# Conformal field theory on the plane

Sylvain Ribault
CEA Saclay, Institut de Physique Théorique
<EMAIL_ADDRESS>
###### Abstract

We provide an introduction to conformal field theory on the plane in the
conformal bootstrap approach. We introduce the main ideas of the bootstrap
approach to quantum field theory, and how they apply to two-dimensional
theories with local conformal symmetry. We describe the mathematical
structures which appear in such theories, from the Virasoro algebra and its
representations, to the BPZ equations and their solutions. As examples, we
study a number of models: Liouville theory, (generalized) minimal models, free
bosonic theories, the $H_{3}^{+}$ model, and the $SU_{2}$ and
$\widetilde{SL}_{2}(\mathbb{R})$ WZW models.

Keywords

conformal field theory, conformal bootstrap, Virasoro algebra, BPZ equations,
Liouville theory, minimal models, free bosons, KZ equations, KZ-BPZ relation,
$H_{3}^{+}$ model, WZW models

Public domain notice

To the extent possible under law, Sylvain Ribault has waived all copyright and
related or neighboring rights to this text.

###### Contents

  1. 0 Preliminaries
    1. 0.1 Context and approach
    2. 0.2 Plan of the article
    3. 0.3 Why one more review on conformal field theory?
    4. 0.4 User’s manual
    5. 0.5 Acknowledgements
  2. 1 Introduction
    1. 1.1 Quantum field theory
      1. 1.1.1 Definition
      2. 1.1.2 Observables
    2. 1.2 The bootstrap approach
      1. 1.2.1 Principles
      2. 1.2.2 Symmetry assumptions
      3. 1.2.3 Consistency conditions
      4. 1.2.4 So what is the Lagrangian of your model?
    3. 1.3 Conformal symmetry in two dimensions
      1. 1.3.1 Global conformal transformations
      2. 1.3.2 Local conformal transformations
      3. 1.3.3 The Virasoro algebra
    4. 1.4 Basic consequences of conformal symmetry
      1. 1.4.1 Structure of the spectrum
      2. 1.4.2 Conformal bootstrap
      3. 1.4.3 Sketching the space of simple models
    5. 1.5 Exercises
  3. 2 From representation theory to differential equations
    1. 2.1 Representations of the Virasoro algebra
      1. 2.1.1 Highest-weight representations
      2. 2.1.2 Verma modules and degenerate representations
      3. 2.1.3 Unitarity
    2. 2.2 Virasoro symmetry in two-dimensional quantum field theory
      1. 2.2.1 Primary fields, decendent fields
      2. 2.2.2 The Virasoro field
      3. 2.2.3 Ward identities
      4. 2.2.4 Virasoro-symmetric OPEs
      5. 2.2.5 Null vector equations, and the fusion rules
    3. 2.3 Consequences of conformal symmetry
      1. 2.3.1 The energy-momentum tensor
      2. 2.3.2 Global conformal symmetry
      3. 2.3.3 Conformal blocks
      4. 2.3.4 BPZ differential equations
      5. 2.3.5 Hypergeometric conformal blocks
    4. 2.4 Exercises
  4. 3 Liouville theory and minimal models
    1. 3.1 Liouville theory
      1. 3.1.1 Definition and spectrum
      2. 3.1.2 The reflection relation and its consequences
      3. 3.1.3 Conformal bootstrap equations
      4. 3.1.4 The three-point function
      5. 3.1.5 Existence and uniqueness of Liouville theory
    2. 3.2 Minimal models
      1. 3.2.1 Definition
      2. 3.2.2 Generalized minimal models
      3. 3.2.3 A-series minimal models
      4. 3.2.4 Examples of A-series minimal models
    3. 3.3 Exercises
  5. 4 Affine symmetry
    1. 4.1 Free bosons
      1. 4.1.1 Symmetry algebra and representations
      2. 4.1.2 Ward identities and Seiberg-Witten equations
      3. 4.1.3 Spectrum and correlation functions
      4. 4.1.4 Free bosons and Liouville theory
    2. 4.2 Nonabelian affine symmetry
      1. 4.2.1 Symmetry algebra
      2. 4.2.2 Fields, representations and fusion rules
      3. 4.2.3 Ward identities and Knizhnik-Zamolodchikov equations
      4. 4.2.4 $\mathfrak{sl}_{2}$ case: the KZ-BPZ relation
    3. 4.3 The $H_{3}^{+}$ model
      1. 4.3.1 Spectrum and correlation functions
      2. 4.3.2 Large level limit and geometrical interpretation
    4. 4.4 WZW models
      1. 4.4.1 Definition and general properties
      2. 4.4.2 The $SU_{2}$ WZW model
      3. 4.4.3 The $\widetilde{SL}_{2}(\mathbb{R})$ WZW model
    5. 4.5 Exercises

##  Chapter 0 Preliminaries

###  0.1 Context and approach

As an introduction to two-dimensional conformal field theory, this text has
the particularity of using exclusively the bootstrap approach, which consists
in systematically exploiting symmetry and consistency assumptions. While this
approach is widely recognized as very powerful, most of the literature uses it
alongside the Lagrangian approach, which can be quite useful for heuristics
and for some computations. Renouncing Lagrangians however has the advantages
of simplicity and consistency, and we will try to show that we do not lose
much in terms of heuristics and computational power – we will only have to
deplore the loss of a simple definition for WZW models.

Another advantage of the bootstrap approach is that by making assumptions
explicit, we can better understand them, and if necessary lift them. For
example, results such as the C-theorem, and the fact that scale invariance
implies conformal invariance in two-dimensional unitary quantum field
theories, rely on the strong assumption that there exists an energy-momentum
tensor – a conserved field with spin two and conformal dimension two, which
generates conformal transformations. We will not lift this assumption, as we
will study theories with local (i.e. not just global) conformal invariance,
which therefore always have an energy-momentum tensor.

We will however lift a number of other common assumptions, in particular the
existence of a vacuum state, and the existence of theories on any Riemann
surface (and not just on the plane). It would be interesting to investigate
how this affects the set of possible models, and in particular the
classification of minimal models. It is however mainly for the sake of
generality and simplicity that we will lift these assumptions. We will indeed
be studying not only rational, but also non-rational theories such as
Liouville theory. Non-rational theories do not necessarily have a vacuum
state, and assuming their consistency on the torus is of little help for
determining their spectrums [1].

By definition, rational conformal field theories are have spectrums made of
finitely many representations. However, in order to ensure closure under
fusion, these representations must have complicated structures. In Liouville
theory, the spectrum is made of a continuum of representations, but these
representations are merely Verma modules, so we will study this theory first.
Then we will study minimal models, which are rational and therefore less
generic and more complicated. Even later will come free theories: since we do
not do use a perturbative approach, they do not play any fundamental role.

###  0.2 Plan of the article

We will do a systematic exposition of the ideas and techniques of the
conformal bootstrap approach in Chapters 1 and 2, before studying particular
models in Chapters 3 and 4.

In Chapter 1, we introduce the bootstrap approach to quantum field theory,
before focusing on the particular case of conformal field theories with their
Virasoro symmetry algebra. We then give a preview of some of the simplest
models: Liouville theory and minimal models.

In Chapter 2, we work out the technical consequences of these ideas for the
spectrum and correlation functions. We study the highest-weight
representations of the Virasoro algebra, and the fields which correspond to
states in these representations. We take some care to distinguish Virasoro
symmetry (Section 2.2), which is an algebraic structure, from conformal
symmetry (Section 2.3), which in addition has a geometrical interpretation.
Virasoro symmetry is encoded in a Virasoro field, while conformal symmetry is
encoded in an energy-momentum tensor – a Virasoro field which obeys the
additional assumption (2.3.1). Virasoro symmetry is enough for deriving fusion
rules, and even the non-trivial factors in conformal blocks, although for
convenience we study conformal blocks only under the assumption of conformal
symmetry.

In Chapter 3, we introduce and solve the simplest non-trivial two-dimensional
conformal field theories: Liouville theory, generalized minimal models, and
A-series minimal models, in that order. We define these theories by a few
simple assumptions on their spectrums and correlation functions, which
uniquely determine these quantities. We pay particular attention to the three-
point structure constants, whose explicit expressions we derive.

In Chapter 4, we introduce extended symmetry algebras, starting with the
affine $\hat{\mathfrak{u}}_{1}$ algebra. This not only allows us to study free
bosonic theories, but also provides alternative perspectives on Liouville
theory – perspectives which were originally obtained using the Lagrangian
approach. We then consider nonabelian affine symmetry algebras, and in
particular $\widehat{\mathfrak{sl}}_{2}$. We deduce the definition and
solution of the $H_{3}^{+}$ model from the KZ-BPZ relation between
differential equations satisfied by $\widehat{\mathfrak{sl}}_{2}$-symmetric
and Virasoro-symmetric correlation functions. We then consider WZW models, and
in particular the $SU_{2}$ and $\widetilde{SL}_{2}(\mathbb{R})$ WZW models.

In each Chapter, the last Section is devoted to exercises. Some exercises are
intended to test and improve one’s comprehension of the material, but such
exercises become scarcer in later Chapters. Some exercises sketch proofs of
results from the main text, in cases when hints and guidance seem warranted.
And some exercises provide supplementary material about:

  * •

singular vectors of Verma modules (Exercise 2.5),

  * •

logarithmic conformal field theory (Exercise 2.13),

  * •

fusion rules of unitary Verma modules (Exercise 3.3),

  * •

Liouville theory with finite multiplicities (Exercise 3.2),

  * •

generalization of the Sugawara construction (Exercise 4.3),

  * •

the Wakimoto free-field representation of $\widehat{\mathfrak{sl}}_{2}$
(Exercise 4.6).

This text comes with an index, which points to the one or two places in the
text where each term is defined. At these places, the terms in question appear
in bold.

At a few places in the text, possibilities to jump forward and omit some
sections are indicated. Section 2.2.4 on OPEs can be omitted. After studying
fusion rules at the end of Section 2.2, it is possible to jump to the
derivation of the spectrums of minimal models in Section 3.2. Section 2.3.3 on
conformal blocks can be omitted.

###  0.3 Why one more review on conformal field theory?

Let us argue that this review article fills an available ecological niche, by
considering which neighbouring niches are already filled:

  * •

The book [2] by A. and Al. Zamoldochikov is in spirit quite close to this
article. That book is a concise exposition of conformal field theory, from the
basic principles to advanced results, mostly in the conformal bootstrap
approach. However, the depth and breadth of the ideas may be too much for
beginning students. Non-rational theories such as Liouville theory are not
covered. And misprints are frequent.

  * •

Non-rational theories are covered in a review by Schomerus [3]. That article
uses multiple approaches, and pays particular attention to boundary conformal
field theory, with applications to string theory in mind.

  * •

The Big Yellow Book [4] is a useful reference, especially on rational
conformal field theories. As its epic length suggests, it is rather
encyclopedic.

  * •

The encyclopedic approach was extended in the direction of Liouville theory by
Nakayama’s review [5], which includes systematic guides to the literature.

  * •

On the more pedagogical side, Gaberdiel’s review [6] provides a consistent
exposition using vertex operators, which mathematically formalize the
conformal bootstrap approach. That text deals more with the symmetry algebras
and their representations, than with correlation functions. Examples include
logarithmic theories, but no non-rational theories.

  * •

Motivations and applications are mostly absent from the present text. On the
statistical mechanics side, they are given in Cardy’s lecture notes [7], which
also provide a concise introduction to the formalism.

  * •

$W$-symmetry is reviewed in depth in Bouwknegt and Schoutens’s article [8],
where $W$-symmetric conformal field theories are however not solved
systematically. An exposition of the $W$-symmetric minimal models and
conformal Toda theories would be a natural extension of the present article,
although these theories have not been fully solved so far.

In contrast to these texts, we will treat both rational and non-rational
theories. Moreover, our exclusive use of the bootstrap approach means that we
need not assume any previous knowledge of quantum field theory. The
prerequisites for this text are limited to elementary complex analysis, and
some familiarity with Lie algebras and their representations.

While this review article mostly presents already known results, the following
aspects may be original:

  * •

in Section 2.3.3, the formulation of Al. Zamolodchikov’s recursion
representation of conformal blocks as an explicit formula,

  * •

in Section 3.1, a complete and correct solution of Liouville theory, using
only the bootstrap approach,

  * •

in Section 3.2, the logical sequence “Liouville theory $\rightarrow$
generalized minimal models $\rightarrow$ minimal models”,

  * •

in Section 4.1.3, the remark that compactified free bosons exist for any value
of $c$ and not just if $c=1$,

  * •

in Section 4.1.4, the derivation of the light and heavy asymptotic limits of
Liouville theory, without using the Lagrangian definition,

  * •

in Section 4.4.2, the definition of the generalized $SU_{2}$ WZW model,

  * •

in Section 4.4.3, a concise derivation of the spectrum and fusion rules of the
$\widetilde{SL}_{2}(\mathbb{R})$ WZW model.

###  0.4 User’s manual

This review article is intended as a tool for learning, and as a collection of
technical results. It provides neither a history of the subject, nor a guide
to the literature. Accordingly, citations to the existing literature are not
meant to distribute credit to researchers in the field. Citations are solely
meant to be helpful to the reader, and are therefore used sparingly. Each
cited work comes with a hopefully clear and precise indication of what could
be useful therein. In choosing cited works, the criterions have been

  1. 1.

ease of access (only freely and if possible legally available texts are
cited),

  2. 2.

clarity and completeness (this often favours review articles over original
works),

  3. 3.

my familiarity with the cited works (admittedly this favours my own works).

Citations are only one tool for finding more information on the subject. The
reader is assumed to have access to other tools such as Google and Wikipedia.
To facilitate searches, an effort has been made to use standard terminology
and notations.

Traditional scientific articles, with their frozen texts and authors, are
obviously a poor way of conveying the ever-evolving knowledge of a community
of researchers. They may become obsolete, but it is not clear what will
replace them. Meanwhile, it seems prudent to distribute the present text so
that it can easily be reused and modified by others. For this purpose:

  * •

This text is in the public domain, in order to eliminate legal restrictions to
its reuse.

  * •

This article is distributed not only using Arxiv, but also using GitHub, so
that it can be collaboratively modified.

  * •

In the Latex file, there are no user-defined global macros. This makes it
easier to reuse excerpts of that file.

I have tried to follow some stylistic good practices, such as: providing a
clickable table of contents, not clogging the bibliography with superfluous
information such as journal data, avoiding Latin plurals when regular plurals
are available, and numbering all equations while boxing the important ones.

###  0.5 Acknowledgements

I am grateful to my colleagues at IPhT Saclay, and in particular to Antoine
Duval, Riccardo Guida, Santiago Migliaccio, Vincent Pasquier, and Pierre
Ronceray, for the opportunity to give lectures on this subject. I am also
grateful to the organizers and participants to the 10th APCTP Focus Program on
Liouville, Integrability and Branes, and in particular to Soojong Rey, for
including my lectures in an otherwise very advanced workshop.

I wish to thank Philippe Di Francesco, Matthias Gaberdiel, Christoph Keller,
Slava Rychkov, Vincent Vargas, and Gérard Watts, for useful discussions and
correspondence.

I am grateful to Connor Behan, François David, Quang-Dien Duong, Bruno Le
Floch, and Raoul Santachiara, for helpful comments on this text.

##  Chapter 1 Introduction

###  1.1 Quantum field theory

Conformal field theory is a special case of quantum field theory, and we start
with a brief reminder of what quantum field theory is.

####  1.1.1 Definition

A theory can be either a general framework such as quantum field theory or
general relativity, or a more specific model such as the standard model of
particle physics or Liouville theory. The standard model is formulated in the
framework of quantum field theory, so we can call it a model of quantum field
theory or simply a quantum field theory. Similarly, Liouville theory is a
conformal field theory. A model need not be directly related to a physical
system: a given physical system can be described by several models, and a
given model can describe a number (possibly zero) of physical systems. This
holds not only for physical systems, but also for what could be called
theoretical systems, such as quantum gravity in $d$ dimensions. These notions
are illustrated by the following examples:

(1.1.1)

Now, what is a quantum theory? First, this is a probabilistic theory, which
does not predict the outcome of a given experiment, but the probabilities of
different outcomes. Second, a quantum theory does not predict just
probabilities, but actually probability amplitudes. Such amplitudes can be
added, giving rise to interferences.

Last, the field in field theory is a variable defined on some space, which can
describe an arbitrary number of particles propagating in that space. In
particular, field theories can describe the creation and annihilation of
particles. For example, a “height of water” field can be defined on the
surface of the ocean, in order to describe arbitrary numbers of water waves. A
field can in some respects be thought of as an infinite collection of
elementary objects: in our example, individual waves of definite wavelengths.
Field theory constrains how these objects behave in relation with the
underlying space, for instance by requiring interactions to be local.

So, quantum field theory is particularly well-suited to predicting the
outcomes of collisions of particles, whether in the cosmos or in a particle
accelerator. Such collisions can indeed create or destroy particles, so we
need a field theory, and repeating the same collision can produce various
outcomes, so we need a probabilistic theory.

####  1.1.2 Observables

In a model, the observables are the quantities which can in principle be
measured. The nature of the observables is in general dictated by the theory.
For example, in the theory of general relativity, the observables are light
signals and motions of objects, and do not include the space-time metric.
Which observables are relevant may depend on the physical system. For example,
in two-dimensional conformal field theory, some observables are functions on
the two-dimensional space. Such observables are relevant in condensed-matter
physics and in quantum geometry, where the space has a physical
interpretation. They are not relevant in string theory, where the two-
dimensional space is the world-sheet of a string. In quantum field theory in
general, and in conformal field theory in particular, we define the
observables to be

  * •

the spectrum and

  * •

the correlation functions.

In quantum theory, the spectrum or space of states is a vector space whose
elements represent the states of a system. The vector space structure is what
leads to interferences. A vector $\sigma$ and the action $A(\sigma)$ of an
operator $A$ on $\sigma$ may be written as

$\displaystyle\left\\{\begin{array}[]{l}\sigma=|\sigma\rangle\ ,\\\
A(\sigma)=A|\sigma\rangle=|A\sigma\rangle\ .\end{array}\right.$ (1.1.4)

Moreover, the spectrum has a sesquilinear scalar product, so that the product
$\langle\tau|\sigma\rangle$ of two states can be interpreted as a probability
amplitude. This product satisfies

$\displaystyle\langle\tau|\lambda_{1}\sigma_{1}+\lambda_{2}\sigma_{2}\rangle$
$\displaystyle=\lambda_{1}\langle\tau|\sigma_{1}\rangle+\lambda_{2}\langle\tau|\sigma_{2}\rangle\
,$ (1.1.5) $\displaystyle\langle\tau|\sigma\rangle$
$\displaystyle=\overline{\langle\sigma|\tau\rangle}\ .$ (1.1.6)

The hermitian conjugate $A^{\dagger}$ of a linear operator $A$ is then defined
by

$\displaystyle\langle\tau|A\sigma\rangle=\langle
A^{\dagger}\tau|\sigma\rangle\ .$ (1.1.7)

If the scalar product is positive definite, then the spectrum is a Hilbert
space. This is a necessary condition for the theory to have a direct
probabilistic interpretation.

Now, in certain quantum field theories (including conformal field theories),
to each state $|\sigma\rangle$ and each point $x$ in space we associate a
linear operator on the spectrum $V_{\sigma}(x)$, and the collection of these
operators when $x$ varies is a field. The linear map

$\displaystyle|\sigma\rangle\ \mapsto\ V_{\sigma}(x)\ ,$ (1.1.8)

is called the state-field correspondence or state-operator correspondence.
Thinking of the state as a type of particles, the field $V_{\sigma}(x)$ can be
thought of as measuring the presence of such particles at point $x$. Then, a
correlation function, here an $N$-point function

$\displaystyle\left\langle\prod_{i=1}^{N}V_{\sigma_{i}}(x_{i})\right\rangle$
(1.1.9)

is the probability amplitude for the interaction of $N$ particles of types
$\sigma_{1},\cdots\sigma_{N}$, located at points $x_{1},\cdots x_{N}$. For the
moment, an $N$-point function is a mere notation for a number that linearly
depends on $N$ fields. Correlation functions will be defined by the equations
that they obey, which will be deduced from the assumptions of the bootstrap
approach.

To construct a model of quantum field theory is therefore to give principles
which uniquely determine a spectrum and a set of correlation functions. To
solve a model is to actually compute the spectrum and correlation functions.

###  1.2 The bootstrap approach

####  1.2.1 Principles

The bootstrap approach is a method for constructing and solving theories,
based on the systematic exploitation of

  1. 1.

symmetry assumptions,

  2. 2.

consistency conditions,

For example, the symmetry assumption of general covariance is the basis of the
theory of general relativity, and the further symmetry assumptions of
homogeneity and isotropy of the universe are used in constructing models of
cosmology. In quantum theories, an essential consistency condition is the
condition that the sum of the probabilities of all possible events is one.

Given a set of assumptions, there may exist any number of models which obey
them:

  * •

no model at all, if the assumptions are too restrictive,

  * •

one model, in which case we may be able to solve it,

  * •

a manageable number of models, in which case we may be able to classify them,
and to focus on one of them by making further assumptions,

  * •

a huge number of models, if the assumptions are not restrictive enough.

We will now explore in turn the symmetry assumptions and consistency
conditions in quantum field theory.

####  1.2.2 Symmetry assumptions

If a model of quantum field theory has a symmetry algebra, then the spectrum
$S$ must be a representation of this algebra, and can therefore be written as

$\displaystyle S=\bigoplus_{R}m_{R}R\ .\ $ (1.2.1)

Here we have decomposed $S$ as a combination of some basic representations $R$
of the symmetry algebra, and the number $m_{R}\in{\mathbb{N}}\cup\\{\infty\\}$
is the multiplicity of the representation $R$ in the spectrum $S$. A state
$\sigma$ in the spectrum can then be written as

$\displaystyle\sigma=(R,v)\ ,$ (1.2.2)

where $v$ is a state in the representation $R$, and the corresponding field
can be written as $V_{\sigma}(x)=V_{(R,v)}(x)$. For example, for any spin
$j\in\frac{1}{2}\mathbb{N}$ the algebra $\mathfrak{sl}_{2}$ has a
representation of dimension $2j+1$, whose states $\sigma=(j,m)$ can be
labelled by their magnetic momentums $m\in\\{-j,-j+1,\cdots j\\}$.

The idea is now that the dependence of a correlation function
$\left\langle\prod_{i=1}^{n}V_{(R_{i},v_{i})}(x_{i})\right\rangle$ on $v_{i}$
is determined by symmetry considerations, whereas the dependence on $R_{i}$ is
constrained by consistency conditions. This idea however comes with a number
of technical assumptions. In particular, for the dependence on $v_{i}$ to be
completely determined by symmetry, $R_{i}$ must be an indecomposable
representation, which cannot be written as a direct sum of smaller
representations.

$\displaystyle R\ \ \text{indecomposable}\quad\Leftrightarrow\quad\nexists
R^{\prime},R^{\prime\prime}\neq 0\quad|\quad R=R^{\prime}\oplus
R^{\prime\prime}\ .$ (1.2.3)

In particular, any irreducible representation, which by definition does not
have a nontrivial subrepresentation, is indecomposable,

$\displaystyle R\ \ \text{irreducible}\quad\Leftrightarrow\quad\nexists
R^{\prime}\neq 0,R\quad|\quad R^{\prime}\subset R\ .$ (1.2.4)

(See Exercise 1.1 for an example of an indecomposable representation which is
not irreducible.)

We will therefore distinguish two types of data:

  * •

universal data, which are also called model-independent or algebraic data.
This is the information on representations of the symmetry algebra, which
determines in particular the $v_{i}$-dependence of correlation functions.
Universal data in conformal field theory, and in particular conformal blocks,
will be studied in Chapter 2.

  * •

model-dependent data, which encode how the spectrum and correlation functions
of a particular model are built from the universal data. This includes the
multiplicities $m_{R}$, which encode how the spectrum is built from
representations, and the structure constants, which encode how correlation
functions are built from conformal blocks. We will study these data in Chapter
3.

Which data are universal or model-dependent depends on the choice of a
symmetry algebra. If a model’s symmetry algebra $\mathcal{A}$ is actually a
subalgebra of a larger symmetry algebra $\mathcal{A}^{\prime}$, then the
representations of $\mathcal{A}$ which appear in the spectrum must combine
into representations of the larger algebra $\mathcal{A}^{\prime}$. Which
combinations can appear is universal data of $\mathcal{A}^{\prime}$, but
model-dependent data from the point of view of the smaller algebra
$\mathcal{A}$.

Since a field theory is defined on some space, we can distinguish two types of
symmetries:

  * •

space-time symmetries, which act on that space,

  * •

internal symmetries, which do not.

Space-time symmetries include in particular the invariances under rotations
and translations. Internal symmetries include gauge invariances. The
distinction between space-time and internal symmetries depends on the choice
of the space: for example, supersymmetry can be an internal or space-time
symmetry depending on whether the theory is formulated on an ordinary space or
on a superspace.

####  1.2.3 Consistency conditions

The consistency conditions on the correlation functions
$\left\langle\prod_{i=1}^{n}V_{\sigma_{i}}(x_{i})\right\rangle$ are derived
from assumptions on the fields $V_{\sigma_{i}}(x_{i})$. These assumptions are:

  1. 1.

Commutativity or locality:

$\displaystyle\boxed{V_{\sigma_{1}}(x_{1})V_{\sigma_{2}}(x_{2})=V_{\sigma_{2}}(x_{2})V_{\sigma_{1}}(x_{1})}\
,$ (1.2.5)

is supposed to hold provided $(x_{1},x_{2})$ obey some condition. In Minkowski
space for instance, $x_{1}-x_{2}$ is supposed to be space-like. In the two-
dimensional Euclidean space, the condition is $x_{1}\neq x_{2}$.

  2. 2.

Associativity:

$\displaystyle\boxed{\left(V_{\sigma_{1}}(x_{1})V_{\sigma_{2}}(x_{2})\right)V_{\sigma_{3}}(x_{3})=V_{\sigma_{1}}(x_{1})\left(V_{\sigma_{2}}(x_{2})V_{\sigma_{3}}(x_{3})\right)}\
.$ (1.2.6)

Together with commutativity, this implies that the ordering of fields in
correlation functions is indifferent.

  3. 3.

Existence of an OPE (operator product expansion):

$\displaystyle\boxed{V_{\sigma_{1}}(x_{1})V_{\sigma_{2}}(x_{2})=\sum_{\sigma\in
S}C_{\sigma_{1},\sigma_{2}}^{\sigma}(x_{1},x_{2})V_{\sigma}(x_{2})}\ ,$
(1.2.7)

where the OPE coefficient $C_{\sigma_{1},\sigma_{2}}^{\sigma}(x_{1},x_{2})$ is
a function (i.e. not a field), and the sum runs over a basis of the spectrum
$S$. The sum is supposed to converge if $x_{1}$ and $x_{2}$ are sufficiently
close. Since the spectrum $S$ is in general much smaller than the space of
linear operators on $S$, the assumption that the operator
$V_{\sigma_{1}}(x_{1})V_{\sigma_{2}}(x_{2})$ is a linear combination of
$(V_{\sigma}(x_{2}))_{\sigma\in S}$ (or more generally
$(V_{\sigma}(x))_{\sigma\in S}$) is a nontrivial assumption, which does not
follow from the state-field correspondence.

Our assumptions do not include the existence of a vacuum state, a frequently
encountered axiom of quantum field theory in flat space. In Liouville theory
this axiom is not satisfied, and in minimal models we can do without it.

Let us now study how these assumptions constrain the correlation functions. By
performing multiple OPEs, an $N$-point function (with $N\geq 3$) can always be
reduced to a combination of two-point functions, for instance

$\displaystyle\left\langle\prod_{i=1}^{4}V_{\sigma_{i}}(x_{i})\right\rangle=\sum_{\sigma\in
S}C_{\sigma_{1},\sigma_{2}}^{\sigma}(x_{1},x_{2})\sum_{\sigma^{\prime}\in
S}C_{\sigma,\sigma_{3}}^{\sigma^{\prime}}(x_{2},x_{3})\left\langle
V_{\sigma^{\prime}}(x_{3})V_{\sigma_{4}}(x_{4})\right\rangle\ .$ (1.2.8)

We could go further and reduce the two-point function to a sum of one-point
functions. This is however unnecessary, because conformal symmetry will
determine the two-point function, so that we can consider it a known quantity.
So, in order to compute correlation functions, all we need to know (in
addition to the spectrum $S$) is the OPE coefficient
$C_{\sigma_{1},\sigma_{2}}^{\sigma}(x_{1},x_{2})$. But this quantity is itself
strongly constrained by the associativity assumption (1.2.6), which implies

$\displaystyle\sum_{\sigma_{s}\in
S}C_{\sigma_{1},\sigma_{2}}^{\sigma_{s}}(x_{1},x_{2})C_{\sigma_{s},\sigma_{3}}^{\sigma_{4}}(x_{2},x_{3})=\sum_{\sigma_{t}\in
S}C_{\sigma_{1},\sigma_{t}}^{\sigma_{4}}(x_{1},x_{3})C_{\sigma_{2},\sigma_{3}}^{\sigma_{t}}(x_{2},x_{3})\
,$ (1.2.9)

for any choice of the four states
$(\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4})$. (See Exercise 1.2 a similar
calculation in a technically simpler context.) This condition can be
represented schematically as

$\displaystyle\sum_{s}\leavevmode\hbox to57.19pt{\vbox
to67.25pt{\pgfpicture\makeatletter\hbox{\hskip 28.59486pt\lower-36.4723pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ } {}{{}}{}{{}} {}{}
{}{}{{}}{}\pgfsys@moveto{-22.76186pt}{17.07138pt}\pgfsys@lineto{0.0pt}{-5.69046pt}\pgfsys@lineto{22.76186pt}{17.07138pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-25.26186pt}{21.0044pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{}} {}{}
{}{}{}{}{{{}{}}}
{}{}{{}}{}\pgfsys@moveto{0.0pt}{17.07138pt}\pgfsys@lineto{-11.38092pt}{5.69046pt}\pgfsys@lineto{0.0pt}{-5.69046pt}\pgfsys@lineto{0.0pt}{-22.76186pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$s$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-33.1393pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\quad\sum_{t}\leavevmode\hbox
to57.19pt{\vbox to67.25pt{\pgfpicture\makeatletter\hbox{\hskip
28.59486pt\lower-36.4723pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ } {}{{}}{}{{}} {}{}
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}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{}{}{}{{{}{}}}
{}{}{{}}{}\pgfsys@moveto{0.0pt}{17.07138pt}\pgfsys@lineto{11.38092pt}{5.69046pt}\pgfsys@lineto{0.0pt}{-5.69046pt}\pgfsys@lineto{0.0pt}{-22.76186pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{21.0044pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.62347pt}{-10.0838pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$t$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-33.1393pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (1.2.10)

where each node corresponds to an OPE coefficient.

The OPE coefficients can be determined by solving the associativity condition
(1.2.9), provided the model has enough symmetry for

  * •

constraining the dependence of the OPE coefficient
$C_{\sigma_{1},\sigma_{2}}^{\sigma}(x_{1},x_{2})$ on $(x_{1},x_{2})$, and

  * •

decomposing the spectrum $S$ into a reasonable number of representations.

These conditions are fulfilled in certain conformal field theories. In
particular, as we will see in Chapter 2, conformal symmetry is enough for
determining the $x$-dependence of OPE coefficients.

For the moment, let us show how symmetry constraints can simplify the
associativity condition. We recall that a state $\sigma\in S$ can be written
as $\sigma=(R,v)$ as in eq. (1.2.2), where $R$ is an indecomposable
representation of the symmetry algebra and $v\in R$. Symmetry constraints lead
to equations for the OPE coefficient
$C_{\sigma_{1},\sigma_{2}}^{\sigma_{3}}(x_{1},x_{2})$, which have two general
properties:

  * •

The symmetry equations for the OPE coefficient are linear.

  * •

A symmetry equation may have terms with different values of the parameters
$v_{i}$ and $x_{i}$, but it involves only one value of each of the parameters
$R_{i}$, as by definition the action of a symmetry transformation on a state
$v_{i}\in R_{i}$ is another state in the same representation.

Therefore, an OPE coefficient can be decomposed into universal and model-
dependent data as

$\displaystyle
C_{\sigma_{1},\sigma_{2}}^{\sigma_{3}}(x_{1},x_{2})=\sum_{\lambda=1}^{N_{R_{1}R_{2}}^{R_{3}}}C_{R_{1},R_{2}}^{R_{3},\lambda}(x_{1},x_{2})f_{R_{1},R_{2}}^{R_{3},\lambda}(v_{1},v_{2},v_{3}|x_{1},x_{2})\
,$ (1.2.11)

where $C_{R_{1},R_{2}}^{R_{3},\lambda}(x_{1},x_{2})$ is a model-dependent
structure function, and the universal quantity
$f_{R_{1},R_{2}}^{R_{3},\lambda}(v_{i}|x_{j})$ is a solution of the symmetry
equations, which for given values of $R_{i}$ should be understood as equations
on the space of functions of $(v_{i},x_{j})$. The index $\lambda$ labels
elements of a basis of solutions of the symmetry equations. The number
$N_{R_{1}R_{2}}^{R_{3}}\in{\mathbb{N}}\cup\\{\infty\\}$ of independent
solutions is the fusion multiplicity, which can be encoded in the fusion rule

$\displaystyle R_{1}\times R_{2}=\sum_{R_{3}}N_{R_{1}R_{2}}^{R_{3}}R_{3}\ .$
(1.2.12)

The fusion rule is a representation-theoretic notion which encodes some of the
universal properties of the OPE. In particular, the field $V_{\sigma_{3}}$ can
appear in a $V_{\sigma_{1}}V_{\sigma_{2}}$ OPE only if
$N_{R_{1}R_{2}}^{R_{3}}\neq 0$.

It is actually possible to define an associative product of representations,
called the fusion product, such that eq. (1.2.12) is the decomposition of the
representation $R_{1}\times R_{2}$ into indecomposable representations. (See
[6] for the case of two-dimensional conformal field theory.) The fusion
product $R_{1}\times R_{2}$ is said to have nontrivial multiplicities if
$N_{R_{1}R_{2}}^{R_{3}}\geq 2$ for some $R_{3}$, and trivial multiplicities if
$N_{R_{1}R_{2}}^{R_{3}}\in\\{0,1\\}$, in which case the sum over $\lambda$ in
eq. (1.2.11) can be omitted. A field is called a simple current if the
corresponding representation $R$ is such that $R\times R^{\prime}$ is
indecomposable for any $R^{\prime}$, so that
$\sum_{R^{\prime\prime}}N_{RR^{\prime}}^{R^{\prime\prime}}=1$. The fusion
product may seem quite similar to a tensor product, but there is an essential
difference: while the tensor product is a purely algebraic notion, the
definition of the fusion product involves the state-field correspondence, and
therefore the space on which the field theory lives.

Using the decomposition (1.2.11) of the OPE coefficients, where we assume that
$f_{R_{1},R_{2}}^{R_{3},\lambda}(v_{i}|x_{j})$ is known, the associativity
equation (1.2.9) reduces to an equation for the structure function
$C_{R_{1},R_{2}}^{R_{3},\lambda}(x_{1},x_{2})$, and the sums over states
$\sigma_{s}$ and $\sigma_{t}$ reduce to sums over representations $R_{s}$ and
$R_{t}$. If the symmetry algebra is large enough or the model simple enough,
the number of representations in the spectrum can be small enough for the
associativity equation to become tractable. To summarize,

OPE coefficients obey linear equations from symmetry and quadratic equations
(1.2.9) from consistency. The numbers of OPE coefficients, consistency
equations, and terms in these equations, are determined by the dimension of
the spectrum and are therefore in general very large. With the help of
symmetry, these numbers can be reduced to the number of representations in the
spectrum.

####  1.2.4 So what is the Lagrangian of your model?

In order to construct and to solve conformal field theories, we will only use
the bootstrap approach. In quantum field theory, this method is much less
widely used than the Lagrangian method, which is based on functional integrals
over spaces of fields. In two-dimensional conformal field theory, however, the
bootstrap approach is particularly effective, while the Lagrangian method is
often needlessly complicated, lacking in rigor, or downright not applicable.
There are nevertheless cases when the Lagrangian method is useful, in
particular for

  1. 1.

suggesting, and sometimes proving, the existence of conformal field theories,

  2. 2.

and analyzing their classical limits, when such limits exist.

In the Lagrangian method, correlation functions are represented as functional
integrals

$\displaystyle\left\langle\prod_{i=1}^{n}V_{\sigma_{i}}(x_{i})\right\rangle=\int
D\phi\ e^{\int dxL[\phi](x)}\
\prod_{i=1}^{n}\tilde{V}_{\sigma_{i}}[\phi](x_{i})\ ,$ (1.2.13)

where

  * •

the integration variable is the fundamental field (or set of fields)
$\phi(x)$,

  * •

the Lagrangian $L[\phi](x)$ of the model is a functional of $\phi(x)$,

  * •

the functional $\tilde{V}_{\sigma_{i}}[\phi](x_{i})$ of $\phi(x)$ corresponds
to the field $V_{\sigma_{i}}(x_{i})$,

  * •

the integration measure $D\phi$ is usually characterized by axioms on
functional integrals.

An important advantage of the Lagrangian method is the existence of the
Lagrangian itself, a relatively simple object which is not tied to any
specific correlation function and encodes much information about the model.
The Lagrangian is the most common, but not the only object of this type: in
certain integrable models or supersymmetric quantum field theories, much
information is encoded in the geometry of a spectral curve.

In the Lagrangian method, we are thus in principle given all the correlation
functions of a specific model from the start. What we do not a priori know are
the symmetry properties of the model. Of course, we may try to choose the
Lagrangian in order to ensure that certain symmetries are present. However,
the symmetries of the model also depend on the integration measure. Axioms
requiring functional integrals to behave as ordinary integrals often leave no
choice for the integration measure, and it can happen that a transformation
leaves the Lagrangian invariant, but changes the integration measure. This
transformation is then called anomalous, and is not a symmetry of the model.

Such an anomaly can also be described as a symmetry of the classical theory,
which is not present in the corresponding quantum theory. This is because the
Lagrangian method can be interpreted as the quantization of the classical
theory which is defined by the Lagrangian. The classical theory consists in
the dynamics of the fundamental field, subject to the equations of motion
associated with the Lagrangian,

$\displaystyle\frac{\delta}{\delta\phi(x^{\prime})}\int dxL[\phi](x)=0\ ,$
(1.2.14)

where $\frac{\delta}{\delta\phi(x^{\prime})}$ is a functional derivative. The
integration measure then contains the additional information which is
necessary for building a quantum theory form the classical theory.

This contrasts with the bootstrap approach, which does not assume a classical
theory to exist. Models constructed with the bootstrap approach may
nevertheless have one or several classical limits. (See Section 4.1.4 for the
case of Liouville theory.)

###  1.3 Conformal symmetry in two dimensions

Our subject of two-dimensional conformal field theory is defined by the
existence of a space-time symmetry, namely local conformal invariance. We
consider a two-dimensional space, and assume it comes equipped with a metric.
The metric plays very little role in conformal field theory, because conformal
invariance will allow us to fix it once and for all. But we need the metric in
order to explain what conformal invariance is. We assume that the metric has
Euclidean signature, not considering the other option that it is Minkowskian.
The Euclidean signature brings important technical simplifications, and does
not necessarily prevent the observables from having an interpretation in
Minkowski space after analytic continuation.

####  1.3.1 Global conformal transformations

Let us assume that the metric is the flat two-dimensional Euclidean metric,
which we write in terms of complex coordinates $x=(z,\bar{z})$,

$\displaystyle ds^{2}=dzd\bar{z}\ .$ (1.3.1)

A transformation $(z,\bar{z})\mapsto(f(z),\overline{f(z)})$ leaves the metric
invariant, provided $ds^{2}=dfd\bar{f}$. Our metric is invariant under the
familiar translations and rotations,

$\displaystyle f_{\text{translation}}(z)$ $\displaystyle=z+b\ ,$ (1.3.2)
$\displaystyle f_{\text{rotation}}(z)$ $\displaystyle=az\ ,\ |a|=1\ ,$ (1.3.3)

where $a$ and $b$ are complex constants. Moreover, if we lift the restriction
$|a|=1$ in rotations, we obtain transformations which do not leave the metric
invariant, but rescale it by a factor of $|a|^{2}$. For $a\in{\mathbb{R}}$,
these transformations are called dilatations or scale transformations,

$\displaystyle f_{\text{dilatation}}(z)$ $\displaystyle=az\ ,\
a\in{\mathbb{R}}\ .$ (1.3.4)

While the laws of physics are (mostly) invariant under translations and
rotations, they are not invariant under dilatations, and the structures of
matter and of the universe are strongly scale-dependent. It is nevertheless
interesting to study scale-invariant models, for at least two reasons.
Firstly, some particular systems are invariant under scale transformations,
for instance certain materials at critical points. Secondly, by rescaling a
quantum field theory to very small or large scales in a controlled way, one
often reaches a scale-invariant theory (called a fixed point of the
renormalization group flow). That new theory can be nontrivial, and helpful
for the understanding of the original theory.

It may seem reasonable to study quantum field theories which are invariant
under translations, rotations and dilatations. But in order to have access to
the powerful methods and results which are the subject of this review, we
actually need much more symmetry assumptions. The surprising fact is that many
interesting systems do obey these further assumptions, which follow neither
from the invariances under translations, rotations and dilatations, nor in
many cases from physically compelling reasons. We will mostly justify these
further assumptions by formal considerations, which are not physically or
mathematically unavoidable.

A first formal consideration is that translations, rotations and dilatations
are all conformal transformations, that is transformations which preserve
angles, and therefore which rescale the metric by a real factor. But they are
not the only conformal transformations: another one is the inversion

$\displaystyle f_{\text{inversion}}(z)=\frac{1}{z}\ .$ (1.3.5)

It may seem unwise to take this transformation as a symmetry, as it is
singular at $z=0$. This is however not a problem, if we enlarge our space with
the addition of a point at $z=\infty$. This actually amounts to compactifying
the space, and working not on the complex plane but on the Riemann sphere.
This does not even prevent us from studying models on the complex plane, as
such models are equivalent to models on the sphere, with one field inserted at
$z=\infty$. Combining the inversion with translations, rotations and
dilatations, we obtain the group of the global conformal transformations of
the Riemann sphere,

$\displaystyle\boxed{f_{\text{global conformal}}(z)=\frac{az+b}{cz+d}}\ ,$
(1.3.6)

and this group is isomorphic to the group
$PSL_{2}({\mathbb{C}})=\frac{SL_{2}({\mathbb{C}})}{{\mathbb{Z}}_{2}}$ of the
complex matrices $g$ of size $2$ and determinant one where $g$ and $-g$ are
identified, through the map

$\displaystyle g=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in
SL_{2}({\mathbb{C}})\quad\longmapsto\quad f_{g}(z)=\frac{az+b}{cz+d}\ .$
(1.3.9)

(See Exercise 1.3.)

####  1.3.2 Local conformal transformations

Actually, any holomorphic function $h(z)$ defines a conformal transformation,

$\displaystyle f_{\text{local conformal}}(z)=h(z)\ ,\ h(z)\
\text{holomorphic}\ ,$ (1.3.10)

which transforms the metric into $dfd\bar{f}=|h^{\prime}(z)|^{2}dzd\bar{z}$,
and is therefore angle-preserving. One may object that the only holomorphic
functions on the sphere are the functions $f_{g}$ (1.3.9) which encode the
global conformal transformations. And indeed, any further conformal
transformation must have singularities, and cannot be one-to-one. We could be
content with considering transformations which would be defined only on some
region. We will adopt a different point of view, and interpret a singularity
of $h(z)$ at $z=z_{0}$ as meaning that a field $V_{\sigma}(z_{0})$ transforms
non-trivially. This makes sense not only with a correlation function
$\left\langle V_{\sigma}(z_{0})\cdots\right\rangle$ which manifestly involves
a field at $z=z_{0}$, but also with arbitrary correlation functions. We can
indeed always assume the presence at $z_{0}$ of a trivial field, called the
identity field, which can however become non-trivial after a conformal
transformation is performed. To summarize,

Global conformal transformations only move fields around, while local
conformal transformations also modify them.

So, we assume that the symmetries include the infinitely many independent
local conformal transformations, in addition to the global conformal
transformations, which are parametrized by the six real coordinates of
$PSL_{2}({\mathbb{C}})$. Let us now describe the mathematical structure of the
resulting space of all conformal transformations. This space is an infinite-
dimensional Lie group, which must include $PSL_{2}({\mathbb{C}})$. For
technical simplicity, we will instead describe the associated Lie algebra of
infinitesimal transformations. The structure of this algebra is worked out by
considering transformations close to the identity,

$\displaystyle
f_{\epsilon}(z)=z+\epsilon(z)\quad\text{with}\quad\epsilon(z)=\sum_{n\in{\mathbb{Z}}}\epsilon_{n}z^{n+1}\
.$ (1.3.11)

To each transformation $f_{\epsilon}$ we associate an infinitesimal
differential operator $D_{\epsilon}$ such that for any function $F(z,\bar{z})$
we have

$\displaystyle
F(f_{\epsilon}(z),\overline{f_{\epsilon}(z)})=\left(1-D_{\epsilon}-\overline{D_{\epsilon}}\right)F(z,\bar{z})+O(\epsilon^{2})\
,$ (1.3.12)

and we find

$\displaystyle
D_{\epsilon}=\sum_{n\in{\mathbb{Z}}}\epsilon_{n}\ell_{n}\quad\text{where}\quad\boxed{\ell_{n}=-z^{n+1}{\frac{\partial}{\partial
z}}}\ .$ (1.3.13)

The algebra of local conformal transformations, called the Witt algebra, has
the generators $(\ell_{n})_{n\in{\mathbb{Z}}}$ and the commutation relations

$\displaystyle\boxed{[\ell_{n},\ell_{m}]=(n-m)\ell_{n+m}}\ .$ (1.3.14)

The generators $(\ell_{-1},\ell_{0},\ell_{1})$ correspond to global
transformations, and the rest to local transformations. (See Exercise 1.4.)

The assumption of local conformal symmetry drastically restricts the
dependence of the theory on the metric. Let us restrict our attention to
theories on compact Riemann surfaces, where a Riemann surface is a two-
dimensional orientable smooth manifold. The topology of a compact Riemann
surface is characterized by a natural number $g$ called the genus, which is
the number of holes. In the case $g=0$, the uniformization theorem states that
all simply-connected compact Riemann surfaces are conformally equivalent. This
implies that the observables of a conformal field theory on any such manifold
can be obtained from their values on the sphere by a change of coordinates. In
the next simplest case $g=1$, manifolds are not all conformally equivalent,
but any manifold with $g=1$ is conformally equivalent to a torus
$\frac{{\mathbb{C}}}{{\mathbb{Z}}+\tau{\mathbb{Z}}}$ for some value of the
complex structure modulus $\tau\in{\mathbb{C}}$. Similarly, for $g\geq 2$, a
conformal field theory depends on the metric only through $3g-3$ complex
structure moduli.

Local conformal symmetry is a natural symmetry assumption in string theory,
where the world-sheet metric is an unphysical variable, and in two-dimensional
gravity, where we expect background independence. The assumption is less
natural in models of condensed-matter physics, where only flat metrics are
usually considered. In any case, theories with local conformal symmetry, which
we will simply call conformal field theories, should be thought of as
exceptional points in the vast and unchartered space of the global conformal
field theories. We will encounter a global conformal field theory in Section
4.1.4, namely the light asymptotic limit of Liouville theory. Another example
is the long-range Ising model.

####  1.3.3 The Virasoro algebra

In a quantum field theory with conformal symmetry, the spectrum is not a
representation of the Witt algebra, but of its central extension, called the
Virasoro algebra $\mathfrak{V}$. This is a general phenomenon in quantum
theories: since states are defined up to scalar factors, the action of a
symmetry group on space, which obeys $g\cdot(g^{\prime}\cdot
x)=(gg^{\prime})\cdot x$, need only translate into a projective action on
states, which obeys
$g\cdot(g^{\prime}\cdot|\sigma\rangle)=\lambda(g,g^{\prime})(gg^{\prime})\cdot|\sigma\rangle$
for some scalar factor $\lambda(g,g^{\prime})$. And a projective action of a
symmetry algebra is equivalent to an action of the corresponding centrally-
extended algebra. So, the Virasoro algebra is defined by the generators
$(L_{n})_{n\in{\mathbb{Z}}}$, which correspond to the generators
$(\ell_{n})_{n\in\mathbb{Z}}$ of the Witt algebra, and the centrally-extended
commutation relations

$\displaystyle\boxed{[L_{n},L_{m}]=(n-m)L_{n+m}+\frac{c}{12}(n-1)n(n+1)\delta_{n+m,0}}\
.$ (1.3.15)

The parameter $c$ is called the central charge, and the central term
$\frac{c}{12}(n-1)n(n+1)\delta_{n+m,0}$ should be understood as the
coefficient of an extra generator, which commutes with all $L_{n}$. It can be
checked that these commutation relations define a Lie algebra, and that the
Virasoro algebra is the only central extension of the Witt algebra. (See
Exercise 1.5.) Notice that the presence of the central term does not affect
the generators $L_{-1},L_{0},L_{1}$ of global conformal transformations.

We will still make one further symmetry assumption. Until now, the coordinates
$z$ and $\bar{z}$ of our two-dimensional space were complex conjugates, so
that any transformation which acted on $z$ automatically acted in a conjugate
way on $\bar{z}$. Now it is possible to consider $z$ and $\bar{z}$ as
independent coordinates, and allow transformations to act separately on $z$
and $\bar{z}$. At the level of the conformal transformations (1.3.12), this
means considering the generators $\ell_{n}$ and their complex conjugates
$\bar{\ell}_{n}$ as independent. Generalizing to a nonzero central charge, we
therefore assume that the symmetry algebra has

  * •

left-moving, chiral or holomorphic generators $L_{n}$, and

  * •

right-moving, anti-chiral or anti-holomorphic generators $\bar{L}_{n}$,

with $[L_{n},\bar{L}_{m}]=0$. The generators $(L_{n})_{n\in{\mathbb{Z}}}$ and
$(\bar{L}_{n})_{n\in{\mathbb{Z}}}$ are supposed to span two distinct Virasoro
algebras, which we assume have the same central charge, called the central
charge of the model. We recapitulate these symmetry algebras in the following
table:

(1.3.20)

Considering $z$ and $\bar{z}$ as independent amounts to complexifying our two-
dimensional real space ${\mathbb{C}}={\mathbb{R}}^{2}$ into the two-
dimensional complex space ${\mathbb{C}}^{2}$. We must however insist that the
correlation functions, which are the $z$-dependent observables of the theory,
do live on the original space and not on ${\mathbb{C}}^{2}$. Functions such as
$z^{\frac{1}{2}}$ may appear at intermediate steps of calculations, but
correlation functions can only involve single-valued functions such as $|z|$.
So, although the space ${\mathbb{C}}^{2}$ and the symmetry algebra
$\mathfrak{V}\times\overline{\mathfrak{V}}$ have factorized forms, the theory
itself is not the mere juxtaposition of two independent theories. We thus
reach a definition of

Two-dimensional conformal field theory: the study of two-dimensional quantum
field theories which possess a $\mathfrak{V}\times\overline{\mathfrak{V}}$
symmetry algebra such that $L_{n}$ and $\bar{L}_{n}$ generate conformal
transformations.

The assumption about the geometrical interpretation of $L_{n}$ and
$\bar{L}_{n}$ is very important, as it is the only link between the symmetry
algebra and the two-dimensional space. Some conformal field theories have
several Virasoro symmetry algebras, only one of which corresponds to conformal
symmetry. In the free boson theory, it is even possible to find Virasoro
algebras with all possible values of the central charge. (See Section 4.1.1.)

###  1.4 Basic consequences of conformal symmetry

####  1.4.1 Structure of the spectrum

We know that the spectrum $S$ must be a representation of the
$\mathfrak{V}\times\overline{\mathfrak{V}}$ symmetry algebra, and let us
further assume that $S$ is a combination of factorized representations of the
type $R\otimes\bar{R^{\prime}}$, where $R$ and $R^{\prime}$ are
representations of $\mathfrak{V}$, and the bar in $\overline{\mathfrak{V}}$
and $\bar{R^{\prime}}$ is only meant to distinguish right-moving from left-
moving objects. Then the decomposition (1.2.1) takes the form

$\displaystyle\boxed{S=\bigoplus_{(R,R^{\prime})\in
Rep(\mathfrak{V})^{2}}m_{R,R^{\prime}}R\otimes\bar{R^{\prime}}}\ ,$ (1.4.1)

where $Rep(\mathfrak{V})$ is some set of indecomposable representations of the
Virasoro algebra. We introduce two important special cases:

  * •

rational models, where that sum involves only finitely many nonvanishing
terms,

  * •

diagonal models, where $m_{R,R^{\prime}}\neq 0\Rightarrow R=R^{\prime}$, so
that $S=\bigoplus_{R\in Rep(\mathfrak{V})}m_{R}R\otimes\bar{R}$.

Let us say that the indecomposable representation $R$ of $\mathfrak{V}$
appears in the spectrum if there is some $R^{\prime}$ such that
$m_{R,R^{\prime}}\neq 0$. We introduce the important axiom of

closure under fusion: if the representations $R_{1}$ and $R_{2}$ appear in the
spectrum, then any representation $R_{3}$ with a non-vanishing fusion
multiplicity $N_{R_{1}R_{2}}^{R_{3}}\neq 0$ also appears in the spectrum.

In other words, we assume that if Virasoro symmetry allows a representation of
$\mathfrak{V}$ to appear in an OPE, then that representation does appear.
There are two similar-looking axioms, which should not be adopted:

  * •

Closure under fusion of $\mathfrak{V}\times\overline{\mathfrak{V}}$
representations, instead of $\mathfrak{V}$ representations, would rule out
most non-trivial models, and in particular diagonal models.

  * •

Closure under tensor product, instead of fusion, would not make sense, as the
tensor product of two representations of $\mathfrak{V}_{c}$ is a
representation of $\mathfrak{V}_{2c}$ – central charges behave additively in a
tensor product.

In order to characterize the action of the symmetry algebra on the spectrum,
we need not only decompose the spectrum into representations of that algebra,
but also know how the symmetry generators $L_{n}$ behave under the scalar
product, in other words what are their hermitian conjugates $L_{n}^{\dagger}$.
Let us first consider the case of the generators $\ell_{n}$ (1.3.13) of the
Witt algebra. As a differential operator, $\ell_{n}$ acts on the space of
analytic functions on ${\mathbb{C}}-\\{0\\}$, whose basis is
$\\{z^{p}\\}_{p\in{\mathbb{Z}}}$. The natural scalar product on this space is

$\displaystyle\langle f|g\rangle=-\frac{1}{2\pi i}\int_{0}^{2\pi}d\theta\
\overline{f(e^{i\theta})}{\frac{\partial}{\partial\theta}}g(e^{i\theta})\quad\Rightarrow\quad\langle
z^{p}|z^{q}\rangle=-p\delta_{p,q}\ .$ (1.4.2)

This scalar product, which is a fixed radius integral, comes from the idea of
radial quantization: the radial direction is considered as the time direction,
and the dilatation operator $\ell_{0}$ as the Hamiltonian, which is hermitian
i.e. $\ell_{0}^{\dagger}=\ell_{0}$. We also have
$\ell_{n}^{\dagger}=\ell_{-n}$ (see Exercise 1.6), and we therefore assume

$\displaystyle\boxed{L^{\dagger}_{n}=L_{-n}}\ .$ (1.4.3)

Conjugates of product operators $\prod_{i}L_{n_{i}}$ can then be deduced using
the formula

$\displaystyle(AB)^{\dagger}=B^{\dagger}A^{\dagger}\ ,$ (1.4.4)

which follows from the definition (1.1.7) of the hermitian conjugate. At the
level of the commutator $[A,B]=AB-BA$, this implies the rule
$[A,B]^{\dagger}=-[A^{\dagger},B^{\dagger}]$. Using this rule, it is possible
to check that eq. (1.4.3) is compatible with the commutation relations
(1.3.15) provided

$\displaystyle c\in{\mathbb{R}}\ .$ (1.4.5)

The decomposition of the spectrum $S$ as a sum of
$\mathfrak{V}\times\overline{\mathfrak{V}}$ representations respects the
scalar product, and two different irreducible representations belonging to the
spectrum must be orthogonal to each other. So the question of the positivity
of the scalar product can be examined within each representation. A
representation is called unitary if it has a positive definite scalar product
such that $L^{\dagger}_{n}=L_{-n}$, and a model whose spectrum is unitary is
itself called unitary.

The interpretation of $L_{0}$ as a Hamiltonian in radial quantization suggests
that we study the eigenvalues of $L_{0}$ when acting on the spectrum $S$ of
the model. The $L_{0}$-eigenvalue of an $L_{0}$-eigenvector is called its
conformal dimension or conformal weight. Interpreting conformal dimensions as
energies of states suggests that they must be bounded from below. Moreover, in
any indecomposable representation of the Virasoro algebra, all conformal
dimensions differ by integers. (See Exercise 1.7.) This follows from the fact
that if two basis states $v_{1}$ and $v_{2}$ are related by the action of
Virasoro generators,

$\displaystyle v_{2}=\left(\prod_{i}L_{n_{i}}\right)v_{1}\ ,$ (1.4.6)

and if $v_{1}$ is an $L_{0}$-eigenstate with conformal dimension $\Delta_{1}$,
then $v_{2}$ is also an eigenstate with the conformal dimension

$\displaystyle\Delta_{2}=\Delta_{1}-\sum_{i}n_{i}\ ,$ (1.4.7)

as a consequence of the Virasoro algebra’s commutation relations (1.3.15).

Finally, let us illustrate the power of local conformal symmetry by thinking
of a model in terms of global conformal symmetry only. We then decompose each
Virasoro representation $R$ into representations of the $\mathfrak{sl}_{2}$
algebra of global conformal transformations with generators
$(L_{0},L_{1},L_{-1})$,

$\displaystyle R=\bigoplus_{n\in{\mathbb{N}}}m_{R,n}D_{\Delta+n}\ ,$ (1.4.8)

where $m_{R,n}\in{\mathbb{N}}$ is a multiplicity, $\Delta$ is the lowest
$L_{0}$-eigenvalue of the indecomposable Virasoro representation $R$, and
$D_{\Delta+n}$ is a representation of $\mathfrak{sl}_{2}$ whose lowest
$L_{0}$-eigenvalue is $\Delta+n$. (See Exercise 1.8.) The above decomposition
is universal data of the Virasoro algebra, and model-dependent information
from the point of view of global conformal symmetry. This shows how
constraining local conformal symmetry is, and how much more freedom there is
in the spectrum of global conformal field theories. In particular, in a global
conformal field theory, there is no reason for the spectrum to contain series
of $\mathfrak{sl}_{2}$ representations, whose lowest $L_{0}$-eigenvalues
differ by integers.

####  1.4.2 Conformal bootstrap

The application of the bootstrap approach to conformal field theories, called
the conformal bootstrap, is particularly powerful in two dimensions. This is
because the algebra of conformal transformations is infinite dimensional, so
that the spectrum can be decomposed into a small number of large
representations. Moreover, the factorization of the symmetry algebra into two
copies of the Virasoro algebra implies that symmetry equations factorize into
$z$-dependent equations from the left-moving Virasoro algebra, and
$\bar{z}$-dependent equations from the right-moving Virasoro algebra. It
follows that all universal quantities, for example
$f_{R_{1},R_{2}}^{R_{3},\lambda}(v_{i}|x_{j})$ from (1.2.11), can be
factorized into $z$-dependent and $\bar{z}$-dependent functions. This is
called holomorphic factorization, where the term “holomorphic” should be
understood as locally holomorphic: the $z$-dependent factors are supposed to
be $\bar{z}$-independent, but they can have singularities such as poles and
branch cuts. We will see in more detail in Section 2.3.3 how holomorphic
factorization simplifies the condition (1.2.9) of OPE associativity.

In two-dimensional conformal field theory, the assumption of the existence of
an OPE has a natural generalization: the assumption that on any closed contour
$C$, one can insert a decomposition of the identity operator,

$\displaystyle\mathbf{1}=\sum_{\sigma\in S}|\sigma\rangle\langle\sigma|\ ,$
(1.4.9)

where the sum runs over an orthonormal basis of the spectrum. Let us
schematically explain why this implies the existence of an OPE. Considering a
contour $C$ around the positions $x_{1},x_{2}$ of two fields
$V_{\sigma_{1}}(x_{1}),V_{\sigma_{2}}(x_{2})$, and inserting the decomposition
of the identity, we obtain

$\displaystyle V_{\sigma_{1}}(x_{1})V_{\sigma_{2}}(x_{2})=\sum_{\sigma\in
S}\left\langle\sigma\left|V_{\sigma_{1}}(x_{1})V_{\sigma_{2}}(x_{2})\right.\right\rangle
V_{\sigma}(x_{2})\ ,$ (1.4.10)

where we used the state-field correspondence in a cavalier way: we replaced
the field $V_{\sigma_{1}}(x_{1})V_{\sigma_{2}}(x_{2})$ with the corresponding
state $|V_{\sigma_{1}}(x_{1})V_{\sigma_{2}}(x_{2})\rangle$, and the state
$|\sigma\rangle$ with the corresponding field $V_{\sigma}(x_{2})$. This can be
drawn as follows:

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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$V_{\sigma}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{102.42923pt}{0.0pt}\pgfsys@moveto{128.03653pt}{0.0pt}\pgfsys@curveto{128.03653pt}{14.1427pt}{116.57193pt}{25.6073pt}{102.42923pt}{25.6073pt}\pgfsys@curveto{88.28653pt}{25.6073pt}{76.82193pt}{14.1427pt}{76.82193pt}{0.0pt}\pgfsys@curveto{76.82193pt}{-14.1427pt}{88.28653pt}{-25.6073pt}{102.42923pt}{-25.6073pt}\pgfsys@curveto{116.57193pt}{-25.6073pt}{128.03653pt}{-14.1427pt}{128.03653pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{102.42923pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{{}}{}\pgfsys@moveto{120.35426pt}{12.80365pt}\pgfsys@moveto{121.70425pt}{12.80365pt}\pgfsys@curveto{121.70425pt}{13.54922pt}{121.09984pt}{14.15364pt}{120.35426pt}{14.15364pt}\pgfsys@curveto{119.60869pt}{14.15364pt}{119.00427pt}{13.54922pt}{119.00427pt}{12.80365pt}\pgfsys@curveto{119.00427pt}{12.05807pt}{119.60869pt}{11.45366pt}{120.35426pt}{11.45366pt}\pgfsys@curveto{121.09984pt}{11.45366pt}{121.70425pt}{12.05807pt}{121.70425pt}{12.80365pt}\pgfsys@closepath\pgfsys@moveto{120.35426pt}{12.80365pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{103.96579pt}{10.50533pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$V_{\sigma_{1}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{{}}
{}{}{}{}{}{}{}{}
}{{}}{}\pgfsys@moveto{120.35426pt}{-12.80365pt}\pgfsys@moveto{121.70425pt}{-12.80365pt}\pgfsys@curveto{121.70425pt}{-12.05807pt}{121.09984pt}{-11.45366pt}{120.35426pt}{-11.45366pt}\pgfsys@curveto{119.60869pt}{-11.45366pt}{119.00427pt}{-12.05807pt}{119.00427pt}{-12.80365pt}\pgfsys@curveto{119.00427pt}{-13.54922pt}{119.60869pt}{-14.15364pt}{120.35426pt}{-14.15364pt}\pgfsys@curveto{121.09984pt}{-14.15364pt}{121.70425pt}{-13.54922pt}{121.70425pt}{-12.80365pt}\pgfsys@closepath\pgfsys@moveto{120.35426pt}{-12.80365pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{103.96579pt}{-15.10197pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$V_{\sigma_{2}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{{}}
{}{}{}{}{}{}{}{}
}{{}}{}\pgfsys@moveto{80.66306pt}{0.0pt}\pgfsys@moveto{82.01305pt}{0.0pt}\pgfsys@curveto{82.01305pt}{0.74557pt}{81.40863pt}{1.34999pt}{80.66306pt}{1.34999pt}\pgfsys@curveto{79.91748pt}{1.34999pt}{79.31306pt}{0.74557pt}{79.31306pt}{0.0pt}\pgfsys@curveto{79.31306pt}{-0.74557pt}{79.91748pt}{-1.34999pt}{80.66306pt}{-1.34999pt}\pgfsys@curveto{81.40863pt}{-1.34999pt}{82.01305pt}{-0.74557pt}{82.01305pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{80.66306pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{84.19606pt}{-2.81389pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$V_{\sigma}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (1.4.11)

We thus obtain an OPE whose coefficient
$\left\langle\sigma\left|V_{\sigma_{1}}(x_{1})V_{\sigma_{2}}(x_{2})\right.\right\rangle$
is essentially a three-point function. We will see the relation between OPE
coefficients and three-point functions in more detail in Section 2.2.4.

Inserting the decomposition of the identity (1.4.9) on a contour $C$ amounts
to cutting the space into two pieces, and in each piece replacing the
resulting hole with a puncture – an insertion of a field $V_{\sigma}(x)$ for
some $x$. If we view the space near the contour $C$ as a cylinder, this can be
drawn as follows:

$\displaystyle\leavevmode\hbox to154.04pt{\vbox
to51.61pt{\pgfpicture\makeatletter\hbox{\hskip 25.8073pt\lower-25.8073pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{25.6073pt}\pgfsys@curveto{19.97221pt}{25.6073pt}{19.97221pt}{-25.6073pt}{0.0pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{25.6073pt}\pgfsys@curveto{-19.97221pt}{25.6073pt}{-19.97221pt}{-25.6073pt}{0.0pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} ; {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{51.21461pt}{25.6073pt}\pgfsys@curveto{71.18684pt}{25.6073pt}{71.18684pt}{-25.6073pt}{51.21461pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{51.21461pt}{25.6073pt}\pgfsys@curveto{31.24239pt}{25.6073pt}{31.24239pt}{-25.6073pt}{51.21461pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} ; {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{102.42923pt}{25.6073pt}\pgfsys@curveto{122.40146pt}{25.6073pt}{122.40146pt}{-25.6073pt}{102.42923pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{102.42923pt}{25.6073pt}\pgfsys@curveto{82.457pt}{25.6073pt}{82.457pt}{-25.6073pt}{102.42923pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} ; {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.79744pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$C$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{25.6073pt}\pgfsys@lineto{102.42923pt}{25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{-25.6073pt}\pgfsys@lineto{102.42923pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{25.6073pt}\pgfsys@lineto{-25.6073pt}{25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-25.6073pt}\pgfsys@lineto{-25.6073pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{102.42923pt}{25.6073pt}\pgfsys@lineto{128.03654pt}{25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{102.42923pt}{-25.6073pt}\pgfsys@lineto{128.03654pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad=\ \
\sum_{\sigma\in S}\quad\leavevmode\hbox to154.04pt{\vbox
to51.61pt{\pgfpicture\makeatletter\hbox{\hskip 25.8073pt\lower-25.8073pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{25.6073pt}\pgfsys@curveto{19.97221pt}{25.6073pt}{19.97221pt}{-25.6073pt}{0.0pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{25.6073pt}\pgfsys@curveto{-19.97221pt}{25.6073pt}{-19.97221pt}{-25.6073pt}{0.0pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} ; {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{102.42923pt}{25.6073pt}\pgfsys@curveto{122.40146pt}{25.6073pt}{122.40146pt}{-25.6073pt}{102.42923pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{102.42923pt}{25.6073pt}\pgfsys@curveto{82.457pt}{25.6073pt}{82.457pt}{-25.6073pt}{102.42923pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} ; {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{25.6073pt}\pgfsys@lineto{-25.6073pt}{25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-25.6073pt}\pgfsys@lineto{-25.6073pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{102.42923pt}{25.6073pt}\pgfsys@lineto{128.03654pt}{25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{102.42923pt}{-25.6073pt}\pgfsys@lineto{128.03654pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}{}\pgfsys@moveto{19.20547pt}{25.6073pt}\pgfsys@curveto{33.34818pt}{25.6073pt}{44.81277pt}{14.1427pt}{44.81277pt}{0.0pt}\pgfsys@curveto{44.81277pt}{-14.1427pt}{33.34818pt}{-25.6073pt}{19.20547pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}{}\pgfsys@moveto{83.22375pt}{-25.6073pt}\pgfsys@curveto{69.08105pt}{-25.6073pt}{57.61646pt}{-14.1427pt}{57.61646pt}{0.0pt}\pgfsys@curveto{57.61646pt}{14.1427pt}{69.08105pt}{25.6073pt}{83.22375pt}{25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{25.6073pt}\pgfsys@lineto{19.20547pt}{25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{-25.6073pt}\pgfsys@lineto{19.20547pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{83.22375pt}{25.6073pt}\pgfsys@lineto{102.42923pt}{25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{83.22375pt}{-25.6073pt}\pgfsys@lineto{102.42923pt}{-25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{{}}{}\pgfsys@moveto{61.45757pt}{0.0pt}\pgfsys@moveto{62.80756pt}{0.0pt}\pgfsys@curveto{62.80756pt}{0.74557pt}{62.20314pt}{1.34999pt}{61.45757pt}{1.34999pt}\pgfsys@curveto{60.71199pt}{1.34999pt}{60.10757pt}{0.74557pt}{60.10757pt}{0.0pt}\pgfsys@curveto{60.10757pt}{-0.74557pt}{60.71199pt}{-1.34999pt}{61.45757pt}{-1.34999pt}\pgfsys@curveto{62.20314pt}{-1.34999pt}{62.80756pt}{-0.74557pt}{62.80756pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{61.45757pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{64.99057pt}{-2.81389pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$V_{\sigma}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{{}}
{}{}{}{}{}{}{}{}
}{{}}{}\pgfsys@moveto{40.97165pt}{0.0pt}\pgfsys@moveto{42.32164pt}{0.0pt}\pgfsys@curveto{42.32164pt}{0.74557pt}{41.71722pt}{1.34999pt}{40.97165pt}{1.34999pt}\pgfsys@curveto{40.22607pt}{1.34999pt}{39.62166pt}{0.74557pt}{39.62166pt}{0.0pt}\pgfsys@curveto{39.62166pt}{-0.74557pt}{40.22607pt}{-1.34999pt}{40.97165pt}{-1.34999pt}\pgfsys@curveto{41.71722pt}{-1.34999pt}{42.32164pt}{-0.74557pt}{42.32164pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{40.97165pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.18318pt}{-2.81389pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$V_{\sigma}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (1.4.12)

In the case of a topologically trivial contour $C$, this equality amounts to
using OPEs for replacing all fields within $C$ with one field $V_{\sigma}(x)$.
If on the other hand $C$ is a non-contractible cycle, then this equality is
nontrivial even in the absence of fields. If we studied a model on a torus
with complex modulus $\tau$, then we could obtain a nontrivial constraint on
the spectrum by applying eq. (1.4.12) to the partition function (the zero-
point function) using two topologically inequivalent contours. The study of
constraints of this type is called the modular bootstrap. We will however only
consider topologically trivial spaces (that is, surfaces of genus $g=0$),
without assuming our models to be defined on spaces of nonzero genus. For our
purposes, the axiom of the existence of the OPE will therefore be general
enough.

####  1.4.3 Sketching the space of simple models

After discussing the general properties of two-dimensional conformal field
theory, we will now describe some models which are known to satisfy the
consistency conditions of the conformal bootstrap, and which are simple in the
sense that their spectrums decompose into reasonable numbers of
representations. These simple models also turn out to be the most commonly
encountered in applications.

We introduce the following models, in whose spectrums representations of
$\mathfrak{V}\times\overline{\mathfrak{V}}$ appear with multiplicities no
higher than one:

  * •

For any value $c\geq 1$ of the central charge, Liouville theory is
characterized as a model whose spectrum is diagonal and made of a continuum of
unitary representations. Under certain assumptions, this model can be shown to
be unique. Liouville theory can be analytically continued to complex values of
$c$, but is then no longer unitary.

  * •

A-series minimal models are characterized as diagonal, rational models. Such a
model exists only if the central charge is of the type

$\displaystyle
c=1-6\frac{(p-q)^{2}}{pq}\quad\text{with}\quad\left\\{\begin{array}[]{l}p,q\
\text{coprime integers}\ ,\\\ 2\leq p<q\ ,\end{array}\right.$ (1.4.15)

so that $0\leq c<1$. There are $\frac{(p-1)(q-1)}{2}$ distinct representations
in the spectrum. The model is unitary if and only if $q=p+1$. Then the central
charge is $c=1-\frac{6}{p(p+1)}=0,\frac{1}{2},\frac{7}{10},\cdots$. The
unitary minimal model with $c=\frac{1}{2}$ is called the Ising model.

On the $c\in{\mathbb{R}}$ line, the unitary diagonal models are therefore
located as follows:

(1.4.16)

Other simple models include:

  * •

the linear dilaton theory, which like Liouville theory has a diagonal,
continuous spectrum and is defined for any $c\in{\mathbb{C}}$, being unitary
if $c\geq 1$. The difference with Liouville theory is that the symmetry
algebra is larger than $\mathfrak{V}\times\overline{\mathfrak{V}}$, so that
representations of $\mathfrak{V}\times\overline{\mathfrak{V}}$ appear with
multiplicity two in the spectrum, and the correlation functions are much
simpler.

  * •

(Virasoro) minimal models are by definition rational models, and beyond the
A-series there exist two other series, called the D- and E-series, which are
not diagonal. The corresponding values of the central charge are still of the
type of eq. (1.4.15).

Therefore, the properties of conformal field theories, starting with their
very existence, crucially depend on the value of the central charge $c$ of the
underlying Virasoro algebra. In the case of unitary theories, we observe that
the size of the spectrum (as measured by the number of representations)
increases with $c$, from the $c=0$ minimal model whose spectrum contains only
one representation, to $c\geq 1$ Liouville theory whose spectrum contains
infinitely many. The central charge is actually additive: given two theories
with spectrums $S$ and $S^{\prime}$ and central charges $c$ and $c^{\prime}$,
there is a natural product theory with central charge $c+c^{\prime}$ and
spectrum $S\otimes S^{\prime}$. Moreover, for any rational theory which is
consistent on a torus, the behaviour of the number $N(\Delta)$ of states with
conformal dimension $\Delta$ is given by

$\displaystyle\text{Cardy's formula:}\qquad\log
N(\Delta)\underset{\Delta\to\infty}{\propto}\sqrt{c\Delta}\ ,$ (1.4.17)

which quantitatively shows how the central charge $c$ controls the size of the
spectrum. It can be checked that Cardy’s formula is compatible with the
additivity of the central charge. (See Exercise 1.9.)

###  1.5 Exercises

######  Exercise 1.1 (Irreducible and indecomposable representations)

Consider the algebra with one generator $a$ such that $a^{2}=0$. Consider a
two-dimensional representation $R$ with a basis $(v_{1},v_{2})$ such that
$av_{1}=v_{2}$. Write the matrix of size two which corresponds to $a$ in the
basis $(v_{1},v_{2})$. Show that $R$ is indecomposable, but not irreducible.

######  Exercise 1.2 (Associativity in a tensor category)

Consider a set of objects $R_{i}$ with two binary operations $\oplus$
(addition) and $\otimes$ (multiplication), such that

$\displaystyle R_{i}\otimes R_{j}=\bigoplus_{k}c_{ij}^{k}R_{k}\ ,$ (1.5.1)

where $c_{ij}^{k}$ is a number. Assuming the objects $R_{i}$ are linearly
independent, show that the associativity of the multiplication $\otimes$
amounts to a quadratic condition on the coefficients $c_{ij}^{k}$,

$\displaystyle\sum_{m}c_{ij}^{m}c_{mk}^{\ell}=\sum_{n}c_{in}^{\ell}c_{jk}^{n}\
.$ (1.5.2)

######  Exercise 1.3 (Group of global conformal transformations)

Show that the map (1.3.9) is a group morphism whose kernel is the center of
$SL_{2}({\mathbb{C}})$. Show that the group of global conformal
transformations of the sphere is isomorphic to $PSL_{2}({\mathbb{C}})$.

######  Exercise 1.4 (Algebra of infinitesimal global conformal
transformations)

Show that the generators $(\ell_{-1},\ell_{0},\ell_{1})$ of conformal
transformations defined in eq. (1.3.13) correspond to infinitesimal global
conformal transformations. To do this, expand the map $f_{g}$ of eq. (1.3.9)
near the identity, following eq. (1.3.11).

######  Exercise 1.5 (Central term of the Virasoro algebra)

Show that the commutation relations (1.3.15) of the Virasoro algebra define a
Lie algebra, which is the only possible central extension of the Witt algebra.
To do this, consider commutation relations of the type

$\displaystyle[L_{n},L_{m}]=(n-m)L_{n+m}+f(n,m)C\quad,\quad[C,L_{n}]=0\ ,$
(1.5.3)

where $C$ is a central generator and $f(n,m)$ an arbitrary antisymmetric
function. Show that these commutation relations obey the Jacobi identities if
and only if

$\displaystyle f(n,m)=\lambda(n^{3}-n)\delta_{m+n,0}+(n-m)g(n+m)\ ,$ (1.5.4)

for some constant $\lambda$ and function $g(n)$. Observe that the second term
can be absorbed by the redefinition $L_{n}\mapsto L_{n}-g(n)C$.

######  Exercise 1.6 (Hermitian conjugates of conformal transformations)

Show that the generators $\ell_{n}$ (1.3.13) of infinitesimal conformal
transformations satisfy $\ell_{n}^{\dagger}=\ell_{-n}$ with respect to the
scalar product (1.4.2). Show that the states $z^{p>0}$ have negative norms and
also negative energies ($\ell_{0}$-eigenvalues).

######  Exercise 1.7 (Spectrum of $L_{0}$ in an indecomposable representation)

Show that the eigenvalues of $L_{0}$ in an indecomposable representation $R$
of the Virasoro algebra differ by integers. To do this, use the decomposition
$R=\oplus_{\Delta}R_{\Delta}$ where $R_{\Delta}$ is the generalized eigenspace
associated to the eigenvalue $\Delta$. Given any value of $\Delta$, show that
$\oplus_{n\in{\mathbb{Z}}}R_{\Delta+n}$ is a subrepresentation, and conclude.

######  Exercise 1.8 (Comparing Virasoro and $\mathfrak{sl}_{2}$
representations)

Consider the $\mathfrak{sl}_{2}$ algebra with generators
$(L_{0},L_{1},L_{-1})$, and a vector $|v\rangle$ such that $L_{1}|v\rangle=0$
and $L_{0}|v\rangle=\Delta|v\rangle$. Repeatedly acting with $L_{-1}$ on
$|v\rangle$, show that you obtain a representation of $\mathfrak{sl}_{2}$,
where the eigenvalues of $L_{0}$ belong to $\Delta+{\mathbb{N}}$, and that
this representation is irreducible if $\Delta\notin-\frac{1}{2}\mathbb{N}$. If
$R$ is an indecomposable representation of the Virasoro algebra where $L_{0}$
is diagonalizable with eigenvalues in $\Delta_{0}+\mathbb{N}$ for
$\Delta_{0}\notin\frac{1}{2}\mathbb{Z}$, show that $R$ can be decomposed into
irreducible $\mathfrak{sl}_{2}$ representations as in eq. (1.4.8).

######  Exercise 1.9 (Number of states and additivity of the central charge)

Show that Cardy’s formula (1.4.17) is compatible with the additivity of the
central charge. To do this, construct the action of the Virasoro algebra on
the spectrum $S\otimes S^{\prime}$ of the product theory by

$\displaystyle L_{n}^{\text{product}}=L_{n}\otimes 1^{\prime}+1\otimes
L_{n}^{\prime}\ ,$ (1.5.5)

where $L_{n}$ and $L_{n}^{\prime}$ act on $S$ and $S^{\prime}$ respectively.
Check the relation $c^{\text{product}}=c+c^{\prime}$, and show
$\Delta^{\text{product}}=\Delta+\Delta^{\prime}$. Assuming that Cardy’s
formula holds for $S$ and $S^{\prime}$, show that computing the large $\Delta$
behaviour of the number of states $N^{\text{product}}(\Delta)$ in the product
theory amounts to finding the maximum of the function
$f(\Delta^{\prime})=\sqrt{c\Delta^{\prime}}+\sqrt{c^{\prime}(\Delta-\Delta^{\prime})}$.
Conclude that Cardy’s formula holds for $S\otimes S^{\prime}$.

##  Chapter 2 From representation theory to differential equations

This chapter is devoted to studying the linear equations which symmetry
constraints impose on correlation functions. Such equations depend on the
action of the symmetry algebra on the fields – in mathematical terms, on the
representations to which the corresponding states belong.

###  2.1 Representations of the Virasoro algebra

####  2.1.1 Highest-weight representations

We will make several standard assumptions about the representations which
appear in the spectrum. We have already assumed that the spectrum is made of
factorized representations of the symmetry algebra
$\mathfrak{V}\times\overline{\mathfrak{V}}$ as in eq. (1.4.1), so that we can
restrict our attention to the representations of the left-moving Virasoro
algebra. In Section 1.4.1, we have assumed that $L_{0}$ is hermitian, and that
its eigenvalues are bounded from below. Since acting with $L_{n>0}$ decreases
$L_{0}$-eigenvalues (see eq. (1.4.7)), an $L_{0}$-eigenstate $|\Delta\rangle$
for the lowest $L_{0}$-eigenvalue $\Delta$ must be a primary state, that is

$\displaystyle\boxed{\left\\{\begin{array}[]{l}L_{n>0}|\Delta\rangle=0\ ,\\\
L_{0}|\Delta\rangle=\Delta|\Delta\rangle\ .\end{array}\right.}$ (2.1.3)

A representation generated by a primary state is called a highest-weight
representation. The representation generated by a state $|v\rangle$ is
$U(\mathfrak{V})|v\rangle$, where $U(\mathfrak{V})$ is the universal
enveloping algebra of the Virasoro algebra – the associative algebra which is
generated by the Virasoro generators $L_{n}$. If $|\Delta\rangle$ is a primary
state, it is enough to consider the algebra $U(\mathfrak{V}^{+})$ of the
creation operators which is generated by the creation modes
$\\{L_{n}\\}_{n<0}$, and the corresponding highest-weight representation is

$\displaystyle
R=U(\mathfrak{V})|\Delta\rangle=U(\mathfrak{V}^{+})|\Delta\rangle\ .$ (2.1.4)

(See Exercise 2.1.) A state of $R$ of the type $U|\Delta\rangle$ with $U\in
U(\mathfrak{V}^{+})$ is called a descendent state if it is linearly
independent from $|\Delta\rangle$.

We assume that the spectrum can be decomposed into such highest-weight
representations. This assumption in particular implies that $L_{0}$ is
diagonalizable, whereas in principle $L_{0}$ could have nontrivial Jordan
cells. There exist theories called logarithmic conformal field theories, where
the action of $L_{0}$ on the spectrum is not diagonalizable, but we will not
consider such theories (except in Exercise 2.13).

Let us explore the structure of a highest-weight representation $R$. By
definition, such a representation comes with a natural surjective map

$\displaystyle\begin{array}[]{cclcl}\varphi_{R}&:&U(\mathfrak{V}^{+})&\rightarrow&R\\\
&&U&\mapsto&U|\Delta\rangle\ .\end{array}$ (2.1.7)

So we had better study the structure of $U(\mathfrak{V}^{+})$. Let us
introduce a basis $\mathcal{L}$ of $U(\mathfrak{V}^{+})$, which is
parametrized by the ordered $p$-uples $(-n_{1},\cdots-n_{p})$ of decreasing,
strictly negative integers:

$\displaystyle\mathcal{L}=\left\\{L_{-n_{1}}\cdots L_{-n_{p}}\right\\}_{1\leq
n_{1}\leq n_{2}\leq\cdots n_{p}}\ .$ (2.1.8)

The natural integer

$\displaystyle N=\left|L_{-n_{1}}\cdots L_{-n_{p}}\right|=\sum_{i=1}^{p}n_{i}\
,$ (2.1.9)

is called the level of the basis element $L_{-n_{1}}\cdots L_{-n_{p}}$, and
also of the corresponding state $L_{-n_{1}}\cdots L_{-n_{p}}|\Delta\rangle$ in
a highest-weight representation. The conformal dimension of such a state is
then $\Delta+N$.

The dimension of the level-$N$ subspace of $U(\mathfrak{V}^{+})$ is the number
$p(N)$ of partitions of $N$. The basis of $U(\mathfrak{V}^{+})$ up to the
level $N=5$ can be plotted as follows:

(2.1.10)

In this diagram, each arrow stands for the action of a Virasoro generator
$L_{n}$ with $n<0$. Acting with generators which are not depicted on the
diagram would produce linear combinations of our basis states, for example
$L_{-2}L_{-1}=L_{-1}L_{-2}-L_{-3}$.

Our basis (2.1.8), while convenient for enumerating the states, is not
distinguished by any particularly useful property. Other bases can be used,
starting with the “reverse-ordered” basis $\\{L_{-n_{1}}\cdots
L_{-n_{p}}\\}_{n_{1}\geq n_{2}\geq\cdots n_{p}\geq 1}$. It can be tempting to
use the fact that the two operators $\\{L_{-1},L_{-2}\\}$ algebraically
generate $\\{L_{n}\\}_{n<0}$, and to consider
$\\{\prod_{i=1}^{p}L_{-n_{i}}\\}_{n_{i}\in\\{1,2\\}}$. While this set does
span $U(\mathfrak{V}^{+})$, it however does not provide a basis. (See Exercise
2.2.)

####  2.1.2 Verma modules and degenerate representations

We define the Verma module $\mathcal{V}_{\Delta}$ with conformal dimension
$\Delta$ as the highest-weight representation which contains a primary state
$|\Delta\rangle$ with conformal dimension $\Delta$, and which is linearly
isomorphic to $U(\mathfrak{V}^{+})$ via the map
$\varphi_{\mathcal{V}_{\Delta}}$ of eq. (2.1.7). In other words,
$\mathcal{V}_{\Delta}$ is the representation whose basis is given by the
states $\left\\{L_{-n_{1}}\cdots L_{-n_{p}}|\Delta\rangle\right\\}_{1\leq
n_{1}\leq n_{2}\leq\cdots n_{p}}$. Therefore, $\mathcal{V}_{\Delta}$ is the
largest possible highest-weight representation with lowest conformal dimension
$\Delta$.

Any highest-weight representation $R$ which is not a Verma module is called a
degenerate representation. If $\Delta$ is the conformal dimension of the
highest-weight state of $R$, we have a natural surjective morphism of
representations from $\mathcal{V}_{\Delta}$ to $R$,

$\displaystyle\varphi_{R}\varphi_{\mathcal{V}_{\Delta}}^{-1}\ :\
\mathcal{V}_{\Delta}\ \rightarrow\ R\ .$ (2.1.11)

Therefore, $R$ is a coset of $\mathcal{V}_{\Delta}$ by some subrepresentation
$R^{\prime}$,

$\displaystyle R=\frac{\mathcal{V}_{\Delta}}{R^{\prime}}\ .$ (2.1.12)

In other words, a degenerate representation is associated to a nontrivial
subrepresentation of a Verma module. Now, in any nontrivial subrepresentation
of a highest-weight representation, the $L_{0}$ eigenvalues are bounded from
below, and there is therefore a primary state $|\chi\rangle$. This primary
state $|\chi\rangle$ is also a descendent – it is then called a singular
vector or null vector of the highest-weight representation. While a highest-
weight representation is by construction always indecomposable, it is
irreducible if and only it has no singular vectors.

Coming back to our Verma module $\mathcal{V}_{\Delta}$, and assuming that it
has a singular vector $|\chi\rangle$, we can define the degenerate
representation
$R=\frac{\mathcal{V}_{\Delta}}{U(\mathfrak{V}^{+})|\chi\rangle}$, which is a
coset of two Verma modules. If a Verma module has several singular vectors,
then it also has several corresponding subrepresentations, and therefore a
number of possible cosets, depending on which combination of these
subrepresentations is chosen as the denominator representation $R^{\prime}$ in
eq. (2.1.12).

Let us look for singular vectors in Verma modules. We start at the level
$N=1$: is $|\chi\rangle=L_{-1}|\Delta\rangle$ a singular vector? The states
$L_{n}|\chi\rangle$ with $n\geq 2$ automatically vanish because they have
negative levels, and we are left with computing

$\displaystyle
L_{1}|\chi\rangle=L_{1}L_{-1}|\Delta\rangle=[L_{1},L_{-1}]|\Delta\rangle=2L_{0}|\Delta\rangle=2\Delta|\Delta\rangle\
.$ (2.1.13)

So the Verma module $\mathcal{V}_{\Delta}$ has a singular vector at the level
$N=1$ if and only if $\Delta=0$. Let us now look for singular vectors at the
level $N=2$. The level-$2$ descendents of a primary state $|\Delta\rangle$ are
of the type

$\displaystyle|\chi\rangle=\left(a_{1,1}L_{-1}^{2}+a_{2}L_{-2}\right)|\Delta\rangle\
,$ (2.1.14)

where $a_{1,1}$ and $a_{2}$ are complex coefficients. We compute

$\displaystyle L_{1}|\chi\rangle$
$\displaystyle=\left((4\Delta+2)a_{1,1}+3a_{2}\right)L_{-1}|\Delta\rangle\ ,$
(2.1.15) $\displaystyle L_{2}|\chi\rangle$ $\displaystyle=\left(6\Delta
a_{1,1}+(4\Delta+\tfrac{1}{2}c)a_{2}\right)|\Delta\rangle\ .$ (2.1.16)

The vanishing of these two vectors leads to a system of two linear equations
for the two unknowns $(a_{1,1},a_{2})$, whose determinant is

$\displaystyle D_{2}(\Delta)=4(2\Delta+1)^{2}+(c-13)(2\Delta+1)+9\ .$ (2.1.17)

Singular vectors at the level $N=2$ exist if and only if

$\displaystyle
D_{2}(\Delta)=0\quad\Leftrightarrow\quad\Delta=\frac{5-c\pm\sqrt{(c-25)(c-1)}}{16}\
.$ (2.1.18)

As our last explicit example, we look for a singular vector at the level
$N=3$,

$\displaystyle|\chi\rangle=\left(a_{1,1,1}L_{-1}^{3}+a_{1,2}L_{-1}L_{-2}+a_{3}L_{-3}\right)|\Delta\rangle\
.$ (2.1.19)

The nontrivial relations which $|\chi\rangle$ must obey in order to be a
singular vector are $L_{1}|\chi\rangle=L_{2}|\chi\rangle=L_{3}|\chi\rangle=0$.
However, since $L_{3}=[L_{2},L_{1}]$, the relation $L_{3}|\chi\rangle=0$ is
actually redundant, and we need only compute

$\displaystyle L_{1}|\chi\rangle$
$\displaystyle=\left((6\Delta+6)a_{1,1,1}+3a_{1,2}\right)L_{-1}^{2}|\Delta\rangle+\left((2\Delta+4)a_{1,2}+4a_{3}\right)L_{-2}|\Delta\rangle\
,$ (2.1.20) $\displaystyle L_{2}|\chi\rangle$
$\displaystyle=\left((18\Delta+6)a_{1,1,1}+(4\Delta+\tfrac{1}{2}c+9)a_{1,2}+5a_{3}\right)L_{-1}|\Delta\rangle\
.$ (2.1.21)

This leads to a system of three linear equations for the three unknowns
$(a_{1,1,1},a_{1,2},a_{3})$, whose determinant is

$\displaystyle
D_{3}(\Delta)=12\Big{(}3(\Delta+1)^{2}+(c-13)(\Delta+1)+12\Big{)}\ .$ (2.1.22)

Singular vectors at the level $N=3$ exist if and only if

$\displaystyle
D_{3}(\Delta)=0\quad\Leftrightarrow\quad\Delta=\frac{7-c\pm\sqrt{(c-25)(c-1)}}{6}\
.$ (2.1.23)

Such formulas can be simplified by the introduction of the Liouville notations
for $c$ and $\Delta$. We actually introduce two new notations for the central
charge $c$: the background charge $Q$ and the Liouville coupling constant $b$,

$\displaystyle\boxed{c=1+6Q^{2}}\quad,\quad\boxed{Q=b+\frac{1}{b}}\ ,$
(2.1.24)

or equivalently

$\displaystyle b=\sqrt{\frac{c-1}{24}}+\sqrt{\frac{c-25}{24}}\ .$ (2.1.25)

The correspondences between the values of $c,Q$ and $b$ are

$\displaystyle\begin{array}[]{|l|c||c|c|c|c|c|c|}\hline\cr\text{central\
charge}&c&\leq 1&1-6\frac{(p-q)^{2}}{pq}&1&[1,25]&25&\geq 25\\\
\hline\cr\text{background\
charge}&Q&i{\mathbb{R}}&i\frac{p-q}{\sqrt{pq}}&0&[0,2]&2&\geq 2\\\
\hline\cr\text{Liouville\ coupling\
constant}&b&i{\mathbb{R}}&i\sqrt{\frac{p}{q}}&i&e^{i\mathbb{R}}&1&{\mathbb{R}}\\\
\hline\cr\end{array}$ (2.1.29)

where for each generic value of $c$ we choose among two corresponding values
$\pm Q$ for $Q$ and four corresponding values $\pm b^{\pm 1}$ for $b$. The
conformal dimension $\Delta$ can be written in terms of a new parameter
$\alpha$ called the momentum, such that

$\displaystyle\boxed{\Delta(\alpha)=\alpha(Q-\alpha)}\ ,$ (2.1.30)

which is defined up to the reflection

$\displaystyle\alpha\mapsto Q-\alpha\ ,$ (2.1.31)

so that $\mathcal{V}_{\alpha}=\mathcal{V}_{Q-\alpha}$.

In terms of the Liouville notations, the singular vectors at the levels
$N=1,2,3$, and the corresponding conformal dimensions $\Delta$ and momentums
$\alpha$, are

$\displaystyle\begin{array}[]{|c|c|c|c|c|}\hline\cr N&\langle
r,s\rangle&\Delta_{\langle r,s\rangle}&\alpha_{\langle r,s\rangle}&L_{\langle
r,s\rangle}\\\ \hline\cr\hline\cr 1&\langle 1,1\rangle&0&0&L_{-1}\\\
\hline\cr\hbox{\multirowsetup 2}&\langle
2,1\rangle&-\frac{1}{2}-\frac{3}{4}b^{2}&-\frac{b}{2}&\frac{1}{b^{2}}L_{-1}^{2}+L_{-2}\\\
\cline{2-5}\cr&\langle
1,2\rangle&-\frac{1}{2}-\frac{3}{4b^{2}}&-\frac{1}{2b}&b^{2}L_{-1}^{2}+L_{-2}\\\
\hline\cr\hbox{\multirowsetup 3}&\langle
3,1\rangle&-1-2b^{2}&-b&\frac{1}{4b^{2}}L_{-1}^{3}+L_{-1}L_{-2}+(b^{2}-\frac{1}{2})L_{-3}\\\
\cline{2-5}\cr&\langle
1,3\rangle&-1-\frac{2}{b^{2}}&-\frac{1}{b}&\frac{1}{4}b^{2}L_{-1}^{3}+L_{-1}L_{-2}+(\frac{1}{b^{2}}-\frac{1}{2})L_{-3}\\\
\hline\cr\end{array}$ (2.1.38)

Singular vectors at levels $N\geq 4$ can similarly be computed. (See Exercises
2.3 and 2.4.) The general result is [4]: for any factorization $N=rs$ of $N$
into two positive integers, there is a number $\Delta_{\langle r,s\rangle}$
such that the Verma module $\mathcal{V}_{\Delta_{\langle r,s\rangle}}$ has a
singular vector at the level $N$, which we denote as

$\displaystyle|\chi_{\langle r,s\rangle}\rangle=L_{\langle
r,s\rangle}|\Delta_{\langle r,s\rangle}\rangle\ .$ (2.1.39)

The general formula for $\Delta_{\langle r,s\rangle}$ is known,

$\displaystyle\Delta_{\langle
r,s\rangle}=\frac{1}{4}\left(Q^{2}-(rb+sb^{-1})^{2}\right)\ .$ (2.1.40)

This corresponds to the momemtums $\alpha_{\langle r,s\rangle}$ and
$Q-\alpha_{\langle r,s\rangle}$, where

$\displaystyle\boxed{\alpha_{\langle
r,s\rangle}=\frac{Q}{2}-\frac{r}{2}b-\frac{s}{2}b^{-1}}\ .$ (2.1.41)

Conversely, any singular vector which is not itself a descendent of another
singular vector is of the type $|\chi_{\langle r,s\rangle}\rangle$. However,
singular vectors which are descendents of other singular vectors are not
necessarily of this type. (See Exercise 2.5.) We define the maximally
degenerate representation $R_{\langle r,s\rangle}$ as the coset of
$\mathcal{V}_{\Delta_{\langle r,s\rangle}}$ by the subrepresentation generated
by all the singular vectors. For generic values of $c$, $|\chi_{\langle
r,s\rangle}\rangle$ is the only singular vector of
$\mathcal{V}_{\Delta_{\langle r,s\rangle}}$, and

$\displaystyle R_{\langle r,s\rangle}=\frac{\mathcal{V}_{\Delta_{\langle
r,s\rangle}}}{U(\mathfrak{V}^{+})|\chi_{\langle
r,s\rangle}\rangle}=\frac{\mathcal{V}_{\Delta_{\langle
r,s\rangle}}}{\mathcal{V}_{\Delta_{\langle-r,s\rangle}}}\ ,$ (2.1.42)

where $\Delta_{\langle-r,s\rangle}$ is actually the conformal dimension of
$|\chi_{\langle r,s\rangle}\rangle$, due to the identity

$\displaystyle\Delta_{\langle r,s\rangle}+rs=\Delta_{\langle-r,s\rangle}\ .$
(2.1.43)

####  2.1.3 Unitarity

Let us discuss which highest-weight representations of the Virasoro algebra
are unitary. A highest-weight representation $R$ is generated by a primary
state $|\Delta\rangle$, and we assume $\langle\Delta|\Delta\rangle=1$. Since
all other states are descendents of $|\Delta\rangle$, their scalar products
can be computed using the assumption $L_{n}^{\dagger}=L_{-n}$ from eq.
(1.4.3). For example,

$\displaystyle\langle
L_{-1}\Delta|L_{-1}\Delta\rangle=\langle\Delta|L_{1}L_{-1}|\Delta\rangle=2\langle\Delta|L_{0}|\Delta\rangle=2\Delta\
.$ (2.1.44)

Since $L_{0}$ is self-conjugate, different $L_{0}$-eigenspaces are orthogonal,
and a representation is unitary if and only if the scalar product is positive
definite on each eigenspace. In a highest-weight representation $R$, the
$L_{0}$-eigenspace which corresponds to the eigenvalue $\Delta+N$ is the
finite-dimensional level-$N$ subspace $R_{N}$. Given a basis $\\{v_{i}\\}$ of
$R_{N}$, the scalar product on $R_{N}$ is characterized by the Gram matrix
$M^{(N)}$ defined by $M^{(N)}_{ij}=\langle v_{i}|v_{j}\rangle$, and in
particular the scalar product is positive definite if and only if all the
eigenvalues of $M^{(N)}$ are strictly positive. To keep track of the signs of
these eigenvalues, it is actually enough to know the determinant of $M^{(N)}$,
which vanishes whenever an eigenvalue vanishes. We now make a useful remark:

In a Verma module, $\det M^{(N)}=0$ if and only if there is a singular vector
at a level $N^{\prime}\leq N$.

To prove this remark, notice that $\det M^{(N)}=0$ if and only if there exists
a nonzero state at the level $N$ which is orthogonal to all states at the
level $N$, and therefore to all states in the representation. Assuming $\det
M^{(N)}=0$, the space of states which are orthogonal to all states is a
nontrivial subrepresentation, which therefore contains a singular vector at a
level $N^{\prime}\leq N$. Conversely, notice that any singular vector and its
descendents are orthogonal to all states. If there is such a singular vector
at a level $N^{\prime}\leq N$, then this singular vector or one of its
descendents lives at the level $N$, and $\det M^{(N)}=0$.

It is easy to see that $\det M^{(N)}$ must be a polynomial function of
$\Delta$ and $c$. Knowing the zeros $\Delta_{\langle r,s\rangle}$ of $\det
M^{(N)}$ as a function of $\Delta$ eq. (2.1.40), we obtain the Kac formula

$\displaystyle\boxed{\det M^{(N)}\propto\prod_{\begin{smallmatrix}r,s\geq 1\\\
rs\leq N\end{smallmatrix}}(\Delta-\Delta_{\langle r,s\rangle})^{p(N-rs)}}\ ,$
(2.1.45)

where the constant of proportionality is a $\Delta,c$-independent positive
number, and $p(N^{\prime})$ is the number of partitions of $N^{\prime}$. The
multiplicity of the zero $\Delta_{\langle r,s\rangle}$ is $p(N-rs)$ because
this is the number of linearly independent descendents at the level $N$ of a
singular vector at the level $rs$.

We can now draw conclusions about the unitarity of highest-weight
representations. According to eq. (2.1.44), the positivity of $M^{(1)}$
implies $\Delta>0$. The unitarity of the Verma module $\mathcal{V}_{\Delta}$
will now depend on the value of the central charge $c$, which must be real as
we already saw in Section 1.4.1:

  1. 1.

If $c>1$, then $\Delta_{\langle r,s\rangle}$ (2.1.40) cannot be a real,
strictly positive number. Therefore, eigenvalues of $\det M^{(N)}$ do not
change signs on the half-line $c>1$. In the limit $c\to\infty$, the matrix
$M^{(N)}$ is positive definite. (See Exercise 2.6.) Therefore,
$\mathcal{V}_{\Delta}$ is unitary.

  2. 2.

If $c=1$, the Verma module $\mathcal{V}_{\Delta}$ is unitary if $\Delta>0$ by
continuity from the $c>1$ case, unless $\Delta=\Delta_{\langle r,s\rangle}$
for some $\langle r,s\rangle$, that is unless $\Delta=\frac{1}{4}n^{2}$ with
$n\in{\mathbb{N}}$.

  3. 3.

If $c<1$, then the Verma module $\mathcal{V}_{\Delta}$ is never unitary. This
is because for a given $\Delta>0$, the values of $c$ such that
$\Delta=\Delta_{\langle r,s\rangle}$ for some $\langle r,s\rangle$ accumulate
at $c=1^{-}$.

We still have to consider the unitarity of highest-weight representations
which are not Verma modules. Such degenerate representations are cosets of the
type $R=\frac{\mathcal{V}_{\Delta}}{R^{\prime}}$, where the subrepresentation
$R^{\prime}$ contains at least one singular vector. For $R$ to be unitary, the
subrepresentation $R^{\prime}$ must include all negative-norm states, that is
all eigenstates of Gram matrices which correspond to negative eigenvalues.

If $c>1$, the only degnererate representation with a positive conformal
dimension is $R_{\langle 1,1\rangle}$, and actually $\Delta_{\langle
1,1\rangle}=0$. This representation is unitary, because all the zero-norm
states in $\mathcal{V}_{0}=\underset{\Delta\to
0^{+}}{\lim}\mathcal{V}_{\Delta}$ are descendents of the level one singular
vector. If $c=1$, all degenerate representations have positive conformal
dimensions, and again they are unitary.

If now $c<1$, it turns out that in order to have a unitary coset, a Verma
module must have not only one but actually two independent singular vectors
[4]. Since the existence of a singular vector in $\mathcal{V}_{\Delta}$
implies a relation between $\Delta$ and the central charge $c$, the existence
of two singular vectors implies a constraint on $c$. As we will see in Section
3.2.3, the corresponding values of $c$ correspond to the minimal models, whose
central charges are given in eq. (1.4.15). However, the existence of two
singular vectors is a necessary, but not yet a sufficient condition for a
Verma module to have a unitary coset. The final result is that for $c<1$
unitary cosets exist only if the central charge $c$ takes values which
correspond to the unitary minimal models,

$\displaystyle c=1-\frac{6}{p(p+1)}\quad\text{with}\quad 2\leq p\ ,$ (2.1.46)

For these values of $c$, the conformal dimensions of Verma modules which have
unitary cosets are $\Delta=\Delta_{\langle r,s\rangle}=\Delta_{\langle
p-r,p+1-s\rangle}$ where

$\displaystyle 1\leq s\leq r\leq p-1\ .$ (2.1.47)

These unitary cosets are the maximally degenerate representations $R_{\langle
r,s\rangle}$, which take the form

$\displaystyle R_{\langle r,s\rangle}=\frac{\mathcal{V}_{\Delta_{\langle
r,s\rangle}}}{U(\mathfrak{V}^{+})|\chi_{\langle
r,s\rangle}\rangle+U(\mathfrak{V}^{+})|\chi_{\langle
p-r,p+1-s\rangle}\rangle}\ ,$ (2.1.48)

where the sum of the two subrepresentations is not a direct sum, because their
intersection is nonzero.

These results are summarized in the following table:

(2.1.52)

###  2.2 Virasoro symmetry in two-dimensional quantum field theory

####  2.2.1 Primary fields, decendent fields

Let us translate the algebraic considerations of Section 2.1 into
considerations on fields, using the state-field correspondence, which
associates a field $V_{\sigma}(z_{0})$ to a state $|\sigma\rangle$. Virasoro
generators $L_{n}$ act on states, and we denote their corresponding actions on
fields as

$\displaystyle
L_{n}V_{\sigma}(z_{0})=L_{n}^{(z_{0})}V_{\sigma}(z_{0})=V_{L_{n}\sigma}(z_{0})\
.$ (2.2.1)

The apparently redundant notation $L_{n}^{(z_{0})}$ is introduced in order to
better distinguish the representations of the Virasoro algebra at different
positions $z_{0}$, which is useful when several fields are involved, for
instance

$\displaystyle
L_{n}^{(z_{2})}V_{\sigma_{1}}(z_{1})V_{\sigma_{2}}(z_{2})=V_{\sigma_{1}}(z_{1})L_{n}V_{\sigma_{2}}(z_{2})\
.$ (2.2.2)

Let a primary field be a field which corresponds to a primary state. Assuming
there is only one primary state $|\Delta\rangle$ of conformal dimension
$\Delta$, the corresponding primary field will be denoted $V_{\Delta}(z_{0})$.
The definition (2.1.3) of a primary state leads to

$\displaystyle\boxed{\left\\{\begin{array}[]{l}L_{n>0}V_{\Delta}(z_{0})=0\
,\\\ L_{0}V_{\Delta}(z_{0})=\Delta V_{\Delta}(z_{0})\ .\end{array}\right.}$
(2.2.5)

Fields such as $\left(\prod_{i=1}^{p}L_{-n_{i}}\right)V_{\Delta}(z_{0})$ which
correspond to descendents of a primary state are then called descendent
fields. And a primary field which has a vanishing descendent is called a
degenerate field. In other words, a degenerate field corresponds to the
highest-weight state of a degenerate representation. The degenerate field
which corresponds to the highest-weight state of the maximally degenerate
representation $R_{\langle r,s\rangle}$ will be denoted $V_{\langle
r,s\rangle}(z_{0})$, and obeys the equation

$\displaystyle L_{\langle r,s\rangle}V_{\langle r,s\rangle}(z_{0})=0\ ,$
(2.2.6)

where $L_{\langle r,s\rangle}$ is defined in eq. (2.1.39). This field differs
from the field $V_{\Delta_{\langle r,s\rangle}}(z_{0})$, which corresponds to
the highest-weight state of the Verma module $\mathcal{V}_{\Delta_{\langle
r,s\rangle}}$, and obeys no such equation.

Assuming that we know the spectrum of a model, and that this spectrum is made
of highest-weight representations, the observables which we should compute in
order to solve the model are the correlation functions of primary and
descendent fields. An $N$-point function of primary fields
$\left\langle\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\right\rangle$ is a function
of their positions and conformal dimensions. The $N$-point functions of the
corresponding descendent fields are of the type
$\left\langle\prod_{i=1}^{N}\left(\left(\prod_{j=1}^{p_{i}}L_{-n_{i,j}}\right)V_{\Delta_{i}}(z_{i})\right)\right\rangle$,
with $p_{i}\geq 0$ and $n_{i,j}\geq 1$. In order to derive the equations which
constrain and relate these correlation functions, we must first explain how
the action of the Virasoro symmetry is encoded in a field, the Virasoro field.

####  2.2.2 The Virasoro field

We have called $L_{n}^{(z_{0})}$ the action of the Virasoro generator $L_{n}$
on a field at the point $z_{0}$, thereby defining a family
$(L_{n}^{(z_{0})})_{z_{0}\in{\mathbb{C}}}$ of representations of the Virasoro
algebra. But we should only have one left-moving Virasoro symmetry algebra:
$(L_{n}^{(z_{1})})_{n\in\mathbb{Z}}$ and $(L_{n}^{(z_{2})})_{n\in\mathbb{Z}}$
should be understood of two bases of the same space of symmetry generators,
and should be linearly related. We will now determine this linear relation,
which in practice will mean determining the dependence of $L_{n}^{(z_{0})}$ on
$z_{0}$.

Let us first consider the Witt algebra, whose generators $\ell_{n}$ have a
representation (1.3.13) as differential operators. This representation
corresponds to radial quantization with the reference point $z_{0}=0$.
Performing a translation, we obtain a representation which corresponds to an
arbitrary reference point $z_{0}$,

$\displaystyle\ell_{n}^{(z_{0})}=-(z-z_{0})^{n+1}{\frac{\partial}{\partial
z}}\ .$ (2.2.7)

We now see that the dependence of $\ell_{n}^{(z_{0})}$ on the reference point
$z_{0}$ is such that

$\displaystyle{\frac{\partial}{\partial
z_{0}}}\ell_{n}^{(z_{0})}=-(n+1)\ell_{n-1}^{(z_{0})}\ .$ (2.2.8)

We assume that the same equation applies to $L_{n}^{(z_{0})}$,

$\displaystyle{\frac{\partial}{\partial
z_{0}}}L_{n}^{(z_{0})}=-(n+1)L_{n-1}^{(z_{0})}\ .$ (2.2.9)

This is equivalent to $\frac{\partial}{\partial z_{0}}T(y)=0$, where we define
the Virasoro field $T(y)$ as

$\displaystyle\boxed{T(y)=\sum_{n\in{\mathbb{Z}}}\frac{L_{n}^{(z_{0})}}{(y-z_{0})^{n+2}}}\
.$ (2.2.10)

(The series is supposed to converge for $y$ in some neighbourhood of $z_{0}$.)
The Virasoro field encodes the action of the Virasoro algebra at all points
$z_{0}$. From $T(y)$, we can indeed recover the Virasoro generators at any
point,

$\displaystyle\boxed{L_{n}^{(z_{0})}=\frac{1}{2\pi
i}\oint_{z_{0}}dy(y-z_{0})^{n+1}T(y)}\ .$ (2.2.11)

So, for any field $V_{\sigma}(z_{0})$, where we do not indicate the dependence
on $\bar{z}_{0}$ which plays no role here, we have

$\displaystyle\boxed{T(y)V_{\sigma}(z_{0})=\sum_{n\in{\mathbb{Z}}}\frac{L_{n}V_{\sigma}(z_{0})}{(y-z_{0})^{n+2}}}\
.$ (2.2.12)

This is actually a special case of the general OPE (1.2.7), where the sum over
the spectrum of the theory reduces to a combination of fields of the type of
$L_{n}V_{\sigma}(z_{0})$, as follows from the definition of $T(y)$ as a
combination of symmetry generators. This OPE simplifies if $V_{\sigma}(z_{0})$
is a primary field,

$\displaystyle T(y)V_{\Delta}(z_{0})=\frac{\Delta
V_{\Delta}(z_{0})}{(y-z_{0})^{2}}+\frac{L_{-1}V_{\Delta}(z_{0})}{y-z_{0}}+\sum_{n=0}^{\infty}(y-z_{0})^{n}L_{-n-2}V_{\Delta}(z_{0})\
.$ (2.2.13)

The field $T(y)$ is a symmetry field: a field which encodes the action of
symmetry transformations, but does not necessarily correspond to a state in
the spectrum. That field nevertheless obeys the same assumptions of
commutativity, associativity, and existence of an OPE, as the fields
$V_{\sigma}(z_{0})$.

The $T(y)V_{\sigma}(z_{0})$ OPE determines the behaviour of $T(y)$ near a
field $V_{\sigma}(z_{0})$. On the other hand, $T(y)$ is assumed to be
holomorphic at any point where no field is present. Since our field theory
lives on the Riemann sphere, we expect $T(y)$ to be holomorphic at $y=\infty$
as well. We will accept that the holomorphy of $T(y)$ at $y=\infty$ means