Conformal field theory

A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, 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] to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.

Scale invariance vs conformal invariance

In quantum field theory, scale invariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and it is less obvious why it occurs in nature.

Under some assumptions it is possible to prove that scale invariance implies conformal invariance in a quantum field theory, for example in unitary compact conformal field theories in two dimensions.

While it is possible for a quantum field theory to be scale invariant but not conformally invariant, examples are rare.[2] 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), than in 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.[3] 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.[4] 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 transformations ${\displaystyle PSL_{2}(\mathbb {C} )}$ , which is finite-dimensional. On the other hand, infinitesimal conformal transformations form the infinite-dimensional Witt algebra: the conformal Killing equations in two dimensions, ${\displaystyle \partial _{\mu }\xi _{\nu }+\partial _{\nu }\xi _{\mu }=\partial \cdot \xi \eta _{\mu \nu },~}$  reduce to just the Cauchy-Riemann equations, ${\displaystyle \partial _{\bar {z}}\xi (z)=0=\partial _{z}\xi ({\bar {z}})}$ , the infinity of modes of arbitrary analytic coordinate transformations ${\displaystyle \xi (z)}$  yield the infinity of Killing vector fields ${\displaystyle z^{n}\partial _{z}}$ .

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.

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.

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 cental charge.

The space of states of a theory is a 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 ${\displaystyle L_{n\geq -1}}$  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.

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 ${\displaystyle d}$ -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 ${\displaystyle J_{\nu }^{\mu }(x)={\frac {\partial f^{\mu }(x)}{\partial x^{\nu }}}}$  is of the form

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

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

Conformal group

The conformal group is locally isomorphic to ${\displaystyle SO(1,d+1)}$  (Euclidean) or ${\displaystyle SO(2,d)}$  (Minkowski). 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 a special conformal transformation

${\displaystyle S_{a}=I\circ T_{a}\circ I,}$

where ${\displaystyle I}$  is the inversion such that

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

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

Conformal algebra

The commutation relations of the corresponding Lie algebra are

${\displaystyle [P_{\mu },P_{\nu }]=0,}$
${\displaystyle [D,K_{\mu }]=-K_{\mu },}$
${\displaystyle [D,P_{\mu }]=P_{\mu },}$
${\displaystyle [K_{\mu },K_{\nu }]=0,}$
${\displaystyle [K_{\mu },P_{\nu }]=\eta _{\mu \nu }D-iM_{\mu \nu },}$

where ${\displaystyle P}$  generate translations, ${\displaystyle D}$  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.

Global issues in Minkowski space

In Minkowski space, the conformal group does not preserve causality. 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.[5]. 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 ${\displaystyle n}$ -point correlation function ${\displaystyle \left\langle O_{1}(x_{1})\cdots O_{n}(x_{n})\right\rangle }$  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 ${\displaystyle \partial _{x_{1}}\left\langle O_{1}(x_{1})\cdots \right\rangle =\left\langle \partial _{x_{1}}O_{1}(x_{1})\cdots \right\rangle }$ .

We focus on CFT on the Euclidean space ${\displaystyle \mathbb {R} ^{d}}$ . In this case, correlation functions are Schwinger functions. They are defined for ${\displaystyle x_{i}\neq x_{j}}$ , 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, 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 ${\displaystyle x\to f(x)}$  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:

${\displaystyle \left\langle \pi _{f}(O_{1})(x_{1})\cdots \pi _{f}(O_{n})(x_{n})\right\rangle =\left\langle O_{1}(x_{1})\cdots O_{n}(x_{n})\right\rangle .}$

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

${\displaystyle \pi _{f}(O)(x)=\Omega (x')^{-\Delta }\rho (R(x'))O(x'),\quad {\text{where}}\ x'=f^{-1}(x).}$

Here ${\displaystyle \Omega (x)}$  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 ${\displaystyle \pi _{f}(O)(x)=\Omega (x')^{-\Delta }O(x')}$  . For vector fields, the representation ${\displaystyle \rho }$  is the fundamental representation, and we would have ${\displaystyle \pi _{f}(O_{\mu })(x)=\Omega (x')^{-\Delta }R_{\mu }^{\nu }(x')O_{\nu }(x')}$ .

A primary field that is characterized by the conformal dimension ${\displaystyle \Delta }$  and representation ${\displaystyle \rho }$  behaves as a highest-weight vector in an induced representation of the conformal group from the subgroup generated by dilations and rotations. In particular, the conformal dimension ${\displaystyle \Delta }$  characterizes a representation of the subgroup of dilations.

Derivatives (of any order) of primary fields are called descendant fields. Their behaviour under conformal transformations is more complicated. For example, if ${\displaystyle O}$  is a primary field, then ${\displaystyle \pi _{f}(\partial _{\mu }O)(x)=\partial _{\mu }\left(\pi _{f}(O)(x)\right)}$  is a linear combination of ${\displaystyle \partial _{\mu }O}$  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.

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.

${\displaystyle \Delta _{1}\neq \Delta _{2}\implies \left\langle O_{1}(x_{1})O_{2}(x_{2})\right\rangle =0.}$

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. ${\displaystyle i\neq j\implies \left\langle O_{i}O_{j}\right\rangle =0}$ . In this case, the two-point function of a scalar primary field is

${\displaystyle \left\langle O(x_{1})O(x_{2})\right\rangle ={\frac {1}{|x_{1}-x_{2}|^{2\Delta }}},}$

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

${\displaystyle \left\langle O_{\mu _{1},\dots ,\mu _{\ell }}(x_{1})O_{\nu _{1},\dots ,\nu _{\ell }}(x_{2})\right\rangle ={\frac {\prod _{i=1}^{\ell }I_{\mu _{i},\nu _{i}}(x_{1}-x_{2})-{\text{traces}}}{|x_{1}-x_{2}|^{2\Delta }}},}$

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

${\displaystyle \left\langle O_{1}(x_{1})O_{2}(x_{2})O_{3}(x_{3})\right\rangle ={\frac {C_{123}}{|x_{12}|^{\Delta _{1}+\Delta _{2}-\Delta _{3}}|x_{13}|^{\Delta _{1}+\Delta _{3}-\Delta _{2}}|x_{23}|^{\Delta _{2}+\Delta _{3}-\Delta _{1}}}},}$

where ${\displaystyle x_{ij}=x_{i}-x_{j}}$ , and ${\displaystyle C_{123}}$  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 ${\displaystyle \ell }$ , there is only one tensor structure, and the three-point function is

${\displaystyle \left\langle O_{1}(x_{1})O_{2}(x_{2})O_{\mu _{1},\dots ,\mu _{\ell }}(x_{3})\right\rangle ={\frac {C_{123}\left(\prod _{i=1}^{\ell }V_{\mu _{i}}-{\text{traces}}\right)}{|x_{12}|^{\Delta _{1}+\Delta _{2}-\Delta _{3}}|x_{13}|^{\Delta _{1}+\Delta _{3}-\Delta _{2}}|x_{23}|^{\Delta _{2}+\Delta _{3}-\Delta _{1}}}},}$

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[6]

${\displaystyle \left\langle \prod _{i=1}^{4}O_{i}(x_{i})\right\rangle ={\frac {\left({\frac {|x_{24}|}{|x_{14}|}}\right)^{\Delta _{1}-\Delta _{2}}\left({\frac {|x_{14}|}{|x_{13}|}}\right)^{\Delta _{3}-\Delta _{4}}}{|x_{12}|^{\Delta _{1}+\Delta _{2}}|x_{34}|^{\Delta _{3}+\Delta _{4}}}}g(u,v).}$

Operator product expansion

The operator 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). 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 the function ${\displaystyle c_{12k}(x)}$  is an OPE coefficient, 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:

${\displaystyle O_{1}(x_{1})O_{2}(x_{2})=\sum _{p}C_{12p}P_{p}(x_{1}-x_{2},\partial _{x_{2}})O_{p}(x_{2}),}$

where the fields ${\displaystyle O_{p}}$  are all primary, and ${\displaystyle C_{12p}}$  is a three-point structure constant. The differential operator ${\displaystyle P_{p}(x_{1}-x_{2},\partial _{x_{2}})}$  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 ${\displaystyle P_{p}(x_{1}-x_{2},\partial _{x_{2}})}$ . 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.

Conformal blocks and crossing symmetry

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 and s-channel conformal blocks,

${\displaystyle \left\langle \prod _{i=1}^{4}O_{i}(x_{i})\right\rangle =\sum _{p}C_{12p}C_{p34}G_{p}^{(s)}(x_{i}).}$

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 ${\displaystyle \left\langle O_{1}O_{2}O_{p}\right\rangle }$  or ${\displaystyle \left\langle O_{3}O_{4}O_{p}\right\rangle }$  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[7] and integrable techniques.[8]

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 of t-channel conformal blocks or u-channel conformal blocks,

${\displaystyle \left\langle \prod _{i=1}^{4}O_{i}(x_{i})\right\rangle =\sum _{p}C_{14p}C_{p23}G_{p}^{(t)}(x_{i})=\sum _{p}C_{13p}C_{p24}G_{p}^{(u)}(x_{i}).}$

The equality of the s-, t- and u-channel decompositions is called crossing 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 ${\displaystyle |x_{12}|<\min(|x_{23}|,|x_{24}|)}$ , conformal blocks can be analytically continued to all (non pairwise coinciding) values of the positions, and the decomposition into conformal blocks converges whenever the four point ${\displaystyle x_{i}}$  do not lie on the same circle. However, conformal blocks are not single-valued functions of the positions, and in general have nontrivial monodromies when points move around one another.

A Euclidean conformal field theory is a spectrum ${\displaystyle \{p\}}$  and three-point structure constants ${\displaystyle \{C_{pp'p''}\}}$  such that all four-point functions are crossing-symmetric. From these data, arbitrary correlation functions can be computed.

Applications

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 = 4, N = 4 supersymmetric Yang–Mills theory, which is dual to Type IIB string theory on AdS5 × S5, and d = 3, N = 6 super-Chern–Simons theory, which is dual to M-theory on AdS4 × S7. (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.)