# The torus one-point block of 2d CFT and null vectors in
$\hat{\mathfrak{sl_{2}}}$

Dario Stocco<EMAIL_ADDRESS>École Polytechnique Paris, ETH Zürich
Advisor: Sylvain Ribault<EMAIL_ADDRESS>IPhT, CEA Saclay

(August 27, 2024)

###### Abstract

This thesis is divided into two parts, where in the first part we investigate
the computation of Virasoro 1-point blocks on the torus in the framework of
Zamolodchikov’s recursion relation. It is widely accepted that this recursion
relation contains unphysical poles in the central charge $c$. At each order we
conjecture how the pole free expressions depend on the internal and external
conformal dimensions and central charge, and propose how to compute it
numerically. In this thesis, we have calculated the pole free expression up to
order 4. In the second part we introduce a conformal field theory with an
extra symmetry, described by highest weight representations of the affine Lie
algebra $\hat{\mathfrak{sl}}_{2}$. At level 1, we determined a universal
$\mathfrak{sl}_{2}$ basis-independent ’null operator’, which generates null
vectors in the usual sense. The ’null operators’ are generalized objects and
can be applied to any state of the horizontal representation, yielding null
vectors and are therefore independent from the choice of horizontal
representations.

###### Contents

  1. I Overview: conformal field theories
    1. I.1 Symmetry algebra
    2. I.2 Representations and fields
      1. I.2.1 Highest weight representation
      2. I.2.2 Degenerate representations
      3. I.2.3 State field correspondence
    3. I.3 Observables
      1. I.3.1 Correlation functions
      2. I.3.2 Global Ward identities
      3. I.3.3 Local Ward identities
      4. I.3.4 Operator product expansion
    4. I.4 Conformal blocks
      1. I.4.1 2- and 3-point blocks
      2. I.4.2 4-point blocks
      3. I.4.3 Torus: 1-point block
    5. I.5 Fusion algebra and fusion rules
  2. II 1-point block on the torus
    1. II.1 Preliminaries
      1. II.1.1 Recursion in powers of $q$
    2. II.2 Singularity free at order 2: hands on calculation
    3. II.3 Proposal: Singularity free at each order
      1. II.3.1 Large $c$ limit
      2. II.3.2 Tracking $\Delta$ in (II.12)
      3. II.3.3 Calculation of pole free expression
  3. III Affine symmetry
    1. III.1 Affine primary fields and Ward identities
      1. III.1.1 Isospin variables
      2. III.1.2 Global Ward identity
      3. III.1.3 Local Ward identity
    2. III.2 $\hat{\mathfrak{sl}}_{2}$ degenerate representations
      1. III.2.1 Affine highest weight representation
      2. III.2.2 Level 0 null vectors
      3. III.2.3 Level 1 null vectors
    3. III.3 $\hat{\mathfrak{sl}}_{2}$ fusion rules
      1. III.3.1 Finite level 0
      2. III.3.2 Generic representation level 1
      3. III.3.3 Generalized Fusion Rule
  4. A Singularity free at order 3 and 4
  5. B Irreducible representations of $\mathfrak{sl}_{2}$

Introduction

Two dimensional conformal field theories (shortand CFT or 2d CFT) are examples
of quantum field theories with a very large amount of symmetry. This large
amount of symmetry occurs in two dimensions because of the connection to
holomorphic functions and drastically restricts the theory. Those restrictions
allows, under certain circumstances, to calculate the correlation functions
non-pertubartively and without a Lagrangian description. Therefore 2d CFTs are
of large interest and widely used in modern theoretical physics. For instance
on one side, it is used to describe statistical phenomenas, like the critical
Ising model or percolation theory and on the other side it is very important
to study string theories.

Let me give an overview how the thesis is structured. In the first section (I)
we introduce into important concepts and results of the bootstrap approach to
2d CFTs. Section (II) studies the poles in the central charge $c$ of the
recursive representation of 1-point blocks on the torus. I conjecture the
functional form of the pole-free expression and propose a method to calculate
the unknown coefficients. In the last section (III) CFTs with an additional
symmetry, described by the affine algebra $\hat{\mathfrak{sl}}_{2}$, are
introduced and the corresponding representation theory is dicussed. I focus on
the study of degenerate representations, i.e. on null vectors within highest
weight representation. Then the null vectors are applied to three point
functions to calculate fusion rules between affine degenerate fields and
affine primary fields (III.3).

##  I Overview: conformal field theories

Let us introduce in this section the main parts of 2d CFTs. We will give an
axiomatic based introduction and follow closely the book of Sylvain Ribault
[1] and the introductory notes of Bert Schellekens [2].
The underlying space of the CFTs, discussed in this thesis, are the
compactified complex plane $\mathbb{C}\cup\\{\infty\\}$, known as Riemann
sphere and the complex torus $\frac{\mathbb{C}}{\mathbb{Z}+\tau\mathbb{Z}}$
with modular parameter $\tau\in\mathbb{C}-\mathbb{R}$. The discussion mainly
concentrates on the Riemann sphere unless stated otherwise.

###  I.1 Symmetry algebra

In the euclidean case we introduce the metric
$\mathrm{d}s^{2}=\mathrm{d}z\mathrm{d}\bar{z}$ on the Riemann sphere. We
consider field theories invariant under conformal transformations, which
prominently includes scale transformations $z\mapsto\lambda z$, translations
$z\mapsto z+a$ and rotations $z\mapsto e^{i\phi}z$. In general conformal
transformations contain any angle preserving coordinate transformations i.e.
$g\mapsto\Omega(z)g$ when $z\mapsto f(z)$. Locally this is fullfilled by any
holomorphic function $(z,\bar{z})\mapsto(f(z),\bar{f}(z))$, with
$\Omega(z)=f^{\prime}(z)\bar{f}^{\prime}(z)$. Here $\bar{z}$ is the complex
conjugate of $z$. Expanding a conformal transformation $f(z)$ around the
identity map $f(z)=z+\epsilon(z)$, yields the infinite dimensional Witt
algebra identified by the holomorphic/antiholomorphic generators
$l_{n}:=-z^{n+1}\partial_{z}$/$\bar{l}_{n}:=-\bar{z}^{n+1}\partial_{\bar{z}}$,
with commutation relations,

$\displaystyle[l_{n},l_{m}]=(n-m)l_{n+m},\quad[\bar{l}_{n},\bar{l}_{m}]=(n-m)\bar{l}_{n+m},\quad[l_{n},\bar{l}_{m}]=0.$
(I.1)

The global, angle preserving transformations are given by the Möbius group
$\mathrm{PSL}_{2}(\mathbb{C})=\frac{SL_{2}(\mathbb{C})}{\\{id,-id\\}}$ and the
identification,

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\mapsto\frac{az+b}{cz+d}.$ (I.2)

A common rescaling of the matrix elements doesn’t change the transformation.
Hence it is enough to consider only matrices with determinant one and identify
$[g]=\\{g,-g\\}$ i.e. $g\in\mathrm{PSL}_{2}$. A change of phase doesn’t change
obersvables in quantum field theories and therefore we consider Lie group
representations $R$ with an additional phase, depending on the multiplication
of group elements $R(g)R(g^{\prime})=e^{i\phi(g,g^{\prime})}R(gg^{\prime})$.
At the level of the Lie algebra this extra phase is accounted by its central
extension. In our case the unique central extended Witt algebra is the
Virasoro algebra $\mathfrak{V}$. Formally the Virasoro algebra is defined as
the complex vector space spanned by $\\{L_{n\in\mathbb{Z}},c\\}$ with Lie
brackets,

$\displaystyle[L_{n},L_{m}]=(n-m)L_{n+m}+\frac{c}{12}m(m^{2}-1),\quad[L_{n},c]=0.$
(I.3)

$c\in\mathbb{C}$ is the central charge of the theory and commutes with all
other elements. Vice versa we have a copy of $\mathfrak{V}$ with the same
central charge, to account for the anti-holomorphic part with
$[\bar{L}_{n},L_{m}]=0$. The complete symmetry algebra of space-time
transformations of CFTs is the direct sum
$\mathfrak{V}\oplus\bar{\mathfrak{V}}$.

###  I.2 Representations and fields

So far we equipped the CFT with a space and a symmetry algebra. In order to be
able to calculate observables, we equip it further with representations of
$\mathfrak{V}\oplus\bar{\mathfrak{V}}$ and fields transforming in this
representations. Vaguely spoken, the space of representations is called the
spectrum $\mathbb{S}$ of the CFT. For simplicity we only investigate
representations of one copy of the Virasoro algebra and tensor them to get
representations of the complete symmetry algebra
$\mathfrak{V}\oplus\bar{\mathfrak{V}}$.

Remark: From now on we do not distinguish between the action $R(g)$ of the
representation and the algebra element $g\in\mathfrak{g}$. Similiarly we do
not distinguish between the representation $R$ and the vector space $V_{R}$ it
acts on.

####  I.2.1 Highest weight representation

Axiom: We demand that $L_{0}$ is diagonalizable.

Let’s introduce the Verma module with highest weight vector $|\Delta\rangle$,

$\mathcal{V}_{\Delta}:=U(\mathfrak{B}^{-})|\Delta\rangle,$ (I.4)

where $U(\mathfrak{B}^{-})$ is the universal envelope of the Virasoro algebra
with only negative indices $L_{n<0}$. We equip the space $U(\mathfrak{B^{-}})$
with the basis $\mathfrak{L}:=\\{L_{N}:=L_{-n_{1}}\cdot...\cdot
L_{-n_{M}}\\}_{1\leq n_{1}\leq...\leq n_{M}}$ and call the integer
$N:=-\sum_{i=1}^{M}n_{i}$ the level. Correspondingly we say that the vector
$L_{-n_{1}}\cdot...\cdot L_{-n_{M}}|\Delta\rangle$ is at level $N$. The action
of $\mathfrak{V}$ on the Verma module is then given by,

$\displaystyle L_{n>0}|\Delta\rangle=0,\quad
L_{0}|\Delta\rangle=(\Delta\in\mathbb{C})|\Delta\rangle.$ (I.5)

The highest weight vector $|\Delta\rangle$ of the representation (I.5) is also
called primary state and $\Delta$ its conformal dimension. We will see in
section (I.3.2) the role of the conformal dimension in conformal
transformations of primary fields. With respect to $L_{0}$ the descendant
states at level $N>0$, have eigenvalue
$L_{0}L_{N}|\Delta\rangle=(\Delta+N)L_{N}|\Delta\rangle$. This implies that
the real-part of the $L_{0}$-eigenvalues is bounded from below:
$Re(\Delta+N)\geq Re(\Delta)$. We interpret the operator $H\sim
L_{0}+\bar{L_{0}}$ as the hamiltonian of the theory and therefore the spectrum
is stable, because of the lower bound on the energies $\Delta$.

####  I.2.2 Degenerate representations

In this section we work out irreducible Verma modules and degenerate
representations. In section (I.5) we will then see how degenerate
representations constrain correlation functions. Verma modules (I.4) are
reducible if they contain null vectors,

######  Definition 1 (Null vector).

A vector $|\chi,N\rangle:=\sum_{|M|=N}a_{M}L_{-M}|\Delta\rangle$, fullfilling
the condition $L_{n>0}|\chi,N\rangle=0$ is called a null vector at level
$N>0$.

A null vector $|\chi,N\rangle$ generates an invariant subspace
$U(\mathfrak{B}^{-})|\chi,N\rangle\subset\mathcal{V}_{\Delta}$ and therefore
the representation is decomposable. There is a very helpful theorem,
connecting null vectors and irreducibility,

######  Theorem 1.

The Verma module is irreducible if and only if it contains no null vectors.

Conversely we build irreducible representations (shorthand irreps) by taking
the quotient of the Verma module with all invariant subspaces generated by all
null vectors. The quotient space is called degenerate representation. For any
two positive integers $r,s$ forming the product $rs=N$, exist
$p_{N}:=\sum_{rs=N}1$ inequivalent null vectors $|\chi_{r,s}\rangle$ at level
$N=rs$,

$\displaystyle|\chi_{r,s}\rangle=L_{r,s}|\Delta_{r,s}\rangle,\quad
L_{r,s}=\sum_{|M|=N}d_{M}(c,r,s)L_{M}.$ (I.6)

The coefficients $d_{M}$ depend on the central charge $c$ and the integers
$r,s$. For example in the case of level 2 null vectors we have,

$L_{2,1}=L_{-1}^{2}+b^{2}L_{-2},\quad L_{1,2}=L_{-1}^{2}+b^{-2}L_{-2},$ (I.7)

where $b$ is introduced in (I.8). Moreover the conformal dimension
$\Delta_{<r,s>}=\Delta(P_{<r,s>})$ in (I.6) is determined by the null vector
condition on $|\chi_{r,s}\rangle$ and given in terms of the central charge $c$
by the parametrization,

$\displaystyle\Delta(P):=\frac{c-1}{24}-P^{2},\quad
P_{<r,s>}:=\frac{1}{2}(rb+sb^{-1}),\quad c:=13+6b^{2}+6b^{-2}.$ (I.8)

$P$ is called the momentum and $b$ the coupling constant. A widely used
relation is that $\Delta_{<r,-s>}=\Delta_{<r,s>}+rs=\Delta_{<-r,s>}$, where
$\Delta_{<r,-s>}$ is the conformal dimension of the null vector
$|\chi_{r,s}\rangle$. In terms of the coupling constant $P_{<r,s>}(b)$ and
$\Delta_{<r,s>}(b)$ are invariant under the simultaneous exchange of
$r\leftrightarrow s$ and $b\leftrightarrow b^{-1}$ (Let’s name the
transformation $b\rightarrow b^{-1}$ $b$-symmetry). For generic values of $c$,
the null vector $|\chi_{r,s}\rangle$ is the only one in the Verma module
$\mathcal{V}_{\Delta_{<r,s>}}$ and thus the quotient space,

$R_{r,s}:=\frac{\mathcal{V}_{\Delta_{<r,s>}}}{\mathcal{V}_{\Delta_{<r,-s>}}},$
(I.9)

forms a single degenerate representation.

A-series minimal models: A very prominent example of CFTs are the A-series
minimal models, including the critical Ising model. Their spectra are made of
doubly degenerate representations, who contain two null vectors and therefore
satisfy $\Delta_{<r,s>}=\Delta_{<r^{\prime},s^{\prime}>}$ for some
$(r,s)\neq(r^{\prime},s^{\prime})$. This condition on the conformal dimension
determines the central charge $c_{p,q}=1-6\frac{(q-p)^{2}}{pq}$ for $p,q$
coprime integers. In particulary A-series minimal models demand the extra
condition $2\leq p,q$ otherwise their spectrum is empty. For example the
critical Ising model has central charge $c=\frac{1}{2}$ with $p=4,q=3$.

####  I.2.3 State field correspondence

We demand that there exists a one to one map $\mathfrak{i}$ of fields $V(z)$
and states in the spectrum $\mathbb{S}$,

$\displaystyle\mathfrak{i}:|v\rangle\mapsto V_{|v\rangle}(z).$ (I.10)

$\mathfrak{i}$ extends the action of $\mathfrak{V}$ onto fields by
$L^{(z)}_{n}V_{|v\rangle}(z):=V_{L_{n}|v\rangle}(z)$. In string theory the
correspondence can be thought of as having the space of states at the origin
of the Riemann sphere $z=0$, corresponding to the far past on the string
worldsheet. The state is translated by
$V_{|v\rangle}(z)=e^{-izL_{-1}}V_{|v\rangle}(0)e^{izL_{-1}}$ to any point on
the sphere.
A general representation of the symmetry algebra
$\mathfrak{V}\oplus\bar{\mathfrak{V}}$ is realized by tensoring two Verma
modules (or equivalently degenerate representations),

$\displaystyle\begin{aligned} \mathfrak{V}\oplus\bar{\mathfrak{V}}&\rightarrow
End(\mathcal{V}_{\Delta}\otimes\mathcal{V}_{\bar{\Delta}})\\\
L_{n}+\bar{L_{m}}&\mapsto(L_{n}+\bar{L_{m}})V_{\Delta,\bar{\Delta}}(z):=V_{L_{n}|\Delta\rangle\otimes|\bar{\Delta}\rangle+|\Delta\rangle\otimes\bar{L_{m}}|\bar{\Delta}\rangle}(z).\end{aligned}$
(I.11)

The field $V_{\Delta,\bar{\Delta}}(z)$ is called primary field and corresponds
to the highest weight state
$|\Delta\rangle\otimes|\bar{\Delta}\rangle\in\mathcal{V}_{\Delta}\otimes\mathcal{V}_{\bar{\Delta}}$.
We call fields diagonal if $\Delta=\bar{\Delta}$ and write $V_{\Delta}$. On
the other side a CFT is diagonal if the spectrum contains only diagonal
fields. Be careful with the expression $\bar{\Delta}$, as it is not the
complex conjugate of $\Delta$ but the conformal dimension corresponding to
$\bar{\mathfrak{V}}$.

Axiom: We introduce the additional axiom, that the generator
$L_{-1}$/$\bar{L}_{-1}$ acts on fields as the derivative in $z$/$\bar{z}$.

There exists an exceptional field, the Virasoro field $T(y)$, which generates
the symmetry algebra $\mathfrak{V}$. The Virasoro field is identified as the
energy momentum tensor of conformal symmetries. It is defined implicitly, such
that its $n+1$’th moment at $z$ is the generator $L^{(z)}_{n}$ (conversely for
$\bar{T}$ generating $\bar{L}^{(z)}_{n}$),

$L^{(z)}_{n}=\frac{1}{2\pi i}\oint_{z}dy(y-z)^{n+1}T(y).$ (I.12)

Therefore $T(y)$ can be written in terms of its moments,

$T(y)=\sum_{i\in\mathbb{Z}}\frac{L_{n}^{(z)}}{(y-z)^{n+2}}.$ (I.13)

Axiom: $T(y)$ is holomorphic and behaves as $O(y^{-4})$ at $y=\infty$.

On the quantum level the fields are operators, where we are interested in the
operator product expansion (shorthand OPE) of $T$ with a primary field
$V_{\Delta}$ and $T$ with itself,

$\displaystyle\begin{aligned}
T(y)V_{\Delta,\bar{\Delta}}(z)&\underset{y\rightarrow z}{=}\frac{\Delta
V_{\Delta,\bar{\Delta}}(z)}{(y-z)^{2}}+\frac{\partial_{z}V_{\Delta,\bar{\Delta}}(z)}{(y-z)}+O(1)\\\
T(y)T(z)&\underset{y\rightarrow
z}{=}\frac{c/2}{(y-z)^{4}}+\frac{2T(z)}{(y-z)^{2}}+\frac{\partial_{z}T(z)}{(y-z)}+O(y-z).\end{aligned}$
(I.14)

In our case the OPEs (I.14) have finite radius of convergence and is therefore
well defined. The $TT$-OPE is equivalent to the Virasoro commutation relation
(I.3) and the $TV$-OPE is generic for primary fields.

###  I.3 Observables

Up to this point we have introduced the underlying space, the symmetry
algebra, the representations and fields. Now its time to discuss the
observables of the CFT.

Beforehand we introduce a generic notation of the complex modulus, which is
often used because of the factorizable left and right chiral representation
$\mathfrak{V}\oplus\bar{\mathfrak{V}}$,

$|f(\Delta,z,L_{n})|^{2}:=f(\Delta,z,L_{n})f(\bar{\Delta},\bar{z},\bar{L}_{n}).$
(I.15)

####  I.3.1 Correlation functions

We define the observables to be correlation function of N fields (shorthand
N-point function). The correlator is a multilinear function of fields,
depending on the conformal dimensions and their position. In the bosonic case
the correlator transforms under the one dimensional symmetric representation
of the symmetric group $S_{N}$. Diagonal CFTs are bosonic and contain only
spin zero fields (see I.3.2). We write the N-point function as,

$\langle\prod_{i=1}^{N}V_{\sigma_{i},\bar{\sigma}_{i}}(z_{i})\rangle,$ (I.16)

where we denoted by $(\sigma,\bar{\sigma})$ to indicate any field in the
representation (I.11). The N-point functions are restricted by Ward
identities, which are determined by the Virasoro symmetry of conformal field
theory. Ward identities can be obtained by a weighted contour integral of the
meromorphic correlator $Z(y):=\langle
T(y)\prod_{i}V_{\sigma_{i}}(z_{i})\rangle$ around all poles $z=z_{i}$. The
weight $\epsilon(y)$ is such that it has no poles outside $\\{z_{i}\\}_{i}$,

$\oint_{\infty}\mathrm{d}y\epsilon(y)Z(y)=0,\quad\epsilon\underset{y\rightarrow\infty}{\leq}O(y^{2}).$
(I.17)

This integral is evaluated using the $TV$-OPE (I.14) and the residue theorem
of complex analysis.

Infinitesimal transformations: A general result of quantum field theory is
that an infinitesimal transformation $f(z)=z+\epsilon(z)$ of a generic field
$\Psi(z)$ is given by [2],

$\delta_{\epsilon}\Psi(z)=\frac{1}{2\pi i}\oint_{z}dy\epsilon(y)T(y)\Psi(z).$
(I.18)

This statement motivates the use of $Z(y)$ and connects it to conformal
transformations of N-point functions.

####  I.3.2 Global Ward identities

Let’s consider $g\in\mathrm{PSL_{2}(\mathbb{C})}$ a Möbius transformation and
the corresponding infinitesimal transformations $\epsilon\in\\{1,y,y^{2}\\}$
in equation (I.17). We can deduce how a primary field transforms under $g$,

$V_{\Delta,\bar{\Delta}}^{\prime}(z):=T_{g}V_{\Delta,\bar{\Delta}}(z)=|(cz+d)^{-2\Delta}|^{2}V_{\Delta,\bar{\Delta}}(gz),$
(I.19)

with the action of $PSL_{2}$ on the space $gz:=\frac{az+b}{cz+d}$. Under
global conformal transformation N-point functions stay invariant, as a
consequence of exponentiating the Ward identities (I.17) with
$\epsilon\in\\{1,y,y^{2}\\}$. Therefore the classical conformal invariance
transmits anomaly free to the quantum theory,

$\langle\prod_{i=1}^{N}V_{\Delta_{i},\bar{\Delta_{i}}}(z_{i})\rangle=\langle\prod_{i=1}^{N}T_{g}V_{\Delta_{i},\bar{\Delta_{i}}}(z_{i})\rangle.$
(I.20)

The $\mathrm{PSL}_{2}$ invariance of N-point functions imply, that within
correlation functions three coordinates can be choosen freely. Especially 2-
and 3-point functions are determined on the whole sphere, if they are known at
some fixed coordinates.

The invariance equation (I.20) is generalized to local conformal i.e.
holomorphic transformations $z\rightarrow h(z)$,

$\langle\prod_{i=1}^{N}V_{\Delta_{i},\bar{\Delta_{i}}}(z_{i})\rangle=\prod_{i=1}^{N}h^{\prime}(z)\langle\prod_{i=1}^{N}V_{\Delta_{i},\bar{\Delta_{i}}}(h(z_{i}))\rangle.$
(I.21)

Outside the domain of biholomorphism of $h(z)$, especially at
$h^{{}^{\prime}}(z_{0})\in\\{0,\infty\\}$, descendant states
$L_{n\leq-2}V_{\Delta}$ appear in the correlation function. But we do not go
into more details.

Spin of a primary: Under space rotations $f(z)=e^{i\phi}z$ the field
transforms as $V_{\Delta,\bar{\Delta}}(z)\rightarrow
e^{i\phi(\Delta-\bar{\Delta})}V_{\Delta,\bar{\Delta}}(z)$. Let
$S:=\Delta-\bar{\Delta}$ be the spin of the primary field. For
$S\in\frac{1}{2}+\mathbb{Z}$ the field flips sign under $2\pi=\phi$ rotation
and is thus fermionic. For $S\in\mathbb{Z}$ the field is bosonic. Generally
the spin can take any value but for the ongoing discussion we dont specify it
or consider diagonal fields i.e. spin zero.

Fields at the north pole: Another subtlety is how the primary fields and its
descendents behave analytically at the point $z=\infty$. From the
transformation properties (I.19) we deduce that
$V_{\Delta}(\infty)=\mathrm{lim}_{z\rightarrow\infty}|z^{2\Delta}|^{2}V_{\Delta}(z)$
(choose $gz=-\frac{1}{z}$ and use equation (I.20)). From the axiom identifying
$L_{-1}=\partial_{z}$, we can argue that
$L_{-1}V_{\Delta}(\infty)=\mathrm{lim}_{z\rightarrow\infty}z^{2\Delta+1}\bar{z}^{2\bar{\Delta}}L_{-1}V_{\Delta}(z)$,
and generally,

$L\bar{L}V_{\Delta}(\infty):=\mathrm{lim}_{z\rightarrow\infty}|z^{2\Delta+|L|}|^{2}L\bar{L}V_{\Delta}(z).$
(I.22)

####  I.3.3 Local Ward identities

If we choose $\epsilon=\frac{1}{(z-z_{i})^{n-1}},\quad n>1$ in equation (I.17)
we get a set of local Ward identities. For simplicity we concentrate only on
the case where there is at most one descendant field in the N-point function,

$\langle L^{(z_{i})}_{-n}V_{\sigma_{i}}(z_{i})\prod_{j\neq
i}V_{\Delta_{j}}(z_{j})\rangle=\sum_{j\neq
i}(-\frac{\partial_{z_{i}}}{z_{ji}^{n-1}}+\frac{(n-1)\Delta_{j}}{z_{ji}^{n}})\langle
V_{\sigma_{i}}(z_{i})\prod_{j\neq i}V_{\Delta_{j}}(z_{j})\rangle,$ (I.23)

where we introduce the shorthand notation $z_{ij}:=z_{i}-z_{j}$. In our setup
it can be shown by induction that any N-point function with descendant fields,
can be related to N-point functions with lower level descendant fields only by
differential operators in $z$.

The global and local Ward identities of the right chiral symmetry
$\bar{\mathfrak{V}}$ take a similiar form.

####  I.3.4 Operator product expansion

In the introduction we discussed how conformal field theories are especially
interesting because they possess non-perturbative solutions. Hence the
$VV$-OPE of primary fields is introduced to be able to solve the CFT exactly.
By axiom the $VV$-OPE takes the form,

$V_{\sigma_{1},\bar{\sigma_{1}}}(z_{1})V_{\sigma_{2},\bar{\sigma_{2}}}(z_{2})\underset{z_{1}\rightarrow
z_{2}}{=}\sum_{\sigma_{3},\bar{\sigma_{3}}\in\mathbb{S}}C^{12}_{3}(z_{1},z_{2})V_{\sigma_{3},\bar{\sigma_{3}}}(z_{2}).$
(I.24)

The sum on the right side sums over all fields in the spectrum $\mathbb{S}$ of
the CFT. In the case of Virasoro symmetry the $VV$-OPE has finite radius of
convergence. Applying $\oint_{z_{1},z_{2}}\mathrm{d}z(z-z_{2})^{n+1}T(z)$ on
both sides of (I.24), yields the OPE Ward identities,

$\bigg{(}L_{n}^{(z_{2})}+\sum_{m=-1}^{n}\binom{m+1}{n+1}z_{12}^{n-m}L_{m}^{(z_{1})}\bigg{)}V_{\sigma_{1}}(z_{1})V_{\sigma_{2}}(z_{2})=\sum_{\sigma_{3}\in\mathbb{S}}C^{12}_{3}(z_{1},z_{2})L^{(z_{2})}_{n}V_{\sigma_{3}}(z_{2}).$
(I.25)

The solution of the OPE Ward identity reads,

$V_{\Delta_{1},\bar{\Delta_{1}}}(z_{1})V_{\Delta_{2},\bar{\Delta_{2}}}(z_{2})=\sum_{\Delta_{3},\bar{\Delta_{3}}\in\mathbb{S}}\frac{C_{123}}{B_{3}}\bigg{|}z_{12}^{\Delta_{3}^{12}}\sum_{L\in
U(\mathfrak{B^{-}})}z_{12}^{|L|}f^{L}_{1,2,3}L\bigg{|}^{2}V_{\Delta_{3},\bar{\Delta_{3}}}(z_{2}),$
(I.26)

where we use the notation $\Delta_{I}^{J}=\sum_{i\in I}\Delta_{i}-\sum_{j\in
J}\Delta_{j}$ and the structure constant $B_{2}$ and $C_{123}$ of the 2- and
3-point function (see equation (I.29) and (I.30)). The coefficients
$f^{L}_{1,2,3}$ are the solutions of the linear equations,

$\sum_{|L|=N-n}f^{L}_{1,2,3}(\Delta_{3}+N-n+n\Delta_{1}-\Delta_{2})LV_{\Delta_{3}}(z)=\sum_{|L|=N}f^{L}_{1,2,3}L_{n}LV_{\Delta_{3}}(z).$
(I.27)

Up to the factor $\frac{C_{123}}{B_{3}}$ and the spectrum $\mathbb{S}$, the
$VV$-OPE of primaries is known only by symmetry considerations. Therefore
N-point functions of primaries can be reduced to (N-1)-point functions by the
$VV$-OPE.

###  I.4 Conformal blocks

In general N-point functions can be written as a sum of spectrum dependent
structure constants $C_{J}$ and universal $z$-dependent functions
$\mathfrak{F}_{J}^{(N)}(z)$,

$\displaystyle\langle\Pi_{i}V_{i}(z_{i})\rangle=\sum_{J}C_{J}\mathfrak{F}_{J}^{(N)}(z)\mathfrak{F}_{J}^{(N)}(\bar{z}).$
(I.28)

The functions $\mathfrak{F}^{(N)}$ are the conformal blocks of N-point
functions (shorthand blocks). Conformal blocks are universal objects of CFTs,
determined only by symmetry considerations and build the theory, as they
determine up to the structure constants, all correlation functions.

####  I.4.1 2- and 3-point blocks

For the 2- and 3-point functions there is only one unknown structure constant
and the blocks take a simple form. In both cases the correlator is determined
by the global Ward identity (I.20). In the 2-point case the correlator and
2-point block is given by,

$\displaystyle\begin{aligned} \langle
V_{\Delta_{1},\bar{\Delta}_{1}}(z_{1})V_{\Delta_{2},\bar{\Delta}_{2}}(z_{2})\rangle&=B_{3}|\mathfrak{F}^{(2)}(\Delta_{1},\Delta_{2}|z_{1},z_{2})|^{2}\\\
\mathfrak{F}^{(2)}&=\delta_{\Delta_{1},\Delta_{2}}z_{12}^{-2\Delta_{1}},\end{aligned}$
(I.29)

and the 3-point correlator and block by,

$\displaystyle\begin{aligned} \langle
V_{\Delta_{1},\bar{\Delta}_{1}}(z_{1})V_{\Delta_{2},\bar{\Delta}_{2}}(z_{2})V_{\Delta_{3},\bar{\Delta}_{3}}(z_{3})\rangle&=C_{123}|\mathfrak{F}^{(3)}(\Delta_{1},\Delta_{2},\Delta_{3}|z_{1},z_{2},z_{3})|^{2}\\\
\mathfrak{F}^{(3)}&=z^{\Delta^{12}_{3}}_{12}z^{\Delta^{23}_{1}}_{23}z^{\Delta^{31}_{2}}_{31}.\end{aligned}$
(I.30)

####  I.4.2 4-point blocks

The four point function,

$\langle
V_{\Delta_{1},\bar{\Delta}_{1}}(z_{1})V_{\Delta_{2},\bar{\Delta}_{2}}(z_{2})V_{\Delta_{3},\bar{\Delta}_{3}}(z_{3})V_{\Delta_{4},\bar{\Delta}_{4}}(z_{4})\rangle$
(I.31)

seems to make more problems. First, the global Ward identities (I.20) only
determines three degree of freedom. Therefore we are obliged to use local Ward
identities i.e. the $VV$-OPE (I.26) to determine the undetermined degree of
freedom from global Ward identities. Second, in the 4-point case there are 3
possibilities to apply the $VV$-OPE (field 1 with field 2, 3 or 4). Each
possibility corresponds to one of the three channel s, t and u. Those channels
are related to each other by a permutation of the conformal dimensions
$\\{\Delta_{i}\\}_{i=1,..,4}$ and complex coordinates
$\\{z_{i}\\}_{i=1,...,4}$. In the s-channel case, where the primaries $V_{1}$
and $V_{2}$ are OPE’d, the result reads,

$\displaystyle\mathfrak{F}_{\Delta_{s}}^{(4,s)}(\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}|z_{1},z_{2},z_{3},z_{4})=z_{13}^{-2\Delta_{1}}z_{23}^{\Delta^{14}_{23}}z_{34}^{\Delta^{12}_{34}}z_{24}^{\Delta^{3}_{124}}\bigg{(}x^{\Delta^{s}_{12}}\sum_{L\in
L}x^{|L|}f_{\Delta_{1},\Delta_{2},\Delta_{s}}^{L}g_{\Delta_{s},\Delta_{3},\Delta_{4}}^{L}\bigg{)},$
(I.32)

with coefficients $g^{L}_{\Delta_{s},\Delta_{3},\Delta_{4}}=\frac{\langle
LV_{\Delta_{s}}(z_{s})V_{\Delta_{3}}(z_{3})V_{\Delta_{4}}(z_{4})\rangle}{C_{s34}}=D_{z_{s}}(L)\mathfrak{F}^{(3)}(\Delta_{s},\Delta_{3},\Delta_{4}|z_{s},z_{3},z_{4})$,
where $D_{z_{s}}(L)$ is the differential operator in $z_{s}$ associated to the
descendant $L\in\mathfrak{L}$.

Remark: The term in the bracket of equation (I.32) correponds to the limit
$(z_{i})_{i=1,...,4}\rightarrow(x,0,\infty,1)$ i.e.
$\mathfrak{F}^{(4)}_{\Delta_{s}}(...|x,0,\infty,1)$.

For the sake of completeness, we show how the u-channel block is related to
the s-channel block. In the u-channel we apply the OPE to the fields $V_{1}$
and $V_{3}$. The resulting u-channel block differs by a permutation from the
s-channel block,

$\mathfrak{F}^{(4,u)}_{\Delta_{u}}(\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}|z_{1},z_{2},z_{3},z_{4})=\mathfrak{F}^{(4,s)}_{\Delta_{u}}(\Delta_{1},\Delta_{3},\Delta_{2},\Delta_{4}|z_{1},z_{3},z_{2},z_{4}).$
(I.33)

In the limit $(z_{i})_{i=1,...,4}\rightarrow(x,0,\infty,1)$ and using the
definition
$\mathfrak{F}^{(4)}(\Delta_{i}|x):=\mathfrak{F}^{(4)}(\Delta_{i}|x,0,\infty,1)$,
we get

$\mathfrak{F}^{(4,u)}_{\Delta_{u}}(\Delta_{i}|x)=x^{-2\Delta_{1}}\mathfrak{F}^{(4,s)}_{\Delta_{u}}(\Delta_{i}|\frac{1}{x}),$
(I.34)

where we used the conformal transformation $x\mapsto\frac{0x+i}{ix+0}$. A
similiar statement can be worked out for the t- and s- channel block and the
t- and u- channel block.

####  I.4.3 Torus: 1-point block

If the underlying space is any Riemann surface, in our example the torus, we
need to account for the extra structure to calculate the correlation
functions. A torus can be described by a 2d lattice on the complex plane. We
identify the points $z\sim w$ if and only if $z=w+nw_{1}+mw_{2}$, where $n,m$
are integers and $w_{1},w_{2}$ linearly independent complex numbers, spanning
the lattice. Global conformal invariance allows us to rescale the lattice
vectors to be $w_{1}=1$ and $w_{2}=\tau\in\mathbb{H}$ in the upper halfplane.
Fields must respect the identification $\phi(z)=\phi(w)$ if $w\sim z$. The
torus is then the quotient space $\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ :=
$\mathbb{C}/\sim$.

In the path integral formalism it is easier to implement the identification
$\phi(z)=\phi(w)$ if $z\sim w$. The path integral formalism needs a Lagrangian
description, but the result can be generalized and only needs a well defined
spectrum $\mathbb{S}$ to trace over,

$\langle
G[\phi](x_{i})\rangle\propto\int_{\phi(z)=\phi(w)}\mathrm{D}fG[\phi](x_{i})e^{-S_{E}(\phi)}=\mathrm{tr}_{\mathbb{S}}(G[\phi](x_{i})e^{2\pi
i\tau(L_{0}-\frac{c}{24})}e^{-2\pi i\bar{\tau}(\bar{L}_{0}-\frac{c}{24})}).$
(I.35)

For $G:=1$, (I.35) is known as the torus partition function
$q^{\Delta-\frac{c-1}{24}}\eta(q)^{-1}$, where
$\eta(q)=q^{\frac{1}{24}}\prod_{i\in\mathbb{N}_{*}}(1-q^{i})$ is the Dedekind
eta function [3]. The partition function is invariant under modular
transformations $\mathrm{PSL}(\mathbb{Z})$ of $\tau$, which corresponds to the
freedom of choosing a lattice base.

Evaluating the 1-point function
$G[\phi_{\Delta,\bar{\Delta}}]=\phi_{\Delta,\bar{\Delta}}$ in equation (I.35),
in terms of blocks yields ($q:=e^{2\pi i\tau}$),

$\langle\phi_{\Delta_{1},\bar{\Delta}_{1}}(0)\rangle=\sum_{\Delta,\bar{\Delta}}C_{\Delta_{1},\bar{\Delta}_{1},\Delta,\bar{\Delta}}|\mathfrak{T}^{(1)}_{\Delta}(\Delta_{1}|q)|^{2},$
(I.36)

where we introduced the torus 1-point block $\mathfrak{T}^{(1)}$.

A $\Delta$-recursive presentation of the torus 1-point block was proposed by
Fateev and Litvinov [4],

$\displaystyle\begin{aligned}
\mathfrak{T}^{(1)}_{\Delta}(\Delta_{1}|q)&=\frac{q^{\Delta-\frac{c-1}{24}}}{\eta(q)}H_{\Delta}(\Delta_{1}|q)\\\
H_{\Delta}&=1+\sum_{m,n=1}^{\infty}\frac{q^{mn}R_{m,n}(\Delta_{1},c)}{\Delta-\Delta_{m,n}}H_{\Delta_{m,-n}},\end{aligned}$
(I.37)

where $R_{m,n}$ is defined in equation (II.1). This recursion, seen as a
series in $q$, makes the poles $\Delta=\Delta_{m,n}$ manifest. The main
disadvantage this recursion relation entails, are the additional singularities
in the central charge $c$. They are considered unphysical in the sense, that
they are expected to vanish after summation. In section (II) the recursion
relation is expressed in an explicit form, to make the $c$-singularities
present and a method is proposed to calculate a $c$-singularity free
expression at each order in $q$.

###  I.5 Fusion algebra and fusion rules

An important feature of the $VV$-OPE (I.26) is the fusion algebra. It encodes
the rules which representations (in our case Verma modules (I.4) or degenerate
representations (I.2.2)) occur in the OPE and is thus similiar to the
decomposition of tensor products of representations. It is usually denoted as
$R_{1}\times R_{2}=\oplus_{i}N^{i}_{12}R_{i},N\in\mathbb{N}$ and shares the
same property as the OPE, namely it is bilinear, associative and commutative.
In this work we are more interested if a certain representation occurs in the
fusion product and thus tend to not specify the value of $N^{i}_{12}$ if it is
non-zero. The most prominent example is the fusion of degenerate
representations $R_{r,s}$ with Verma modules $\mathcal{V}_{\Delta}$. Consider
the action $L_{r,s}$ on the degenerate field $L_{r,s}V_{\Delta_{<r,s>}}=0$
within a 3-point function of diagonal fields,

$\langle(L_{r,s}V_{\Delta_{<r,s>}})V_{\Delta_{2}}(z_{2})V_{\Delta_{3}}(z_{3})\rangle=0.$
(I.38)

It turns out that this condition constrains the conformal dimension
$\Delta_{3}$ as follow,

$\Delta(P_{3})=\Delta(P_{2}+ib+jb^{-1}),$ (I.39)

for $i\in I:=\\{-\frac{r-1}{2},...,\frac{r-1}{2}\\}$ and $j\in
J:=\\{-\frac{s-1}{2},...,\frac{s-1}{2}\\}$. The condition (I.39) is equivalent
to the fusion rule,

$R_{r,s}\times\mathcal{V}_{\Delta(P)}=\sum_{i\in I}\sum_{j\in
J}\mathcal{V}_{\Delta(P+ib+jb^{-1})}.$ (I.40)

Fusion rules in a CFT with the larger symmerty algebra
$\hat{\mathfrak{sl}}_{2}$ will be discussed and derived in chapter (III.3).

##  II 1-point block on the torus

We have seen in section (I.4.3) a recursive formulation (I.37) to calculate
the 1-point block of a primary on the torus. As already pointed out there will
arrise poles in the central charge $c$. In this section we show explicitly
that the $c$-poles, of the first four orders in fact cancel and assume they
cancel at each order i.e. they are a mathematical artefact of the recursive
formulation. To start the discussion we introduce the factors $R_{m,n}$,
useful reformulations and definitions. It is of importance to keep in mind the
three variables we are going to work with: the central charge $c$ in terms of
$b$ (I.8), the external conformal dimension $\Delta_{1}$ and the internal
conformal dimension $\Delta$.

###  II.1 Preliminaries

The $R_{m,n}$-factors in equation (I.37) are defined with help of the momentum
$P_{<r,s>}$ (recall I.8),

$\displaystyle
R_{m,n}:=\frac{2P_{<0,0>}P_{<m,n>}}{\prod_{r=1-m}^{m}\prod_{s=1-n}^{n}2P_{<r,s>}}\prod_{r\underset{2}{=}1-2m}^{2m-1}\prod_{s\underset{2}{=}1-2n}^{2n-1}(P_{1}+P_{<r,s>}),$
(II.1)

where the subscript 2 under the equation sign indicates, to increment the
steps in the product by two instead of one.

Note: We have seen that $P_{<r,s>}$ under $b$-symmetry exchanges
$r\leftrightarrow s$ and therefore $R_{r,s}$ has the same feature
$R_{r,s}(b)=R_{s,r}(b^{-1})$.

For convenience we split $R_{m,n}=E_{m,n}F_{m,n}$ and write it explicitly in
terms of $\beta:=b^{2}$, $\Delta_{1}$ and $\Delta$,

$\displaystyle\begin{aligned}
F_{m,n}&:=\prod_{r=^{2}1}^{2m-1}\prod_{s=^{2}1}^{2n-1}(\Delta_{<r,s>}-\Delta_{1})(\Delta_{<r,-s>}-\Delta_{1})\\\
E_{m,n}&:=\frac{mn}{2m!^{2}n!^{2}}\beta^{2m(n-1)}\prod_{r=1}^{m-1}\frac{-n^{-2}}{1-\frac{r^{2}}{n^{2}}\beta^{2}}\prod_{s=1}^{n-1}\frac{s^{-2}}{1-\frac{m^{2}}{s^{2}}\beta^{2}}\prod_{r=1}^{m-1}\prod_{s=1}^{n-1}(\frac{s^{-2}}{1-\frac{r^{2}}{s^{2}}\beta^{2}})^{2}.\end{aligned}$
(II.2)

The reason of this reformulation is to be able to expand it easily around
$\beta=0$, using $\frac{1}{1-ax}=\sum_{i\geq 0}(ax)^{i}$. And it makes the
poles of $E_{m,n}$ at
$\beta\in\\{\pm\frac{1}{1,...,m},...,\pm\frac{n}{1,...,m-1}\\}$ visible.
Whereas $F_{mn}$ is regular in $\beta$ outside $0,\infty$.

We have already discussed the $b$-symmetry in connection with degenerate
representations. $b$-symmetry is manifest with respect to observables because
it leaves the central charge (I.8) invariant. Thus we define the $b$-symmetric
Laurent-polynomials,

$\displaystyle B_{0}:=1,\quad B_{j}:=\beta^{j}+\beta^{-j},\quad
j\in\mathbb{N}_{>0},$ (II.3)

because they will help us to formulate the results in a nice way. Since the
recursion sums up terms in the range $1\leq mn\leq N$, it is convenient to
define the number $P_{N}:=\sum_{mn\leq N}1$. It counts how many possibilities
$(m,n)\in\mathbb{N}^{2}_{*}$ exist with product $mn\leq N$. As an example we
give the first four cases,

$P_{1}=1,\quad P_{2}=3,\quad P_{3}=5,\quad P_{4}=8.$ (II.4)

Another common expression at fixed $(m,n)\in\mathbb{N}_{*}^{2}$, which is used
to calculate a $c$-pole free recursion is,

$\frac{1}{\prod_{rs\leq
N-mn}(\Delta_{m,-n}-\Delta_{r,s})}=(4\beta)^{P_{N-mn}}\prod_{rs\leq
N-mn}\frac{1}{s-n+(r+m)\beta}\prod_{rs\leq N-mn}\frac{1}{s+n+(r-m)\beta},$
(II.5)

where we rewrote it, to be able to expand it easily around $\beta=0$. In this
expansion the smallest power of the $\beta$-series appearing in (II.5), is
$P_{N-mn}-(\big{\lfloor}\frac{N}{n}\big{\rfloor}-m)$, where we counted the
amount of times the condition $s=n$ within the left product on the right-hand-
side in (II.5) is fulfilled. For example in the case $m=1=n$ and $N=4$, we
have the smallest $\beta$-power
$P_{4-1}-(\big{\lfloor}\frac{4}{1}\big{\rfloor}-1)=5-(4-1)=2$.

####  II.1.1 Recursion in powers of $q$

We rewrite the recursion (I.37) in a power series expansion of $q$,

$H_{\Delta}(P_{1}|q)=1+\sum_{i=1}^{\infty}q^{i}H^{(i)}_{\Delta}(P_{1}).$
(II.6)

At each order $i$ (II.6) the $H^{(i)}_{\Delta}$ can be reexpressed as:

$\displaystyle\begin{aligned}
H^{(1)}_{\Delta}(P_{1})&=\frac{R_{1,1}}{\Delta}\\\
H^{(2)}_{\Delta}(P_{1})&=H^{(1)}_{\Delta=1}H^{(1)}_{\Delta}+\frac{R_{2,1}}{\Delta-\Delta_{<2,1>}}+\frac{R_{1,2}}{\Delta-\Delta_{<1,2>}}=\frac{R_{1,1}^{2}}{\Delta}+\frac{R_{2,1}}{\Delta-\Delta_{<2,1>}}+\frac{R_{1,2}}{\Delta-\Delta_{<1,2>}}\\\
H^{(3)}_{\Delta}(P_{1})&=H^{(2)}_{\Delta=1}H^{(1)}_{\Delta}+\frac{R_{2,1}R_{1,1}}{(\Delta-\Delta_{<2,1>})\Delta_{<2,-1>}}+\frac{R_{1,2}R_{1,1}}{(\Delta-\Delta_{<1,2>})\Delta_{<1,-2>}}+\frac{R_{3,1}}{\Delta-\Delta_{<3,1>}}+\frac{R_{1,3}}{\Delta-\Delta_{<1,3>}}\\\
H^{(4)}_{\Delta}(P_{1})&=\frac{R_{1,1}}{\Delta}H^{(3)}_{\Delta=1}+\frac{R_{2,1}}{\Delta-\Delta_{<2,1>}}H^{(2)}_{\Delta=\Delta_{<2,-1>}}+\frac{R_{1,2}}{\Delta-\Delta_{<1,2>}}H^{(2)}_{\Delta=\Delta_{<1,-2>}}+\frac{R_{3,1}}{\Delta-\Delta_{<3,1>}}H^{(1)}_{\Delta=\Delta_{<3,-1>}}\\\
&+\frac{R_{1,3}}{\Delta-\Delta_{<1,3>}}H^{(1)}_{\Delta=\Delta_{<1,-3>}}+\frac{R_{4,1}}{\Delta-\Delta_{<4,1>}}+\frac{R_{2,2}}{\Delta-\Delta_{<2,2>}}+\frac{R_{1,4}}{\Delta-\Delta_{<1,4>}}\\\
&\vdots\\\ H^{(N)}_{\Delta}(P_{1})&=\sum_{mn\leq
N}\frac{R_{m,n}}{\Delta-\Delta_{m,n}}H^{(N-mn)}_{\Delta=\Delta_{<m,-n>}}(P_{1}),\end{aligned}$
(II.7)

where we define the zero’th order $H^{(0)}_{\Delta}:=1$. Using equation (II.1)
we calculate $R_{1,1}=\frac{\Delta_{1}(\Delta_{1}-1)}{2}$ and therefore see
that there are no poles in the central charge at order 1.

###  II.2 Singularity free at order 2: hands on calculation

The poles in $c$ start to make trouble at order 2 and we go through the steps
to get rid of it. Let’s consider the expression of $H^{(2)}_{\Delta}$,

$H^{(2)}_{\Delta}(P_{1})=\frac{R_{1,1}^{2}}{\Delta}+\frac{R_{2,1}}{\Delta-\Delta_{<2,1>}}+\frac{R_{1,2}}{\Delta-\Delta_{<1,2>}}.$
(II.8)

Altough $R_{1,1}$ is regular, $R_{2,1}(\beta)$ is not and we calculate using
(II.2),

$E_{2,1}(\beta)=\frac{1}{4}\frac{1}{\beta^{2}-1},\quad
E_{1,2}(\beta)=\beta^{2}E_{2,1}(\beta),$ (II.9)

and therefore the extra poles $\beta=\pm 1$ arise. To cancel the extra poles,
we pair the second and third term in (II.8) and bring them to the same
denominator,

$\frac{\beta
E_{1,2}}{(\Delta-\Delta_{<1,2>})(\Delta-\Delta_{<2,1>})}\bigg{(}\beta^{-1}F_{1,2}(\Delta-\Delta_{<2,1>})-\beta
F_{2,1}(\Delta-\Delta_{<1,2>})\bigg{)}.$ (II.10)

We observe that the factor in brackets in (II.10) changes sign under
$b$-symmetry transformation. Laurent-polynomials with this property can be
written as a sum of
$\beta^{n}-\beta^{-n}=(\beta-\beta^{-1})\sum_{i\underset{2}{=}-(n-1)}^{n-1}\beta^{i}$
and therefore the poles (II.9) cancel i.e. the poles are unphysical.

A more carful analysis and bringing all terms in (II.8) down to a common
denominator, yields the pole free expression at order 2,

$\displaystyle
H^{(2)}_{\Delta}=\frac{1}{32}\frac{R_{1,1}}{\Delta(\Delta-\Delta_{<1,2>})(\Delta-\Delta_{<2,1>})}\big{(}96\Delta^{2}+4\Delta(-14\Delta_{1}+2\Delta_{1}^{2}+5+c)+c\Delta_{1}(\Delta_{1}-1)\big{)}.$
(II.11)

This procedure can be applied hypothetically at any order $N$, where we
identify terms with the same poles, bring it down to a common denominator and
finally cancel the pole. At higher order there arise more and more pole in
each term because of the expression $E_{mn}$ in (II.2). Therefore it is
getting out of control to cancel pole by pole. Nevertheless it has been done
up to order 3 and 4, where the reader find the solutions in the appendix (A).

###  II.3 Proposal: Singularity free at each order

To start the discussion, we state first the following conjecture,

######  Conjecture 1.

At each order $H^{(N)}_{\Delta}$ the poles in the central charge $c$ cancel in
between terms and therefore a singularity free expression exists.

Second we define the functions $K_{N}(\Delta,\Delta_{1},\beta)$ in the sense
of the recursion (I.37),

$\displaystyle\begin{aligned} H^{(N)}_{\Delta}&=\frac{R_{11}}{\prod_{kl\leq
N}(\Delta-\Delta_{<k,l>})}K_{N}\\\ K_{N}&:=\sum_{mn\leq
N}\bigg{(}\frac{E_{mn}F_{mn}}{R_{11}}\prod_{\underset{k,l\neq m,n}{kl\leq
N}}(\Delta-\Delta_{<k,l>})H^{(N-mn)}_{\Delta_{<m,-n>}}\bigg{)}.\end{aligned}$
(II.12)

Investigating the algebraic form of the pole free expression of the first four
orders (II.11 and appendix A) leads us to conjecture the form of the pole free
expression,

######  Conjecture 2.

$K_{N}$ is a $b$-symmetric Laurent-polynomial in $\beta$ of degree
$Q_{N}:=P_{N}-N$, with coefficients polynomial in $\Delta,\Delta_{1}$.
Concretely the algebraic form of $K_{N}$ is given by (recall the definition of
$B_{s}$ (II.3)),

$\displaystyle\boxed{\begin{aligned}
K_{N}(\Delta,\Delta_{1},\beta)=&\biggl{\\{}\sum_{i=0}^{P_{N}-2}\Delta^{P_{N}-1-i}\biggl{(}\sum_{s=0}^{min(Q_{N},i)}\sum_{j=0}^{min(2N-2,2i-2s)}(-1)^{j}C^{N}_{ijs}\Delta_{1}^{j}B_{s}\biggr{)}\biggr{\\}}\\\
&+\Delta_{1}(\Delta_{1}-1)\sum_{s=0}^{Q_{N}}\sum_{j=0}^{2N-4}(-1)^{j}C^{N}_{(P_{N}-1)js}\Delta_{1}^{j}B_{s},\end{aligned}}$
(II.13)

with positive rational coefficients $C^{N}_{ijs}\in\mathbb{Q}_{+}$.

Note that in the first line of (II.13) the power of $\Delta$ is at least one.
Whereas the second line, not included in the sum over $i$, has no
$\Delta$-dependence and the power of $\Delta_{1}$ is at least one.

####  II.3.1 Large $c$ limit

In the large central charge limit of the torus 1-point block, an upper bound
on the degree of the Laurent-polynomial $K_{N}$ can be calculated. Let’s work
for now with a hypothetical degree $\tilde{Q}_{N}$ of the Laurent-polynom
$K_{N}$. The large $c$ limit of the block exists and is represented as [9],

$q^{\frac{c}{24}-\Delta}\mathfrak{T}^{(1)}_{\Delta}(\Delta_{1}|q)=\mathcal{L}_{\Delta}(\Delta_{1}|q)+\mathcal{O}(c^{-1}),$
(II.14)

where $\mathcal{L}_{\Delta}$ is known as the light torus block.

We observe that in the $\beta\rightarrow 0$ limit, the central charge behaves
as $c=6\beta^{-1}\rightarrow\infty$. And conclude that in the recursive
representation (I.37), along with the assumption that $K_{N}$ is a Laurent-
polynom of degree $\tilde{Q}_{N}$, the limit $\beta\rightarrow 0$ exists and
is given by,

$\displaystyle
q^{\frac{c}{24}-\Delta}\mathfrak{T}^{(1)}_{\Delta}(\Delta_{1}|q)\propto\sum_{N\geq
0}\frac{q^{N}}{\prod_{rs\leq
N}(\Delta-\Delta_{r,s})}K_{N}(\beta)\underset{\beta\rightarrow
0}{\sim}\sum_{N\geq 0}q^{N}\mathcal{O}(\beta^{Q_{N}-\tilde{Q}_{N}}),$ (II.15)

where the dominant behaviour of $\prod_{rs\leq N}(\Delta-\Delta_{r,s})^{-1}$
is given by $\beta^{Q_{N}}(1+\mathcal{O}(\beta))$. Therefore we conclude that
the limit (II.15) exists only if $\tilde{Q}_{N}\leq Q_{N}$.

####  II.3.2 Tracking $\Delta$ in (II.12)

By multiplying out the product $\prod_{\underset{k,l\neq m,n}{kl\leq
N}}(\Delta-\Delta_{k,l})$ in (II.12), we inspect the prefactors of the
$\Delta^{i}$-monomials for $i\in\\{1,...,P_{N}-1\\}$ in $K_{N}$ and compare it
to the conjecture (II.13). Those prefactors are manifest $b$-symmetry
invariant 111The $b$-symmetry invariance can be seen if we pair $(m,n)$\- with
$(n,m)$-terms in (II.12). Due to the expected, very restricted form of
$K_{N}$, there will arise many non-trivial constraints.

($\Delta^{0}$): The coefficient in front of $\Delta^{0}$ takes a very simple
form,

$K_{N}(\Delta=0,\Delta_{1})=\Delta^{0}\times\biggl{\\{}(-1)^{P_{N}-1}\frac{\big{(}\prod_{\underset{k,l\neq
1,1}{kl\leq N}}\Delta_{<k,l>}\big{)}\cdot R_{11}}{\prod_{rs\leq
N-1}\big{(}1-\Delta_{<r,s>}\big{)}}K_{N-1}(\Delta=1,\Delta_{1})\biggr{\\}}.$
(II.16)

Thus we can check if the power $Q_{N}$ is consistent and indeed given by the
upper bound. We apply the conjecture on $K_{N-1}$, i.e. it is a Laurent-
polynomial of degree $Q_{N-1}$ and expand everything else in (II.16) around
$\beta=0$ (see II.5). In the resulting product of $\beta$-series, we observe
that the smallest power of $\beta$ appearing is $-Q_{N}$ and therefore get the
expected power of $Q_{N}$.

In fact, the factor $2R_{11}=\Delta_{1}(\Delta_{1}-1)$ in (II.16) implies the
zeroes
$K_{N}(\Delta=0,\Delta_{1}=0,\beta)=0=K_{N}(\Delta=0,\Delta_{1}=1,\beta)$,
because $K_{N}$ is regular in $\Delta_{1}$. Thats why we conclude that no
$\Delta^{0}\Delta_{1}^{0}\beta^{i}$-terms appear as conjectured, but the
factor $R_{11}\propto\Delta_{1}(\Delta_{1}-1)$ appears in the second line of
(II.13).

($\Delta^{P_{N}-1}$): Another interesting identity arises from the prefactor
of $\Delta^{P_{N}-1}$. Using the conjecture we infer that,

$\sum_{mn\leq N}\frac{R_{mn}}{R_{11}}H^{(N-mn)}_{\Delta_{<m,-n>}},$ (II.17)

is just a number, independent of $\beta$, $\Delta$ and $\Delta_{1}$ and thus
equal to $C^{N}_{000}$. Assuming this number exists, we are able to determine
a closed formula for it,

$C^{N}_{000}=\sum_{mn=N}n,$ (II.18)

where we took in (II.17) the limit $\Delta_{1}\rightarrow 0$ and evaluated the
term proportional to $\beta^{0}$. In the $\Delta_{1}$-limit only the $mn=N$ \-
terms contribute, because $R_{mn}\propto\Delta_{1}(\Delta_{1}-1)$ and
$H^{(N-mn)}\propto\Delta_{1}(\Delta_{1}-1)$ are simply zero (for
$\Delta_{1}=0$ the 1-point function (I.36) is simply equal to the torus
partition function and therefore $H_{\Delta}(\Delta_{1}=0,\beta)=1$ (II.6)).
And in the final step we determined the $\beta=0$ \- expansion of
$E_{mn}F_{mn}/R_{11}\underset{\Delta_{1}=0}{=}n+\mathcal{O}(\beta)$, using the
expressions of $E_{mn}$ and $F_{mn}$ (II.2). For example $C^{N}_{000}$ for the
first 6 orders is given by,

$C^{1}_{000}=1,\quad C^{2}_{000}=3,\quad C^{3}_{000}=4,\quad
C^{4}_{000}=7,\quad C^{5}_{000}=6,\quad C^{6}_{000}=12,$ (II.19)

which coincides with the results of the pole free expressions at order 1, 2, 3
and 4 (II.11), (appendix A).

Tracking in (II.17) the smallest $\beta$-power in the $\beta=0$ expansion
(which occurs at $(m,n)=(1,2)$ 222The number $\lfloor\frac{N}{n}\rfloor-m$ is
maximum for positive integers $(1,1)\neq(m,n)=(1,2)$.) leads to the
constraint,

$0=\big{(}\beta^{\lfloor
N/2\rfloor-2}H^{N-2}_{\Delta_{1,-2}}\big{)}\big{|}_{\beta=0}$ (II.20)

if $2-\lfloor N/2\rfloor<0$. For example at order $N=6$, we might have naively
expected a non-zero contribution at $\beta^{2-\lfloor 6/2\rfloor}=\beta^{-1}$
but it has been numerically checked that (II.20) is indeed fullfilled for
$N=6$.

####  II.3.3 Calculation of pole free expression

We have conjectured the existence of a pole free expression and deduced the
algebraic form of the $K_{N}$’s (II.13). Here we propose a way to compute the
pole free expression from the definition of $K_{N}$ (II.12),

######  Corollary 1.

The pole free expression in conjecture (II.13) can be calculated by expanding
$K_{N}$ (II.12) around $\beta=0$,

$\underset{\beta\rightarrow
0}{\mathrm{lim}}K_{N}=K_{N}(\Delta,\Delta_{1})_{Q_{N}}\beta^{-Q_{N}}+K_{N}(\Delta,\Delta_{1})_{1-Q_{N}}\beta^{-Q_{N}+1}+...+K_{N}(\Delta,\Delta_{1})_{0}\beta^{0}+\mathcal{O}(\beta^{1}),$
(II.21)

where
$K_{N}(\Delta,\Delta_{1},\beta)=\sum_{j=0}^{Q_{N}}K_{N}(\Delta,\Delta_{1})_{j}B_{j}$
is the pole free expression.

Expanding (II.21) around $\beta=0$ seems hard to evaluate and thus we want to
give a short description how to do it. Recall the definition of $K_{N}$,

$K_{N}=\sum_{mn\leq N}\bigg{\\{}\frac{E_{mn}F_{mn}}{R_{11}}\prod_{k,l\neq
m,n}(\Delta-\Delta_{<k,l>})\bigg{(}\frac{R_{11}}{\prod_{kl\leq
N-mn}(\Delta_{<m,-n>}-\Delta_{<k,l>})}K_{N-mn}\bigg{)}^{1-\delta_{N,mn}}\bigg{\\}},$
(II.22)

where we recursively replaced $H^{(N-mn)}_{\Delta_{<m,-n>}}$ by the expression
(II.12). The power $1-\delta_{N,mn}$ needs to be added because
$H^{(N-mn)}_{\Delta}=1$, if $mn=N$. Equation (II.22) is enough to calculate
the $\beta=0$ expansion. We can simply replace the factors $E_{mn}$ by
expression (II.2) and the factor $\prod_{rs\leq
N-mn}(\Delta_{m,-n}-\Delta_{r,s})^{-1}$ by (II.5). In this substitution
(II.22) is completely written as a sum of products of $\beta$-series 333The
series are either finite or infinite, but with a lower bound $\gamma>-\infty$
on the powers of $\beta$, $\sum_{i=\gamma}^{\infty}a_{i}\beta^{i}$.. This is
because we observe that any other factor, for example $K_{N-mn}$ or $F_{mn}$
(II.2), arising in $K_{N}$, is already a Laurent-polynom in $\beta$. The last
step is to evaluate the product of all series and Laurent-polynomials in
$\beta$ and to pick up the prefactors of $\beta^{-Q_{N}}$ up to $\beta^{0}$.

Remark: There is no need to treat $\Delta_{1}$ as a variable, since the
recursion (II.7) does not need informations concering $\Delta_{1}$ and thus
can be fixed. It is nevertheless useful to know the $\Delta_{1}$-dependence,
otherwise the $K_{N}$’s for fixed $\Delta_{1}$ have to be calculated again for
a different choice of $\Delta_{1}$.

##  III Affine symmetry

In this section we introduce a CFT with an additional symmetry described by
the affine Lie algebra $\hat{\mathfrak{g}}$. For a general Lie algebra
$\mathfrak{g}$, we define holomorphic currents $J^{a}(z)$ implicitly through
their OPE,

$J^{a}(y)J^{b}(z)=\frac{kK^{ab}}{(y-z)^{2}}+\frac{f^{ab}_{c}J^{c}(z)}{(y-z)}+O(1),$
(III.1)

where we introduced the level $k$, the structure constant $f^{ab}_{c}$ of the
Lie algebra $\mathfrak{g}$ and the Killing form
$K^{ab}:=\frac{1}{2g}f^{ac}_{d}f^{bd}_{c}$ ($g$ is the dual coxeter number of
the Lie algebra $\mathfrak{g}$, in the case of $\mathfrak{sl}_{N}$ we have
$g=N$). By extracting the modes of the current $J^{a}(y)$,

$J^{a,(z)}_{n}:=\oint_{z}dy(y-z)^{n}J^{a}(y),$ (III.2)

we get the generators $J^{a}_{n}$ of the affine Lie algebra
$\hat{\mathfrak{g}}$, which obey the commutation relations,

$[J^{a}_{n},J^{b}_{m}]=f^{ab}_{c}J^{c}_{n+m}+nkK^{ab}\delta_{n+m,0}.$ (III.3)

Here the level $k$ becomes the central element of the algebra 222For
indecomposable representations of $\hat{\mathfrak{g}}$, the level $k$ acts as
a number.. Do not confuse the level $k$ with the level of descendence $N$. The
level of descendence of an element
$J^{A}_{N}:=(J^{a_{1}}_{-n_{1}}\cdot...\cdot J^{a_{M}}_{-n_{M}})_{a_{i}\in
A}\in U(\hat{\mathfrak{g}})$, where $A$ is some discrete set, is defined to be
the integer $N:=\sum_{i=1}^{M}n_{i}$.

######  Definition 2 (Horizontal algebra).

The subalgebra of $\hat{\mathfrak{g}}$ generated by the set
$\\{J^{a}_{0}\\}_{a}$ (we check that the commutations relations (III.3) are
invariant for $n=0$), is isomorph to the underlying Lie algebra
$\mathfrak{g}$. Let’s call the subalgebra in this context ’horizontal
algebra’.

Similiar to the Virasoro case a Virasoro field can be constructed to generate
the conformal symmetry of the affine symmetric CFT,

$T(z):=\frac{K_{ab}(J^{a}J^{b})(z)}{2(k+g)},$ (III.4)

where we introduced the normal ordered product,

$(AB)(z)=\frac{1}{2\pi i}\oint_{z}\frac{\mathrm{d}z}{y-z}A(y)B(z).$ (III.5)

Equation (III.4) is better known as the Sugawara construction and is
consistent only if $k\neq-2$. The modes of $T(z)$ fulfill the Virasoro
commutation relation (I.3) with central charge
$c=\frac{k\mathrm{dim(\mathfrak{g})}}{k+g}$.

###  III.1 Affine primary fields and Ward identities

An affine primary field $\Phi^{R}(z)$ is defined by the OPE with the current
$J^{a}(z)$,

$J^{a}(y)\Phi^{R}(z)=\frac{-R(t^{a})^{T}\Phi^{R}(z)}{(y-z)}+O(1),$ (III.6)

where $R$ is an arbritrary representation of the horizontal algebra. For
convenience lets call $R$ the horizontal representation111The reader is
advised to read appendix (B) to get a short introduction to
$\mathfrak{sl}_{2}$-irreducible representations, which are prominent examples
of horizontal representations., where the state $|\Phi^{R}\rangle$
corresponding to $\Phi^{R}(z)$, is a state in the horizontal representation.
And the action of $R$ (III.6) is better understood if written in a basis of
the horizontal representation,
$\Phi_{i}\mapsto-R(t^{a})^{T}\Phi_{i}=-R(t^{a})_{ij}\Phi_{j}$. Note that we
expect the OPE $J^{a}J^{b}\Phi$ to be associative and therefore need the minus
sign in (III.6). The primary field $\Phi^{R}(z)$ and the Virasoro field $T(z)$
(III.4) satisfies the OPE (I.14) and is thus a primary in the sense of the
Virasoro algebra. Therefore correlators of affine primaries fulfill the
Virasoro Ward identities and the results from section I can be used.

####  III.1.1 Isospin variables

Affine primary fields $\Phi^{R}$ transform linearly under the representation
$R$ of the horizontal algebra $\mathfrak{g}$. Moreover the primaries can be
represented as functions $\Phi^{R}_{x}$ depending on the isospin variable $x$.
$R$ acts then on the isospin variable $x$ as differential operators
$D^{R}_{x}(t^{a})\Phi^{R}_{x}:=R(t^{a})\Phi^{R}_{x}$. In the
$\mathfrak{sl}_{2}$-case a common choice of basis, the $x$-basis, is given by
the differential operators,

$\displaystyle D^{j}_{x}(t^{-})=-\partial_{x},\quad
D^{j}_{x}(t^{0})=x\partial_{x}-j\quad D^{j}_{x}(t^{+})=x^{2}\partial_{x}-2jx.$
(III.7)

In the initial definition (III.6) the action is defined by its transposed,
such that in the isospin formalism the consecutive action acts as
$J^{a}_{0}J^{b}_{0}\Phi^{R}_{x}(z)=D^{R}_{x}(t^{b})D^{R}_{x}(t^{a})\Phi^{R}_{x}(z)$.

In the same way, as in the Virasoro case, we can derive affine Ward identities
to constrain the N-point functions.

####  III.1.2 Global Ward identity

The global Ward identities for a given basis $\\{t^{a}\\}_{a}$ of the
horizontal algebra read,

$0=\sum_{i=1}^{N}D^{R_{i}}_{X_{i}}(t^{a})\langle\prod^{N}_{i=1}\Phi^{R_{i}}_{X_{i}}(z_{i})\rangle.$
(III.8)

In the $x$-basis and the case of $\mathfrak{sl}_{2}$, the solution of the
global Ward identities (III.8) for the three point function is given by,

$\langle\Phi^{j_{1}}_{x_{1}}(z_{1})\Phi^{j_{2}}_{x_{2}}(z_{2})\Phi^{j_{3}}_{x_{3}}(z_{3})\rangle\propto
x^{j^{3}_{12}}_{12}x^{j^{1}_{23}}_{23}x^{j^{2}_{31}}_{31},$ (III.9)

where we again used the short hand notation $j_{I}^{J}=\sum_{i\in
I}j_{i}-\sum_{j\in J}j_{j}$ and $x_{ij}:=x_{i}-x_{j}$.

####  III.1.3 Local Ward identity

If all except one field in the N-point functions are affine primaries, a
useful local Ward identity reads,

$\langle J^{a}_{n<0}\Phi^{\sigma_{i}}(z_{i})\prod_{j\neq
i}\Phi^{R_{j}}_{X_{j}}(z_{j})\rangle=\sum_{j\neq
i}\frac{D^{R_{j}}_{X_{j}}(t^{a})}{z_{ji}^{n}}\langle\Phi^{\sigma_{i}}(z_{i})\prod_{j\neq
i}\Phi^{R_{j}}_{X_{j}}(z_{j})\rangle.$ (III.10)

Both, global and local Ward identities, along with the three point function
(III.9), will be used in section (III.3) to calculate fusion rules between
affine primary fields and degenerate fields.

###  III.2 $\hat{\mathfrak{sl}}_{2}$ degenerate representations

In this section we proceed to discuss degenerate representations and determine
null vectors in the case of general $\mathfrak{sl}_{2}$ horizontal
representations. Remember a degenerate representation is constructed by
modding out the sub-representations, generated by the null vectors. We work
with the variable $t:=k+2$.

As for our purpose ($\mathfrak{sl_{2}}$) we use the $(0,\pm)$-basis with Lie
bracket,

$\displaystyle[t^{0},t^{\pm}]=\pm t^{\pm},\quad[t^{+},t^{-}]=2t^{0}.$ (III.11)

The corresponding symmetric Killing form is given by $K_{00}=2,K_{+-}=1$, with
inverse $K^{00}=\frac{1}{2},K^{+-}=1$ and is used to raise and lower indices.
Specifically in the $\mathfrak{sl}_{2}$ case, the structure constant fulfills
the non-trivial identity
$f^{ab}_{i}f^{ic}_{d}=2(K^{a}_{d}K^{bc}-K^{ac}K^{b}_{d})$. Furthermore the
quadratic Casimir element $C_{2}=K_{ab}J^{a}_{0}J^{b}_{0}$ in the
$(0,\pm)$-basis is given by
$C_{2}=2((J^{0}_{0})^{2}+J^{+}_{0}J^{-}_{0}-J^{0}_{0})$.

####  III.2.1 Affine highest weight representation

Let’s take an affine Lie algebra $\hat{\mathfrak{g}}$ and a representation
$R(t^{a})$ of the horizontal algebra
$t^{a}\in\mathfrak{g}\subset\hat{\mathfrak{g}}$. The highest weight
representation $\hat{R}$, equivalent to the OPE (III.6)111The equivalence can
be worked out by determining the action of $J^{a}_{n}$ on the primary field
$\Phi^{R}$, which is indeed a highest weight representation., is constructed
by generalizing the action from $\mathfrak{g}$ to $\hat{\mathfrak{g}}$ on any
state $|v\rangle\in R$,

$\displaystyle J^{a}_{n>0}|v\rangle=0,\quad
J^{a}_{0}|v\rangle=-(t_{a})^{T}|v\rangle.$ (III.12)

The resulting, generalized action is a highest weight representation of
$\hat{\mathfrak{sl}}_{2}$. Compared to the Virasoro case (I.4), the horizontal
subspace is in general not one dimensional and spanned by a basis of the
horizontal vector space.

Null vectors in highest weight representations (III.12) of affine Lie algebras
are defined generally,

######  Definition 3 (Affine null vector).

A null vector at level $N$
$|\chi,N\rangle=\sum_{i,A}c_{A}^{i}J^{A}_{N}|v_{i}\rangle$ in a highest weight
representation (III.12) fulfills the conditions $J^{a}_{n>0}|\chi,N\rangle=0$
for any $a$.

It is enough to check the null vector condition only for $J^{a}_{1}$ because
of the commutation $[J^{a}_{1},J^{b}_{1}]=f^{ab}_{c}J_{2}^{c}$ and by
induction on the level.

The connection to irreducibility is given by the following theorem [7],

######  Theorem 2.

The highest weight representation (III.12) is irreducible if and only if it
contains no null vectors.

Moreover the existence of null vectors imply an $\mathfrak{sl}_{2}$-module,

######  Proposition 1.

The left-action of $\mathfrak{sl}_{2}$ on null vectors yields other null
vectors at the same level.

###### Proof.

Let $|\chi,N\rangle$ be a null vector. We check for the null vector condition,
$J^{m}_{1}J^{a}_{0}|\chi,N\rangle=([J^{m}_{1},J^{a}_{0}]+J^{a}_{0}J^{m}_{1})|\chi,N\rangle$
= 0, where we use $[J^{m}_{1},J^{a}_{0}]=f^{ma}_{e}J^{e}_{1}$. ∎

The goal in the next sections is to find null vectors at level 0 and 1, to be
able to reproduce the fusion rules of degenerate representations at any level
similiar to section (I.5).

####  III.2.2 Level 0 null vectors

The highest weight representation extension of the finite irreducible
representation with half integer spin $j=j_{0,s}:=\frac{s-1}{2},\quad
s\in\mathbb{N}_{+}$ (see appendix B) of $\mathfrak{sl}_{2}$ contains a well
known null vector at level 0 [5],

$(J^{-}_{0})^{2j+1}|j,j\rangle=0.$ (III.13)

This null vector satisifes trivially the null vector condition and is actually
of different nature, since it is equal the $0$-element of the representation.
Consequently, we don’t need to mod out the subspace, in order to derive fusion
rules.

####  III.2.3 Level 1 null vectors

Malikov, Feigin and Fuks already determined examples of null vectors at each
level $N\geq 1$, but they worked them out in a specific $\mathfrak{sl}_{2}$
basis and horizontal representation [6]. It is therefore of interest to work
out Lie algebra basis independent null vectors in a horizontal representation-
independent framework. Hence the question arise, if ideas in this framework
around $\mathfrak{sl}_{2}$ can be extended to the Lie algebra
$\mathfrak{sl}_{N}$ or general simple Lie algebras. We go to a general setting
and focus on horizontal representations, which fulfill two properties,

(1) The horizontal representation is indecomposable and thus the quadratic
Casimir $C_{2}$ is proportional to the identity. In the case of
$\mathfrak{sl}_{2}$ we use the parametrization $C_{2}=2j(j+1)$ with reflection
symmetry $j\mapsto-1-j$.
(2) The horizontal representation $R$/$V_{R}$ canonically extends to a highest
weight representation $\hat{R}$/ $\hat{V}_{R}$.

This leads to the natural definition,

######  Definition 4 (Null vector space).

Consider $U(\hat{\mathfrak{sl}}_{2})$ where we mod out the subvectorspaces
$W_{1}:=\mathrm{span}_{\mathbb{C}}(U(\hat{\mathfrak{sl}}_{2})J^{a}_{n>0})$ and
$W_{2}:=\mathrm{span}_{\mathbb{C}}(U(\hat{\mathfrak{sl}}_{2})(C_{2}-2j(j+1)))$.
We call the resulting vector space
$\frac{U(\hat{\mathfrak{sl}}_{2})}{W_{1}+W_{2}}$, the null vector space.

Remember in the universal enveloping framework two elements are equal if they
differ by commutation. For example in $W_{2}$ we have
$J^{+}_{0}J^{0}_{0}J^{-}_{0}=J^{0}_{0}J^{+}_{0}J^{-}_{0}-J^{+}_{0}J^{-}_{0}$
and thus can be replaced by
$J^{+}_{0}J^{-}_{0}=j(j+1)-(J^{0}_{0})^{2}+J^{0}_{0}$. And in $W_{1}$ the
element $J^{+}_{1}J^{-}_{0}=2J^{0}_{1}+J^{-}_{0}J^{+}_{1}$ is simply zero.

The null vector space encodes the action of the affine highest weight
representation, i.e. for any
$T\in\frac{U(\hat{\mathfrak{sl}}_{2})}{W_{1}+W_{2}}$ we define the canonical
map,

$T:V_{R}\rightarrow\hat{V}_{R},\quad|v\rangle\mapsto T|v\rangle.$ (III.14)

We also need to redefine the null vector condition accordingly,

######  Definition 5 (Null operator).

Within the null vector space, the null vector condition transforms into
$\forall a:J^{a}_{1}\hat{T}=0$ or equivalently $\forall
a:[J^{a}_{1},\hat{T}]=0$. $\hat{T}$ is called a null operator and maps any
horizontal state to a null vector via the corresponding map (III.14).

Left-action of $U(\mathfrak{sl}_{2})$ on $\hat{T}$ and the freedom of choice
of $|v\rangle\in V_{R}$ forms a set of null operators
$U(\mathfrak{sl}_{2})\hat{T}U(\mathfrak{sl}_{2})$. This space actually forms a
vector space over $\mathbb{C}$ and is an $\mathfrak{sl}_{2}$-module by means
of the adjoint action. The main advantage of the null vector algebra is that
we are able to search for null operators, without taking care about the affine
highest weight representation.

Eigenspaces:

The null vector space forms an $\mathfrak{sl}_{2}$-module and therefore we can
diagonalize it with respect to the adjoint action $[J^{0}_{0},\circ]$. The
resulting eigenspaces $E^{Q}$ contains elements of constant charge $Q$, where
the charge of an element
$\prod_{i=1}^{M}J^{a_{i}}_{n_{i}}\in\frac{U(\hat{\mathfrak{sl}}_{2})}{W_{1}+W_{2}}$
is given by $Q:=\sum_{i=1}^{M}a_{i}$. Such that for an element $J^{Q}\in
E^{Q}$ we have $[J^{0}_{0},J^{Q}]=QJ^{Q}$.

Universal null vector:

At level 1 we have determined a basis independent null operator,

$\boxed{\hat{T}^{c}_{1}:=2K_{ab}J^{a}_{-1}J^{b}_{0}J_{0}^{c}-tf^{c}_{ab}J^{a}_{-1}J^{b}_{0}-t^{2}J^{c}_{-1},}$
(III.15)

where the quadratic Casimir takes the value $C_{2}=\frac{t^{2}}{2}-t$. We want
to show explicitly that $\hat{T}^{c}_{1}$ fulfills the null vector condition.
To do so let’s act with $J^{d}_{1}$ and use the commutation relation (III.3),

$\displaystyle\begin{aligned}
J^{d}_{1}\hat{T}^{c}_{1}&=2K_{ab}[J^{d}_{1},J^{a}_{-1}]J^{b}_{0}J^{c}_{0}-tf^{c}_{ab}[J^{d}_{1},J^{a}_{-1}]J^{b}_{0}-t^{2}[J^{d}_{1},J^{c}_{-1}]\\\
&=(f^{da}_{e}J^{e}_{0}+kK^{da})(2K_{ab}J^{b}_{0}J^{c}_{0}-tf^{c}_{ab}J^{b}_{0})-t^{2}(f^{dc}_{e}J^{e}_{0}+kK^{dc})=...\end{aligned}$
(III.16)

Then we use the identities $f^{d}_{be}J^{e}_{0}J^{b}_{0}=2J^{d}_{0}$ and
$f^{ab}_{i}f^{ic}_{d}=2(K^{a}_{d}K^{bc}-K^{ac}K^{b}_{d})$ (this identity only
applies to $\mathfrak{sl}_{2}$) and group together similiar terms,

$\displaystyle\begin{aligned}
...=J^{d}_{0}J^{c}_{0}(4+2k)-2tJ^{c}_{0}J^{d}_{0}+K^{cd}(2tC_{2}-t^{2}k)+f^{cd}_{e}J^{e}_{0}(\underbrace{t^{2}-tk-2t}_{=0}).\end{aligned}$
(III.17)

Finally we commute the term
$-2tJ^{c}_{0}J^{d}_{0}=-2t(J^{d}_{0}J^{c}_{0}+f^{cd}_{e}J^{e}_{0})$ and get
that (III.17) is zero if and only if $C_{2}=\frac{tk}{2}=\frac{t^{2}}{2}-t$.
Interestingly the null vector operators $T^{c}_{1}$ satisfy the commutation
relation,

$[J^{a}_{0},T^{b}_{1}]=f^{ab}_{c}T^{c}_{1}.$ (III.18)

Set of null operators:

The main goal is to give a sketch of the proof of the following conjecture,
where we work for simplicity in the $(0,\pm)$-basis,

######  Conjecture 3 (Universal null operator).

The null operator $\hat{T}^{0}_{1}$ at level 1 and $C_{2}=\frac{t^{2}}{2}-t$
is universal, such that the subspace
$\mathrm{span}_{\mathbb{C}}(U(\mathfrak{sl}_{2})\hat{T}^{0}_{1}U(\mathfrak{sl}_{2}))$
contains the set of all null operators in the null vector space
$\\{\hat{T}\in\frac{U(\hat{\mathfrak{sl}}_{2})}{W_{1}+W_{2}}|J^{a}_{1}\hat{T}=0\\}$.

To begin with, we want to show the weaker proposition,

######  Proposition 2.

$\hat{T}^{0}_{1}$ generates any finite null operator in $E^{0}$. The null
operators are generated by
$\hat{T}^{0}_{1}\mathrm{span}_{\mathbb{C}}(J_{0}^{0})$.

In general we expect, that every null operator can be written as a linear
combination of products $\prod^{N}_{i=1}J^{a_{i}}_{n_{i}}$ and therefore the
following proposition helps,

######  Proposition 3.

Every product $\prod^{N}_{i=1}J^{a_{i}}_{n_{i}}$ at level $(-\sum_{i}n_{i})=1$
can be rewritten as a linear combinations of products
$J^{c}_{-1}\prod_{j}J^{b_{j}}_{0}$ such that the charge
$c+\sum_{j}b_{j}=\sum_{i}a_{i}$ stays invariant.

###### Proof.

Identify all the generators $J^{a}_{n>0}$ and commute them to the right, to be
left with generators at level 0 and one generator at level 1. Commute the
level 1 generator to the left. All this operations conserve the level and
charge due to the commutation relation (III.3). ∎

In addition we conclude that if
$\hat{T}=\lambda_{1}\prod_{i}J^{a_{i}}_{n_{i}}+\lambda_{2}\prod_{i}J^{b_{i}}_{m_{i}}$
is a null operator with charge $A\sum a_{i}\neq\sum b_{i}$, then each term is
a null operator by its own, because of linear independence of vectors in
different eigenspaces $E^{Q}$. Therefore at each charge and level, we can
rearrange the null vector candidate into a simple ansatz. The ansatz at level
one and in the Eigenspace of zero charge $E^{0}$ can generally be written as,

$\hat{T}=J^{+}_{-1}J^{-}_{0}(\sum_{i\geq
0}a_{i}(J^{0}_{0})^{i})+J^{0}_{-1}(\sum_{i\geq
0}b_{i}(J^{0}_{0})^{i})+J^{-}_{-1}J^{+}_{0}(\sum_{i\geq
0}c_{i}(J^{0}_{0})^{i}).$ (III.19)

Applying the null operator condition $J^{a}_{1}\hat{T}=0$ to this ansatz
yields the solution,

$b_{i+1}=2a_{i}-ta_{i+1},\quad c_{i+1}=\frac{2}{t}(a_{i}-c_{i})-a_{i+1},$
(III.20)

where we define that quantities with negative indices vanish.

Finite solutions:

A finite solution of (III.20) is defined by the condition $a_{N+(i\geq 1)}=0$
with $N\in\mathbb{N}_{*}$, which implies $b_{N+1+(i\geq 1)}=0$ and
$c_{N+(i\geq 1)}=0$. The recursion (III.20) constrains
$a_{N}=\frac{2}{t}(a_{N-1}-c_{N-1}(a_{i\leq N-1}))-a_{N}$ and therefore we can
freely choose $A_{N}:=(a_{0},...,a_{N-1})$, to determine any finite solutions
for any $N$. We write $\hat{T}_{N}(A_{N})$ for the null vector operator
corresponding to the solution $A_{N}$. The space of finite solutions
$\\{\hat{T}_{N}(A_{N})|A_{N}\in\mathbb{C}^{N},N\in\mathbb{N}\\}$ forms a
vector space because the recursion (III.20) is linear.

We will give two examples of finite solutions at $N=1,2$,

$\displaystyle\begin{aligned}
\hat{T}_{1}(t,2)=&J^{0}_{-1}(4(J^{0}_{0})^{2}-t^{2})+J^{-}_{-1}J^{+}_{0}(2J_{0}^{0}-t)+J_{-1}^{+}J_{0}^{-}(2J_{0}^{0}+t)\\\
\hat{T}_{2}(t,4,4/t)=&J^{0}_{-1}(-t^{2}-2tJ^{0}_{0}+4(J^{0}_{0})^{2}+8/t(J^{0}_{0})^{3})+J^{-}_{-1}J^{+}_{0}(-t+4/t(J^{0}_{0})^{2})\\\
&+J_{-1}^{+}J_{0}^{-}(t+4J^{0}_{0}+4/t(J^{0}_{0})^{2}).\end{aligned}$ (III.21)

Actually we have the redundancy $\hat{T}_{2}=\hat{T_{1}}(2J_{0}^{0}+t)/t$. The
solution $\hat{T}_{1}(t,2)$ corresponds to the initial result (III.15) with
$c=0$. After working out the finite solutions in $E^{0}$, we are ready to
proof proposition 2,

###### Proof.

The proof is done by induction on the positive integer $N$, such that
$a_{N+i}=0$ for $i\geq 1$. We have already seen that at $N=2$ it can be
related to $N=1$ and thus assume it is possible up to $N-1$.
We start by noticing that, due to the linear character of the solution
$(A_{N},a_{N})$ we can choose without loss of generality $a_{0}=1$ and write
$(A_{N},a_{N})=(1,0,...,0,a_{N}^{\prime})+(0,a_{1},...,a_{N}-a_{N}^{\prime})$.
Because of $0=a_{0}=c_{0}=b_{0}$ in the second term, we can factor out
$a_{1}J^{0}_{0}$ in the ansatz (III.15) and relate it to the $N-1$ case.
To decompose the first term $(1,0,...,0,a_{N}^{\prime})$, we notice first that
the recursion (III.20) implies $a_{N}^{\prime}=(-1)^{N+1}(\frac{2}{t})^{N}$.
Therefore we decompose it further
$(1,0,...,0,a_{N}^{\prime})=(1,0...,0,a_{N-1}^{\prime})+(-1)^{N+1}(\frac{2}{t})^{N-1}(0,...,0,1,\frac{2}{t})$
and factor out $(J_{0}^{0})^{N-1}$ in the second term. We see both terms can
be related to lower $N$-cases. This construction is equivalent to
$\hat{T}(1,0,...,a_{N})=\hat{T}(1,0,...,a_{N-1})+(-1)^{N+1}(\frac{2}{t})^{N-1}\hat{T}(1,\frac{2}{t})(J_{0}^{0})^{N-1}$.
Using the induction assumption again, we conclude that we can decompose every
finite solution to $\hat{T}^{0}_{1}\sum_{i}a_{i}(J_{0}^{0})^{i}$. ∎

In different eigenspaces with non-zero charge $E^{Q}$ we can create null
operators by acting multiple time with $J^{\pm}_{0}$ on $\hat{T}^{0}_{1}$ from
the left or right. The last step would be to proof that at each charge $Q\neq
0$ all finite solutions can be generated by $\hat{T}^{0}_{1}$. We haven’t
found a promising proof, but it should closely follow the proof of proposition
2 and therefore leave it as a conjecture,

######  Conjecture 4.

The null vectors in each eigenspace $E^{Q}$ with charge $0\neq Q\in\mathbb{Z}$
are generated by the unique element $T^{0}_{1}$.

Remark: Note that indecomposable horizontal representations do not contain a
unique vector to generate the whole vector space by repeatedly applying the
action, whereas an irreducible does. Therefore we expect that every null
vector in the irreducible case, can be reached with the map (III.14) and the
subspace
$\mathrm{span}_{\mathbb{C}}(U(\mathfrak{sl}_{2})\hat{T}^{0}_{1}U(\mathfrak{sl}_{2}))$
from conjecture 3. It is not entirely clear if the same holds for
indecomposable representations.

###  III.3 $\hat{\mathfrak{sl}}_{2}$ fusion rules

As in the Virasoro case, we deduce the analog of fusion but with primary
fields in $\hat{\mathfrak{sl}}_{2}$-degenerate representations (shorthand
degenerate field). Generally we name the null operator $\hat{T}^{r,s}$ and the
degenerate representation $\hat{R}^{r,s}$, if they correspond to the spin
$j_{r,s}=\frac{s-1}{2}-\frac{t}{2}r$. On the other side we name the highest
weight representation $R_{j}$ with well defined qadratic Casimir $2j(j+1)$ and
spin $j$. The idea is then to use the prominent relation
$0=\hat{T}^{r,s}|v\rangle$ of degenerate representations within N-point
functions,

$0=\langle\hat{T}^{r,s}\Phi_{j_{r,s}}\Phi^{R_{1}}_{j_{1}}\cdot...\cdot\Phi^{R_{N-1}}_{j_{N-1}}\rangle,$
(III.22)

where the degenerate field $\Phi_{r,s}$ corresponds to the state $|v\rangle$.
Then we apply the affine Ward identities (III.8) resp. (III.10) to deduce the
differential equation of (III.22) in the isospin framework. We call these
equations (III.22) null vector equations. Take care that the consecutive
action acts as $J^{a}_{0}J^{b}_{0}=(-D(t^{b}))(-D(t^{a}))$.

####  III.3.1 Finite level 0

For the finite irreducible horizontal representation (III.2.2), we get the
null vector equation $(J^{-}_{0})^{2j+1}|j,j\rangle=0$ with spin $j=j_{0,s}$.
Using the global Ward identity (III.8) and the $x$-basis (III.7), this
constrains the 3-point function (III.9),

$0=(\partial_{x})^{2j+1}\langle\Phi^{j}_{x}(z)\Phi^{j_{2}}_{x_{2}}(z_{2})\Phi^{j_{3}}_{x_{3}}(z_{3})\rangle,$
(III.23)

For a non-zero 3-point function the condition on the spins read,

$\prod_{i=0}^{2j}(j_{3}-j_{2}-j_{0,s}+i)=0.$ (III.24)

This can be identified as the fusion rule of a generic field with a degenerate
field with spin $j_{0,s}$,

$\hat{R}^{0,s}\times\hat{R}_{j}=\hat{R}_{j+j_{0,s}}+\hat{R}_{j+j_{0,s-2}}+...+\hat{R}_{j-j_{0,s}}.$
(III.25)

Thus in general the field with spin $j_{0,1}=0\Rightarrow\Delta_{j=0}=0$
111The conformal dimension of an $\mathfrak{sl}_{2}$ primary is given by
$\Delta_{j}=\frac{j(j+1)}{t}$ acts as the identity field $\Phi^{0}(z)\propto
id$.

####  III.3.2 Generic representation level 1

Altough Malikov, Feigin and Fuks have determined a null vector at level 1, we
would not be able to use it directly to determine the null vector equation.
This is because it is written in a basis which diagonalizes $J^{0}_{0}$.
Therefore we take the degenerate representation with null operator
$\hat{T}_{1}^{0}$ (III.15) on any horizontal state $\hat{T}^{0}_{1}|v\rangle$
and spin $j_{1,1}$. This approach yields the following null vector equation in
the case of the 3-point function,

$\displaystyle\begin{aligned}
0=\sum_{s=2,3}\frac{1}{z_{s1}}\bigg{(}2K_{ab}D_{x_{1}}(t^{0})D_{x_{s}}(t^{a})D_{x_{1}}(t^{b})+tf^{0}_{ab}D_{x_{s}}(t^{a})D_{x_{1}}(t^{b})&\\\
-t^{2}D_{x_{s}}(t^{0})\bigg{)}\langle\Phi^{j_{1,1}}_{x_{1}}(z_{1})\Phi^{j_{2}}_{x_{2}}(z_{2})\Phi^{j_{3}}_{x_{3}}(z_{3})\rangle&.\end{aligned}$
(III.26)

In the limit $z_{1},z_{3}\rightarrow 0,\infty$ only the term proportional to
$\frac{1}{z_{21}}$ stays alive and in the $x$-basis it becomes the
differential equation,

$\displaystyle\begin{aligned}
0=\big{(}&-2x_{1}x_{12}^{2}\partial_{2}(\partial_{1})^{2}+4j_{2}x_{1}x_{21}(\partial_{1})^{2}+4x_{1}(1+t)x_{21}\partial_{2}\partial_{1}\\\
&-4j_{2}x_{1}(1+t)\partial_{1}-2x_{1}t(1+t)\partial_{2}\big{)}\langle\Phi^{j_{1,1}}_{x_{1}}(z_{1})\Phi^{j_{2}}_{x_{2}}(z_{2})\Phi^{j_{3}}_{x_{3}}(z_{3})\rangle.\end{aligned}$
(III.27)

We use the solution of the 3-point function (III.9) and get the following
condition on the spins,

$\displaystyle\begin{aligned}
0=\big{(}&8j_{2}^{3}+8j_{3}^{3}-4j_{3}^{2}(t-2)-2j_{3}t^{2}+(t-2)t^{2}\\\
&-4j_{2}^{2}(2j_{3}+t-2)-2j_{2}(4j_{3}^{2}-4j_{3}(t-2)+t^{2})\big{)}.\end{aligned}$
(III.28)

If the 3-point function is non-zero, this equation has the three solutions
$j_{3}\in\\{j_{2}\pm j_{1,1},-1-j_{2}+j_{1,1}\\}$ 222The parametrization
$C_{2}=2j(j+1)$ contains the symmetry $j\rightarrow-1-j$ and we get equivalent
solutions up to this symmetry.. Therefore the fusion rules can be determined,

$\hat{R}^{1,1}\times\hat{R}_{j}=\hat{R}_{j+j_{1,1}}+\hat{R}_{j-j_{1,1}}.$
(III.29)

####  III.3.3 Generalized Fusion Rule

The Fusion algebra is associative and commutative, such that we are able to
generate the fusion rule of a degenerate field with spin $j_{r,s}$ by
repeatedly fusing fields with spin $j_{0,2}$ and $j_{1,1}$ (the field
corresponding to $j_{0,1}=0$ is the identity field
$\hat{R}^{0,1}\times\hat{R}_{j}=\hat{R}_{j}$). To see that these two spins are
enough to generate the fusion rules, we take a look at the fusions,

$\hat{R}^{1,1}\times\hat{R}^{1,1}=\hat{R}^{2,1}\oplus\hat{R}^{0,1},\quad\hat{R}^{0,2}\times\hat{R}^{0,2}=\hat{R}^{0,3}\oplus\hat{R}^{0,1},\quad\hat{R}^{0,2}\times\hat{R}^{1,1}=\hat{R}^{1,2}.$
(III.30)

Thus $\hat{R}^{1,1}$ increases or decreases the left index by one unit and
$\hat{R}^{0,2}$ increases the right index by one unit. Using this we can write
down, how a generic highest weight representation $\hat{R}_{j}$ fusions with a
degenerate representation $\hat{R}^{r,s}$:

$\hat{R}^{r,s}\times\hat{R}_{j}=\hat{R}_{j-j_{r,s}}+\hat{R}_{j-j_{r-2,s}}+\hat{R}_{j-j_{r,s-2}}+...+\hat{R}_{j-j_{-r,-s+2}}$
(III.31)

The fusion between two degenerate fields, respecting associativity and
commutativity, yields the closed formula,

$\hat{R}^{r,s}\times\hat{R}^{a,b}=\sum_{i\underset{2}{=}|r-a|}^{r+a}\sum_{j\underset{2}{=}|s-b|+1}^{s+b-1}\hat{R}^{i,j}.$
(III.32)

The derived fusion rule (III.31) coincide with previous work, for example in
[1], [7], [8].

## Conclusion

In this thesis we have investigated the poles in the central charge the torus
1-point block and null vectors in highest weight representation of
$\hat{\mathfrak{sl}}_{2}$.

By calculating the first four orders of the recursion relation of the 1-point
block (II.7), we have found that there seems to be a very specific algebraic
form of the $c$-pole free expression $K_{N}$ (II.12). It is a laurent
polynomial in the variable $\beta:=b^{2}$ of degree $Q_{N}:=\big{(}\sum_{1\leq
mn\leq N}1\big{)}-N$, with prefactors depending on the external $\Delta_{1}$
and internal $\Delta$ conformal dimension. Altough it seems hard to get rid of
the poles in the central charge, we have found that an expansion around
$\beta=0$ of $K_{N}$, can be used to find the unknown prefactors of the
Laurent-polynomial. The possibility to calculate a pole-free torus 1-point
block, helps to determine numerically 1-point functions of CFTs with rational
central charge. For future work, calculations and checks at higher order will
help to affirm that the algebraic form holds for orders higher than four.

Within the second part, we moved to CFTs with affine Lie algebras as
underlying symmetry, where we specially focused on null vectors and degenerate
representations. In the case of $\hat{\mathfrak{sl}}_{2}$, we have found a
$\mathfrak{sl}_{2}$-basis independent and horizontal representation
independent (up to indecomposability) null operator at level 1 with quadratic
Casimir $C_{2}=\frac{t^{2}}{2}-t$,

$\hat{T}^{c}_{1}:=2K_{ab}J^{a}_{-1}J^{b}_{0}J_{0}^{c}-tf^{c}_{ab}J^{a}_{-1}J^{b}_{0}-t^{2}J^{c}_{-1}.$
(III.33)

Applying the null operator $\hat{T}^{c}_{0}$ on any state of the horizontal
representation, generates a null vector in the usual sense. In this framework
we derived the fusion rules of degenerate representations with general highest
weight representations, which coincides with earlier work.

## Acknowledgement

I would like to express my deepest gratitude to my supervisor Dr. Sylvain
Ribault, who invested a lot of time to discuss the content of this work and
for his all-time valuable advice. A special thanks goes to the Institute de
Physique Théoretique for hosting me during the M2-internship and providing me
this very special opportunity. Last but not least I want to thank my family
without their help I wouldn’t be at this point where I am.

## References

  * [1] S. Ribault, “Conformal field theory on the plane,” [arXiv:1406.4290 [hep-th]].
  * [2] A. N. Schellekens, “Introduction to conformal field theory,” Fortsch. Phys. 44 (1996), 605-705. Retrieved from https://www.nikhef.nl/~t58/CFT.pdf (07/2022)
  * [3] P. Di Francesco, P. Mathieu and D. Sénéchal, “Conformal Field Theory,” (1997). doi:10.1007/978-1-4612-2256-9
  * [4] V. A. Fateev and A. V. Litvinov, “On AGT conjecture,” JHEP 14 (2010) [arXiv:1406.4290 [hep-th]].
  * [5] L. Eberhardt, “Wess-Zumino-Witten Models,” YRISW PhD School in Vienna, 2 (2019). Retrieved from https://conf.itp.phys.ethz.ch/esi-school/Lecture_notes/WZW%20models.pdf (07/2022)
  * [6] F. G. Malikov, B. L. Feigin and D. B. Fuks, “Singular vectors in Verma modules over Kac-Moody algebras,” Funct. Anal. Its Appl. 20 (1986), 103-113. doi:10.1007/BF01077264
  * [7] M. Bauer and N. sochen, “Fusion and singular vectors in $A1(1)$ highest weight cyclic modules,” Commun. Math. Phys. 152 (1993), 127-160. doi:10.1007/BF02097060
  * [8] H. Awata and Y. Yamada, “Fusion Rules for the Fractional Level $\hat{\mathfrak{sl}}_{2}$ Algebra,” Mod. Phys. Lett. A 7 (1992), 1185-1196. doi:10.1142/S0217732392003645
  * [9] K. B. Alkalev, R. V. Geiko and V. A. Rappoport, “Various semiclassical limits of torus conformal blocks,” JHEP 04 (2017) 070. https://doi.org/10.1007/JHEP04(2017)070

##  Appendix A Singularity free at order 3 and 4

The $c$-pole free expression $K_{N}$ (II.12) at order 3 and 4 have been
calculated and nummerically checked to coincide with the expressions (II.7)
outside the poles in the central charge. To express the solutions, we recall
the $b$-symmetric Laurent-polynomials,

$B_{0}=1,\quad B_{j}=\beta^{j}+\beta^{-j}.$ (A.1)

At order 3 the solutions reads,

$\displaystyle\begin{aligned}
K_{3}&=\frac{1}{192}\bigg{\\{}\bigg{(}144\Delta+288\Delta^{2}-144\Delta\Delta_{1}+48(\Delta_{1}-1)\Delta_{1}+144\Delta\Delta_{1}^{2}-12(\Delta_{1}-1)\Delta_{1}^{2}\\\
&+12(\Delta_{1}-1)\Delta_{1}^{3}\bigg{)}B_{2}+\bigg{(}888\Delta+960\Delta^{3}-1840\Delta\Delta_{1}+440(\Delta_{1}-1)\Delta_{1}+1574\Delta\Delta_{1}^{2}\\\
&-164(\Delta_{1}-1)\Delta_{1}^{2}-332\Delta\Delta_{1}^{3}+68(\Delta_{1}-1)\Delta_{1}^{3}+22\Delta\Delta_{1}^{4}+24\Delta^{2}(76-75\Delta_{1}+27\Delta_{1}^{2})\bigg{)}B_{1}\\\
&+\bigg{(}1536\Delta+768\Delta^{4}-3232\Delta\Delta_{1}+824(\Delta_{1}-1)\Delta_{1}+2535\Delta\Delta_{1}^{2}-323(\Delta_{1}-1)\Delta_{1}^{2}\\\
&-590\Delta\Delta_{1}^{3}+115(\Delta_{1}-1)\Delta_{1}^{3}+39\Delta\Delta_{1}^{4}+96\Delta^{3}(24-19\Delta_{1}+3\Delta_{1}^{2})+8\Delta^{2}(342-411\Delta_{1}\\\
&+238\Delta_{1}^{2}-20\Delta_{1}^{3}+\Delta_{1}^{4})\bigg{)}\bigg{\\}}\end{aligned}$
(A.2)

At order 4 the solutions reads,

$\displaystyle K_{4}$
$\displaystyle=\frac{45}{2048}\bigg{(}96\Delta+288\Delta^{2}+192\Delta^{3}-36\Delta_{1}-168\Delta\Delta_{1}-144\Delta^{2}\Delta_{1}+52\Delta_{1}^{2}+192\Delta\Delta_{1}^{2}+144\Delta^{2}\Delta_{1}^{2}$
$\displaystyle-33\Delta_{1}^{3}-48\Delta\Delta_{1}^{3}+19\Delta_{1}^{4}+24\Delta\Delta_{1}^{4}-3\Delta_{1}^{5}+\Delta_{1}^{6}\bigg{)}B_{4}+\frac{3}{4096}\bigg{(}45120\Delta+141600\Delta^{2}$
$\displaystyle+132000\Delta^{3}+41280\Delta^{4}-14652\Delta_{1}-92604\Delta\Delta_{1}-136152\Delta^{2}\Delta_{1}-61392\Delta^{3}\Delta_{1}+27284\Delta_{1}^{2}$
$\displaystyle+109180\Delta\Delta_{1}^{2}+105984\Delta^{2}\Delta_{1}^{2}+26832\Delta^{3}\Delta_{1}^{2}-22281\Delta_{1}^{3}-49095\Delta\Delta_{1}^{3}-26064\Delta^{2}\Delta_{1}^{3}$
$\displaystyle+11363\Delta_{1}^{4}+16885\Delta\Delta_{1}^{4}+4392\Delta^{2}\Delta_{1}^{4}-2091\Delta_{1}^{5}-1749\Delta\Delta_{1}^{5}+377\Delta_{1}^{6}+103\Delta\Delta_{1}^{6}\bigg{)}B_{3}$
$\displaystyle+\frac{1}{8192}\bigg{(}1423008\Delta+4167168\Delta^{2}+4189536\Delta^{3}+2013312\Delta^{4}+446976\Delta^{5}-442908\Delta_{1}$
$\displaystyle-3093480\Delta\Delta_{1}-5225328\Delta^{2}\Delta_{1}-3529776\Delta^{3}\Delta_{1}-993408\Delta^{4}\Delta_{1}+914364\Delta_{1}^{2}+3900600\Delta\Delta_{1}^{2}$
$\displaystyle+4702504\Delta^{2}\Delta_{1}^{2}+2296944\Delta^{3}\Delta_{1}^{2}+336768\Delta^{4}\Delta_{1}^{2}-799689\Delta_{1}^{3}-2092518\Delta\Delta_{1}^{3}-1684440\Delta^{2}\Delta_{1}^{3}$
$\displaystyle-450048\Delta^{3}\Delta_{1}^{3}+391899\Delta_{1}^{4}+673314\Delta\Delta_{1}^{4}+343312\Delta^{2}\Delta_{1}^{4}+41280\Delta^{3}\Delta_{1}^{4}-74427\Delta_{1}^{5}$
$\displaystyle-84018\Delta\Delta_{1}^{5}-20280\Delta^{2}\Delta_{1}^{5}+10761\Delta_{1}^{6}+4902\Delta\Delta_{1}^{6}+712\Delta^{2}\Delta_{1}^{6}\bigg{)}B_{2}+\frac{1}{49152}\bigg{(}1683456\Delta^{6}$
$\displaystyle+3072\Delta^{5}(3292-1669\Delta_{1}+409\Delta_{1}^{2})+768\Delta^{4}(45703-32773\Delta_{1}+17503\Delta_{1}^{2}-2532\Delta_{1}^{3}$
$\displaystyle+162\Delta_{1}^{4})+9\Delta_{1}(-734868+1587868\Delta_{1}-1423299\Delta_{1}^{2}+684065\Delta_{1}^{3}-130937\Delta_{1}^{4}+17171\Delta_{1}^{5})$
$\displaystyle+64\Delta^{3}(978855-1016103\Delta_{1}+729839\Delta_{1}^{2}-190791\Delta_{1}^{3}+25799\Delta_{1}^{4}-1023\Delta_{1}^{5}+29\Delta_{1}^{6})$
$\displaystyle+8\Delta^{2}(7271424-10546749\Delta_{1}+10186433\Delta_{1}^{2}-3976128\Delta_{1}^{3}+840752\Delta_{1}^{4}-61824\Delta_{1}^{5}$
$\displaystyle+2156\Delta_{1}^{6})+3\Delta(7167168-15881724\Delta_{1}+20837492\Delta_{1}^{2}-11935235\Delta_{1}^{3}+3761009\Delta_{1}^{4}$
$\displaystyle-491001\Delta_{1}^{5}+28819\Delta_{1}^{6})\bigg{)}B_{1}+\frac{1}{49152}\bigg{(}28999296\Delta+75605760\Delta^{2}+85087104\Delta^{3}+47781120\Delta^{4}$
$\displaystyle+15793152\Delta^{5}+2617344\Delta^{6}+344064\Delta^{7}-8901576\Delta_{1}-64431288\Delta\Delta_{1}-115515792\Delta^{2}\Delta_{1}$
$\displaystyle-92818272\Delta^{3}\Delta_{1}-38587392\Delta^{4}\Delta_{1}-8169984\Delta^{5}\Delta_{1}-1314816\Delta^{6}\Delta_{1}+19491840\Delta_{1}^{2}$
$\displaystyle+85511472\Delta\Delta_{1}^{2}+114226816\Delta^{2}\Delta_{1}^{2}+68972000\Delta^{3}\Delta_{1}^{2}+20325120\Delta^{4}\Delta_{1}^{2}+3518976\Delta^{5}\Delta_{1}^{2}$
$\displaystyle+208896\Delta^{6}\Delta_{1}^{2}-17583363\Delta_{1}^{3}-49771617\Delta\Delta_{1}^{3}-45926220\Delta^{2}\Delta_{1}^{3}-18616256\Delta^{3}\Delta_{1}^{3}$
$\displaystyle-3539200\Delta^{4}\Delta_{1}^{3}-344064\Delta^{5}\Delta_{1}^{3}+8397441\Delta_{1}^{4}+15592539\Delta\Delta_{1}^{4}+9878804\Delta^{2}\Delta_{1}^{4}$
$\displaystyle+2576704\Delta^{3}\Delta_{1}^{4}+314880\Delta^{4}\Delta_{1}^{4}+18432\Delta^{5}\Delta_{1}^{4}-1609353\Delta_{1}^{5}-2056035\Delta\Delta_{1}^{5}-775652\Delta^{2}\Delta_{1}^{5}$
$\displaystyle-102976\Delta^{3}\Delta_{1}^{5}-9984\Delta^{4}\Delta_{1}^{5}+205011\Delta_{1}^{6}+121329\Delta\Delta_{1}^{6}+26924\Delta^{2}\Delta_{1}^{6}+2880\Delta^{3}\Delta_{1}^{6}+256\Delta^{4}\Delta_{1}^{6}\bigg{)}$

##  Appendix B Irreducible representations of $\mathfrak{sl}_{2}$

We give a short overview of irreducible representations (shorthand irreps) of
$\mathfrak{sl}_{2}$, as examples of horizontal representations of highest
weight representations. For simplicity we work in the $m$-basis, which
diagonalizes the $J^{0}_{0}$ operator. Hence the states are given by
$|j,m\rangle$, with diagonal quadratic Casimir operator $C_{2}=2j(j+1)$. There
exist 4 different types of irreps, one finite type with half integer spin
$j\in\mathbb{N}/2$ and three infinite types with complex spin
$j\in\mathbb{C}$. The three infinite types have either one of the state
$|j,\pm j\rangle$ in their space or none, whereas the finite type has both
states in its space. In a diagramatic way,

Figure 1: Diagramatic representation of the $J^{0}_{0}$-eigenvalue of the four
irreducible representations. The yellow coloured line corresponds to half
integer spin and is finite (contains $|j,\pm j\rangle$), the red coloured
lines correspond to discrete type representations (upper line contains
$|j,j\rangle$, lower line $|j,-j\rangle$) and the orange coloured to the
continuous type representation (contains none $|j,\pm j\rangle$).

First we define a general action of the subalgebra
$\mathfrak{sl_{2}}\subset\hat{\mathfrak{sl_{2}}}$ on the descending vector
space,

$V^{-}_{j}:=\mathrm{span}_{\mathbb{C}}(\\{|j,m\rangle;j\in\mathbb{C},m\in
j-\mathbb{N}\\}).$ (B.1)

The action is defined as,

$\displaystyle J^{0}_{0}|j,m\rangle=m|j,m\rangle,\quad
J^{\pm}_{0}|j,m\rangle=(j\mp m)|j,m\pm 1\rangle.$ (B.2)

This action can be generated by the highest weight state $|j,j\rangle$, which
satisfies $J^{+}_{0}|j,j\rangle=0$.

With the help of the self-inverse automorphism
$(\circ)^{*}:J^{0}_{0}\mapsto-J^{0}_{0},J^{\pm}_{0}=-J^{\mp}_{0}$ we define
the ascending vector space implicit via the action
$V^{+}_{j}:=(V^{-}_{j})^{*}$. Here we use the same notation for the vector
space and its representation but mean that we apply the automorphism action on
the descending vector space and identify $|j,-m\rangle^{*}=|j,m\rangle$. It is
characterised by $m\in-j+\mathbb{N}$ and generated by the lowest weight
$|j,-j\rangle^{*}$ because
$0=(J^{-}_{0})^{*}|j,j\rangle=J^{-}_{0}|j,-j\rangle^{*}$.

If we restrict on $j\in(-\infty,-1/2)$ we call $D^{\pm}_{j}:=V^{\pm}_{j}$ the
discrete series representations, both are unitary, infinite and irreducible.

The continuous series representation is less constrained and defined via,

$C_{j}^{\alpha}:=\mathrm{span}_{\mathbb{C}}(\\{|j,m\rangle;j\in-1/2+i\mathbb{R}_{+},\alpha\in\mathbb{R}/\mathbb{Z},m\in\alpha+\mathbb{Z}\\}).$
(B.3)

Where we use the same action as for the discrete series and notice that it has
not a highest nor a lowest weight state. It is unitary, infinite and
irreducible. Under conjugation we have $(C^{\alpha}_{j})^{*}=C^{-\alpha}_{j}$.