Conformal field theory

                 A conformal field theory (CFT) is a quantum field theory, also recognized as a model of statistical mechanics at a thermodynamic critical point, that is invariant under conformal transformations. Conformal field theory is often studied in two dimensions where there is an infinite-dimensional group of local conformal transformations, described by the holomorphic functions.

Conformal field theory has important applications in string theory, statistical mechanics, and condensed matter physics. The theory was first proposed in 1936 by Leigh Page and Norman I. Adams.

Dimensional considerations 

There are two versions of 2D CFT: 1) Euclidean, and 2) Lorentzian. The former applies to statistical mechanics, and the latter toquantum 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

and

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 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 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, L_i, . This contains the Möbius subgroup. The rest of the conformal group is spontaneously broken.