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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 string theory, statistical mechanics, and condensed matter physics. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic critical points.

Scale invariance vs. conformal invariance

While it is possible for a quantum field theory to be scale invariant but not conformally-invariant, examples are rare.[1] For this reason, the terms are often used interchangeably in the context of quantum field theory, even though the scale symmetry is larger.

In some particular cases 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.
Dimensional considerations
Two dimensions

There are two versions of 2D CFT: 1) Euclidean, and 2) Lorentzian. The former applies to statistical mechanics, and the latter to quantum field theory. The two versions are related by a Wick rotation.

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 and only the primary fields (or chiral fields) are invariant with respect to the full infinitesimal conformal group.

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 modified to become the Virasoro algebra.

In Euclidean CFT, we have a holomorphic and an antiholomorphic copy of the Virasoro algebra. In Lorentzian CFT, we have 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

\( \bar L_n W(z)=0.\, \)

Similarly for an antichiral field. Δ is 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.

Two-dimensional conformal field theories play an important role in statistical mechanics, where they describe critical points of many lattice models.
More than two dimensions

In d > 2 dimensions, the conformal group is isomorphic to SO(d+1, 1 ) in Euclidean signature, or SO(d, 2 ) in Minkowski space.

Higher-dimensional conformal field theories are prominent 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 x S5, and d=3 N=6 super-Chern–Simons theory, which is dual to M-theory on AdS4 x 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.)
Conformal symmetry

Conformal symmetry is a symmetry under scale invariance and under the special conformal transformations having the following relations.

\( [P_\mu,P_\nu]=0, \)
\( [D,K_\mu]=-K_\mu, \)
\( [D,P_\mu]=P_\mu, \)
\( [K_\mu,K_\nu]=0, \)
\( [K_\mu,P_\nu]=\eta_{\mu\nu}D-iM_{\mu\nu}, \)

where P generates translations, D generates scaling transformations as a scalar and \( K_\mu \_ generates the special conformal transformations as a covariant vector under Lorentz transformation.