# Minimal lectures on two-dimensional
conformal field theory

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

We provide a brief but self-contained review of conformal field theory on the
Riemann sphere. We first introduce general axioms such as local conformal
invariance, and derive Ward identities and BPZ equations. We then define
minimal models and Liouville theory by specific axioms on their spectrums and
degenerate fields. We solve these theories by computing three- and four-point
functions, and discuss their existence and uniqueness.

Lectures given at the Cargèse school on “Quantum integrable systems, conformal
field theories and stochastic processes” in September 2016, under the title
“Conformal bootstrap approach to Liouville theory”.

###### Contents

  1. 0 Introduction
  2. 1 The Virasoro algebra and its representations
    1. 1.1 Algebra
    2. 1.2 Representations
    3. 1.3 Null vectors and degenerate representations
  3. 2 Conformal field theory
    1. 2.1 Fields
    2. 2.2 Correlation functions and Ward identities
    3. 2.3 Belavin–Polyakov–Zamolodchikov equations
  4. 3 Conformal bootstrap
    1. 3.1 Single-valuedness
    2. 3.2 Operator product expansion and crossing symmetry
    3. 3.3 Degenerate fields and the fusion product
  5. 4 Minimal models and Liouville theory
    1. 4.1 Diagonal minimal models
    2. 4.2 Liouville theory
  6. 5 Four-point functions
    1. 5.1 Single-valued four-point functions
    2. 5.2 Determining three-point structure constants
    3. 5.3 Crossing symmetry

##  0 Introduction

Since the time of Euclid, mathematical objects are defined by axioms.
Axiomatic definitions focus on the basic features of the defined objects,
thereby avoiding alternative constructions that may be less fundamental. For
example, in conformal field theory, the axiomatic approach (also called the
conformal bootstrap approach) makes functional integrals unnecessary. We will
define Liouville theory and minimal models by a sequence of axioms, starting
with local conformal symmetry. Our axioms are necessary conditions. Their
mutual consistency, in other words the existence of Liouville theory and
minimal models, will be tested but not proved.

In the first three sections, most axioms are common to all two-dimensional
conformal field theories. These axioms specify in particular how the Virasoro
symmetry algebra acts on fields, and the existence and properties of the
operator product expansion. Next, we introduce additional axioms that single
out either Liouville theory, or minimal models. In particular, these axioms
determine the three-point functions. Finally we check that these uniquely
defined theories do exist, by studying their four-point functions. It is the
success of such checks, much more than a priori considerations, that justifies
our choices of axioms.

This text aims to be self-contained, except at the very end when we will refer
to [1] for the properties of generic conformal blocks. For a more detailed
text in the same spirit, see the review article [2]. For a wider and more
advanced review, and a guide to the recent literature, see Teschner’s text
[3]. The Bible of rational conformal field theory is of course the epic
textbook [4]. And Cardy’s lecture notes [5] provide an introduction to the
statistical physics applications of conformal field theory.

That these lectures are minimal does not just mean that they are relatively
brief. This also means that they omit many concepts and assumptions that are
usually introduced in two-dimensional conformal field theory, but that are not
necessary for our purposes. Among these concepts and assumptions, let us
mention the existence of a vacuum state, the notion of unitarity, the
construction of fields as operators on the space of states, radial
quantization, and consistency on Riemann surfaces other than the sphere.
Minimalism is valuable not only for pedagogy, but also for research: we will
follow shortcuts that may have been hard to see when originally solving
Liouville theory and minimal models, but that can now be used when exploring
other theories, for example non-diagonal theories [6].

##  1 The Virasoro algebra and its representations

###  1.1 Algebra

By definition, conformal transformations are transformations that preserve
angles. In two dimensions with a complex coordinate $z$, any holomorphic
transformation preserves angles. Infinitesimal conformal transformations are
holomorphic functions close to the identity function,

$\displaystyle z\mapsto z+\epsilon z^{n+1}\qquad(n\in\mathbb{Z}\ ,\
\epsilon\ll 1)\ .$ (1.1)

These transformations act on functions of $z$ via the differential operators

$\displaystyle\ell_{n}=-z^{n+1}\frac{\partial}{\partial z}\ ,$ (1.2)

and these operators generate the Witt algebra, with commutation relations

$\displaystyle[\ell_{n},\ell_{m}]=(n-m)\ell_{m+n}\ .$ (1.3)

The generators $(\ell_{-1},\ell_{0},\ell_{1})$ generate an $s\ell_{2}$
subalgebra, called the algebra of infinitesimal global conformal
transformations. The corresponding Lie group is the group of conformal
transformations of the Riemann sphere $\mathbb{C}\cup\\{\infty\\}$,

$\displaystyle z\mapsto\frac{az+b}{cz+d}\ .$ (1.4)

######  Exercise 1.1 (Global conformal group of the sphere)

Show that the global conformal group of the sphere is $PSL_{2}(\mathbb{C})$,
and includes translations, rotations, and dilatations.

In a quantum theory, symmetry transformations act projectively on states.
Projective representations of an algebra are equivalent to representations of
a centrally extended algebra. This is why we always look for central
extensions of symmetry algebras.

######  Definition 1.2 (Virasoro algebra)

The central extension of the Witt algebra is called the Virasoro algebra. It
has the generators $(L_{n})_{n\in\mathbb{Z}}$ and $\mathbf{1}$, and the
commutation relations

$\displaystyle[\mathbf{1},L_{n}]=0\quad,\quad[L_{n},L_{m}]=(n-m)L_{n+m}+\frac{c}{12}(n-1)n(n+1)\delta_{n+m,0}\mathbf{1}\
,$ (1.5)

where the number $c$ is called the central charge. (The notation $c\mathbf{1}$
stands for a central generator that always has the same eigenvalue $c$ within
a given conformal field theory.)

######  Exercise 1.3 (Uniqueness of the Virasoro algebra)

Show that the Virasoro algebra is the unique central extension of the Witt
algebra.

###  1.2 Representations

The spectrum, i.e. the space of states, must be a representation of the
Virasoro algebra. Let us now make assumptions on what type of representation
it can be.

######  Axiom 1.4 (Representations that can appear in the spectrum)

The spectrum is a direct sum of irreducible representations. In the spectrum,
$L_{0}$ is diagonalizable, and the real part of its eigenvalues is bounded
from below.

Why this special role for $L_{0}$? Because we want to interpret it as the
energy operator. Since the corresponding Witt algebra generator $\ell_{0}$
generates dilatations, considering it as the energy operator amounts to
consider the radial coordinate as the time. We however do not assume that
$L_{0}$ eigenvalues are real or that the spectrum is a Hilbert space, as this
would restrict the central charge to be real. The $L_{0}$ eigenvalue of an
$L_{0}$ eigenvector is called its conformal dimension.

Let us consider an irreducible representation that is allowed by our axiom.
There must be an $L_{0}$ eigenvector $|v\rangle$ with the smallest eigenvalue
$\Delta$. Then $L_{n}|v\rangle$ is also an $L_{0}$ eigenvector,

$\displaystyle
L_{0}L_{n}|v\rangle=L_{n}L_{0}|v\rangle+[L_{0},L_{n}]|v\rangle=(\Delta-n)L_{n}|v\rangle\
.$ (1.6)

If $n>0$ we must have $L_{n}|v\rangle=0$, and $|v\rangle$ is called a primary
state.

######  Definition 1.5 (Primary and descendent states, level, Verma module)

A primary state with conformal dimension $\Delta$ is a state $|v\rangle$ such
that

$\displaystyle L_{0}|v\rangle=\Delta|v\rangle\quad,\quad L_{n>0}|v\rangle=0\
.$ (1.7)

The Verma module $\mathcal{V}_{\Delta}$ is the representation whose basis is
$\left\\{\prod_{i=1}^{k}L_{-n_{i}}|v\rangle\right\\}_{0<n_{1}\leq\dots\leq
n_{k}}$. The level of the state $\prod_{i=1}^{k}L_{-n_{i}}|v\rangle$ is
$N=\sum_{i=1}^{k}n_{i}\geq 0$. A state of level $N\geq 1$ is called a
descendent state.

Let us plot a basis of primary and descendent states up to the level $3$:

(1.8)

We need not include the state $L_{-2}L_{-1}|v\rangle$, due to
$L_{-2}L_{-1}=L_{-1}L_{-2}-L_{-3}$.

Are Verma module irreducible representations? i.e. do they have nontrivial
subrepresentations? In any subrepresentation of a Verma module, $L_{0}$ is
again diagonalizable and bounded from below, so there must be a primary state
$|\chi\rangle$. If the subrepresentation differs from the Verma module, that
primary state must differ from $|v\rangle$, and therefore be a descendent of
$|v\rangle$.

###  1.3 Null vectors and degenerate representations

######  Definition 1.6 (Null vectors)

A descendent state that is also primary is called a null vector or singular
vector.

In the Verma module $\mathcal{V}_{\Delta}$, let us look for null vectors at
the level $N=1$. For $n\geq 1$ we have

$\displaystyle
L_{n}L_{-1}|v\rangle=[L_{n},L_{-1}]|v\rangle=(n+1)L_{n-1}|v\rangle=\left\\{\begin{array}[]{ll}0&\quad\text{if
}n\geq 2\ ,\\\ 2\Delta|v\rangle&\quad\text{if }n=1\ .\end{array}\right.$
(1.11)

So $L_{-1}|v\rangle$ is a null vector if and only if $\Delta=0$, and the Verma
module $\mathcal{V}_{0}$ is reducible. Let us now look for null vectors at the
level $N=2$. Let $|\chi\rangle=(L_{-1}^{2}+aL_{-2})|v\rangle$, then $L_{n\geq
3}|\chi\rangle=0$.

###### Exercise 1.7

Compute $L_{1}|\chi\rangle$ and $L_{2}|\chi\rangle$, and find

$\displaystyle
L_{1}|\chi\rangle=\left((4\Delta+2)+3a\right)L_{-1}|v\rangle\quad,\quad
L_{2}|\chi\rangle=\left(6\Delta+(4\Delta+\tfrac{1}{2}c)a\right)|v\rangle\ .$
(1.12)

Requiring that $L_{1}|\chi\rangle$ and $L_{2}|\chi\rangle$ vanish, find the
coefficient $a$, and show that

$\displaystyle\Delta=\frac{1}{16}\left(5-c\pm\sqrt{(c-1)(c-25)}\right)\ .$
(1.13)

In order to simplify this formula, let us introduce other notations for $c$
and $\Delta$. We define

$\displaystyle\text{the background charge }Q\ ,$ $\displaystyle\quad
c=1+6Q^{2}\ ,\quad\text{up to }Q\to-Q\ ,$ (1.14) $\displaystyle\text{the
coupling constant }b\ ,$ $\displaystyle\quad Q=b+\frac{1}{b}\ ,\quad\text{up
to }b\to\pm b^{\pm 1}\ ,$ (1.15) $\displaystyle\text{the momentum }\alpha\ ,$
$\displaystyle\quad\Delta=\alpha(Q-\alpha)\ ,\quad\text{up to reflections
}\alpha\to Q-\alpha\ .$ (1.16)

The condition (1.13) for the existence of a level two null vector becomes

$\displaystyle\alpha=-\frac{1}{2}b^{\pm 1}\ .$ (1.17)

To summarize, null vectors at levels $1$ and $2$ occur for particular values
of $\Delta$. The null vectors at levels $N\leq 2$ can be written as
$L_{\langle r,s\rangle}|v\rangle$ where $r,s$ are strictly positive integers
such that $rs=N$,

$\displaystyle\begin{array}[]{|c|c|c|c|c|}\hline\cr N&\langle
r,s\rangle&\Delta_{\langle r,s\rangle}&\alpha_{\langle r,s\rangle}&L_{\langle
r,s\rangle}\\\ \hline\cr\hline\cr 1&\langle 1,1\rangle&0&0&L_{-1}\\\
\hline\cr\hbox{\multirowsetup 2}&\langle
2,1\rangle&-\frac{1}{2}-\frac{3}{4}b^{2}&-\frac{b}{2}&L_{-1}^{2}+b^{2}L_{-2}\\\
\cline{2-5}\cr&\langle
1,2\rangle&-\frac{1}{2}-\frac{3}{4b^{2}}&-\frac{1}{2b}&L_{-1}^{2}+b^{-2}L_{-2}\\\
\hline\cr\end{array}$ (1.22)

The pattern goes on at higher levels [4]: null vectors occur at level $N$ for
finitely many dimensions $\Delta_{\langle r,s\rangle}$, with

$\displaystyle\alpha_{\langle r,s\rangle}=\frac{Q}{2}-\frac{1}{2}(rb+sb^{-1})\
.$ (1.23)

(See Exercise 3.10 for a derivation.) If $\Delta\notin\\{\Delta_{\langle
r,s\rangle}\\}_{r,s\in\mathbb{N}^{*}}$, then $\mathcal{V}_{\Delta}$ is
irreducible. If $\Delta=\Delta_{\langle r,s\rangle}$, then
$\mathcal{V}_{\Delta}$ contains a nontrivial submodule, generated by the null
vector and its descendent states. For generic values of the central charge
$c$, this submodule is the Verma module $\mathcal{V}_{\Delta_{\langle
r,s\rangle}+rs}$.

######  Definition 1.8 (Degenerate representation)

The coset of the reducible Verma module $\mathcal{V}_{\Delta_{\langle
r,s\rangle}}$, by its Verma submodule $\mathcal{V}_{\Delta_{\langle
r,s\rangle}+rs}$, is an irreducible module $\mathcal{R}_{\langle r,s\rangle}$,
that is called a degenerate representation:

$\displaystyle\mathcal{R}_{\langle
r,s\rangle}=\frac{\mathcal{V}_{\Delta_{\langle
r,s\rangle}}}{\mathcal{V}_{\Delta_{\langle r,s\rangle}+rs}}\ .$ (1.24)

In this representation, the null vector vanishes,

$\displaystyle L_{\langle r,s\rangle}|v\rangle=0\ .$ (1.25)

The vanishing of null vectors will be crucial for solving Liouville theory and
minimal models.

##  2 Conformal field theory

Now that we understand the algebraic structure of conformal symmetry in two
dimensions, let us study how the Virasoro algebra acts on objects that live on
the Riemann sphere – the fields of conformal field theory. (Technically,
fields are sections of vector bundles over the sphere.)

###  2.1 Fields

######  Axiom 2.1 (State-field correspondence)

For any state $|w\rangle$ in the spectrum, there is an associated field
$V_{|w\rangle}(z)$. The map $|w\rangle\mapsto V_{|w\rangle}(z)$ is linear and
injective. We define the action of the Virasoro algebra on such fields as

$\displaystyle
L_{n}V_{|w\rangle}(z)=L_{n}^{(z)}V_{|w\rangle}(z)=V_{L_{n}|w\rangle}(z)\ .$
(2.1)

######  Definition 2.2 (Primary field, descendent field, degenerate field)

Let $|v\rangle$ be the primary state of the Verma module
$\mathcal{V}_{\Delta}$. We define the primary field
$V_{\Delta}(z)=V_{|v\rangle}(z)$. This field obeys

$\displaystyle L_{n\geq 0}V_{\Delta}(z)=0\quad,\quad L_{0}V_{\Delta}(z)=\Delta
V_{\Delta}(z)\ .$ (2.2)

Similarly, descendent fields correspond to descendent states. And the
degenerate field $V_{\langle r,s\rangle}(z)$ corresponds to the primary state
of the degenerate representation $\mathcal{R}_{\langle r,s\rangle}$, and
therefore obeys

$\displaystyle L_{\langle r,s\rangle}V_{\langle r,s\rangle}(z)=0\qquad\text{--
for example, \ }L_{-1}V_{\langle 1,1\rangle}(z)=0\ .$ (2.3)

Now let us specify how fields depend on $z$.

######  Axiom 2.3 (Dependence of fields on $z$)

For any field $V(z)$, we have

$\displaystyle\frac{\partial}{\partial z}V(z)=L_{-1}V(z)\ .$ (2.4)

Moreover, in this Section 2 we tacitly assume that all our fields are locally
holomorphic i.e. $\frac{\partial}{\partial\bar{z}}V(z)=0$. The dependence on
$\bar{z}$ will be reintroduced and studied in Section 3.1.

Let us derive consequences of this axiom, starting with the $z$-dependence of
the action $L_{n}^{(z)}$ of Virasoro generators on fields. On the one hand,

$\displaystyle\frac{\partial}{\partial
z}\left(L_{n}^{(z)}V(z)\right)=\left(\frac{\partial}{\partial
z}L_{n}^{(z)}\right)V(z)+L_{n}^{(z)}\frac{\partial}{\partial z}V(z)\ .$ (2.5)

On the other hand, using our axiom, we find

$\displaystyle\frac{\partial}{\partial
z}\left(L_{n}^{(z)}V(z)\right)=L_{-1}^{(z)}L_{n}^{(z)}V(z)=-(n+1)L_{n-1}^{(z)}V(z)+L_{n}^{(z)}L_{-1}^{(z)}V(z)\
.$ (2.6)

This implies

$\displaystyle\frac{\partial}{\partial z}L_{n}^{(z)}=-(n+1)L_{n-1}^{(z)}\
,\qquad(\forall n\in\mathbb{Z})\ .$ (2.7)

These infinitely many equations can be encoded into one functional equation,

$\displaystyle\frac{\partial}{\partial
z}\sum_{n\in\mathbb{Z}}\frac{L_{n}^{(z)}}{(y-z)^{n+2}}=0\ .$ (2.8)

######  Definition 2.4 (Energy-momentum tensor)

The energy-momentum tensor is a field, that we define by the formal Laurent
series

$\displaystyle T(y)=\sum_{n\in\mathbb{Z}}\frac{L_{n}^{(z)}}{(y-z)^{n+2}}\ .$
(2.9)

In other words, for any field $V(z)$, we have

$\displaystyle
T(y)V(z)=\sum_{n\in\mathbb{Z}}\frac{L_{n}V(z)}{(y-z)^{n+2}}\quad,\quad
L_{n}V(z)=\frac{1}{2\pi i}\oint_{z}dy\ (y-z)^{n+1}T(y)V(z)\ .$ (2.10)

In the case of a primary field $V_{\Delta}(z)$, using eq. (2.4) and writing
regular terms as $O(1)$, this definition reduces to

$\displaystyle
T(y)V_{\Delta}(z)=\frac{\Delta}{(y-z)^{2}}V_{\Delta}(z)+\frac{1}{y-z}\frac{\partial}{\partial
z}V_{\Delta}(z)+O(1)\ .$ (2.11)

This is our first example of an operator product expansion.

The energy-momentum tensor $T(y)$ is locally holomorphic as a function of $y$,
and acquires poles in the presence of other fields. Since we are on the
Riemann sphere, it must also be holomorphic at $y=\infty$.

######  Axiom 2.5 (Behaviour of $T(y)$ at infinity)

$\displaystyle T(y)\underset{y\to\infty}{=}O\left(\frac{1}{y^{4}}\right)\ .$
(2.12)

To motivate this axiom, let us do some dimensional analysis. If $z$ has
dimension $-1$, then according to eq. (2.4) $L_{-1}$ has dimension $1$, and
$T(y)$ has dimension $2$. The dimensionless quantity that should be
holomorphic at infinity is therefore the differential $T(y)dy^{2}$. At
infinity a holomorphic coordinate is $\frac{1}{y}$ and a holomorphic
differential is $d(\frac{1}{y})=-\frac{dy}{y^{2}}$, so our axiom amounts to
$T(y)dy^{2}=O(\frac{dy^{2}}{y^{4}})$ being holomorphic.

###  2.2 Correlation functions and Ward identities

######  Definition 2.6 (Correlation function)

To $N$ fields $V_{1}(z_{1}),\dots,V_{N}(z_{N})$ with $i\neq j\implies
z_{i}\neq z_{j}$, we associate a number called their correlation function or
$N$-point function, and denoted as

$\displaystyle\Big{<}V_{1}(z_{1})\cdots V_{N}(z_{N})\Big{>}\ .$ (2.13)

For example, $\left<\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\right>$ is a function
of $\\{z_{i}\\},\\{\Delta_{i}\\}$ and $c$. Correlation functions depend
linearly on fields, and in particular $\frac{\partial}{\partial
z_{1}}\left<V_{1}(z_{1})\cdots
V_{N}(z_{N})\right>=\left<\frac{\partial}{\partial z_{1}}V_{1}(z_{1})\cdots
V_{N}(z_{N})\right>$.

######  Axiom 2.7 (Commutativity of fields)

Correlation functions do not depend on the order of the fields,

$\displaystyle V_{1}(z_{1})V_{2}(z_{2})=V_{2}(z_{2})V_{1}(z_{1})\ .$ (2.14)

Let us work out the consequences of conformal symmetry for correlation
functions. In order to study an $N$-point function $Z$ of primary fields, we
introduce an auxiliary $(N+1)$-point function $Z(y)$ where we insert the
energy-momentum tensor,

$\displaystyle Z=\left<\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\right>\quad,\quad
Z(y)=\left<T(y)\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\right>\ .$ (2.15)

$Z(y)$ is a meromorphic function of $y$, with poles at $y=z_{i}$, whose
residues are given by eq. (2.11) (using the commutativity of fields). Moreover
$T(y)$, and therefore also $Z(y)$, vanish in the limit $y\to\infty$. So $Z(y)$
is completely determined by its poles and residues,

$\displaystyle
Z(y)=\sum_{i=1}^{N}\left(\frac{\Delta_{i}}{(y-z_{i})^{2}}+\frac{1}{y-z_{i}}\frac{\partial}{\partial
z_{i}}\right)Z\ .$ (2.16)

But $T(y)$ does not just vanish for $y\to\infty$, it behaves as
$O(\frac{1}{y^{4}})$. So the coefficients of $y^{-1},y^{-2},y^{-3}$ in the
large $y$ expansion of $Z(y)$ must vanish,

$\displaystyle\sum_{i=1}^{N}\partial_{z_{i}}Z=\sum_{i=1}^{N}\left(z_{i}\partial_{z_{i}}+\Delta_{i}\right)Z=\sum_{i=1}^{N}\left(z_{i}^{2}\partial_{z_{i}}+2\Delta_{i}z_{i}\right)Z=0\
.$ (2.17)

These three equations are called global Ward identities. They determine how
$Z$ behaves under global conformal transformations of the Riemann sphere,

$\displaystyle\left<\prod_{i=1}^{N}V_{\Delta_{i}}\left(\frac{az_{i}+b}{cz_{i}+d}\right)\right>=\prod_{i=1}^{N}(cz_{i}+d)^{2\Delta_{i}}\left<\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\right>\
.$ (2.18)

Let us solve the global Ward identities in the cases of one, two, three and
four-point functions. For a one-point function, we have

$\displaystyle\partial_{z}\Big{<}V_{\Delta}(z)\Big{>}=0\quad,\quad\Delta\Big{<}V_{\Delta}(z)\Big{>}=0\
.$ (2.19)

So one-point functions are constant, and non-vanishing only if $\Delta=0$.
Similarly, two-point functions must obey

$\displaystyle\Big{<}V_{\Delta_{1}}(z_{1})V_{\Delta_{2}}(z_{2})\Big{>}\propto(z_{1}-z_{2})^{-2\Delta_{1}}\quad,\quad(\Delta_{1}-\Delta_{2})\Big{<}V_{\Delta_{1}}(z_{1})V_{\Delta_{2}}(z_{2})\Big{>}=0\
.$ (2.20)

So a two-point function can be non-vanishing only if the two fields have the
same dimension. For three-point functions, there are as many equations (2.17)
as unknowns $z_{1},z_{2},z_{3}$, and therefore a unique solution with no
constraints on $\Delta_{i}$,

$\displaystyle\left<\prod_{i=1}^{3}V_{\Delta_{i}}(z_{i})\right>\propto(z_{1}-z_{2})^{\Delta_{3}-\Delta_{1}-\Delta_{2}}(z_{1}-z_{3})^{\Delta_{2}-\Delta_{1}-\Delta_{3}}(z_{2}-z_{3})^{\Delta_{1}-\Delta_{2}-\Delta_{3}}\
,$ (2.21)

with an unknown proportionality coefficient that does not depend on $z_{i}$.
For four-point functions, the general solution is

$\displaystyle\left<\prod_{i=1}^{4}V_{\Delta_{i}}(z_{i})\right>=\left(\prod_{i<j}(z_{i}-z_{j})^{\delta_{ij}}\right)G\left(\frac{(z_{1}-z_{2})(z_{3}-z_{4})}{(z_{1}-z_{3})(z_{2}-z_{4})}\right)\
,$ (2.22)

where $G$ is an arbitrary function, and $\delta_{ij}$ are such that
$\sum_{j\neq i}\delta_{ij}=-2\Delta_{i}$, where
$\delta_{ij}\underset{i>j}{=}\delta_{ji}$. The six numbers
$(\delta_{ij})_{i<j}$ are subject to only four equations, leaving two
undetermined combinations. Changing these combinations amounts to a
redefinition $G(z)\to z^{\lambda}(1-z)^{\mu}G(z)$.

So the three global Ward identities effectively reduce the four-point function
to a function of one variable $G$ – equivalently, we can set
$z_{2},z_{3},z_{4}$ to fixed values, and recover the four-point function from
its dependence on $z_{1}$ alone.

######  Exercise 2.8 (Conformal symmetry of four-point functions)

Let us define $V_{\Delta}(\infty)=\lim_{z\to\infty}z^{2\Delta}V_{\Delta}(z)$.
Check that this is finite when inserted into a two- or three-point function.
More generally, show that this is finite using the behaviour (2.18) of
correlation functions under $z\to-\frac{1}{z}$. Show that there is a (unique)
choice of $\delta_{ij}$ such that

$\displaystyle
G(z)=\Big{<}V_{\Delta_{1}}(z)V_{\Delta_{2}}(0)V_{\Delta_{3}}(\infty)V_{\Delta_{4}}(1)\Big{>}\
.$ (2.23)

We have been studying global conformal invariance of correlation functions of
primary fields, rather than more general fields. This was not only for making
things simpler, but also because correlation functions of descendents can be
deduced from correlation functions of primaries. For example,

$\displaystyle\Big{<}L_{-2}V_{\Delta_{1}}(z_{1})V_{\Delta_{2}}(z_{2})\cdots\Big{>}=\frac{1}{2\pi
i}\oint_{z_{1}}\frac{dy}{y-z_{1}}Z(y)=\sum_{i=2}^{N}\left(\frac{1}{z_{1}-z_{i}}\frac{\partial}{\partial
z_{i}}+\frac{\Delta_{i}}{(z_{i}-z_{1})^{2}}\right)Z\ ,$ (2.24)

where we used first eq. (2.10) for $L_{-2}V_{\Delta_{1}}(z_{1})$, and then eq.
(2.16) for $Z(y)$. This can be generalized to any correlation function of
descendent fields. The resulting equations are called local Ward identities.

###  2.3 Belavin–Polyakov–Zamolodchikov equations

Local and global Ward identities are all we can deduce from conformal
symmetry. But correlation functions that involve degenerate fields obey
additional equations.

For example, let us replace $V_{\Delta_{1}}(z_{1})$ with the degenerate
primary field $V_{\langle 1,1\rangle}(z_{1})$ in our $N$-point function $Z$.
Since $\frac{\partial}{\partial z_{1}}V_{\langle
1,1\rangle}(z_{1})=L_{-1}V_{\langle 1,1\rangle}(z_{1})=0$, we obtain
$\frac{\partial}{\partial z_{1}}Z=0$. In the case $N=3$, having
$\Delta_{1}=\Delta_{\langle 1,1\rangle}=0$ in the three-point function (2.21)
leads to

$\displaystyle\left<V_{\langle
1,1\rangle}(z_{1})V_{\Delta_{2}}(z_{2})V_{\Delta_{3}}(z_{3})\right>\propto(z_{1}-z_{2})^{\Delta_{3}-\Delta_{2}}(z_{1}-z_{3})^{\Delta_{2}-\Delta_{3}}(z_{2}-z_{3})^{-\Delta_{2}-\Delta_{3}}\
,$ (2.25)

and further imposing $z_{1}$-independence leads to

$\displaystyle\left<V_{\langle
1,1\rangle}(z_{1})V_{\Delta_{2}}(z_{2})V_{\Delta_{3}}(z_{3})\right>\neq
0\quad\implies\quad\Delta_{2}=\Delta_{3}\ .$ (2.26)

This coincides with the condition (2.20) that the two-point function
$\left<V_{\Delta_{2}}(z_{2})V_{\Delta_{3}}(z_{3})\right>$ does not vanish.
Actually, the field $V_{\langle 1,1\rangle}$ is an identity field, i.e. a
field whose presence does not affect correlation functions. (See Exercise
3.6.)

In the case of $V_{\langle 2,1\rangle}(z_{1})$, we have

$\displaystyle\left(L_{-1}^{2}+b^{2}L_{-2}\right)V_{\langle
2,1\rangle}(z_{1})=0\qquad\text{so that}\qquad L_{-2}V_{\langle
2,1\rangle}(z_{1})=-\frac{1}{b^{2}}\frac{\partial^{2}}{\partial
z_{1}^{2}}V_{\langle 2,1\rangle}(z_{1})\ .$ (2.27)

Using the local Ward identity (2.24), this leads to the second-order
Belavin–Polyakov–Zamolodchikov partial differential equation

$\displaystyle\left(\frac{1}{b^{2}}\frac{\partial^{2}}{\partial
z_{1}^{2}}+\sum_{i=2}^{N}\left(\frac{1}{z_{1}-z_{i}}\frac{\partial}{\partial
z_{i}}+\frac{\Delta_{i}}{(z_{1}-z_{i})^{2}}\right)\right)\left<V_{\langle
2,1\rangle}(z_{1})\prod_{i=2}^{N}V_{\Delta_{i}}(z_{i})\right>=0\ .$ (2.28)

More generally, a correlation function with the degenerate field $V_{\langle
r,s\rangle}$ obeys a partial differential equation of order $rs$.

######  Exercise 2.9 (Second-order BPZ equation for a three-point function)

Show that

$\displaystyle\left<V_{\langle
2,1\rangle}V_{\Delta_{2}}V_{\Delta_{3}}\right>\neq 0\quad\implies\quad
2(\Delta_{2}-\Delta_{3})^{2}+b^{2}(\Delta_{2}+\Delta_{3})-2\Delta_{\langle
2,1\rangle}^{2}-b^{2}\Delta_{\langle 2,1\rangle}=0\ .$ (2.29)

Up to reflections of momentums, show that this is equivalent to

$\displaystyle\alpha_{2}=\alpha_{3}\pm\frac{b}{2}\ .$ (2.30)

In the case of a four-point function, the BPZ equation amounts to a
differential equation for the function of one variable $G(z)$, which therefore
belongs to a finite-dimensional space of solutions.

######  Exercise 2.10 (BPZ second-order differential equation)

Show that the second-order BPZ equation for $G(z)=\Big{<}V_{\langle
2,1\rangle}(z)V_{\Delta_{1}}(0)V_{\Delta_{2}}(\infty)V_{\Delta_{3}}(1)\Big{>}$
is

$\displaystyle\left\\{\frac{z(1-z)}{b^{2}}\frac{\partial^{2}}{\partial
z^{2}}+(2z-1){\frac{\partial}{\partial z}}+\Delta_{\langle
2,1\rangle}+\frac{\Delta_{1}}{z}-\Delta_{2}+\frac{\Delta_{3}}{1-z}\right\\}G(z)=0\
,$ (2.31)

##  3 Conformal bootstrap

We have seen how conformal symmetry leads to linear equations for correlation
functions: Ward identities and BPZ equations. In order to fully determine
correlation functions, we need additional, nonlinear equations, and therefore
additional axioms: single-valuedness of correlation functions, and existence
of operator product expansions. Using these axioms for studying conformal
field theories is called the conformal bootstrap method.

###  3.1 Single-valuedness

######  Axiom 3.1 (Single-valuedness)

Correlation functions are single-valued functions of the positions, i.e. they
have trivial monodromies.

Our two-point function (2.20) however has nontrivial monodromy unless
$\Delta_{1}\in\frac{1}{2}\mathbb{Z}$, as a result of solving holomorphic Ward
identities. We would rather have a single-valued function of the type
$|z_{1}-z_{2}|^{-4\Delta_{1}}=(z_{1}-z_{2})^{-2\Delta_{1}}(\bar{z}_{1}-\bar{z}_{2})^{-2\Delta_{1}}$.
This suggests that we need antiholomorphic Ward identities as well, and
therefore a second copy of the Virasoro algebra.

######  Axiom 3.2 (Left and right Virasoro algebras)

We have two mutually commuting Virasoro symmetry algebras with the same
central charge, called left-moving or holomorphic, and right-moving or
antiholomorphic. Their generators are written $L_{n},\bar{L}_{n}$, with in
particular

$\displaystyle\frac{\partial}{\partial
z}V(z)=L_{-1}V(z)\quad,\quad\frac{\partial}{\partial\bar{z}}V(z)=\bar{L}_{-1}V(z)\
.$ (3.1)

The generators of conformal transformations are the diagonal combinations
$L_{n}+\bar{L}_{n}$.

Let us point out that we do not use the widespread notation $f(z,\bar{z})$ for
a generic function, and $f(z)$ for a locally holomorphic function (i.e. a
function such that $\frac{\partial}{\partial\bar{z}}f(z)=0$). This notation
comes from a complexification of the Riemann sphere that makes $z$ and
$\bar{z}$ independent. For us, $\bar{z}$ is always the complex conjugate of
$z$, so the notation $f(z,\bar{z})$ would be redundant.

Let us now consider left- and right-primary fields
$V_{\Delta_{i},\bar{\Delta}_{i}}(z_{i})$. According to eq. (2.21), the three-
point function of such fields is

$\displaystyle\left<\prod_{i=1}^{3}V_{\Delta_{i},\bar{\Delta}_{i}}(z_{i})\right>\propto(z_{1}-z_{2})^{\Delta_{3}-\Delta_{1}-\Delta_{2}}(\bar{z}_{1}-\bar{z}_{2})^{\bar{\Delta}_{3}-\bar{\Delta}_{1}-\bar{\Delta}_{2}}\times\cdots\
.$ (3.2)

Single-valuedness as a function of $z_{i}$ constrains the spins
$s_{i}=\Delta_{i}-\bar{\Delta}_{i}$,

$\displaystyle s_{i}+s_{j}-s_{k}\in\mathbb{Z}\ ,\qquad(i\neq j\neq k)\ .$
(3.3)

This implies in particular $2s_{i}\in\mathbb{Z}$. In other words, any primary
field $V_{\Delta,\bar{\Delta}}(z)$ must obey

$\displaystyle\Delta-\bar{\Delta}\in\frac{1}{2}\mathbb{Z}\ .$ (3.4)

The simplest case is $\Delta=\bar{\Delta}$, which leads to the definition

######  Definition 3.3 (Diagonal states, diagonal fields and diagonal
spectrums)

A primary state or field is called diagonal if it has the same left and right
conformal dimensions. A spectrum is called diagonal if all primary states are
diagonal.

From now on we will use the notation $V_{\Delta}(z)$ for the diagonal field
$V_{\Delta,\Delta}(z)$.

###  3.2 Operator product expansion and crossing symmetry

######  Axiom 3.4 (Operator product expansion)

Let $(|w_{i}\rangle)$ be a basis of the spectrum. There exist coefficients
$C^{i}_{12}(z_{1},z_{2})$ such that we have the operator product expansion
(OPE)

$\displaystyle
V_{|w_{1}\rangle}(z_{1})V_{|w_{2}\rangle}(z_{2})=\sum_{i}C^{i}_{12}(z_{1},z_{2})V_{|w_{i}\rangle}(z_{2})\
.$ (3.5)

In a correlation function, this sum converges for $z_{1}$ sufficiently close
to $z_{2}$.

While the linear equations of Section 2 relate $N$-point functions to other
$N$-point function, OPEs allow us to reduce $N$-point functions to
$(N-1)$-point functions. The price to pay for this reduction is the
introduction of OPE coefficients. Using OPEs iteratively, we can actually
reduce any correlation function to a combination of OPE coefficients, and two-
point functions. (We stop at two-point functions because they are simple
enough for being considered as known quantities.)

Let us study some properties of OPE coefficients. Assuming that the spectrum
is made of diagonal primary states and their descendent states, the OPE of two
primary fields is

$\displaystyle V_{\Delta_{1}}(z_{1})V_{\Delta_{2}}(z_{2})=\sum_{\Delta\in
S}C_{\Delta_{1},\Delta_{2},\Delta}|z_{1}-z_{2}|^{2(\Delta-\Delta_{1}-\Delta_{2})}\Big{(}V_{\Delta}(z_{2})+O(z_{1}-z_{2})\Big{)}\
,$ (3.6)

where the subleading terms are contributions of descendents fields. In
particular, the $z_{1},z_{2}$-dependence of the coefficients is dictated by
the behaviour (2.18) of correlation functions under translations $z_{i}\to
z_{i}+c$ and dilatations $z_{i}\to\lambda z_{i}$, leaving a
$z_{i}$-independent unknown factor $C_{\Delta_{1},\Delta_{2},\Delta}$. Then,
as in correlation functions, contributions of descendents are deduced from
contributions of primaries via local Ward identitites.

######  Exercise 3.5 (Computing the OPE of primary fields)

Compute the first subleading term in the OPE (3.6), and find

$\displaystyle
O(z_{1}-z_{2})=\frac{\Delta+\Delta_{1}-\Delta_{2}}{2\Delta}\Big{(}(z_{1}-z_{2})L_{-1}+(\bar{z}_{1}-\bar{z}_{2})\bar{L}_{-1}\Big{)}V_{\Delta}(z_{2})+O((z_{1}-z_{2})^{2})\
.$ (3.7)

Hints: Insert $\oint_{C}dz(z-z_{2})^{2}T(z)$ on both sides of the OPE, for a
contour $C$ that encloses both $z_{1}$ and $z_{2}$. Compute the relevant
contour integrals with the help of eq. (2.11).

######  Exercise 3.6 ($V_{\langle 1,1\rangle}$ is an identity field)

Using $\frac{\partial}{\partial z_{1}}V_{\langle 1,1\rangle}(z_{1})=0$, show
that the OPE of $V_{\langle 1,1\rangle}$ with another primary field is of the
form

$\displaystyle V_{\langle
1,1\rangle}(z_{1})V_{\Delta}(z_{2})=C_{\Delta}V_{\Delta}(z_{2})\ ,$ (3.8)

where the subleading terms vanish. Inserting this OPE in a correlation
function, show that the constant $C_{\Delta}$ actually does not depend on
$\Delta$. Deduce that, up to a factor $C=C_{\Delta}$, the field $V_{\langle
1,1\rangle}$ is an identity field.

We will now express three- and four-point functions in terms of OPE
coefficients, and deduce constraints on these coefficients. Inserting the OPE
in a three-point function of primary fields, we find

$\displaystyle\left<\prod_{i=1}^{3}V_{\Delta_{i}}(z_{i})\right>$
$\displaystyle=\sum_{\Delta\in
S}C_{\Delta_{1},\Delta_{2},\Delta}|z_{1}-z_{2}|^{2(\Delta-\Delta_{1}-\Delta_{2})}\Big{(}\left<V_{\Delta}(z_{2})V_{\Delta_{3}}(z_{3})\right>+O(z_{1}-z_{2})\Big{)}\
,$ (3.9)
$\displaystyle=C_{\Delta_{1},\Delta_{2},\Delta_{3}}|z_{1}-z_{2}|^{2(\Delta_{3}-\Delta_{1}-\Delta_{2})}\Big{(}|z_{2}-z_{3}|^{-4\Delta_{3}}+O(z_{1}-z_{2})\Big{)}\
,$ (3.10)

assuming the two-point function is normalized as
$\left<V_{\Delta}(z_{1})V_{\Delta}(z_{2})\right>=|z_{1}-z_{2}|^{-4\Delta}$. It
follows that $C_{\Delta_{1},\Delta_{2},\Delta_{3}}$ coincides with the
undertermined constant prefactor of the three-point function (3.2). This
factor is called the three-point structure constant, and we have

$\displaystyle\left<\prod_{i=1}^{3}V_{\Delta_{i}}(z_{i})\right>=C_{\Delta_{1},\Delta_{2},\Delta_{3}}|z_{1}-z_{2}|^{2(\Delta_{3}-\Delta_{1}-\Delta_{2})}|z_{1}-z_{3}|^{2(\Delta_{2}-\Delta_{1}-\Delta_{3})}|z_{2}-z_{3}|^{2(\Delta_{1}-\Delta_{2}-\Delta_{3})}\
.$ (3.11)

Let us now insert the OPE in a four-point function of primary fields:

$\displaystyle\Big{<}V_{\Delta_{1}}(z)V_{\Delta_{2}}(0)V_{\Delta_{3}}(\infty)V_{\Delta_{4}}(1)\Big{>}$
$\displaystyle=\sum_{\Delta\in
S}C_{\Delta_{1},\Delta_{2},\Delta}|z|^{2(\Delta-\Delta_{1}-\Delta_{2})}$
$\displaystyle\qquad\qquad\times\left(\Big{<}V_{\Delta}(0)V_{\Delta_{3}}(\infty)V_{\Delta_{4}}(1)\Big{>}+O(z)\right)\
,$ (3.12) $\displaystyle=\sum_{\Delta\in
S}C_{\Delta_{1},\Delta_{2},\Delta}C_{\Delta,\Delta_{3},\Delta_{4}}|z|^{2(\Delta-\Delta_{1}-\Delta_{2})}\Big{(}1+O(z)\Big{)}\
.$ (3.13)

The contributions of descendents factorize into those of left-moving
descendents, generated by the operators $L_{n<0}$, and right-moving
descendents, generated by $\bar{L}_{n<0}$. So the last factor has a
holomorphic factorization such that

$\displaystyle\Big{<}V_{\Delta_{1}}(z)V_{\Delta_{2}}(0)V_{\Delta_{3}}(\infty)V_{\Delta_{4}}(1)\Big{>}=\sum_{\Delta\in
S}C_{\Delta_{1},\Delta_{2},\Delta}C_{\Delta,\Delta_{3},\Delta_{4}}\mathcal{F}^{(s)}_{\Delta}(z)\mathcal{F}^{(s)}_{\Delta}(\bar{z})\
.$ (3.14)

######  Definition 3.7 (Conformal block)

The four-point conformal block on the sphere,

$\displaystyle\mathcal{F}^{(s)}_{\Delta}(z)=z^{\Delta-\Delta_{1}-\Delta_{2}}\Big{(}1+O(z)\Big{)}\
,$ (3.15)

is the normalized contribution of the Verma module $\mathcal{V}_{\Delta}$ to a
four-point function, obtained by summing over left-moving descendents. Its
dependence on $c,\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}$ are kept
implicit. The label $(s)$ stands for $s$-channel, we will soon see what this
means.

Conformal blocks are in principle known, as they are universal functions,
entirely determined by conformal symmetry. This is analogous to characters of
representations, also known as zero-point conformal blocks on the torus.

######  Exercise 3.8 (Computing conformal blocks)

Compute the conformal block $\mathcal{F}^{(s)}_{\Delta}(z)$ up to the order
$O(z)$, and find

$\displaystyle\mathcal{F}^{(s)}_{\Delta}(z)=z^{\Delta-\Delta_{1}-\Delta_{2}}\left(1+\frac{(\Delta+\Delta_{1}-\Delta_{2})(\Delta+\Delta_{4}-\Delta_{3})}{2\Delta}z+O(z^{2})\right)\
.$ (3.16)

Show that the first-order term has a pole when the Verma module
$\mathcal{V}_{\Delta}$ has a null vector at level one. Compute the residue of
this pole. Compare the condition that this residue vanishes with the condition
(2.26) that three-point functions involving $V_{\langle 1,1\rangle}$ exist.

Our axiom 2.7 on the commutativity of fields implies that the OPE is
associative, and that we can use the OPE of any two fields in a four-point
function. In particular, using the OPE of the first and fourth fields, we
obtain

$\displaystyle\Big{<}V_{\Delta_{1}}(z)V_{\Delta_{2}}(0)V_{\Delta_{3}}(\infty)V_{\Delta_{4}}(1)\Big{>}=\sum_{\Delta\in
S}C_{\Delta,\Delta_{1},\Delta_{4}}C_{\Delta_{2},\Delta_{3},\Delta}\mathcal{F}^{(t)}_{\Delta}(z)\mathcal{F}^{(t)}_{\Delta}(\bar{z})\
,$ (3.17)

where
$\mathcal{F}^{(t)}_{\Delta}(z)=(z-1)^{\Delta-\Delta_{1}-\Delta_{4}}\Big{(}1+O(z-1)\Big{)}$
is a $t$-channel conformal block. The equality of our two decompositions
(3.14) and (3.17) of the four-point function is called crossing symmetry,
schematically

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This equation holds if the sums on both sides converge, that is if $z$ is
sufficiently close to both $0$ and $1$,

(3.19)

Given the spectrum $S$, crossing symmetry is a system of quadratic equations
for the structure constant $C_{\Delta_{1},\Delta_{2},\Delta_{3}}$. Requiring
that this system has solutions is a strong constraint on the spectrum. In
diagonal theories, crossing symmetry implies the existence of arbitrary
correlation functions on the sphere [7], and not just of four-point functions.

###  3.3 Degenerate fields and the fusion product

Crossing symmetry equations are powerful, but typically involve infinite sums,
which makes them difficult to solve. However, if at least one field is
degenerate, then the four-point function belongs to the finite-dimensional
space of solutions of a BPZ equation, and is therefore a combination of
finitely many conformal blocks. For example, $\Big{<}V_{\langle
2,1\rangle}(z)V_{\Delta_{1}}(0)V_{\Delta_{2}}(\infty)V_{\Delta_{3}}(1)\Big{>}$
is a combination of only two holomorphic $s$-channel conformal blocks. These
two blocks are a particular basis of solutions of the BPZ equation (2.31).
They are fully characterized by their asymptotic behaviour near $z=0$ (3.15),
where the BPZ equation allows only two values of $\Delta$, namely
$\Delta\in\\{\Delta(\alpha_{1}-\frac{b}{2}),\Delta(\alpha_{1}+\frac{b}{2})\\}$.
The explicit expressions of these blocks are

$\displaystyle\mathcal{F}^{(s)}_{\alpha_{1}-\frac{b}{2}}(z)=z^{b\alpha_{1}}(1-z)^{b\alpha_{3}}F(A,B,C,z)\quad,\quad\mathcal{F}^{(s)}_{\alpha_{1}+\frac{b}{2}}(z)=\left.\mathcal{F}^{(s)}_{\alpha_{1}-\frac{b}{2}}(z)\right|_{\alpha_{1}\to
Q-\alpha_{1}}\ ,$ (3.20)

where $F(A,B,C,z)$ is the hypergeometric function with parameters

$\displaystyle\left\\{\begin{array}[]{l}A=\frac{1}{2}+b(\alpha_{1}+\alpha_{3}-Q)+b(\alpha_{2}-\tfrac{Q}{2})\
,\\\ B=\frac{1}{2}+b(\alpha_{1}+\alpha_{3}-Q)-b(\alpha_{2}-\tfrac{Q}{2})\ ,\\\
C=1+b(2\alpha_{1}-Q)\ .\end{array}\right.$ (3.24)

Another basis of solutions of the same BPZ equation is given by two
$t$-channel blocks, which are characterized by their power-like behaviour near
$z=1$. Their explicit expressions are

$\displaystyle\mathcal{F}^{(t)}_{\alpha_{3}-\frac{b}{2}}(z)$
$\displaystyle=z^{b\alpha_{1}}(1-z)^{b\alpha_{3}}F(A,B,A+B-C+1,1-z)\ ,$
$\displaystyle\mathcal{F}^{(t)}_{\alpha_{3}+\frac{b}{2}}(z)$
$\displaystyle=\left.\mathcal{F}^{(t)}_{\alpha_{3}-\frac{b}{2}}(z)\right|_{\alpha_{3}\to
Q-\alpha_{3}}\ .$ (3.25)

The presence of only two $s$-channel fields with momentums
$\alpha_{1}\pm\frac{b}{2}$, and the constraint (2.30) on momentums in the
three-point function $\left<V_{\langle
2,1\rangle}V_{\alpha_{2}}V_{\alpha_{3}}\right>$, both mean that the operator
product expansion $V_{\langle 2,1\rangle}(z)V_{\alpha_{1}}(0)$ involves only
two primary fields $V_{\alpha_{1}\pm\frac{b}{2}}(0)$.

######  Axiom 3.9 (Fusion product)

There is a bilinear, associative product of representations of the Virasoro
algebra, that encodes the constraints on OPEs from Virasoro symmetry and null
vectors. In particular,

$\displaystyle\mathcal{R}_{\langle
1,1\rangle}\times\mathcal{V}_{\alpha}=\mathcal{V}_{\alpha}\quad,\quad\mathcal{R}_{\langle
2,1\rangle}\times\mathcal{V}_{\alpha}=\sum_{\pm}\mathcal{V}_{\alpha\pm\frac{b}{2}}\quad,\quad\mathcal{R}_{\langle
1,2\rangle}\times\mathcal{V}_{\alpha}=\sum_{\pm}\mathcal{V}_{\alpha\pm\frac{1}{2b}}\
.$ (3.26)

The fusion product can be defined algebraically [8]: the fusion product of two
representations coincides with their tensor product as a vector space, where
however the Virasoro algebra does not act as it would in the tensor product.
(In the tensor product, central charges add.)

Using the associativity of the fusion product, we have

$\displaystyle\mathcal{R}_{\langle 2,1\rangle}\times\mathcal{R}_{\langle
2,1\rangle}\times\mathcal{V}_{\alpha}=\mathcal{R}_{\langle
2,1\rangle}\times\left(\sum_{\pm}\mathcal{V}_{\alpha\pm\frac{b}{2}}\right)=\mathcal{V}_{\alpha-b}+2\cdot\mathcal{V}_{\alpha}+\mathcal{V}_{\alpha+b}\
.$ (3.27)

Since the fusion product of $\mathcal{R}_{\langle
2,1\rangle}\times\mathcal{R}_{\langle 2,1\rangle}$ with $\mathcal{V}_{\alpha}$
has finitely many terms, $\mathcal{R}_{\langle
2,1\rangle}\times\mathcal{R}_{\langle 2,1\rangle}$ must be a degenerate
representation. On the other hand, eq. (3.26) implies that
$\mathcal{R}_{\langle 2,1\rangle}\times\mathcal{R}_{\langle 2,1\rangle}$ is
made of representations with momentums $\alpha_{\langle
2,1\rangle}\pm\frac{b}{2}=0,-b$. The degenerate representation with momentum
$0$ is $\mathcal{R}_{\langle 1,1\rangle}$. Calling $\mathcal{R}_{\langle
3,1\rangle}$ the degenerate representation with momentum $-b$, we just found

$\displaystyle\mathcal{R}_{\langle 2,1\rangle}\times\mathcal{R}_{\langle
2,1\rangle}=\mathcal{R}_{\langle 1,1\rangle}+\mathcal{R}_{\langle
3,1\rangle}\quad,\quad\mathcal{R}_{\langle
3,1\rangle}\times\mathcal{V}_{\alpha}=\mathcal{V}_{\alpha-b}+\mathcal{V}_{\alpha}+\mathcal{V}_{\alpha+b}\
.$ (3.28)

It can be checked that $\mathcal{R}_{\langle 3,1\rangle}$ has a vanishing null
vector at level $3$, so that our definition of $\mathcal{R}_{\langle
3,1\rangle}$ from fusion agrees with the definition from representation theory
in Section 1.3.

######  Exercise 3.10 (Higher degenerate representations)

Show that there exist degenerate representations $\mathcal{R}_{\langle
r,s\rangle}$ (for $r,s\in\mathbb{N}^{*}$) with momentums $\alpha_{\langle
r,s\rangle}$ (1.23), such that

$\displaystyle\mathcal{R}_{\langle r,s\rangle}\times\mathcal{V}_{\alpha}$
$\displaystyle=\sum_{i=0}^{r-1}\sum_{j=0}^{s-1}\mathcal{V}_{\alpha+\alpha_{\langle
r,s\rangle}+ib+jb^{-1}}\ ,$ (3.29) $\displaystyle\mathcal{R}_{\langle
r_{1},s_{1}\rangle}\times\mathcal{R}_{\langle r_{2},s_{2}\rangle}$
$\displaystyle=\sum_{r_{3}\overset{2}{=}|r_{1}-r_{2}|+1}^{r_{1}+r_{2}-1}\
\sum_{s_{3}\overset{2}{=}|s_{1}-s_{2}|+1}^{s_{1}+s_{2}-1}\mathcal{R}_{\langle
r_{3},s_{3}\rangle}\ ,$ (3.30)

where the superscript in $\overset{2}{=}$ indicates that the corresponding sum
runs by increments of $2$.

The knowledge of such fusion products will be crucial for defining and
studying minimal models, whose spectrums are made of degenerate
representations.

##  4 Minimal models and Liouville theory

Let us start the investigation of specific conformal field theories.

######  Definition 4.1 (Conformal field theory)

A (model of) conformal field theory on the Riemann sphere is a spectrum $S$
and a set of correlation functions
$\left<\prod_{i=1}^{N}V_{|w_{i}\rangle}(z_{i})\right>$ with $|w_{i}\rangle\in
S$ that obey all our axioms, in particular crossing symmetry.

######  Definition 4.2 (Defining and solving)

To define a conformal field theory is to give principles that uniquely
determine its spectrum and correlation functions. To solve a conformal field
theory is to actually compute them.

In this Section we will define minimal models and Liouville theory. In Section
5 we will solve them.

###  4.1 Diagonal minimal models

######  Definition 4.3 (Minimal model)

A minimal model is a conformal field theory whose spectrum is made of finitely
many irreducible representations of the product of the left and the right
Virasoro algebras.

Although there exist non-diagonal minimal models [4], we focus on diagonal
minimal models, whose spectrums are of the type

$\displaystyle S=\bigoplus_{\mathcal{R}}\mathcal{R}\otimes\mathcal{\bar{R}}\
,$ (4.1)

where $\mathcal{R}$ and $\mathcal{\bar{R}}$ denote the same Virasoro
representation, viewed as a representation of the left- or right-moving
Virasoro algebra respectively.

######  Axiom 4.4 (Degenerate spectrum)

All representations that appear in the spectrum of a minimal model are
degenerate.

It is natural to use degenerate representations, because in an OPE of
degenerate fields, only finitely many representations can appear. Conversely,
we now assume that all representations that are allowed by fusion do appear in
the spectrum, in other words

######  Axiom 4.5 (Closure under fusion)

The spectrum is closed under fusion.

Let us assume that the spectrum contains a nontrivial degenerate
representation such as $\mathcal{R}_{\langle 2,1\rangle}$. Fusing it with
itself, we get $\mathcal{R}_{\langle 1,1\rangle}$ and $\mathcal{R}_{\langle
3,1\rangle}$. Fusing multiple times, we get $(\mathcal{R}_{\langle
r,1\rangle})_{r\in\mathbb{N}^{*}}$ due to $\mathcal{R}_{\langle
2,1\rangle}\times\mathcal{R}_{\langle r,1\rangle}=\mathcal{R}_{\langle
r-1,1\rangle}+\mathcal{R}_{\langle r+1,1\rangle}$. If the spectrum moreover
contains $\mathcal{R}_{\langle 1,2\rangle}$, then it must contain all
degenerate representations.

######  Definition 4.6 (Generalized minimal model)

For any value of the central charge $c\in\mathbb{C}$, the generalized minimal
model is the conformal field theory whose spectrum is

$\displaystyle
S^{\mathrm{GMM}}=\bigoplus_{r=1}^{\infty}\bigoplus_{s=1}^{\infty}\mathcal{R}_{\langle
r,s\rangle}\otimes\mathcal{\bar{R}}_{\langle r,s\rangle}\ ,$ (4.2)

assuming it exists and is unique.

So, using only degenerate representations is not sufficient for building
minimal models. In order to have even fewer fields in fusion products, let us
consider representations that are multiply degenerate. For example, if
$\mathcal{R}_{\langle 2,1\rangle}=\mathcal{R}_{\langle 1,3\rangle}$ has two
vanishing null vectors, then $\mathcal{R}_{\langle
2,1\rangle}\times\mathcal{R}_{\langle 2,1\rangle}=\mathcal{R}_{\langle
1,1\rangle}$ has only one term, as the term $\mathcal{R}_{\langle 3,1\rangle}$
is not allowed by the fusion rules of $\mathcal{R}_{\langle 1,3\rangle}$.

In order for a representation to have two null vectors, we however need a
coincidence of the type $\Delta_{\langle r,s\rangle}=\Delta_{\langle
r^{\prime},s^{\prime}\rangle}$. This is equivalent to $\alpha_{\langle
r,s\rangle}\in\\{\alpha_{\langle
r^{\prime},s^{\prime}\rangle},Q-\alpha_{\langle
r^{\prime},s^{\prime}\rangle}\\}$, and it follows that $b^{2}$ is rational,

$\displaystyle
b^{2}=-\frac{q}{p}\qquad\text{with}\qquad\left\\{\begin{array}[]{l}(p,q)\in\mathbb{N}^{*}\times\mathbb{Z}^{*}\\\
p\wedge q=1\end{array}\right.\qquad\text{i.e.}\qquad
c=1-6\frac{(q-p)^{2}}{pq}\ .$ (4.5)

For any integers $r,s$, we then have the coincidence

$\displaystyle\Delta_{\langle r,s\rangle}=\Delta_{\langle p-r,q-s\rangle}\ .$
(4.6)

In particular, for $1\leq r\leq p-1$ and $1\leq s\leq q-1$, there exists a
doubly degenerate representation $\mathcal{R}_{\langle
r,s\rangle}=\mathcal{R}_{\langle p-r,q-s\rangle}$. The diagonal spectrum built
from representations of this type is

$\displaystyle
S_{p,q}=\frac{1}{2}\bigoplus_{r=1}^{p-1}\bigoplus_{s=1}^{q-1}\mathcal{R}_{\langle
r,s\rangle}\otimes\mathcal{\bar{R}}_{\langle r,s\rangle}\ ,$ (4.7)

where the factor $\frac{1}{2}$ is here to avoid counting the same
representation twice. This spectrum is not empty provided the coprime integers
$p,q$ are both greater than $2$,

$\displaystyle p,q\geq 2\ ,$ (4.8)

which implies in particular $b,Q\in i\mathbb{R}$ and $c<1$. For other values
of $p,q$, it turns out that no minimal models exist.

######  Exercise 4.7 (Closure of minimal model spectrums under fusion)

Show that $S_{p,q}$ is closed under fusion. If you are brave, compute the
fusion products of the representations that appear in $S_{p,q}$. If you are
very brave, show that $p,q\geq 2$ is a necessary condition for the existence
of a finite, nontrivial set of multiply degenerate representations that closes
under fusion.

######  Definition 4.8 (Diagonal minimal model)

For $p,q\geq 2$ coprime integers, the $(p,q)$ minimal model is the conformal
field theory whose spectrum is $S_{p,q}$, assuming it exists and is unique.

For example, the minimal model with the central charge $c=\frac{1}{2}$ has the
spectrum $S_{4,3}$,

$\displaystyle\left\\{\begin{array}[]{l}\Delta_{\langle
1,1\rangle}=\Delta_{\langle 3,2\rangle}=0\ ,\\\ \Delta_{\langle
1,2\rangle}=\Delta_{\langle 3,1\rangle}=\frac{1}{2}\ ,\\\ \Delta_{\langle
2,1\rangle}=\Delta_{\langle 2,2\rangle}=\frac{1}{16}\
.\end{array}\right.\qquad\iff\quad\text{the Kac
table}\quad\begin{array}[]{c|ccc}2&\frac{1}{2}&\frac{1}{16}&0\\\
1&0&\frac{1}{16}&\frac{1}{2}\\\ \hline\cr&1&2&3\end{array}$ (4.15)

###  4.2 Liouville theory

######  Definition 4.9 (Liouville theory)

For any value of the central charge $c\in\mathbb{C}$, Liouville theory is the
conformal field theory whose spectrum is

$\displaystyle
S^{\mathrm{Liouville}}=\int_{\frac{Q}{2}+i\mathbb{R}_{+}}d\alpha\
\mathcal{V}_{\alpha}\otimes\bar{\mathcal{V}}_{\alpha}\ ,$ (4.16)

and whose correlation functions are smooth functions of $b$ and $\alpha$,
assuming it exists and its unique.

Let us give some justification for this definition. We are looking for a
diagonal theory whose spectrum is a continuum of representations of the
Virasoro algebra. For $c\in\mathbb{R}$ it is natural to assume
$\Delta\in\mathbb{R}$. Let us write this condition in terms of the momentum
$\alpha$,

$\displaystyle\Delta\in\mathbb{R}\iff\alpha\in\mathbb{R}\cup\left(\frac{Q}{2}+i\mathbb{R}\right)\
,\qquad\leavevmode\hbox to97.68pt{\vbox
to69.89pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{}{{}} {}{}{{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\frac{Q}{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
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}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
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}{}\pgfsys@moveto{-18.20955pt}{0.0pt}\pgfsys@lineto{62.59511pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (4.17)

From Axiom 1.4, we need $\Delta$ to be bounded from below, and the natural
bound is

$\displaystyle\Delta_{\text{min}}=\Delta\left(\alpha=\frac{Q}{2}\right)=\frac{Q^{2}}{4}=\frac{c-1}{24}\
.$ (4.18)

This leads to $\alpha\in\frac{Q}{2}+i\mathbb{R}$. Assuming that each allowed
representation appears only once in the spectrum, we actually restrict the
momentums to $\alpha\in\frac{Q}{2}+i\mathbb{R}_{+}$, due to the reflection
symmetry (1.16). We then obtain our guess (4.16) for the spectrum,
equivalently $S^{\mathrm{Liouville}}=\int_{\frac{c-1}{24}}^{\infty}d\Delta\
\mathcal{V}_{\Delta}\otimes\bar{\mathcal{V}}_{\Delta}$. We take this guess to
hold not only for $c\in\mathbb{R}$, but also for $c\in\mathbb{C}$ by
analyticity.

Other guesses for the lower bound may seem equally plausible, in particular
$\Delta_{\text{min}}=0$. In the spirit of the axiomatic method, the arbiter
for such guesses is the consistency of the resulting theory. This will be
tested in Section 5.3, and the spectrum $S^{\mathrm{Liouville}}$ will turn out
to be correct.

Let us schematically write two- and three-point functions in Liouville theory,
as well as OPEs:

$\displaystyle\Big{<}V_{\alpha_{1}}V_{\alpha_{2}}\Big{>}$
$\displaystyle=B(\alpha_{1})\delta(\alpha_{1}-\alpha_{2})\ ,$ (4.19)
$\displaystyle\Big{<}V_{\alpha_{1}}V_{\alpha_{2}}V_{\alpha_{3}}\Big{>}$
$\displaystyle=C_{\alpha_{1},\alpha_{2},\alpha_{3}}\ ,$ (4.20) $\displaystyle
V_{\alpha_{1}}V_{\alpha_{2}}$
$\displaystyle=\int_{\frac{Q}{2}+i\mathbb{R}_{+}}d\alpha\,\frac{C_{\alpha_{1},\alpha_{2},\alpha}}{B(\alpha)}\Big{(}V_{\alpha}+\cdots\Big{)}\
,$ (4.21)

where the expression for the OPE coefficient
$\frac{C_{\alpha_{1},\alpha_{2},\alpha}}{B(\alpha)}$ in terms of two- and
three-point structure constants is obtained by inserting the OPE into a three-
point function. It would be possible to set $B(\alpha)=1$ by renormalizing the
primary fields $V_{\alpha}$, but this would prevent
$C_{\alpha_{1},\alpha_{2},\alpha_{3}}$ from being a meromorphic function of
the momentums, as we will see in Section 5.2.

In order to have reasonably simple crossing symmetry equations, we need
degenerate fields. But the spectrum of Liouville theory is made of Verma
modules, and does not involve any degenerate representations. In order to have
degenerate fields, we need a special axiom:

######  Axiom 4.10 (Degenerate fields in Liouville theory)

The degenerate fields $V_{\langle r,s\rangle}$, and their correlation
functions, exist.

By the existence of degenerate fields, we also mean that such fields and their
correlation functions obey suitable generalizations of our axioms. In
particular, we generalize Axiom 3.4 by assuming that there exists an OPE
between the degenerate field $V_{\langle 2,1\rangle}$, and a field
$V_{\alpha}$. However, according to the fusion rules (3.26), this OPE leads to
fields with momentums $\alpha\pm\frac{b}{2}$, and in general
$\alpha\in\frac{Q}{2}+i\mathbb{R}\centernot\implies(\alpha\pm\frac{b}{2})\in\frac{Q}{2}+i\mathbb{R}$.
We resort to the assumption in Definition 4.9 that correlation functions are
smooth functions of $\alpha$, and take $V_{\alpha}$ to actually be defined for
$\alpha\in\mathbb{C}$ by analytic continuation. (Or for $\alpha$ in a dense
open subset of $\mathbb{C}$, if there are singularities.) This allows us to
write the OPE

$\displaystyle V_{\langle 2,1\rangle}V_{\alpha}\sim
C_{-}(\alpha)V_{\alpha-\frac{b}{2}}+C_{+}(\alpha)V_{\alpha+\frac{b}{2}}\ ,$
(4.22)

where we introduced the degenerate OPE coefficients $C_{\pm}(\alpha)$.

##  5 Four-point functions

Let us determine the three-point structure constant by solving crossing
symmetry equations. We begin with the equations that come from four-point
functions with degenerate fields. These equations are enough for uniquely
determining the three-point structure constant.

###  5.1 Single-valued four-point functions

The four-point function $G(z)=\Big{<}V_{\langle
2,1\rangle}(z)V_{\Delta_{1}}(0)V_{\Delta_{2}}(\infty)V_{\Delta_{3}}(1)\Big{>}$
obeys second-order BPZ equations in $z$ and $\bar{z}$. The most general
solution of these two equations can be written in terms of the $s$-channel
conformal blocks $\mathcal{F}^{(s)}_{\alpha_{1}\pm\frac{b}{2}}$ of Section 3.3
as

$\displaystyle
G(z)=\sum_{i,j=\pm}c^{(s)}_{ij}\mathcal{F}^{(s)}_{\alpha_{1}+i\frac{b}{2}}(z)\mathcal{F}^{(s)}_{\alpha_{1}+j\frac{b}{2}}(\bar{z})\
.$ (5.1)

Let us determine how single-valuedness constrains the coefficients
$c^{(s)}_{ij}$. Our conformal blocks have singularities at $z=0,1,\infty$, and
single-valuedness is equivalent to $G(z)$ having trivial monodromy around
$z=0,1$. Our $s$-channel decomposition is convenient for studying the
monodromy around $z=0$: near this point, each one of the two $s$-channel
conformal blocks $\mathcal{F}^{(s)}_{\alpha_{1}\pm\frac{b}{2}}(z)$ behaves as
a power of $z$ (3.15), with its own exponent. For a generic value of
$\alpha_{1}$, the difference of their two exponents is not integer, and
$\mathcal{F}^{(s)}_{\alpha_{1}-\frac{b}{2}}(z)\mathcal{F}^{(s)}_{\alpha_{1}+\frac{b}{2}}(\bar{z})$
is not single-valued near $z=0$. On the other hand, the two terms
$\mathcal{F}^{(s)}_{\alpha_{1}\pm\frac{b}{2}}(z)\mathcal{F}^{(s)}_{\alpha_{1}\pm\frac{b}{2}}(\bar{z})$
are single-valued. We conclude that $c^{(s)}_{+-}=c^{(s)}_{-+}=0$. Similarly,
if we decompose the same four-point function in the $t$-channel basis of
conformal blocks,

$\displaystyle
G(z)=\sum_{i,j=\pm}c^{(t)}_{ij}\mathcal{F}^{(t)}_{\alpha_{1}+i\frac{b}{2}}(z)\mathcal{F}^{(t)}_{\alpha_{1}+j\frac{b}{2}}(\bar{z})\
,$ (5.2)

then single-valuedness near $z=1$ requires $c^{(t)}_{+-}=c^{(t)}_{-+}=0$. Now
the $s$\- and $t$-channel bases of solutions of the BPZ equation must be
related by a change of basis,

$\displaystyle\mathcal{F}^{(s)}_{\alpha_{1}+i\frac{b}{2}}(z)=\sum_{j=\pm}F_{ij}\mathcal{F}^{(t)}_{\alpha_{1}+j\frac{b}{2}}(z)\
,$ (5.3)

for some matrix $F_{ij}$ that is well-known in the case of our hypergeometric
conformal blocks (3.20). Inserting this change of basis in our $s$-channel
decomposition (5.1) of $G(z)$, we find a $t$-channel decomposition, with the
coefficients

$\displaystyle
c^{(t)}_{i^{\prime}j^{\prime}}=\sum_{i,j=\pm}c^{(s)}_{ij}F_{ii^{\prime}}F_{jj^{\prime}}\
.$ (5.4)

Inserting our single-valuedness conditions $c^{(t)}_{+-}=0$ and
$c^{(s)}_{-+}=c^{(s)}_{+-}=0$ in this relation, we deduce

$\displaystyle\frac{c_{++}^{(s)}}{c_{--}^{(s)}}=-\frac{F_{-+}F_{--}}{F_{++}F_{+-}}\
.$ (5.5)

Using the known formula for $F_{ij}$, this is explicitly

$\displaystyle\frac{c_{++}^{(s)}}{c_{--}^{(s)}}=\frac{\gamma(C)\gamma(C-1)}{\gamma(A)\gamma(B)\gamma(C-A)\gamma(C-B)}\quad\text{with}\quad\gamma(x)=\frac{\Gamma(x)}{\Gamma(1-x)}\
,$ (5.6)

where the combinations $A,B,C$ of the momentums
$\alpha_{1},\alpha_{2},\alpha_{3}$ are given in eq. (3.24).

This concludes the mathematical exercise of finding single-valued solutions of
the hypergeometric equation. Now, in the case of Liouville theory, let us
determine the coefficients $c^{(s)}_{ij}$ in terms of OPE coefficients and
three-point structure constants. Using the degenerate OPE (4.22) and the
three-point function (4.20), we find

(5.7)

Then eq. (5.6) becomes a shift equation for the dependence of
$C_{\alpha_{1},\alpha_{2},\alpha_{3}}$ on $\alpha_{1}$,

$\displaystyle\frac{C_{+}(\alpha_{1})C_{\alpha_{1}+\frac{b}{2},\alpha_{2},\alpha_{3}}}{C_{-}(\alpha_{1})C_{\alpha_{1}-\frac{b}{2},\alpha_{2},\alpha_{3}}}=\frac{\gamma(b(2\alpha_{1}-Q))\gamma(1+b(2\alpha_{1}-Q))}{\prod_{\pm,\pm}\gamma(\frac{1}{2}+b(\alpha_{1}-\frac{Q}{2})\pm
b(\alpha_{2}-\frac{Q}{2})\pm b(\alpha_{3}-\frac{Q}{2}))}\ .$ (5.8)

In order to find the three-point structure constant
$C_{\alpha_{1},\alpha_{2},\alpha_{3}}$, we need to constrain the degenerate
OPE coefficients $C_{\pm}(\alpha)$. To do this, we consider the special case
where the last field is degenerate too, i.e. the four-point function
$\Big{\langle}V_{\langle
2,1\rangle}(z)V_{\alpha}(0)V_{\alpha}(\infty)V_{\langle
2,1\rangle}(1)\Big{\rangle}$. In this case, using the degenerate OPE (4.22)
twice, and the two-point function (4.19), we find

(5.9)

Then eq. (5.6) boils down to

$\displaystyle\frac{C_{+}(\alpha)^{2}B(\alpha+\tfrac{b}{2})}{C_{-}(\alpha)^{2}B(\alpha-\tfrac{b}{2})}=\frac{\gamma(b(2\alpha-Q))}{\gamma(-b(2\alpha-Q))}\frac{\gamma(-b^{2}-b(2\alpha-Q))}{\gamma(-b^{2}+b(2\alpha-Q))}\
.$ (5.10)

Moreover, if we had the degenerate field $V_{\langle 1,2\rangle}$ instead of
$V_{\langle 2,1\rangle}$ in our four-point functions, we would obtain the
equations (5.8) and (5.10) with $b\to\frac{1}{b}$. Next, we will solve these
equations.

###  5.2 Determining three-point structure constants

In order to solve the shift equations for
$C_{\alpha_{1},\alpha_{2},\alpha_{3}}$, we need a function that produces Gamma
functions when its argument is shifted by $b$ or $\frac{1}{b}$.

######  Exercise 5.1 (Upsilon function)

For $b>0$, show that there is a unique (up to a constant factor) holomorphic
function $\Upsilon_{b}(x)$ that obeys the shift equations

$\displaystyle\frac{\Upsilon_{b}(x+b)}{\Upsilon_{b}(x)}=b^{1-2bx}\gamma(bx)\qquad\text{and}\qquad\frac{\Upsilon_{b}(x+\frac{1}{b})}{\Upsilon_{b}(x)}=b^{\frac{2x}{b}-1}\gamma(\tfrac{x}{b})\
.$ (5.11)

For $ib>0$, show that the meromorphic function

$\displaystyle\hat{\Upsilon}_{b}(x)=\frac{1}{\Upsilon_{ib}(-ix+ib)}\ ,$ (5.12)

obeys shift equations that differ from eq. (5.11) by
$b^{\cdots}\to(ib)^{\cdots}$.

The function that solves the exercise can be written as

$\displaystyle\Upsilon_{b}(x)=\lambda_{b}^{(\frac{Q}{2}-x)^{2}}\prod_{m,n=0}^{\infty}f\left(\frac{\frac{Q}{2}-x}{\frac{Q}{2}+mb+nb^{-1}}\right)\quad\text{with}\quad
f(x)=(1-x^{2})e^{x^{2}}\ ,$ (5.13)

where $\lambda_{b}$ is an unimportant $b$-dependent constant. The solution is
unique for $b>0$, because the ratio of two solutions would be a continuous
function with aligned periods $b$ and $\frac{1}{b}$, and such a function must
be constant if $b^{2}\notin\mathbb{Q}$. In the complex plane, the periods $b$
and $\frac{1}{b}$ indeed look as follows:

$i$$0$$1$$\begin{array}[]{c}b>0\\\ c\geq
25\end{array}$$\begin{array}[]{c}b\in\mathbb{C}\\\
c\in\mathbb{C}\end{array}$$\begin{array}[]{c}ib>0\\\ c\leq 1\end{array}$
(5.14)

The formula (5.13) for $\Upsilon_{b}(x)$ actually makes sense not only for
$b>0$, but for $\Re b>0$. This defines the functions $\Upsilon_{b}(x)$ and
$\hat{\Upsilon}_{b}(x)$ for $\Re b>0$ and $\Re ib>0$ respectively, such that
they are analytic in $b$.

Let us now solve the shift equation (5.8) using the function $\Upsilon_{b}$.
We write the ansatz

$\displaystyle
C_{\alpha_{1},\alpha_{2},\alpha_{3}}=\frac{N_{0}\prod_{i=1}^{3}N(\alpha_{i})}{\Upsilon_{b}(\alpha_{1}+\alpha_{2}+\alpha_{3}-Q)\Upsilon_{b}(\alpha_{1}+\alpha_{2}-\alpha_{3})\Upsilon_{b}(\alpha_{2}+\alpha_{3}-\alpha_{1})\Upsilon_{b}(\alpha_{3}+\alpha_{1}-\alpha_{2})}\
,$ (5.15)

where $N_{0}$ is a function of $b$, and $N(\alpha)$ is a function of $b$ and
$\alpha$. The denominator of this ansatz takes care of the denominator of the
shift equation, which therefore reduces to an equation that involves the
dependence on $\alpha_{1}$ only,

$\displaystyle\frac{C_{-}(\alpha_{1})N(\alpha_{1}-\frac{b}{2})}{C_{+}(\alpha_{1})N(\alpha_{1}+\frac{b}{2})}=\frac{b^{4b(2\alpha_{1}-Q)}}{\gamma(b(2\alpha_{1}-Q))\gamma(1+b(2\alpha_{1}-Q))}\
.$ (5.16)

Combining this equation with the shift equation for $B(\alpha)$ (5.10), we can
eliminate the unknown degenerate OPE coefficients $C_{\pm}(\alpha)$, and we
obtain

$\displaystyle\frac{\left(N^{2}B^{-1}\right)(\alpha-\frac{b}{2})}{\left(N^{2}B^{-1}\right)(\alpha+\frac{b}{2})}=b^{8b(2\alpha-Q)}\frac{\gamma(-b(2\alpha-Q))}{\gamma(b(2\alpha-Q))}\frac{\gamma(-b^{2}-b(2\alpha-Q))}{\gamma(-b^{2}+b(2\alpha-Q))}\
.$ (5.17)

Together with its image under $b\to b^{-1}$, this equation has the solution

$\displaystyle\left(N^{2}B^{-1}\right)(\alpha)=\Upsilon_{b}(2\alpha)\Upsilon_{b}(2\alpha-Q)\
.$ (5.18)

Therefore, we have only determined the combination $N^{2}B^{-1}$, and not the
individual functions $B$ and $N$ that appear in the two- and three-point
functions. This is because we still have the freedom of performing changes of
field normalization $V_{\alpha}(z)\to\lambda(\alpha)V_{\alpha}(z)$. Under such
changes, we have $B\to\lambda^{2}B$ and $N\to\lambda N$, while the combination
$N^{2}B^{-1}$ is invariant. Invariant quantities are the only ones that we can
determine without choosing a normalization, and the only ones that will be
needed for checking crossing symmetry.

It can nevertheless be convenient to choose a particular field normalization,
when comparing different approaches to Liouville theory, or when considering
specific applications. The normalization should then be chosen so that the
structure constants have nice properties. For example, we should not choose
the normalization such that $B(\alpha)=1$, which would cause $N(\alpha)$ (and
therefore $C_{\alpha_{1},\alpha_{2},\alpha_{3}}$) to have square-root branch
cuts. We could rather adopt the field normalization such that
$N(\alpha)=\Upsilon_{b}(2\alpha)$, which makes the three-point structure
constant meromorphic as a function of the momentums. We also choose $N_{0}=1$,
and we obtain the DOZZ formula (for Dorn, Otto, A. Zamolodchikov and Al.
Zamolodchikov),

$\displaystyle
C_{\alpha_{1},\alpha_{2},\alpha_{3}}=\frac{\Upsilon_{b}(2\alpha_{1})\Upsilon_{b}(2\alpha_{2})\Upsilon_{b}(2\alpha_{3})}{\Upsilon_{b}(\alpha_{1}+\alpha_{2}+\alpha_{3}-Q)\Upsilon_{b}(\alpha_{1}+\alpha_{2}-\alpha_{3})\Upsilon_{b}(\alpha_{2}+\alpha_{3}-\alpha_{1})\Upsilon_{b}(\alpha_{3}+\alpha_{1}-\alpha_{2})}\
.$ (5.19)

For the original derivation of this formula from the Lagrangian formulation of
Liouville theory, and the discussion of the formula in various limits, see
[9].

The DOZZ formula holds if $c\notin]-\infty,1]$ i.e. $\Re b>0$. On the other
hand, doing the replacement $\Upsilon_{b}\to\hat{\Upsilon}_{b}$, we obtain a
solution $\hat{C}$ that holds if $c\notin[25,\infty[$ i.e. $\Re ib>0$,
together with the corresponding functions $\hat{B}$ and $\hat{N}$. The
solution of the shift equations is unique if $b$ and $b^{-1}$ are aligned,
i.e. if $b^{2}\in\mathbb{R}$. Therefore, for generic values of the central
charge, both $C$ and $\hat{C}$ are solutions, and there are actually
infinitely many other solutions. In order to prove the existence and
uniqueness of Liouville theory, we will have to determine which solutions lead
to crossing-symmetric four-point functions.

Our shift equations for three-point structure constants, and their solutions,
are valid not only for Liouville theory, but also for (generalized) minimal
models. In such models, the momentums $\alpha_{\langle r,s\rangle}$ belong to
a lattice with periods $b$ and $\frac{1}{b}$, so they are uniquely determined
by the shift equations. The solution is given by $C$ or $\hat{C}$, which
coincide. (Actually $C$ has poles when $\alpha_{i}$ take degenerate values,
one should take the residues.) This shows that (generalized) minimal models
are unique, but not yet that they exist.

###  5.3 Crossing symmetry

We have found that (generalized) minimal models are unique, while Liouville
theory is unique at least if $b^{2}\in\mathbb{R}$. We will now address the
question of their existence.

Using the $V_{\alpha_{1}}V_{\alpha_{2}}$ OPE (4.21), let us write the
$s$-channel decomposition of a Liouville four-point function,

(5.20)

(We have a similar expression with $B,C\to\hat{B},\hat{C}$ whenever the
solutions $\hat{B},\hat{C}$ exist.) Let us accept for a moment that Liouville
theory is crossing-symmetric if the central charge is such that the shift
equations have unique solutions $C$ or $\hat{C}$, i.e. for $c\geq 25$ or
$c\leq 1$ respectively. The integrand of our $s$-channel decomposition is
well-defined, and analytic as a function of $b$, in the much larger regions
$c\notin]-\infty,1]$ and $c\notin[25,\infty[$ respectively. If the integral
itself was analytic as well, then crossing symmetry would hold in these
regions by analyticity.

In order to investigate the analytic properties of the integral, let us first
extend the integration half-line to a line,
$\int_{\frac{Q}{2}+i\mathbb{R}_{+}}\to\frac{1}{2}\int_{\frac{Q}{2}+i\mathbb{R}}$.
This is possible because the integrand is invariant under $\alpha\to
Q-\alpha$: the conformal blocks are invariant because they are functions of
the conformal dimension (1.16), and the combination
$B(\alpha)^{-1}C_{\alpha_{1},\alpha_{2},\alpha}C_{\alpha,\alpha_{3},\alpha_{4}}$
of structure constants is invariant as a consequence of
$\Upsilon_{b}(\alpha)=\Upsilon_{b}(Q-\alpha)$. Let us then study the
singularities of the integrand. We accept that the conformal blocks
$\mathcal{F}_{\alpha}^{(s)}(z)$ have poles when $\alpha=\alpha_{\langle
r,s\rangle}$ (1.23), the momentums for which the $s$-channel representation
becomes reducible [1]. We now plot the positions of these poles (blue regions)
relative to the integration line (red), depending on the central charge:

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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.43855pt}{22.54881pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{3mm}\\\ c\geq
25&c\notin]-\infty,1]\cup[25,\infty[&c\leq 1\end{array}$ (5.23)

When $c$ varies in the region $c\notin]-\infty,1]$, the poles never cross the
integration line. Therefore, the four-point function built from $C$ is
analytic on $c\notin]-\infty,1]$. So if Liouville theory exists for $c\geq
25$, then it also exists for $c\notin]-\infty,1]$, with the same structure
constant $C$. On the other hand, if $c\leq 1$, then the poles are on the
integration line, and actually the line has to be slightly shifted in order to
avoid the poles. We cannot analytically continue the four-point function from
the region $c\leq 1$ to complex values of $c$, because this would make
infinitely many poles cross the integration line. So the structure constant
$\hat{C}$ is expected to be valid for $c\leq 1$ only.

That is how far we can easily get with analytic considerations. Let us now
seek input from numerical tests of crossing symmetry, using Al.
Zamolodchikov’s recursive formula for computing conformal blocks [1]. (See the
associated Jupyter notebook, and the article [10].) We find that Liouville
theory exists for all values of $c\in\mathbb{C}$, with the three-point
structure constants $\hat{C}$ for $c\leq 1$, and $C$ otherwise. We also find
that generalized minimal models exist for all values of $c$, and minimal
models exist at the discrete values (4.5) of $c\leq 1$ where they are defined.
And we can numerically compute correlation functions with a good precision.

Historically, Liouville theory was first defined by quantizing a classical
theory whose equation of motion is Liouville’s equation. That definition
actually gave its name to the theory. (That definition does not cover the case
$c\leq 1$: using the name Liouville theory in this case, while natural in our
approach, is not universally done at the time of this writing.) It can be
shown that our definition of Liouville theory agrees with the historical
definition, either by proving that the originally defined theory obeys our
axioms, or by checking that both definitions lead to the same correlation
functions, in particular the same three-point structure constants. See [3] for
a guide to the literature on the construction of Liouville theory by
quantization.

## Acknowledgements

I am grateful to the organizers of the Cargèse school, for challenging me to
explain Liouville theory in about four hours. I with to thank Bertrand Eynard,
Riccardo Guida and André Voros for helpful suggestions and comments. I am
grateful to the participants of the Cargèse school, for their stimulating
participation in the lectures. I wish to thank the SciPost editor and
reviewers for their feedback and suggestions, which led to many improvements,
both perturbative and non-perturbative.

## References

  * [1] A. Zamolodchikov, A. Zamolodchikov (1990 book)
Conformal field theory and 2-D critical phenomena

  * [2] S. Ribault (2014 review) [1406.4290]
Conformal field theory on the plane

  * [3] J. Teschner (2017 review) [1708.00680]
A guide to two-dimensional conformal field theory

  * [4] P. Di Francesco, P. Mathieu, D. Sénéchal (1997 book)
Conformal field theory

  * [5] J. Cardy (2008 review) [0807.3472]
Conformal Field Theory and Statistical Mechanics

  * [6] S. Migliaccio, S. Ribault (2017) [1711.08916]
The analytic bootstrap equations of non-diagonal two-dimensional CFT

  * [7] G. W. Moore, N. Seiberg (1989)
Classical and Quantum Conformal Field Theory

  * [8] M. R. Gaberdiel (2000 review) [hep-th/9910156]
An Introduction to conformal field theory

  * [9] A. B. Zamolodchikov, A. B. Zamolodchikov (1996) [hep-th/9506136]
Structure constants and conformal bootstrap in Liouville field theory

  * [10] S. Ribault, R. Santachiara (2015) [1503.02067]
Liouville theory with a central charge less than one