# Two-dimensional conformal field theory

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.

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.[1] Given a CFT, it is indeed possible to glue two Riemann surfaces where it exists, and obtain the CFT on the glued surface.[1][2] On the other hand, some CFTs exist only on the sphere. Unless stated otherwise, we consider CFT on the sphere in this article.

The space of states, also called the spectrum, of a CFT, is a representation of the product of the two Virasoro algebras.

A CFT is called rational if its space of states decomposes into finitely many irreducible representations of the product of the two Virasoro algebras.

In a two-dimensional conformal field theory, properties are called chiral if they follow from the action of one of the two Virasoro algebras. If the space of states can be decomposed into factorized representations of the product of the two Virasoro algebras, then all consequences of conformal symmetry are chiral. In other words, the actions of the two Virasoro algebras can be studied separately.

Conformal Ward identities are linear equations that correlation functions obey as a consequence of conformal symmetry.[3] They can be derived by studying correlation functions that involve insertions of the energy–momentum tensor. Their solutions are conformal blocks.

A correlation function that involves a degenerate field satisfies a linear partial differential equation called a Belavin–Polyakov–Zamolodchikov equation after Alexander Belavin, Alexander Polyakov and Alexander Zamolodchikov.[5] The order of this equation is the level of the null vector in the corresponding degenerate representation.

Alternatively, fusion rules have an algebraic definition in terms of an associative fusion product of representations of the Virasoro algebra at a given central charge. The fusion product differs from the tensor product of representations. (In a tensor product, the central charges add.) In certain finite cases, this leads to the structure of a fusion category.

A conformal field theory is quasi-rational is the fusion product of two indecomposable representations is a sum of finitely many indecomposable representations.[7] For example, generalized minimal models are quasi-rational without being rational.

The conformal bootstrap method consists in defining and solving CFTs using only symmetry and consistency assumptions, by reducing all correlation functions to combinations of structure constants and conformal blocks. In two dimensions, this method leads to exact solutions of certain CFTs, and to classifications of rational theories.

The contributions of descendant fields, denoted by the dots, are completely determined by conformal symmetry.[3]

Any correlation function can be written as a linear combination of conformal blocks: functions that are determined by conformal symmetry, and labelled by representations of the symmetry algebra. The coefficients of the linear combination are products of structure constants.[5]

In two-dimensional CFT, the symmetry algebra is factorized into two copies of the Virasoro algebra, and a conformal block that involves primary fields has a holomorphic factorization: it is a product of a locally holomorphic factor that is determined by the left-moving Virasoro algebra, and a locally antiholomorphic factor that is determined by the right-moving Virasoro algebra. These factors are themselves called conformal blocks.

For example, using the OPE of the first two fields in a four-point function of primary fields yields

As first explained by Witten,[8] the space of conformal blocks of a two-dimensional CFT can be identified with the quantum Hilbert space of a 2+1 dimensional Chern-Simons theory, which is an example of a topological field theory. This connection has been very fruitful in the theory of the fractional quantum Hall effect.

When a correlation function can be written in terms of conformal blocks in several different ways, the equality of the resulting expressions provides constraints on the space of states and on three-point structure constants. These constraints are called the conformal bootstrap equations. While the Ward identities are linear equations for correlation functions, the conformal bootstrap equations depend non-linearly on the three-point structure constants.

The consistency of a CFT on the sphere is equivalent to crossing symmetry of the four-point function. The consistency of a CFT on all Riemann surfaces also requires modular invariance of the torus one-point function.[1] Modular invariance of the torus partition function is therefore neither necessary, nor sufficient, for a CFT to exist. It has however been widely studied in rational CFTs, because characters of representations are simpler than other kinds of conformal blocks, such as sphere four-point conformal blocks.

A minimal model is a CFT whose spectrum is built from finitely many irreducible representations of the Virasoro algebra. Minimal models only exist for particular values of the central charge,[3]

Liouville theory has been solved, in the sense that its three-point structure constants are explicitly known. Liouville theory has applications to string theory, and to two-dimensional quantum gravity.

In some CFTs, the symmetry algebra is not just the Virasoro algebra, but an associative algebra (i.e. not necessarily a Lie algebra) that contains the Virasoro algebra. The spectrum is then decomposed into representations of that algebra, and the notions of diagonal and rational CFTs are defined with respect to that algebra.[3]

The symmetry algebra of a supersymmetric CFT is a super Virasoro algebra, or a larger algebra. Supersymmetric CFTs are in particular relevant to superstring theory.

W-algebras are natural extensions of the Virasoro algebra. CFTs based on W-algebras include generalizations of minimal models and Liouville theory, respectively called W-minimal models and conformal Toda theories. Conformal Toda theories are more complicated than Liouville theory, and less well understood.

In two dimensions, classical sigma models are conformally invariant, but only some target manifolds lead to quantum sigma models that are conformally invariant. Examples of such target manifolds include toruses, and Calabi–Yau manifolds.

The known torus partition function[12] suggests that the model is non-rational with a discrete spectrum.