory, the conformal dimension is conserved modulo integers. Similarly, it is possible to construct mean field theories starting from a field with non-trivial Lorentz spin. For example, the 4d Maxwell theory (in the absence of charged matter fields) is a mean field theory built out of an antisymmetric tensor field F_ with scaling dimension \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).


Critical Ising model

The critical Ising model is the critical point of the
Ising model The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
on a hypercubic lattice in two or three dimensions. It has a \mathbb_2 global symmetry, corresponding to flipping all spins. The
two-dimensional critical Ising model The two-dimensional critical Ising model is the critical limit of the Ising model in two dimensions. It is a two-dimensional conformal field theory whose symmetry algebra is the Virasoro algebra with the central charge c=\tfrac12. Correlation fu ...
includes the \mathcal(4,3) Virasoro minimal model, which can be solved exactly. There is no Ising CFT in d \geq 4 dimensions.


Critical Potts model

The critical Potts model with q=2,3,4,\cdots colors is a unitary CFT that is invariant under the
permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...
S_q. It is a generalization of the critical Ising model, which corresponds to q=2. The critical Potts model exists in a range of dimensions depending on q. The critical Potts model may be constructed as the
continuum limit In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world processe ...
of the Potts model on ''d''-dimensional hypercubic lattice. In the Fortuin-Kasteleyn reformulation in terms of clusters, the Potts model can be defined for q\in\mathbb, but it is not unitary if q is not integer.


Critical O(N) model

The critical O(N) model is a CFT invariant under the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
. For any integer N, it exists as a interacting, unitary and compact CFT in d=3 dimensions (and for N=1 also in two dimensions). It is a generalization of the critical Ising model, which corresponds to the O(N) CFT at N=1. The O(N) CFT can be constructed as the
continuum limit In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world processe ...
of a lattice model with spins that are ''N''-vectors, discussed
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Television * Here TV (formerly "here!"), a ...
. Alternatively, the critical O(N) model can be constructed as the \varepsilon \to 1 limit of Wilson-Fisher fixed point in d=4-\varepsilon dimensions. At \varepsilon = 0, the Wilson-Fisher fixed point becomes the tensor product of N free scalars with dimension \Delta = 1. For 0 < \varepsilon < 1 the model in question is non-unitary. When N is large, the O(N) model can be solved perturbatively in a 1/N expansion by means of the
Hubbard–Stratonovich transformation The Hubbard–Stratonovich (HS) transformation is an exact mathematical transformation invented by Russian physicist Ruslan L. Stratonovich and popularized by British physicist John Hubbard. It is used to convert a particle theory into its respect ...
. In particular, the N \to \infty limit of the critical O(N) model is well-understood.


Conformal gauge theories

Some conformal field theories in three and four dimensions admit a Lagrangian description in the form of a
gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie grou ...
, either abelian or non-abelian. Examples of such CFTs are conformal QED with sufficiently many charged fields in d=3 or the Banks-Zaks fixed point in d=4.


Applications


Continuous phase transitions

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 transitions in condensed matter systems with ''D'' spatial dimensions may be described by Lorentzian ''D+1'' dimensional conformal field theories (related by
Wick rotation In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that s ...
to Euclidean CFTs in ''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

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


AdS/CFT correspondence

Conformal field theories play a prominent role in the
AdS/CFT correspondence In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter ...
, 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'' = 4,
N = 4 supersymmetric Yang–Mills theory ''N'' = 4 supersymmetric Yang–Mills (SYM) theory is a mathematical and physical model created to study particles through a simple system, similar to string theory, with conformal symmetry. It is a simplified toy theory based on Ya ...
, which is dual to
Type IIB string theory In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings theories. Type II string theory accounts for two of the five consistent superstring theories in ten dimensions. Both theorie ...
on AdS5 × S5, and ''d'' = 3, ''N'' = 6 super-
Chern–Simons theory The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type developed by Edward Witten. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and Jam ...
, which is dual to
M-theory M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witt ...
on AdS4 × S7. (The prefix "super" denotes
supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
, ''N'' denotes the degree of
extended supersymmetry In theoretical physics, extended supersymmetry is supersymmetry whose infinitesimal generators Q_i^\alpha carry not only a spinor index \alpha, but also an additional index i=1,2 \dots \mathcal where \mathcal is integer (such as 2 or 4). Extended ...
possessed by the theory, and d the number of space-time dimensions on the boundary.)


See also

* Logarithmic conformal field theory *
AdS/CFT correspondence In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter ...
*
Operator product expansion In quantum field theory, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. As an axiom, it offers a non-perturbative approach to quantum field theory. One example is the ver ...
* Critical point * Boundary conformal field theory *
Primary field In theoretical physics, a primary field, also called a primary operator, or simply a primary, is a local operator in a conformal field theory which is annihilated by the part of the conformal algebra consisting of the lowering generators. From the ...
* Superconformal algebra *
Conformal algebra In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry ...
* Conformal bootstrap * History of conformal field theory


References


Further reading

* * Martin Schottenloher, ''A Mathematical Introduction to Conformal Field Theory'', Springer-Verlag, Berlin, Heidelberg, 1997. , 2nd edition 2008, .


External links

* {{DEFAULTSORT:Conformal Field Theory Symmetry Scaling symmetries Mathematical physics