A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations.
In contrast to other types of conformal field theories, two-dimensional conformal field theories have infinite-dimensional symmetry algebras. In some cases, this allows them to be solved exactly, using the conformal bootstrap method.
Notable two-dimensional conformal field theories include minimal models, Liouville theory, massless free bosonic theories, Wess–Zumino–Witten models, and certain sigma models.
Basic structures
Geometry
Two-dimensional conformal field theories (CFTs) are defined on Riemann surfaces, where local conformal maps are holomorphic functions. While a CFT might conceivably exist only on a given Riemann surface, its existence on any surface other than the sphere implies its existence on all surfaces. Given a CFT, it is indeed possible to glue two Riemann surfaces where it exists, and obtain the CFT on the glued surface. On the other hand, some CFTs exist only on the sphere. Unless stated otherwise, we consider CFT on the sphere in this article.
Symmetries and integrability
Given a local complex coordinate , the real vector space of infinitesimal conformal maps has the basis , with . (For example, and generate translations.) Relaxing the assumption that is the complex conjugate of , i.e. complexifying the space of infinitesimal conformal maps, one obtains a complex vector space with the basis .
With their natural commutators, the differential operators generate a Witt algebra. By standard quantum-mechanical arguments, the symmetry algebra of conformal field theory must be the central extension of the Witt algebra, i.e. the Virasoro algebra, whose generators are , plus a central generator. In a given CFT, the central generator takes a constant value , called the central charge.
The symmetry algebra is therefore the product of two copies of the Virasoro algebra: the left-moving or holomorphic algebra, with generators , and the right-moving or antiholomorphic algebra, with generators .
In the universal enveloping algebra of the Virasoro algebra, it is possible to construct an infinite set of mutually commuting charges. The first charge is , the second charge is quadratic in the Virasoro generators, the third charge is cubic, and so on. This shows that any two-dimensional conformal field theory is also a quantum integrable system.
Space of states
The space of states, also called the spectrum, of a CFT, is a representation of the product of the two Virasoro algebras.
For a state that is an eigenvector of and with the eigenvalues and , is the left conformal dimension, is the right conformal dimension, is the total conformal dimension or the energy, is the conformal spin.
A CFT is called rational if its space of states decomposes into finitely many irreducible representations of the product of the two Virasoro algebras.
A CFT is called diagonal if its space of states is a direct sum of representations of the type , where is an indecomposable representation of the left Virasoro algebra, and is the same representation of the right Virasoro algebra.
The CFT is called unitary if the space of states has a positive definite Hermitian form such that and are self-adjoint, and . This implies in particular that , and that the central charge is real. The space of states is then a Hilbert space. While unitarity is necessary for a CFT to be a proper quantum system with a probabilistic interpretation, many interesting CFTs are nevertheless non-unitary, including minimal models and Liouville theory for most allowed values of the central charge.
Fields and correlation functions
The state-field correspondence is a linear map from the space of states to the space of fields, which commutes with the action of the symmetry algebra.
In particular, the image of a primary state of a lowest weight representation of the Virasoro algebra is a primary field , such that
Descendant fields are obtained from primary fields by acting with creation modes . Degenerate fields correspond to primary states of degenerate representations. For example, the degenerate field obeys , due to the presence of a null vector in the corresponding degenerate representation.
An -point correlation function is a number that depends linearly on fields, denoted as with . In the path integral formulation of conformal field theory, correlation functions are defined as functional integrals. In the conformal bootstrap approach, correlation functions are defined by axioms. In particular, it is assumed that there exists an operator product expansion (OPE),