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. Unfortunately the Witt algebra on its own always generates a space of particle states which has infinitely many negative energy states with the energy of each state getting progressively lower. For a physical theory to make sensible predictions in the sense of having a stationary phase approximation of the action to expand about, there must be a lowest energy state called the vacuum. The energy of the vacuum is completely arbitrary since a central scalar constant may be added to the Hamiltonian to globally shift the phase without changing the observable dynamics, and so the vacuum energy may take negative values so long as it is bounded below. This requirement is for instance what prompted the Dirac sea interpretation to address the Dirac equation's prediction of negative energy solutions, precisely because they generate an algebra of creation operators that can lower the energy ad infinitum.
To rectify this situation, the Witt algebra is centrally extended to provide a richer variety of Hilbert space modules to choose from, including the so-called positive energy representations, while leaving intact almost all of the Lie bracket relations between operators. This new algebra is called the Virasoro algebra, whose generators are , plus a central generator. The central generator takes a constant value , called the central charge, and the values of for which there is a positive energy representation is known (either or for ).
The symmetry algebra of a CFT is the product of two commuting copies of the Virasoro algebra: the left-moving or holomorphic algebra, with generators , and the right-moving or antiholomorphic algebra, with generators . These two copies are also known as the chiral algebras.
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. In a rational CFT that is defined on all Riemann surfaces, the central charge and conformal dimensions are rational numbers.
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),
where is a basis of the space of states, and the numbers are called OPE coefficients. Moreover, correlation functions are assumed to be invariant under permutations on the fields, in other words the OPE is assumed to be associative and commutative. (OPE commutativity does not imply that OPE coefficients are invariant under , because expanding on fields breaks that symmetry.)
OPE commutativity implies that primary fields have integer conformal spins . A primary field with zero conformal spin is called a diagonal field. There also exist fermionic CFTs that include fermionic fields with half-integer conformal spins , which anticommute. There also exist parafermionic CFTs that include fields with more general rational spins . Not only parafermions do not commute, but also their correlation functions are multivalued.
The torus partition function is a particular correlation function that depends solely on the spectrum , and not on the OPE coefficients. For a complex torus with modulus , the partition function is
where . The torus partition function coincides with the character of the spectrum, considered as a representation of the symmetry algebra.
Chiral conformal field theory 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.
Energy–momentum tensor The dependence of a field on its position is assumed to be determined by
It follows that the OPE
defines a locally holomorphic field that does not depend on This field is identified with (a component of) the energy–momentum tensor. In particular, the OPE of the energy–momentum tensor with a primary field is
The OPE of the energy–momentum tensor with itself is
where is the central charge. (This OPE is equivalent to the commutation relations of the Virasoro algebra.)
Conformal Ward identities Conformal Ward identities are linear equations that correlation functions obey as a consequence of conformal symmetry. They can be derived by studying correlation functions that involve insertions of the energy–momentum tensor. Their solutions are conformal blocks.
For example, consider conformal Ward identities on the sphere. Let be a global complex coordinate on the sphere, viewed as Holomorphy of the energy–momentum tensor at is equivalent to
Moreover, inserting in an -point function of primary fields yields
From the last two equations, it is possible to deduce local Ward identities that express -point functions of descendant fields in terms of -point functions of primary fields. Moreover, it is possible to deduce three differential equations for any -point function of primary fields, called global conformal Ward identities:
These identities determine how two- and three-point functions depend on
where the undetermined proportionality coefficients are functions of
BPZ equations 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. The order of this equation is the level of the null vector in the corresponding degenerate representation.
A trivial example is the order one BPZ equation
which follows from
The first nontrivial example involves a degenerate field with a vanishing null vector at the level two,
where is related to the central charge by
Then an -point function of and other primary fields obeys:
A BPZ equation of order for a correlation function that involve the degenerate field can be deduced from the vanishing of the null vector, and the local Ward identities. Thanks to global Ward identities, four-point functions can be written in terms of one variable instead of four, and BPZ equations for four-point functions can be reduced to ordinary differential equations.
Fusion rules In an OPE that involves a degenerate field, the vanishing of the null vector (plus conformal symmetry) constrains which primary fields can appear. The resulting constraints are called fusion rules. Using the momentum such that
instead of the conformal dimension for parametrizing primary fields, the fusion rules are
in particular
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 if the fusion product of two indecomposable representations is a sum of finitely many indecomposable representations. For example, generalized minimal models are quasi-rational without being rational.
Conformal bootstrap 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.
Structure constants Let be a left- and right-primary field with left- and right-conformal dimensions and . According to the left and right global Ward identities, three-point functions of such fields are of the type
where the -independent number is called a three-point structure constant. For the three-point function to be single-valued, the left- and right-conformal dimensions of primary fields must obey