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 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. 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. 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. 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.

Two dimensions

Two-dimensional CFTs are (in some way) invariant under an infinite-dimensional symmetry group. For example, consider a CFT on the Riemann sphere. It has the Möbius transformations as the conformal group, which is isomorphic to (the finite-dimensional) PSL(2,C).

However, the infinitesimal conformal transformations form an infinite-dimensional algebra, called the Witt algebra, but this infinity of conformal transformations do not have global inverses on ℂ. Only the primary fields (or chiral fields) are invariant with respect to this full infinitesimal conformal group. Its generators are indexed by integers n,

$L_{n}=\oint _{z=0}{\frac {dz}{2\pi i}}z^{n+1}T_{zz}~,$

where Tzz is the holomorphic part of the non-trace piece of the energy momentum tensor of the theory. E.g., for a free scalar field,

$T_{zz}={\tfrac {1}{2}}(\partial _{z}\phi )^{2}~.$

In most conformal field theories, a conformal anomaly, also known as a Weyl anomaly, arises in the quantum theory. This results in the appearance of a nontrivial central charge, and the Witt algebra is extended to the Virasoro algebra.

In Euclidean CFT, one has both a holomorphic and an antiholomorphic copy of the Virasoro algebra. In Lorentzian CFT, one has a left-moving and a right moving copy of the Virasoro algebra (spacetime is a cylinder, with space being a circle, and time a line).

This symmetry makes it possible to classify two-dimensional CFTs much more precisely than in higher dimensions. In particular, it is possible to relate the spectrum of primary operators in a theory to the value of the central charge, c.

The Hilbert space of physical states is a unitary module of the Virasoro algebra corresponding to a fixed value of c. Stability requires that the energy spectrum of the Hamiltonian be nonnegative. The modules of interest are the highest weight modules of the Virasoro algebra.

A chiral field is a holomorphic field W(z) which transforms as

$L_{n}W(z)=-z^{n+1}{\frac {\partial }{\partial z}}W(z)-(n+1)\Delta z^{n}W(z)$

and

${\bar {L}}_{n}W(z)=0~.$

Analogously, mutatis mutandis, for an antichiral field. Δ is called the conformal weight of the chiral field W.

Furthermore, it was shown by Alexander Zamolodchikov that there exists a function, C, 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.

Frequently, we are not just interested in the operators, but we are also interested in the vacuum state, or in statistical mechanics, the thermal state. Unless c=0, there can't possibly be any state which leaves the entire infinite dimensional conformal symmetry unbroken. The best we can come up with is a state which is invariant under L−1, L0, L1, Li, $i>1$ . This contains the Möbius subgroup. 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 $\mathbb {R} ^{d}$  or of the Minkowski space $\mathbb {R} ^{1,d-1}$ .

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

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

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

Conformal group

The conformal group of $\mathbb {R} ^{d}$  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

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

In the sphere $S^{d}=\mathbb {R} ^{d}\cup \{\infty \}$ , 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

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

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

Correlation functions

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.

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

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

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

A primary field that is characterized by the conformal dimension $\Delta$  and representation $\rho$  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 $\Delta$  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 $O$  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 quantities such as 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.

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

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

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

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 $\ell$ , the two-point function is

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

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

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