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\begin{document}

\centerline{\bf Lecture~7}
\centerline{\bf Operator product expansions, crossing symmetry and conformal blocks}
\bigskip

Return for a while to a general field theory. Let $\{\cO_I\}$ be a complete system of mutually local linearly independent operators. The completeness will be understood in the sense of operator product expansions (OPE) in the Euclidean theory:%
%
\footnote{In the Minkowski space we shall consider their analytic continuations.}
%
\eq$$
\cO_I(x')\cO_J(x)=\sum_KD^K_{IJ}(x',x)\cO_K(x).
\label{OPE-gen}
$$
We will assume the series
$$
\sum_KD^K_{IJ}(x',x)\langle\cO_K(x)\cO_{M_1}(x_1)\ldots\cO_{M_N}(x_N)\rangle
$$
converge for small enough $|x'-x|$. The functions $D^K_{IJ}(x',x)$ are called \emph{structure functions}. Consider the product of three operators $\cO_{I_3}(x_3)\cO_{I_2}(x_2)\cO_{I_1}(x_1)$. We may do it in several ways. We will consider two of them. We may first fuse $\cO_{I_2}$ with $\cO_{I_1}$ and then the result with $\cO_{I_3}$. Alternatively, we may first fuse $\cO_{I_3}$ with $\cO_{I_2}$ and then with $\cO_{I_1}$. Let us decompose about the point~$x_2$. In the first manner we have
\Multline*$$
\overbrace{\cO_{I_3}(x_3)\overbrace{\cO_{I_2}(x_2)\cO_{I_1}(x_1)}}
=(-)^{(I_1,I_2)}\overbrace{\cO_{I_3}(x_3)\overbrace{\cO_{I_1}(x_1)\cO_{I_2}(x_2)}}
\\
=(-)^{(I_1,I_2)}\sum_JD^J_{I_1I_2}(x_1,x_2)\cO_{I_3}(x_3)\cO_J(x_2)
=(-)^{(I_1,I_2)}\sum_{J,I_4}D^{I_4}_{I_3J}(x_3,x_2)D^J_{I_1I_2}(x_1,x_2)\cO_{I_4}(x_2).
$$
Here $(-)^{(I,J)}=(-1)^{4s_Is_J}$ ($s_I$ being the spin of~$\cO_I$) is $-1$, if both operators are fermionic and $1$ otherwise. Alternatively, we have
\Multline*$$
\overbrace{\overbrace{\cO_{I_3}(x_3)\cO_{I_2}(x_2)}\cO_{I_1}(x_1)}
=(-)^{(I_1,I_2)+(I_1,I_3)}\overbrace{\cO_{I_1}(x_1)\overbrace{\cO_{I_3}(x_3)\cO_{I_2}(x_2)}}
\\
=(-)^{(I_1,I_2)+(I_1,I_3)}\sum_JD^J_{I_3I_2}(x_3,x_2)\cO_{I_1}(x_1)\cO_J(x_2)
=(-)^{(I_1,I_2)+(I_1,I_3)}\sum_{J,I_4}D^{I_4}_{I_1J}(x_1,x_2)D^J_{I_3I_2}(x_3,x_2)\cO_{I_4}(x_2).
$$
Comparison of these two expansions provides the \emph{crossing symmetry} equation
\eq$$
\sum_JD^{I_4}_{I_3J}(x_3,x_2)D^J_{I_1I_2}(x_1,x_2)
=(-)^{(I_1,I_3)}\sum_JD^{I_4}_{I_1J}(x_1,x_2)D^J_{I_3I_2}(x_3,x_2),
\label{crossing-gen}
$$
which can be illustrated graphically as
$$
\sum_J\vcenter{\hbox{
\begin{tikzpicture} [>={Stealth[scale=1.2]}]
\draw[->] (-1,1) node[left] {$I_3$} -- (-.7,.4); \draw (-.7,.4) -- (-.5,0);
\draw[->] (-.5,0) -- (-.8,-.6); \draw (-.8,-.6) -- (-1,-1) node[left] {$I_4$};
\draw[->] (.5,0) -- (-.1,0); \draw (-.1,0) -- (-.5,0);
\draw[->] (1,1) node[right] {$I_2$} -- (.7,.4); \draw (.7,.4) -- (.5,0);
\draw[->] (1,-1) node[right] {$I_1$} -- (.7,-.4); \draw (.7,-.4) -- (.5,0);
\node[above=3pt] at (0,0) {$J$};
\end{tikzpicture}
}}
\quad=\quad
(-)^{(I_1,I_3)}\sum_J\vcenter{\hbox{
\begin{tikzpicture}[>={Stealth[scale=1.2]}]
\draw[->] (-1,1) node[left] {$I_3$} -- (-.4,.7); \draw (-.4,.7) -- (0,.5);
\draw[->] (0,-.5) -- (-.6,-.8); \draw (-.6,-.8) -- (-1,-1) node[left] {$I_4$};
\draw[->] (0,.5) -- (0,-.1); \draw (0,-.1) -- (0,-.5);
\draw[->] (1,1) node[right] {$I_2$} -- (.4,.7); \draw (.4,.7) -- (0,.5);
\draw[->] (1,-1) node[right] {$I_1$} -- (.4,-.7); \draw (.4,-.7) -- (0,-.5);
\node[left=2pt] at (0,0) {$J$};
\end{tikzpicture}
}}
$$
The correlation function are expressed in terms of operator product expansions and vacuum expectation values $\langle\cO_I(x)\rangle$. For example, the two-, three- and four-point correlation functions read
\Align$$
\langle\cO_{I_2}(x_2)\cO_{I_1}(x_1)\rangle
&=\sum_JD^J_{I_1I_2}(x_2,x_1)\langle\cO_J(x_1)\rangle,
\label{2p-gen}
\\
\langle\cO_{I_3}(x_3)\cO_{I_2}(x_2)\cO_{I_1}(x_1)\rangle
&=\sum_{JK}D^K_{I_3J}(x_3,x_1)D^J_{I_2I_1}(x_2,x_1)\langle\cO_K(x_1)\rangle,
\label{3p-gen}
\\
\langle\cO_{I_4}(x_4)\cO_{I_3}(x_3)\cO_{I_2}(x_2)\cO_{I_1}(x_1)\rangle
&=\sum_{JKL}D^L_{I_4K}(x_4,x_1)D^K_{I_3J}(x_3,x_1)D^J_{I_2I_1}(x_2,x_1)\langle\cO_L(x_1)\rangle.
\label{4p-gen}
$$

Now consider a conformal field theory with a set of local operators $\{\cO_I\}$ of fixed conformal dimensions $(\Delta_I,\bDelta_I)$. Stress that the operators $\cO_I$ are not necessarily primary. They are simply eigenvectors of the operators $L_0,\bL_0$. From the translation, rotation and dilatation invariance we immediately conclude that
\eq$$
D^K_{IJ}(x',x)=C^K_{IJ}(z'-z)^{\Delta_K-\Delta_I-\Delta_J}(\bz'-\bz)^{\bDelta_K-\bDelta_I-\bDelta_J}.
\label{DC-rel}
$$
Another simplification of CFT is that (see Problem~2 to the last lecture)
\eq$$
\langle\cO_I(x)\rangle=0,
\quad\text{if $\Delta_I\ne0$ or $\bDelta_I\ne0$,}
\label{VEV-CFT}
$$
for any field $\cO_I$, and
\eq$$
\langle\cO_I(x')\cO_J(x)\rangle=0,
\quad\text{if $\Delta_I\ne\Delta_J$ or $\bDelta_I\ne\bDelta_J$,}
\label{2p-select-CFT}
$$
and if the fields $\cO_I(x),\cO_J(x)$ are Möbius invariant.

Consider for simplicity a model of CFT that possesses the only field of zero dimension $\cO_0=1$. In this case the formulas (\ref{2p-gen})--(\ref{4p-gen}) simplify:
\Align$$
\langle\cO_{I_2}(x_2)\cO_{I_1}(x_1)\rangle
&=C^0_{I_2I_2}(z'-z)^{-\Delta_{I_1}-\Delta_{I_2}}(\bz'-\bz)^{-\bDelta_{I_1}-\bDelta_{I_2}},
\label{2p-CFT}
\\
\langle\cO_{I_3}(x_3)\cO_{I_2}(x_2)\cO_{I_1}(x_1)\rangle
&=\sum_JC^0_{I_3J}C^J_{I_2I_1}
(z_3-z_1)^{-\Delta_{I_3}-\Delta_J}(z_2-z_1)^{\Delta_J-\Delta_{I_2}-\Delta_{I_1}}
\notag
\\
&\quad\times
(\bz_3-\bz_1)^{-\bDelta_{I_3}-\bDelta_J}(\bz_2-\bz_1)^{\bDelta_J-\bDelta_{I_2}-\bDelta_{I_1}},
\label{3p-CFT}
\\
\langle\cO_{I_4}(x_4)\cO_{I_3}(x_3)\cO_{I_2}(x_2)\cO_{I_1}(x_1)\rangle
&=\sum_{JK}C^0_{I_4K}C^K_{I_3J}C^J_{I_2I_1}
\notag
\\
&\quad\times
(z_4-z_1)^{-\Delta_{I_4}-\Delta_K}(z_3-z_1)^{\Delta_K-\Delta_{I_3}-\Delta_J}(z_2-z_1)^{\Delta_J-\Delta_{I_2}-\Delta_{I_1}}
\notag
\\
&\quad\times
(\bz_4-\bz_1)^{-\bDelta_{I_4}-\bDelta_K}(\bz_3-\bz_1)^{\bDelta_K-\bDelta_{I_3}-\bDelta_J}
(\bz_2-\bz_1)^{\bDelta_J-\bDelta_{I_2}-\bDelta_{I_1}}.
\label{4p-CFT}
$$
Now let us recall two important facts. First, the operators in CFT are classified by highest weight representations of the Virasoro algebra:
\eq$$
\cO_I(x)
=(L_{-k_1}\cdots L_{-k_r}\bL_{-l_1}\cdots\bL_{-l_s}\Phi_i)(x),
\qquad
k_1\le\cdots\le k_r,\ l_1\le\cdots\le l_s,
\qquad
I=(i,\vec k,\vec l).
\label{OI-Vir}
$$
Such set is generally linearly dependent, since the Verma modules are, in general, degenerated and must be reduced to irreducible representations by factorization over submodules. Nevertheless, keeping in mind this fact, we continue.

For the sake of simplicity we will write $\Phi_1,\Phi_2,\ldots$ instead of $\Phi_{i_1},\Phi_{i_2},\ldots$ from now on. The same simplified notation will be used for $\Delta_i$, $C^k_{ij}=C^{(k,\varnothing,\varnothing)}_{(i,\varnothing,\varnothing)(j,\varnothing,\varnothing)}$.

Second, the operators $L_{-k},\bL_{-k}$ act on correlation functions as differential operators. Hence, if we only find the correlation functions of primary operators, we are able to compute any correlation function. Thus consider the functions
\eq$$
\langle\Phi_2(x_2)\Phi_1(x_1)\rangle
={g_{21}\over(z_2-z_1)^{2\Delta_1}(\bz_2-\bz_1)^{2\bDelta_1}},
\qquad
g_{12}=\Cases{C^0_{21},&\text{if $\Delta_1=\Delta_2$, $\bDelta_1=\bDelta_2$;}\\
0&\text{otherwise.}}
\label{2p-prim-CFT}
$$
Here $g_{21}$ is a matrix, which plays the role of metrics on the space of primary fields of a given dimension. In principle, by taking appropriate linear transformations we can make it any fixed non-degenerate matrix, e.g.\ the unit matrix. But for some time we retain it for generality. Nevertheless, we must remember that this matrix is \emph{given and fixed}, it is not something we mean to compute.

Now consider three-point functions. Now we have an infinite sum
\Multline*$$
\langle\Phi_{i_3}(x_3)\Phi_{i_2}(x_2)\Phi_{i_1}(x_1)\rangle
=\sum_{j,\vec k,\vec l}C^0_{i_3(j,\vec k,\vec l)}C^{(j,\vec k,\vec l)}_{i_2i_1}
(z_3-z_1)^{-2\Delta_{i_3}}(z_2-z_1)^{\Delta_{i_3}-\Delta_{i_1}-\Delta_{i_2}}
\left(z_2-z_1\over z_3-z_1\right)^{\lVert\vec k\rVert}
\\
\times(\bz_3-\bz_1)^{-2\bDelta_{i_3}}(\bz_2-\bz_1)^{\bDelta_{i_3}-\bDelta_{i_1}-\bDelta_{i_2}}
\left(\bz_2-\bz_1\over\bz_3-\bz_1\right)^{\lVert\vec l\rVert}.
$$
Here $\lVert\vec k\rVert=\sum_ik_i$. Now consider the limit $x_3\to\infty$ or, more precisely, $|z_3-z_1|\gg|z_2-z_1|$. In this limit the leading contribution comes from the primary intermediate operator: $\lVert\vec k\rVert=\lVert\vec l\rVert=0$. Hence,
\Multline*$$
\langle\Phi_3(x_3)\Phi_2(x_2)\Phi_1(x_1)\rangle
=\sum_jg_{i_3j}C^j_{i_2i_1}(z_3-z_1)^{-2\Delta_3}(z_2-z_1)^{\Delta_3-\Delta_1-\Delta_2}
\\
\times
(\bz_3-\bz_1)^{-2\bDelta_3}(\bz_2-\bz_1)^{\bDelta_3-\bDelta_1-\bDelta_2}
\quad
\text{as $|z_3-z_1|\gg|z_2-z_1|$.}
$$
By comparing with the formula from Problem~3 of the last lecture, we obtain
\Align$$
\langle\Phi_3(x_3)\Phi_2(x_2)\Phi_1(x_1)\rangle
&=C_{321}
(z_3-z_2)^{\Delta_1-\Delta_2-\Delta_3}(z_3-z_1)^{\Delta_2-\Delta_1-\Delta_3}(z_2-z_1)^{\Delta_3-\Delta_1-\Delta_2}
\notag
\\
&\quad\times
(\bz_3-\bz_2)^{\bDelta_1-\bDelta_2-\bDelta_3}(\bz_3-\bz_1)^{\bDelta_2-\bDelta_1-\bDelta_3}
(\bz_2-\bz_1)^{\bDelta_3-\bDelta_1-\bDelta_2},
\label{3p-prim-CFT}
$$
where
\eq$$
C_{321}=\sum_jg_{3j}C^j_{21}.
\label{Csub-def}
$$
We see that the knowledge of all structure constants related to primary operators in a conformal field theory is equivalent to the knowledge of all three-point correlation functions. This provides an evident symmetry property
\eq$$
C_{321}=(-1)^{(2,3)}C_{231}=(-1)^{(1,2)}C_{312}.
\label{C-symm}
$$

Now turn to the most complicated case: the four-point function. Without loss of generality we will consider the limit $x_4\to\infty$ and fix $z_3=1$, $z_1=0$. Indeed, consider the Möbius transformation
\eq$$
w=f(z)\equiv{z_3-z_4\over z_3-z_1}{z-z_1\over z-z_4}.
\label{01inf-trans}
$$
This transformation maps points $z=z_1,z_2,z_4$ to $w=0,1,\infty$. If we define
\eq$$
\langle\Phi_i(\infty)X\rangle=\langle\Phi_i|X|0\rangle=\lim_{z,\bz\to\infty}z^{2\Delta_i}\bz^{2\bDelta_i}\langle\Phi_i(z,\bz)X\rangle,
\label{Phi-inf-def}
$$
we obtain
\eq$$
\Aligned{
\langle\Phi_4(x_4)\Phi_3(x_3)\Phi_2(x_2)\Phi_1(x_1)\rangle
&=K(\vec z,\vec\Delta)K(\vec\bz,\vec\bDelta)
\langle\Phi_4(\infty)\Phi_3(1)\Phi_2(w,\bw)\Phi_1(0)\rangle,
\\
K(\vec z,\vec\Delta)
&={(z_1-z_3)^{\sum\Delta_i}(z_4-z_2)^{2\Delta_2}\over(z_4-z_1)^{\sum\Delta_i-2\Delta_1}(z_4-z_3)^{\sum\Delta_i-2\Delta_3}}.
}\label{4p-inf1w0}
$$
For brevity we wrote here $\Phi(0)$ instead of $\Phi(0,0)$ etc.

From (\ref{4p-CFT}) we obtain
$$
\langle\Phi_4(\infty)\Phi_3(1)\Phi_2(z,\bz)\Phi_1(0)\rangle
=\sum_{J\equiv(j,\vec k,\vec l)}C_{43J}C^J_{21}z^{\Delta_j-\Delta_1-\Delta_2+\lVert\vec k\rVert}\bz^{\bDelta_j-\bDelta_1-\bDelta_2+\lVert\vec l\rVert}.
$$
Now I want to show you that the structure constants $C_{43J}$ and $C^J_{21}$ factorize as follows
\eq$$
\Aligned{
C_{43J}
&=C_{43j}\gamma_{43j}^{\vec k}\bgamma_{43j}^{\vec l},
\\
C^J_{21}
&=C^j_{21}\beta_{j21}^{\vec k}\bbeta_{j21}^{\vec l},
}
$$
so that the constants $\gamma,\bgamma,\beta,\bbeta$ are uniquely defined by the conformal dimensions of the corresponding primary fields. For the constants $C_{43J}$ it is rather easy. Indeed,
\Multline*$$
C_{43J}=\langle\Phi_4|\Phi_3(z,\bz)L_{-k_1}\cdots L_{-k_r}\bL_{-l_1}\cdots\bL_{-k_s}|\Phi_j\rangle|_{z=\bz=1}
\\
=\left.(-1)^{r+s}C_{43j}
(\cL_{-k_1}\cdots\cL_{-k_r}z^{\Delta_4-\Delta_3-\Delta_j})
(\bar\cL_{-l_1}\cdots\bar\cL_{-l_s}\bz^{\bDelta_4-\bDelta_3-\bDelta_j})\right|_{z=\bz=1},
$$
where
$$
\cL_k=(k+1)z^k\Delta_3+z^{k+1}\d,
\qquad
\bar\cL_k=(k+1)z^k\bDelta_3+\bz^{k+1}\d
$$
Evidently,
$$
\Aligned{
\gamma_{43j}^{\vec k}
&=(-1)^r\left.\cL_{-k_1}\cdots\cL_{-k_r}z^{\Delta_4-\Delta_3-\Delta_j}\right|_{z=1},
\\
\bgamma_{43j}^{\vec k}
&=(-1)^s\left.\bar\cL_{-k_1}\cdots\bar\cL_{-k_r}\bz^{\bDelta_4-\bDelta_3-\bDelta_j}\right|_{\bz=1}.
}
$$

The situation with $C^J_{21}$ is more complicated. Consider the product $\Phi_2(z,\bz)|\Phi_1\rangle$ and act on it by $L_k$ with $k>0$:
$$
L_k\Phi_2(z,\bz)|\Phi_1\rangle
=((k+1)z^k\Delta_2+z^{k+1}\d)\Phi_2(z,\bz)|\Phi_1\rangle.
$$
Now apply the operator product expansion to both sides:
\Multline*$$
\sum_JC^J_{21}z^{\Delta_j+\lVert\vec k\rVert-\Delta_1-\Delta_2}\bz^{\bDelta_j+\lVert\vec l\rVert-\bDelta_1-\bDelta_2}
L_kL_{-k_1}\cdots L_{-k_r}\bL_{-l_1}\cdots\bL_{-l_s}|\Phi_j\rangle
\\
=\sum_JC^J_{21}(k\Delta_2-\Delta_1+\Delta_j+\lVert\vec k\rVert)z^{\Delta_j+\lVert\vec k\rVert+k-\Delta_1-\Delta_2}
\bz^{\bDelta_j+\lVert\vec l\rVert-\bDelta_1-\bDelta_2}
L_{-k_1}\cdots L_{-k_r}\bL_{-l_1}\cdots\bL_{-l_s}|\Phi_j\rangle.
$$
We see that these equations do not involve the $\bz$ variable and cannot fix the overall normalization. Let
\eq$$
\Aligned{
|\beta_{j21},K\rangle
&=\sum_{\vec k,\ \lVert\vec k\rVert=K}\beta_{j21}^{\vec k}L_{-k_1}\cdots L_{-k_r}|\Delta_j\rangle,
\\
|\bbeta_{j21},K\rangle
&=\sum_{\vec k,\ \lVert\vec k\rVert=K}\bbeta_{j21}^{\vec k}\bL_{-k_1}\cdots\bL_{-k_r}|\bDelta_j\rangle
}\label{beta-vec-def}
$$
Then we have
\eq$$
\Aligned{
L_k|\beta_{j21},K\rangle
&=(k\Delta_2-\Delta_1+\Delta_j+K-k)|\beta_{j21},K-k\rangle,
\\
\bL_k|\bbeta_{j21},K\rangle
&=(k\bDelta_2-\bDelta_1+\bDelta_j+K-k)|\bbeta_{j21},K-k\rangle.
}\label{beta-eqs}
$$
In fact, it is sufficient to only impose these equations for $k=1,2$. All other equations follow from these two due to the commutation relations. The number of equations on each level is greater than the number of variables so that the system is overdetermined. It can be shown that for generic values of $\Delta_j$ the coefficients are uniquely determined by the system. For degenerate values $\Delta_j=\Delta_{mn}$ for $K\ge mn$ the solution exists only if the fusion rule holds, and the vectors $|\beta_{j21},K\rangle$ are defined modulo the corresponding submodule.

Define the function
\eq$$
F\left(\left.\Matrix{\Delta_1&\Delta_4\\\Delta_2&\Delta_3}\quad\Delta_j\right|z\right)
=\sum_{\vec k}\gamma^{\vec k}_{43j}\beta^{\vec k}_{j21}z^{\Delta_j+\lVert\vec k\rVert-\Delta_1-\Delta_2},
\label{confblock-def}
$$
which is called the (4-point) \emph{conformal block}. I wand to stress that the conformal block is uniquely defined in the representation theory of the Virasoro algebra, and does not involve any details of the field theory. It means that it is a well-defined mathematical object. In fact, there exist several approaches for calculating it, such as free field representation (for some special values of dimensions), Al.~Zamolodchikov's recurrence relations and, the most recent, the AGT construction.

The four-point correlation function is expressed in terms of the conformal blocks as follows:
\eq$$
\langle\Phi_4(\infty)\Phi_3(1)\Phi_2(z,\bz)\Phi_1(0)\rangle
=\sum_jC_{43j}C^j_{21}
F\left(\left.\Matrix{\Delta_1&\Delta_4\\\Delta_2&\Delta_3}\quad\Delta_j\right|z\right)
F\left(\left.\Matrix{\bDelta_1&\bDelta_4\\\bDelta_2&\bDelta_3}\quad\bDelta_j\right|\bz\right).
\label{4p-prim-CFT}
$$
Analogously,
\eq$$
\langle\Phi_4(\infty)\Phi_1(0)\Phi_2(z,\bz)\Phi_3(1)\rangle
=\sum_jC_{41j}C^j_{23}
F\left(\left.\Matrix{\Delta_1&\Delta_4\\\Delta_2&\Delta_3}\quad\Delta_j\right|1-z\right)
F\left(\left.\Matrix{\bDelta_1&\bDelta_4\\\bDelta_2&\bDelta_3}\quad\bDelta_j\right|1-\bz\right).
\label{4p-prim-CFT-cross}
$$
Comparing them, we obtain the crossing symmetry equation~\cite{Belavin:1984vu}%
%
\footnote{In the derivation here we used slightly different expansion: we expanded the four-point functions about $x_1=(0,0)$ in the l.h.s.\ and about $x_3=(1,1)$  rather than about $x_2=(z,\bz)$ in both side, and used the translation invariance. We have done it to get rid of numerous extra minus signs in the intermediate formulas. You can easily check that it is completely equivalent to~(\ref{crossing-gen}) in the conformal case.}
%
\Multline$$
\sum_jC_{43j}C^j_{21}
F\left(\left.\Matrix{\Delta_1&\Delta_4\\\Delta_2&\Delta_3}\quad\Delta_j\right|z\right)
F\left(\left.\Matrix{\bDelta_1&\bDelta_4\\\bDelta_2&\bDelta_3}\quad\bDelta_j\right|\bz\right)
\\*
=\sum_jC_{{\red14}j}C^j_{23}
F\left(\left.\Matrix{\Delta_3&\Delta_4\\\Delta_2&\Delta_1}\quad\Delta_j\right|1-z\right)
F\left(\left.\Matrix{\bDelta_3&\bDelta_4\\\bDelta_2&\bDelta_1}\quad\bDelta_j\right|1-\bz\right).
\label{crossing}
$$
(We used the fact that $\sum s_i=0$ and (\ref{C-symm}).) This is an equation for the structure constants $C^i_{jk}$. Generally, it has multiple solutions, but in the case of minimal and rational minimal models with spinless fields the solution turns out to be unique. The structure constants in this case were calculated by Vl.~Dotsenko and V.~Fateev~\cite{Dotsenko:1984nm,Dotsenko:1984ad,Dotsenko:1985hi}. There solution was based on the fact that the correlation functions of minimal conformal models can be represented in terms of the free field theory:
\Multline$$
\left\langle\prod^N_{i=1}\Phi_{m_in_i}(x_i)\right\rangle
=\const\times
\\
\times\left\langle\lcolon\e^{-(b+b^{-1})\varphi(\infty)}\rcolon\lcolon\e^{\alpha_{-m_N,-n_N}\varphi(x_N)}\rcolon
\prod^{N-1}_{i=1}\lcolon\e^{\alpha_{m_in_i}\varphi(x_i)}\rcolon
\left(\int d^2u\,\lcolon\e^{b^{-1}\varphi(u)}\rcolon\right)^r
\left(\int d^2v\,\lcolon\e^{b\varphi(v)}\rcolon\right)^s\right\rangle,
\label{ff-rep}
$$
where
$$
r={1\over2}\left(\sum^{N-1}_{i=1}m_i-m_N\right),
\qquad
s={1\over2}\left(\sum^{N-1}_{i=1}n_i-n_N\right).
$$
The surface integrals can be calculated by reducing to products of pairs of contour integrals in complex variables, and this decomposition turns out to be just decomposition of the form~(\ref{4p-prim-CFT}). It is very interesting subject, but I have to omit it due to the lack of time.

Nevertheless, I want to explain one important point. Let $\Delta_2=\Delta_{mn}$, while other dimension will be left arbitrary. Then the correlation function (\ref{4p-prim-CFT}) satisfies a linear differential equation of the order~$mn$. Since this equation is analytic in $z$, the conformal block $F\left(\left.\Matrix{\Delta_1&\Delta_4\\\Delta_{mn}&\Delta_3}\quad\Delta_j\right|z\right)$ satisfies the same equation. This equation possesses just $mn$ independent solution. On the other hand according to the fusion rule we know that
$$
\alpha_j=\alpha_1+b^{-1}k+bl,
\qquad k=-m,-m+2,\ldots,m,
\qquad l=-n,-n+2,\ldots,n.
$$
We have $mn$ independent conformal blocks with different monodromy properties. Indeed, if we take $z$ and move them around $0$ we get
\eq$$
F\left(\left.\Matrix{\Delta_1&\Delta_4\\\Delta_{mn}&\Delta_3}\quad\Delta_j\right|\e^{2\pi\i}z\right)
=\e^{2\pi\i(\Delta_j-\Delta_{mn}-\Delta_1)}
F\left(\left.\Matrix{\Delta_1&\Delta_4\\\Delta_{mn}&\Delta_3}\quad\Delta_j\right|z\right).
\label{F0-monodromy}
$$
These conformal blocks form a basis in the space of solution to the differential equation. The expression (\ref{4p-prim-CFT}) guarantees us that the correlation functions is invariant under this monodromy, if $\bz=z^*$. But to be sure that it is unique-valued we have to guarantee that the monodromy invariant if we mode the variable $z$ around~$1$. This is not clear, because the monodromy properties of conformal blocks may be given by a non-diagonal matrix about $z=1$. And at this point the crossing symmetry plays role. Indeed, the r.h.s.\ of the crossing equation (\ref{crossing}) is invariant under such monodromy. The conformal blocks $F\left(\left.\Matrix{\Delta_3&\Delta_4\\\Delta_{mn}&\Delta_1}\quad\Delta_j\right|1-z\right)$ form another basis in the space of solution to the differential equation with diagonal monodromy about $z=1$. The two basis are related by a linear transformation:
\eq$$
F\left(\left.\Matrix{\Delta_1&\Delta_4\\\Delta_{mn}&\Delta_3}\quad\Delta_j\right|z\right)
=X\left(\left.\Matrix{\Delta_1&\Delta_4\\\Delta_{mn}&\Delta_3}\right|\Matrix{\Delta_{j'}\\\Delta_j}\right)
F\left(\left.\Matrix{\Delta_3&\Delta_4\\\Delta_{mn}&\Delta_1}\quad\Delta_{j'}\right|1-z\right).
\label{FF-trans}
$$
The structure constants then satisfy a system of bilinear algebraic equations
\eq$$
\sum_jC_{43j}C^j_{21}
X\left(\left.\Matrix{\Delta_1&\Delta_4\\\Delta_{mn}&\Delta_3}\right|\Matrix{\Delta_{j'}\\\Delta_j}\right)
X\left(\left.\Matrix{\Delta_1&\Delta_4\\\Delta_{mn}&\Delta_3}\right|\Matrix{\Delta_{j''}\\\Delta_j}\right)
=C_{14j'}C^{j'}_{23}\delta_{j'j''}.
\label{CC-eq}
$$
Together with the normalization, e.g.\ $C^0_{ij}=g_{ij}=\delta_{ij}$ it gives the structure constants, if we know the $X$ constants. Just the $X$ constants can be fixed by means of the free field representation.

It is conjectured that the linear transformation like (\ref{FF-trans}) holds for arbitrary values of $\Delta_2$, where there is no differential equation. In the case of the Liouville theory $c>25$ it was conjectured to be an integral transformation. Nevertheless, no method other than the free field representation makes it possible to compute the $X$ coefficients precisely. But the free field representation only works for special values of $\Delta_i$ since the numbers $r$ and $s$ must be integer. Thus it is an open question.





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\bigbreak
\bigskip
\centerline{\bf Problems}
\medskip


{\bf 1.}~Find the coefficients $\beta^1_{j21}$, $\beta^2_{j21}$ and $\beta^{1,1}_{j21}$ by solving equations (\ref{beta-eqs}) explicitly. Show that for $\Delta_j=\Delta_{11},\Delta_{21},\Delta_{12}$ the corresponding fusion rules follow from the equations.


\end{document}
