$\displaystyle\boxed{T(y)\underset{y\to\infty}{=}O\left(\frac{1}{y^{4}}\right)}\
.$ (2.2.14)

This assumption will be justified in Section 2.3.2 in the case when $T(y)$
generates conformal transformations.

Now the commutation relations (1.3.15) of the Virasoro algebra are equivalent
to the following OPE of the field $T(y)$ with itself,

$\displaystyle\boxed{T(y)T(z)=\frac{\frac{c}{2}}{(y-z)^{4}}+\frac{2T(z)}{(y-z)^{2}}+\frac{\partial
T(z)}{y-z}+O(1)}\ .$ (2.2.15)

The proof of this equivalence is sketched in Exercise 2.7. The regular terms
of this OPE, which involve $L_{n\leq-2}T(z)$, do not appear in that proof, and
are therefore omitted. Regular terms are generally omitted when writing OPEs
of $T(y)$, because that field is locally holomorphic and is therefore
completely determined by its singularities.

Notice also that the term $\frac{c}{2}$ should be understood as $\frac{c}{2}I$
where $I$ is the identity field, a field whose presence does not affect
correlation functions, and which is in particular $z$-independent,

$\displaystyle\left\langle
I\cdots\right\rangle=\left\langle\cdots\right\rangle\ .$ (2.2.16)

This field $I$ corresponds to the central generator of the Virasoro algebra,
the generator which is implicitly present in the central term of the
commutation relations (1.3.15), and which acts as the identity in any
representation. The Virasoro field itself can be seen as a descendent of the
identity field. (See Exercise 2.8.)

Finally, notice that all our formulas are consistent with dimensional
analysis, if we adopt the following dimensions:

$\displaystyle\boxed{[z]=-1,\quad[T]=2,\quad[L_{n}]=-n}\ .$ (2.2.17)

####  2.2.3 Ward identities

We now derive linear equations for correlation functions, which are called the
Ward identities associated to the Virasoro symmetry. These identities follow
from the properties of the Virasoro field $T(z)$, which translate into the
following properties of the correlation function $\left\langle
T(z)\prod_{i=1}^{N}V_{\sigma_{i}}(z_{i})\right\rangle$ as a function of $z$:

  * •

it is holomorphic on ${\mathbb{C}}-\\{z_{1},\cdots z_{N}\\}$,

  * •

its behaviour at $z=z_{i}$ is controlled by the OPE (2.2.12),

  * •

its behaviour at $z=\infty$ is controlled by the assumption (2.2.14).

For any meromorphic function $\epsilon(z)$, with no poles outside
$\\{z_{1},\cdots z_{N}\\}$, we therefore have

$\displaystyle\oint_{\infty}dz\ \epsilon(z)\left\langle
T(z)\prod_{i=1}^{N}V_{\sigma_{i}}(z_{i})\right\rangle=0\quad\text{provided}\quad\epsilon(z)\underset{z\to\infty}{=}O(z^{2})\
,$ (2.2.18)

where the contour of integration encloses all the points $z_{1},\cdots z_{N}$.
We will distinguish two types of symmetry equations:

  * •

global Ward identities, which are obtained if $\epsilon(z)$ is holomorphic,
thus a polynomial of degree two,

  * •

local Ward identities, which are obtained if
$\epsilon(z)\underset{z\to\infty}{=}O(\frac{1}{z})$, so that $\epsilon(z)$
must have poles.

All symmetry equations can be obtained as linear combinations of such local
and global Ward identities.

The three global Ward identities are obtained by taking
$\epsilon(z)\in\\{1,z,z^{2}\\}$ in eq. (2.2.18). Since the integrand has
singularities only at $z_{1},\cdots z_{N}$, we have
$\oint_{\infty}=\sum_{i=1}^{N}\oint_{z_{i}}$. Using the
$T(y)V_{\sigma}(z_{0})$ OPE (2.2.12), we obtain

$\displaystyle\left\langle\sum_{i=1}^{N}L_{-1}^{(z_{i})}\prod_{i=1}^{N}V_{\sigma_{i}}(z_{i})\right\rangle$
$\displaystyle=0\ ,$ (2.2.19)
$\displaystyle\left\langle\sum_{i=1}^{N}\left(L_{0}^{(z_{i})}+z_{i}L_{-1}^{(z_{i})}\right)\prod_{i=1}^{N}V_{\sigma_{i}}(z_{i})\right\rangle$
$\displaystyle=0\ ,$ (2.2.20)
$\displaystyle\left\langle\sum_{i=1}^{N}\left(L_{1}^{(z_{i})}+2z_{i}L_{0}^{(z_{i})}+z_{i}^{2}L_{-1}^{(z_{i})}\right)\prod_{i=1}^{N}V_{\sigma_{i}}(z_{i})\right\rangle$
$\displaystyle=0\ .$ (2.2.21)

These equations simplify if the fields are primary, so that
$L_{1}^{(z_{i})}=0$ and $L_{0}^{(z_{i})}=\Delta_{i}$. These conditions are
necessary but not sufficient for fields to be primary, which suggests that we
define a quasi-primary field with conformal dimension $\Delta$ as a field
$V_{\Delta}(z)$ such that

$\displaystyle\left\\{\begin{array}[]{l}L_{1}V_{\Delta}(z)=0\ ,\\\
L_{0}V_{\Delta}(z)=\Delta V_{\Delta}(z)\ .\end{array}\right.$ (2.2.24)

We have already met a quasi-primary field which is not primary: the Virasoro
field $T(z)$ itself, which is quasi-primary with conformal dimension two. (See
Exercise (2.8).)

Let us now write the local Ward identities. A spanning set of these identities
can be obtained by taking $\epsilon(z)=\frac{1}{(z-z_{i})^{n-1}}$ with $n\geq
2$ and $i=1,\cdots N$ in eq. (2.2.18), leading to

$\displaystyle\left\langle\left(L_{-n}^{(z_{i})}+(-1)^{n+1}\sum_{j\neq
i}\sum_{p=-1}^{\infty}\frac{\binom{p+n-1}{p+1}}{(z_{i}-z_{j})^{n+p}}L_{p}^{(z_{j})}\right)\prod_{j=1}^{N}V_{\sigma_{j}}(z_{j})\right\rangle=0\
,$ (2.2.25)

where $\binom{p+n-1}{p+1}$ is a binomial coefficient. This simplifies if the
fields with indices $j\neq i$ are primary:

$\displaystyle\left\langle\left(L_{-n}^{(z_{i})}+\sum_{j\neq
i}\left(\frac{1}{(z_{j}-z_{i})^{n-1}}L_{-1}^{(z_{j})}+\frac{1-n}{(z_{j}-z_{i})^{n}}\Delta_{j}\right)\right)V_{\sigma_{i}}(z_{i})\prod_{j\neq
i}V_{\Delta_{j}}(z_{j})\right\rangle=0\ .$ (2.2.26)

Let us discuss the solution of the Ward identities for an $N$-point function
of primary fields
$\left\langle\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\right\rangle,$ together with
the $N$-point functions of the corresponding descendent fields. Such
descendent fields are in principle obtained by acting with operators
$L_{-n}^{(z_{i})}$ with $n\geq 1$, but due to the local Ward identities
(2.2.25) we can eliminate the operators $L_{-n}^{(z_{i})}$ with $n\geq 2$. So
we restrict our attention to the global Ward identities (2.2.21), which
involve only $L_{-1}^{(z_{i})}$-descendents. There are three global Ward
identities for $N$ “unknowns” $L_{-1}^{(z_{i})}$, and the situation crucially
depends on the value of $N$:

  * •

$\boxed{N=0}$ : the global Ward identities are trivial ($0=0$). The zero-point
function, which may be called the sphere partition function, is actually a
number rather than a function, and carries no useful information.

  * •

$\boxed{N=1}$ : the global Ward identities amount to $\left\langle
L_{-1}V_{\Delta_{1}}(z_{1})\right\rangle=\Delta_{1}\left\langle
V_{\Delta_{1}}(z_{1})\right\rangle=0$. This implies $\left\langle
V_{\Delta_{1}}(z_{1})\right\rangle=0$ or $\Delta_{1}=0$. In any case, the one-
point functions of $L_{-1}V_{\Delta_{1}}(z_{1})$ and of all other descendent
fields must vanish.

  * •

$\boxed{N=2}$ : we still have more Ward identities than unknowns, and we can
obtain a condition on $\Delta_{i}$ by eliminating $L_{-1}^{(z_{i})}$ from the
Ward identities. The elegant way to do this is to use
$\epsilon(z)=(z-z_{1})(z-z_{2})$ in eq. (2.2.18), which leads to

$\displaystyle(z_{1}-z_{2})(\Delta_{1}-\Delta_{2})\left\langle
V_{\Delta_{1}}(z_{1})V_{\Delta_{2}}(z_{2})\right\rangle=0\ .$ (2.2.27)

Assuming $\left\langle
V_{\Delta_{1}}(z_{1})V_{\Delta_{2}}(z_{2})\right\rangle\neq 0$ and $z_{1}\neq
z_{2}$, we must have

$\displaystyle\Delta_{1}=\Delta_{2}\ .$ (2.2.28)

The Ward identities then allow us to compute correlation functions involving
arbitrary descendent fields, for instance

$\displaystyle\left\langle
L_{-1}^{(z_{1})}V_{\Delta_{1}}(z_{1})V_{\Delta_{2}}(z_{2})\right\rangle=-\frac{2\Delta_{1}}{z_{1}-z_{2}}\left\langle
V_{\Delta_{1}}(z_{1})V_{\Delta_{2}}(z_{2})\right\rangle\ .$ (2.2.29)

If $\Delta_{1}\neq\Delta_{2}$ then not only $\left\langle
V_{\Delta_{1}}(z_{1})V_{\Delta_{2}}(z_{2})\right\rangle$ but also the
correlation functions of all corresponding descendent fields vanish.

  * •

$\boxed{N=3}$ : we have as many Ward identities as unknowns. So, while the
Ward identities do not lead to constraints on $\Delta_{i}$, they allow us to
deduce three-point functions involving arbitrary descendent fields, if we know
the three-point function of primary fields. In particular, for any creation
operator $L\in U(\mathfrak{V}^{+})$, there exists a universal, $c$-independent
function

$\displaystyle
g^{L}_{\Delta_{1},\Delta_{2},\Delta_{3}}(z_{1},z_{2},z_{3})=\frac{\left\langle
L^{(z_{1})}V_{\Delta_{1}}(z_{1})V_{\Delta_{2}}(z_{2})V_{\Delta_{3}}(z_{3})\right\rangle}{\left\langle
V_{\Delta_{1}}(z_{1})V_{\Delta_{2}}(z_{2})V_{\Delta_{3}}(z_{3})\right\rangle}\
.$ (2.2.30)

For example, using $\epsilon(z)=(z-z_{2})(z-z_{3})$ in eq. (2.2.18) we obtain

$\displaystyle
g^{L_{-1}}_{\Delta_{1},\Delta_{2},\Delta_{3}}(z_{1},z_{2},z_{3})=\frac{\Delta_{3}-\Delta_{1}-\Delta_{2}}{z_{1}-z_{2}}+\frac{\Delta_{2}-\Delta_{1}-\Delta_{3}}{z_{1}-z_{3}}\
.$ (2.2.31)

The next examples are

$\displaystyle g^{L_{-1}^{2}}$
$\displaystyle=\frac{(\Delta_{1}+\Delta_{2}-\Delta_{3})_{2}}{(z_{1}-z_{2})^{2}}+2\frac{(\Delta_{1}+\Delta_{2}-\Delta_{3})}{(z_{1}-z_{2})}\frac{(\Delta_{1}+\Delta_{3}-\Delta_{2})}{(z_{1}-z_{3})}+\frac{(\Delta_{1}+\Delta_{3}-\Delta_{2})_{2}}{(z_{1}-z_{3})^{2}}\
,$ (2.2.32) $\displaystyle g^{L_{-2}}$
$\displaystyle=\frac{\Delta_{1}+2\Delta_{2}-\Delta_{3}}{(z_{1}-z_{2})^{2}}+\frac{\Delta_{1}-\Delta_{2}-\Delta_{3}}{(z_{1}-z_{2})(z_{1}-z_{3})}+\frac{\Delta_{1}+2\Delta_{3}-\Delta_{2}}{(z_{1}-z_{3})^{2}}\
,$ (2.2.33)

where we use the notation

$\displaystyle(x)_{n}=\frac{\Gamma(x+n)}{\Gamma(x)}=\prod_{i=0}^{n-1}(x+i)\ .$
(2.2.34)

  * •

$\boxed{N\geq 4}$ : the Ward identities constrain the correlation functions of
descendents, but do not determine them completely.

The coming Section on Virasoro-symmetric OPEs is not essential for the
following, and can be skipped, provided eqs. (2.2.35) and (2.2.47) are
accepted.

####  2.2.4 Virasoro-symmetric OPEs

We make the fundamental assumption that the OPE of primary or descendent
fields only involves primary and descendent fields – in other words, that the
set of highest-weight representations is closed under fusion. This assumption
is not as innocent as it may seem: we will see in Section 4.4.3 that the set
of affine highest-weight representations is not closed under fusion. The
fields which appear in OPEs will then be denoted as
$V_{s,\bar{s}}(z,\bar{z})=L\bar{L}V_{\Delta,\bar{\Delta}}(z,\bar{z})$, where
$s=(\Delta,L)$ is made of a conformal dimension $\Delta$, and an element $L$
of the basis $\mathcal{L}$ (2.1.8) of $U(\mathfrak{V}^{+})$.

We already know the general form (1.2.7) of OPEs, and we will now see how
Virasoro symmetry constrains OPE coefficients. First, we anticipate that
fusion multiplicities are trivial, as we will check below. Then the OPE’s
universal parts, like the Ward identities, should respect holomorphic
factorization, invariance under translations, and dimensional analysis
(2.2.17). This leads to

$V_{s_{1},\bar{s}_{1}}(z_{1},\bar{z}_{1})V_{s_{2},\bar{s}_{2}}(z_{2},\bar{z}_{2})=\sum_{\Delta,\bar{\Delta}}\sum_{L,\bar{L}\in\mathcal{L}}C_{\Delta_{1},\bar{\Delta}_{1},\Delta_{2},\bar{\Delta}_{2}}^{\Delta,\bar{\Delta}}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})\\\
\times
f_{s_{1},s_{2}}^{s}f_{\bar{s}_{1},\bar{s}_{2}}^{\bar{s}}(z_{1}-z_{2})^{|L|-|L_{1}|-|L_{2}|}(\bar{z}_{1}-\bar{z}_{2})^{|\bar{L}|-|\bar{L}_{1}|-|\bar{L}_{2}|}L\bar{L}V_{\Delta,\bar{\Delta}}(z_{2},\bar{z}_{2})\
,$ (2.2.35)

where $f_{s_{1},s_{2}}^{s}$ is a universal, $z_{i}$-independent coefficient.
In order to study this coefficient, we will focus on the contributions of the
left-moving descendents of $V_{\Delta}(z_{2})$ for a given $\Delta$, which we
schematically write as

$\displaystyle
V_{s_{1}}(z_{1})V_{s_{2}}(z_{2})\sim\sum_{L\in\mathcal{L}}f_{s_{1},s_{2}}^{\Delta,L}(z_{1}-z_{2})^{|L|-|L_{1}|-|L_{2}|}LV_{\Delta}(z_{2})\
.$ (2.2.36)

Constraints from Virasoro symmetry can be obtained by inserting $\oint_{C}dz\
(z-z_{2})^{n+1}T(z)$ on both sides of the OPE, where the contour $C$ encircles
both $z_{1}$ and $z_{2}$. If $n\geq-1$, the resulting constraint is

$\displaystyle\left(L_{n}^{(z_{2})}+\sum_{m=-1}^{n}\binom{m+1}{n+1}z_{12}^{n-m}L_{m}^{(z_{1})}\right)V_{s_{1}}(z_{1})V_{s_{2}}(z_{2})\sim
L_{n}\sum_{L\in\mathcal{L}}f_{s_{1},s_{2}}^{\Delta,L}z_{12}^{|L|-|L_{1}|-|L_{2}|}LV_{\Delta}(z_{2})\
,$ (2.2.37)

where we use the notation $z_{12}=z_{1}-z_{2}$.

We will now compute the universal coefficients
$f_{\Delta_{1},\Delta_{2}}^{\Delta,L}$ in the case when
$V_{s_{1}}(z_{1})=V_{\Delta_{1}}(z_{1})$ and
$V_{s_{2}}(z_{2})=V_{\Delta_{2}}(z_{2})$ are primary fields. The case $n=0$ of
eq. (2.2.37) leads to

$\displaystyle
z_{12}L_{-1}V_{\Delta_{1}}(z_{1})V_{\Delta_{2}}(z_{2})\sim\sum_{L\in\mathcal{L}}f_{\Delta_{1},\Delta_{2}}^{\Delta,L}z_{12}^{|L|}(\Delta+|L|-\Delta_{1}-\Delta_{2})LV_{\Delta}(z_{2})\
.$ (2.2.38)

In the case $n\geq 1$, the contribution of $L_{n}^{(z_{2})}$ on the left-hand
side of eq. (2.2.37) vanishes, and the contribution of $L_{-1}^{(z_{1})}$ can
be eliminated with the help of the $n=0$ case, leading to

$\displaystyle\sum_{L\in\mathcal{L}}f_{\Delta_{1},\Delta_{2}}^{\Delta,L}z_{12}^{|L|}\big{(}L_{n}-z_{12}^{n}(\Delta+|L|+n\Delta_{1}-\Delta_{2})\big{)}LV_{\Delta}(z_{2})=0\
,\quad(n\geq 1)\ .$ (2.2.39)

Since the descendent fields $LV_{\Delta}(z_{2})$ are all linearly independent,
this amounts to a system of linear equations for
$f_{\Delta_{1},\Delta_{2}}^{\Delta,L}$, which determines them all in terms of
the first coefficient $f_{\Delta_{1},\Delta_{2}}^{\Delta,1}$. Let us
demonstrate this by writing these linear equations for low values of $n$ and
$|L|$. Up to the order $z_{12}^{2}$, the $n=1$ equation reads

$\Big{\\{}f^{1}z_{12}(\Delta_{2}-\Delta-\Delta_{1})+f^{L_{-1}}z_{12}\left(L_{1}-z_{12}(\Delta+1+\Delta_{1}-\Delta_{2})\right)L_{-1}\\\
+f^{L_{-1}^{2}}z_{12}^{2}L_{1}L_{-1}^{2}+f^{L_{-2}}z_{12}^{2}L_{1}L_{-2}\Big{\\}}V_{\Delta}(z_{2})=0\
,$ (2.2.40)

where we use the simplified notation $f^{L}$ for
$f_{\Delta_{1},\Delta_{2}}^{\Delta,L}$. With the help of the Virasoro
commutation relations (1.3.15), we obtain

$\displaystyle 2\Delta f^{L_{-1}}$
$\displaystyle=(\Delta+\Delta_{1}-\Delta_{2})f^{1}\ ,$ (2.2.41) $\displaystyle
2(2\Delta+1)f^{L_{-1}^{2}}+3f^{L_{-2}}$
$\displaystyle=(\Delta+1+\Delta_{1}-\Delta_{2})f^{L_{-1}}\ .$ (2.2.42)

Similarly, using the $n=2$ equation up to the order $z_{12}^{2}$ leads to

$\displaystyle 6\Delta
f^{L_{-1}^{2}}+(4\Delta+\tfrac{c}{2})f^{L_{-2}}=(\Delta+2\Delta_{1}-\Delta_{2})f^{1}\
.$ (2.2.43)

Let us discuss the solutions of these equations, assuming the normalization
condition

$\displaystyle f^{\Delta,1}_{\Delta_{1},\Delta_{2}}=1\ .$ (2.2.44)

The first equation (2.2.41) has a unique solution $f^{L_{-1}}$, unless
$\Delta=0$, in which case it has no solution. If $\Delta\neq 0$, the equations
for $f^{L_{-1}^{2}}$ and $f^{L_{-2}}$ have a unique solution, unless
$\Delta\in\\{\Delta_{\langle 1,2\rangle},\Delta_{\langle 2,1\rangle}\\}$, in
which case they have no solution. More generally, eq. (2.2.39) has a unique
solution $\\{f^{\Delta,L}_{\Delta_{1},\Delta_{2}}\\}_{L\in\mathcal{L}}$,
unless $\Delta\in\\{\Delta_{\langle r,s\rangle}\\}_{r,s\in{\mathbb{N}}^{*}}$.
(See Exercise 2.9.) This means that fusion multiplicities are trivial – they
do not exceed one. This is equivalent to the statement in Section 2.2.3, that
a three-point function of primary fields uniquely determines the three-point
functions of the corresponding descendent fields.

Finally, let us show that the structure function
$C_{\Delta_{1},\bar{\Delta}_{1},\Delta_{2},\bar{\Delta}_{2}}^{\Delta,\bar{\Delta}}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})$,
which appears in the OPE (2.2.35), is directly related to the three-point
function of primary fields. Inserting the OPE inside the three-point function,
we have

$\left\langle\prod_{i=1}^{3}V_{\Delta_{i},\bar{\Delta}_{i}}(z_{i},\bar{z}_{i})\right\rangle=\sum_{\Delta,\bar{\Delta}}\sum_{L,\bar{L}\in\mathcal{L}}C_{\Delta_{1},\bar{\Delta}_{1},\Delta_{2},\bar{\Delta}_{2}}^{\Delta,\bar{\Delta}}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})\\\
\times
f_{\Delta_{1},\Delta_{2}}^{\Delta,L}f_{\bar{\Delta}_{1},\bar{\Delta}_{2}}^{\bar{\Delta},\bar{L}}(z_{1}-z_{2})^{|L|}(\bar{z}_{1}-\bar{z}_{2})^{|\bar{L}|}\left\langle
L\bar{L}V_{\Delta,\bar{\Delta}}(z_{2},\bar{z}_{2})V_{\Delta_{3},\bar{\Delta}_{3}}(z_{3},\bar{z}_{3})\right\rangle\
.$ (2.2.45)

But we know that the two-point function $\left\langle
L\bar{L}V_{\Delta,\bar{\Delta}}(z_{2},\bar{z}_{2})V_{\Delta_{3},\bar{\Delta}_{3}}(z_{3},\bar{z}_{3})\right\rangle$
vanishes unless $\Delta=\Delta_{3}$ and $\bar{\Delta}=\bar{\Delta}_{3}$.
Moreover, dimensional analysis (2.2.17) implies

$\displaystyle\frac{\left\langle
L\bar{L}V_{\Delta,\bar{\Delta}}(z_{2},\bar{z}_{2})V_{\Delta_{3},\bar{\Delta}_{3}}(z_{3},\bar{z}_{3})\right\rangle}{\left\langle
V_{\Delta,\bar{\Delta}}(z_{2},\bar{z}_{2})V_{\Delta_{3},\bar{\Delta}_{3}}(z_{3},\bar{z}_{3})\right\rangle}\underset{z_{3}\to\infty}{=}O\left(z_{3}^{-|L|}\bar{z}_{3}^{-|\bar{L}|}\right)\
.$ (2.2.46)

Sending $z_{3}$ to infinity therefore reduces the sum over descendents to the
first term $(L,\bar{L})=(1,1)$, and we obtain

$\displaystyle
C_{\Delta_{1},\bar{\Delta}_{1},\Delta_{2},\bar{\Delta}_{2}}^{\Delta_{3},\bar{\Delta}_{3}}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})=\underset{z_{3}\to\infty}{\lim}\frac{\left\langle
V_{\Delta_{1},\bar{\Delta}_{1}}(z_{1},\bar{z}_{1})V_{\Delta_{2},\bar{\Delta}_{2}}(z_{2},\bar{z}_{2})V_{\Delta_{3},\bar{\Delta}_{3}}(z_{3},\bar{z}_{3})\right\rangle}{\left\langle
V_{\Delta_{3},\bar{\Delta}_{3}}(z_{2},\bar{z}_{2})V_{\Delta_{3},\bar{\Delta}_{3}}(z_{3},\bar{z}_{3})\right\rangle}\
.$ (2.2.47)

####  2.2.5 Null vector equations, and the fusion rules

The Ward identities are the only linear equations which are obeyed by general
correlation functions. However, more equations appear if a correlation
function involves degenerate fields, as a consequence of $L_{\langle
r,s\rangle}V_{\langle r,s\rangle}=0$ eq. (2.2.6). Inserting this into
correlation functions, we obtain linear equations called null vector
equations.

In the case of three-point functions, correlation functions of descendent
fields are related to correlation functions of primary fields by universal
ratios, such as the function
$g^{L}_{\Delta_{1},\Delta_{2},\Delta_{3}}(z_{1},z_{2},z_{3})$ of eq. (2.2.30).
We thus have

$\displaystyle g^{L_{\langle r,s\rangle}}_{\Delta_{\langle
r,s\rangle},\Delta_{2},\Delta_{3}}(z_{1},z_{2},z_{3})=0\ .$ (2.2.48)

This must hold for any values of the positions $z_{i}$, which leads to
constraints on $\Delta_{2}$ and $\Delta_{3}$. These constraints can be
interpreted as the fusion rules of the degenerate representation $R_{\langle
r,s\rangle}$, using the relation (2.2.47) between OPE structure functions and
three-point functions.

For example, let us consider the case $\langle r,s\rangle=\langle 1,1\rangle$.
Using eq. (2.1.38) for $\Delta_{\langle 1,1\rangle}$ and $L_{\langle
1,1\rangle}$, we obtain $g^{L_{-1}}_{0,\Delta_{2},\Delta_{3}}=0$. Using eq.
(2.2.31) for $g^{L_{-1}}$, this implies $\Delta_{2}=\Delta_{3}$, which is
equivalent to the fusion rule

$\displaystyle\boxed{R_{\langle
1,1\rangle}\times\mathcal{V}_{\Delta}=\mathcal{V}_{\Delta}}\ .$ (2.2.49)

Next, let us consider the case $\langle r,s\rangle=\langle 2,1\rangle$. Using
eq. (2.1.38), we obtain
$g^{L_{-2}+\frac{1}{b^{2}}L_{-1}^{2}}_{-\frac{1}{2}-\frac{3}{4}b^{2},\Delta_{2},\Delta_{3}}=0$.
Using eqs. (2.2.32) and (2.2.33) for $g^{L_{-1}^{2}}$ and $g^{L_{-2}}$, this
implies

$\displaystyle
2(\Delta_{2}-\Delta_{3})^{2}+b^{2}(\Delta_{2}+\Delta_{3})-\frac{1}{2}-b^{2}-\frac{3}{8}b^{4}=0\
.$ (2.2.50)

Given a value of $\Delta_{2}$, we therefore have two solutions for
$\Delta_{3}$. The actual solutions are most easily found by rewriting the
equation in terms of the momentum $\alpha$ (2.1.30) instead of the conformal
dimension $\Delta$,

$\displaystyle\left((\alpha_{2}-\alpha_{3})^{2}-\frac{1}{4}b^{2}\right)\left((Q-\alpha_{2}-\alpha_{3})^{2}-\frac{1}{4}b^{2}\right)=0\
.$ (2.2.51)

Up to reflections of $\alpha_{2}$ or $\alpha_{3}$, we must therefore have
$\alpha_{3}=\alpha_{2}\pm\frac{b}{2}$. We would similarly find
$\alpha_{3}=\alpha_{2}\pm\frac{1}{2b}$ if we considered the case $\langle
r,s\rangle=\langle 1,2\rangle$. The resulting fusion rules are

$\displaystyle\boxed{R_{\langle
2,1\rangle}\times\mathcal{V}_{\alpha}=\sum_{\pm}\mathcal{V}_{\alpha\pm\frac{b}{2}}}\quad,\quad\boxed{R_{\langle
1,2\rangle}\times\mathcal{V}_{\alpha}=\sum_{\pm}\mathcal{V}_{\alpha\pm\frac{1}{2b}}}\
.$ (2.2.52)

These fusion rules could be derived by directly analyzing OPEs, instead of
three-point functions. (See Exercise 2.10.)

We now assume that the fusion rules are associative, and proceed to deduce
$R_{\langle r,s\rangle}\times\mathcal{V}_{\alpha}$. Due to null vector
equations, we expect this product to be a sum of a finite number of Verma
modules. By associativity, this implies that the fusion product of two
degenerate representations must itself be a sum of degenerate representations.
In the case of $R_{\langle 2,1\rangle}\times R_{\langle r,s\rangle}$, the
conformal dimensions of the resulting degenerate representations are given by
eq. (2.2.52), and we find

$\displaystyle R_{\langle 2,1\rangle}\times R_{\langle r,s\rangle}=R_{\langle
r-1,s\rangle}+R_{\langle r+1,s\rangle}\ ,\quad(r\geq 1)\ .$ (2.2.53)

We could similarly compute $R_{\langle 1,2\rangle}\times R_{\langle
r,s\rangle}$. Then, using the associativity of the fusion products $R_{\langle
2,1\rangle}\times R_{\langle r,s\rangle}\times\mathcal{V}_{\alpha}$ and
$R_{\langle 1,2\rangle}\times R_{\langle
r,s\rangle}\times\mathcal{V}_{\alpha}$, a recursion on $r$ and $s$ leads to

$\displaystyle\boxed{R_{\langle
r,s\rangle}\times\mathcal{V}_{\alpha}=\sum_{i=0}^{r-1}\sum_{j=0}^{s-1}\mathcal{V}_{\alpha+\alpha_{\langle
r,s\rangle}+ib+jb^{-1}}}\ ,$ (2.2.54)

where the momentum $\alpha_{\langle r,s\rangle}$ is given by eq. (2.1.41). In
particular, the two “extremal” representations, which are obtained for
$(i,j)=(0,0)$ and $(i,j)=(r-1,s-1)$, have the momentums
$\alpha\pm\alpha_{\langle r,s\rangle}$.

The fusion product of two degenerate representations must of course obey the
fusion rules of both, and we find

$\displaystyle\boxed{R_{\langle r_{1},s_{1}\rangle}\times R_{\langle
r_{2},s_{2}\rangle}=\sum_{r_{3}\overset{2}{=}|r_{1}-r_{2}|+1}^{r_{1}+r_{2}-1}\
\sum_{s_{3}\overset{2}{=}|s_{1}-s_{2}|+1}^{s_{1}+s_{2}-1}R_{\langle
r_{3},s_{3}\rangle}}\ ,$ (2.2.55)

where a superscript $\overset{2}{=}$ indicates that the corresponding sums run
by increments of $2$. So the fusion product $R_{\langle
r_{1},s_{1}\rangle}\times R_{\langle r_{2},s_{2}\rangle}$ is a sum of
$\min(r_{1},r_{2})\min(s_{1},s_{2})$ degenerate representations. This can be
rewritten as

$\displaystyle R_{\langle r_{1},s_{1}\rangle}\times R_{\langle
r_{2},s_{2}\rangle}=\sum_{r_{3},s_{3}=1}^{\infty}f_{r_{1},r_{2},r_{3}}f_{s_{1},s_{2},s_{3}}\
R_{\langle r_{3},s_{3}\rangle}\ ,$ (2.2.56)

using the permutation-symmetric coefficients

$\displaystyle
f_{r_{1},r_{2},r_{3}}=\left\\{\begin{array}[]{l}1\quad\text{if}\quad\forall
i\neq j\neq k\ ,\quad r_{i}+r_{j}-r_{k}\in 1+2{\mathbb{N}}\ .\\\
0\quad\text{otherwise}\ .\end{array}\right.$ (2.2.59)

We could already derive the spectrums and fusion rules of minimal models, and
readers who are impatient to learn about particular models may proceed to
Section 3.2, where they will however have to skip the bits about correlation
functions.

###  2.3 Consequences of conformal symmetry

In Section 2.2, we have studied the linear equations which are associated to a
Virasoro symmetry algebra. We now further assume that this Virasoro algebra
corresponds to conformal transformations, rather than internal symmetries. The
associated Virasoro field will then be called the energy-momentum tensor.

####  2.3.1 The energy-momentum tensor

We define the energy-momentum tensor as a Virasoro field, which in addition
obeys

$\displaystyle\boxed{L_{-1}V_{\sigma}(z)={\frac{\partial}{\partial
z}}V_{\sigma}(z)}\ ,$ (2.3.1)

for any field $V_{\sigma}(z)$. This amounts to identifying $L_{-1}^{(z)}$ with
the differential operator ${\frac{\partial}{\partial z}}$, an identification
which is allowed by the particular $z$-dependence of the generators
$L_{n}^{(z)}$. Using eq. (2.2.9) and the Virasoro commutation relations
(1.3.15), we indeed have $[L_{-1}^{(z)}-{\frac{\partial}{\partial
z}},L_{n}^{(z)}]=0$.

Since $L_{-1}^{(z)}$ does depend on $z$, the assumption (2.3.1) entails
infinitely many identifications of symmetry generators with differential
operators. This can lead to a representation of generators $L_{n}^{(z)}$ as
differential operators, depending however on the correlation function where
such generators are inserted. For example, the local Ward identity (2.2.26)
for an $N$-point function involving $N-1$ primary fields becomes

$\displaystyle\boxed{\left\langle
L_{-n}^{(z_{i})}V_{\sigma_{i}}(z_{i})\prod_{j\neq
i}V_{\Delta_{j}}(z_{j})\right\rangle=\sum_{j\neq
i}\left(-\frac{1}{z_{ji}^{n-1}}{\frac{\partial}{\partial
z_{j}}}+\frac{n-1}{z_{ji}^{n}}\Delta_{j}\right)\left\langle
V_{\sigma_{i}}(z_{i})\prod_{j\neq i}V_{\Delta_{j}}(z_{j})\right\rangle}\ ,$
(2.3.2)

where we assume $n\geq 1$, and we use the notation $z_{ji}=z_{j}-z_{i}$. (See
Exercise 2.11.) Now let us insert the energy-momentum tensor itself into an
$N$-point function of primary fields, obtaining the $N+1$-point function
$\left\langle T(z)\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\right\rangle$. This
$N+1$-point function is locally holomorphic as a function of $z$, with a
double pole at $z=z_{i}$ as dictated by the OPE (2.2.13), and a behaviour at
$z=\infty$ dictated by the assumption (2.2.14). Together with our assumption
(2.3.1), this implies

$\displaystyle\boxed{\left\langle
T(z)\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\right\rangle=\sum_{i=1}^{N}\left(\frac{\Delta_{i}}{(z-z_{i})^{2}}+\frac{1}{z-z_{i}}{\frac{\partial}{\partial
z_{i}}}\right)\left\langle\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\right\rangle}\
.$ (2.3.3)

In more general correlation functions, where fields are not necessarily
primary, insertions of $T(z)$ or creation operators are represented as
combinations of differential operators and annihilation operators. Consider
then the vector space generated by an $N$-point function of descendent fields
$\left\langle\prod_{i=1}^{N}V_{\sigma_{i}}(z_{i})\right\rangle$, and the
$N$-point functions obtained by repeatedly acting with all possible
annihilation operators $L_{n}^{(z_{i})}$ with $n>0$. This space is finite-
dimensional, and in this space insertions of $T(z)$ or any generator
$L_{n}^{(z_{i})}$ with $n\in{\mathbb{Z}}$ can be represented as matrix
differential operators.

As a consequence, all Ward identities and null vector equations will become
differential equations if only primary fields are involved, and matrix
differential equations if descendent fields are involved. Let us show this
explicitly in the case of the global Ward identities.

####  2.3.2 Global conformal symmetry

This Section is devoted to solving the global Ward identities
(2.2.19)-(2.2.21) for an $N$-point function
$\left\langle\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\right\rangle$ of quasi-
primary fields, in the case when $L_{-1}$ is the generator of translations and
obeys eq. (2.3.1). In this case, the global Ward identities reduce to a system
of three differential equations,

$\displaystyle\forall
a\in\\{0,+,-\\}\quad,\quad\left(\sum_{i=1}^{N}D_{z_{i}}^{-\Delta_{i}}(t^{a})\right)\left\langle\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\right\rangle$
$\displaystyle=0\ ,$ (2.3.4)

where we define the differential operators

$\displaystyle\left\\{\begin{array}[]{rl}D_{x}^{j}(t^{-})&=-{\frac{\partial}{\partial
x}}\ ,\\\ D_{x}^{j}(t^{0})&=x{\frac{\partial}{\partial x}}-j\ ,\\\
D_{x}^{j}(t^{+})&=x^{2}{\frac{\partial}{\partial x}}-2jx\ .\end{array}\right.$
(2.3.8)

These operators form a representation of the Lie algebra $\mathfrak{sl}_{2}$
with the generators $(t^{0},t^{+},t^{-})$ and commutation relations

$\displaystyle[t^{0},t^{\pm}]=\pm t^{\pm}\quad,\quad[t^{+},t^{-}]=2t^{0}\ .$
(2.3.9)

The algebra $\mathfrak{sl}_{2}$ also has a representation in terms of
traceless matrices of size two,

$\displaystyle\left\\{\begin{array}[]{rl}M(t^{+})&=\left(\begin{smallmatrix}0&1\\\
0&0\end{smallmatrix}\right)\ ,\\\
M(t^{0})&=\frac{1}{2}\left(\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}\right)\ ,\\\ M(t^{-})&=\left(\begin{smallmatrix}0&0\\\
1&0\end{smallmatrix}\right)\ .\end{array}\right.$ (2.3.13)

In the case $j=0$, the differential operators (2.3.8) correspond to the
generators $(\ell_{-1},\ell_{0},\ell_{1})$ (1.3.13) of global conformal
transformations. This shows that, as their names suggest, global Ward
identities correspond to global conformal transformations. In this
correspondence, the holomorphic function $\epsilon(z)\in\\{1,z,z^{2}\\}$ which
appears in the derivation (2.2.18) of the global Ward identities, corresponds
to the function $\epsilon(z)$ which parametrizes an infinitesimal conformal
transformation (1.3.11).

So $\mathfrak{sl}_{2}$ matrices $M(t^{a})$ correspond to infinitesimal global
conformal transformations, and to the infinitesimal global Ward identities
(2.3.4). Then $SL_{2}({\mathbb{C}})$ matrices correspond to finite global
conformal transformations (1.3.9), and the finite global Ward identities are

$\displaystyle\left\langle\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\right\rangle=\left\langle\prod_{i=1}^{N}T_{g}V_{\Delta_{i}}(z_{i})\right\rangle\
,$ (2.3.14)

where we define the image $T_{g}V_{\Delta}(z)$ of a quasi-primary field as

$\displaystyle\boxed{T_{g}V_{\Delta}(z)=(cz+d)^{-2\Delta}V_{\Delta}\left(\frac{az+b}{cz+d}\right)}\quad\text{with}\quad
g=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in
SL_{2}({\mathbb{C}})\ .$ (2.3.17)

(See Exercise 2.12 for an interpretation in terms of representations of
$\mathfrak{sl}_{2}$.) This involves the expression $(cz+d)^{-2\Delta}$, which
is a multivalued function of $z$ unless $2\Delta\in{\mathbb{Z}}$. Correlation
functions are however supposed to be single-valued when their dependence on
$\bar{z}$ is taken into account. Since
$T_{g}V_{\Delta,\bar{\Delta}}(z,\bar{z})=(cz+d)^{-2\Delta}(\bar{c}\bar{z}+\bar{d})^{-2\bar{\Delta}}V_{\Delta}\left(\frac{az+b}{cz+d},\frac{\bar{a}\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{d}}\right)$,
a necessary condition for the single-valuedness of correlation functions
involving $V_{\Delta,\bar{\Delta}}(z,\bar{z})$ is

$\displaystyle\boxed{\Delta-\bar{\Delta}\in\frac{1}{2}{\mathbb{Z}}}\ .$
(2.3.18)

The number $\Delta-\bar{\Delta}$ is called the conformal spin of the field
$V_{\Delta,\bar{\Delta}}(z,\bar{z})$, as it controls how that field behaves
under rotations, $T_{\left(\begin{smallmatrix}e^{i\theta}&0\\\
0&e^{-i\theta}\end{smallmatrix}\right)}V_{\Delta,\bar{\Delta}}(z,\bar{z})=e^{2i\theta(\Delta-\bar{\Delta})}V_{\Delta,\bar{\Delta}}(e^{2i\theta}z,e^{-2i\theta}\bar{z})$.
Then $V_{\Delta,\bar{\Delta}}(z,\bar{z})$ is called a fermionic field if
$\Delta-\bar{\Delta}\in\frac{1}{2}+{\mathbb{Z}}$, and a bosonic field if

$\displaystyle\Delta-\bar{\Delta}\in{\mathbb{Z}}\ .$ (2.3.19)

In the following, we will only study models whose fields are all bosonic.

We can now determine the behaviour of a quasi-primary fields $V_{\Delta}(z)$
at $z=\infty$. By covariance, this is the same as the behaviour of
$T_{\left(\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right)}V_{\Delta}(z)=(-z)^{-2\Delta}V_{\Delta}(-\frac{1}{z})$.
But $V_{\Delta}(-\frac{1}{z})$ is smooth at $z=\infty$, and we obtain

$\displaystyle\boxed{V_{\Delta}(z)\underset{z\to\infty}{=}O\left(z^{-2\Delta}\right)}\
.$ (2.3.20)

This is consistent with the assumed behaviour (2.2.14) of the energy-momentum
tensor, which is a quasi-primary field of dimension $\Delta=2$. Moreover, this
suggests that we may define a field at $z=\infty$ by

$\displaystyle
V_{\Delta}(\infty)=\underset{z\to\infty}{\lim}z^{2\Delta}V_{\Delta}(z)\ .$
(2.3.21)

Of course, the behaviour (2.3.20) of $V_{\Delta}(z)$ at $z=\infty$ holds only
provided no other field is present at $z=\infty$. To determine the behaviour
of a correlation function $\left\langle
V_{\Delta}(z)V_{\Delta^{\prime}}(\infty)\cdots\right\rangle$ at $z=\infty$, we
would need to know the OPE $V_{\Delta}(z)V_{\Delta^{\prime}}(\infty)$.

One may then wonder whether local Ward identities are infinitesimal
expressions of the covariance of correlation functions under local conformal
transformations (1.3.10). For any holomorphic function $h(z)$, which we
interpret as a local conformal transformation, we can define the image
$T_{h}V_{\Delta}(z)$ of a primary field as

$\displaystyle\boxed{T_{h}V_{\Delta}(z)=(h^{\prime}(z))^{\Delta}V_{\Delta}(h(z))}\
,$ (2.3.22)

which generalizes the image of a quasi-primary field under a global conformal
transformation (2.3.17). However, unlike a global transformation, a local
transformation $h(z)$ must have a singularity at some point $z_{0}$, in which
case our formula for $T_{h}V_{\Delta}(z_{0})$ does not make sense. But we know
from Section 1.3.2 that local transformations modify fields, in addition to
moving them around. This is manifest in the local Ward identities, which
involve not only $V_{\Delta}(z_{0})$ and its derivatives, but also its
$L_{n\leq-2}$ descendents. Therefore, a covariance equation of the type of eq.
(2.3.14) holds for correlation functions of primary fields, provided we take
$T_{h}V_{\Delta}(z_{0})$ to involve $L_{n\leq-2}$ descendents whenever $h(z)$
is singular at $z=z_{0}$.

Let us now solve the differential equations (2.3.4). As we already saw in
Section 2.2.3, solving the three global Ward identities for an $N$-point
function crucially depends on the value of $N$:

  * •

$\boxed{N=1}$ : we already know that the one-point function $\left\langle
V_{\Delta}(z)\right\rangle$ is non-vanishing only if $V_{\Delta}(z)$ is a
degenerate field, with a vanishing descendent at level one. The null vector
equation $L_{-1}V_{\Delta}(z)=0$ now implies that $\langle
V_{\Delta}(z)\rangle$ is $z$-independent.

  * •

$\boxed{N=2}$ : the equation (2.2.29) becomes a linear differential equation
for our two-point function, whose solution is

$\displaystyle\boxed{\mathcal{F}^{(2)}(\Delta_{1}|z_{1},z_{2})=z_{12}^{-2\Delta_{1}}}\
.$ (2.3.23)

The $z_{i}$-independent constant in a two-point function can be set to one by
a choice of normalization of the fields $V_{\Delta,\bar{\Delta}}(z,\bar{z})$,
so that

$\displaystyle\left\langle
V_{\Delta_{1},\bar{\Delta}_{1}}(z_{1},\bar{z}_{1})V_{\Delta_{2},\bar{\Delta}_{2}}(z_{2},\bar{z}_{2})\right\rangle=\delta_{\Delta_{1},\Delta_{2}}\delta_{\bar{\Delta}_{1},\bar{\Delta}_{2}}\left|\mathcal{F}^{(2)}(\Delta_{1}|z_{1},z_{2})\right|^{2}\
,$ (2.3.24)

where the symbol $\delta_{\Delta_{1},\Delta_{2}}$ is a Kronecker delta or a
Dirac delta function, depending on whether the conformal dimensions take
discrete or continuous values, and we use the modulus square notation in the
sense $|f(\Delta,z)|^{2}=f(\Delta,z)f(\bar{\Delta},\bar{z})$.

  * •

$\boxed{N=3}$ : the equation (2.2.30) with $L=L_{-1}$ becomes a linear
differential equation for our three-point function, whose solution is

$\displaystyle\boxed{\mathcal{F}^{(3)}(\Delta_{1},\Delta_{2},\Delta_{3}|z_{1},z_{2},z_{3})=z_{12}^{\Delta_{3}-\Delta_{1}-\Delta_{2}}z_{23}^{\Delta_{1}-\Delta_{2}-\Delta_{3}}z_{31}^{\Delta_{2}-\Delta_{3}-\Delta_{1}}}\
.$ (2.3.25)

(See Exercise 2.13 for the generalization to logarithmic conformal field
theory.) So, the dependence of the three-point function of quasi-primary
fields on $z_{i}$ is fully determined by global conformal symmetry, and we
have

$\displaystyle\left\langle\prod_{i=1}^{3}V_{\Delta_{i},\bar{\Delta}_{i}}(z_{i},\bar{z}_{i})\right\rangle=C_{\Delta_{1},\bar{\Delta}_{1},\Delta_{2},\bar{\Delta}_{2},\Delta_{3},\bar{\Delta}_{3}}\left|\mathcal{F}^{(3)}(\Delta_{1},\Delta_{2},\Delta_{3}|z_{1},z_{2},z_{3})\right|^{2}\
,$ (2.3.26)

where
$C_{\Delta_{1},\bar{\Delta}_{1},\Delta_{2},\bar{\Delta}_{2},\Delta_{3},\bar{\Delta}_{3}}$
is called the three-point structure constant.

  * •

$\boxed{N\geq 4}$ : the three differential equations are not enough for
controlling the dependence of an $N$-point function on the $N\geq 4$ variables
$z_{i}$. Omitting the dependence on $\bar{z}_{i}$, the general solution can be
written as

$\displaystyle\left\langle\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\right\rangle=\prod_{i<j}z_{ij}^{\delta_{ij}}\
F(x_{1},x_{2},\cdots x_{N-3})\quad\text{with}\ \
\left\\{\begin{array}[]{l}\sum_{j<i}\delta_{ji}+\sum_{i<j}\delta_{ij}=-2\Delta_{i}\
,\\\
x_{i}=\frac{(z_{i}-z_{N-2})(z_{N-1}-z_{N})}{(z_{i}-z_{N-1})(z_{N-2}-z_{N})}\
,\end{array}\right.$ (2.3.29)

where $F(x_{1},x_{2},\cdots x_{N-3})$ is an arbitrary function of the cross-
ratios $x_{i}$, which is therefore invariant under the global conformal
transformations (1.3.9). This function can be interpreted as an $N$-point
function with $(z_{N-2},z_{N-1},z_{N})=(0,\infty,1)$, as we now illustrate in
the case $N=4$.

  * •

$\boxed{N=4}$ : in this case there is only one cross-ratio

$\displaystyle x=\frac{z_{12}z_{34}}{z_{13}z_{24}}\ ,$ (2.3.30)

and the general solution can be written as

$\displaystyle\left\langle\prod_{i=1}^{4}V_{\Delta_{i}}(z_{i})\right\rangle=\mathcal{F}^{(3)}(\Delta_{1},\Delta_{2},0|z_{1},z_{2},z_{3})\mathcal{F}^{(3)}(\Delta_{3},\Delta_{4},0|z_{3},z_{4},z_{2})\
F(x)\ ,$ (2.3.31)

where we built the prefactor as a combination of three-point functions, such
that each variable $z_{i}$ has the correct total exponent $-2\Delta_{i}$. With
this choice of prefactor, we notice that $F(x)$ is related to a four-point
function with $(z_{2},z_{3},z_{4})=(0,\infty,1)$ as

$\displaystyle F(x)=x^{\Delta_{1}+\Delta_{2}}\left\langle
V_{\Delta_{1}}(x)V_{\Delta_{2}}(0)V_{\Delta_{3}}(\infty)V_{\Delta_{4}}(1)\right\rangle\
.$ (2.3.32)

Knowing the $z_{i}$-dependences of the two- and three-point functions, we can
find the $z_{i}$-dependence of the OPE structure functions. Using eq.
(2.2.47), we find

$\displaystyle\boxed{C_{\Delta_{1},\bar{\Delta}_{1},\Delta_{2},\bar{\Delta}_{2}}^{\Delta_{3},\bar{\Delta}_{3}}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})=C_{\Delta_{1},\bar{\Delta}_{1},\Delta_{2},\bar{\Delta}_{2},\Delta_{3},\bar{\Delta}_{3}}\
z_{12}^{\Delta_{3}-\Delta_{1}-\Delta_{2}}\bar{z}_{12}^{\bar{\Delta}_{3}-\bar{\Delta}_{1}-\bar{\Delta}_{2}}}\
,$ (2.3.33)

where the three-point structure constant
$C_{\Delta_{1},\bar{\Delta}_{1},\Delta_{2},\bar{\Delta}_{2},\Delta_{3},\bar{\Delta}_{3}}$
was defined by eq. (2.3.26). Alternatively, the $z_{i}$-dependence of the OPE
structure functions can be derived from a direct analysis of the OPE itself.
(See Exercise 2.14.)

The coming Section introduces general conformal blocks and their properties.
Only special cases of conformal blocks will be needed in the sequel, and the
coming Section can be skipped.

####  2.3.3 Conformal blocks

We will now define conformal blocks as the universal constituents of
correlation functions. We want correlation functions to be decomposable as
combinations of the model-dependent three-point structure constant, and
conformal blocks. In this sense, we already encountered simple examples of
conformal blocks: two-point blocks $\mathcal{F}^{(2)}(\Delta_{1}|z_{1},z_{2})$
(2.3.23), and three-point blocks
$\mathcal{F}^{(3)}(\Delta_{1},\Delta_{2},\Delta_{3}|z_{1},z_{2},z_{3})$
(2.3.25). (On the torus, there exist nontrivial zero-point blocks, which are
also known as the characters of representations.)

Using the OPE (2.2.35), together with the results of Section 2.3.2, a four-
point function of descendent fields of the type $V_{s,\bar{s}}(z,\bar{z})$
where $s=(\Delta,L)$ can be decomposed as

$\displaystyle\left\langle\prod_{i=1}^{4}V_{s_{i},\bar{s}_{i}}(z_{i},\bar{z}_{i})\right\rangle=\sum_{\Delta_{s},\bar{\Delta}_{s}}C_{\Delta_{1},\bar{\Delta}_{1},\Delta_{2},\bar{\Delta}_{2},\Delta_{s},\bar{\Delta}_{s}}C_{\Delta_{s},\bar{\Delta}_{s},\Delta_{3},\bar{\Delta}_{3},\Delta_{4},\bar{\Delta}_{4}}\left|\mathcal{F}^{(s)}_{\Delta_{s}}(s_{i}|z_{i})\right|^{2}\
,$ (2.3.34)

where all quantities which are not three-point structure constants have been
combined into the $s$-channel four-point conformal block
$\mathcal{F}^{(s)}_{\Delta_{s}}(s_{i}|z_{i})$. The appearance of the
factorized quantity
$\left|\mathcal{F}^{(s)}_{\Delta_{s}}(s_{i}|z_{i})\right|^{2}$ is a
manifestation of holomophic factorization, and the conformal block
$\mathcal{F}^{(s)}_{\Delta_{s}}(s_{i}|z_{i})$ is a locally holomorphic
function of $z_{i}$.

Our decomposition of the four-point function followed from the $V_{1}V_{2}$
OPE: what if we used the $V_{2}V_{3}$ OPE instead? We would then obtain a
decomposition involving $t$-channel four-point conformal blocks
$\mathcal{F}^{(t)}_{\Delta_{t}}(s_{i}|z_{i})$. The equality of the two
decompositions is called the crossing symmetry of the four-point function.
Introducing two diagrammatic notations for conformal blocks,

$\displaystyle\mathcal{F}^{(s)}_{\Delta_{s}}=\leavevmode\hbox to75.75pt{\vbox
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}{{}}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ } {}{{}}{} {}{}{}{}{{{}{}}}
{}{}{}{}{{{}{}}} {}{}{}{}{{{}{}}} {}{}{}{}{{{}{}}}
{}{}{}{}{{{}{}}}{}\pgfsys@moveto{0.0pt}{-22.76186pt}\pgfsys@lineto{-22.76186pt}{0.0pt}\pgfsys@lineto{0.0pt}{22.76186pt}\pgfsys@lineto{22.76186pt}{0.0pt}\pgfsys@lineto{0.0pt}{-22.76186pt}\pgfsys@lineto{0.0pt}{22.76186pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-20.31393pt}{-21.75836pt}\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}}}
\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}{-20.31393pt}{15.31393pt}\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}{15.31393pt}{15.31393pt}\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{{$3$}}
}}\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}{15.31393pt}{-21.75836pt}\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}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.62051pt}{-2.15277pt}\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{{$s$}}
}}\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\ ,\quad\
\mathcal{F}^{(t)}_{\Delta_{t}}=\leavevmode\hbox to58.68pt{\vbox
to64.33pt{\pgfpicture\makeatletter\hbox{\hskip
29.33784pt\lower-32.16296pt\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{-17.07182pt}{25.60774pt}\pgfsys@lineto{0.0pt}{17.07182pt}\pgfsys@lineto{0.0pt}{-17.07182pt}\pgfsys@lineto{-17.07182pt}{-25.60774pt}\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}{-26.00484pt}{22.38553pt}\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}{-7.54413pt}{-3.0754pt}\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}{-26.00484pt}{-28.82996pt}\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}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{{}}{}\pgfsys@moveto{0.0pt}{17.07182pt}\pgfsys@lineto{17.07182pt}{25.60774pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.00484pt}{22.38553pt}\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{{$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{{}}{}\pgfsys@moveto{0.0pt}{-17.07182pt}\pgfsys@lineto{17.07182pt}{-25.60774pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.00484pt}{-28.82996pt}\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}}=\leavevmode\hbox
to47.29pt{\vbox to50.18pt{\pgfpicture\makeatletter\hbox{\hskip
23.64694pt\lower-25.09137pt\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}{0.0pt}\pgfsys@lineto{0.0pt}{22.76186pt}\pgfsys@lineto{22.76186pt}{0.0pt}\pgfsys@lineto{0.0pt}{-22.76186pt}\pgfsys@lineto{-22.76186pt}{0.0pt}\pgfsys@lineto{22.76186pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-20.31393pt}{15.31393pt}\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}{15.31393pt}{15.31393pt}\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{{$3$}}
}}\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}{15.31393pt}{-21.75836pt}\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}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-20.31393pt}{-21.75836pt}\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}}}
\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}{-1.80556pt}{-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}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ \ ,$ (2.3.35)

we can write crossing symmetry of the four-point function as the equation

$\displaystyle\sum_{\Delta_{s},\bar{\Delta}_{s}}C_{12s}C_{s34}\left|\leavevmode\hbox
to75.75pt{\vbox to47.25pt{\pgfpicture\makeatletter\hbox{\hskip
20.80193pt\lower-23.62704pt\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{-8.5359pt}{17.07182pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{34.14365pt}{0.0pt}\pgfsys@lineto{42.67957pt}{17.07182pt}\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}{-17.46892pt}{13.84961pt}\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}{14.72807pt}{3.93301pt}\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{{$s$}}
}}\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}{46.61258pt}{13.84961pt}\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{{$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{}}
{}{}{}\pgfsys@moveto{-8.5359pt}{-17.07182pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-17.46892pt}{-20.29404pt}\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}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{{}}{}\pgfsys@moveto{34.14365pt}{0.0pt}\pgfsys@lineto{42.67957pt}{-17.07182pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{46.61258pt}{-20.29404pt}\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}}\right|^{2}=\sum_{\Delta_{t},\bar{\Delta}_{t}}C_{23t}C_{t41}\left|\leavevmode\hbox
to58.68pt{\vbox to64.33pt{\pgfpicture\makeatletter\hbox{\hskip
29.33784pt\lower-32.16296pt\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{-17.07182pt}{25.60774pt}\pgfsys@lineto{0.0pt}{17.07182pt}\pgfsys@lineto{0.0pt}{-17.07182pt}\pgfsys@lineto{-17.07182pt}{-25.60774pt}\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}{-26.00484pt}{22.38553pt}\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}{-7.54413pt}{-3.0754pt}\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}{-26.00484pt}{-28.82996pt}\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}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{{}}{}\pgfsys@moveto{0.0pt}{17.07182pt}\pgfsys@lineto{17.07182pt}{25.60774pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.00484pt}{22.38553pt}\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{{$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{{}}{}\pgfsys@moveto{0.0pt}{-17.07182pt}\pgfsys@lineto{17.07182pt}{-25.60774pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.00484pt}{-28.82996pt}\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}}\right|^{2}\ .$
(2.3.36)

Assuming the conformal blocks to be known, this is a quadratic equation for
the three-point structure constant. This equation is equivalent to the
associativity of the OPE (1.2.9). $N$-point functions on the sphere can
similarly be decomposed into three-point structure constants and $N$-point
conformal blocks, but the cases $N\geq 5$ do not yield more equations for the
three-point structure constant.

Conformal blocks obey the same local and global Ward identities as the
corresponding correlation functions. In particular, given a four-point
function of primary fields, the holomorphic global Ward identities (2.3.4) are
linear differential equations for locally holomorphic functions of
$(z_{1},z_{2},z_{3},z_{4})$. Then $s$-channel and $t$-channel four-point
blocks each provide a basis of solutions of these equations, whose elements
are respectively parametrized by $\Delta_{s}$ and $\Delta_{t}$. So there must
exist a linear, invertible fusing relation between the two bases,

$\displaystyle\mathcal{F}^{(s)}_{\Delta_{s}}(s_{i}|z_{i})=\sum_{\Delta_{t}}\
F_{\Delta_{s},\Delta_{t}}\begin{bmatrix}\Delta_{2}&\Delta_{3}\\\
\Delta_{1}&\Delta_{4}\end{bmatrix}\
\mathcal{F}^{(t)}_{\Delta_{t}}(s_{i}|z_{i})\ ,$ (2.3.37)

whose $z_{i}$-independent kernel
$F_{\Delta_{s},\Delta_{t}}\begin{bmatrix}\Delta_{2}&\Delta_{3}\\\
\Delta_{1}&\Delta_{4}\end{bmatrix}$ is called the fusing matrix of the
Virasoro algebra. The parameters of the fusing matrix are in principle
representations of the Virasoro algebra: the conformal dimensions which we
wrote stand for the corresponding Verma modules. A diagrammatic notation for
the fusing matrix is

$\displaystyle F_{s,t}\begin{bmatrix}2&3\\\ 1&4\end{bmatrix}=\ \
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{}{{}}{}\pgfsys@moveto{-31.2982pt}{0.0pt}\pgfsys@lineto{-3.12976pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{3.12976pt}{0.0pt}\pgfsys@lineto{31.2982pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ \ .$ (2.3.38)

It is possible to eliminate the conformal blocks from the crossing symmetry
equation (2.3.36), using the fusing relation and the linear independence of
blocks within a given basis,

$\displaystyle\sum_{\Delta_{s},\bar{\Delta}_{s}}C_{12s}C_{s34}F_{\Delta_{s},\Delta_{t}}\begin{bmatrix}\Delta_{2}&\Delta_{3}\\\
\Delta_{1}&\Delta_{4}\end{bmatrix}F_{\bar{\Delta}_{s},\bar{\Delta}_{t}}\begin{bmatrix}\bar{\Delta}_{2}&\bar{\Delta}_{3}\\\
\bar{\Delta}_{1}&\bar{\Delta}_{4}\end{bmatrix}=C_{23t}C_{t41}\ .$ (2.3.39)

This can be further simplified if the theory is diagonal. In this case, we
have $\Delta_{i}=\bar{\Delta}_{i}$ for $i=1,2,3,4$, and we should insert
factors $\delta_{\Delta_{s},\bar{\Delta}_{s}}$ (left) and
$\delta_{\Delta_{t},\bar{\Delta}_{t}}$ (right) in the above formula. Using the
inverse fusing matrix, we obtain

$\displaystyle
C_{12s}C_{s34}F_{\Delta_{s},\Delta_{t}}\begin{bmatrix}\Delta_{2}&\Delta_{3}\\\
\Delta_{1}&\Delta_{4}\end{bmatrix}=C_{23t}C_{t41}F^{-1}_{\Delta_{t},\Delta_{s}}\begin{bmatrix}\Delta_{2}&\Delta_{3}\\\
\Delta_{1}&\Delta_{4}\end{bmatrix}\ .$ (2.3.40)

This can be useful, because the fusing matrix is usually a simpler object than
the conformal blocks, as we will see in a special case in Section 2.3.5.
Explicit formulas for
$F_{\Delta_{s},\Delta_{t}}\begin{bmatrix}\Delta_{2}&\Delta_{3}\\\
\Delta_{1}&\Delta_{4}\end{bmatrix}$ with generic values of $\Delta_{i}$ can be
found in [9].

Let us sketch how conformal blocks can be computed. To begin with, let us show
how an $s$-channel conformal block can be expressed in terms of previously
introduced universal quantities. For simplicity, we assume that our four
fields are primary. Using the OPE (2.2.35) together with eq. (2.3.33), the
expression (2.2.30) for three-point functions of descendent fields in terms of
three-point functions of primary fields, and the expression (2.3.26) for
three-point functions of primary fields, we obtain

$\displaystyle\mathcal{F}^{(s)}_{\Delta_{s}}(\Delta_{i}|z_{i})=\sum_{L_{s}\in\mathcal{L}}f_{\Delta_{1},\Delta_{2}}^{\Delta_{s},L_{s}}g^{L_{s}}_{\Delta_{s},\Delta_{3},\Delta_{4}}(z_{2},z_{3},z_{4})z_{12}^{|L_{s}|+\Delta_{s}-\Delta_{1}-\Delta_{2}}\mathcal{F}^{(3)}(\Delta_{s},\Delta_{3},\Delta_{4}|z_{2},z_{3},z_{4})\
.$ (2.3.41)

Thanks to global conformal symmetry, we can restrict our attention to

$\displaystyle\mathcal{F}^{(s)}_{\Delta_{s}}(\Delta_{i}|x)=\mathcal{F}^{(s)}_{\Delta_{s}}(\Delta_{i}|x,0,\infty,1)\
,$ (2.3.42)

from which $\mathcal{F}^{(s)}_{\Delta_{s}}(\Delta_{i}|z_{i})$ can be
reconstructed using eqs. (2.3.31) and (2.3.32). This simplifies the
computation of the conformal block:

$\displaystyle\mathcal{F}^{(s)}_{\Delta_{s}}(\Delta_{i}|x)=\sum_{L_{s}\in\mathcal{L}}f_{\Delta_{1},\Delta_{2}}^{\Delta_{s},L_{s}}g^{L_{s}}_{\Delta_{s},\Delta_{3},\Delta_{4}}x^{|L_{s}|+\Delta_{s}-\Delta_{1}-\Delta_{2}}\
,$ (2.3.43)

where we defined
$g^{L_{s}}_{\Delta_{s},\Delta_{3},\Delta_{4}}=g^{L_{s}}_{\Delta_{s},\Delta_{3},\Delta_{4}}(0,\infty,1)$.
Explicitly, using eqs. (2.2.41)-(2.2.44) for
$f_{\Delta_{1},\Delta_{2}}^{\Delta_{s},L_{s}}$ and eqs. (2.2.31)-(2.2.33) for
$g^{L_{s}}_{\Delta_{s},\Delta_{3},\Delta_{4}}$, we find

$\mathcal{F}^{(s)}_{\Delta_{s}}(\Delta_{i}|x)=x^{\Delta_{s}-\Delta_{1}-\Delta_{2}}\Bigg{\\{}1+\frac{(\Delta_{s}+\Delta_{1}-\Delta_{2})(\Delta_{s}+\Delta_{4}-\Delta_{3})}{2\Delta_{s}}x\\\
+\frac{1}{D_{2}(\Delta_{s})}\begin{bmatrix}(\Delta_{s}+\Delta_{1}-\Delta_{2})_{2}\\\
\Delta_{s}+2\Delta_{1}-\Delta_{2}\end{bmatrix}^{T}\begin{bmatrix}2+\frac{c}{4\Delta_{s}}&-3\\\
-3&4\Delta_{s}+2\end{bmatrix}\begin{bmatrix}(\Delta_{s}+\Delta_{4}-\Delta_{3})_{2}\\\
\Delta_{s}+2\Delta_{4}-\Delta_{3}\end{bmatrix}x^{2}+O(x^{3})\Bigg{\\}}\ ,$
(2.3.44)

where the level-$2$ determinant $D_{2}(\Delta_{s})$ was defined in eq.
(2.1.17), and $(x)_{2}=x(x+1)$. Once $s$-channel blocks are known, $t$-channel
blocks can be obtained by a permutation of the arguments, as suggested by the
diagrammatic notations,

$\displaystyle\mathcal{F}^{(t)}_{\Delta_{t}}(\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}|z_{1},z_{2},z_{3},z_{4})=\mathcal{F}^{(s)}_{\Delta_{t}}(\Delta_{2},\Delta_{3},\Delta_{4},\Delta_{1}|z_{2},z_{3},z_{4},z_{1})\
.$ (2.3.45)

Taking global conformal symmetry into account, this becomes

$\displaystyle\mathcal{F}^{(t)}_{\Delta_{t}}(\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}|x)=(1-x)^{\Delta_{2}+\Delta_{3}-\Delta_{1}-\Delta_{4}}\mathcal{F}^{(s)}_{\Delta_{t}}(\Delta_{2},\Delta_{3},\Delta_{4},\Delta_{1}|1-x)\
.$ (2.3.46)

For practical computations of conformal blocks, the formula (2.3.43) is not
very efficient: the number of terms grows quickly with the level $|L_{s}|$,
and the cofficients
$f_{\Delta_{1},\Delta_{2}}^{\Delta_{s},L_{s}}g^{L_{s}}_{\Delta_{s},\Delta_{3},\Delta_{4}}$
are not known explicitly. Fortunately, there is also Al. Zamolodchikov’s
recursion representation [2]

$\displaystyle\mathcal{F}^{(s)}_{\Delta_{s}}(\Delta_{i}|x)=(16q)^{\Delta_{s}-\frac{Q^{2}}{4}}x^{\frac{Q^{2}}{4}-\Delta_{1}-\Delta_{2}}(1-x)^{\frac{Q^{2}}{4}-\Delta_{1}-\Delta_{4}}\theta_{3}(q)^{3Q^{2}-4(\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4})}H_{\Delta_{s}}(\Delta_{i}|q)\
,$ (2.3.47)

which uses the following elliptic variable $q$ and elliptic function
$\theta_{3}(q)$,

$\displaystyle
q=\exp-\pi\frac{F(\frac{1}{2},\frac{1}{2},1,1-x)}{F(\frac{1}{2},\frac{1}{2},1,x)}\quad,\quad\theta_{3}(q)=\sum_{n\in{\mathbb{Z}}}q^{n^{2}}\
,$ (2.3.48)

where $F(\frac{1}{2},\frac{1}{2},1,x)$ is a special case of the hypergeometric
function (2.3.74), and $|q|<1$ for any $x\in\mathbb{C}$. The nontrivial factor
is the function

$\displaystyle
H_{\Delta_{s}}(\Delta_{i}|q)=\sum_{k=0}^{\infty}\prod_{j=1}^{k}\sum_{m_{j},n_{j}=1}^{\infty}\frac{(16q)^{m_{j}n_{j}}R_{m_{j},n_{j}}}{\Delta_{\langle
m_{j-1},-n_{j-1}\rangle}-\Delta_{\langle m_{j},n_{j}\rangle}}$ (2.3.49)

where by convention $\Delta_{\langle m_{0},-n_{0}\rangle}=\Delta_{s}$, and we
introduce the coefficients

$\displaystyle R_{m,n}=\frac{-2P_{\langle 0,0\rangle}P_{\langle
m,n\rangle}}{\prod_{r=1-m}^{m}\prod_{s=1-n}^{n}2P_{\langle
r,s\rangle}}\prod_{r\overset{2}{=}1-m}^{m-1}\prod_{s\overset{2}{=}1-n}^{n-1}\prod_{\pm}(P_{2}\pm
P_{1}+P_{\langle r,s\rangle})(P_{3}\pm P_{4}+P_{\langle r,s\rangle})\ .$
(2.3.50)

In this formula $P$ is yet another notation for the momentum, which is related
to the notations $\Delta$ and $\alpha$ by

$\displaystyle\Delta=\frac{Q^{2}}{4}+P^{2}\quad\text{and}\quad\alpha=\frac{Q}{2}+iP\
,\quad\text{so that}\quad P_{\langle
r,s\rangle}=\frac{i}{2}\left(rb+sb^{-1}\right)\ .$ (2.3.51)

The recursion representation owes its name to the original definition of the
function $H_{\Delta_{s}}(\Delta_{i}|q)$ (2.3.49) by a recursive formula,

$\displaystyle
H_{\Delta_{s}}(\Delta_{i}|q)=1+\sum_{m,n=1}^{\infty}\frac{(16q)^{mn}}{\Delta_{s}-\Delta_{\langle
m,n\rangle}}R_{m,n}H_{\Delta_{\langle m,-n\rangle}}(\Delta_{i}|q)\ .$ (2.3.52)

The recursion representation has the following properties:

  * •

it converges very fast,

  * •

it uses an elliptic variable $q$ instead of the natural variable $x$,

  * •

it manifestly has the correct poles as a function of $\Delta_{s}$, namely
$\Delta_{s}\in\\{\Delta_{\langle r,s\rangle}\\}_{r,s\in{\mathbb{N}}^{*}}$,

  * •

it is manifestly invariant under reflections of all momentums,

  * •

it has no known interpretation as a sum over descendents in some basis.

Alternatively, conformal blocks have a combinatorial representation which
follows from the AGT relation [10], and which has the opposite properties.

The general crossing symmetry equation (2.3.36), while it may in principle
determine the three-point structure constant, is in practice often
intractable, because

  1. 1.

it can involve sums over large sets of possible conformal dimensions in the
$s$\- and $t$-channels,

  2. 2.

the general conformal blocks are rather complicated.

So, while the general crossing equation is useful for testing the consistency
of proposals for the spectrum and three-point structure constant, it is often
useless for deriving specific proposals in the first place. However, it is
possible to restrict the $s$\- and $t$-channels conformal dimensions to finite
sets, and to simplify the conformal blocks, by taking one of the four fields
to be degenerate. If the other three fields remain generic, this still leads
to constraints on the generic three-point structure constant.

So, in the remainder of Section 2.3, we will study conformal blocks which
involve degenerate fields, and therefore obey differential equations. Notice
however that the conformal block
$\mathcal{F}^{(s)}_{\Delta_{s}}(\Delta_{i}|z_{i})$ is a smooth function of
$\Delta_{i}$, which has a finite value for $\Delta_{1}=\Delta_{\langle
r,s\rangle}$ – irrespective of the values of
$\Delta_{2},\Delta_{3},\Delta_{4},\Delta_{s}$. If however $\Delta_{2}$ and
$\Delta_{s}$ happen to be related as dictated by the fusion rule of the
degenerate field $V_{\langle r,s\rangle}$, then this value obeys a
differential equation of order $rs$. In other words, a conformal block
involving a degenerate representation coincides with the conformal block
involving the corresponding Verma module. The degenerate representation
manifests itself in the decompositions of correlation functions, where fusion
rules dictate which conformal blocks contribute. On the other hand, conformal
blocks involving a degenerate representation in the $s$-channel are not mere
limits of $\mathcal{F}^{(s)}_{\Delta_{s}}(\Delta_{i}|z_{i})$, which is
actually singular at $\Delta_{s}=\Delta_{\langle r,s\rangle}$. The sum over
descendents in eq. (2.3.43) is valid only if the $s$-channel representation is
a Verma module, and must be modified otherwise.

####  2.3.4 BPZ differential equations

Using the representation (2.3.2) of creation operators as differential
operators, null vector equations for correlation functions of primary fields
become differential equations, called BPZ equations after Belavin, Polyakov
and Zamolodchikov. For example, in the cases of the degenerate fields
$V_{\langle 1,1\rangle}(x)$ and $V_{\langle 2,1\rangle}(x)$, eq. (2.2.6) gives
rise to the following BPZ equations:

$\displaystyle{\frac{\partial}{\partial x}}\left\langle V_{\langle
1,1\rangle}(x)\prod_{i=1}^{N-1}V_{\Delta_{i}}(z_{i})\right\rangle=0\ ,$
(2.3.53) $\displaystyle\boxed{\left(\frac{\partial^{2}}{\partial
x^{2}}+b^{2}\sum_{j=1}^{N-1}\left[\frac{1}{x-z_{j}}{\frac{\partial}{\partial
z_{j}}}+\frac{\Delta_{j}}{(x-z_{j})^{2}}\right]\right)\left\langle V_{\langle
2,1\rangle}(x)\prod_{i=1}^{N-1}V_{\Delta_{i}}(z_{i})\right\rangle=0}\ .$
(2.3.54)

More generally, a degenerate field $V_{\langle r,s\rangle}(x)$ with a
vanishing descendent at level $rs$ leads to a BPZ equation of order $rs$.

In the case of a three-point function ($N=3$), the dependence on $z_{i}$ is
already completely determined by global conformal symmetry, see eq. (2.3.25).
The BPZ equation therefore leads to constraints on the conformal dimensions
$\Delta_{1}$ and $\Delta_{2}$ – that is, to the fusion rules. (See Exercise
2.15.) If $N\geq 4$, taking the global conformal symmetry into account as in
eq. (2.3.29), it is possible to rewrite a BPZ equation as a differential
equation for a function of $N-3$ variables. In particular, a BPZ equation for
a four-point function boils down to an ordinary differential equation – a
differential equation for a function of a single variable.

Let us derive the ordinary differential equation for $\left\langle V_{\langle
2,1\rangle}(x)\prod_{i=1}^{3}V_{\Delta_{i}}(z_{i})\right\rangle$. Inserting
the identity $\oint_{\infty}dz\ \epsilon(z)T(z)=0$ with
$\epsilon(z)=\frac{(z-z_{1})(z-z_{2})(z-z_{3})}{z-x}$, and using eq. (2.2.6),
we obtain a differential equation with derivatives only with respect to $x$,

$\left\\{\prod_{i=1}^{3}(x-z_{i})\left(-\frac{1}{b^{2}}\frac{\partial^{2}}{\partial
x^{2}}+\sum_{i=1}^{3}\frac{1}{x-z_{i}}{\frac{\partial}{\partial
x}}\right)+(3x-z_{1}-z_{2}-z_{3})\Delta_{\langle 2,1\rangle}\right.\\\
\left.+\frac{z_{12}z_{13}}{z_{1}-x}\Delta_{1}+\frac{z_{21}z_{23}}{z_{2}-x}\Delta_{2}+\frac{z_{31}z_{32}}{z_{3}-x}\Delta_{3}\right\\}\left\langle
V_{\langle 2,1\rangle}(x)\prod_{i=1}^{3}V_{\Delta_{i}}(z_{i})\right\rangle=0\
.$ (2.3.55)

Setting $(z_{1},z_{2},z_{3})=(0,\infty,1)$, this amounts to the equation

$\displaystyle\left\\{\frac{x(1-x)}{b^{2}}\frac{\partial^{2}}{\partial
x^{2}}+(2x-1){\frac{\partial}{\partial x}}+\Delta_{\langle
2,1\rangle}+\frac{\Delta_{1}}{x}-\Delta_{2}+\frac{\Delta_{3}}{1-x}\right\\}\mathcal{F}(x)=0\
,$ (2.3.56)

for

$\displaystyle\mathcal{F}(x)=\left\langle V_{\langle
2,1\rangle}(x)V_{\Delta_{1}}(0)V_{\Delta_{2}}(\infty)V_{\Delta_{3}}(1)\right\rangle\
.$ (2.3.57)

(The $s$-th order differential equation for $\left\langle V_{\langle
s,1\rangle}(x)V_{\Delta_{1}}(0)V_{\Delta_{2}}(\infty)V_{\Delta_{3}}(1)\right\rangle$
can be found in [11].)

Before solving our second-order differential equation, let us discuss its
qualitative properties. The coefficients of the equation diverge, and the
solutions can therefore be non-analytic, at the singularities $x=0,1,\infty$.
These singularities are regular singular points, where a regular singular
point $x_{0}$ is characterized by the existence of a basis of solutions of the
type

$\displaystyle\mathcal{F}(x)=(x-x_{0})^{\lambda}\left(1+\sum_{n=0}^{\infty}c_{n}(x-x_{0})^{n}\right)\
,$ (2.3.58)

where $\lambda$ is called a characteristic exponent. The characteristic
exponents of the equation (2.3.56) at the regular singular point $x=0$ are
obtained by inserting the ansatz $\mathcal{F}(x)=x^{\lambda}(1+O(x))$, which
leads to

$\displaystyle\lambda^{2}-bQ\lambda+b^{2}\Delta_{1}=0\ .$ (2.3.59)

The roots of this equation are

$\displaystyle\lambda_{\pm}=b\alpha_{1}^{\pm}\ ,$ (2.3.60)

where for a given momentum $\alpha$ we define

$\displaystyle\alpha^{+}=\alpha\quad,\quad\alpha^{-}=Q-\alpha\ .$ (2.3.61)

In conformal field theory, singularities of a correlation function correspond
to coincidences of fields, and characteristic exponents are constrained by the
corresponding OPEs. In the case of our four-point function (2.3.57) at $x=0$,
the fusion rule (2.2.52) leads to an OPE of the type

$\displaystyle V_{\langle
2,1\rangle}(x)V_{\alpha}(0)=\sum_{\pm}C_{\pm}(\alpha)x^{\Delta\left(\alpha\mp\frac{b}{2}\right)-\Delta(\alpha)-\Delta_{\langle
2,1\rangle}}\left(V_{\alpha\mp\frac{b}{2}}(0)+O(x)\right)\ ,$ (2.3.62)

where $C_{\pm}(\alpha)$ are some constants, and the powers of $x$ are dictated
by eq. (2.3.33). Computing these powers,

$\displaystyle\Delta\left(\alpha\mp\frac{b}{2}\right)-\Delta(\alpha)-\Delta_{\langle
2,1\rangle}=b\alpha^{\pm}\ ,$ (2.3.63)

we recover the characteristic exponents $\lambda_{\pm}$, so that each
characteristic exponent corresponds to a primary field in the OPE. (See also
Exercise 2.16.) The corresponding solutions are called $s$-channel degenerate
conformal blocks, denoted $\mathcal{F}^{(s)}_{\pm}(x)$, and depicted

$\displaystyle\mathcal{F}^{(s)}_{\pm}(x)=\leavevmode\hbox to114.24pt{\vbox
to60.24pt{\pgfpicture\makeatletter\hbox{\hskip
40.86917pt\lower-31.09486pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ } {}{{}}{}{{}} {}{}
{}{}{}{}{{{}{}}}
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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{}{{}}{}\pgfsys@moveto{45.52371pt}{0.0pt}\pgfsys@lineto{56.90465pt}{-22.76186pt}\pgfsys@stroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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}\hbox{\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{
} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\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}{23.84985pt}{25.6543pt}\pgfsys@invoke{
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}}\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}{23.84985pt}{-28.1554pt}\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{{$\alpha_{3}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-21.07458pt}{-2.43054pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\alpha_{1}\mp\frac{b}{2}\\!\\!$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ \ ,$ (2.3.64)

where the degenerate field is represented as a dashed line. These degenerate
conformal blocks are special cases of the more general conformal blocks of
Section 2.3.3.

####  2.3.5 Hypergeometric conformal blocks

Let us define the hypergeometric equation with parameters $A,B,C$,

$\displaystyle\left\\{x(1-x)\frac{\partial^{2}}{\partial
x^{2}}+\left[C-(A+B+1)x\right]{\frac{\partial}{\partial
x}}-AB\right\\}\mathcal{G}(x)=0\ .$ (2.3.65)

Using a change of notations and unknown function,

$\displaystyle\mathcal{H}(x)=x^{\frac{C}{2}}(x-1)^{\frac{A+B+1-C}{2}}\mathcal{G}(x)\
,\quad\left\\{\begin{array}[]{l}j_{1}=-\frac{C}{2}\ ,\\\
j_{2}=\frac{B-A-1}{2}\ ,\\\ j_{3}=\frac{C-A-B-1}{2}\
,\end{array}\right.\quad\left\\{\begin{array}[]{l}A=-1-j_{1}-j_{2}-j_{3}\ ,\\\
B=-j_{1}+j_{2}-j_{3}\ ,\\\ C=-2j_{1}\ ,\end{array}\right.$ (2.3.72)

the hypergeometric equation is equivalent to the twisted hypergeometric
equation,

$\displaystyle\left\\{x(1-x)\frac{\partial^{2}}{\partial
x^{2}}-\frac{j_{1}(j_{1}+1)}{x}+j_{2}(j_{2}+1)+\frac{j_{3}(j_{3}+1)}{x-1}\right\\}\mathcal{H}(x)=0\
.$ (2.3.73)

The characteristic exponents of the hypergeometric equation at the regular
singular point $x=0$ are $\lambda\in\\{0,1-C\\}$, and the solution
$\mathcal{G}(x)$ which corresponds to $\lambda=0$ is the hypergeometric
function

$\displaystyle
F(A,B,C,x)=\sum_{n=0}^{\infty}\frac{(A)_{n}(B)_{n}}{n!(C)_{n}}x^{n}\ ,$
(2.3.74)

where $(A)_{n}=\frac{\Gamma(A+n)}{\Gamma(A)}$. The solution with the
characteristic exponent $\lambda=1-C$ is $x^{1-C}F(A-C+1,B-C+1,2-C,x)$. The
hypergeometric function obeys the identities

$\displaystyle F(A,B,C,x)=F(B,A,C,x)=(1-x)^{C-A-B}F(C-A,C-B,C,x)\ .$ (2.3.75)

In the case of four-point functions, we have the following equivalences of
differential equations:

BPZ equation (2.3.56) for $\displaystyle\mathcal{F}(x)$ (2.3.76)
$\displaystyle\Leftrightarrow\quad\text{hypergeometric\ equation\ for\quad}$
$\displaystyle\mathcal{G}(x)=x^{-b\alpha_{1}}(1-x)^{-b\alpha_{3}}\mathcal{F}(x)$
(2.3.77) $\displaystyle\Leftrightarrow\quad\text{twisted\ hypergeometric\
equation\ for\quad}$
$\displaystyle\mathcal{H}(x)=(x(1-x))^{-\frac{b^{2}}{2}}\mathcal{F}(x)\ ,$
(2.3.78)

provided the parameters are related as

$\displaystyle\left\\{\begin{array}[]{l}A=\frac{1}{2}+b(\alpha_{1}+\alpha_{3}-Q)+b(\alpha_{2}-\tfrac{Q}{2})\
,\\\ B=\frac{1}{2}+b(\alpha_{1}+\alpha_{3}-Q)-b(\alpha_{2}-\tfrac{Q}{2})\ ,\\\
C=1+b(2\alpha_{1}-Q)\ ,\end{array}\right.$ (2.3.82)

or equivalently $\alpha_{i}=-b^{-1}j_{i}+\frac{b}{2}$. (See Exercise 2.17.)
Therefore, the solutions of the equation (2.3.56) which correspond to the
characteristic exponents $\lambda_{\pm}$ (2.3.60) at $x=0$ are

$\displaystyle\left\\{\begin{array}[]{l}\mathcal{F}^{(s)}_{+}(x)=x^{b\alpha_{1}}(1-x)^{b\alpha_{3}}F(A,B,C,x)\
,\\\
\mathcal{F}^{(s)}_{-}(x)=x^{b(Q-\alpha_{1})}(1-x)^{b\alpha_{3}}F(A-C+1,B-C+1,2-C,x)=\left.\mathcal{F}^{(s)}_{+}(x)\right|_{\alpha_{1}\to
Q-\alpha_{1}}\ ,\end{array}\right.$ (2.3.85)

where the superscript in $\mathcal{F}^{(s)}_{\pm}(x)$ stands for the
$s$-channel basis of solutions. These formulas are consistent with the
invariance of the original differential equation (2.3.56) under reflections of
$\alpha_{i}$: reflection of $\alpha_{1}$ exchanges the two solutions, and
reflections of $\alpha_{2}$ and $\alpha_{3}$ leave each solution invariant, as
follows from eq. (2.3.75). The corresponding solutions of the twisted equation
(2.3.73) are

$\displaystyle\left\\{\begin{array}[]{l}\mathcal{H}^{(s)}_{+}(x)=x^{-j_{1}}(1-x)^{-j_{3}}F(-j_{1}-j_{2}-j_{3}-1,-j_{1}+j_{2}-j_{3},-2j_{1},x)\
,\\\
\mathcal{H}^{(s)}_{-}(x)=x^{j_{1}+1}(1-x)^{-j_{3}}F(j_{1}-j_{2}-j_{3},j_{1}+j_{2}-j_{3}+1,2j_{1}+2,x)\
.\end{array}\right.$ (2.3.88)

We can also define the $t$-channel basis of solutions, whose elements are
associated to the characteristic exponents at the regular singular point
$x=1$. Such solutions are obtained from the $s$-channel solutions by the
exchange $\left\\{\begin{smallmatrix}1\leftrightarrow 3\\\ x\leftrightarrow
1-x\end{smallmatrix}\right.$, and we find

$\displaystyle\left\\{\begin{array}[]{l}\mathcal{F}^{(t)}_{+}(x)=x^{b\alpha_{1}}(1-x)^{b\alpha_{3}}F(A,B,A+B-C+1,1-x)\
,\\\
\mathcal{F}^{(t)}_{-}(x)=x^{b\alpha_{1}}(1-x)^{b(Q-\alpha_{3})}F(C-A,C-B,C-A-B+1,1-x)\
,\end{array}\right.$ (2.3.91)

whose diagrammatic representations are

$\displaystyle\mathcal{F}^{(t)}_{\pm}(x)=\leavevmode\hbox to91.48pt{\vbox
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.23741pt}{-11.04965pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\alpha_{3}\mp\frac{b}{2}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ \ .$ (2.3.92)

In order to find the relation between our two bases of solutions, we need the
formula

$F(A,B,C,x)=\frac{\Gamma(C)\Gamma(C-A-B)}{\Gamma(C-A)\Gamma(C-B)}F(A,B,A+B-C+1,1-x)\\\
+\frac{\Gamma(C)\Gamma(A+B-C)}{\Gamma(A)\Gamma(B)}(1-x)^{C-A-B}F(C-A,C-B,C-A-B+1,1-x)\
.$ (2.3.93)

Together with eq. (2.3.75), this implies

$\displaystyle\mathcal{F}^{(s)}_{i}(x)=\sum_{j=\pm}F_{ij}\mathcal{F}^{(t)}_{j}(x)\
,$ (2.3.94)

where we introduce the fusing matrix

$\displaystyle F=\left[\begin{array}[]{cc}F_{++}&F_{+-}\\\
F_{-+}&F_{--}\end{array}\right]=\left[\begin{array}[]{cc}\frac{\Gamma(C)\Gamma(C-A-B)}{\Gamma(C-A)\Gamma(C-B)}&\frac{\Gamma(C)\Gamma(A+B-C)}{\Gamma(A)\Gamma(B)}\\\
\frac{\Gamma(2-C)\Gamma(C-A-B)}{\Gamma(1-A)\Gamma(1-B)}&\frac{\Gamma(2-C)\Gamma(A+B-C)}{\Gamma(A-C+1)\Gamma(B-C+1)}\end{array}\right]\
,$ (2.3.99)

whose determinant is

$\displaystyle\det F=\frac{1-C}{A+B-C}\ ,$ (2.3.100)

and whose elements can be represented diagrammatically as

$\displaystyle F_{ij}=\leavevmode\hbox to89.84pt{\vbox
to80.87pt{\pgfpicture\makeatletter\hbox{\hskip
49.40509pt\lower-40.43367pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ } {}{{}}{} {}{}{}{}{{{}{}}}
{}{}{}{}{{{}{}}} {}{}{}{}{{{}{}}}
{}{}{}\pgfsys@moveto{-39.83368pt}{0.0pt}\pgfsys@lineto{0.0pt}{39.83368pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@lineto{0.0pt}{-39.83368pt}\pgfsys@lineto{0.0pt}{39.83368pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-33.04688pt}{25.6543pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\alpha_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{23.84985pt}{25.6543pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\alpha_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{23.84985pt}{-28.1554pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\alpha_{3}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-39.83368pt}\pgfsys@lineto{-39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-46.07208pt}{-31.34985pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\langle 2,1\rangle$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}{{}}{}\pgfsys@moveto{-39.83368pt}{0.0pt}\pgfsys@lineto{-3.98329pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-12.6068pt}{-10.52826pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$j$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{3.98329pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.37814pt}{7.9163pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$i$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ .$ (2.3.101)

Now how can the single-valued four-point function $\left\langle V_{\langle
2,1\rangle}(x,\bar{x})V_{\Delta_{1}}(0)V_{\Delta_{2}}(\infty)V_{\Delta_{3}}(1)\right\rangle$
be decomposed as a combination of our solutions? We assume that we are dealing
with a model whose spectrum is diagonal, so that the four-point function obeys
not only the BPZ equation (2.3.56), but also the complex conjugate equation,
and must be of the type

$\displaystyle\left\langle V_{\langle
2,1\rangle}(x,\bar{x})V_{\Delta_{1}}(0)V_{\Delta_{2}}(\infty)V_{\Delta_{3}}(1)\right\rangle=\sum_{i,j=\pm}c^{(s)}_{ij}\mathcal{F}_{i}^{(s)}(x)\mathcal{F}_{j}^{(s)}(\bar{x})=\sum_{i,j=\pm}c^{(t)}_{ij}\mathcal{F}_{i}^{(t)}(x)\mathcal{F}_{j}^{(t)}(\bar{x})\
,$ (2.3.102)

where $c^{(s)}_{ij}$ and $c^{(t)}_{ij}$ are $x,\bar{x}$-independent
coefficients. These two sets of coefficients are related by the change of
basis (2.3.94),

$\displaystyle\sum_{i,j=\pm}c^{(s)}_{ij}F_{ii^{\prime}}F_{jj^{\prime}}=c^{(t)}_{i^{\prime}j^{\prime}}\
.$ (2.3.103)

Now, single-valuedness near $x=0$ implies $c^{(s)}_{+-}=c^{(s)}_{-+}=0$, and
single-valuedness near $x=1$ similarly implies $c^{(t)}_{+-}=c^{(t)}_{-+}=0$.
Thus we can write

$\displaystyle\left\langle V_{\langle
2,1\rangle}(x,\bar{x})V_{\Delta_{1}}(0)V_{\Delta_{2}}(\infty)V_{\Delta_{3}}(1)\right\rangle=\sum_{i=\pm}c^{(s)}_{i}\mathcal{F}_{i}^{(s)}(x)\mathcal{F}_{i}^{(s)}(\bar{x})=\sum_{i=\pm}c^{(t)}_{i}\mathcal{F}^{(t)}_{i}(x)\mathcal{F}^{(t)}_{i}(\bar{x})\
,$ (2.3.104)

where the coefficients $c^{(s)}_{\pm}$ and $c^{(t)}_{\pm}$ obey the relations

$\displaystyle\frac{c^{(s)}_{+}}{c^{(t)}_{-}}$
$\displaystyle=-\frac{F_{-+}}{F_{+-}}\frac{1}{\det
F}=\frac{\gamma(A)\gamma(B)}{\gamma(C)\gamma(A+B-C)}\ ,$ (2.3.105)
$\displaystyle\frac{c^{(s)}_{-}}{c^{(t)}_{-}}$
$\displaystyle=\frac{F_{++}}{F_{--}}\frac{1}{\det
F}=\frac{\gamma(C-1)\gamma(C-A-B+1)}{\gamma(C-A)\gamma(C-B)}\ ,$ (2.3.106)
$\displaystyle\frac{c^{(s)}_{+}}{c^{(s)}_{-}}$
$\displaystyle=-\frac{F_{-+}F_{--}}{F_{++}F_{+-}}=\frac{\gamma(A)\gamma(B)\gamma(C-A)\gamma(C-B)}{\gamma(C)\gamma(C-1)}\
,$ (2.3.107)

where we introduce the function

$\displaystyle\gamma(x)=\frac{\Gamma(x)}{\Gamma(1-x)}\ .$ (2.3.108)

This determines $\left\langle V_{\langle
2,1\rangle}(x,\bar{x})V_{\Delta_{1}}(0)V_{\Delta_{2}}(\infty)V_{\Delta_{3}}(1)\right\rangle$
up to an $x,\bar{x}$-independent factor.

###  2.4 Exercises

######  Exercise 2.1 (Spanning set for a highest-weight representation)

Assuming $|\Delta\rangle$ to be a primary state, prove the equality
$U(\mathfrak{V})|\Delta\rangle=U(\mathfrak{V}^{+})|\Delta\rangle$ in eq.
(2.1.4). To do this, prove that any state of the type
$\prod_{i=1}^{p}L_{n_{i}}|\Delta\rangle$ with $n_{1},\cdots
n_{p}\in{\mathbb{Z}}$ belongs to $U(\mathfrak{V}^{+})|\Delta\rangle$. The
proof can be done by hand in the cases $p=0,1,2$, and then by induction on
$p$.

######  Exercise 2.2 (Alternative spanning set for a Verma module)

In $U(\mathfrak{V}^{+})$, write the states belonging to
$\\{\prod_{i=1}^{p}L_{-n_{i}}\\}_{n_{i}\in\\{1,2\\}}$ up to the level $N=5$,
and compare with the known basis (2.1.8). For which levels do we obtain a
basis? Explain the observed results by studying whether $L_{-1}$ and $L_{-2}$
are algebraically independent. In particular, show that the Virasoro
commutation relations (1.3.15) imply a relation of the type

$\displaystyle[L_{-1},[L_{-1},[L_{-1},L_{-2}]]]\propto[L_{-2},[L_{-1},L_{-2}]]\
.$ (2.4.1)

######  Exercise 2.3 (Singular vectors at the level $N=4$)

Compute the singular vectors at the level $N=4$, and write the results in a
table analogous to (2.1.38). In particular, show that the five coefficients of
the singular vectors obey five linear equations, and that the determinant of
the system is a polynomial of degree three in the conformal dimension
$\Delta$. If $\Delta_{\langle 2,2\rangle}=\Delta_{\langle 4,1\rangle}$, do the
corresponding singular vectors coincide?

######  Exercise 2.4 (Singular vectors at higher levels)

For a null vector $|\chi\rangle$ at level $N$ with $p(N)$ coefficients, how
many equations do the constraints $L_{1}|\chi\rangle=L_{2}|\chi\rangle=0$
provide? Count the excess equations using the pentagonal number theorem, and
explain their existence using eq. (2.4.1).

Write a computer program for computing singular vectors at arbitrary levels,
and check that at level $N$ the number of Verma modules which have a singular
vector is the number of factorizations of $N$ into two positive integers. The
program should preferably be written using free, open-source software.

######  Exercise 2.5 (Singular vectors that are not of the type
$|\chi_{\langle r,s\rangle}\rangle$)

Show that the existence of two singular vectors in the same Verma module
implies the rationality of $b^{2}$. From now on we assume $b^{2}=-\frac{q}{p}$
where $p,q$ are strictly positive integers. Prove the identities

$\displaystyle\Delta_{\langle r,s\rangle}=\Delta_{\langle
r+p,s+q\rangle}=\Delta_{\langle p-r,q-s\rangle}\ .$ (2.4.2)

Under suitable assumptions on $r$ and $s$, show that
$\mathcal{V}_{\Delta_{\langle r,s\rangle}}$ has two singular vectors
$|\chi_{\langle r,s\rangle}\rangle$ and $|\chi_{\langle
p-r,q-s\rangle}\rangle$. Show that each one of the two states $|\chi_{\langle
r,s\rangle}\rangle$ and $|\chi_{\langle p-r,q-s\rangle}\rangle$ has a
descendent which is itself a singular vector at the level $pq+qr-ps$ in
$\mathcal{V}_{\Delta_{\langle r,s\rangle}}$. Assuming that these two singular
vectors, which live at the same level $pq+qr-ps$, are in fact identical,
enumerate all the singular vectors of $\mathcal{V}_{\Delta_{\langle
r,s\rangle}}$. Which ones are of the type $|\chi_{\langle
r^{\prime},s^{\prime}\rangle}\rangle$? In particular, show that the singular
vector at the level $pq+qr-ps$ is not of this type in
$\mathcal{V}_{\Delta_{\langle r,s\rangle}}$, although it is of this type when
considered as a singular vector of the Verma modules generated by
$|\chi_{\langle r,s\rangle}\rangle$ and $|\chi_{\langle
p-r,q-s\rangle}\rangle$.

######  Exercise 2.6 (Unitarity of Virasoro representations)

Show that the Gram matrix $M^{(N)}$ of the level-$N$ subspace of the Verma
module $\mathcal{V}_{\Delta}$ is positive definite in the limit $c\to\infty$,
provided $\Delta>0$. To do this, show that in this limit the diagonal elements
of $M^{(N)}$ coincide with its eigenvalues, provided $M^{(N)}$ is written in a
basis of creation operators of the type $(\prod_{i}L_{-n_{i}})L_{-1}^{p}$ with
$n_{i}\geq 2$. In other words, the Virasoro algebra (1.3.15) effectively
reduces to a sum of commuting finite-dimensional subalgebras,

$\displaystyle\underset{c\to\infty}{\lim}\mathfrak{V}=\operatorname{Span}(L_{-1},L_{0},L_{1})\oplus\bigoplus_{n=2}^{\infty}\operatorname{Span}(L_{-n},L_{n})\
.$ (2.4.3)

######  Exercise 2.7 (Virasoro algebra and OPE)

Show that the $T(y)T(z)$ OPE (2.2.15), the commutativity axiom
$T(y)T(z)=T(z)T(y)$, and the expansion (2.2.10) of $T(y)$ into modes
$L_{n}^{(z_{0})}$, imply that such modes obey the Virasoro commutation
relations (1.3.15) for any choice of $z_{0}$. To do this, write

$\displaystyle[L_{n}^{(z_{0})},L_{m}^{(z_{0})}]=-\frac{1}{4\pi^{2}}\left(\oint_{z_{0}}dy\oint_{z_{0}}dz-\oint_{z_{0}}dz\oint_{z_{0}}dy\right)(y-z_{0})^{n+1}(z-z_{0})^{m+1}T(y)T(z)\
,$ (2.4.4)

and use contour manipulations to show that

$\displaystyle\oint_{z_{0}}dy\oint_{z_{0}}dz-\oint_{z_{0}}dz\oint_{z_{0}}dy=\oint_{z_{0}}dy\oint_{y}dz\
.$ (2.4.5)

Explain why regular terms in the $T(y)T(z)$ OPE do not contribute to the
result.

######  Exercise 2.8 (From the identity field to the Virasoro field)

By comparing it with the general $T(y)V_{\sigma}(z_{0})$ OPE (2.2.12), show
that the $T(y)T(z)$ OPE (2.2.15) encodes the equations

$\displaystyle L_{-1}T(z)$ $\displaystyle=\partial T(z)\ ,$ (2.4.6)
$\displaystyle L_{0}T(z)$ $\displaystyle=2T(z)\ ,$ (2.4.7) $\displaystyle
L_{1}T(z)$ $\displaystyle=0\ ,$ (2.4.8) $\displaystyle L_{2}T(z)$
$\displaystyle=\frac{c}{2}I\ ,$ (2.4.9) $\displaystyle L_{n\geq 3}T(z)$
$\displaystyle=0\ .$ (2.4.10)

With the help of eq. (2.2.9), show that these equations follow from

$\displaystyle\partial I$ $\displaystyle=0\ ,$ (2.4.11) $\displaystyle
L_{n\geq-1}I$ $\displaystyle=0\ ,$ (2.4.12) $\displaystyle L_{-2}^{(z)}I$
$\displaystyle=T(z)\ ,$ (2.4.13)

######  Exercise 2.9 (Computing OPE coefficients)

Compute the OPE coefficients $f^{\Delta,L}_{\Delta_{1},\Delta_{2}}$ at the
level $|L|=2$ by solving the linear equations (2.2.39). Compare your results
with the available literature. Write the equations for the OPE coefficients at
the level $|L|=3$. More generally, show that
$f^{\Delta,L}_{\Delta_{1},\Delta_{2}}$ is uniquely determined, by counting the
equations and unknowns in eq. (2.2.39) up to a given order $z_{12}^{k}$, as in
Exercise 2.4.

######  Exercise 2.10 (Fusion rules from OPEs)

Rederive the fusion rule (2.2.52) by analyzing the corresponding OPE. To do
this, write the OPE $V_{\Delta_{1}}(z_{1})V_{\langle 2,1\rangle}(z_{2})$ as in
eq. (2.2.36). If $C$ is a contour around both $z_{1}$ and $z_{2}$, insert
$\oint_{C}dz\ T(z)$ and $\oint_{C}dz\ \frac{1}{z-z_{2}}T(z)$ on both sides of
that OPE, and compute the leading terms of the OPEs
$V_{\Delta_{1}}(z_{1})LV_{\langle 2,1\rangle}(z_{2})$ with
$L\in\\{L_{-1},L_{-1}^{2},L_{-2}\\}$, before using $L_{\langle
2,1\rangle}V_{\langle 2,1\rangle}(z_{2})=0$.

######  Exercise 2.11 (Creation operators as differential operators)

Check that the representation (2.3.2) of creation operators $L_{-n}^{(z_{i})}$
(with $n\geq 1$) as differential operators in $z_{1},\cdots z_{N}$, is
consistent with the commutation relations of the Virasoro algebra.

######  Exercise 2.12 (Quasi-primary fields and representations of
$\mathfrak{sl}_{2}$)

By the state-field correspondence, a quasi-primary field $V_{\Delta}(z)$
corresponds to a representation of the algebra of global conformal
transformations
$\mathfrak{sl}_{2}^{(z)}=\operatorname{Span}(L_{-1}^{(z)},L_{0}^{(z)},L_{1}^{(z)})$.
Using the definition (2.2.24), identify this representation as a highest-
weight representation. How is this compatible with eq. (2.3.17), which
suggests that $V_{\Delta}(z)$ transforms in a generic representation of
$SL_{2}({\mathbb{C}})$? To solve the apparent contradiction, consider how the
algebra $\mathfrak{sl}_{2}^{(z)}$ depends on the choice of $z$, and how
$\mathfrak{sl}_{2}^{(z^{\prime})}$ with $z^{\prime}\neq z$ acts on
representations of $\mathfrak{sl}_{2}^{(z)}$.

######  Exercise 2.13 (Logarithmic conformal field theory)

Consider a finite-dimensional vector space $E$ with a linear action of
$L_{0}$, and the representation $R=U(\mathfrak{V}^{+})E$ of the Virasoro
algebra obtained by assuming $L_{n>0}E=0$. Assume that $R$ is indecomposable
but reducible: what does this mean for the action of $L_{0}$ on $E$, and the
corresponding matrix? Let $V_{v}(z)$ be the field which corresponds to a
vector $v\in E$: write the global Ward identities for correlation functions of
such fields, and prove

$\displaystyle{\frac{\partial}{\partial z_{i}}}\left\langle
z_{12}^{L_{0}^{(1)}+L_{0}^{(2)}-L_{0}^{(3)}}z_{23}^{L_{0}^{(2)}+L_{0}^{(3)}-L_{0}^{(1)}}z_{13}^{L_{0}^{(1)}+L_{0}^{(3)}-L_{0}^{(2)}}\prod_{i=1}^{3}V_{v_{i}}(z_{i})\right\rangle=0\
.$ (2.4.14)

Compare the three-point function
$\left\langle\prod_{i=1}^{3}V_{v_{i}}(z_{i})\right\rangle$ with the three-
point function (2.3.25) of primary fields. In the simplest example when $E$ is
two-dimensional, show that the three-point function involves not only powers
of $z_{ij}$, but also logarithms. Finally, cite an example of a reducible
Verma module, and conclude that a conformal field theory can involve
indecomposable, reducible representations without being logarithmic.

######  Exercise 2.14 (Dependence of OPE structure constants on $z_{i}$)

Rederive the dependence of OPE structure functions on $z_{i}$, by combining
the Virasoro symmetry constraint eq. (2.2.38) with the assumption (2.3.1).

######  Exercise 2.15 (Third-order BPZ equation)

Write the BPZ equation for an $N$-point function involving a degenerate field
$V_{\langle 1,3\rangle}(x)$. In the case $N=3$, rederive the relevant fusion
rule.

######  Exercise 2.16 (Third-order BPZ equation for a four-point function)

Write the third-order BPZ ordinary differential equation for the four-point
function $\left\langle V_{\langle
1,3\rangle}(x)V_{\Delta_{1}}(0)V_{\Delta_{2}}(1)V_{\Delta_{3}}(\infty)\right\rangle$.
Check that the characteristic exponents at $x=0,1,\infty$ are consistent with
the fusion rules.

######  Exercise 2.17 (Hypergeometric form of the second-order BPZ equation
for a four-point function)

Check that the change of unknown function
$\mathcal{F}(x)=x^{\lambda}(1-x)^{\mu}\mathcal{G}(x)$ in the second-order BPZ
equation (2.3.56) leads to a hypergeometric equation (2.3.65) for
$\mathcal{G}(x)$, provided $\lambda$ and $\mu$ are suitably chosen. What are
the four possible choices for the pair $(\lambda,\mu)$? Show that the choice
$\lambda=b\alpha_{1},\ \mu=b\alpha_{3}$ leads to the values of $A,B,C$ given
in eq. (2.3.82).

##  Chapter 3 Liouville theory and minimal models

In this Chapter we introduce and solve Liouville theory, generalized minimal
models, and A-series minimal models, which are the simplest nontrivial
families of two-dimensional conformal field theories.

###  3.1 Liouville theory

####  3.1.1 Definition and spectrum

We assume

  1. 1.

that Liouville theory is a family of conformal field theories, parametrized by
the central charge $c\in\mathbb{C}$;

  2. 2.

that each theory has a continuous spectrum, where representations of the
symmetry algebra $\mathfrak{V}\times\overline{\mathfrak{V}}$ have
multiplicities zero or one;

  3. 3.

that Liouville theory with $c>1$ is unitary;

  4. 4.

and that correlation functions are smooth as functions of $c$ and of the
conformal dimensions of the fields, in a sense which we will make more
precise.

Since the spectrum is continuous, it must involve Verma modules – the other
type of highest-weight representations, degenerate representations, form a
discrete set. From Section 2.1.3, we know that unitary Verma modules exist
only if $c\geq 1$. So we can assume unitarity only if $c\geq 1$, and to avoid
subtleties due to the existence of degenerate representations with positive
dimensions for $c=1$, we actually assume unitarity only if $c>1$.

When deriving the fusion rules (2.2.54) involving degenerate fields, we have
seen that the natural parameters are not the conformal dimension $\Delta$ and
the central charge $c$ – rather, the momentum $\alpha$ (2.1.30) and the
Liouville coupling constant $b$ (2.1.24). We assume that correlation functions
are analytic as functions of $\alpha$ and $b$, with possibly poles, but no
branch cuts. We will however see that analyticity in $b$ breaks down on the
$c\leq 1$ line.

Let us assume $c>1$ for a moment, and determine the spectrum from the
unitarity and analyticity assumptions. The set of values of $\alpha$ which
obey the unitarity condition $\Delta(\alpha)>0$ is the union of the segment
$\alpha\in]0,Q[$ and the line $\alpha\in\frac{Q}{2}+i{\mathbb{R}}$:

(3.1.1)

We interpret our analyticity assumptions as implying that the set of values of
$\alpha$ which appear in the spectrum has no endpoints. We must then leave the
segment $]0,Q[$ out, and build the spectrum from the line

$\displaystyle\alpha\in\frac{Q}{2}+i{\mathbb{R}}\quad\Leftrightarrow\quad\Delta\in\left[\frac{c-1}{24},\infty\right[\
.$ (3.1.2)

The representation
$\mathcal{V}_{\Delta}\otimes\overline{\mathcal{V}}_{\bar{\Delta}}$ can appear
in the spectrum only if both $\Delta$ and $\bar{\Delta}$ obey this condition.
Moreover, single-valuedness of correlation functions requires
$\Delta-\bar{\Delta}\in\frac{1}{2}{\mathbb{Z}}$, see eq. (2.3.18). By
continuity, $\Delta-\bar{\Delta}$ must be a constant, and this constant must
be zero for both $\Delta$ and $\bar{\Delta}$ to span the whole half-line
$\left[\frac{c-1}{24},\infty\right[$. Therefore, the spectrum must be
diagonal,

$\displaystyle\boxed{S=\frac{1}{2}\int_{\frac{Q}{2}+i{\mathbb{R}}}d\alpha\
\mathcal{V}_{\alpha}\otimes\overline{\mathcal{V}}_{\alpha}=\int_{\frac{c-1}{24}}^{\infty}d\Delta\
\mathcal{V}_{\Delta}\otimes\overline{\mathcal{V}}_{\Delta}}\ ,$ (3.1.3)

where the $\frac{1}{2}$ factor in front of the integral over $\alpha$ is meant
to eliminate the redundancy which comes from the reflection relation
$\mathcal{V}_{\alpha}=\mathcal{V}_{Q-\alpha}$.

We assume that the spectrum (3.1.3) that we obtained for $c>1$ also holds for
any $c\in\mathbb{C}$ by analyticity. (If $c\notin\mathbb{R}$ this gives us
$\Delta\in\frac{c-1}{24}+\mathbb{R}_{+}$ as the correct interpretation of
$\Delta\in\left[\frac{c-1}{24},\infty\right[$.) A subtlety occurs if the line
$\alpha\in\frac{Q}{2}+i\mathbb{R}$ contains degenerate values $\alpha_{\langle
r,s\rangle}$, which happens if and only if $c\leq 1$ i.e. $Q\in i\mathbb{R}$.
These degenerate values would lead to singularities in the decomposition
(2.3.34) of four-point functions into conformal blocks, as the conformal block
$\mathcal{F}_{\Delta_{s}}^{(s)}(s_{i}|z_{i})$ has poles at
$\Delta_{s}=\Delta_{\langle r,s\rangle}$. The solution is simply to replace
the line $\alpha\in\frac{Q}{2}+i\mathbb{R}=i\mathbb{R}$ with the slightly
shifted line

$\displaystyle(c\leq 1)\qquad\alpha\in i\mathbb{R}+\epsilon\ ,$ (3.1.4)

and the four-point function will turn out to be independent of the value of
the regularizing parameter $\epsilon\in\mathbb{R}^{*}$. This subtlety with the
spectrum is a first hint that something special happens if $c\leq 1$ – we will
find subtleties with the three-point structure constants too.

So the spectrum of Liouville theory does not contain any degenerate
representation. Nevertheless, we make the crucial assumption

  1. 5.

that degenerate fields exist, and in particular obey the axioms of
associativity of OPEs, and single-valuedness of correlation functions.

This assumption will allow us to consider four-point functions that obey BPZ
differential equations, and lead to simple crossing symmetry equations.

####  3.1.2 The reflection relation and its consequences

Since we use the momentum $\alpha$ instead of the conformal dimension $\Delta$
for parametrizing representations, we a priori have two notations
$V_{\alpha}(z,\bar{z})$ and $V_{Q-\alpha}(z,\bar{z})$ for the primary field
which corresponds to the representation
$\mathcal{V}_{\alpha}\otimes\overline{\mathcal{V}}_{\alpha}$. We must
therefore have a reflection relation

$\displaystyle\boxed{V_{\alpha}(z,\bar{z})=R(\alpha)V_{Q-\alpha}(z,\bar{z})}\
,$ (3.1.5)

where $R(\alpha)$ is the reflection coefficient, which must obey

$\displaystyle R(\alpha)R(Q-\alpha)=1\ ,$ (3.1.6)

so that in particular $R(\frac{Q}{2})\in\\{1,-1\\}$. By the reflection
relation, commutativity (1.2.5), and conformal symmetry (2.3.23), a two-point
function must be of the type

$\displaystyle\Big{\langle}V_{\alpha_{1}}(z_{1},\bar{z}_{1})V_{\alpha_{2}}(z_{2},\bar{z}_{2})\Big{\rangle}=c(\alpha_{1})\Big{\\{}\delta(Q-\alpha_{1}-\alpha_{2})+R(\alpha_{1})\delta(\alpha_{1}-\alpha_{2})\Big{\\}}|z_{12}|^{-4\Delta(\alpha_{1})}\
,$ (3.1.7)

where the coefficient $c(\alpha)$ must obey $c(\alpha)=c(Q-\alpha)$, and is
otherwise undetermined. Since both $c(\alpha)$ and $R(\alpha)$ can be absorbed
in a field renormalization,

$\displaystyle V_{\alpha}(z,\bar{z})\to\lambda(\alpha)V_{\alpha}(z,\bar{z})\
,$ (3.1.8)

we are free to assume

$\displaystyle c(\alpha)=1\ .$ (3.1.9)

We however keep $R(\alpha)$ undetermined, as this will later be necessary for
ensuring the analyticity of correlation functions. This leaves us with the
freedom to perform a field renormalization, provided the factor
$\lambda(\alpha)$ obeys

$\displaystyle\lambda(\alpha)\lambda(Q-\alpha)=1\ .$ (3.1.10)

Omitting the dependence on $z_{i}$, we schematically rewrite the two-point
function as

$\displaystyle\boxed{\left\langle
V_{\alpha_{1}}V_{\alpha_{2}}\right\rangle=\delta(Q-\alpha_{1}-\alpha_{2})+R(\alpha_{1})\delta(\alpha_{1}-\alpha_{2})}\
.$ (3.1.11)

According to eq. (2.3.26), the three-point function takes the form

$\displaystyle\left\langle\prod_{i=1}^{3}V_{\alpha_{i}}(z_{i},\bar{z}_{i})\right\rangle=C_{\alpha_{1},\alpha_{2},\alpha_{3}}\
|z_{12}|^{2(\Delta_{3}-\Delta_{1}-\Delta_{2})}|z_{23}|^{2(\Delta_{1}-\Delta_{2}-\Delta_{3})}|z_{31}|^{2(\Delta_{3}-\Delta_{2}-\Delta_{1})}\
,$ (3.1.12)

where $\Delta_{i}=\Delta(\alpha_{i})$ is given in eq. (2.1.30). We
schematically rewrite this formula as

$\displaystyle\boxed{\left\langle
V_{\alpha_{1}}V_{\alpha_{2}}V_{\alpha_{3}}\right\rangle=C_{\alpha_{1},\alpha_{2},\alpha_{3}}}\
.$ (3.1.13)

The three-point structure constant $C_{\alpha_{1},\alpha_{2},\alpha_{3}}$ is
constrained by the reflection relation, for instance

$\displaystyle
C_{\alpha_{1},\alpha_{2},\alpha_{3}}=R(\alpha_{1})C_{Q-\alpha_{1},\alpha_{2},\alpha_{3}}\
.$ (3.1.14)