A two-dimensional conformal field theory is a
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
on a Euclidean
two-dimensional space
In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
, that is invariant under local
conformal transformation
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
s.
In contrast to other types of
conformal field theories
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 sometime ...
, two-dimensional conformal field theories have
infinite-dimensional symmetry algebras. In some cases, this allows them to be solved exactly, using the
conformal bootstrap
The conformal bootstrap is a non-perturbative mathematical method to constrain and solve Conformal field theory, conformal field theories, i.e. models of particle physics or statistical physics that exhibit similar properties at different levels of ...
method.
Notable two-dimensional conformal field theories include
minimal models,
Liouville theory
In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation.
Liouville theory is defined for all complex values of the ...
,
massless free bosonic theories,
Wess–Zumino–Witten model
In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, Bruno Zumino, Sergei Novikov and Edwa ...
s, and certain
sigma model
In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold, although it is most commonly taken to be either a Lie group or ...
s.
Basic structures
Geometry
Two-dimensional conformal field theories (CFTs) are defined on
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s, where local
conformal map
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
s are
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s.
While a CFT might conceivably exist only on a given Riemann surface, its existence on any
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
other than the
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
of infinitesimal conformal maps
has the basis
, with
. (For example,
and
generate translations.) Relaxing the assumption that
is the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
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
In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie 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
In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string the ...
, whose
generator
Generator may refer to:
* Signal generator, electronic devices that generate repeating or non-repeating electronic signals
* Electric generator, a device that converts mechanical energy to electrical energy.
* Generator (circuit theory), an eleme ...
s 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 Virasoro generators, the third charge is cubic, etc. 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
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigrou ...
if the space of states has a positive definite
Hermitian form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
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
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. 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
In theoretical physics, a primary field, also called a primary operator, or simply a primary, is a local operator in a conformal field theory which is annihilated by the part of the conformal algebra consisting of the lowering generators. From the ...
, 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 mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which .
In the theory of real number, real bilinear forms, definite quadrat ...
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
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional i ...
of conformal field theory, correlation functions are defined as functional integrals. In the
conformal bootstrap
The conformal bootstrap is a non-perturbative mathematical method to constrain and solve Conformal field theory, conformal field theories, i.e. models of particle physics or statistical physics that exhibit similar properties at different levels of ...
approach, correlation functions are defined by axioms. In particular, it is assumed that there exists an
operator product expansion
In quantum field theory, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. As an axiom, it offers a non-perturbative approach to quantum field theory. One example is the vert ...
(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
. 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
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
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 Energy–momentum may refer to:
* Four-momentum
* Stress–energy tensor
* Energy–momentum relation
{{dab ...
.
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 "Sasha" Abramovich Belavin (russian: Алекса́ндр Абра́мович Бела́вин, born 1942) is a Russian physicist, known for his contributions to string theory.
He is a professor at the Independent University of Moscow a ...
,
Alexander Polyakov and
Alexander Zamolodchikov
Alexander Borisovich Zamolodchikov (russian: Алекса́ндр Бори́сович Замоло́дчиков; born September 18, 1952) is a Russian physicist, known for his contributions to condensed matter physics, two-dimensional conform ...
.
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
Ward may refer to:
Division or unit
* Hospital ward, a hospital division, floor, or room set aside for a particular class or group of patients, for example the psychiatric ward
* Prison ward, a division of a penal institution such as a priso ...
. 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
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
of representations. (In a tensor product, the central charges add.) In certain finite cases, this leads to the structure of a
fusion category
In mathematics, a fusion category is a category that is rigid, semisimple, k-linear, monoidal and has only finitely many isomorphism classes of simple objects, such that the monoidal unit is simple. If the ground field k is algebraically closed ...
.
A conformal field theory is quasi-rational is 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
The conformal bootstrap is a non-perturbative mathematical method to constrain and solve Conformal field theory, conformal field theories, i.e. models of particle physics or statistical physics that exhibit similar properties at different levels of ...
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
:
This condition is satisfied by bosonic (
) and fermionic (
) fields. It is however violated by parafermionic fields (
), whose correlation functions are therefore not single-valued on the Riemann sphere.
Three-point structure constants also appear in OPEs,
:
The contributions of descendant fields, denoted by the dots, are completely determined by conformal symmetry.
Conformal blocks
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.
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
:
where
is an s-channel four-point conformal block. Four-point conformal blocks are complicated functions that can be efficiently computed using
Alexei Zamolodchikov's recursion relations. If one of the four fields is degenerate, then the corresponding conformal blocks obey BPZ equations. If in particular one the four fields is
, then the corresponding conformal blocks can be written in terms of the
hypergeometric function.
As first explained by Witten, 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
The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of e^2/h. It is a property of a collective state in which elec ...
.
Conformal bootstrap equations
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.
For example, a four-point function
can be written in terms of conformal blocks in three inequivalent ways, corresponding to using the OPEs
(s-channel),
(t-channel) or
(u-channel). The equality of the three resulting expressions is called
crossing symmetry
In quantum field theory, a branch of theoretical physics, crossing is the property of scattering amplitudes that allows antiparticles to be interpreted as particles going backwards in time.
Crossing states that the same formula that determines t ...
of the four-point function, and is equivalent to the associativity of the OPE.
For example, the torus partition function is invariant under the action of the
modular group on the modulus of the torus, equivalently
. This invariance is a constraint on the space of states. The study of modular invariant torus partition functions is sometimes called the modular bootstrap.
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.
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.
Examples
Minimal models
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,
:
There is an
ADE classification
In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, r ...
of minimal models.
[Andrea Cappelli and Jean-Bernard Zuber (2010)]
"A-D-E Classification of Conformal Field Theories"
Scholarpedia 5(4):10314. In particular, the A-series minimal model with the central charge
is a diagonal CFT whose spectrum is built from
degenerate
lowest weight representations of the Virasoro algebra. These degenerate representations are labelled by pairs of integers that form the Kac table,
:
For example, the A-series minimal model with
describes spin and energy correlators of the
two-dimensional critical Ising model
The two-dimensional critical Ising model is the critical limit of the Ising model in two dimensions. It is a two-dimensional conformal field theory whose symmetry algebra is the Virasoro algebra with the central charge c=\tfrac12.
Correlation fu ...
.
Liouville theory
For any
Liouville theory is a diagonal CFT whose spectrum is built from Verma modules with conformal dimensions
:
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.
Extended symmetry algebras
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.
Massless free bosonic theories
In two dimensions, massless free bosonic theories are conformally invariant. Their symmetry algebra is the
affine Lie algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...
built from the abelian, rank one Lie algebra. The fusion product of any two representations of this symmetry algebra yields only one representation, and this makes correlation functions very simple.
Viewing minimal models and Liouville theory as perturbed free bosonic theories leads to the Coulomb gas method for computing their correlation functions. Moreover, for
there is a one-parameter family of free bosonic theories with infinite discrete spectrums, which describe compactified free bosons, with the parameter being the compactification radius.
Wess–Zumino–Witten models
Given a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
the corresponding Wess–Zumino–Witten model is a CFT whose symmetry algebra is the
affine Lie algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...
built from the Lie algebra of
If
is compact, then this CFT is rational, its central charge takes discrete values, and its spectrum is known.
Superconformal field theories
The symmetry algebra of a supersymmetric CFT is a
super Virasoro algebra
In mathematical physics, a super Virasoro algebra is an extension of the Virasoro algebra (named after Miguel Ángel Virasoro) to a Lie superalgebra. There are two extensions with particular importance in superstring theory: the Ramond algebra (n ...
, or a larger algebra. Supersymmetric CFTs are in particular relevant to superstring theory.
Theories based on W-algebras
W-algebra
In conformal field theory and representation theory, a W-algebra is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov, and the name "W-algebra" comes from the fact that Zamolodchi ...
s 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.
Sigma models
In two dimensions, classical
sigma model
In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold, although it is most commonly taken to be either a Lie group or ...
s 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 manifold
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring ...
s.
Logarithmic conformal field theories
Logarithmic conformal field theories are two-dimensional CFTs such that the action of the Virasoro algebra generator
on the spectrum is not diagonalizable. In particular, the spectrum cannot be built solely from
lowest weight representations. As a consequence, the dependence of correlation functions on the positions of the fields can be logarithmic. This contrasts with the power-like dependence of the two- and three-point functions that are associated to lowest weight representations.
Critical Q-state Potts model
The critical
-state Potts model or critical
random cluster model In statistical mechanics, probability theory, graph theory, etc. the random cluster model is a random graph that generalizes and unifies the Ising model, Potts model, and percolation model. It is used to study random combinatorial structures, elec ...
is a conformal field theory that generalizes and unifies the critical
Ising model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
,
Potts model
In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenom ...
, and
percolation
Percolation (from Latin ''percolare'', "to filter" or "trickle through"), in physics, chemistry and materials science, refers to the movement and filtering of fluids through porous materials.
It is described by Darcy's law.
Broader applicatio ...
. The model has a parameter
, which must be integer in the Potts model, but which can take any complex value in the random cluster model.
This parameter is related to the central charge by
:
Special values of
include:
The known torus partition function
suggests that the model is non-rational with a discrete spectrum.
References
Further reading
* P. Di Francesco, P. Mathieu, and D. Sénéchal, ''Conformal Field Theory'', Springer-Verlag, New York, 1997. .
Conformal Field Theorypage i
String Theory Wikilists books and reviews.
* {{cite arXiv , eprint=1406.4290, last1=Ribault, first1=Sylvain, title=Conformal field theory on the plane, year=2014, class=hep-th
Conformal field theory