$\mathfrak{g}$ has positive roots. In particular, an
$\mathfrak{g}=\mathfrak{sl}_{n}$ isospin variable has $\frac{n(n-1)}{2}$
components, and an $\mathfrak{g}=\mathfrak{sl}_{2}$ isospin variable has one
component. We already know differential operators $D^{j}_{x}(t^{a})$ (2.3.8)
which obey the $\mathfrak{sl}_{2}$ commutation relations (2.3.9) for any
choice of the spin $j$. The eigenvalue of the quadratic Casimir operator in
the corresponding representation of $\mathfrak{sl}_{2}$ is

$\displaystyle C_{2}(j)=K_{ab}D_{x}^{j}(t^{a})D_{x}^{j}(t^{b})=2j(j+1)\ ,$
(4.2.23)

and the conformal dimension of the corresponding field is

$\displaystyle\Delta_{j}=\frac{j(j+1)}{k+2}\ .$ (4.2.24)

Another triplet of $\mathfrak{sl}_{2}$ differential operators is given by

$\displaystyle\left\\{\begin{array}[]{l}D_{\mu}^{j}(t^{-})=-\mu\ ,\\\
D_{\mu}^{j}(t^{0})=-\mu{\frac{\partial}{\partial\mu}}\ ,\\\
D_{\mu}^{j}(t^{+})=\mu\frac{\partial^{2}}{\partial\mu^{2}}-\frac{j(j+1)}{\mu}\
.\end{array}\right.$ (4.2.28)

The $x$-basis field $\Phi^{j}_{x}(z_{0})$ and $\mu$-basis field
$\Phi^{j}_{\mu}(z_{0})$ are related by the formal Fourier transform

$\displaystyle\Phi_{x}^{j}(z_{0})=\int d\mu\ \mu^{-j-1}e^{\mu
x}\Phi_{\mu}^{j}(z_{0})\ .$ (4.2.29)

The $\widehat{\mathfrak{sl}}_{2}$ currents and primary fields can be
represented in terms of free fields, in the same way as the Virasoro algebra
and primary fields were represented in terms of the $\hat{\mathfrak{u}}_{1}$
current in Section 4.1.1. This Wakimoto free-field representation of
$\widehat{\mathfrak{sl}}_{2}$ is described in Exercise 4.6.

Let us now consider the states $|v^{R}\rangle$ which correspond to the
$R$-valued field $\Phi^{R}(z_{0})$ by the state-field correspondence. The
definition (4.2.18) of $\Phi^{R}(z_{0})$ is equivalent to

$\displaystyle\left\\{\begin{array}[]{l}J^{a}_{n>0}|v^{R}\rangle=0\ ,\\\
J^{a}_{0}|v^{R}\rangle=-t^{a}|v^{R}\rangle\ .\end{array}\right.$ (4.2.32)

The states $|v^{R}\rangle$ are killed by the annihilation operators, and are
called affine primary states. They transform in the representation $R$ of the
horizontal subalgebra $\mathfrak{g}\subset\hat{\mathfrak{g}}$ generated by
$\\{J^{a}_{0}\\}$. Acting on the affine primary states with creation operators
$J^{a}_{n<0}$ generates affine descendent states, which form an affine
highest-weight representation $\hat{R}$ of the affine Lie algebra
$\hat{\mathfrak{g}}$.

Let us study the fusion rules of affine highest-weight representations,
equivalently the OPEs of affine primary fields. To write such OPEs, we will
omit the dependences on the positions $z_{i}$ of the fields, which are
dictated by conformal symmetry since affine primary fields are also primary
fields. We will also omit the affine descendent fields because, as in the
cases of the Virasoro and $\hat{\mathfrak{u}}_{1}$ algebras, the contributions
of affine descendent fields are determined by the contributions of the affine
primary fields. (This would however not be true in the case of a $W$ algebra.)
So we write a generic OPE as

$\displaystyle\Phi^{R_{1}}_{X_{1}}\Phi^{R_{2}}_{X_{2}}\sim\sum_{R_{3}}\int
dX_{3}\ C^{R_{1},R_{2}}_{R_{3}}(X_{1},X_{2}|X_{3})\ \Phi^{R_{3}}_{X_{3}}\ .$
(4.2.33)

Inserting $\oint dz\ J^{a}(z)$ on both sides, and using the linear
independence of the operators $\Phi^{R_{3}}_{X_{3}}$, we obtain an equation
for the structure function $C^{R_{1},R_{2}}_{R_{3}}(X_{1},X_{2}|X_{3})$,

$\displaystyle\left(D_{X_{1}}^{R_{1}}(t^{a})+D_{X_{2}}^{R_{2}}(t^{a})-\left(D_{X_{3}}^{R_{3}}(t^{a})\right)^{\dagger}\right)C^{R_{1},R_{2}}_{R_{3}}(X_{1},X_{2}|X_{3})=0\
,$ (4.2.34)

where the dagger denotes the hermitian conjugate for $X_{3}$-differential
operators, such that for any functions $f,g$ we have $\int fDg=\int
gD^{\dagger}f$. This equation characterizes
$C^{R_{1},R_{2}}_{R_{3}}(X_{1},X_{2}|X_{3})$ as an intertwiner between the
representations $R_{1}\otimes R_{2}$ and $R_{3}$ of the Lie algebra
$\mathfrak{g}$. This shows that the fusion multiplicities of affine highest-
weight representations of $\hat{\mathfrak{g}}$ are bounded by the tensor
product multiplicities of the underlying representations of $\mathfrak{g}$,

$\displaystyle m_{\hat{R}_{1},\hat{R}_{2}}^{\hat{R}_{3}}\leq
m_{R_{1},R_{2}}^{R_{3}}\ .$ (4.2.35)

The presence of null vectors in the representations $\hat{R}_{i}$ can lead to
extra conditions on the structure function
$C^{R_{1},R_{2}}_{R_{3}}(X_{1},X_{2}|X_{3})$, in which case
$m_{\hat{R}_{1},\hat{R}_{2}}^{\hat{R}_{3}}<m_{R_{1},R_{2}}^{R_{3}}$. And
nothing guarantees that only highest-weight representations appear in the
fusion product of two highest-weight representations. In the case of the
$\widetilde{SL}_{2}(\mathbb{R})$ WZW model, other types of representations do
appear. (See Section 4.4.3.)

Finally, let us point out that the maximum multiplicity
$m_{R_{1},R_{2}}^{R_{3}}$ in a fusion rule – that is, the number of linearly
independent solutions of eq. (4.2.34) in the absence of further constraints,
is

$\displaystyle m_{\mathrm{max}}=\left\\{\begin{array}[]{l}2\quad\text{if}\
\mathfrak{g}=\mathfrak{sl}_{2}\ ,\\\ \infty\quad\text{if}\
\mathfrak{g}=\mathfrak{sl}_{n\geq 3}\ .\end{array}\right.$ (4.2.38)

We indeed have a set of $\dim\mathfrak{sl}_{n}=n^{2}-1$ equations, for a
function $C^{R_{1},R_{2}}_{R_{3}}(X_{1},X_{2}|X_{3})$ of $3\frac{n(n-1)}{2}$
variables – the components of $X_{1},X_{2}$ and $X_{3}$. If $n=2$ there are
three equations and three variables, and this can actually be reduced to one
second-order differential equation for a function of one variable. (See eq.
(4.2.46).) If $n\geq 3$ there are more variables than equations, so that
$C^{R_{1},R_{2}}_{R_{3}}(X_{1},X_{2}|X_{3})$ has an arbitrary dependence on a
number of variables. In the case of finite-dimensional representations of
$\mathfrak{sl}_{n\geq 3}$, the multiplicities are of course finite, but they
can take arbitrarily high values, depending on the involved representations.

####  4.2.3 Ward identities and Knizhnik-Zamolodchikov equations

Let us study the $\hat{\mathfrak{g}}$ Ward identities for correlation
functions. To do this, we need to know how the $\hat{\mathfrak{g}}$ currents
$J^{a}(y)$ behave as $y\to\infty$. The relation (4.2.11) with the energy-
momentum tensor $T(y)$, and the behaviour (2.2.14) of $T(y)$, suggest

$\displaystyle\boxed{J^{a}(y)\underset{y\to\infty}{=}O\left(\frac{1}{y^{2}}\right)}\
.$ (4.2.39)

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

$\displaystyle\oint_{\infty}dy\ \epsilon(y)\left\langle
J^{a}(y)\prod_{i=1}^{N}\Phi^{\sigma_{i}}(z_{i})\right\rangle=0\quad\text{provided}\quad\epsilon(y)\underset{y\to\infty}{=}O(1)\
,$ (4.2.40)

In the case $\epsilon(y)=1$, we obtain the $\hat{\mathfrak{g}}$ global Ward
identities,

$\displaystyle\left\langle\sum_{i=1}^{N}(J_{0}^{a})^{(z_{i})}\prod_{i=1}^{N}\Phi^{\sigma_{i}}(z_{i})\right\rangle=0\
.$ (4.2.41)

In the case $\epsilon(y)=\frac{1}{(y-z_{i})^{n}}$ with $n\geq 1$, we obtain
the $\hat{\mathfrak{g}}$ local Ward identities, which are formally identical
to the $\hat{\mathfrak{u}}_{1}$ local Ward identities (4.1.8).

Let us specialize to correlation functions involving affine primary fields.
Knowing the poles (4.2.18) of $J^{a}(y)$ and its behaviour near $y=\infty$, we
have

$\displaystyle\left\langle
J^{a}(y)\prod_{i=1}^{N}\Phi^{R_{i}}_{X_{i}}(z_{i})\right\rangle=-\sum_{i=1}^{N}\frac{D^{R_{i}}_{X_{i}}(t^{a})}{y-z_{i}}\left\langle\prod_{i=1}^{N}\Phi^{R_{i}}_{X_{i}}(z_{i})\right\rangle\
.$ (4.2.42)

In the case of an $N$-point function where the fields with indices $j\neq i$
are affine primaries, the local Ward identities become

$\displaystyle\left\langle\left(J^{a}_{-n}\right)^{(z_{i})}\Phi^{\sigma_{i}}(z_{i})\prod_{j\neq
i}\Phi^{R_{j}}_{X_{j}}(z_{j})\right\rangle$ $\displaystyle=\sum_{j\neq
i}\frac{D_{X_{j}}^{R_{j}}(t^{a})}{(z_{j}-z_{i})^{n}}\left\langle\Phi^{\sigma_{i}}(z_{i})\prod_{j\neq
i}\Phi^{R_{j}}_{X_{j}}(z_{j})\right\rangle\ .$ (4.2.43)

In the case of an $N$-point function of affine primary fields, the global Ward
identites become

$\displaystyle\sum_{i=1}^{N}D_{X_{i}}^{R_{i}}(t^{a})\
\left\langle\prod_{i=1}^{N}\Phi^{R_{i}}_{X_{i}}(z_{i})\right\rangle=0\ .$
(4.2.44)

For example, let us solve the $\mathfrak{sl}_{2}$ global Ward identities for a
three-point function. In the $x$-basis, these identities are formally
identical to the global Ward identities (2.3.4), whose solution is eq.
(2.3.25). We thus find

$\displaystyle\left\langle\prod_{i=1}^{3}\Phi^{j_{i}}_{x_{i}}\right\rangle\propto\
x_{12}^{j_{1}+j_{2}-j_{3}}x_{23}^{j_{2}+j_{3}-j_{1}}x_{31}^{j_{3}+j_{1}-j_{2}}\
,$ (4.2.45)

where we omitted the dependence on $z_{i}$. In the $\mu$-basis, we find

$\displaystyle\left\langle\prod_{i=1}^{3}\Phi^{j_{i}}_{\mu_{i}}\right\rangle\propto\
\mu_{2}\delta(\mu_{1}+\mu_{2}+\mu_{3})\
\mathcal{H}\left(-\frac{\mu_{1}}{\mu_{2}}\right)\ ,$ (4.2.46)

where the function $\mathcal{H}(x)$, which parametrizes the general solution
of the $t^{-}$ and $t^{0}$ equations, is constrained by the $t^{+}$ equation
to obey the twisted hypergeometric differential equation (2.3.73). It may seem
strange that in the $\mu$-basis we obtain a second-order differential
equation, whereas in the $x$-basis the solution appeared to be unique. The
number of solutions of the global Ward identities has the algebraic
interpretation of a tensor product multiplicity for $\mathfrak{sl}_{2}$
representations, and this should not depend on our choice of isospin variable.
Actually it is the $x$-basis calculation which is misleading: analyticity
conditions on the $x_{i}$-dependence of
$\left\langle\prod_{i=1}^{3}\Phi^{j_{i}}_{x_{i}}\right\rangle$ in general
allow the existence of two solutions which differ globally, although they are
locally identical [18]. The tensor product multiplicity for generic
$\mathfrak{sl}_{2}$ representations is two, as correctly suggested by the
$\mu$-basis calculation.

Let us now assume that the Virasoro field, which we defined by the Sugawara
construction (4.2.11), is actually the energy-momentum tensor, so that eq.
(2.3.1) is obeyed. Inserting the equation (4.2.20) in an $N$-point function of
affine primary fields, and applying the local Ward identity (4.2.43) to
$J^{b}_{-1}\Phi^{R}_{X}=(J^{b}\Phi^{R}_{X})$, we obtain the Knizhnik-
Zamolodchikov equations or KZ equations,

$\displaystyle\boxed{\left\\{(k+g){\frac{\partial}{\partial
z_{i}}}+\sum_{j\neq
i}\frac{K_{ab}D_{X_{i}}^{R_{i}}(t^{a})D_{X_{j}}^{R_{j}}(t^{b})}{z_{i}-z_{j}}\right\\}\left\langle\prod_{i=1}^{N}\Phi^{R_{i}}_{X_{i}}(z_{i})\right\rangle=0}\
.$ (4.2.47)

These are first-order differential equations in $z_{i}$, and like the
analogous equations (4.1.13) for free boson correlation functions, they
determine the dependence on $z_{i}$ of correlation functions of primary
fields. However, unlike the free boson equations, the KZ equations do not have
simple solutions in general. It can be checked that the KZ equations imply the
conformal global Ward identities. (See Exercise 4.7.)

The KZ equations can be rewritten as

$\displaystyle\left\\{(k+g){\frac{\partial}{\partial
z_{i}}}+H_{i}\right\\}\left\langle\prod_{i=1}^{N}\Phi^{R_{i}}_{X_{i}}(z_{i})\right\rangle=0\
,$ (4.2.48)

where $H_{i}$ are the mutually commuting Gaudin Hamiltonians, the Hamiltonians
of the Gaudin model – an integrable model associated to the Lie algebra
$\mathfrak{g}$. Techniques developed for studying the $\mathfrak{g}$ Gaudin
model can be useful for solving the $\mathfrak{g}$ KZ equations, as we will
now see in the case $\mathfrak{g}=\mathfrak{sl}_{2}$.

####  4.2.4 $\mathfrak{sl}_{2}$ case: the KZ-BPZ relation

We will now show that the $\mathfrak{sl}_{2}$ KZ equations are equivalent to
certain BPZ equations via Sklynanin’s separation of variables for the
$\mathfrak{sl}_{2}$ Gaudin model.

Our derivation of the KZ equations amounted to inserting the Sugawara
construction (4.2.11) at the points $z_{i}$ in the $N$-point function
$\left\langle\prod_{i=1}^{N}\Phi^{j_{i}}_{X_{i}}(z_{i})\right\rangle$ of
affine primary fields. Inserting the Sugawara construction at an arbitrary
point $y$ instead, we obtain the identity

$\displaystyle\left\langle\left(T(y)-\frac{1}{2(k+2)}\left[2(J^{0}J^{0})(y)+(J^{+}J^{-})(y)+(J^{-}J^{+})(y)\right]\right)\prod_{i=1}^{N}\Phi^{j_{i}}_{X_{i}}(z_{i})\right\rangle=0\
.$ (4.2.49)

According to eqs. (2.3.3) and (4.2.42), inserting the fields $T(y)$ and
$J^{a}(y)$ amounts to acting with the differential operators

$\displaystyle\hat{T}(y)$
$\displaystyle=\sum_{i=1}^{N}\left(\frac{\Delta_{j_{i}}}{(z-z_{i})^{2}}+\frac{1}{z-z_{i}}{\frac{\partial}{\partial
z_{i}}}\right)\ ,$ (4.2.50) $\displaystyle\hat{J}^{a}(y)$
$\displaystyle=-\sum_{i=1}^{N}\frac{D^{R_{i}}_{X_{i}}(t^{a})}{y-z_{i}}\ ,$
(4.2.51)

where $\Delta_{j}$ is defined in eq. (4.2.24). These differential operators
obey the commutation relations

$\displaystyle\left[\hat{T}(y),\hat{J}^{a}(z)\right]$
$\displaystyle={\frac{\partial}{\partial
z}}\frac{\hat{J}^{a}(y)-\hat{J}^{a}(z)}{y-z}\ ,$ (4.2.52)
$\displaystyle\left[\hat{J}^{a}(y),\hat{J}^{b}(z)\right]$
$\displaystyle=f^{ab}_{c}\frac{\hat{J}^{c}(y)-\hat{J}^{c}(z)}{y-z}\ ,$
(4.2.53)

where $f^{ab}_{c}$ are the structure constants of the Lie algebra
$\mathfrak{sl}_{2}$, as encoded in the commutation relations (2.3.9). And one
can easily show that inserting a normal-ordered product, for instance
$(J^{-}J^{+})(y)$, amounts to acting with the product of the corresponding
differential operators. Therefore, the identity (4.2.49) amounts to the
differential equation

$\displaystyle\left(\hat{T}(y)-\frac{1}{2(k+2)}\left[2\hat{J}^{0}(y)\hat{J}^{0}(y)+\hat{J}^{+}(y)\hat{J}^{-}(y)+\hat{J}^{-}(y)\hat{J}^{+}(y)\right]\right)\left\langle\prod_{i=1}^{N}\Phi^{j_{i}}_{X_{i}}(z_{i})\right\rangle=0\
.$ (4.2.54)

The idea is to simplify this identity by taking $y$ to be one of the zeros
$\hat{y}_{j}$ of $\hat{J}^{-}(y)$. These zeros are differential operators, and
the value of a $y$-dependent differential operator $\hat{O}(y)$ at
$y=\hat{y}_{j}$ is defined by inserting $\hat{y}_{j}$ from the left, that is
$\hat{O}(\hat{y}_{j})=\frac{1}{2\pi
i}\oint_{\hat{y}_{j}}\frac{dy}{y-\hat{y}_{j}}\hat{O}(y)$. Using eq. (4.2.53)
for bringing the $\hat{J}^{-}(y)$ factors to the left, we obtain

$\displaystyle\left(\hat{T}(\hat{y}_{j})-\frac{1}{k+2}\left[(\hat{J}^{0})^{2}(\hat{y}_{j})+\partial\hat{J}^{0}(\hat{y}_{j})\right]\right)\left\langle\prod_{i=1}^{N}\Phi^{j_{i}}_{X_{i}}(z_{i})\right\rangle=0\
.$ (4.2.55)

Let us further study the differential operators $\hat{y}_{j}$ and
$\hat{J}^{0}(\hat{y}_{j})$. According to eq. (4.2.53), we have
$[\hat{J}^{-}(y),\hat{J}^{-}(z)]=0$. Therefore $[\hat{y}_{j},\hat{y}_{k}]=0$,
and

$\displaystyle\boxed{\hat{J}^{-}(y)=\hat{Y}_{2}\frac{\prod_{j}(y-\hat{y}_{j})}{\prod_{i}(y-z_{i})}}\
,$ (4.2.56)

where $\hat{Y}_{2}$ is a differential operator such that
$[\hat{Y}_{2},\hat{y}_{j}]=0$. Using eq. (4.2.53) we find
$[\hat{J}^{0}(\hat{y}_{j}),\hat{J}^{-}(z)]=\frac{\hat{J}^{-}(z)}{\hat{y}_{j}-z}$,
and deduce

$\displaystyle[\hat{p}_{j},\hat{Y}_{2}]=0\quad\text{and}\quad[\hat{p}_{j},\hat{y}_{k}]=\delta_{j,k}\quad\text{where}\quad\hat{p}_{j}=\hat{J}^{0}(\hat{y}_{j})\
.$ (4.2.57)

We can now simplify the second term of eq. (4.2.55),

$(\hat{J}^{0})^{2}(\hat{y}_{j})+\partial\hat{J}^{0}(\hat{y}_{j})=\frac{1}{2\pi
i}\oint_{\hat{y}_{j}}\frac{dy}{y-\hat{y}_{j}}\left(\hat{J}^{0}(y)+{\frac{\partial}{\partial
y}}\right)\hat{J}^{0}(y)\\\ =\frac{1}{2\pi
i}\oint_{\hat{y}_{j}}\frac{dy}{y-\hat{y}_{j}}\left(\hat{p}_{j}+{\frac{\partial}{\partial
y}}\right)\hat{J}^{0}(y)=\hat{p}_{j}\frac{1}{2\pi
i}\oint_{\hat{y}_{j}}\frac{dy}{y-\hat{y}_{j}}\hat{J}^{0}(y)=\hat{p}_{j}^{2}\
.$ (4.2.58)

So eq. (4.2.55) becomes

$\displaystyle\boxed{\left(\frac{1}{k+2}\hat{p}_{j}^{2}-\hat{T}(\hat{y}_{j})\right)\left\langle\prod_{i=1}^{N}\Phi^{j_{i}}_{X_{i}}(z_{i})\right\rangle=0}\
.$ (4.2.59)

We have therefore replaced the differential operators
$D^{R_{i}}_{X_{i}}(t^{a})$, which appeared in $\hat{J}^{a}(y)$ eq. (4.2.51),
with the operators $\hat{y}_{j}$ and $\hat{p}_{j}$. Therefore, it is natural
to introduce the Sklyanin variables $y_{j}$ as the eigenvalues of the mutually
commuting operators $\hat{y}_{j}$. But how many Sklyanin variables do we have?
The $\widehat{\mathfrak{sl}}_{2}$ global Ward identity (4.2.44) associated to
the current $J^{a}$ can be written as

$\displaystyle\underset{y\to\infty}{\lim}y\hat{J}^{a}(y)\left\langle\prod_{i=1}^{N}\Phi^{j_{i}}_{X_{i}}(z_{i})\right\rangle=0\
.$ (4.2.60)

When combined with the definition (4.2.56) of $\hat{y}_{j}$, the identity
associated to $J^{-}$ suggests that we have $N-2$ Sklyanin variables $y_{j}$.
Let us define two more variables as the eigenvalues $Y_{1}$ and $Y_{2}$ of the
operators $\hat{Y}_{1}$ and $\hat{Y}_{2}$, where we define

$\displaystyle\hat{Y}_{1}=\underset{y\to\infty}{\lim}y\hat{J}^{-}(y)\ .$
(4.2.61)

We can now define Sklyanin’s separation of variables for the
$\mathfrak{sl}_{2}$ Gaudin model as the linear map $\mathcal{K}$ from
functions of $(X_{1},\cdots X_{N})$ to functions of $(Y_{1},Y_{2},y_{1},\cdots
y_{N-2})$, which diagonalizes the operators
$(\hat{Y}_{1},\hat{Y}_{2},\hat{y}_{1},\cdots\hat{y}_{N-2})$, so that in
particular $\mathcal{K}\hat{y}_{j}f(X_{1},\cdots
X_{N})=y_{j}\mathcal{K}f(X_{1},\cdots X_{N})$. We actually only define this
map on functions which obey the global Ward identity associated with $J^{-}$
i.e. which are killed by $\hat{Y}_{1}$, whose images therefore have a
$\delta(Y_{1})$ prefactor. Using in addition the global Ward identity
associated to $J^{0}$, we find that the combined dependence of
$\mathcal{K}\left\langle\prod_{i=1}^{N}\Phi^{j_{i}}_{X_{i}}(z_{i})\right\rangle$
on $Y_{1}$ and $Y_{2}$ is a $\delta(Y_{1})Y_{2}$ prefactor. (This is done with
the help of eq. (4.2.53), which implies
$[\underset{y\to\infty}{\lim}y\hat{J}^{0}(y),\hat{J}^{-}(z)]=-\hat{J}^{-}(z)$.)

Let us rewrite our equation (4.2.59) in terms of Sklyanin variables. Using eq.
(4.2.57), we find
$\mathcal{K}\hat{p}_{j}\mathcal{K}^{-1}={\frac{\partial}{\partial y_{j}}}$.
Therefore, eq. (4.2.59) is equivalent to the following equation for
$\mathcal{K}\left\langle\prod_{i=1}^{N}\Phi^{j_{i}}_{X_{i}}(z_{i})\right\rangle$,

$\displaystyle\left(\frac{1}{k+2}\frac{\partial^{2}}{\partial
y_{j}^{2}}-\mathcal{K}\hat{T}(\hat{y}_{j})\mathcal{K}^{-1}\right)\mathcal{K}\left\langle\prod_{i=1}^{N}\Phi^{j_{i}}_{X_{i}}(z_{i})\right\rangle=0\
.$ (4.2.62)

This equation apparently only involves $y_{j}$ for a given index $j$, whereas
each KZ equation involves $(X_{1},\cdots X_{N})$. So it may seem that using
Sklyanin variables leads to a separation of variables no only in the Gaudin
model, but also in the KZ equations. This is however not true, because the
differential operator $\hat{T}(y)$ (4.2.50) involves $z_{i}$-derivatives at
fixed isospin variables $X_{i}$. When writing
$\mathcal{K}\hat{T}(\hat{y}_{j})\mathcal{K}^{-1}$, we have to use
$z_{i}$-derivatives at fixed Sklyanin variables $y_{k}$, and we find

$\displaystyle\mathcal{K}\hat{T}(\hat{y}_{j})\mathcal{K}^{-1}=\sum_{i}\left[\frac{\Delta_{j_{i}}}{(y_{j}-z_{i})^{2}}+\frac{1}{y_{j}-z_{i}}\left({\frac{\partial}{\partial
z_{i}}}+{\frac{\partial}{\partial y_{j}}}\right)\right]-\sum_{k\neq
j}\frac{1}{y_{jk}}\left({\frac{\partial}{\partial
y_{j}}}-{\frac{\partial}{\partial y_{k}}}\right)\ .$ (4.2.63)

(See Exercise 4.8.) Introducing the function

$\displaystyle\Theta=\frac{\prod_{i<i^{\prime}}(z_{i}-z_{i^{\prime}})\prod_{j<j^{\prime}}(y_{j}-y_{j^{\prime}})}{\prod_{i,j}(z_{i}-y_{j})}\
,$ (4.2.64)

the equation (4.2.62) is equivalent to

$\left\\{\frac{1}{k+2}\frac{\partial^{2}}{\partial
y_{j}^{2}}-\sum_{i}\frac{1}{y_{j}-z_{i}}{\frac{\partial}{\partial
z_{i}}}-\sum_{k\neq j}\frac{1}{y_{j}-y_{k}}{\frac{\partial}{\partial
y_{k}}}\right.\\\
\left.-\sum_{i}\frac{\Delta_{j_{i}}-\frac{k}{4}}{(y_{j}-z_{i})^{2}}-\sum_{k\neq
j}\frac{\frac{3k}{4}+1}{(y_{j}-y_{k})^{2}}\right\\}\Theta^{\frac{k+2}{2}}\mathcal{K}\left\langle\prod_{i=1}^{N}\Phi^{j_{i}}_{X_{i}}(z_{i})\right\rangle=0\
.$ (4.2.65)

As an equation for
$\Theta^{\frac{k+2}{2}}\mathcal{K}\left\langle\prod_{i=1}^{N}\Phi^{j_{i}}_{X_{i}}(z_{i})\right\rangle$,
this coincides with the BPZ equation (2.3.54) for the correlation function of
$\left\langle\prod_{i=1}^{N}V_{\alpha_{i}}(z_{i})\prod_{j=1}^{N-2}V_{\langle
2,1\rangle}(y_{j})\right\rangle$, provided the parameter $b$ of the Virasoro
algebra is given by

$\displaystyle\boxed{b^{2}=-k-2}\ ,$ (4.2.66)

and the momentums $\alpha_{i}$ are given in terms of the spins $j_{i}$ by

$\displaystyle\boxed{\alpha=-b^{-1}j+\frac{b}{2}}\quad\Rightarrow\quad\boxed{\Delta(\alpha(j))=\Delta_{j}-\frac{k}{4}}\
,$ (4.2.67)

where $\Delta(\alpha)$ is given by eq. (2.1.30). The BPZ equations only
constrain the dependence on $y_{j}$, but we already determined the dependence
of
$\mathcal{K}\left\langle\prod_{i=1}^{N}\Phi^{j_{i}}_{X_{i}}(z_{i})\right\rangle$
on the variables $Y_{1}$ and $Y_{2}$. This leads to Feigin, Frenkel and
Stoyanovsky’s KZ-BPZ relation,

$\displaystyle\boxed{\mathcal{K}\left\langle\prod_{i=1}^{N}\Phi^{j_{i}}_{X_{i},\bar{X}_{i}}(z_{i},\bar{z}_{i})\right\rangle=\delta^{(2)}(Y_{1})|Y_{2}|^{2}|\Theta|^{b^{2}}\left\langle\prod_{i=1}^{N}V_{\alpha_{i}}(z_{i},\bar{z}_{i})\prod_{j=1}^{N-2}V_{\langle
2,1\rangle}(y_{j},\bar{y}_{j})\right\rangle}\ .$ (4.2.68)

This equation means that the differential equations obeyed by both sides are
related by Sklyanin’s separation of variables. Solutions of the KZ equations,
which are called $\widehat{\mathfrak{sl}}_{2}$ conformal blocks, are therefore
related to certain Virasoro $2N-2$-point conformal blocks. In the diagrammatic
representation of conformal blocks as trees, the relevant Virasoro block is
obtained from an $\widehat{\mathfrak{sl}}_{2}$ conformal block by adding a
degenerate field near each node, and identifying the two fusion channels of
that degenerate field with $\widehat{\mathfrak{sl}}_{2}$ fusion
multiplicities:

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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (4.2.69)

The separation of variables $\mathcal{K}$ is in general an integral
transformation, but in the case of the $\mu$-basis (4.2.28) of isospin
variables, we have

$\displaystyle\hat{J}^{-}(y)=\sum_{i=1}^{N}\frac{\mu_{i}}{y-z_{i}}\quad\Rightarrow\quad\left\\{\begin{array}[]{l}Y_{1}=\sum_{i=1}^{N}\mu_{i}\
,\\\ Y_{2}=\sum_{i=1}^{N}\mu_{i}z_{i}\ ,\\\
\mu_{i}=Y_{2}\frac{\prod_{j=1}^{N-2}(z_{i}-y_{j})}{\prod_{i^{\prime}\neq
i}(z_{i}-z_{i^{\prime}})}\ ,\end{array}\right.$ (4.2.73)

and the variables on both sides of the relation are functions of one another.

The KZ-BPZ relation plays in important role in the study of
$\widehat{\mathfrak{sl}}_{2}$-symmetric theories such as the $H_{3}^{+}$
model, and one may wonder whether the $\mathfrak{g}$ KZ equations are involved
in similar relations for more general choices of the Lie algebra
$\mathfrak{g}$. While Sklyanin’s separation of variables for the
$\mathfrak{sl}_{3}$ Gaudin model exists, writing the $\mathfrak{sl}_{3}$ KZ
equations in Sklyanin variables however does not lead to the expected
generalizations of the BPZ equations [19], and the reason for this discrepancy
is not known. Another tentative generalization of the KZ-BPZ relation (4.2.68)
is to replace the field $V_{\langle 2,1\rangle}$ on the right-hand side with
another primary field (degenerate or not). Then the resulting expression can
be interpreted as an $N$-point function in a theory whose symmetry algebra is
a generalization of $\widehat{\mathfrak{sl}}_{2}$, as argued in [20].

###  4.3 The $H_{3}^{+}$ model

The $H_{3}^{+}$ model is to the $\widehat{\mathfrak{sl}}_{2}$ algebra what
Liouville theory is to the Virasoro algebra: the simplest non-rational model
with the given symmetry. However, in contrast to Liouville theory, there is no
value of the central charge such that the $H_{3}^{+}$ model is unitary.

As we did with Liouville theory, we will define the $H_{3}^{+}$ model by its
symmetries, and other assumptions on the correlation functions. We could
follow the same route as with Liouville theory, of studying the
representations of $\widehat{\mathfrak{sl}}_{2}$, and deducing the three-point
function from the associativity of OPEs involving degenerate fields, as was
done by Teschner [21]. Rather, we will follow the shortcut of assuming that
correlation functions of the $H_{3}^{+}$ model and Liouville theory are
related as suggested by the KZ-BPZ relation (4.2.68) – this relation, when
applied to correlation functions (instead of differential equations or
conformal blocks), is then called the $H_{3}^{+}$-Liouville relation.

####  4.3.1 Spectrum and correlation functions

Let us study the correlation functions in the $\mu$-basis, taking advantage of
the simplicity of the relation (4.2.73) between the variables $(\mu_{i})$ and
$(Y_{1},Y_{2},y_{j})$. Let us first consider the two-point function. In this
case there are no Sklyanin variables $y_{j}$, and the separation of variables
is simply

$\displaystyle
N=2\quad\Rightarrow\quad\left\\{\begin{array}[]{l}Y_{1}=\mu_{1}+\mu_{2}\ ,\\\
Y_{2}=\mu_{1}(z_{1}-z_{2})\ .\end{array}\right.$ (4.3.3)

Using the two-point function (3.1.11) of Liouville theory, we obtain

$\displaystyle\left\langle\Phi^{j_{1}}_{\mu_{1},\bar{\mu}_{1}}\Phi^{j_{2}}_{\mu_{2},\bar{\mu}_{2}}\right\rangle=\delta^{(2)}(\mu_{1}+\mu_{2})|\mu_{1}|^{2}b\Big{(}\delta(j_{1}+j_{2}+1)+R\left(-b^{-1}j_{1}+\tfrac{b}{2}\right)\delta(j_{2}-j_{1})\Big{)}\
,$ (4.3.4)

where we omitted the dependence on $z_{i}$, and we used the parameter $b$
(4.2.66) and reflection coefficient $R(\alpha)$ (3.1.42) of Liouville theory.
Actually, the quantity $R\left(-b^{-1}j_{1}+\frac{b}{2}\right)$ is the
reflection coefficent of the $H_{3}^{+}$ model in the $\mu$-basis,

$\displaystyle\Phi^{j}_{\mu,\bar{\mu}}=R(-b^{-1}j+\tfrac{b}{2})\
\Phi^{-j-1}_{\mu,\bar{\mu}}\ .$ (4.3.5)

Then let us consider the three-point function. In this case there is one
Sklyanin variables $y_{1}$, and we have

$\displaystyle
N=3\quad\Rightarrow\quad\left\\{\begin{array}[]{l}y_{1}=-\frac{\mu_{1}z_{2}z_{3}+\mu_{2}z_{3}z_{1}+\mu_{3}z_{1}z_{2}}{Y_{2}}\
,\\\
\frac{(y_{1}-z_{1})(z_{2}-z_{3})}{(y_{1}-z_{2})(z_{1}-z_{3})}=-\frac{\mu_{1}}{\mu_{2}}\
.\end{array}\right.$ (4.3.8)

The relation (4.2.68) then reads

$\displaystyle\left\langle\prod_{i=1}^{3}\Phi^{j_{i}}_{\mu_{i},\bar{\mu}_{i}}\right\rangle=\delta^{(2)}(\textstyle{\sum}_{i=1}^{3}\mu_{i})|\mu_{2}|^{2}\sum_{\pm}C_{\pm}(\alpha_{1})C_{\alpha_{1}\mp\frac{b}{2},\alpha_{2},\alpha_{3}}\left|\mathcal{H}^{(s)}_{\pm}(-\tfrac{\mu_{1}}{\mu_{2}})\right|^{2}\
,$ (4.3.9)

where we used the momentums $\alpha_{i}$ (4.2.67), the Liouville theory
structure constants $C_{\pm}(\alpha)$ and
$C_{\alpha_{1},\alpha_{2},\alpha_{3}}$, and the functions
$\mathcal{H}^{(s)}_{\pm}(x)$, which are related to the $s$-channel Virasoro
conformal blocks $\mathcal{F}^{(s)}_{\pm}(x)=\leavevmode\hbox to97.17pt{\vbox
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ by eq. (2.3.78).

However, the three-point function is simpler in the $x$-basis, see eq.
(4.2.45). Assuming correlation functions are single-valued as functions of the
isospin variables, we must have

$\displaystyle\left\langle\prod_{i=1}^{3}\Phi^{j_{i}}_{x_{i},\bar{x}_{i}}\right\rangle=C^{H_{3}^{+}}_{j_{1},j_{2},j_{3}}\
|x_{12}|^{2(j_{1}+j_{2}-j_{3})}|x_{23}|^{2(j_{2}+j_{3}-j_{1})}|x_{31}|^{2(j_{3}+j_{1}-j_{2})}\
,$ (4.3.10)

which provides a natural definition of the three-point structure constant
$C^{H_{3}^{+}}_{j_{1},j_{2},j_{3}}$. Let us compute this structure constant,
by relating the $x$-basis and $\mu$-basis three-point functions. The relation
between the two bases is a single-valued version of the holomorphic Fourier
transformation of eq. (4.2.29),

$\displaystyle\Phi^{j}_{x,\bar{x}}=\gamma(2j+1)\int_{{\mathbb{C}}}d^{2}\mu\
|\mu|^{-2j-2}e^{\mu x-\bar{\mu}\bar{x}}\Phi^{j}_{\mu,\bar{\mu}}\ .$ (4.3.11)

Here we introduce a prefactor $\gamma(2j+1)$, which will be needed in Section
4.4.2 for ensuring that the $H_{3}^{+}$ OPE has a finite limit when the spin
becomes half-integer. Moreover, we flipped the sign of the second term of the
exponent of $e^{\mu x-\bar{\mu}\bar{x}}$, as compared to the expected $e^{\mu
x+\bar{\mu}\bar{x}}$, in order to ensure the convergence of the integral when
$x$ and $\bar{x}$ are complex conjugates. This affects the relation between
the differential operators $\bar{D}^{j}_{\bar{x}}(t^{a})$ and
$\bar{D}^{j}_{\bar{\mu}}(t^{a})$, which we define in terms of their
holomorphic counterparts $D^{j}_{x}(t^{a}),D^{j}_{\mu}(t^{a})$ as

$\displaystyle\bar{D}^{j}_{\bar{x}}(t^{a})=D^{j}_{\bar{x}}(t^{a})\quad,\quad\bar{D}^{j}_{\bar{\mu}}(t^{a})=D^{j}_{-\bar{\mu}}(t^{a})\
.$ (4.3.12)

Notice that the minus sign in the operators $D^{j}_{-\bar{\mu}}(t^{a})$
modifies neither their commutation relations, nor the KZ equations.

The $x$-basis and $\mu$-basis three-point functions can be compared using the
formula [22]

$\prod_{i=1}^{3}\left(\frac{|\mu_{i}|^{2j_{i}+2}}{\pi^{2}}\int_{{\mathbb{C}}}d^{2}x_{i}\
e^{-\mu_{i}x_{i}+\bar{\mu}_{i}\bar{x}_{i}}\right)|x_{12}|^{2(j_{1}+j_{2}-j_{3})}|x_{23}|^{2(j_{2}+j_{3}-j_{1})}|x_{31}|^{2(j_{3}+j_{1}-j_{2})}\\\
=\frac{1}{\pi^{2}}\delta^{(2)}(\textstyle{\sum}_{i=1}^{3}\mu_{i})|\mu_{2}|^{2}\sum_{\pm}d_{\pm}\left|\mathcal{H}^{(s)}_{\pm}(-\tfrac{\mu_{1}}{\mu_{2}})\right|^{2}\
,$ (4.3.13)

where we define

$\displaystyle
d_{+}=\frac{\gamma(-j_{1}+j_{2}+j_{3}+1)}{\gamma(-2j_{1})}\quad,\quad
d_{-}=\frac{\gamma(j_{1}+j_{2}-j_{3}+1)\gamma(j_{1}-j_{2}+j_{3}+1)}{\gamma(-j_{1}-j_{2}-j_{3}-1)\gamma(2j_{1}+2)}\
.$ (4.3.14)

The right-hand side of eq. (4.3.13) is a combination of solutions (4.2.46) of
the $\mu$-basis global Ward identities, written using the functions
$\mathcal{H}^{(s)}_{\pm}(x)$ of eq. (2.3.88). Comparing with eq. (4.3.9), we
obtain the three-point structure constant in terms of Liouville theory
structure constants,

$\displaystyle
C^{H_{3}^{+}}_{j_{1},j_{2},j_{3}}=\frac{\pi^{2}\prod_{i=1}^{3}\gamma(2j_{i}+1)}{d_{\pm}}C_{\pm}(\alpha_{1})C_{\alpha_{1}\mp\frac{b}{2},\alpha_{2},\alpha_{3}}\
.$ (4.3.15)

The conformal boostrap equations of Liouville theory guarantee that this does
not depend on the sign $\pm$, see Section 3.1.3. Using eqs. (3.1.43) and
(3.1.59) for $C_{\pm}(\alpha_{1})$ and
$C_{\alpha_{1}\mp\frac{b}{2},\alpha_{2},\alpha_{3}}$ respectively, we find

$\displaystyle\boxed{C^{H_{3}^{+}}_{j_{1},j_{2},j_{3}}=\frac{\pi^{2}b^{-1}\left[b^{\frac{2}{b^{2}}}\mu^{\frac{1}{b}}\right]^{j_{1}+j_{2}+j_{3}+1}\Upsilon^{\prime}_{b}(0)\Upsilon_{b}(-\frac{2j_{1}}{b})\Upsilon_{b}(-\frac{2j_{2}}{b})\Upsilon_{b}(-\frac{2j_{3}}{b})}{\Upsilon_{b}(-\frac{j_{1}+j_{2}+j_{3}+1}{b})\Upsilon_{b}(-\frac{j_{1}+j_{2}-j_{3}}{b})\Upsilon_{b}(-\frac{j_{1}-j_{2}+j_{3}}{b})\Upsilon_{b}(-\frac{-j_{1}+j_{2}+j_{3}}{b})}}\
,$ (4.3.16)

where the parameter $b$ is given by eq. (4.2.66), and the cosmological
constant $\mu$ is inherited from Liouville theory. (We would have a different
formula in the case $b\in i\mathbb{R}$, based on the alternative Liouville
structure constant (3.1.60).)

The $x$-basis two-point function can be obtained from the $\mu$-basis two-
point function with the help of the formula

$\displaystyle\int_{{\mathbb{C}}}d^{2}\mu\ e^{\mu
x-\bar{\mu}\bar{x}}|\mu|^{-4j-2}=|x|^{4j}\pi\gamma(-2j)\ ,$ (4.3.17)

and we find

$\displaystyle\left\langle\Phi^{j_{1}}_{x_{1},\bar{x}_{1}}\Phi^{j_{2}}_{x_{2},\bar{x}_{2}}\right\rangle=\frac{-\pi^{2}b}{(2j_{1}+1)^{2}}\,\delta(j_{1}+j_{2}+1)\delta^{(2)}(x_{12})+\frac{\pi}{b}\mu^{\frac{2j_{1}+1}{b}}\gamma(-\tfrac{2j_{1}+1}{b^{2}})\,\delta(j_{1}-j_{2})|x_{12}|^{4j_{1}}\,.$
(4.3.18)

Using the same formula, we also obtain the $x$-basis reflection relation from
the $\mu$-basis reflection relation,

$\displaystyle\Phi^{j}_{x,\bar{x}}=\frac{b^{2}}{\pi}\mu^{\frac{2j+1}{b}}\gamma(1-\tfrac{2j+1}{b^{2}})\int_{{\mathbb{C}}}d^{2}x^{\prime}\
|x-x^{\prime}|^{4j}\Phi^{-j-1}_{x^{\prime},\bar{x}^{\prime}}\ .$ (4.3.19)

The OPEs of affine primary fields are

$\displaystyle\Phi^{j_{1}}_{x_{1},\bar{x}_{1}}\Phi^{j_{2}}_{x_{2},\bar{x}_{2}}$
$\displaystyle\sim\frac{1}{2}\int_{-\frac{1}{2}+ib\mathbb{R}}\frac{(2j+1)^{2}dj}{-\pi^{2}b}\int_{{\mathbb{C}}}d^{2}x\left\langle\Phi^{j_{1}}_{x_{1},\bar{x}_{1}}\Phi^{j_{2}}_{x_{2},\bar{x}_{2}}\Phi^{-j-1}_{x,\bar{x}}\right\rangle\Phi^{j}_{x,\bar{x}}\
,$ (4.3.20)
$\displaystyle\Phi^{j_{1}}_{\mu_{1},\bar{\mu}_{1}}\Phi^{j_{2}}_{\mu_{2},\bar{\mu}_{2}}$
$\displaystyle\sim\frac{1}{2}\int_{-\frac{1}{2}+ib\mathbb{R}}dj\int_{{\mathbb{C}}}\frac{d^{2}\mu}{b|\mu|^{2}}\left\langle\Phi^{j_{1}}_{\mu_{1},\bar{\mu}_{1}}\Phi^{j_{2}}_{\mu_{2},\bar{\mu}_{2}}\Phi^{-j-1}_{-\mu,-\bar{\mu}}\right\rangle\Phi^{j}_{\mu,\bar{\mu}}\
.$ (4.3.21)

Notice that the measures of integration on the spin and isospin variables are
dictated by the $\delta(j_{1}+j_{2}+1)$ term in the two-point function. These
equations are analogous to eq. (3.1.16) for the OPE in Liouville theory. The
values of the spin $j$,

$\displaystyle\boxed{j\in-\frac{1}{2}+ib\mathbb{R}}\ ,$ (4.3.22)

are deduced from Liouville theory via the relation (4.2.67). These values are
the same for the left- and right-moving $\widehat{\mathfrak{sl}}_{2}$ algebras
– the spectrum is diagonal.

Our treatment of the $H_{3}^{+}$ model is valid for levels $k\neq-2$. If
$k=-2$, the Sugawara construction breaks down, and the limit of the model is
no longer a conformal field theory. The central charges of the $H_{3}^{+}$
model (4.2.12) and of the corresponding Liouville theory (4.2.66) take the
values

$\displaystyle\boxed{c^{H_{3}^{+}}=\frac{3k}{k+2}\in\mathbb{C}-\\{3\\}}\
,\quad c^{\text{Liouville}}=1-6k-\frac{6}{k+2}\in\mathbb{C}\ .$ (4.3.23)

####  4.3.2 Large level limit and geometrical interpretation

Having defined the $H_{3}^{+}$ model by its spectrum and correlation
functions, we still have to understand these objects and their interpretation.

To do this, we will consider the large level limit $k\to\infty$ of the model,
which is sometimes called the minisuperspace limit. The KZ equations (4.2.47)
imply that correlation functions do not depend on the positions $z_{i}$ in
this limit, and are functions of the sole isospin variables. Moreover, given a
generator $J^{a}_{m}$, the commutator $[J^{a}_{m},J^{b}_{-m}]$ (4.2.14) tends
to infinity for some index $b$, unless $m=0$. So the generators $J^{a}_{m\neq
0}$, and the descendent states they create, disappear from the theory, and
only the horizontal subalgebra of $\widehat{\mathfrak{sl}}_{2}$ survives. This
corresponds to an $SL_{2}({\mathbb{C}})$ symmetry group, which acts on a field
of spin $j$ as

$\displaystyle
U_{g}\Phi^{j}_{x,\bar{x}}=|cx+d|^{4j}\Phi^{j}_{\frac{ax+b}{cx+d},\frac{\bar{a}\bar{x}+\bar{b}}{\bar{c}\bar{x}+\bar{d}}}\quad\text{with}\quad
g=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in
SL_{2}({\mathbb{C}})\ .$ (4.3.26)

(This action is formally identical to the action (2.3.17) of global conformal
transformations on quasi-primary fields.) Now we notice that the
transformation $U_{g}$ also describes the action of $SL_{2}({\mathbb{C}})$ on
functions on the space $H_{3}^{+}$ of hermitian matrices of size two and
determinant one. Consider indeed the function

$\displaystyle\Phi^{j}_{x,\bar{x}}(h)=\left(\begin{bmatrix}x\\\
1\end{bmatrix}^{\dagger}h\begin{bmatrix}x\\\ 1\end{bmatrix}\right)^{2j}\ ,$
(4.3.27)

and the natural action of $g\in SL_{2}({\mathbb{C}})$ on the space
$\mathcal{F}(H_{3}^{+})$ of functions on $H_{3}^{+}$,

$\displaystyle g\cdot f(h)=f(g^{\dagger}hg)\ ,$ (4.3.28)

then we have

$\displaystyle g\cdot\Phi^{j}_{x,\bar{x}}(h)=U_{g}\Phi^{j}_{x,\bar{x}}(h)\ .$
(4.3.29)

This suggests that we identitfy the large level limit of the field
$\Phi^{j}_{x,\bar{x}}$ with the function $\Phi^{j}_{x,\bar{x}}(h)$. We can
then define a scalar product on the large level limit of the spectrum, using
the natural, positive definite scalar product on $\mathcal{F}(H_{3}^{+})$,

$\displaystyle\left\langle
f\middle|f^{\prime}\right\rangle=\int_{H_{3}^{+}}dh\
\overline{f(h)}f^{\prime}(h)\ ,$ (4.3.30)

where $dh$ is the $SL_{2}({\mathbb{C}})$-invariant measure on $H_{3}^{+}$. For
this scalar product, the action of $g\in SL_{2}({\mathbb{C}})$ is a unitary
tranformation:

$\displaystyle\left\langle f\middle|g\cdot
f^{\prime}\right\rangle=\int_{H_{3}^{+}}dh\
\overline{f(h)}f^{\prime}(g^{\dagger}hg)=\int_{H_{3}^{+}}dh\
\overline{f((g^{-1})^{\dagger}hg^{-1})}f^{\prime}(h)=\left\langle g^{-1}\cdot
f\middle|f^{\prime}\right\rangle\ .$ (4.3.31)

Let us interpret this at the level of the symmetry algebra. The Lie algebra
$\mathfrak{sl}_{2}({\mathbb{C}})$ of the symmetry group $SL_{2}({\mathbb{C}})$
can be viewed as a six-dimensional real space, whose generators are related to
$J^{a}_{0},\bar{J}^{a}_{0}$ by the ${\mathbb{R}}$-linear map

$\displaystyle\left\\{\begin{array}[]{lcl}t^{a}&\mapsto&J_{0}^{a}+\bar{J}_{0}^{a}\
,\\\ it^{a}&\mapsto&i(J^{a}_{0}-\bar{J}^{a}_{0})\ .\end{array}\right.$
(4.3.34)

(The minus sign in the image of $it^{a}$ comes from the complex conjugation of
the elements of $g$ in $U_{g}\Phi^{j}_{x,\bar{x}}$ (4.3.26).) Our scalar
product is such that $t^{a}$ and $it^{a}$ are antihermitian, which is
equivalent to

$\displaystyle(J^{a}_{0})^{\dagger}=-\bar{J}^{a}_{0}\ .$ (4.3.35)

In the large level limit, the OPE of the fields $\Phi^{j}_{x,\bar{x}}$ must
remain associative and commutative. The product of functions in
$\mathcal{F}(H_{3}^{+})$ has these properties, which suggests that it
coincides with the large level limit of the OPE – given how stringent the
condition of associativity is, we do not expect alternative products to exist.
Then the large level limit of the spectrum is the subspace of
$\mathcal{F}(H_{3}^{+})$ which is spanned by the functions
$\Phi^{j}_{x,\bar{x}}$. The OPE (4.3.20) implies that this subspace is closed
under multiplication, which suggests that the subspace in question is
$\mathcal{F}(H_{3}^{+})$ itself. Calling $C^{j}$ the principal series
representation of $\mathfrak{sl}_{2}({\mathbb{C}})$ of spin
$j\in-\frac{1}{2}+i{\mathbb{R}}$, which is defined by the action of
$\mathfrak{sl}_{2}({\mathbb{C}})$ on $\Phi^{j}_{x,\bar{x}}$, we obtain

$\displaystyle\mathcal{F}(H_{3}^{+})=\bigoplus_{j\in-\frac{1}{2}+i{\mathbb{R}}_{+}}C^{j}\
.$ (4.3.36)

(For a more rigorous definition of $\mathcal{F}(H_{3}^{+})$, and proof of its
decomposition into representations of $\mathfrak{sl}_{2}({\mathbb{C}})$, see
[23].) Furthermore, the identification of the limit of the OPE with the
product of functions suggests

$\displaystyle\underset{k\to\infty}{\lim}\left\langle\prod_{i=1}^{N}\Phi^{j_{i}}_{x_{i},\bar{x}_{i}}(z_{i},\bar{z}_{i})\right\rangle\propto\int_{H_{3}^{+}}dh\
\prod_{i=1}^{N}\Phi^{j_{i}}_{x_{i},\bar{x}_{i}}(h)\ ,$ (4.3.37)

where the unknown proportionality factor is an $x,\bar{x}$-independent field
normalization. Both sides of this equation can be computed explicitly in the
case $N=3$, and are found to agree [23]. Notice that the large level limit of
the $H_{3}^{+}$ $N$-point function is formally identical to the light
asymptotic limit of the Liouville $N$-point function (4.1.54), whose
interpretation is however quite different as it depends on positions
$z,\bar{z}$ instead of isospin variables $x,\bar{x}$. So the large level limit
of our two-dimensional conformal field theory is the quantum mechanics of a
point particle on the space $H_{3}^{+}$, which justifies naming the theory the
$H_{3}^{+}$ model. Other names for the same theory include the $H_{3}^{+}$ WZW
model and the $SL_{2}({\mathbb{C}})/SU_{2}$ WZW model; the latter name comes
from the realization of the space $H_{3}^{+}$ as a coset.

We can now investigate the unitarity of the $H_{3}^{+}$ model, using the
conjugation rule (4.3.35) for the horizontal subalgebra, which we now extend
to the entire affine Lie algebra as

$\displaystyle\boxed{(J^{a}_{n})^{\dagger}=-\bar{J}^{a}_{-n}}\ .$ (4.3.38)

This is compatible with the structure (4.2.14) of the affine Lie algebra
$\hat{\mathfrak{sl}}_{2}$ provided

$\displaystyle k\in\mathbb{R}\ .$ (4.3.39)

Via the Sugawara construction (4.2.16), the conjugation rule for $J^{a}_{n}$
implies $L_{n}^{\dagger}=\bar{L}_{-n}$, which differs from the conjugation
rule (1.4.3) which we previously assumed. We refrain from discussing how this
affects the geometrical interpretation of conformal transformations. Let us
rather investigate the unitarity of the theory. Given a primary state
$|v\rangle$, the norm square of some of the level one descendents of
$|v\rangle$ is

$\displaystyle\left\langle(J^{0}_{-1}+\lambda\bar{J}^{0}_{-1})v\middle|(J^{0}_{-1}+\lambda\bar{J}^{0}_{-1})v\right\rangle=-\left\langle
v\middle|(\bar{\lambda}J^{0}_{1}+\bar{J}^{0}_{1})(J^{0}_{-1}+\lambda\bar{J}^{0}_{-1})v\right\rangle=-\frac{k}{2}(\lambda+\bar{\lambda})\langle
v|v\rangle\ .$ (4.3.40)

This cannot be positive for all values of $\lambda$, so the theory cannot be
unitary – except of course in the large level limit, where descendent modes
disappear.

###  4.4 WZW models

####  4.4.1 Definition and general properties

Given a simple Lie group $G$, the $G$ Wess–Zumino–Witten model or $G$ WZW
model is usually defined by a Lagrangian, which depends on a parameter $k$.
This model can then be shown to be a conformal field theory with a
$\hat{\mathfrak{g}}$ symmetry algebra, where $\mathfrak{g}$ is the Lie algebra
of $G$. The parameter $k$ coincides with the level of $\hat{\mathfrak{g}}$,
and may have to be quantized for the model to be consistent, depending on the
group $G$. In any case, the allowed values of $k$ accumulate near $k=\infty$.

We will of course not use the Lagrangian definition of WZW models, which
raises the question of characterizing these models in the conformal bootstrap
approach. The fundamental axiom is the presence of the symmetry algebra
$\hat{\mathfrak{g}}$, and some authors call all models with this symmetry
algebra WZW models. Here we will insist that, among models with this symmetry,
a WZW model can be unambiguously associated to a Lie group $G$, such that its
spectrum $S$ obeys

$\displaystyle\boxed{\underset{k\to\infty}{\lim}S=\mathcal{F}(G)}\ ,$ (4.4.1)

where $\mathcal{F}(G)$ is the space of functions on $G$. This property brings
the group $G$, and not just the Lie algebra $\mathfrak{g}$, into the
definition of a WZW model. This property still does not fully characterize WZW
models: in particular, nothing forces the level $k$ to be quantized whenever
the Lagrangian definition dictates it. It is plausible that the spectrum of
WZW models can be characterized in terms of functions on a manifold related to
the loop group of $G$, but such a characterization is not known. Instead of a
proper definition of WZW models, we will limit ourselves to giving a few known
properties of the spectrum of the $G$ WZW model:

  1. 1.

If $G$ is compact, then the $G$ WZW model is rational.

  2. 2.

If $G$ is compact, then the level $k$ takes positive integer values.

  3. 3.

The $G$ WZW model is diagonal if and only if $G$ is simply connected.

The relation between the simple connectedness of $G$ and the diagonality of
the associated model is already manifest in the case of abelian groups. The
free bosonic theory whose spectrum was given in eq. (4.1.17) (with $Q=0$) may
be called the ${\mathbb{R}}$ WZW model, and it is diagonal. The compactified
free boson, whose spectrum was given in eq. (4.1.21), may be called the $U(1)$
WZW model, and it is not diagonal.

Let us study the consequences of the axiom (4.4.1). The space $\mathcal{F}(G)$
of functions on $G$ has a natural action of $G\times\bar{G}$, where the bar is
here for distinguishing the two copies of $G$, such that for
$f\in\mathcal{F}(G)$ we have

$\displaystyle\left((g,\bar{g})\cdot f\right)(h)=f(g^{-1}h\bar{g})\ .$ (4.4.2)

We identify the corresponding infinitesimal symmetry algebra
$\mathfrak{g}\times\bar{\mathfrak{g}}$ with the large level limit of the
symmetry algebra $\hat{\mathfrak{g}}\times\bar{\hat{\mathfrak{g}}}$ of the WZW
model. That is, $\mathfrak{g}$ and its generators $t^{a}$ are identified with
the horizontal subalgebra of $\hat{\mathfrak{g}}$ and its generators
$J^{a}_{0}$. The natural scalar product on $\mathcal{F}(G)$ is

$\displaystyle\langle f|f^{\prime}\rangle=\int_{G}dh\
\overline{f(h)}f^{\prime}(h)\ ,$ (4.4.3)

where $dh$ is the Haar measure, which is invariant under the left and right
actions of $G$ on itself. As in the case of the $H_{3}^{+}$ model, the
generators of the symmetry algebra are antihermitian for this scalar product,
which now implies

$\displaystyle(J^{a}_{0})^{\dagger}=-J^{a}_{0}\ .$ (4.4.4)

This conjugation rule is naturally extended to the following conjugation rule
on the affine Lie algebra,

$\displaystyle\boxed{(J^{a}_{n})^{\dagger}=-J^{a}_{-n}}\ .$ (4.4.5)

This is compatible with the commutation relations (4.2.14) of the affine Lie
algebra $\hat{\mathfrak{g}}$, provided the structure constants $f^{ab}_{c}$
and level $k$ are real. This is also compatible with the conjugation rule
(1.4.3) for the generators of the Virasoro algebra, via the Sugawara
construction (4.2.16).

If $G$ is compact, then the space $\mathcal{F}(G)$ of functions on $G$ can be
decomposed into irreducible representations of the symmetry group
$G\times\bar{G}$ using the Peter-Weyl theorem,

$\displaystyle\mathcal{F}(G)=\bigoplus_{R\in\mathcal{R}}R\otimes\bar{R}\ ,$
(4.4.6)

where $\mathcal{R}$ is the set of irreducible unitary representations of $G$,
which coincides with the set of irreducible finite-dimensional representations
of $G$. Since the scalar product (4.4.3) is positive definite, it is obvious
that only unitary representations can appear. The nontrivial statement is that
all unitary irreducible representations do appear, and that they couple
diagonally. If $G$ is simply connected, then the spectrum of the WZW model is
still diagonal for all positive integer values of $k$, [4]

$\displaystyle S=\bigoplus_{R\in\mathcal{R}_{k}}\hat{R}\otimes\bar{\hat{R}}\
,$ (4.4.7)

where $\hat{R}$ is the affine highest-weight representation of
$\hat{\mathfrak{g}}$ which is built from $R$ by acting with the creation modes
and removing the null vectors, and the finite subset $\mathcal{R}_{k}$ of
$\mathcal{R}$ is defined by certain $k$-dependent conditions. The resulting
representations $\hat{R}$ are called the integrable highest-weight
representations of $\hat{\mathfrak{g}}$. We have
$\underset{k\to\infty}{\lim}\mathcal{R}_{k}=\mathcal{R}$ and
$\underset{k\to\infty}{\lim}\hat{R}=R$, so that the spectrum $S$ has the
desired large level limit (4.4.1).

These results on the spectrum of the $G$ WZW model when $G$ is compact can be
derived by looking for rational models, whose spectrums are be made of
multiply degenerate representations – the logic we followed with Virasoro
minimal models. We will now do this in the case $G=SU_{2}$.

####  4.4.2 The $SU_{2}$ WZW model

We want to build theories with the $\widehat{\mathfrak{sl}}_{2}$ symmetry
algebra, whose spectrums are discrete or even rational. These spectrums must
therefore be made of degenerate representations. Rather than studying the
structure and null vectors of the representations of
$\widehat{\mathfrak{sl}}_{2}$, as we did in Section 2.1.2 in the case of the
Virasoro algebra, we will use a shortcut. Remembering that we recovered the
Virasoro degenerate fields and their fusion rules from the Liouville OPE in
Section 3.1.4, we will use the $H_{3}^{+}$ OPE for studying
$\widehat{\mathfrak{sl}}_{2}$ degenerate fields.

According to our axiom (4.4.1), and to the Peter-Weyl theorem, the fields
which are relevant for the $SU_{2}$ WZW model correspond to the finite-
dimensional representations of $SU_{2}$. We will admit the well-known result
that such representations have spins
$j\in\\{0,\frac{1}{2},1,\frac{3}{2}\cdots\\}$ and dimensions $2j+1$. We will
soon see that the corresponding fields are degenerate, in the sense that their
OPEs with other fields involve finitely many primary fields. Experience with
Virasoro minimal models however suggests that we should consider multiply
degenerate fields, and it will turn out that we also need fields with spins
$j\in\frac{1}{2}{\mathbb{N}}+\frac{1}{2}b^{2}=\frac{1}{2}{\mathbb{N}}-1-\frac{k}{2}$,
which will also turn out to be degenerate. Let us introduce notations for the
limits of $H_{3}^{+}$ field when the spin takes such particular values,

$\displaystyle
J\in\frac{1}{2}{\mathbb{N}}\quad\Rightarrow\quad\left\\{\begin{array}[]{ccl}\Phi^{J}&=&\underset{j\to
J}{\lim}\ \Phi^{j}\ ,\\\ \Phi^{(J,1)}&=&\underset{j\to
J-1-\frac{k}{2}}{\lim}\Phi^{j}\ .\end{array}\right.$ (4.4.10)

Let us now consider the OPE
$\Phi^{j_{1}}_{x_{1},\bar{x}_{1}}\Phi^{j_{2}}_{x_{2},\bar{x}_{2}}$ (4.3.20)
where we initially assume $j_{1},j_{2}\in-\frac{1}{2}+ib{\mathbb{R}}$. The
positions of the line of integration $j\in-\frac{1}{2}+ib{\mathbb{R}}$ and the
eight cones of poles of the OPE coefficient are

(4.4.11)

Taking the limit $j\to J$ or $j\to J-1-\frac{k}{2}$, the OPE coefficient
vanishes, and the only surviving contributions are from poles that cross the
line of integration. (See Section 3.1.5 for more explanations.) The resulting
OPEs are schematically written as

$\displaystyle\Phi^{J}\Phi^{j_{2}}$
$\displaystyle\sim\sum_{j=j_{2}-J}^{j_{2}+J}\Phi^{j}\ ,$ (4.4.12)
$\displaystyle\Phi^{(J,1)}\Phi^{j_{2}}$
$\displaystyle\sim\sum_{j=j_{2}-1-\frac{k}{2}-J}^{j_{2}-1-\frac{k}{2}+J}\Phi^{j}+\sum_{j=j_{2}+1+\frac{k}{2}-J}^{j_{2}+1+\frac{k}{2}+J}\Phi^{j}\
,$ (4.4.13)

where the sums run by increments of $1$, and we have taken the reflection
relation $\Phi^{j}\propto\Phi^{-j-1}$ into account.

For generic values of the level $k$, the OPE of two fields with half-integer
spins is therefore

$\displaystyle\boxed{\Phi^{J_{1}}\Phi^{J_{2}}\sim\sum_{J=|J_{1}-J_{2}|}^{J_{1}+J_{2}}\Phi^{J}}\
.$ (4.4.14)

This suggests that there exists a generalized $SU_{2}$ WZW model, whose
spectrum is

$\displaystyle\boxed{S=\bigoplus_{J=0,\frac{1}{2},1,\frac{3}{2},\cdots}\hat{R}_{J}\otimes\bar{\hat{R}}_{J}}\
,$ (4.4.15)

where $\hat{R}_{J}$ is the affine highest-weight representation of
$\widehat{\mathfrak{sl}}_{2}$ built from the finite-dimensional representation
of $\mathfrak{sl}_{2}$ of spin $J$. The two-point function of the generalized
$SU_{2}$ WZW model is obtained from the corresponding $H_{3}^{+}$ two-point
function,

$\displaystyle\left\langle\Phi^{J_{1}}_{x_{1},\bar{x}_{1}}\Phi^{J_{2}}_{x_{2},\bar{x}_{2}}\right\rangle=\frac{\pi}{\sqrt{-k-2}}\mu^{\frac{2J_{1}+1}{\sqrt{-k-2}}}\gamma(\tfrac{2J_{1}+1}{k+2})\,\delta_{J_{1},J_{2}}|x_{12}|^{4J_{1}}\,.$
(4.4.16)

In complete analogy with the case of the generalized minimal models, the
three-point function of a generalized $SU_{2}$ WZW model is obtained from the
three-point function (4.3.16) of the $H_{3}^{+}$ model by sending the spins to
their half-integer values, and replacing the (vanishing) $\Upsilon_{b}$
functions with their derivatives.

Let us now look for rational WZW models, by considering doubly degenerate
fields. We restrict our attention to the case when $\Phi^{J}$ and
$\Phi^{(J^{\prime},1)}$ are related by reflection, so that
$J+J^{\prime}=\frac{k}{2}$. This restricts the level $k$ to integer values,

$\displaystyle\boxed{k\in{\mathbb{N}}}\ ,$ (4.4.17)

and the spin $J$ to the finite set

$\displaystyle J\in\left\\{0,\frac{1}{2},\cdots\frac{k}{2}\right\\}\ .$
(4.4.18)

The OPE of the doubly degenerate field
$\Phi^{J}\propto\Phi^{(\frac{k}{2}-J,1)}$ with $\Phi^{j_{2}}$ is constrained
by both equations (4.4.12) and (4.4.13). Taking the reflection relation
$\Phi^{j}\propto\Phi^{-j-1}$ into account, we find that the OPE is
nonvanishing only if $j_{2}$ itself belongs to the set (4.4.18). In this case,
we have

$\displaystyle\boxed{\Phi^{J_{1}}\Phi^{J_{2}}=\sum_{J=|J_{1}-J_{2}|}^{\operatorname{min}(J_{1}+J_{2},k-J_{1}-J_{2})}\Phi^{J}}\
.$ (4.4.19)

This suggests that the set of our doubly degenerate fields is closed under
fusion, and that there exists a rational model whose spectrum is

$\displaystyle\boxed{S=\bigoplus_{J=0,\frac{1}{2},\cdots\frac{k}{2}}\hat{R}_{J}\otimes\bar{\hat{R}}_{J}}\
.$ (4.4.20)

This model, which exists for any positive integer value of $k$, is the
$SU_{2}$ WZW model. The two- and three-point functions of this model are
special cases of corresponding correlation functions of the generalized
$SU_{2}$ WZW model.

The space of models with the $\widehat{\mathfrak{sl}}_{2}$ symmetry algebra is
therefore similar to the space of models with Virasoro symmetry algebra,
according to the following table:

(4.4.25)

In addition, for both symmetry algebras, there exist rational, non-diagonal
models, which in both cases fit in an A-D-E classification [4]. Non-diagonal
models with $\widehat{\mathfrak{sl}}_{2}$ symmetry include the $SO_{3}$ WZW
models, where $SO_{3}=\frac{SU_{2}}{\mathbb{Z}_{2}}$ is not simply connected.

####  4.4.3 The $\widetilde{SL}_{2}(\mathbb{R})$ WZW model

After the $H_{3}^{+}$ model, the $\widetilde{SL}_{2}(\mathbb{R})$ WZW model is
our second example of a non-rational model with an
$\widehat{\mathfrak{sl}}_{2}$ symmetry algebra. Work on the
$\widetilde{SL}_{2}(\mathbb{R})$ WZW model has been motivated by its relevance
to string theory in $AdS_{3}$. The model has not been fully solved: the three-
point function is known only partially, and crossing symmetry has not been
proved. We will limit ourselves to working out the fusion rules, and deriving
Maldacena and Ooguri’s widely believed and well-tested conjecture for the
spectrum of the model. This spectrum is more complicated than the spectrum of
the $H_{3}^{+}$ model, which is why solving the
$\widetilde{SL}_{2}(\mathbb{R})$ WZW model is more difficult.

The Lie group $\widetilde{SL}_{2}(\mathbb{R})$ is defined as the universal
covering group of the group $SL_{2}({\mathbb{R}})$ of matrices of size two
with real coefficients and determinant one. The group $SL_{2}({\mathbb{R}})$
is not simply-connected, as the matrices of the type
$\left(\begin{smallmatrix}\cos\tau&\sin\tau\\\
-\sin\tau&\cos\tau\end{smallmatrix}\right)$ form a non-contractible loop, and
the first homotopy group of $SL_{2}({\mathbb{R}})$ is ${\mathbb{Z}}$. So
$\widetilde{SL}_{2}(\mathbb{R})$ is obtained from $SL_{2}({\mathbb{R}})$ by
decompactifying the $\tau$ direction, and we have
$SL_{2}({\mathbb{R}})=\widetilde{SL}_{2}(\mathbb{R})/{\mathbb{Z}}$.

Let us study the space $\mathcal{F}(\widetilde{SL}_{2}(\mathbb{R}))$ of
functions on $\widetilde{SL}_{2}(\mathbb{R})$, which is assumed to be the
large level limit of the spectrum of our model. In this limit, a basis of
symmetry generators is $(J^{a}_{0})$, and the elements of this basis are
antihermitian for the natural scalar product (4.4.3). We will however work
with a basis that includes a hermitian generator, whose eigenvalues are
therefore real in unitary representations. We still call this basis
$(J^{a}_{0})$, although it is obtained from the original basis by a complex
change of bases, such that the conjugation rule becomes

$\displaystyle\left\\{\begin{array}[]{l}(J_{0}^{0})^{\dagger}=J_{0}^{0}\ ,\\\
(J_{0}^{\pm})^{\dagger}=-J_{0}^{\mp}\ .\end{array}\right.$ (4.4.28)

In the matrix representation (2.3.13) of $\mathfrak{sl}_{2}$, this corresponds
to the $\mathfrak{su}_{1,1}$ conjugation rule
$M^{\dagger}=\left(\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}\right)\bar{M}^{T}\left(\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}\right)$. (The matrices which satisfy $M^{\dagger}=-M$
for the $\mathfrak{su}_{1,1}$ conjugation rule are by definition the elements
of $\mathfrak{su}_{1,1}$ itself).

The decomposition of $\mathcal{F}(\widetilde{SL}_{2}(\mathbb{R}))$ into
irreducible unitary representations of the global symmetry algebra
$\mathfrak{sl}_{2}\times\overline{\mathfrak{sl}}_{2}$ involves two types of
representations: the principal series representation of $\mathfrak{sl}_{2}$
$C^{j}_{\alpha}$ and the discrete series representation of $\mathfrak{sl}_{2}$
$D^{j,\pm}$, where $j$ is the spin. These representations can be characterized
by the eigenvalues of the hermitian generator $J_{0}^{0}$,

(4.4.33)

Given a representation $R$, let $R^{*}$ be the dual representation, that is
the representation with opposite $J_{0}^{0}$ eigenvalues:

$\displaystyle(C^{j}_{\alpha})^{*}$ $\displaystyle=C^{j}_{-\alpha}\ ,$
(4.4.34) $\displaystyle(D^{j,\pm})^{*}$ $\displaystyle=D^{j,\mp}\ ,$ (4.4.35)

So, according to [18] and references therein, we have

$\displaystyle\mathcal{F}(\widetilde{SL}_{2}(\mathbb{R}))$
$\displaystyle=\int^{\oplus}_{-\frac{1}{2}+i{\mathbb{R}}_{+}}dj\int^{\oplus}_{]0,1[}d\alpha\
C^{j}_{\alpha}\otimes\bar{C}^{j}_{\alpha}\oplus\bigoplus_{\pm}\int^{\oplus}_{]-\frac{1}{2},\infty[}dj\
D^{j,\pm}\otimes\bar{D}^{j,\pm}\ ,$ (4.4.36)
$\displaystyle\mathcal{F}(SL_{2}(\mathbb{R}))$
$\displaystyle=\int^{\oplus}_{-\frac{1}{2}+i{\mathbb{R}}_{+}}dj\bigoplus_{\alpha\in\\{0,\frac{1}{2}\\}}C^{j}_{\alpha}\otimes\bar{C}^{j}_{\alpha}\oplus\bigoplus_{\pm}\bigoplus_{j=-\frac{1}{2},0,\frac{1}{2}\cdots}D^{j,\pm}\otimes\bar{D}^{j,\pm}\
,$ (4.4.37)

where the identification of $\mathcal{F}(SL_{2}(\mathbb{R}))$ with the space
of $\tau$-periodic functions on $\widetilde{SL}_{2}(\mathbb{R})$ amounts to
restricting the eigenvalues of our hermitian generator to be half-integers.
Tensor products of principal and discrete series representations can be
written as

$\displaystyle R_{1}\otimes
R_{2}=\bigoplus_{R_{3}}N_{R_{1},R_{2},R_{3}^{*}}R_{3}\ ,$ (4.4.38)

where $N_{R_{1},R_{2},R_{3}}$ is an integer-valued function, which obeys
$N_{R_{1}^{*},R_{2}^{*},R_{3}^{*}}=N_{R_{1},R_{2},R_{3}}$ and is symmetric
under permutations. Modulo these symmetries, the only nonzero values of
$N_{R_{1},R_{2},R_{3}}$ are

$\displaystyle N_{D^{j_{1},+},D^{j_{2},+},D^{j_{3},-}}$
$\displaystyle=1\quad\text{if}\ j_{3}\in j_{1}+j_{2}+1+{\mathbb{N}}\ ,$
(4.4.39) $\displaystyle N_{C^{j_{1}}_{\alpha_{1}},D^{j_{2},+},D^{j_{3},-}}$
$\displaystyle=1\quad\text{if}\ \alpha_{1}+j_{2}-j_{3}\in{\mathbb{Z}}\ ,$
(4.4.40) $\displaystyle
N_{C^{j_{1}}_{\alpha_{1}},C^{j_{2}}_{\alpha_{2}},D^{j_{3},+}}$
$\displaystyle=1\quad\text{if}\ \alpha_{1}+\alpha_{2}+j_{3}\in{\mathbb{Z}}\ ,$
(4.4.41) $\displaystyle
N_{C^{j_{1}}_{\alpha_{1}},C^{j_{2}}_{\alpha_{2}},C^{j_{3}}_{\alpha_{3}}}$
$\displaystyle=2\quad\text{if}\
\alpha_{1}+\alpha_{2}+\alpha_{3}\in{\mathbb{Z}}\ .$ (4.4.42)

For example, the tensor product of two principal series representations is

$\displaystyle C^{j_{1}}_{\alpha_{1}}\otimes C^{j_{2}}_{\alpha_{2}}$
$\displaystyle=2\int^{\oplus}_{-\frac{1}{2}+i{\mathbb{R}}_{+}}dj\
C^{j}_{\alpha_{1}+\alpha_{2}}\oplus\bigoplus_{\pm}\bigoplus_{\begin{subarray}{c}j\in\pm\alpha_{1}\pm\alpha_{2}+{\mathbb{Z}}\\\
j\in]-\frac{1}{2},\infty[\end{subarray}}D^{j,\pm}\ ,$ (4.4.43)

where the factor of $2$ is the tensor product multiplicity which we already
encountered in Section 4.2.3. Actually, all the values of
$N_{R_{1},R_{2},R_{3}}$ can be deduced from the values of
$N_{D^{j_{1},+},D^{j_{2},+},D^{j_{3},-}}$, using the remark that as far as the
eigenvalues (4.4.33) of $J^{0}_{0}$ are concerned, we have the identification

$\displaystyle C^{j}_{\alpha}\sim D^{\alpha-1,+}\oplus D^{-\alpha,-}\ .$
(4.4.44)

This identification apparently predicts a nonzero value for
$N_{C^{j}_{\alpha},D^{j_{2},+},D^{j_{3},+}}$. That value should be discarded,
because it depends on $\alpha$ whereas $C^{j}_{\alpha}$ only depends on
$\alpha\ \text{mod}\ {\mathbb{Z}}$.

After these reminders on $\widetilde{SL}_{2}(\mathbb{R})$ and its
representations, we are ready to consider the associated WZW model. The
natural extension of the conjugation rule (4.4.28) to the symmetry algebra
$\widehat{\mathfrak{sl}}_{2}$ is

$\displaystyle\left\\{\begin{array}[]{l}(J_{n}^{0})^{\dagger}=J_{-n}^{0}\ ,\\\
(J_{n}^{\pm})^{\dagger}=-J_{-n}^{\mp}\ ,\end{array}\right.$ (4.4.47)

which is compatible with the commutation relations (4.2.14) provided the level
$k$ is real. The principal and discrete series representations of
$\mathfrak{sl}_{2}$ are naturally extended to affine highest-weight
representations of $\widehat{\mathfrak{sl}}_{2}$: the principal series
representations $\hat{C}^{j}_{\alpha}$ and discrete series representations
$\hat{D}^{j,\pm}$. By computing the norms of level one states, it is easy to
see that such representations are not unitary. But the
$\widetilde{SL}_{2}(\mathbb{R})$ WZW model is not expected to be unitary, as
the metric on the underlying group has a mixed signature. (What matters for
applications to string theory is not unitarity, but another property called
the no-ghost theorem.)

So, can we build the spectrum of the model from such affine highest-weight
representations? The answer turns out to be no, as this would be incompatible
with an algebraic feature of $\widehat{\mathfrak{sl}}_{2}$ called the spectral
flow [24]. The spectral flow is a family $(\rho_{w})_{w\in{\mathbb{Z}}}$ of
automorphisms of $\widehat{\mathfrak{sl}}_{2}$, which obey
$\rho_{w}\rho_{w^{\prime}}=\rho_{w+w^{\prime}}$ and act as

$\displaystyle\rho_{w}(J^{0}_{n})$
$\displaystyle=J^{0}_{n}+\frac{1}{2}kw\delta_{n,0}\ ,$ (4.4.48)
$\displaystyle\rho_{w}(J^{\pm}_{n})$ $\displaystyle=J^{\pm}_{n\pm w}\ .$
(4.4.49)

According to eqs. (4.2.16) and (4.2.17), this implies

$\displaystyle\rho_{w}(L_{n})=L_{n}+wJ^{0}_{n}+\frac{1}{4}kw^{2}\delta_{n,0}\
.$ (4.4.50)

Given a representation $R$ of $\widehat{\mathfrak{sl}}_{2}$, that is an action
of the generators $J^{a}_{n}$ on some vectors $|v\rangle$, we define the
spectrally flowed representation $\rho_{w}(R)$ by the action
$\rho_{-w}(J^{a}_{n})|v\rangle$. It follows from this definition that

$\displaystyle\rho_{w}(R)^{*}=\rho_{-w}(R^{*})\ .$ (4.4.51)

Moreover, it is believed that the action of spectral flow commutes with
fusion, in the sense that [25]

$\displaystyle\rho_{w}(R)\times\rho_{w^{\prime}}(R^{\prime})=\rho_{w+w^{\prime}}(R\times
R^{\prime})\ .$ (4.4.52)

We assume that fusion products of representations of
$\widehat{\mathfrak{sl}}_{2}$ have the form

$\displaystyle R_{1}\times
R_{2}=\bigoplus_{R_{3}}N_{R_{1},R_{2},R_{3}^{*}}R_{3}\ ,$ (4.4.53)

where the $N_{R_{1},R_{2},R_{3}}$ are a permutation-symmetric, integer-valued
fusion multiplicities such that
$N_{R_{1}^{*},R_{2}^{*},R_{3}^{*}}=N_{R_{1},R_{2},R_{3}}$. From eq. (4.4.52),
we must then have

$\displaystyle w_{1}+w_{2}+w_{3}=0\quad\Rightarrow\quad
N_{\rho_{w_{1}}(R_{1}),\rho_{w_{2}}(R_{2}),\rho_{w_{3}}(R_{3})}=N_{R_{1},R_{2},R_{3}}\
.$ (4.4.54)

Let us consider the action of spectral flow on our affine highest-weight
representations. We introduce the notations

$\displaystyle\hat{C}^{j,w}_{\alpha}$
$\displaystyle=\rho_{w}(\hat{C}^{j}_{\alpha})\quad,\quad(w\in{\mathbb{Z}})\ ,$
(4.4.55) $\displaystyle\hat{D}^{j,w}$
$\displaystyle=\rho_{w-\frac{1}{2}}(\hat{D}^{j,+})\quad,\quad(w\in\tfrac{1}{2}+{\mathbb{Z}})\
.$ (4.4.56)

If $w\neq 0$, then $\hat{C}^{j,w}_{\alpha}$ cannot be an affine highest-weight
representation, because the eigenvalues of
$\rho_{-w}(L_{0})=L_{0}-wJ^{0}_{0}+\frac{1}{4}kw^{2}$ in
$\hat{C}^{j}_{\alpha}$ are not bounded from below – and actually, the
representations $(\hat{C}^{j,w}_{\alpha})_{w\in{\mathbb{Z}}}$ all differ from
one another. Let us now focus on discrete series representations. The
representation $\hat{D}^{j,\pm}$ can be characterized by the existence of a
state $|v^{j,\pm}\rangle$ such that

$\displaystyle J^{\mp}_{n\geq
0}|v^{j,\pm}\rangle=J^{0}_{n>0}|v^{j,\pm}\rangle=J^{\pm}_{n>0}|v^{j,\pm}\rangle=(J^{0}_{0}\mp(j+1))|v^{j,\pm}\rangle=0\
.$ (4.4.57)

So we can characterize $\rho_{w}(\hat{D}^{j,\pm})$ by the action of
$\rho_{-w}(J^{a}_{n})$ on $|v^{j,\pm}\rangle$. In particular, we notice that
$\rho_{1}(J^{a}_{n})|v^{j,+}\rangle=0\ \Leftrightarrow\
J^{a}_{n}|v^{-j-2-\frac{k}{2},-}\rangle=0$, which leads to

$\displaystyle\rho_{-1}(\hat{D}^{j,+})=\hat{D}^{-j-2-\frac{k}{2},-}\ .$
(4.4.58)

So the spectral flow orbit of $\hat{D}^{j,+}$ contains two affine highest-
weight representations, namely $\hat{D}^{j,\frac{1}{2}}=\hat{D}^{j,+}$ itself
and $\hat{D}^{j,-\frac{1}{2}}=\hat{D}^{-j-2-\frac{k}{2},-}$. The rest of the
orbit is made of representations where the eigenvalues of $L_{0}$ are not
bounded from below. And the dual representations of our spectrally flowed
representations are

$\displaystyle(\hat{C}^{j,w}_{\alpha})^{*}$
$\displaystyle=\hat{C}^{j,-w}_{-\alpha}\ ,$ (4.4.59)
$\displaystyle(\hat{D}^{j,w})^{*}$
$\displaystyle=\hat{D}^{-j-2-\frac{k}{2},-w}\ .$ (4.4.60)

This concludes our discussion of spectral flow. We will now be able to argue
that spectrally flowed representations must appear in the spectrum.

By our definition of WZW models, the large level limit of the spectrum
$\tilde{S}$ of the $\widetilde{SL}_{2}(\mathbb{R})$ WZW model is
$\mathcal{F}(\widetilde{SL}_{2}(\mathbb{R}))$ (4.4.36), and we therefore
expect $\tilde{S}$ to contain affine discrete representations of both series
$\hat{D}^{j,\pm}$. Let us show that $\tilde{S}$ cannot contain only affine
highest-weight representations. Consider spins $j_{1},j_{2},j_{3}$ such that
$N_{D^{j_{1},+},D^{j_{2},-},D^{j_{3},+}}=1$. If the level $k$ is large enough,
we must then have
$N_{\hat{D}^{j_{1},\frac{1}{2}},\hat{D}^{j_{2},-\frac{1}{2}},\hat{D}^{j_{3},\frac{1}{2}}}=1$.
Using the behaviour (4.4.54) of fusion multiplicities under spectral flow,
this implies
$N_{\hat{D}^{j_{1},-\frac{1}{2}},\hat{D}^{j_{2},-\frac{1}{2}},\hat{D}^{j_{3},\frac{3}{2}}}=1$.
So the fusion product
$\hat{D}^{j_{1},-\frac{1}{2}}\times\hat{D}^{j_{2},-\frac{1}{2}}$ contains the
representation $(\hat{D}^{j_{3},\frac{3}{2}})^{*}$, which is not an affine
highest-weight representation. Generalizing this argument, the spectrum must
in fact contain representations of the type $\hat{D}^{j,w}$ for all values of
$w\in\frac{1}{2}+{\mathbb{Z}}$. Then it is natural to assume that the spectral
flow actually leaves the spectrum invariant. But what are the allowed values
of the spin $j$ of $\hat{D}^{j,w}$? We still impose the constraint
$j>-\frac{1}{2}$ which comes from the representation $D^{j,\pm}$ of
$\mathfrak{sl}_{2}$. If this applies to both spins in the relation (4.4.58),
we must then have

$\displaystyle-\frac{1}{2}<j<-\frac{k+3}{2}\ .$ (4.4.61)

This defines a non-empty interval provided

$\displaystyle\boxed{k\in]-\infty,-2[}\ .$ (4.4.62)

(Nevertheless, the model surely exists for $k\in\mathbb{C}-\\{2\\}$.) The
natural conjectures for the spectrums of the $\widetilde{SL}_{2}(\mathbb{R})$
and $SL_{2}(\mathbb{R})$ WZW models are then

$\displaystyle\tilde{S}$
$\displaystyle=\bigoplus_{w\in{\mathbb{Z}}}\int^{\oplus}_{-\frac{1}{2}+i{\mathbb{R}}_{+}}dj\int^{\oplus}_{]0,1[}d\alpha\
\hat{C}^{j,w}_{\alpha}\otimes\bar{\hat{C}}^{j,w}_{\alpha}\oplus\bigoplus_{w\in\frac{1}{2}+{\mathbb{Z}}}\int^{\oplus}_{]-\frac{1}{2},-\frac{k+3}{2}[}dj\
\hat{D}^{j,w}\otimes\bar{\hat{D}}^{j,w}\ ,$ (4.4.63) $\displaystyle S$
$\displaystyle=\bigoplus_{\begin{subarray}{c}w_{L},w_{R}\in{\mathbb{Z}}\\\
w_{L}-w_{R}\in
2{\mathbb{Z}}\end{subarray}}\int^{\oplus}_{-\frac{1}{2}+i{\mathbb{R}}_{+}}dj\bigoplus_{\alpha\in\\{0,\frac{1}{2}\\}}\hat{C}^{j,w_{L}}_{\alpha}\otimes\bar{\hat{C}}^{j,w_{R}}_{\alpha}\oplus\bigoplus_{\begin{subarray}{c}w_{L},w_{R}\in\frac{1}{2}+{\mathbb{Z}}\\\
w_{L}-w_{R}\in 2{\mathbb{Z}}\end{subarray}}\
\bigoplus_{\begin{subarray}{c}j=-\frac{1}{2},0,\frac{1}{2},\cdots\\\
j<-\frac{k+3}{2}\end{subarray}}\hat{D}^{j,w_{L}}\otimes\bar{\hat{D}}^{j,w_{R}}\
.$ (4.4.64)

Notice that the left and right spectral flow numbers $w_{L}$ and $w_{R}$ are
independent in the case of $SL_{2}(\mathbb{R})$, and equal in the case of
$\widetilde{SL}_{2}(\mathbb{R})$, so that the spectrum $\tilde{S}$ is
diagonal. The rule, which can only be heuristic in the absence of a definition
of WZW models based on the corresponding loop groups, is:

> In the $G$ WZW model the spectral flow takes values in the first homotopy
> group of the global symmetry group, that is
> $\pi_{1}(\frac{G\times\bar{G}}{Z(G)})$ where $Z(G)$ is the center of $G$.

For our WZW models, the relevant homotopy groups are
$\pi_{1}(\frac{\widetilde{SL}_{2}(\mathbb{R})\times\overline{\widetilde{SL}_{2}(\mathbb{R})}}{{\mathbb{Z}}})={\mathbb{Z}}$
and
$\pi_{1}(\frac{SL_{2}({\mathbb{R}})\times\overline{SL_{2}({\mathbb{R}})}}{{\mathbb{Z}}_{2}})=\frac{{\mathbb{Z}}\times\overline{{\mathbb{Z}}}}{{\mathbb{Z}}_{2}}$.
The rule also applies to the case $G=U(1)$ of the compactified free boson, if
we consider the winding number as a spectral flow number.

Finally, let us check that we can find fusion rules for the representations of
$\widehat{\mathfrak{sl}}_{2}$, such that

  1. 1.

the rule (4.4.54) is obeyed,

  2. 2.

in the large level limit $k\to-\infty$, the fusion rules reduce to the tensor
product rules for representations of $\mathfrak{sl}_{2}$,

  3. 3.

the conjectured spectrums of the $\widetilde{SL}_{2}(\mathbb{R})$ and
$SL_{2}(\mathbb{R})$ WZW models are closed under fusion.

We first obtain the fusion multiplicities
$N_{\hat{D}^{j_{1},\pm},\hat{D}^{j_{2},\pm},\hat{D}^{j_{3},\pm}}$ for affine
highest-weight representations of the discrete series by assuming that it can
be nonzero only when the corresponding tensor product multiplicity (4.4.39) is
nonzero, and when all spins obey the condition (4.4.61). The rest of the
nonzero fusion multiplicities of the type
$N_{\hat{D}^{j_{1},w_{1}},\hat{D}^{j_{2},w_{2}},\hat{D}^{j_{3},w_{3}}}$ are
obtained by the rule (4.4.54). Then we generalize the relation (4.4.44)
between $\mathfrak{sl}_{2}$ representations of the principal and discrete
series, and obtain

$\displaystyle\hat{C}^{j,w}_{\alpha}\sim\hat{D}^{\alpha-1,w+\frac{1}{2}}\oplus\hat{D}^{\alpha-2-\frac{k}{2},w-\frac{1}{2}}\
.$ (4.4.65)

This enables us to compute fusion multiplicities involving representations of
the type $\hat{C}^{j,w}_{\alpha}$. The only subtlety is that we obtain
aberrant terms in
$N_{\hat{C}^{j_{1},w_{1}}_{\alpha_{1}},\hat{D}^{j_{2},w_{2}},\hat{D}^{j_{3},w_{3}}}$,
which depend on $\alpha_{1}$ instead of $\alpha_{1}\ \text{mod}\
{\mathbb{Z}}$, and must be discarded. Keeping the condition (4.4.61) on spins
of discrete series representations implicit, the results are

$\displaystyle
N_{\hat{D}^{j_{1},w_{1}},\hat{D}^{j_{2},w_{2}},\hat{D}^{j_{3},w_{3}}}$
$\displaystyle=\delta_{\sum w_{i},\frac{1}{2}}\delta_{\sum
j_{i}+3+\frac{k}{2},-{\mathbb{N}}}+\delta_{\sum
w_{i},-\frac{1}{2}}\delta_{\sum j_{i}+3+k,{\mathbb{N}}}\ ,$ (4.4.66)
$\displaystyle
N_{\hat{C}^{j_{1},w_{1}}_{\alpha_{1}},\hat{D}^{j_{2},w_{2}},\hat{D}^{j_{3},w_{3}}}$
$\displaystyle=\delta_{\sum
w_{i},0}\delta_{\alpha_{1}+j_{2}+j_{3}+\frac{k}{2},{\mathbb{Z}}}\ ,$ (4.4.67)
$\displaystyle
N_{\hat{C}^{j_{1},w_{1}}_{\alpha_{1}},\hat{C}^{j_{2},w_{2}}_{\alpha_{2}},\hat{D}^{j_{3},w_{3}}}$
$\displaystyle=\delta_{\sum
w_{i},\frac{1}{2}}\delta_{\alpha_{1}+\alpha_{2}+j_{3},{\mathbb{Z}}}+\delta_{\sum
w_{i},-\frac{1}{2}}\delta_{\alpha_{1}+\alpha_{2}+j_{3}+\frac{k}{2},{\mathbb{Z}}}\
,$ (4.4.68) $\displaystyle
N_{\hat{C}^{j_{1},w_{1}}_{\alpha_{1}},\hat{C}^{j_{2},w_{2}}_{\alpha_{2}},\hat{C}^{j_{3},w_{3}}_{\alpha_{3}}}$
$\displaystyle=\delta_{\sum
w_{i},1}\delta_{\sum\alpha_{i}-\frac{k}{2},{\mathbb{Z}}}+2\,\delta_{\sum
w_{i},0}\delta_{\sum\alpha_{i},{\mathbb{Z}}}+\delta_{\sum
w_{i},-1}\delta_{\sum\alpha_{i}+\frac{k}{2},{\mathbb{Z}}}\ .$ (4.4.69)

This leads to the following fusion rules:

$\hat{D}^{j_{1},w_{1}}\times\hat{D}^{j_{2},w_{2}}=\int^{\oplus}_{-\frac{1}{2}+i{\mathbb{R}}_{+}}dj\
\hat{C}^{j,w_{1}+w_{2}}_{j_{1}+j_{2}+\frac{k}{2}}\\\
\oplus\bigoplus_{\begin{subarray}{c}j\in j_{1}+j_{2}+1+{\mathbb{N}}\\\
j\in]-\frac{1}{2},-\frac{k+3}{2}[\end{subarray}}\hat{D}^{j,w_{1}+w_{2}-\frac{1}{2}}\oplus\bigoplus_{\begin{subarray}{c}j\in
j_{1}+j_{2}+1+\frac{k}{2}-{\mathbb{N}}\\\
j\in]-\frac{1}{2},-\frac{k+3}{2}[\end{subarray}}\hat{D}^{j,w_{1}+w_{2}+\frac{1}{2}}\
,$ (4.4.70)
$\displaystyle\hat{C}^{j_{1},w_{1}}_{\alpha_{1}}\times\hat{D}^{j_{2},w_{2}}$
$\displaystyle=\int^{\oplus}_{-\frac{1}{2}+i{\mathbb{R}}_{+}}dj\left(\hat{C}^{j,w_{1}+w_{2}-\frac{1}{2}}_{\alpha_{1}+j_{2}}\oplus\hat{C}^{j,w_{1}+w_{2}+\frac{1}{2}}_{\alpha_{1}+j_{2}+\frac{k}{2}}\right)\oplus\bigoplus_{\begin{subarray}{c}j\in\alpha_{1}+j_{2}+{\mathbb{Z}}\\\
j\in]-\frac{1}{2},-\frac{k+3}{2}[\end{subarray}}\hat{D}^{j,w_{1}+w_{2}}\ ,$
(4.4.71)
$\hat{C}^{j_{1},w_{1}}_{\alpha_{1}}\times\hat{C}^{j_{2},w_{2}}_{\alpha_{2}}=\int^{\oplus}_{-\frac{1}{2}+i{\mathbb{R}}_{+}}dj\left(\hat{C}^{j,w_{1}+w_{2}-1}_{\alpha_{1}+\alpha_{2}-\frac{k}{2}}\oplus
2\,\hat{C}^{j,w_{1}+w_{2}}_{\alpha_{1}+\alpha_{2}}\oplus\hat{C}^{j,w_{1}+w_{2}+1}_{\alpha_{1}+\alpha_{2}+\frac{k}{2}}\right)\\\
\oplus\bigoplus_{\begin{subarray}{c}j\in\alpha_{1}+\alpha_{2}-\frac{k}{2}+{\mathbb{Z}}\\\
j\in]-\frac{1}{2},-\frac{k+3}{2}[\end{subarray}}\hat{D}^{j,w_{1}+w_{2}-\frac{1}{2}}\oplus\bigoplus_{\begin{subarray}{c}j\in\alpha_{1}+\alpha_{2}+{\mathbb{Z}}\\\
j\in]-\frac{1}{2},-\frac{k+3}{2}[\end{subarray}}\hat{D}^{j,w_{1}+w_{2}+\frac{1}{2}}\
.$ (4.4.72)

So, while the spectral flow number $w$ is not conserved, fusion violates it by
at most one unit. This can alternatively be shown at the level of correlation
functions by studying how spectral flow affects the Ward identities and
Knizhnik-Zamolodchikov equations [26].

###  4.5 Exercises

######  Exercise 4.1 (Normalization of OPE coefficients in free bosonic
theories)

The aim of this exercise is to show that the OPE coefficient in the OPE of
$\hat{\mathfrak{u}}_{1}$-primary fields (4.1.1) can be set to one by
renormalizing the fields. In other words, calling this coefficient
$C_{\alpha_{1},\alpha_{2}}$, we want to show that there exists a function
$\lambda(\alpha)$ such that

$\displaystyle
C_{\alpha_{1},\alpha_{2}}=\frac{\lambda(\alpha_{1}+\alpha_{2})}{\lambda(\alpha_{1})\lambda(\alpha_{2})}\
.$ (4.5.1)

To begin with, use commutativity (1.2.5) and associativity (1.2.6) of the OPE,
and show that

$\displaystyle C_{\alpha_{1},\alpha_{2}}$
$\displaystyle=C_{\alpha_{2},\alpha_{1}}\ ,$ (4.5.2) $\displaystyle
C_{\alpha_{1},\alpha_{2}}C_{\alpha_{1}+\alpha_{2},\alpha_{3}}$
$\displaystyle=C_{\alpha_{1},\alpha_{2}+\alpha_{3}}C_{\alpha_{2},\alpha_{3}}\
.$ (4.5.3)

Then consider the ansatz

$\displaystyle\lambda(\alpha)=\frac{1}{C_{\alpha,0}}\exp\int_{0}^{\alpha}\varphi\
,\quad\text{where}\quad\varphi(\alpha)$
$\displaystyle=\left.{\frac{\partial}{\partial\alpha_{2}}}\log
C_{\alpha,\alpha_{2}}\right|_{\alpha_{2}=0}\ .$ (4.5.4)

Show that $C_{\alpha,0}$ is actually an $\alpha$-independent constant, and
that the function $\varphi(\alpha)$ is such that

$\displaystyle{\frac{\partial}{\partial\alpha_{2}}}\log
C_{\alpha_{1},\alpha_{2}}=\varphi(\alpha_{1}+\alpha_{2})-\varphi(\alpha_{2})\
.$ (4.5.5)

Prove eq. (4.5.1) by showing that both sides have the same value at
$\alpha_{2}=0$, and the same logarithmic derivative wrt $\alpha_{2}$. (If the
spectrum is discrete, a similar proof can be done using finite differences
instead of derivatives.)

######  Exercise 4.2 (Sugawara construction of a Virasoro field)

Compute the $TJ^{a}$ OPE (4.2.13), by applying Wick’s theorem to  $J^{a}$
$(z)K_{bc}$
$(J^{b}J^{c})$(y)$,goingthroughthefollowingintermediatesteps:\begin{aligned}
\mbox{\rule[-6.5pt]{0.0pt}{1.0pt}\makebox[0.0pt][l]{\hskip
4.73352pt\makebox[0.0pt]{\rule[-6.5pt]{0.3pt}{3.0pt}}\rule[-6.5pt]{4.73352pt}{0.3pt}}\mbox{$J^{a}$}\makebox[0.0pt][l]{\rule[-6.5pt]{12.8681pt}{0.3pt}}\mbox{$(z)$}\makebox[0.0pt][l]{\rule[-6.5pt]{12.79918pt}{0.3pt}\makebox[0.0pt]{\rule[-6.5pt]{0.3pt}{3.0pt}}}\mbox{$(J^{b}J_{b})$}}(y)&=\frac{1}{2\pi
i}\oint_{y}\frac{dx}{x-y}\left(\frac{kJ^{a}(y)}{(x-z)^{2}}+\frac{f^{ab}_{c}J^{c}(x)J_{b}(y)}{z-x}+\frac{kJ^{a}(x)}{(y-z)^{2}}-\frac{f^{ab}_{c}J^{c}(x)J_{b}(y)}{z-y}\right),\\\
&=\frac{2kJ^{a}(y)}{(y-z)^{2}}+\frac{f^{ab}_{c}f^{dc}_{b}J_{d}(y)}{(y-z)^{2}}=2(k+g){\frac{\partial}{\partial
z}}\frac{J^{a}(z)}{y-z}+O(1)\ .\end{aligned}ThenapplyWick^{\prime}stheoremto$
$T$ $(y)K_{ab}$
$(J^{a}J^{b})$(z)$,andcheckthefollowingidentities:\begin{aligned}
\mbox{\rule[-6.5pt]{0.0pt}{1.0pt}\makebox[0.0pt][l]{\hskip
3.61632pt\makebox[0.0pt]{\rule[-6.5pt]{0.3pt}{3.0pt}}\rule[-6.5pt]{3.61632pt}{0.3pt}}\mbox{$T$}\makebox[0.0pt][l]{\rule[-6.5pt]{13.0394pt}{0.3pt}}\mbox{$(y)$}\makebox[0.0pt][l]{\rule[-6.5pt]{13.35594pt}{0.3pt}\makebox[0.0pt]{\rule[-6.5pt]{0.3pt}{3.0pt}}}\mbox{$(J^{a}J_{a})$}}(z)&=\frac{1}{2\pi
i}\oint_{z}\frac{dx}{x-z}\left({\frac{\partial}{\partial
x}}\frac{J^{a}(x)J_{a}(z)}{y-x}+{\frac{\partial}{\partial
z}}\frac{J^{a}(x)J_{a}(z)}{y-z}\right)\ ,\\\ &=\frac{1}{2\pi
i}\oint_{z}dx\frac{J^{a}(x)J_{a}(z)}{(x-z)(y-x)(y-z)}+{\frac{\partial}{\partial
z}}\frac{1}{2\pi i}\oint_{z}dx\frac{J^{a}(x)J_{a}(z)}{(x-z)(y-z)}\ ,\\\
&=\frac{k\dim\mathfrak{g}}{(y-z)^{4}}+\frac{(J^{a}J_{a})(z)}{(y-z)^{2}}+{\frac{\partial}{\partial
z}}\frac{(J^{a}J_{a})(z)}{y-z}\
.\end{aligned}Concludethatthefield$T$satisfiestheVirasorofieldOPE\eqref{tt}withthecentralcharge\eqref{ckg}.$

######  Exercise 4.3 (Modified Sugawara construction)

Compute the $\hat{T}\hat{T}$ OPE for the modified Sugawara field
$\hat{T}=T+Q_{a}\partial J^{a}$. Show that $\hat{T}$ is not a Virasoro field.
Then, relax the assumption that $\mathfrak{g}$ be semi-simple. The Killing
$K^{ab}$ form being degenerate, replace it with a non-degenerate, invariant
symmetric tensor $C^{ab}$ in the $J^{a}J^{b}$ OPE (4.2.7) and in the
definition of $T$ (4.2.11). What are now the conditions for $\hat{T}$ to be a
Virasoro field?

######  Exercise 4.4 (Associativity of the $J^{a}J^{b}\Phi^{R}$ OPE)

Check the associativity of the $J^{a}J^{b}\Phi^{R}$ OPE, by performing two
different computations of the behaviour near $y_{2}=z_{0}$ of

$\displaystyle\mathcal{O}=\frac{1}{2\pi i}\oint_{y_{2}}dy_{1}\
J^{a}(y_{1})J^{b}(y_{2})\Phi^{R}(z_{0})\ .$ (4.5.6)

Firstly, use the $J^{a}J^{b}$ OPE (4.2.7), and check that

$\displaystyle\mathcal{O}=f_{c}^{ab}J^{c}(y_{2})\Phi^{R}(z_{0})=\frac{-f_{c}^{ab}t^{c}\Phi^{R}(z_{0})}{y_{2}-z_{0}}+O(1)\
.$ (4.5.7)

Secondly, split the integration contour in two terms,
$\oint_{y_{2}}=\oint_{y_{2},z_{0}}-\oint_{z_{0}}$. In the first term, use the
$J^{b}(y_{2})\Phi^{R}(z_{0})$ OPE, as no integration contour runs between
these two operators. In the second term, use the $J^{a}(y_{1})\Phi^{R}(z_{0})$
OPE. Check that you obtain the following intermediate steps:

$\displaystyle\mathcal{O}$ $\displaystyle=\frac{1}{2\pi
i}\oint_{y_{2},z_{0}}dy_{1}J^{a}(y_{1})\frac{-t^{b}\Phi^{R}(z_{0})}{y_{2}-z_{0}}-\frac{1}{2\pi
i}\oint_{z_{0}}dy_{1}J^{b}(y_{2})\frac{-t^{a}\Phi^{R}(z_{0})}{y_{1}-z_{0}}+O(1)\
,$ (4.5.8) $\displaystyle=\frac{-t^{b}}{y_{2}-z_{0}}\frac{1}{2\pi
i}\oint_{y_{2},z_{0}}dy_{1}J^{a}(y_{1})\Phi^{R}(z_{0})+t^{a}J^{b}(y_{2})\Phi^{R}(z_{0})+O(1)\
,$ (4.5.9)
$\displaystyle=\frac{(t^{b}t^{a}-t^{a}t^{b})\Phi^{R}(z_{0})}{y_{2}-z_{0}}+O(1)=\frac{-f_{c}^{ab}t^{c}\Phi^{R}(z_{0})}{y_{2}-z_{0}}+O(1)\
.$ (4.5.10)

Notice that this explains the minus sign in the $J^{a}\Phi^{R}$ OPE eq.
(4.2.18).

######  Exercise 4.5 (Affine primary fields are primary fields)

Let $\Phi^{R}$ be an affine primary field. Compute the OPE
$T(y)\Phi^{R}(z_{0})$, and check that this OPE is of the form (2.2.13), with
the conformal dimension and action of $L_{-1}$ given by eqs. (4.2.19) and
(4.2.20) respectively. To do this, apply Wick’s theorem to  $\Phi^{R}$
$(z_{0})$ $(J^{a}J_{a})$(y)$,usingeq.\eqref{jpr}intheform$ $\Phi^{R}$
$(z_{0})$ $J^{a}$(x) = -taΦR(x)x-z0.

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

Consider fields $(J,\beta,\gamma)$ such that

$\displaystyle
J(y)J(z)=\frac{-\frac{1}{2}}{(y-z)^{2}}+O(1)\quad,\quad\beta(y)\gamma(z)=\frac{-1}{y-z}+O(1)\
,$ (4.5.11)

and the OPEs $J\beta,J\gamma,\beta\beta,\beta\gamma$ have no singular terms.
Consider the fields

$\displaystyle J^{-}=-\beta\quad,\quad J^{0}=-(\beta\gamma)-bJ\quad,\quad
J^{+}=(\beta\gamma^{2})+2b(\gamma J)+k\partial\gamma\ ,$ (4.5.12)

where the brackets are normal-ordered products, and the parameters $b$ and $k$
obey the relation (4.2.66). Show that the fields $(J^{-},J^{0},J^{+})$ obey
the OPEs (4.2.10), and are therefore $\widehat{\mathfrak{sl}}_{2}$ currents.
Show that the Sugawara construction yields the Virasoro field

$\displaystyle T=-\beta\partial\gamma-J^{2}+b^{-1}\partial J\ ,$ (4.5.13)

and rederive the central charge from this formula. Then define the field
$\Phi^{j}_{\mu}(z)$ by

$\displaystyle\beta(y)\Phi^{j}_{\mu}(z)=\frac{-\mu}{y-z}\Phi^{j}_{\mu}(z)+O(1)\quad,\quad\gamma(y)\Phi^{j}_{\mu}(z)=O(1)\
,$ (4.5.14) $\displaystyle
J(y)\Phi^{j}_{\mu}(z)=\frac{-b^{-1}(j+1)}{y-z}\Phi^{j}_{\mu}(z)+O(1)\ .$
(4.5.15)

Show that the field
$\left({\frac{\partial}{\partial\mu}}-\gamma(z)\right)\Phi^{j}_{\mu}(z)$
satisfies the same relations, and is therefore proportional to
$\Phi^{j}_{\mu}(z)$. Choosing the coefficient of proportionality such that

$\displaystyle{\frac{\partial}{\partial\mu}}\Phi^{j}_{\mu}(z)=\left(\frac{j+1}{\mu}+\gamma(z)\right)\Phi^{j}_{\mu}(z)\
,$ (4.5.16)

write the OPEs $J^{a}(y)\Phi^{j}_{\mu}(z)$ in terms of differential operators
as in eq. (4.2.21), and conclude that $\Phi^{j}_{\mu}(z)$ is a $\mu$-basis
affine primary field.

######  Exercise 4.7 (Conformal global Ward identities from KZ equations)

Prove that the KZ equations imply the global Ward identities of conformal
symmetry (2.3.4). Use the global Ward identities of the affine symmetry
(4.2.44), the Casimir relation for the isospin differential operators
(4.2.22), and the conformal dimensions of affine primary fields (4.2.19).

######  Exercise 4.8 (Proof of the identity (4.2.63))

Check that the identity holds when applied to positions $z_{i}$, due to
$\mathcal{K}^{-1}z_{i}=z_{i}$. To check that the identity also holds when
applied to Sklyanin variables $y_{j}$, it is enough to check that both sides
have the same commutator with $\mathcal{K}\hat{J}^{-}(y)\mathcal{K}^{-1}$.
Compute the commutator of the left-hand side using
$[\hat{T}(\hat{y}_{j}),\hat{J}^{-}(y)]={\frac{\partial}{\partial
y}}\frac{1}{y-\hat{y}_{j}}\hat{J}^{-}(y)$, which is a consequence of eq.
(4.2.52). Compute the commutator of the right-hand side using the formula
(4.2.56) for $\hat{J}^{-}(y)$, and conclude.

## Index

  * A-series minimal model 2nd item, §3.2.1
  * accessory parameter §4.1.4
  * affine highest-weight representation §4.2.2
  * affine Lie algebra §4.2.1
  * affine Lie algebra $\hat{\mathfrak{u}}_{1}$ §4.1.1
  * affine primary field §4.2.2
  * associativity item 2
  * background charge §2.1.2, §4.1.1
  * bootstrap approach §1.2.1
  * bosonic field §2.3.2
  * BPZ equation §2.3.4
  * central charge §1.3.3
  * characteristic exponent §2.3.4
  * closure under fusion §1.4.1
  * commutativity item 1
  * compactified free boson §4.1.3
  * conformal block §2.3.3, §2.3.4
  * conformal bootstrap §1.4.2
  * conformal dimension §1.4.1
  * conformal field theory §1.3.3
  * conformal perturbation theory §4.1.4
  * conformal transformation §1.3.1
  * correlation function §1.1.2
  * cosmological constant §3.1.3
  * Coulomb gas integral §4.1.4
  * creation operator §2.1.1
  * cross-ratio 4th item
  * crossing symmetry §2.3.3
  * degenerate field §2.2.1
  * degenerate representation §2.1.2
  * descendent field §2.2.1
  * descendent state §2.1.1
  * diagonal model 2nd item
  * dilatation §1.3.1
  * discrete series representation of $\mathfrak{sl}_{2}$ §4.4.3
  * doubly degenerate representation §3.2.1
  * DOZZ formula §3.1.4
  * energy-momentum tensor §2.3
  * field §1.1.1
  * free boson §4.1.3
  * free bosonic theory §4.1.3
  * free energy §4.1.4
  * fusing matrix §2.3.3, §2.3.5
  * fusion multiplicity §1.2.3
  * fusion product §1.2.3
  * fusion rule §1.2.3
  * $\hat{\mathfrak{g}}$ current §4.2.1
  * Gaudin model §4.2.3
  * generalized minimal model §3.2.1
  * generalized $SU_{2}$ WZW model §4.4.2
  * global conformal field theory §1.3.2
  * global conformal transformation §1.3.1
  * global Ward identity 1st item
  * $H_{3}^{+}$ §4.1.4, §4.3.2
  * $H_{3}^{+}$ model §4.3, §4.3.2
  * $H_{3}^{+}$-Liouville relation §4.3
  * heavy asymptotic limit 4.1.45
  * hermitian conjugate §1.1.2
  * highest-weight representation §2.1.1
  * holomorphic factorization §1.4.2
  * horizontal subalgebra §4.2.2
  * hypergeometric equation §2.3.5
  * hypergeometric function §2.3.5
  * identity field §2.2.2
  * indecomposable representation §1.2.2
  * irreducible representation §1.2.2
  * Ising model 2nd item, §3.2.4
  * isospin variable §4.2.2
  * Kac formula §2.1.3
  * Kac table §3.2.3
  * Killing form §4.2.1
  * KZ equations §4.2.3
  * KZ-BPZ relation §4.2.4
  * Lagrangian 2nd item
  * large level limit §4.3.2
  * left-moving 1st item
  * level (affine Lie algebra) §4.2.1
  * level (descendent state) §2.1.1
  * light asymptotic limit 4.1.50
  * linear dilaton theory 1st item, §4.1.3
  * Liouville coupling constant §2.1.2
  * Liouville equation §4.1.4
  * Liouville theory 1st item
  * local conformal transformation §1.3.2
  * local Ward identity 2nd item
  * locally holomorphic §1.4.2
  * logarithmic conformal field theory §2.1.1
  * maximally degenerate representation §2.1.2
  * metric §1.3
  * minimal model 2nd item
  * model §1.1.1
  * model-dependent data 2nd item
  * modular bootstrap §1.4.2
  * momentum §2.1.2, §4.1.1
  * $\mu$-basis field §4.2.2
  * multiplicity §1.2.2
  * $N$-point function §1.1.2
  * normal-ordered product §4.1.1
  * null vector §2.1.2
  * null vector equation §2.2.5
  * observable §1.1.2
  * OPE item 3
  * OPE coefficient item 3
  * Peter-Weyl theorem §4.4.1
  * physical system §1.1.1
  * primary field §2.2.1
  * primary state §2.1.1
  * principal series representation of $\mathfrak{sl}_{2}$ §4.4.3
  * principal series representation of $\mathfrak{sl}_{2}({\mathbb{C}})$ §4.3.2
  * quadratic Casimir operator §4.2.1
  * quantum §1.1.1
  * quantum field theory §1.1
  * quasi-primary field §2.2.3
  * radial quantization §1.4.1
  * rational model 1st item
  * reflection §2.1.2
  * reflection coefficient §3.1.2
  * reflection relation §3.1.2
  * regular singular point §2.3.4
  * Riemann sphere §1.3.1
  * Riemann surface §1.3.2
  * right-moving 2nd item
  * rotation §1.3.1
  * $s$-channel §2.3.3, §2.3.4
  * scalar product §1.1.2
  * Seiberg-Witten equation §4.1.2
  * separation of variables §4.2.4
  * simple current §1.2.3
  * singular vector §2.1.2
  * Sklyanin variable §4.2.4
  * $\mathfrak{sl}_{2}$ §2.3.2
  * $\widehat{\mathfrak{sl}}_{2}$ conformal block §4.2.4
  * $\widetilde{SL}_{2}(\mathbb{R})$ §4.4.3
  * $\widetilde{SL}_{2}(\mathbb{R})$ WZW model §4.4.3
  * solve (model) §1.1.2
  * spectral flow §4.4.3
  * spectrally flowed representation §4.4.3
  * spectrum §1.1.2
  * spin (conformal) §2.3.2
  * spin ($\mathfrak{sl}_{2}$) §4.2.2
  * state-field correspondence §1.1.2
  * structure function §1.2.3
  * $SU_{2}$ WZW model §4.4.2
  * Sugawara construction §4.2.1
  * symmetry field §2.2.2
  * $t$-channel §2.3.3, §2.3.5
  * T-duality §4.1.3
  * theory §1.1.1
  * three-point structure constant 3rd item
  * translation §1.3.1
  * $\hat{\mathfrak{u}}_{1}$ current §4.1.1
  * $\hat{\mathfrak{u}}_{1}$-primary field §4.1.1
  * unitary §1.4.1
  * universal data 1st item
  * Verma module §2.1.2
  * Virasoro algebra §1.3.3
  * Virasoro field §2.2.2
  * Wakimoto free-field representation of $\widehat{\mathfrak{sl}}_{2}$ §4.2.2
  * Ward identity §2.2.3
  * Wick’s theorem §4.1.1
  * Witt algebra §1.3.2
  * WZW model §4.4.1
  * $x$-basis field §4.2.2

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