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\begin{document}

\centerline{\bf Lecture~6}
\centerline{\bf Conformal field theory: fusion rules, conformal blocks and crossing symmetry}
\bigskip

Let us extract some information from the level 2 differential equation
\Multline$$
\left({\d^2\over\d z^2}+\lambda\sum^N_{i=1}\left({\Delta_i\over(z_i-z)^2}-{1\over z_i-z}{\d\over\d z_i}\right)\right)
\langle\Phi_1(z_1,\bz_1)\ldots\Phi_N(z_N,\bz_N)\Phi(z,\bz)\rangle=0
\\
\text{where $\lambda=b^{\mp2}$ for $\Delta_\Phi=\Delta_{21},\Delta_{12}$.}
\label{level2-corr-diffeq-2}
$$
Consider the limit $z\to z_i$ and keep the leading singular terms only:
\eq$$
\left({\d^2\over\d z^2}+{\lambda\Delta_i\over(z-z_i)^2}-{\lambda\over z-z_i}{\d\over\d z}\right)\Phi(z)\Phi_i(z_i)=0.
\label{PhiPhii-diffeq}
$$
For a while we forget about the left variables $\bz,\bz_i$. Later I will explain how to restore them.

The equation
$$
f''(z)-\lambda z^{-1}f'(z)+\lambda\Delta z^{-2}f(z)=0
$$
is homogeneous in $z$, so that its solution is $f(z)=A_1z^{\gamma_1}+A_2z^{\gamma_2}$ with arbitrary $A_i$ and some particular $\gamma_i$. After substituting $z^\gamma$ we obtain $\gamma^2-(\lambda+1)\gamma+\lambda\Delta=0$ so that
$$
\gamma_{1,2}={\lambda+1\pm\sqrt{\lambda^2-2(2\Delta-1)\lambda+1}\over2}.
$$
If we assume the operator product of the form
\eq$$
\Phi(z')\Phi_i(z_i)
=C^{(1)}(z'-z)^{\Delta^{(1)}_i-\Delta_\Phi-\Delta_1}\Phi^{(1)}_i(z_i)
+C^{(2)}(z'-z)^{\Delta^{(2)}_i-\Delta_\Phi-\Delta_1}\Phi^{(2)}_i(z_i)+\cdots.
$$
We obtain
\eq$$
\Delta^{(1,2)}_i=\Delta_i+\Delta+\gamma_{1,2}.
\label{Delta(1,2)-def}
$$
This equation looks cumbersome, but it becomes trivial in the parametrization $\Delta=\alpha(b+b^{-1}-\alpha)$:
\eq$$
\alpha^{(1,2)}_i=\Cases{\alpha_i\pm\alpha_{21},&\text{if $\Delta_\Phi=\Delta_{21}$;}\\
\alpha_i\pm\alpha_{12},&\text{if $\Delta_\Phi=\Delta_{12}$.}}
\label{alpha(1,2)-def}
$$
Let us parametrize the primary field with the parameter $\alpha$: $\Phi_\alpha=\Phi_{b+b^{-1}-\alpha}=\Phi_\Delta$. Then
\eq$$
\Aligned{{}
[\Phi_\alpha][\Phi_{21}]
&=[\Phi_{\alpha+\alpha_{21}}]+[\Phi_{\alpha-\alpha_{21}}],
\\
[\Phi_\alpha][\Phi_{12}]
&=[\Phi_{\alpha+\alpha_{12}}]+[\Phi_{\alpha-\alpha_{12}}].
}\label{21-12-fusion}
$$
Here square bracket means the \emph{conformal families}, i.e.\ sets of all descendants of the corresponding primary operators. 

Now let us try to answer the question: what happens, when $\alpha=\alpha_{mn}$ in~(\ref{21-12-fusion})? Formally, we obtain
\eq$$
\Aligned{{}
[\Phi_{mn}][\Phi_{21}]
&=[\Phi_{m+1,n}]+[\Phi_{m-1,n}],
\\
[\Phi_{mn}][\Phi_{12}]
&=[\Phi_{m,n+1}]+[\Phi_{m,n-1}].
}\label{21-12-spec-fusion}
$$
But there is a subtlety: consider the product $[\Phi_{12}][\Phi_{21}]$. According to the first line of (\ref{21-12-spec-fusion}) we obtain $[\Phi_{22}]+[\Phi_{02}]$. But if we apply the second rule we get $[\Phi_{22}]+[\Phi_{20}]$. Consistency demands that $[\Phi_{12}][\Phi_{21}]=[\Phi_{22}]$. Indeed, the families $[\Phi_{02}]$ and $[\Phi_{20}]$ are nondegenerate, and it would be strange, if they appeared in fusion rules of degenerate families. We see that the rule (\ref{21-12-spec-fusion}) should be improved.

To do it return to general values of $\alpha$. Consider the product $[\Phi_\alpha][\Phi_{21}][\Phi_{21}]$. On one hand it is
$$
[\Phi_\alpha][\Phi_{21}][\Phi_{21}]=[\Phi_{\alpha+\alpha_{21}}][\Phi_{21}]+[\Phi_{\alpha-\alpha_{21}}][\Phi_{21}]
=[\Phi_{\alpha+2\alpha_{21}}]+[\Phi_\alpha]+[\Phi_{\alpha-2\alpha_{21}}].
$$
One the other hand by fusing two last factors we obtain
$$
[\Phi_\alpha][\Phi_{21}][\Phi_{21}]=[\Phi_\alpha][\Phi_{11}]+[\Phi_\alpha][\Phi_{31}].
$$
It is evident that $[\Phi_\alpha][\Phi_{11}]=[\Phi_\alpha]$. We conclude that
\eq$$
[\Phi_\alpha][\Phi_{31}]=[\Phi_{\alpha+2\alpha_{21}}]+[\Phi_\alpha]+[\Phi_{\alpha-2\alpha_{21}}].
\label{31-fusion}
$$
By continuing in tha same manner we obtain
\eq$$
[\Phi_\alpha][\Phi_{mn}]=\sum^{m-1}_{i=0}\sum^{n-1}_{j=0}[\Phi_{\alpha+(m-1-2i)\alpha_{21}+(m-1-2j)\alpha_{12}}].
\label{mn-fusion}
$$
Now specify $\alpha$ to $\alpha_{m'n'}$. Then the last equation reads
$$
[\Phi_{m'n'}][\Phi_{mn}]=\sum^{m-1}_{i=0}\sum^{n-1}_{j=0}[\Phi_{m'+m-1-2i,n'+n-1-2j}]
=\sum^{m+m'-1}_{\substack{m''=m'-m+1\\\pmod2}}\sum^{n+n'-1}_{\substack{n''=n'-n+1\\\pmod2}}[\Phi_{m''n''}].
$$
Well, but we may interchange the pairs $m,n$ and $m',n'$. It imposes another restriction to the terms that may appear in the r.h.s. It results
\eq$$
[\Phi_{m'n'}][\Phi_{mn}]=\sum^{m+m'-1}_{\substack{m''=|m-m'|+1\\\pmod2}}\sum^{n+n'-1}_{\substack{n''=|n-n'|+1\\\pmod2}}[\Phi_{m''n''}].
\label{m'n'-mn-fusion}
$$
The system of operators $\Phi_{mn}$ with $m,n>0$ is closed with respect to the fusion rule~(\ref{m'n'-mn-fusion}). It makes it possible to define the so called \emph{minimal conformal models}, which contain just one primary field $\Phi_{mn}$ of each conformal dimension $\Delta=\bDelta=\Delta_{mn}$.%
%
\footnote{There are also theories with non-diagonal primary fields ($\Delta\ne\bDelta$), but we will not discuss them here except one example.}
%
Nevertheless, generic minimal conformal models possess fields with negative conformal dimensions. This means that norms of some vectors are negative. Indeed,
$$
\lVert L_{-1}|\Delta\rangle\rVert^2=\langle\Delta|L_1L_{-1}|\Delta\rangle=2\langle\Delta|L_0|\Delta\rangle=2\Delta
$$
is negative for $\Delta<0$. Though nonunitary models have some applications, construction of unitary ones is our priority.

Recall the definition of $\alpha_{mn}$:
\eq$$
\alpha_{mn}={1-m\over2}b^{-1}+{1-n\over2}b.
\label{alphamn-def-2}
$$
All these numbers are different for $m,n>0$, if $b^2$ is positive or negative and irrational. But if
\eq$$
b^2=-{r\over s}
\quad\text{or}\quad
c=1-{6(s-r)^2\over rs},
\label{b2rat}
$$
with $r,s$ coprime positive integers (for definiteness we will assume $r<s$), we have
\eq$$
\alpha_{m+r,n+s}=\alpha_{mn}.
\label{alpha-periodicity}
$$
For conformal dimensions
\eq$$
\Delta_{mn}={(sm-rm)^2-(s-r)^2\over4rs}
\label{Delta-rat}
$$
we have
\eq$$
\Aligned{
\Delta_{mn}
&=\Delta_{m+r,n+s}
\\
\Delta_{mn}
&=\Delta_{r-m,s-n}.
}\label{Delta-rat-props}
$$
Due to the last reflection equation the fusion rules are closed on the set of families
\eq$$
[\Phi_{mn}],\quad1\le m\le r-1,\quad1\le n\le s-1.
\label{Kac-table}
$$
The fusion rule consistent with the reflection property is
\eq$$
[\Phi_{m'n'}][\Phi_{mn}]=\sum^{\min(m+m'-1,r-|m-m'|-1)}_{\substack{m''=\max(|m-m'|+1,m+m'-r+1)\\\pmod2}}
\sum^{\min(n+n'-1,s-|n-n'|-1)}_{\substack{n''=\max(|n-n'|+1,n+n'-s+1)\\\pmod2}}[\Phi_{m''n''}].
\label{m'n'-mn-rat-fusion}
$$
The \emph{rational minimal models} are obtained after identification
\eq$$
\Phi_{r-m,s-n}(z,\bz)=\Phi_{mn}(z,\bz)
\label{Phi-reflection}
$$
and bounding the set of primary operators to the \emph{Kac table}~(\ref{Kac-table}). Note that operators off the Kac table, which can be obtained from operators within the Kac table by any composition of the reflections $m\to-m$, $m\to2r-m$, $n\to-n$, $n\to2s-n$ has the same dimensions as appropriate null vectors in the Kac table:
$$
\Delta_{-m,n}=\Delta_{m,-n}=\Delta_{mn}+mn,
\qquad
\Delta_{2r-m,n}=\Delta_{m,2s-n}=\Delta_{mn}+(r-m)(s-n).
$$
In fact, every Verma module in the Kac table contains an infinite number of null vector. The first two are on the levels $mn$ and $(r-m)(s-n)$. But, as we see, they again generate degenerate Verma modules. These two modules intersect and have two null vectors in their intersection, which possess the same property.

Special role is played by the `self-reflected' operators (e.g.~$\Phi_{0n}$), which appear in the so called \emph{logarithmic} conformal field theory.

\begin{theorem*}
All irreducible representations in the Kac table are unitary, if and only if $s=r+1$.%
%
\footnote{We assumed that $s>r$.}
%
\end{theorem*}

We will not prove the theorem. We only make sure that for $s=r+1$ all dimensions are positive. Indeed, $\Delta_{mn}<0$ if and only if
$$
|sm-rn|<s-r.
$$
If $s-r=1$ the only value of $sm-rn$ that satisfies this condition is zero. But $sm-rn\ne0$ since $s$ and $r$ are mutually simple. For $s-r>1$ this argument fails, since $sm-rn$ can be nonzero.

Now we consider some examples.

The simplest one is $r=2,s=3$, which corresponds to $c=0$. Its Kac table looks like
$$
\begin{array}{|c|}
\hline
0
\\
\hline
0
\\
\hline
\end{array}
=
\begin{array}{|c|}\hline
I\\\hline
I\\\hline
\end{array}
$$
In the left table we wrote conformal dimensions, while in the write one the corresponding operators. We have the only primary field $\Phi_{11}=1\equiv I$, and the corresponding Verma module contains two lowest null vectors $L_{-1}|0\rangle$ and $(L_{-2}-{3\over2}L_{-1}^2)|0\rangle$. Since all $L_{-k}$ are obtained as commutators of $L_{-1}$ and $L_{-2}$, these two null vectors generate all vectors in the module except the vacuum one. Hence, the irreducible representation is one-dimensional and the theory is void: its space of operators consists of the unit operator and its space of states only consists of vacuum.%
%
\footnote{Note that it is true just for the minimal model. It admits some extensions. In particular, its extension by logarithmic operators finds an application to the percolation theory (Cardy, 1991).}
%

Now consider the case $r=3,s=4$, which corresponds to $c=1/2$. It is just the central charge of the free fermion. The Kac table reads
$$
\def\arraystretch{1.4}
\begin{array}{|c|c|}\hline
1\over2&0\\\hline
1\over16&1\over16\\\hline
0&1\over2\\\hline
\end{array}
=
\begin{array}{|c|c|}\hline
\varepsilon&\bf1\\\hline
\sigma&\sigma\\\hline
\bf1&\varepsilon\\\hline
\end{array}
=
\begin{array}{|c|c|}\hline
\varepsilon&I\\\hline
\mu&\mu\\\hline
I&\varepsilon\\\hline
\end{array},
$$
which gives the dimensions of the operators $1,\varepsilon,\sigma$ or $1,\varepsilon,\mu$. From the point of view of bootstrap conformal theory there is no difference between these to sets. They form two same operator algebras. The set of mutually local operators $1,\Psi,\bPsi,\varepsilon$ (and their descendants) corresponds to a non-diagonal pairing of two chiral parts of the same fields.

The next unitary minimal conformal model $r=4,s=5$ and $c={7\over10}$ corresponds to the so called tricritical Ising model:
$$
H=-J\sum_{<i,j>}\sigma_i\sigma_j-\mu\sum_i|\sigma_i|,
\qquad
\sigma_i=0,\pm1.
$$
Its Kac table reads
$$
\def\arraystretch{1.4}
\begin{array}{|c|c|c|}\hline
3\over2&7\over16&0\\\hline
3\over5&3\over80&1\over10\\\hline
1\over10&3\over80&3\over5\\\hline
0&7\over16&3\over2\\\hline
\end{array}
=
\begin{array}{|c|c|c|}\hline
\varepsilon''&\sigma'&I\\\hline
v=\varepsilon'&\sigma&\varepsilon\\\hline
\varepsilon&\sigma&v=\varepsilon'\\\hline
I&\sigma'&\varepsilon''\\\hline
\end{array}
$$
The operator $v=\varepsilon'$ is the vacancy ($\sigma_i=0$) density operator or subleading energy density operator. The operator $\varepsilon'$ is the sub-subleading energy density operator. The operator $\sigma'$ is the subleading magnetization operator.

Our last example is the non-unitary $r=2,s=5$ model. Its central charge is $c=-22/5$. It is the so called Lee\--Yang critical point in the Ising model. Its Kac table is
$$
\def\arraystretch{1.4}
\begin{array}{|c|}\hline
0\\\hline
-{1\over5}\\\hline
-{1\over5}\\\hline
0\\\hline
\end{array}
=
\def\arraystretch{1.4}
\begin{array}{|c|}\hline
I\\\hline
\Phi\\\hline
\Phi\\\hline
I\\\hline
\end{array}
$$
Except the unit operator, it has only one primary operator $\Phi=\Phi_{12}=\Phi_{13}$ of negative dimension.




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


{\bf1.}~Assuming that any Verma module with the weight $\Delta_{mn}$ in the Kac table contains two null vectors with dimensions $Delta_{mn}+mn$ and $\Delta_{mn}+(r-m)(s-m)$, prove that it contains an infinite set of null vectors of dimensions
$$
\Aligned{
\Delta_{m,n+2sk}
&=\Delta_{mn}+(rk-m)(sk+n)+mn
&&\quad(k\in\Z\setminus\{0\}),
\\
\Delta_{m,-n+2sk}
&=\Delta_{mn}+(rk-m)(sk-n)
&&\quad(k\in\Z).
}
$$

{\bf2.}~Prove that for the Möbius invariant field theory for any field $\Phi(x)$ of definite dimensions~$(\Delta,\bDelta)$
$$
\langle\Phi(x)\rangle=0,
\quad\text{if $\Delta\ne0$ or $\bDelta\ne0$,}
$$
and for any two Möbius invariant fields $\Phi_1(x)$ and $\Phi_2(x)$ of dimensions $(\Delta_1,\bDelta_1)$ and $(\Delta_2,\bDelta_2)$ respectively
$$
\langle\Phi_1(x')\Phi_2(x)\rangle=0,
\quad\text{if $\Delta_1\ne\Delta_2$ or $\bDelta_1\ne\bDelta_2$.}
$$
The Möbius invariance of the fields means invariance with respect to the transformation
$$
\Phi(z,\bz)\to(f'(z))^\Delta(\bar f'(\bz))^\bDelta\Phi(f(z),\bf(\bz))
$$
with Möbius functions $f,\bar f$. In particular, any primary field is Möbius invariant.

{\bf3.}~Prove that for three Möbius invariant fields $\Phi_i(x)$ ($i=1,2,3$) the following relation holds
\Multline*$$
\langle\Phi_3(x_3)\Phi_2(x_2)\Phi_1(x_1)\rangle
=C_{123}(z_2-z_1)^{\Delta_3-\Delta_1-\Delta_2}(z_3-z_1)^{\Delta_2-\Delta_1-\Delta_3}(z_3-z_2)^{\Delta_1-\Delta_2-\Delta_3}
\\
\times(\bz_2-\bz_1)^{\bDelta_3-\bDelta_1-\bDelta_2}(\bz_3-\bz_1)^{\bDelta_2-\bDelta_1-\bDelta_3}
(\bz_3-\bz_2)^{\bDelta_1-\bDelta_2-\bDelta_3}.
$$



\end{document}
