text:
The Inbetweeners Movie Quotes
The Inbetweeners Movie is a television program that appeared on TV in 1970 . The Inbetweeners Movie ended its run in 1970.
It features Christopher Young as producer, Mike Skinner (musician) in charge of musical score, and Ben Wheeler as head of cinematography.
The Inbetweeners Movie is recorded in English and originally aired in United Kingdom. Each episode of The Inbetweeners Movie is 97 minutes long. The Inbetweeners Movie is distributed by Film4.
The cast includes: James Buckley as Jay Cartwright, Joe Thomas as Simon Cooper, Simon Bird as Will McKenzie, Blake Harrison as Neil Sutherland, Theo James as James, Lydia Rose Bewley as Jane, Greg Davies as Mr. Gilbert, and Tamla Kari as Lucy.
Simon Bird as Will McKenzie
(Simon Bird) "That's it. Neil, you're right. It may not be paradise, but it's time we started enjoying this place for what it is."
(Joe Thomas) "A s***hole?"
(Simon Bird) "Yes, but it's our s***hole. So I say, we get out there, and get royally f***ed up on Jay's dead grandad's money."
(Simon Bird) "First stop the Minoan palace in Knossos."
(James Buckley) "We haven't come half way round the world to look at some f***ing Greek ruins."
(Blake Harrison) "Yeah you can see that s*** anywhere."
(Simon Bird) "So smelling like an industrial accident in a Lynx factory and looking like the world's s***test boyband, we hit the town."
Theo James as James
(Theo James) "You better watch you don't fall overboard."
(Lydia Rose Bewley) "Or what? I'll get harpooned because someone will mistake me for a whale? Someone will think they've discovered a new island? All the water will splash over onto the boat? I've heard them all. Take your pick."
Blake Harrison as Neil Sutherland
(Blake Harrison) "I stopped believing in god when I realised it was just dog spelt backwards."
Joe Thomas as Simon Cooper
(Joe Thomas) "She's also really funny."
(Tamla Kari) "In what way?"
(Joe Thomas) "Well, you know when something's funny, and people get it?"
(Tamla Kari) "Yes --"
(Joe Thomas) "So in that way. And also in a comedy way."
James Buckley as Jay Cartwright
(James Buckley) "This girl's so wet for me I can hear the waves breaking in her fanny."
(James Buckley) "Don't you know about foreign police? They take you up a hill, beat you up and then they bum you."
(Blake Harrison) "Yeah. and if they don't kill you, you kill yourself because of the shame of you getting a boner whilst you was being bummed."
(James Buckley) "Why go for hamburgers, when you have steak at home?"
Greg Davies as Mr. Gilbert
(Greg Davies) "This isn't The Dead Poets Society and I am not that bloke on bbc2 keeps getting kids to sing in choirs. I especially don't want to hear how well you are settling down at uni or how much growing up you have done in the past 12 months. At best I am ambivalent towards most of you, but some of you I actively dislike, for no other reason than your poor personal hygiene or your irritating personalities. I hope I have made myself clear on this point and in case any of you think I am joking, I am not. I assure you, once my legal obligation to look after you best interests is removed, I can be one truly nasty f***er. Good luck with the rest of your lives and try not to kill anyone, it reflects very badly on all of us here."
The Gift of Love Quotes | A Handful of Dust (film) Quotes | Attack of the 50 Ft. Woman (1993 film) Quotes | The Street Singer (1937 film) Quotes | The Traitor (1957 film) Quotes | Beautiful (2009 film) Quotes | Women Are Better Diplomats Quotes | Face in the Night Quotes | A Cold Wind in August Quotes | Krishnadasi Quotes | Grandmother's House (film) Quotes | Right America: Feeling Wronged – Some Voices from the Campaign Trail Quotes | The Englishman and the Girl Quotes | The Law and the Lady (film) Quotes | Look Before You Love Quotes | Today's FBI Quotes | The Amateur Gentleman (1936 film) Quotes | Tomorrow We Live (1936 film) Quotes | Old Mother Riley Detective Quotes | Old Mother Riley Joins Up Quotes | Old Mother Riley's Circus Quotes | Old Mother Riley in Business Quotes | Nine till Six Quotes | The Return of Sherlock Holmes (1929 film) Quotes | Colombiana Quotes |


text:
\section{Introduction}

	Classification of the Rational Conformal Field Theories (RCFTs) is a
	long-standing problem. The unitary minimal models with $c<1$  were
	classified by \cite{Belavin:1984vu, Friedan:1983xq} a long time ago. The $c=1$
	theories have a continuous moduli space corresponding to  A and D type
	symmetries and three isolated points corresponding to E type
	symmetries \cite{Ginsparg:1987eb}.   Comparatively, the classification
	of $c>1$ is still an open problem. The $c>1$ RCFTs, which have a
	finite number of characters, need an extended algebra.  Ka\v c-Moody
	algebras, one of the most popular extended algebras, is a
	symmetry of the Wess-Zumino-Witten(WZW) models\cite{Gepner:1986wi}. The WZW models are
	ubiquitous and appear in many places including, AdS$_3$/CFT$_2$
	correspondence \cite{Maldacena:1997re, Maldacena:2000hw, Maldacena:2000kv, Maldacena:2001km} and the dS$_3$/CFT$_2$ correspondence
	\cite{Hikida:2021ese, Hikida:2022ltr}.  The chiral algebras of
	two-dimensional WZW models are also related to four-dimensional
	$\mathcal{N}=2$ superconformal field theories (SCFTs)
	\cite{Beem:2013sza}.

	Mathur, Mukhi, and Sen(MMS)\cite{Mathur:1988na,
		Mathur:1988gt} developed a classification program for RCFTs in terms
	of the number of Ka\v c-Moody characters of the RCFT without specifying the Ka\v c-Moody
	algebra. They developed the Modular Linear Differential Equation
	(MLDE) approach which provides an $n$-th ordered linear differential equation whose solutions are the
	characters of that RCFT\footnote{See \cite{
			Hampapura:2015cea,Chandra:2018pjq, Das:2020wsi, Das:2021uvd,
			Kaidi:2021ent} for some recent applications of the MLDE approach to
		RCFT classification and \cite{Mukhi:2019xjy} for a nice review.}.
	They obtained a partial classification for two character theories, the
	so-called MMS series.   Usually, not all solutions of an MLDE are
	characters of an RCFT and a check is needed to weed out
	the characters which are not admissible characters.  In particular, not all solutions possess a \textit{q}-expansion with positive integral coefficients.

	The procedural checks have been carried out routinely in the
	past to generate admissible characters \cite{Mathur:1988gt,Chandra:2018pjq,Das:2020wsi,Das:2021uvd,Kaidi:2021ent}. Besides proving a complete classification
	of two character theories, \cite{Chandra:2018pjq} also brought another class of
	interesting solutions into light which they called
	\textit{quasi-characters}.  The quasi-characters are not admissible
	characters of any CFT because their $q$-expansion
	has integer coefficients but not necessarily positive integer
	coefficients.  As a result, we cannot interpret them as a partition
	sum of any CFT.   However, \cite{Chandra:2018pjq, Mukhi:2020gnj} showed that quasi-characters has more applications. Namely, that certain linear combinations of the quasi-characters can be used to build admissible characters by appropriately adding two quasi-characters with the same modular $T$- transformation and appropriate modular $S$-transformation, or by a multiplication of the quasi-character by a suitable power of the  $j$-function.

	Although most of the study of the WZW models has been carried out for
	the integrable representations, that is, when the Ka\v c-Moody level
	is a positive integer, there are interesting rational
	values of the levels where one has a finite number of characters in the
	representation theory of the Ka\v c-Moody algebra. These levels are
	referred to as the admissible levels \cite{Kac:1988qc}.  We will
	concentrate on the representation theory of these admissible levels of
	the $\widehat{su}(2)$ affine Lie-algebra \cite{Kac:1988qc}. These theories have
	finite dimensional modular transformation matrices $S$ and $T$, but a
	naive application of the Verlinde formula \cite{Verlinde:1988sn}
	does not give rise to all positive integer fusion
	coefficients \cite{Koh:1988sm}. This is typically a signature of a wrong
	identification of the vacuum module and can be remedied by correctly
	choosing the vacuum module. While this choice renders all the fusion
	coefficients positive, this does not necessarily imply that the
	resulting theory is a unitary CFT. There is of course, a
	possibility that in the representation theory, the vacuum is not unique up to the
	presence of characters with conformal weights differing by integers
	and may contain logarithmic mode(s).  Gaberdiel \cite{Gaberdiel:2001ny}
	showed that to close the algebraical fusion algebra, $su(2)_{-4/3}$ must be a
	logarithmic Conformal Field Theory ($\ log$ CFT). This has also been shown
	in the $su(2)_{-1/2}$
	case \cite{Lesage:2002ch,Lesage:2003kn,Ridout:2008nh, Ridout:2010jk}.
	Subsequent works by Ridout and Creutzig \cite{ Creutzig:2012sd, Creutzig:2013hma, Creutzig:2013yca} points towards
	the genericity of $\ log$ CFTs at admissible levels.  All of the
	above models of $\ log$ CFTs requires additional infinite family of modules
	which were not part of the original description.

	We find, rather surprisingly, that $\widehat{su}(2)$ CFTs  seem to
	generically possess \textit{quasi-characters}.  Although
	quasi-characters may be useful in building admissible characters
	of some other CFT, they themselves are not suitable to define a CFT.

	Nevertheless at certain admissible levels,\footnote{{\em i.e.}, when the level is $m=t/u$ with $u$ a positive integer, $(t,u)$ relatively prime, and
		$t+h^\vee(u-1)\ge 0$, where $h^{\vee}$ is the dual coxeter number. $h^{\vee}=2$ for $su(2)$.} we
	do find that the representation theory does not contain any
	quasi-characters.  This happens when either
	$(p=2, u=2N+1)$, or $(p=2N+3, u=2)$, or $(p=3,u=4)$, where
	$p=t+h^\vee u$ and $N\in\mathbb{N}$.

	We also establish a relation between the admissible characters found at these levels and the characters of integrable representations of $\widehat{su}(2)$ for the
	infinite series $(p=2, u=2N+1)$, as well as for
	$(p=2N+1, u=2)$, where $N\in\mathbb{N}$, and for the isolated point $(p,u)=(3,4)$ we show that
	 a subset of characters is related to the $\widehat{so}(5)$ characters at level 1 and another subset to $\widehat{su}(2)$ characters at level 1. These also guarantee positivity of the admissible characters we found for the two infinite series and the isolated point.

	\begin{comment}
		These are rich theories which provide a window to more intricate
		and bigger affine Lie-Algebras Wess-Zumino-Witten (WZW) models
		besides providing applications to the recent progress in finding an
		explicit example of $dS_3/CFT_2$ \cite{Hikida:2021ese,Hikida:2022ltr}
		or for example to $AdS_3/CFT_2$. They have been related to Logarithmic
		CFTs in \cite{Gaberdiel:2001ny,Creutzig:2013yca} which are another
		interesting class of theories on their own.

		The $su(2)_m$ Wess-Zumino-Witten (WZW) models are uniquitous in
		physics. Recently, in the large central charge limit these theories
		were realised as the holographic dual to Einstein gravity on $dS_3$
		\cite{Hikida:2021ese, Hikida:2022ltr, Chen:2022ozy}. These are the
		Conformal Field Theories (CFT) with an affine Lie-Algebra symmetry,
		representation of which form the building blocks. An understanding of
		the WZW model at positive integer level, (denoted $m$)\footnote{which
			is the central extension of the underlying lie algebra} is covered in
		standard textbooks, for example, \cite{DiFrancesco:1997nk} but an
		understanding at any arbitrary complex level still eludes us. Amongst
		these levels, there are certain special rational levels which are
		known as \textit{admissible levels} first defined in the seminal work
		\cite{Kac:1988qc}. These are special in the sense that they have a
		finite number of finite dimensional highest weight representations of
		the entire torus group $SL(2,\mathbb{Z})$. These are identified with
		Rational Conformal Field Theories or RCFT. But the coefficients of the
		fusion algebra are not positive semi-definite, not consitent with the
		fusion products of the primaries in a CFT. These results were first
		reported in \cite{Koh:1988sm}.

		There has been considerable effort in developing an understanding of
		the fractional level WZW models in the aftermath since Gaberdiel in
		\cite{Gaberdiel:2001ny} showed that the highest weight
		representations $su(2)_{-4/3}$ do not form a closed algebra under
		fusion and that the theory is a logarithmic conformal field theory
		\cite{Lesage:2002ch, Lesage:2003kn, Ridout:2008nh, Ridout:2010jk,
			Creutzig:2013hma, Creutzig:2013yca}. These examples are limited to
		specific levels and we need to be sure about the structure in
		general. On the other hand, makes sense when we use a
		\textit{unitarisation procedure} to reidentify the characters as we
		will also show. We study the characters of highest weight
		representations of real and rational admissible levels and find that
		in any interpretation, whether non-unitary or unitary
		representations, that there are quasi-characters. Quasi -characters
		were first defined in \cite{Chandra:2018pjq} as unphysical solutions
		of the characters of Modular Linear Differential Equations or
		MLDE\footnote{MLDE were originally defined in\cite{Mathur:1988gt}
			where they used the fact that the characters of a CFT should form a
			set of solutions to a linear differential equation whose coefficient
			functions are set by modular transformation properties of the
			characters}.
		_____________________________________________________________
		_____________________________________________________________
		***************************************************************************
		_____________________________________________________________
		_____________________________________________________________

		Conformal Field Theory (CFT) are ubiquitous in physics. These are
		the fixed point in the renormalisation group flow of any quantum
		field theory hence they also describe all critical phenomena.

		A classification of the CFTs is a very ineteresting problem. A
		program to classify theories with a finite number of characters ,
		i.e., Rational Conformal Field Theories (RCFT) was started in
		\cite{Mathur:1988gt} and recently continued in
		\cite{Chandra:2018pjq, Das:2020wsi, Das:2021uvd, Kaidi:2021ent}. The
		minimal models, with $c<1$, which contains a finite number of
		primaries which are irreducible representations of two copies of the
		Virasoro algebra (denoted by left (right) $\mathcal{V}$),i.e., they
		transform covariantly under conformal transformations are already
		classified but the general problem is still open.
	\end{comment}

	\subsection{Summary of Results and Organisation}
	The $\widehat{su}(2)$ theory at a fractional admissible level $m$ is
	parametrised as $m=p/u-2$ with $p,u \in \mathbb{N}$ and coprime, and
	admissibility implies $p\geq 2$.  Each of these theories has a finite number of characters. To our surprise, we find that many highest weight modules give rise to quasi-characters.  Based on the data obtained from the $q$-expansion (we have carried out series expansions up to
	$q^{100}$) within the domain of
	$(2\leq p \leq 5\, ,\, 2\leq u\leq 30)$ and
	$(5\leq p \leq 10\, ,\, 2\leq u\leq 20)$ we conjecture that all of
	the fractional level $\widehat{su}(2)$ theories contain at least one quasi-character except for the two
	infinite series and the isolated point in the $(p,u)$ plane. At the
	isolated level $m=-5/4\ (p=3,u=4)$ and for two infinite series
	corresponding to $(p=2,u=2N+1)$ (called threshold cases) and $(p=2N+3,u=2)$, where $N\in \mathbb{N}$ we find no
	quasi-characters.  We show that these levels are special since a
	re-identification of these characters takes us to the integrable highest weight characters of $\widehat{su}(2)$ at level
	$4N$ for the two infinite series, $\widehat{su}(2)$ at level 1 and $\widehat{so}(5)$ at level 1 for the two subsectors of the theory at the isolated point. Since the re-identification reduces the space of independent RCFT character
	vectors, these results help in classification of RCFTs using characters. In our search for closed subsets of characters which do not have quasi-character(s) inside the set of characters of the theory which has quasi-character(s), we also find two elimination criteria to exclude certain fractional levels as their characters do not realise an RCFT. These criteria eliminate the levels $(p\in\ prime, u\in\ prime)$ due to the first elimination criterion, and due to the second elimination criterion, possibly the levels $(p, Np-1)$, where $N\in\mathbb{N}$ and $p\geq 3$, which have quasi-characters corresponding to the vacuum modules of the unitarised theory.

	The organisation of the paper is as follows. In section
	\ref{KW_formula} we set up the notation and define the flavoured characters of
	the $\widehat{su}(2)$ algebra as defined by Ka\v c and Wakimoto
	\cite{Kac:1988qc}. We use even and odd combinations of the characters \cite{Mukhi:1989bp} to
	extract unflavoured characters.  In section \ref{physical_characters}
	we study the even admissible $\widehat{su}(2)$ character vectors of the two infinite series
	$(p=2, u=2N+1)$, $(p=2N+3, u=2)$, where $N\in\mathbb{N}$ and the isolated point $(p=3,u=4)$. We relate the characters to integrable highest weight characters of
	$\widehat{su}(2)$ and $\widehat{so}(5)$ at positive integer levels through the process of
	unitarisation of the original non-unitary $\widehat{su}(2)$ theories.  The series $(p=2N+1, u=2)$ possesses
	certain peculiar properties. We will also elaborate on them in section
	\ref{physical_characters}.  It is not entirely clear to us if the
	partition function of the $(p=2N+1, u=2)$ theories corresponds to a partition
	function of any CFT, with possible exceptions of logarithmic CFTs.  In
	section \ref{sub-sector} we will argue that the modular $S$-matrix of these
	theories intertwines the quasi-characters with the admissible characters and propose two elimination criteria based on the modular and conformal properties of the characters.  We
	conclude the paper with a summary of our results and speculate about the possible
	application of our results in section \ref{conclusions}.

	The appendices contain some explicit details of the computation.
	A few explicit examples of quasi-characters are given in appendix
	\ref{examples_quasi} with tabulated data on the
	quasi-characters.  In the appendix \ref{rearrangement} we carry out a
	rearrangement of the $\textit{q}$-expansion of the unflavoured characters to extract the leading behaviour for arbitrary values of $p$ and $u$. We will also see a
	criterion for finding quasi-characters using this rearranged \textit{q}-expansion.
	The results are in complete agreement with our data on explicit
	$q$-expansions.

	%
	\section{Ka\v c-Wakimoto Formula for $\widehat{su}(2)$ }\label{KW_formula}
	%

	It was shown in \cite{Kac:1988qc} that there are only a finite number
	of primaries for admissible rational levels of the $\widehat{su}(2)_m$
	affine Lie algebra ($m=p/u-2$ with $(p,u)$ coprime with $p\ge
	2$).

	The admissible representations are parametrised in terms of two
	integers $1 \leq n+1 \leq p-1$ and $0\leq k \leq u-1$ as,
	\begin{equation}
		2j(n,k)+1=n+1-\frac{kp}{u},
	\end{equation}
	where $j(n,k)$ is the spin of the representation. The conformal dimension
	of the primary with spin $j(n,k)$ is,
	\begin{align}
		h(n,k)=      \frac{j(j+1)u}{p} = \frac{(nu-kp)^2}{4up} +
		\frac{(nu-kp)}{2p}\ .
	\end{align}

	The spectrum remains invariant under the transformation
	$\sigma_0=j\mapsto -1-j$.  This can be stated in terms of $n,k$ as
	\begin{align}
		\bar n+1&=p-(n+1),\\
		\bar k &=u-k,\quad k\neq0,
	\end{align}
	under which
	$j(\bar n,\bar k)=-1-j(n,k)$ and $h(\bar n,\bar k)=h( n, k)$.
	\subsection{Flavoured Characters}
	The closed form expression for the Ka\v c-Wakimoto character formula
	\cite{Kac:1988qc} is defined in terms of
	\begin{equation}
		\label{eq:banda}
		b_{\pm}(n,k)=\pm u(n+1)-kp\ ,\ \quad a=pu\ .
	\end{equation}
	Notice, $\sigma_0:b_{+}\mapsto -b_{+}$.  The spin $j$ character is
	written in terms of theta functions as,
	\begin{align}\label{kac-wakimoto_char}
		\chi_{j(n,k)}(\tau,z)=\frac{\Theta_{b_{+},a}(\tau,z/u)-
			\Theta_{b_{-},a}(\tau,z/u)}{\Theta_{1,2}(\tau,z)-\Theta_{-1,2}(\tau,z)}\ ,
	\end{align}
	where,
	\begin{align}
		\Theta_{b,a}(\tau,z)=\sum_{r=\mathbb{Z}+b/2a}q^{ar^2}\omega^{ar}, \qquad q=e^{2\pi i \tau}.
	\end{align}
	The theta functions in the denominator can be expanded in terms of a
	$z$-series  as,
	\begin{equation}\label{kac_wakimoto_den}
		\Theta_{1,2}(\tau,z)-\Theta_{-1,2}(\tau,z)

		=2\pi iz \eta^3(q)+\mathcal{O}(z^3),
	\end{equation}

	We can also write it schematically as,
	\begin{equation}
		(\Theta_{1,2}(\tau,z)-\Theta_{-1,2}(\tau,z))^{-1}=\frac{1}{2\pi iz}q^{-1/8}\sum_n c_nq^n+\mathcal{O}(z).
	\end{equation}
	A few properties of the coefficients $c_n$ are given in the appendix
	\ref{rearrangement}.  To reduce the Ka\v c-Moody character
	\eqref{kac-wakimoto_char} to unrefined character, which is only a function of $q$, we need to take
	$z\to 0$ limit.  A naive limit
	$z\rightarrow 0 $ is not allowed as even for the case for integrable
	characters. To provide a good limiting procedure we need to expand
	the character in a power series of $z$.  Using the series form of the
	$\Theta$ function,
	\begin{align}\label{Theta_function_sum_in_Z}
		\Theta_{b_{+},a}(\tau,z)&=
		\sum_{s \in \mathbb{Z}}
		q^{as^2+b_{+}^2/4a+sb_{+}}\omega^{as+b_{+}/2}\ ,
	\end{align}
	in the characters and then expanding it in powers of $z$ gives,
	\begin{align}\label{char_expansion1}

		\chi_{j(n,k)}(q,z)&=\frac{1}{2\pi i z\eta^3(q)}\Big(q^{b_{+}^2/4a}
		\sum_{s \in \mathbb{Z}}
		q^{as^2+sb_{+}}-q^{b_{-}^2/4a}
		\sum_{s \in \mathbb{Z}} q^{as^2+sb_{-}}\Big)+
		\mathcal{O}(z)\nonumber\\
		&\quad+\frac{1}{u\eta^3(q)}\Big(q^{b_{+}^2/4a}
		\sum_{s \in \mathbb{Z}}
		q^{as^2+sb_{+}}(as+b_{+}/2)
		-q^{b_{-}^2/4a}\sum_{s \in \mathbb{Z}}
		q^{as^2+sb_{-}}
		(as+b_{-}/2)\Big).
	\end{align}
	For $k=0$, $b=b_{+}=-b_{-}$ the character takes the form as $z\to 0$,
	\begin{equation}\label{k0_char}
		\chi_{j(n,0)}(q)

		=\frac{1}{\eta^3(q)}q^{(n+1)^2/4(m+2)}
		\sum_{s \in \mathbb{Z}} q^{us((n+1)+us(m+2))}
		(n+1+2us(m+2)).
	\end{equation}
	All of the characters labelled by $(n,0)$ are admissible characters.
	These characters do not have singularity in the  $z\to 0$ limit, in
	fact, these characters are analogous to the characters for integrable
	highest weight representations for $\widehat{su}(2)_M$ where
	$M\in\mathbb{N}$ \cite{DiFrancesco:1997nk}.  To put it differently,
	these characters are those of the integrable highest weight
	representation when $u=1$.
	\begin{align}\label{integrable_character}
		\chi^{(M)}_{l}(q)=\frac{q^{(l+1)^2/4(M+2)}}{\eta^{3}(q)}
		\sum_n\big(l+1+2n(M+2)\big)q^{n(l+1+(M+2)n)}\ ,
	\end{align}
	where we have used the notation $\chi^{(M)}_{l}$ to denote the character of spin $l/2$ in the representation of $\widehat{su}(2)$ at
	level $M$.

	\subsection{Even and Odd Characters}
	We find
	that all quasi-characters come from the $(n,k\neq 0)$ sector. For these characters with $k\neq0$, we need to define odd
	$(-)$ and even $(+)$ linear combinations as done in \cite{Mukhi:1989bp}.
	Treating characters as functions of $q$ and $z$ we can write linear
	combinations of the characters in the following manner,
	\begin{align}\label{char_combination}
		\chi^{\pm}_{j(n,k)}(\tau,z)&=\chi_{j(n,k)}(\tau,z)
		\pm \chi_{j(n,k)}(\tau,-z)
	\end{align}
	The odd characters have been shown to be related to the characters of
	minimal models \cite{Mukhi:1989bp} satisfying all conditions of an admissible character.  The even combination of characters do not possess any
	singularity in the $z\to 0$ limit and we will be interested in these
	characters in the paper. Let us note the following properties of the even characters.

	Using $\sigma_0$,
	\begin{equation}
		\chi^{+}_{j(\bar n,\bar k)}(\tau)=\chi_{j(\bar n,\bar k)}
		(\tau,z)-\chi_{j(\bar n,\bar k)}
		(\tau,-z)
		=-\chi^{+}_{j(n,k)}(\tau) \ .
	\end{equation}
	Due to this relation, we now have a pair of identical characters up to a negative sign in the set of even combination of characters.
	While the action of modular transformation $T$ on the characters
	is by a left multiplication by a diagonal matrix
	\begin{align}
		T_{nk,n^{'}k^{'}}=e^{2\pi i h(n,k)}\delta_{nk,n^{'}k^{'}} \ ,
	\end{align}
	the characters transform non-trivially under the modular $S$-transformation.
	The $S$-matrix components \cite{Mukhi:1989bp} are,
	\begin{equation}\label{s}
		S_{nk,n'k'}=\sqrt{\frac{2}{pu}}(-1)^{k'(n+1)+k(n'+1)}
		e^{-i\pi kk'p/u}\sin\left(\pi\frac{(n+1)(n'+1)u}{p}\right)
	\end{equation}
	The modular $S$-transformations of the even characters can then be
	written in the following manner,
	\begin{equation}\label{s_of_+}
		\chi^{+}_{j(n,k)}(-\frac{1}{\tau},\frac{z}{\tau})=\sum_{n',k'\not=0}2S^{+}_{nk,n'k'}\chi^+_{j(n',k')}
		(\tau,z)+\sum_{n'}2S_{nk,n'0}\chi_{j(n',0)}(\tau,z)\ ,
	\end{equation}
	and,
	\begin{equation}\label{s_of_0}
		\chi_{j(n,0)}(-\frac{1}{\tau},\frac{z}{\tau})=\sum_{n',k'\not=0}
		S_{n0,n'k'}\chi^+_{j(n',k')}(\tau,z)
		+\sum_{n'}S_{n0,n'0}\chi_{j(n',0)}(\tau,z)\ ,
	\end{equation}
	where the sum runs over the labels $(n',k')$ of even characters.
	The components of the modular $S^+$-transformation matrix are
	\begin{equation}\label{s+}
		S^{+}_{nk,n'k'}=\frac{1}{2}\left(S_{nk,n'k'}-
		S_{\bar n\bar k,n'k'}\right).
	\end{equation}
	Thus we see that the even characters form a closed modular invariant
	set under the modular $S$ and $T$ transformations which we will refer to as the even sector.  As mentioned above
	we will work with the even characters and in particular we will
	focus on the unflavoured characters, {\em i.e.}, they are functions
	of the modular parameter $\tau$ only.

	%
	\subsection{Unflavoured Even Characters}
	%

	In this subsection we will analyse the $q$-series expansion of even
	characters $\chi^+$ in the $z\to 0$ limit.

	Due to the $\sigma_0$ symmetry, even characters always appear in pairs,
	one with $b_{+} <0$ and the other with $b_{+} > 0$.  We can therefore
	restrict ourselves to $b_{+} < 0$ in this subsection\footnote{We will not restrict ourselves to this condition later and will choose the even sector as per convenience.}.

	The $q$-expansion of even characters is finite in
	$z\rightarrow0$ limit and is given by,
	\begin{eqnarray}\label{even_character}
		\chi^{+}_{j(n,k)}(q)&=(1+3q+9q^2+22q^3+51q^4+108q^5+221q^6+429q^7
		+810q^8+ \mathcal{O}(q^9))\nonumber\\
		&\times \left\{q^{b_{+}^{2}/4a-1/8}\Big[  \frac{b_{+}}{u}+
		\sum_{s} 2spq^{s^2 a+sb_{+}}
		+\frac{b_{+}}{u} \sum_{s\neq 0}q^{s^2 a+sb_{+}}\Big]\right .\nonumber\\
		&\left. -q^{b_{-}^{2}/4a-1/8}\Big[\frac{b_{-}}{u}+\sum_{s} 2spq^{s^2
			a+sb_{-}}+\frac{b_{-}}{u}\sum_{s\neq 0}q^{s^2
			a+sb_{-}}\Big]\right\} \ .
	\end{eqnarray}

	The leading term in the $q$-series in the $\tau\to i\infty$ limit is
	$\frac{b_{+}}{u} q^{b_{+}^2/4a-1/8}$ and the exponent can be identified as,

		\begin{equation}
			\label{eq:b+c}
			b_{+}^2/4a-1/8= -c/24+h(n,k).
		\end{equation}

	where $c$ is the Virasoro central charge and $h(n,k)$ is the
	conformal dimension.  The set of even characters can be multiplied by $u$ which ensures that the coefficients are integers and the modular transformations remain the same, however, as we will see,
	positivity of the coefficients is not obvious.

	Note that the difference between the leading powers in the square brackets
	is always an integer.
	\begin{equation}
		\label{eq:b-k}
		b_{-}^{2}/4a-b_{+}^{2}/4a=k(n+1)\in \mathbb{Z}\ .
	\end{equation}

	The terms can now be rearranged in an ascending order of powers of $q$. There are two
	orderings of the expansion based on the values of $b_{-}$. That is, for
	$|b_{-}|<a$, and $b_{+}<0$ we have,
	\begin{align}\label{arrangement_1}
		\chi^{+}_{j(n,k)}(\tau)&=\frac{b_{+}}{u}q^{b_{+}^{2}/4a-1/8}(1+3q+9q^2+22q^3+51q^4
		+108q^5+221q^6+429q^7 \mathcal{O}(q^8))\times\nonumber\\
		&\quad\Big[1-\frac{b_{-}}{b_{+}} q^{k(n+1)}-\sum_{s\in \mathbb{N}} q^{k(n+1)+s^2a+sb_{-}}\big(\frac{2usp}{b_{+}}+\frac{b_{-}}{b_{+}}\big)+\sum_{s\in \mathbb{N}}q^{s^2a+sb_{+}}\big(1+\frac{2usp}{b_{+}}\big)\nonumber\\
		&\quad+\sum_{s\in \mathbb{N}} q^{s^2a-sb_{+}}\big(1-\frac{2usp}{b_{+}}\big)+\sum_{s\in \mathbb{N}} q^{k(n+1)+s^2 a-sb_{-}}\big(\frac{2usp}{b_{+}}-\frac{b_{-}}{b_{+}}\big)\Big] \ ,
	\end{align}
	or, for $|b_{-}|>a$, and $b_{+}<0$ we have,
	\begin{align}\label{arrangement_2}
		\chi^{+}_{j(n,k)}(\tau)&=\frac{b_{+}}{u}q^{b_{+}^{2}/4a-1/8}(1+3q+9q^2+22q^3+51q^4+108q^5+221q^6+429q^7 \mathcal{O}(q^8))\times\nonumber\\
		&\quad\Big[1+\big(-\frac{2up}{b_{+}}-\frac{b_{-}}{b_{+}}\big)q^{k(n+1)+a+b_{-}} -\frac{b_{-}}{b_{+}} q^{k(n+1)}
		-\sum_{s-1\in\mathbb{N}} q^{k(n+1)+s^2a+sb_{-}}\big(\frac{2usp}{b_{+}}+\frac{b_{-}}{b_{+}}\big)\nonumber\\
		&\quad+\sum_{s\in \mathbb{N}} q^{s^2a+sb_{+}}\big(1+\frac{2usp}{b_{+}}\big)+\sum_{s\in \mathbb{N}} q^{s^2a-sb_{+}}\big(1-\frac{2usp}{b_{+}}\big)\nonumber\\
		&\quad+\sum_{s\in \mathbb{N}} q^{k(n+1)+s^2 a-sb_{-}}\big(\frac{2usp}{b_{+}}-\frac{b_{-}}{b_{+}}\big)\Big] \ .
	\end{align}
	Details of the ordering can be found in the
	appendix \ref{rearrangement}. This rearrangement paves the way for a hierarchical arrangement of the powers of \textit{q}. This is shown in
	eq. \eqref{long_character} for the arrangement with $|b_{-}|<a$. The expansion  makes the coefficients of the
	\textit{q}-series up to $q^{4a+2b_{-}+k(n+1)}$ appear manifestly as a function of $(n,k)$ for a level parametrised as $(p,u)$. To investigate if the character is a quasi-character  we look for a sign flip of the coefficients in the \textit{q}-expansion. \footnote{A
		quasi-character has integer coefficients, but not all of them are positive (or negative), that is, they undergo a sign flip in its $q$-expansion}.  From these $q$-expansions we have found that there are quasi-character(s) at every level except the levels $(p=2,u=2N+1)$, $(p=2N+3,u=2)$, where $N\in\mathbb{N}$ and $(p=3,u=4)$.

	\section{Characters of $\widehat{su}(2)$ algebra at special fractional
		levels}\label{physical_characters}

	In this section, we will study representations of $\widehat{su}(2)$
	algebra at the special fractional levels which only have admissible characters. If we look at the $q$-series expansion of
	\eqref{arrangement_1} we see that quasi-characters are ubiquitous
	except at very special points in the space of admissible
	representations.  These special levels correspond to $(p=2, u=2N+1)$,
	$(p=2N+3,u=2)$, where $N\in\mathbb{N}$ and an isolated point at $(p=3,u=4)$.  We will focus
	on these special points in this section but before we discuss that, a
	comment on the appearance of quasi-characters for all other cases is
	in order.

	It is obvious from the $q$-expansions of the character formula at
	these levels that, except at the special levels mentioned above, we
	always find one quasi-character with labels $(n=0, k=\lceil(u/p)\rceil)$,where$\lceil x\rceil$ denotes the ceiling function of $x$.  Furthermore, for higher $k$ and $n$ values, one may
	find additional quasi-character(s).  While we have studied the
	$q$-expansions for a large but finite set of $p$ and $u$ values,
	existence of a quasi-character for the labels $(n=0,
	k=\lceil(u/p)\rceil)$ guarantees that every $\widehat{su}(2)$ theory at
	fractional levels, except for the special ones, have at least one
	quasi-character.  To get a better picture of this pattern some
	illustrative examples are presented in the appendix
	\ref{examples_quasi}.  In the appendix \ref{quasi_conditions} we show
	that the condition on $(n, k)$ labels of the quasi-character can be motivated analytically as
	well, from the small order \textit{q}-expansion of the unflavoured
	character \eqref{long_character}.

	\subsection{Threshold cases}

	Among the special levels mentioned at the beginning of this section,
	the set with $p=2$ corresponds to the threshold cases.  Any values of
	$p$ smaller than 2 do not lead to admissible representation theory
	{\em a la} Ka\v c-Wakimoto.  In this sub-section, we will look only at
	these threshold level cases.

	These cases have the level characterised by $p=2$ which implies $n=0$
	and $u=2N+1$, because $p$ and $u$ are coprime, therefore
	$k=0,\cdots, 2N$.  The central charge for these theories is,
	\begin{equation}
		c=-6N.
	\end{equation}
	The level $m=-4/3$ is the most widely studied example among the
	threshold cases since it provides the smallest dimensional
	representation theory with only two highest weight characters \cite{Mathur:1988gt, Mukhi:1989bp,
		Gaberdiel:2001ny, Adamovic:2004zi, Creutzig:2013hma,
		Buican:2015ina, Beem:2017ooy, Beem:2019tfp}.

	We follow a procedure, which we will refer to as unitarisation, in
	which we relabel the highest weight representations such that the lowest
	dimensional primary is identified with the vacuum of the unitarised theory. This representation corresponds to a representation theory of another RCFT, with typically positive central charge and conformal dimensions. The relabelling of the
	representations does not affect the character expansions. Thus, if
	the unitarisation procedure gives a representation theory of a unitary RCFT then it automatically guarantees that all the characters are admissible. We find that it is true for every theory at threshold levels, thereby ensuring that all threshold
	cases admit admissible characters corresponding to RCFTs\footnote{It has been argued by
		Gaberdiel\cite{Gaberdiel:2001ny} that the $\widehat{su}(2)_{-4/3}$ is a
		$\ log$ CFT since the fusion algebra will otherwise not close. In
		addition, it was known earlier that the Verlinde formula gives
		negative fusion coefficients \cite{Mathur:1988gt}. The negative
		fusion coefficients might occur for other threshold levels as
		well, but it has no bearing on the unitarisation procedure.  It may
		happen that the unitarised theory have negative fusion
		coefficients, but that indicates that the correct CFT is a
		non-unitary one.}.  Had we failed in identifying an appropriate unitary
	partner for a particular threshold case, it would have cast doubt on
	the well-definedness of these theories, and we would have to check it
	in a case-by-case manner.  Thankfully, this does not happen in the
	threshold cases, but the situation is not as rosy in the case of $u=2$
	theories, which we will consider in section \ref{half_integer}.

	The lowest dimension primary is obtained from the set of highest
	weight representations by minimising the conformal dimension with
	respect to the label $k$,
	\begin{align}
		h_{min}=h_N=-\frac{N}{2}\left(\frac{N+1}{2N+1}\right)\ .
	\end{align}
	We identify this representation with the vacuum representation of
	the unitarised theory. The character of this vacuum representation has
	its leading coefficient $b_{+}(N)=1$, hence it is non-degenerate.  The central charge of
	unitarised CFT is,
	\begin{align}
		c_U=c-24h_{min}
		=\frac{3(4N)}{4N+2}\ ,
	\end{align}
	which is the central charge of $\widehat{su}(2)_{4N}$ CFT.  The spectrum
	of conformal dimensions in this theory is obtained by just shifting
	all the conformal dimensions of the parent fractional level theory by
	$h_{min}$,
	\begin{align}
		h^{U}_{k}=\frac{(N-k)(N-k+1)}{4N+2}\ ,
	\end{align}
	with the spins $l/2=N-k$.

	The unitarisation procedure for the threshold cases provides a map to
	the D-type modular invariants of $\widehat{su}(2)_{4N}$ theories with
	$N\in\mathbb{N}$ \cite{Mukhi:1989bp}.

	Although the fractional level theory has several highest weight
	representations, only the unflavoured characters in the even sector of the
	threshold cases are related to the unflavoured characters of the unitarised CFT.
	In fact, the flavoured characters do not match, but the unflavoured
	even characters are equal to the unflavoured characters of the D-type
	modular invariants of $\widehat{su}(2)$ theory at level $4N$. This can be
	easily shown by manipulating the numerators of the characters, the
	denominator factor is independent of the level and does not play any
	role in the manipulation.  Instead of using the label $k$, we
	choose the spin of the representation $l$ to label the character so as
	to match with the conventional representation of characters of the
	integrable representation.
	\begin{align}\label{dtypechar}
		\chi^{+}_{j(k)} (q)
		&=\frac{2(2N+1)}{\eta^3(q)}q^{(l+1)^2/4(4N+2)}\sum_s\Big[ q^{s^2(4N+2)+s(l+1)}\big(2s+
		\frac{l+1}{4N+2}\big)\nonumber\\
		&\quad-q^{k+s^2(4N+2)+s(-4N+l-1)}\big(2s-\frac{4N-l+1}{4N+2}\big)
		\Big]\nonumber\\

		&=\chi^{(4N)}_{l}(q)+
		\chi^{(4N)}_{4N-l}(q).
	\end{align}

	The common denominator, $u=2N+1$ in the characters is removed by scaling the characters by $u$, thereby ensuring the integrality of all even characters.  Note that the character $\chi_{j(0)}=\chi^{(4N)}_{2N}$.The final expression of the characters can be compared with
	eq. \eqref{integrable_character}, with the identification $4N=M$.  For simplicity, we will call $\chi^{+}_j(n,k) (q)$ from now on to
	denote the characters with integer coefficients, that is, all the even characters are multiplied by $u$, which preserves the form of the modular $S$ transformation.

	This establishes the positivity of the characters of threshold cases.
	Their partition function is also a D-type modular invariant, as
	mentioned above.
	\begin{align}
		\mathcal{Z}^{(4N)}=2|\chi^{(4N)}_{2N}|^2+\sum_{l=0}^{N-1}
		|\chi^{(4N)}_{2l}+\chi^{(4N)}_{4N-2l}|^2\ .
	\end{align}

	Let us now consider the example of $\widehat{su}(2)_{-4/3}$ theory, where we have the A-type modular invariant
	partition function given by
	\begin{align}
		\mathcal{Z}=|\chi_{j(0,0)}|^2+\frac{1}{2}|\chi^{+}_{j(0,1)}|^2\ ,
	\end{align}
	where, $j(n,k)$ represents the highest weight state.  Clearly this
	partition function does not have integer coefficients but we can write
	another modular invariant partition function $\tilde{\mathcal{Z}}$,

	\begin{align}
		\tilde{\mathcal{Z}}=2|\chi_{j(0,0)}|^2+|\chi^{+}_{j(0,1)}|^2\ ,
	\end{align}
	which has a $q$-expansion with integer coefficients.

	If we identify the highest weight state $j(0,0)$ with the vacuum
	of the theory, which may be justified by virtue of it having zero
	conformal dimensions, then we find that we do not have a unique vacuum state. It has a degeneracy $u=3$ and a multiplicity of $2$. This can be
	interpreted in multiple ways, in \cite{Mathur:1988gt} it was shown
	that the fusion coefficients are not positive semidefinite and on
	unitarisation, which gives the partition function of $\widehat{su}(3)$ at
	level 1, does give rise to positive semidefinite fusion coefficients.
	Alternatively, one can interpret the non-uniqueness of the vacuum state as
	the existence of a logarithmic mode \cite{Gaberdiel:2001ny}.  This requires
	enlarging the Hilbert space by adding the logarithmic mode to it.
	Since we are using the unitarisation procedure we will follow the
	method of \cite{Mathur:1988gt}, identifying the vacuum with the labels $(0,1)$. We find that this is a
	recurring theme in the fractional level theories, as evidenced in the
	eq.\eqref{dtypechar}.  For higher values of $u$, we get higher
	degeneracy of the vacuum.  These theories may also have logarithmic extensions
	with $\ln^{k+1}$ type modes with $k=0,\cdots ,u-1$.  We hope to explore
	this possibility in future.

	\subsection{The Curious Case of $m=-5/4$ $(p=3,u=4)$}

	In this subsection, we will look at the isolated case of $m=-5/4$.  This
	is a particularly interesting case because it is an island in the
	two-dimensional space of coprime integers $(p,u)$.  This is a
	$\widehat{su}(2)_{-5/4}$ theory with $p=3$ and $u=4$.  The labels, spins, and their conformal dimensions are listed in table
	\ref{table:fivebyfour}.  As can be seen from the table that the last three entries have conformal dimensions which occur twice.  Their labels are related by
	$(\bar n, \bar k) = (p-2-n, u-k)$.  The subset $\Gamma$, which is the chosen even sector, is given by the shaded region in the table.  The central charge of this theory is $c=-5$.

	\begin{table}[h]
		\centering
		\begin{tabular}{|m {1.5 cm}|m {1.5 cm}|m {1.5 cm}|}
			\hline
			$(n,k)$ & $j(n,k)$ & $h(n,k)$\\
			\hline
			\rowcolor[rgb]{.9,.9,1}
			(0,0) & 0 & 0 \\
			\rowcolor[rgb]{.9,.9,1}
			(1,0) & $1/2$ & 1 \\
			\rowcolor[rgb]{.9,.9,1}
			(0,1) & -$3/8$ & -$5/16$ \\
			\rowcolor[rgb]{.9,.9,1}
			(1,1) & $1/8$ & $3/16$ \\
			\rowcolor[rgb]{.9,.9,1}
			(1,2) & -$3/4$ & -$1/4$ \\
			\hline
			(0,2) & -$1/4$ & -$1/4$ \\
			(0,3) & -$9/8$ & $3/16$ \\
			(1,3) & -$5/8$ & -$5/16$ \\
			\hline
		\end{tabular}
		\caption{The primaries are labelled by $(n,k)$,
			their spins are $j(n,k)$, and conformal dimensions $h(n,k)$. The unitarised conformal dimensions are denoted by $h_U(n,k)$. The
			shaded region is the smaller set $\Gamma$.}
		\label{table:fivebyfour}
	\end{table}

	\begin{comment}
		\begin{align}
			(n,k)\in\left\{ (0,0), (1,0), (0,1), (1,1), (0,2), (1,2),
			(0,3), (1,3)\right\}.
		\end{align}
		Since we will look at even characters we restrict to a smaller set
		\begin{align}
			\Gamma=\left\{ (0,0),(1,0),(0,1),(1,1),(0,2)\right\},
		\end{align}
		with conformal dimensions $h(0,0)=0$, $h(1,0)=1$, $h(0,1)=-5/16$,
		$h(1,1)=3/16$, $h(0,2)=-1/4$ and
	\end{comment}

	Note that the conformal dimensions of $(n=0,k=0)$ and $(n=1,k=0)$ differ
	by an integer which may give an impression that this theory can be a
	logarithmic CFT, however, this is a special case and we will see that
	the sum of the two characters is required to build characters of the
	unitarised theory.  Since $\ log$ CFTs are always non-unitary there is no
	unitarised counterpart of it.

	We can construct A-type modular invariant partition function using
	only the even characters,
	\begin{align}
		\mathcal{Z}(\tau)=\sum_{i,j\in \Gamma}\bar \chi_{i}M_{ij}\chi_{j}\ .
	\end{align}
	Corresponding modular
	$S$-matrix can be read off from eq.\eqref{s} and
	eq.\eqref{s+},
	\begin{align}
		\begin{pmatrix}
			\chi_{j(0,0)}(-1/\tau)\\
			\chi_{j(1,0)}(-1/\tau)\\
			\chi^{+}_{j(0,1)}(-1/\tau)\\
			\chi^{+}_{j(1,1)}(-1/\tau)\\
			\chi^{+}_{j(1,2)}(-1/\tau)\\
		\end{pmatrix}
		=\frac{1}{2\sqrt{2}}
		\begin{pmatrix}
			-1 & 1 & 1 & -1 & 1\\
			1 & -1 & 1 & -1 & -1\\
			2 & 2 & \sqrt{2} & \sqrt{2} & 0\\
			-2 & -2 & \sqrt{2} & \sqrt{2} & 0\\
			2 & -2 & 0 & 0 & 2\\
		\end{pmatrix}
		\begin{pmatrix}
			\chi_{j(0,0)}(\tau)\\
			\chi_{j(1,0)}(\tau)\\
			\chi^{+}_{j(0,1)}(\tau)\\
			\chi^{+}_{j(1,1)}(\tau)\\
			\chi^{+}_{j(1,2)}(\tau)\\
		\end{pmatrix}\ ,
	\end{align}
	where we have rescaled all characters by a factor $u$ so that all
	coefficients in the $q$-expansion are integers.

	The $S$-matrix is not unitary but $S^{\dagger}MS=M$, where
	$M=\hbox{\rm diag}(1,\, 1,\,\frac{1}{2},\,\frac{1}{2},\,\frac{1}{2})$.
	The factors $1/2$ imply that we need to rescale the $\chi_{n,0}$
	characters by $2$ since it is the smallest integer which makes the
	coefficients integers. Hence the actual modular invariant is,
	\begin{align}
		\mathcal{Z}(\tau)
		&=2|\chi_{j(0,0)}|^2 +2 |\chi_{j(1,0)}|^2+ |\chi^{+}_{j(0,1)}|^2
		+|\chi^{+}_{j(1,1)}|^2+|\chi^{+}_{j(1,2)}|^2\ .
	\end{align}
	This implies that the choice of the vacuum for the non-unitary theory is
	not unique.  We can instead choose the lowest dimensional primary as
	the vacuum character.  In this example, it corresponds to the choice of
	character $\chi^{+}_{j(0,1)}$ to belong to the vacuum module. The
	resulting unitarised theory have non-negative conformal dimensions
	and central charge.

	The RCFT description is in terms of the unitarised theory, where the primary with labels $(0,1)$ is identified with the vacuum. Table \ref{unitarised_p3u4} lists the unitarised conformal dimensions. The unitarised theory has the
	central charge,
	\begin{equation}
		\label{eq:cu}
		c_{U} = c-24h_{min} =-5+24(5/16))=5/2
	\end{equation}

	\begin{table}[h]
		\begin{center}
			\begin{tabular}{|m {1.5cm}|m {1.5 cm}|}
				\hline
				$(n,k)$ & $h_U(n,k)$ \\
				\hline
				\rowcolor[rgb]{.9,.9,1}
				(0,0) & $5/16$\\
				\rowcolor[rgb]{.9,.9,1}
				(1,0) & $21/16$\\
				\rowcolor[rgb]{.9,.9,1}
				(0,1) & 0\\
				\rowcolor[rgb]{.9,.9,1}
				(1,1) & $1/2$\\
				\hline
				(1,2) & $1/16$\\
				\hline
			\end{tabular}
			\caption{The unitarised value of conformal dimension is denoted by
				$h_U(n,k)$. The shaded region is the set of character that belongs
				to the unitarised theory.}\label{unitarised_p3u4}
		\end{center}
	\end{table}

	The primary field with conformal dimension $1/2$ indicates the
	presence of a fermion.  Combining this with the fact that the central
	charge $c_U=5/2$ implies that this theory has a free field representation in terms of 5 fermions
	which form a representation of the $\widehat{so}(5)_1$ current algebra.  It is
	convenient to use the matrix $U$ to interchange the position of the
	vacuum character with the character corresponding to $(0,1)$ to
	get the representation of the unitarised theory.
	\begin{align}
		U\overrightarrow{\chi}
		=\begin{pmatrix}
			0 & 0 & 1 & 0 & 0\\
			0 & 1 & 0 & 0 & 0\\
			1 & 0 & 0 & 0 & 0\\
			0 & 0 & 0 & 1 & 0\\
			0 & 0 & 0 & 0 & 1
		\end{pmatrix}
		\begin{pmatrix}
			\chi_{j(0,0)}(\tau)\\
			\chi_{j(1,0)}(\tau)\\
			\chi^{+}_{j(0,1)}(\tau)\\
			\chi^{+}_{j(1,1)}(\tau)\\
			\chi^{+}_{j(1,2)}(\tau)\\
		\end{pmatrix}=
		\begin{pmatrix}
			\chi^{+}_{j(0,1)}(\tau)\\
			\chi_{j(1,0)}(\tau)\\
			\chi_{j(0,0)}(\tau)\\
			\chi^{+}_{j(1,1)}(\tau)\\
			\chi^{+}_{j(1,2)}(\tau)\\
		\end{pmatrix},
	\end{align}
	This $U$ matrix transforms the modular $S$-matrix of the original
	theory by adjoint action to that of the unitarised theory,
	\begin{align}
		S_{U}=U^{-1}SU=\frac{1}{2\sqrt{2}}
		\begin{pmatrix}
			\sqrt{2} & 2& 2 & \sqrt{2}& 0\\
			1 & -1 & 1  & -1 & -1\\
			1 & 1 & -1 & -1 & 1\\
			\sqrt{2} & -2 & -2 & \sqrt{2} & 0\\
			0 & -2 & 2 & 0 & 2\\
		\end{pmatrix}\ .
	\end{align}
	However, note that this unitarised $S$-matrix has an unusual
	property, namely, $S_{05}=0$.  Vanishing matrix elements in the vacuum
	row spells doom for the Verlinde fusion coefficients as $S_{0i}$ appears
	in the denominator.  However, we can restrict ourselves to the $4
	\times 4$ subsector of the $S$-matrix,{\em i.e.},
	\begin{align}
		S_{4}=\frac{1}{2\sqrt{2}}\begin{pmatrix}
			\sqrt{2} & 2 & 2 & \sqrt{2}\\
			1 & -1 & 1 & -1\\
			1 & 1 & -1 & -1\\
			\sqrt{2} & -2 & -2 & \sqrt{2}\\
		\end{pmatrix}\ .
	\end{align}
	However this does not alleviate the problem completely.  In
	particular, there is no $4\times 4$ diagonal matrix left invariant by
	$S_{4}$.  While this rules out the $A$-type modular invariant, we can
	look for a $D$-type modular invariant to accommodate this spectrum since we have
	the characters whose conformal dimensions are separated by an integer
	which can be combined to build the non-diagonal partition function.\footnote{In section \ref{sub-sector} we look at this from the perspective of finding a block-diagonal form of the $S$-matrix to locate sub-sectors within the theory.}.  The non-diagonal modular invariant partition
	function has the form,

	\begin{align}
		\mathcal{Z}(\tau)=|\chi_{j(0,1)}|^2+|\chi_{j(0,0)}+\chi_{j(1,0)}|^2
		+|\chi_{j(1,1)}|^2 \ .
	\end{align}
	The existence of this sub-sector justifies our choice to restrict to the smaller $4\times 4$ S-matrix. Since the vacuum character corresponds to $\chi^{+}_{j(0,1)}$, which has a
	unit leading coefficient in its \textit{q}-expansion, ensures that the vacuum is unique for this
	theory. The modular $S$-transformation of the characters is given by,
	\begin{equation*}
		\begin{pmatrix}
			\chi^{+}_{j(0,1)}(-1/\tau)\\
			\chi_{j(1,0)}(-1/\tau)+\chi_{j(0,0)}(-1/\tau)\\
			\chi^{+}_{j(1,1)}(-1/\tau)
		\end{pmatrix}
		=
		\begin{pmatrix}
			1/2 & 1/\sqrt{2} & 1/2\\
			1/\sqrt{2} & 0 & -1/\sqrt{2}\\
			1/2 & -1/\sqrt{2} & 1/2
		\end{pmatrix}
		\begin{pmatrix}
			\chi^{+}_{j(0,1)}(\tau)\\
			\chi_{j(1,0)}(\tau)+\chi_{j(0,0)}(\tau)\\
			\chi^{+}_{j(1,1)}(\tau)
		\end{pmatrix}.
	\end{equation*}
	Note that this $S$-matrix is now orthogonal
	$S^{T}S=\mathbb{I}$ (and therefore unitary). The Verlinde Fusion rules,
	\begin{align}
		N_{(n_i,k_i),(n_j,k_j)}^{(n_l,k_l)}=\sum_{r\in \Gamma}
		\frac{S_{(n_i,k_i),(n_r,k_r)}S_{(n_j,k_j),(n_r,k_r)}
			S^{*}_{(n_r,k_r),(n_l,k_l)}}{S_{(0,0),(n_r,k_r)}}\ ,
	\end{align}
	can be explicitly calculated from the $S$-matrix and they are given by the fusion matrices $(N_i)^j_k$,
	\begin{align}
		N_{(0,1)}=
		\begin{pmatrix}
			1 & 0 & 0\\
			0 & 1 & 0\\
			0 & 0 & 1
		\end{pmatrix},
		N_{(0,0)+(1,0)}=
		\begin{pmatrix}
			0 & 1 & 0\\
			1 & 0 & 1\\
			0 & 1 & 0
		\end{pmatrix},
		N_{(1,1)}=
		\begin{pmatrix}
			0 & 0 & 1\\
			0 & 1 & 0\\
			1 & 0 & 0
		\end{pmatrix}\ .
	\end{align}
	The resulting unitary CFT is the $\widehat{so}(5)_1$ WZW model which has central
	charge $c=5/2$.  It can also be described in terms of 5 copies of the Ising
	Model.  We can identify the primaries in the following manner,
	\begin{align}
		\left(0,1\right) &\equiv \mathbb{I}\nonumber\\
		\left(0,0\right)+\left(1,0\right)&\equiv \sigma\nonumber\\
		\left(1,1\right) &\equiv \epsilon\ .
	\end{align}
	With this identification these primaries have same fusion rules as the
	Ising model,
	\begin{align}
		\epsilon \times \epsilon &=\mathbb{I}\ ,\nonumber\\
		\sigma \times \sigma &= \mathbb{I}+\epsilon\ ,\\
		\sigma \times \epsilon &=\sigma\ .\nonumber
	\end{align}
	We can explicitly show that these characters are identical to the
	characters of $\widehat{so}(5)_{1}$\cite{DiFrancesco:1997nk}.
	\begin{align}
		\chi_{j(0,1)}&=\chi_{\hat{\omega}_0}=\frac{1}{2}\frac{
			\theta_{3}^{5/2}+\theta_{4}^{5/2}}{\eta^{5/2}}\ ,\\
		\chi_{j(1,1)}  &=\chi_{\hat{\omega}_1}=\frac{1}{2}\frac{
			\theta_{3}^{5/2}-\theta_{4}^{5/2}}{\eta^{5/2}}\
		,\\
		\chi_{j(0,0)}+\chi_{j(1,0)}&=\chi_{\hat{\omega}_2}=\frac{1}{\sqrt{2}}\left(
		\frac{\theta_{2}}{\eta}\right)^{5/2}\ .
	\end{align}

	Recall, we are looking only at the even characters, it is worth
	pointing out that the odd combination of characters for this level
	gives us the characters of the Ising Model.

	There is an interesting alternative to the choice of the sub-sector, with the choice of the
	representation $(1,2)$ as the vacuum representation $\chi_{j(1,2)}$
	and take orthogonal non-diagonal combination of $(0,0)$ and $(1,0)$,
	namely, $\chi_{j(0,0)}-\chi_{j(1,0)}$.  This set forms a modular
	invariant combination with the central charge $c=1$ and corresponds to
	$\widehat{su}(2)_{1}$ WZW model.  The spectrum of conformal dimensions is
	$h_U(1,2)=0$ and $h_U((0,0)-(1,0))=1/4$.  The explicit equivalence of the characters is given below. The common multiplicative factor $u=4$ for the even characters can be reduced to 2 by scaling the characters of the sub-sector $ \chi^{+}_{j(1,2)}$, and $\chi^{+}_{j(0,0)}(q)-\chi^{+}_{j(1,0)}$ by $1/2$, which preserves the modular $S$-transformation of the sub-sector.\footnote{This will be manifest in section \ref{sub-sector} where we write the block-diagonal $S$-matrix.}
	\begin{align}
		\chi^{+}_{j(1,2)}(q)=
		&=\frac{2q^{1/12}}{ \eta^3(q)} \sum_s\left[ q^{12s^2+2s}(6s+\frac{1}{2})-
		q^{12s^2-14s+4}(6s-\frac{7}{2})\right]
		\nonumber\\
		&=\frac{q^{1/12}}{ \eta^3(q)}\sum_s(1+6s)q^{s+3s^2}=\chi^{(1)}_{0}(q) \ .
	\end{align}
	Similarly,
	\begin{align}
		\chi_{j(0,0)}(q)-\chi_{j(1,0)}(q)
		&=\frac{2q^{1/3}}{\eta^3(q)}\sum_s\Big[ q^{12s^2+4s}(6s+1)-q^{12s^2+8s+1}(6s+2)\Big]\nonumber\\
		&=\frac{q^{1/3}}{\eta^3(q)}\sum_s(2+6s)q^{3s^2+2s}
		=\chi^{(1)}_{1}(q)\ .
	\end{align}
	This identification guarantees that all coefficients in the
	$q$-expansion are positive. These relations completes the proof of the positivity of the characters in the representation of $(p=3,u=4)$.

	\subsection{Positive Half-Integer Levels}\label{half_integer}

	In this subsection, we will look at the other infinite series of
	theories without quasi-characters.  These theories are parametrised by
	$(p=2N+3, u=2)$ with $N\in \mathbb{N}$.  The corresponding levels are half odd integers
	$m=p/u-2=(2N-1)/2$.  We have checked that the
	$q$-expansions of all the characters for $5\leq p\leq 29$ have
	positive coefficients (up to $\sim q^{2000}$).  As an aside, the $(p=3,u=2)$ theory has a
	quasi-character but we will deal with this case separately.  For now,
	we focus on the unitarisation of the entire $(p=2N+1, u=2)$ series with $N\in \mathbb{N}$.

	The central charge of these theories is given by,
	\begin{align}
		c=3-\frac{12}{2N+1}\ .
	\end{align}
	Since we have $u=2$, $k$ can take two values $0$ and $1$.  The number
	of independent even characters for $p=2N+1$ is $3N$.  The $2N$
	characters corresponding to $k=0$ are equal to the half-integer spin
	characters of $\widehat{su}(2)_{4N}$ theory.  The remaining
	$N$
	even characters $\chi^{+}_{j(n,1)}$, corresponding to $k=1$ can be written in terms of the
	difference of integer spin characters of the same affine Lie algebra
	$\widehat{su}(2)_{4N}$.

	The spins $j(n,k)$ are
	\begin{align}
		j(n,0)&=\frac{1}{2}n, \nonumber\\
		j(n,1)&=\frac{1}{2}(n-N)-\frac{1}{4},
	\end{align}
	with the conformal dimensions,
	\begin{align}
		h(n,0)&=\frac{n(n+2)}{2(2N+1)},\nonumber\\
		h(n,1)&=\frac{(n-N)^2+(n-N)-3/4}{2(2N+1)}\ .
	\end{align}
	The minimum conformal dimension among them is
	\begin{align}\label{u2_hmin}
		h_{min}=h(N,1)=\frac{-3}{8(2N+1)}\ .
	\end{align}

	Following the usual unitarisation procedure, {\em i.e.}, identifying
	the character $\chi^{+}_{j(N,1)}$ with the character of the identity of the unitary
	theory gives the unitarised central charge,
	\begin{equation}
		c_U=3-\frac{12}{2N+1}+\frac{9}{2N+1}=\frac{3(4N)}{4N+2}\ .
	\end{equation}
	This is the central charge of $\widehat{su}(2)_{4N}$ WZW models from the
	traditional Sugawara construction.

	The unitarised conformal dimensions are obtained by
	adding $h_{min}$ to all the conformal dimensions of the original theory. For $k=0$ set,
	\begin{align}
		h_{U}(n,0)=\frac{(n+1/2)((n+1/2)+1)}{4N+2} \ .
	\end{align}
	We identify the spin $l/2=n+1/2$ with odd $l$ values of the representation of
	$\widehat{su}(2)_{4N}$. The other set of characters (with $k=1$) have dimensions,
	\begin{align}
		h_{U}(n,1)=\frac{(n-N)(n-N+1)}{4N+2}\ .
	\end{align}
	The identification of spins in this case is $l/2=(n-N)$ with even $l$.
	We can choose the set $N\leq n \leq 2N-1$ so that $l$ is positive.

	The $k=0$ unflavoured characters match with the unflavoured characters of
	integrable highest weight representations of $\widehat{su}(2)_{4N}$ with
	spins $l/2=n+1/2$,
	\begin{align}
		\chi_{j(n,0)}(q)

		&=\frac{1}{\eta(q)^3}q^{(2n+2)^2/4(4N+2)}\sum_{s \in \mathbb{Z}}
		q^{(4N+2)s^2+s(2n+2)}(2(4N+2)s+2n+2)\nonumber\\
		&=\chi_{2n+1}^{(4N)}(q)=\chi_{l}^{(4N)}(q)\ .
	\end{align}

	The characters with $k=1$, on the other hand, can be written as the
	difference of the characters $(\chi^{(4N)}_l-\chi^{(4N)}_{4N-l})$ of the integrable
	highest weight representations of the unitarised theory with spins
	$l=2(n-N)$ and $4N-l=6N-2n$ respectively, with $n$ restricted to take
	values in $N\leq n \leq 2N-1$ for both cases. Note that
	\begin{align}
		b_{+}(n,1)&=2(n+1)-(2N+1)=l+1,\nonumber\\
		b_{-}(n,1)&=-2(n+1)-(2N+1)=-(l+1)-(4N+2).\nonumber
	\end{align}
	We will use the variable $l$ instead of $n$ to write the $q$-expansion
	of the character,
	\begin{align}
		\chi^{+}_{j(n,1)}(q)
		&=\frac{2q^{(l+1)^2/4(4N+2)}}{\eta^3(q)}\sum_s\Big[q^{s^2(4N+2)+s(l+1)}
		(s(4N+2)+(l+1)/2)\nonumber\\
		&\quad-q^{s^2(4N+2)-s(l+1)-s(4N+2)+n+1}(s(4N+2)-
		\frac{l+1+4N+2}{2})\Big].
	\end{align}
	Recall the map $\sigma_0$ takes $l/2\rightarrow -l/2-1$ and
	$s\rightarrow -s$.  We use this map to rearrange the characters to bring
	them to a form which is easy to compare with the $\widehat{su}(2)_{4N}$
	characters.
	\begin{align}
		\chi^{+}_{j(\bar n,1)}(q)

		&=\frac{-q^{(l+1)^2/4(4N+2)}}{\eta^3(q)}\Big[\sum_s
		q^{s^2(4N+2)+s(l+1)}(2s(4N+2)+(l+1))\Big]\nonumber\\
		&\quad+\frac{q^{(4N-l+1)^{2}/4(4N+2)}}{\eta^{3}(q)}\Big[
		\sum_s q^{s^2(4N+2)+s(4N-l+1)}(2s(4N+2)+4N-l+1)\Big]
		\nonumber\\
		&=-\left[\chi^{(4N)}_{l}(q)-
		\chi^{(4N)}_{4N-l}(q)\right]\ .
	\end{align}
	Using the relation between the characters with labels
	$(\bar n, \bar k)$ and those with $(n,k)$, we can write the characters
	with the label $(n,1)$ as,
	\begin{equation}
		\chi^{+}_{j(n,1)}(q)= -\chi^{+}_{j(\bar n,1)}(q)=
		\left[\chi^{(4N)}_{l}(q)-\chi^{(4N)}_{4N-l}(q)\right]\ ,
	\end{equation}

	where, $l$ is an even integer.  It is not obvious from this expression for the characters that the resulting character $\chi^{+}_{j(n,1)}$ has manifestly positive coefficients.  In fact, there is a counter-example, at the level $m=-1/2$, {\em i.e.} $N=1$, the character $\chi^{+}_{j(0,1)}=-\chi^{+}_{j(1,1)}$ has the $q$-expansion\footnote{It is tempting to think of
		this expression not as a character but as a Witten Index of a theory
		with broken supersymmetry as the alternating signs are reminiscent of
		the operator $(-1)^{F}$ insertion in the character.  However, we do
		not have a reason to believe supersymmetry is at play here.},
	\begin{align}\label{qc_3_2}
		\chi^{+}_{j(1,1)}(q)=q^{-1/12}\left(1-2q^{1}+q^{2}-2q^{3}+4 q^{4}-4 q^{5}+5 q^{6}-6 q^{7}+9 q^{8}+\mathcal{O}(q^9)\right)\ ,
	\end{align}
	The unitarised theory has 3 independent even characters with the vacuum character given by eq. \eqref{qc_3_2}. We have checked for large powers of $q$,
	that the quasi-character $\chi^{+}_{j(1,1)}$ alternates between positive and negative signs which belongs to the class D of quasi-characters and the three character theory in class DAA as defined in \cite{Mukhi:2020gnj}.

	The $m=-1/2$ model has been studied in
	detail \cite{Lesage:2002ch, Lesage:2003kn, Ridout:2008nh,
		Ridout:2010jk, Creutzig:2012sd, Creutzig:2013hma, Creutzig:2013yca}
	where the logarithmic nature of $\widehat{su}(2)_{-1/2}$ representations has
	been analysed.  In particular, Ridout and Creutzig
	have shown that there exists a continuous family of
	modules \cite{Creutzig:2013hma} giving
	complete but an alternative description of the $m=-1/2$ theory, including positive semidefinite fusion rules using the Verlinde formula.

	The unitarisation procedure we are using may
	fail to produce an RCFT with correct modular data, which do not give us well-defined, positive semidefinite Verlinde fusion coefficients in certain situations, and in that case we might need to reconsider
	the non-unitary theory with the extended modules. In general for
	the half odd integer levels, we find that the unitary theory has a
	unique vacuum but the fusion coefficients are not well-defined on the other
	hand, the original non-unitary avatar does not have a unique vacuum
	and in addition, we do not get positive semi-definite Verlinde fusion coefficients \cite{Koh:1988sm}.
	A possible resolution of this conundrum is to adopt the interpretation of
	these theories as logarithmic CFTs, which is in resonance with the
	above mentioned literature.  It is also proposed that all non-unitary
	fractional level $\widehat{su}(2)$ theories are logarithmic \cite{Creutzig:2012sd, Creutzig:2013yca}.
	\begin{comment}
		They have also written manifestly positive fermionic characters of
		the extended algebra. The extended algebra utilises the zero mode of
		the $\beta\gamma$ representation instead of the
		$\beta\partial\gamma$ small algebra.  The characters of $A1$ have a
		relative minus sign due to the braiding pattern of the singular
		vectors of $\widehat{su}(2)_{-1/2}$ and hence are not necessarily
		positive.Details for the extended algebra and their representations
		can be found in \cite{}.
	\end{comment}

	For the theories with only admissible characters, $N\geq 2$, given the fact that the character $\chi^{+}_{j(n,1)}$ is obtained by
	subtraction of two characters, it is not guaranteed that it will have
	manifestly positive definite coefficients in the $q$-expansion.
	Nevertheless, except in the case of $m=-1/2$, it is possible to show that
	the coefficients of $\chi^{(4N)}_{l}$ are consistently larger than
	those for $\chi^{(4N)}_{4N-l}$ for $N\geq 2$ and hence the
	character $\chi^{+}_{j(n,1)}$ is an admissible character.  Although we
	do not have explicit proof of this assertion, we have checked this
	up to sufficiently high orders in the $q$-expansion.  For example,
	consider the $N=2$ case which has the characters,
	\begin{align}\label{eq:1}
		\chi ^{+}_{j(2,1)}(q)
		&=q^{-1/10}\left(1+3q+6q^{2}+3q^{3}+9 q^{4}+4 q^{5}
		+18q^{6}+9 q^{7}+30q^{8}+12q^9+\mathcal{O}(q^9)\right)\nonumber\\
		\chi ^{+}_{j(3,1)}(q)
		&=q^{1/10}\left(3+2q+6q^{2}+3q^{3}+12 q^{4}
		+6q^{5}+24q^{6}+9q^{7}+39q^{8}+18q^9+\mathcal{O}(q^9)\right).
	\end{align}
	For higher $N$ the coefficients increase at faster rates.  The
	subtraction of the characters also means that the character
	$\chi^{(4N)}_{2N}$ will never appear in the partition function.

	Let us continue with the example of $N=2$.  This is an illustrative example, all higher $N$ values work out essentially in
	identical manner.
	\begin{align}
		\mathcal{Z}=|\chi_{j(0,0)}|^2+|\chi_{j(1,0)}|^2+|\chi_{j(2,0)}|^2
		+|\chi_{j(3,0)}|^2+\frac{1}{2}|\chi^{+}_{j(2,1)}|^2+\frac{1}{2}
		|\chi^{+}_{j(3,1)}|^2\ .
	\end{align}
	The factor of $1/2$ in front of a couple of characters is potentially
	problematic for we need the partition integers to be positive
	integers.  This can be avoided by rescaling the partition function by
	a factor of 2.
	\begin{align}
		\tilde{\mathcal{Z}}= 2\mathcal{Z}=|\chi^{+}_{j(2,1)}|^2+|\chi^{+}_{j(3,1)}|^2
		+2|\chi_{j(0,0)}|^2+2|\chi_{j(1,0)}|^2+2|\chi_{j(2,0)}|^2
		+2|\chi_{j(3,0)}|^2\ .
	\end{align}
	At this point, it appears that we have a couple of choices for the
	vacuum character, namely, it could be $\chi^{+}_{j(2,1)}$ or
	$\chi^{+}_{j(3,1)}$, but a glance at eq.\eqref{eq:1} would convince us
	that it is the former one that is a vacuum character and not the
	latter as predicted from eq. \eqref{u2_hmin}.

	Since the characters are written as subtraction of two integrable
	highest weight characters, the partition function when expanded in
	terms of the integrable characters have negative coefficients. The
	presence of these negative signs does not allow the description of the
	partition function as an ordinary CFT.
	\begin{align}
		\tilde{\mathcal{Z}}= 2\mathcal{Z}=|\chi^{(8)}_{0}-\chi^{(8)}_{8}|^2+|\chi^{(8)}_{2}
		-\chi^{(8)}_{6}|^2+2|\chi^{(8)}_{1}|^2+2|\chi^{(8)}_{3}|^2
		+2|\chi^{(8)}_{5}|^2+2|\chi^{(8)}_{7}|^2
	\end{align}
	However, partition functions with negative coefficients are quite
	common in defect CFT, but the defect partition functions have
	different modular properties.  In this case, we have a partition
	function which is modular invariant under $SL(2,\mathbb{Z})$, and as a
	result, it is not a partition function of a defect CFT.  This
	effectively rules out the possibility of the half-integer level
	$\widehat{su}(2)$ highest weight states being related to defect CFT.  The
	possibility of the fractional level theories being logarithmic CFT
	\cite{Gaberdiel:2001ny, Creutzig:2013hma, Creutzig:2013yca}, however,
	is not ruled out by this analysis.

	Although, we find an interesting correspondence between the characters at
	the fractional level $m=(2N-3)/2$ and the character at the level
	$M=4N$ of the $\widehat{su}(2)$ CFT, the choice of the vacuum in the
	unitary theory poses a new puzzle.

	To understand this better let us look at the modular transformation
	of the characters $\chi^{+}$ in the `unitary' theory,
	\begin{equation}
		\chi^{+}_{j(n,1)}(-1/\tau)=2\left(\sum_{n'\in\Gamma}S^{+}_{n1,n'1}
		\chi^{+}_{j(n',1)}(\tau)+\sum_{n'}S_{n1,n'0}\chi_{j(n',0)}(\tau)\right)\ ,
	\end{equation}
	where the modular $S$-matrix element $S_{n1,n'0}$ is given by eq. \eqref{s} and
	\begin{align}
		S^{+}_{n1,n'1}&=\frac{1}{2}\left(S_{n1,n'1}-S_{\bar n1,n'1}\right)\nonumber\\
		&=\frac{1}{2}\Big[\frac{1}{\sqrt{2N+1}}(-1)^{n+1+n'+1}e^{-i\pi(N+1/2)}
		\sin\big(\frac{2\pi}{2N+1}(n+1)(n'+1)\nonumber \big)\\
		&\quad -\frac{1}{\sqrt{2N+1}}(-1)^{\bar n+1+n'+1}e^{-i\pi(N+1/2)}\sin \big(\frac{2\pi}{2N+1}(\bar n+1)(n'+1)\big)\Big]\ .
	\end{align}
	Writing $\bar n$ in terms of $p$ and $n$, followed by use of a
	trigonometric identity gives,
	\begin{equation}
		S^{+}_{n1,n'1}=\frac{-i(-1)^{n+n'+N}}{\sqrt{2N+1}}\Big[
		\sin\big(\pi(n'+1)\big)\cos\big(\frac{\pi}{2N+1}
		(n'+1)(2n-2N+1)\big)\Big]=0\ .
	\end{equation}
	Notice, $S^{+}_{n1,n'1}$ vanishes for all $n,n'$, and the vacuum
	character of the unitarised theory belongs to this sector.  It, therefore, follows that the $S$-matrix elements involving the vacuum
	character $\chi^{+}_{j(N,1)}$ and any character belonging to the $\chi^{+}_{j(n,1)}$ set vanishes.
	In the Verlinde fusion coefficients, the matrix elements corresponding to the vacuum are in the denominator. Since some matrix elements in the vacuum row of the modular $S$-transformation vanishes, the corresponding fusion coefficients are not well-defined. This leads to the
	conclusion that this theory is not an RCFT.

	The modular $S$-transformation of $\chi^{+}$ is written below,
	\begin{align}\label{eq:3}
		\chi^{+}_{j(n,1)}(-1/\tau)=\frac{2}{\sqrt{2N+1}}\sum_{n'}(-1)^{n'+1}
		\sin\big(\frac{2\pi}{2N+1}(n+1)(n'+1)\big)\chi_{j(n',0)}(\tau).
	\end{align}
	We will now relabel, $n=\frac{l}{2}+N$ and $n'=\frac{l'-1}{2}$ so that
	we can relate them to the integrable representations of $\widehat{su}(2)$
	at level $4N$,
	\begin{equation}
		\label{equality_s_matrix_u=2}
		\chi^{+}_{j(l/2+N,1)}(-1/\tau)

		=\frac{2}{\sqrt{2N+1}}\sum_{l'\in \hbox{\rm odd}}\sin\big(\frac{\pi}{4N+2}(l+1)(l'+1)\big)\chi_{j(l'/2-1/2,0)}(\tau).
	\end{equation}
	Since these characters are written in terms of subtraction of two characters of $\widehat{su}(2)$ at level $M=4N$ theory, we write the modular S-transformation as,
	\begin{equation}
		\label{eq:4}
		(\chi^{(4N)}_{l}-\chi^{(4N)}_{4N-l})(-1/\tau)

		=\frac{2}{\sqrt{2N+1}}\sum_{l'=0}^{4N-1}\big[\sin\big(\frac{\pi(l'+1)}{4N+2}(l-2N)\big)\cos\frac{\pi}{2}(l'+1)\big]\chi^{(4N)}_{l'}(\tau)
	\end{equation}
	It is easy to check that for even $l'$ the RHS of eq.\eqref{eq:4}
	vanishes and for odd $l'$ it reproduces eq. \eqref{equality_s_matrix_u=2}.
	\begin{equation}\label{no-subsector_u2}
		\chi^{(4N)}_{l}-\chi^{(4N)}_{4N-l}(-1/\tau)
		=\frac{2}{\sqrt{2N+1}}\sum_{l'\in \hbox{\rm odd}}\big[\sin\big(\frac{\pi}{4N+2}(l'+1)(l+1)\big)\big]\chi^{(4N)}_{l'}(\tau)\ .
	\end{equation}
	In summary, we find that the unitary theory does not have correct
	fusion coefficients due to vanishing matrix elements involving the
	vacuum character.  It may be interesting to explore the possibility of
	expanding the Hilbert space of the CFT in such a way that the modular
	$S$-matrix has non-vanishing vacuum row and column entries.

	There is an alternate way of salvaging these theories by looking for a suitable subsector of the theory which forms a closed set under the
	fusion algebra and admits a modular invariant partition function comprising only the characters in the subsector. While this seems a worthwhile exercise for the
	theories containing the quasi-characters, it is easy to see that there
	is no such suitable subsector in any theory with $u=2$.
	The argument is as follows. The primaries with the characters $\chi^{(4N)}_{2l}-\chi^{(4N)}_{4N-2l}$ have integer spins. One may be tempted to
	think that the integer spin subsector can very well have closed fusion
	algebra, but for
	$u=2$ case, we have seen that the modular
	$S$-transformation eq. \eqref{no-subsector_u2} relates the integer spin sector to the half odd-integer spin sector. On the other hand, the sector with characters $\chi^{(4N)}_{2l+1}$ have half odd-integer spins, which by themselves do not close under the $su(2)$ algebra. A little generalisation of this argument can show that no
	mixing of half odd-integer and integer spin subsectors can have a closed
	fusion algebra.

	\section{Closed Subsets without Quasi-Characters?}\label{sub-sector}

	In this section, we report our preliminary investigations into the problem of finding sub-sectors which are devoid of
	quasi-characters. So far we have been looking at theories at those
	fractional levels, which only have admissible characters. However, as
	we have seen, there are theories at other fractional levels which
	possess at least one quasi-character. It is easy to see that these
	theories have a modular invariant partition function with the
	inclusion of the quasi-character, but the presence of the quasi-character in the modular invariant
	makes difficult an interpretation of the partition function in terms of state
	counting of a physical system.

	To remedy this problem, we try to find subsectors of these theories. The subsectors are an admissible subset of characters closed under the fusion algebra equipped with a modular invariant partition function.  It is a priori not guaranteed that theories with such
	sub-sectors without quasi-characters exist at all. The reason for
	this suspicion stems from the observation that the modular $S$-matrix for these theories typically does not decompose into a manifestly block-diagonal form
	so that quasi-characters can be isolated. If the
	$S$-transformation matrix is in a block-diagonal form, it is a sufficient
	condition for the existence of a closed sub-sector. To see that it is
	also necessary\footnote{ In this point of view, we are either ignoring the (possible) linear combination of characters, or we can treat the suitable linear combination of the characters as the new basis for the S-matrix. In either case, the block-diagonalisability is a necessary and sufficient condition for the modular invariant of the sub-sector. }, let us consider an example of a theory with three
	characters denoted by $\chi_0$, $\chi_1$, and $\chi_2$ with the $S$-transformation of the $\chi_{0}$ character given by,
	\begin{align}
		\chi_0(-1/\tau)&=S_{00}\chi_0(\tau) +S_{01}\chi_1(\tau) + S_{02}\chi_2(\tau) \ ,\nonumber\\[2mm]
		\bar \chi_0(-1/\bar \tau)\chi_0(-1/\tau)&= |S_{00}|^2|\chi_0 |^2+S^{*}_{00}S_{01}\chi^{*}_{0}(\bar \tau)\chi_1(\tau)+ S^{*}_{00}S_{02}\chi^{*}_{0}(\bar \tau)\chi_2(\tau)\nonumber\\
		&\quad +S^{*}_{01}S_{00}\chi^{*}_{1}(\bar \tau)\chi_{0}(\tau)+|S_{01}|^2|\chi_1 |^2+S^{*}_{01}S_{02}\chi^{*}_{1}(\bar \tau)\chi_{2}(\tau)\nonumber\\
		&\quad +S^{*}_{02}S_{00}\chi^{*}_{2}(\bar \tau)\chi_{0}(\tau)
		+S^{*}_{02}S_{01}\chi^{*}_{2}(\bar \tau)\chi_{1}(\tau)+|S_{02}|^2|\chi_2 |^2 \ .
	\end{align}
	We can similarly write $S$-transformations for $\chi_1$ and $\chi_2$.
	Suppose, the character $\chi_2$ is a quasi-character and suppose that
	a sub-sector exists comprising only of $\chi_0$ and $\chi_1$, then the
	diagonal partition function is in the form,
	\begin{align}
		\mathcal{Z}=a_1|\chi_0 |^2+a_2|\chi_1 |^2 \ ,
	\end{align}
	where $a_1$ and $a_2$ are positive integers.  The condition of modular
	invariance of the partition function gives rise to,
	\begin{align}
		a_1|S_{02}|^2+a_2|S_{12}|^2=0 \ .
	\end{align}
	Since both $a_{1}$ and $a_{2}$ are positive integers, this condition
	is satisfied only when $S_{02}=S_{12}=0$.  All other conditions coming
	from the modular invariance of the partition function are then
	automatically satisfied.  It immediately follows from
	eq.\eqref{s_of_+} and eq.\eqref{s_of_0} that
	$S_{20}=S_{21}=0$\footnote{Up to the factor of 2 due to the relation
		$2S_{n0,n'k'}=S_{n'k',n0}$ for $k'\neq0$ it is a
		symmetric matrix.}.It is also easy to see that a block diagonal $S$-matrix is sufficient for the closure of the Verlinde fusion subalgebra, but
	not necessary.

	We have seen a glimpse of a couple of sub-sectors when we studied the case of $m=-5/4$. Although, there are no quasi-characters in this theory, it serves as an illustrative example.  For convenience, let us
	rewrite the S-matrix of the $m=-5/4$ theory here,
	\begin{equation}\label{eq:2}
		S= \frac{1}{2\sqrt{2}}\begin{pmatrix}
			-1 & 1 & 1 & -1 & 1\\
			1 & -1 & 1 & -1 & -1\\
			2 & 2 & \sqrt{2} & \sqrt{2} & 0\\
			-2 & -2 & \sqrt{2} & \sqrt{2} & 0\\
			2 & -2 & 0 & 0 & 2\\
		\end{pmatrix}
	\end{equation}

	Let us label the characters as $\chi_1$, $\chi_2$, $\chi_3$, $\chi_4$, and $\chi_5$, which form the basis of the above $S$-matrix.  The conformal dimensions of the primary fields corresponding to $\chi_{1}$ and $\chi_{2}$ differ by an integer.  Although the $S$-matrix \eqref{eq:2} is not manifestly block-diagonal, we can utilise the fact that we can take linear combinations of $\chi_{1}$ and $\chi_{2}$ to write new characters.  A rearrangement of the column vector after taking the linear combination gives,
	\begin{equation}\label{eq:5}
		\begin{pmatrix}
			(\chi_1+\chi_2)(-1/\tau)\\
			\chi_3(-1/\tau)\\
			\chi_4(-1/\tau)\\
			(\chi_1-\chi_2)(-1/\tau)\\
			\chi_5(-1/\tau)
		\end{pmatrix}= \frac{1}{2\sqrt{2}}\begin{pmatrix}
			0 & 2 & -2 & 0 & 0\\
			2 & \sqrt{2} & \sqrt{2} & 0 & 0\\
			-2 & \sqrt{2} & \sqrt{2} & 0 & 0\\
			0& 0 & 0 & -2 & 2\\
			0 & 0 & 0 & 2 & 2\\
		\end{pmatrix}
		\begin{pmatrix}
			(\chi_1+\chi_2)(\tau)\\
			\chi_3(\tau)\\
			\chi_4(\tau)\\
			(\chi_1-\chi_2)(\tau)\\
			\chi_5(\tau)
		\end{pmatrix} \ .
	\end{equation}
	This choice of the basis generates two independent sub-sectors, one with the characters $\chi_1-\chi_2$, $\chi_5$, and the other excluding them as we have seen earlier. Note that the linear combination of the characters provided a new basis, in terms of which we could block-diagonalise the $S$-matrix. In general a block-diagonal $S$-matrix is not easy to achieve. It is  even more difficult when the eigenvalues of the modular $T$-transformation are non-degenerate.

	In the remaining part of this section we set up two simple elimination criteria. The admissible fractional level theories with quasi-characters which satisfy the criteria do not have a subsector without the quasi-character.

	\subsection{First Elimination Criterion}

	One simple way a quasi-character with labels $(n_q,k_q)$ would decouple from a subsector is when $S_{(n_{q}k_{q}),(nk)}=0 \ , \forall\ (n,k)\not= (n_{q},k_{q})$\footnote{Here $S_{(n_{q}k_{q}),(nk)}$ denotes the modular $S$-matrix for the entire even sector and not just $k=0$ sector.}. This implies that the quasi-character forms a meromorphic (single-character) CFT imposing strict bootstrap constraints on the conformal dimension of the quasi-character and the central charge of the theory. Namely, $c-24h_q=0 \text{ mod(8)}$.  We performed a simple check for the quasi-character with labels $(0,\lceil u/p \rceil)$ for $(3\leq p \leq 30, 3\leq u\leq 30)$ and did not find any quasi-character of this type which forms a single character CFT. We will therefore consider a weaker criterion of
	imposing the vanishing of the $S$-matrix element condition only on the
	sub-sector labelled by $(n,k)$ within all allowed values of $(n,k)$.
	This will imply that the quasi-character can have non-vanishing
	$S$-matrix entries with the characters outside the chosen sub-sector.

	On the other extreme, if there is no entry of the modular $S$-matrix which vanishes and all characters have non-degenerate eigenvalues of the modular $T$-transformation then it is guaranteed that the $S$-matrix can not be reduced to a block-diagonal form. This implies that a subsector is not possible in such a theory. This is the first elimination criterion we propose in this subsection.

	To explore whether we can find such theories with no subsectors, we need to find the level $(p,u)$ which do not have any vanishing entry of the modular $S$-matrix for the even sector. Since the even characters have two classes, $k=0$ and $k\neq 0$, we have two conditions corresponding to the $S$-matrix elements for the two classes. We will evaluate the constraints separately as they are independent.

	We first set,
	\begin{equation}\label{eq:6}
		S_{nk,n^{'}k^{'}}=0\ ,
	\end{equation}
	to find out what constraints this condition imposes on $(n,k)$, $(n',k')$ and $(p,u)$ \footnote {For a simpler criterion, we can only look at the $(n_{q}=0,k_{q}=\lceil u/p\rceil)$ row of the $S$-matrix, but since there can be more quasi-characters, we
		will look at a more general constraint.}.  The constraint
	coming from eq. \eqref{eq:6} is

	\begin{align}
		\frac{u(n+1)(n'+1)}{p}&=r'\ , r'\in\mathbb{Z}^{+}\nonumber\\
		(n+1)(n'+1)&=rp \ , r \in \mathbb{Z}^{+}.\label{0c-reduced}
	\end{align}
	The second equality follows from the first by using the fact that $p$
	and $u$ are co-prime. This constraint is independent of $k$ and $k'$.  We can therefore derive conditions on $n$
	and $n'$ irrespective of the value of $k,k'$.

	Our first elimination criterion comes from the case when $p$ is a
	prime number.  Since $p$ cannot be factored, eq. \eqref{0c-reduced}
	implies $p$ is either $n+1$ or $n'+1$.  However, values of $n$ and
	$n'$ are bounded, $0\leq n,n' \leq p-2$.  Therefore,
	eq. \eqref{0c-reduced} is not satisfied for any prime number $p$.  This
	criterion will be sharpened momentarily when we consider the second
	condition on the $S$-matrix, but before we do that, let us mention that
	if $p$ is not a prime number then it is possible to obtain solutions
	to eq. \eqref{0c-reduced}.

		The non-prime $p$ can be factorised in at least one way as $p=p_1p_2$,
		where $p_1$ and $p_2$ are (non-unique) integer factors of $p$ at this step.
		As can be easily seen, the equation eq. \eqref{0c-reduced} can have multiple solutions in
		general.
		We easily see that if we choose $p_1$ and $p_2$ such that $p=p_1p_2$ and $(p_1+p_2)$ is minimised, we have $\text{max}(r)=(p-p_1)(p-p_2)/p$. It can also be easily seen that
		for every value of $1\leq r\leq\text{max}(r)$ there exists at least
		one solution to eq. \eqref{0c-reduced}.

		Another place where zeros of $S$-matrix can occur comes from the
		condition $S_{nk,n^{'}k^{'}}=S_{\bar n\bar k,n^{'}k^{'}}$.

		This implies a condition on $k$ and $k'$ and this condition is
		independent of the values $n$ and $n'$,
		\begin{equation}\label{0c-a-reduced}

			2kk'= (2s-1)u \ ,  s \in \mathbb{N}
		\end{equation}
		Since the left-hand side of eq. \eqref{0c-a-reduced} is an even integer,
		it implies that unless $u$ is an even integer, there are no solutions $k,k'$ of this condition.  We then
		reach an interesting conclusion, combining the results for
		this eq. \eqref{0c-a-reduced} and the first condition eq. \eqref{0c-reduced},
		that the S-matrix has no zeros for the level parametrised by
		$(p\in\text{prime}, u\in \text{odd})$ which translates to the statement
		that there are no sub-sectors for these levels, up to the existence of
		characters with degenerate eigenvalues of the modular
		$T$-transformation, which by suitable addition of rows of the $S$-matrix
		could, in principle make a sub-sector
		possible\footnote{The characters which transform in the same
			way under the modular $T$-transform should also transform appropriately
			under the modular $S$-transformations. Thus, an arbitrary linear combination of the characters is not possible.}. We tabulate a few special cases where the dimensions of the
		characters belonging to the same theory are separated by an integer in
		table \ref{tab:degenerate}. A closer look at the table makes it clear that the elimination criteria works well only for $(p\in\text{prime},u\in\text{prime})$.
		\begin{table}
			\begin{center}
				\begin{tabular}{|L|L|}
					\hline
					p&  u\\
					\hline
					\rowcolor[rgb]{.9,.9,1}
					3 & 25\\
					5 & 9,21,27\\
					\rowcolor[rgb]{.9,.9,1}
					7 & 9,15,25,27\\
					11 & 9, 15, 21, 25, 27\\
					\rowcolor[rgb]{.9,.9,1}
					13 & 9, 15, 21, 25, 27\\
					17 & 9, 15, 21, 25, 27\\
					\rowcolor[rgb]{.9,.9,1}
					19 & 9, 15, 21, 25, 27\\
					23 & 9,15,21, 25, 27\\
					\rowcolor[rgb]{.9,.9,1}
					29 & 9,15, 21,25, 27\\
					\hline
				\end{tabular}
				\caption{Theories with $(p\in3\leq \text{prime}\leq 19,u\in3\leq \text{odd}\leq 29)$  which have characters of degenerate eigenvalue of the modular
					$T$-transformation.}
				\label{tab:degenerate}
			\end{center}
		\end{table}
		Since there is no sub-sector, more so without a quasi-character, the
		theories with $(p\in\text{prime}, u\in\text{prime})$ can be eliminated. This is the refined version
		of the first elimination criterion.

		We can continue to find solutions to the condition
		eq. \eqref{0c-a-reduced} for $u=2w, w\in\mathbb{Z}^{+}$.
		This condition can be subdivided into whether $w$ is prime or not. If $w$ is prime, eq. \eqref{0c-a-reduced}
		can only be satisfied when either $k=w$ or $k'=w$ as
		$k,k'\leq u/2=w$. For $k=w$, we have $\lceil w/2 \rceil=(w+1)/2 $ number of
		$k'$ as solutions of this condition. Whereas if $w\not\in$ prime we can write $w=w_1w_2$ as a (non-unique) factorisation of $w$ where $w_1$ and $w_2$ are integer factors of $w$ at this step.
		Again, we have the set of solutions due to $k=w$, with
		$s$ up to the same value $\text{max}(s)$, given by
		$ \text{max}(s)=\lceil w/2 \rceil$. However, we will have additional solutions, as compared with $w\in$ prime, due to different factorisations of $w$.

	For the quasi-character labelled by $(n=0,k=\lceil u/p\rceil)$ we see
	that the first condition eq. \eqref{0c-reduced} does not give any
	solutions due to the bound $0\leq n'\leq p-2$. To locate a subsector with labels $(n',k')$, which decouples the
	quasi-character, they should satisfy
	\begin{equation}
		2\lceil u/p\rceil k'_{s}=(2s-1)u \ , s \in \mathbb{N} \ , \forall n' \ .
	\end{equation}
	Thus, the solutions to the above condition, in addition to the
	conformal data, which may allow for suitable linear combinations of the characters and cause the vanishing of some entries of the modular $S$-matrix, should be enough to provide a necessary condition for the existence of a  sub-sector. We hope to provide more details elsewhere.

	Let us look at an example where both parameters $p$, and $u$ are prime
	numbers, e.g., $(p=3,u=5)$. This theory satisfies the elimination
	criterion.  This theory has 6 characters in the even sector
	corresponding to the set $\Gamma$ tabulated below.
	\begin{table}[h]
		\begin{center}
			\begin{tabular}{| m{2 cm} | m{2 cm} | m{2 cm} | m{2 cm} | }
				\hline
				$(n,k)$ & $j(n,k)$ & $h(n,k)$ & $h_{U}(n,k)$\\
				\hline
				(0,0) &  0 & 0 & $2/5$\\
				(1,0) & $1/2$ & $5/4$ & $33/20$\\
				(0,1) & -$3/10$ & -$7/20$ & $1/20$\\
				(1,1) & $1/5$ & $2/5$ & $4/5$\\
				\rowcolor[rgb]{.9,.9,1}
				(0,2) & -$3/5$ & -$2/5$ & 0\\
				(1,2) & -$1/10$ & -$3/20$ & $1/4$\\
				\hline
			\end{tabular}
			\caption{Conformal data for the even sector of $(p=3,u=5)$.  The shaded entry corresponds to the only quasi-character in the even sector.}\label{p3u5}
		\end{center}
	\end{table}
	The $S$-transformation matrix is given below in the ordered basis of
	the table \ref{p3u5}.  Note that all entries of the S-transformation
	are non-zero and the conformal dimensions are not integer separated to
	allow any simplification.
	\begin{align}
		S=
		\frac{1}{\sqrt{10}}\left(
		\begin{array}{cccccc}
			-1 & -1 & 1 & 1 & -1 & -1 \\
			-1 & 1 & -1 & 1 & -1 & 1 \\
			2 & -2 &  2\cos(2 \pi/5)  & -2 \cos(2 \pi/5)  & - 2\cos(\pi/5) &  2\cos( \pi/5)\\
			2 & 2 & -2 \cos(2 \pi/5) & -2 \cos(2 \pi/5)  & - 2\cos( \pi/5) & -2 \cos(\pi/5) \\
			-2 & -2 & -2 \cos( \pi/5)& -2 \cos(\pi/5) & -2 \cos(2 \pi/5)& -2 \cos(2 \pi/5)  \\
			-2 & 2 & 2\cos( \pi/5) & -2 \cos( \pi/5) & -2 \cos(2 \pi/5) &  2\cos(2 \pi/5) \\
		\end{array}
		\right)
	\end{align}
	This $S$-matrix satisfies $S^{\dagger}MS=M$ where
	$M=\text{diag}(2,2,1,1,1,1)$. We want to identify a possible
	sub-sector without the quasi-character, but a diagonal modular invariant for a possible subsector theory does not exist as the S-matrix can not be written as a block-diagonal matrix, nor does it allow an Awata-Yamada fusion subalgebra\footnote{The Awata-Yamada fusion algebra \cite{Awata:1992sm} gives the fusion coefficients either 0 or 1 for all fractional level $\widehat{su}(2)$ theories.}.

	\subsection{Second Elimination Criterion}

	The $(p=3,u=5)$ example also illustrates a simple feature. For the
	original non-unitary theory, the vacuum is degenerate with the degeneracy
	$u=5$ and a multiplicity in the number of primaries equal to
	2. Since the lowest
	dimension primary has $|b_{+}(0,2)|=1$, it can be identified with the
	vacuum because it is non-degenerate. In this example, the lowest dimension
	primary also happens to be a quasi-character of the type
	$(n=0,k=\lceil5/3\rceil)$. Since the vacuum is the identity element of the fusion algebra, the simple
	identification of the quasi-character with the vacuum character of the unitary theory prohibits any closed fusion subalgebra for the
	theory without the quasi-character, provided we do not change the vacuum identification. A natural option is to look for
	any other vacuum candidate for a sub-sector. But all other characters are degenerate once we have scaled the characters by $u$ to guarantee integrality and preserving the form of the modular $S$-transformation.
	As we had seen for the $(p=3,u=4)$ example, once we have identified a subsector, we could also rescale the characters of the subsector without affecting the form of the modular $S$-transformations. Thus, the candidate vacuum must preserve integrality with leading coefficient unity after rescaling by the common factor and be a part of a sub-sector without the quasi-character. In general, this criterion is hard to satisfy for the subsector, although not outright impossible.

	Before we state the criterion for a generic level let us consider
	another example, that of $(p=4,u=3)$.  The $(p=4,u=3)$ theory has in
	total 9 primaries and 6 independent even characters. The shaded entry
	in the table \ref{p4u3} below belongs to $(n,k)=(0,1)$ which is a quasi-character and happens to be the lowest dimension primary.
	\begin{table}[h]
		\begin{center}
			\begin{tabular}{| m{2 cm}| m{2 cm} | m{2 cm} | m{2 cm} | m{2 cm} }
				\hline
				$(n,k)$ & $j(n,k)$ & $h(n,k)$ & $h_{U}(n,k)$\\
				\hline
				(0,0) &  0 & 0 & $1/6$\\
				(1,0) & $1/2$ & $9/16$ & $35/48$\\
				(2,0) & 1 & $3/2$ & $5/3$\\
				\rowcolor[rgb]{.9,.9,1}
				(0,1) & -$2/3$ & -$1/6$ & $0$\\
				(1,1) & -$1/6$ & -$5/48$ & $1/16$\\
				(2,1) & $1/3$ & $1/3$ & $1/2$\\
				\hline
				(0,2) & -$4/3$ & $1/3$ & $1/2$\\
				(1,2) & -$5/6$ & -$5/48$ & $1/16$\\
				(2,2) & -$1/3$ & -$1/6$ & $0$\\
				\hline
			\end{tabular}
			\caption{Conformal data of $(p=4,u=3)$.  The shaded entry
				corresponds to the only quasi-character in the even sector}\label{p4u3}
		\end{center}
	\end{table}
	From the conformal data it looks like it has the same fate
	as that of $(p=3,u=5)$, but Awata-Yamada (AY) fusion rules
	allows a closed sub-sector with the fusion algebra,
	\begin{align}
		(0,0)\times(2,0)&=(2,0)\nonumber\\
		(2,0)\times(2,0)&=(0,0) \ .
	\end{align}

	These primaries have spins $0$ and $1$ respectively, hence from the
	$su(2)$ point of view, their closure under fusion is expected. For this sub-sector,
	the degeneracy of the vacuum character from the \textit{q}-expansion can be factored out by a
	rescaling. However, their conformal data does not agree with the
	general two primary criteria \cite{Verlinde:1988sn}.  We also find that
	the corresponding $2\times2$ $S$-transformation matrix is singular and
	the unflavoured partition function composed of the two elements does
	not exist\footnote{Although there exists another AY sub-sector which
		excludes only the $(1,k)$ primaries, this subsector is irrelevant because the subsector includes the quasi-character.}.

	It seems like at an arbitrary level, if the primary with
	the lowest conformal dimension has a quasi-character, then the
	corresponding unitarised theory has no closed fusion subalgebra. As discussed above, the natural
	way out is to identify some other primary (not the lowest dimension
	primary) as the vacuum of a subsector theory which excludes the
	quasi-character.  Note that $|b_{+}|=1$ occurs only
	once each \cite{Mukhi:1989bp} in the sets $(n,k=0)$ and $(n,k\neq0)\in\Gamma$ for the lowest dimension
	primary.  Therefore, there are no other modules with non-degenerate grade zero subspace to begin with, unless there is a common factor in the
	\textit{q}-expansion of a set of characters, forming a subsector, which can be factored out as we have seen for the example $(p=3,u=4)$. A closer look at the \textit{q}-expansion given in \eqref{arrangement_1} or \eqref{arrangement_2} implies that the factor $b_{+}$ can be factored out while preserving the integrality of the candidate vacuum character if the following two conditions are simultaneously satisfied,
	\begin{align}
		\frac{b_{-}}{b_{+}}\in\mathbb{Z} \ , \quad \text{ and, } \quad
		\frac{2up}{b_{+}}\in\mathbb{Z} \ .
	\end{align}
	Apart from the known solutions with $|b_{+}|=1$, an easy solution is the character with the labels $(n=1,k=0)$ with $b_{+}=-b_{-}=2u$, which is a candidate vacuum for a subsector theory. Another solution is the character with the common factor $|b_{+}|=2$ and $|b_{-}|=2r$ with $r\in\mathbb{N}$ , among others. If we can locate a vacuum character from the above criteria, it should have a degeneracy which can be factored out from all the characters in the subsector, that is, it should have the lowest factor which is a divisor of all other degeneracies such that its leading coefficient is unity, preserving integrality of every character in the subsector and the form of modular $S$-transformation.

	A great choice to fulfil these criteria is the $(n,k=0)$ sector as it looks plausible from the point of view of
	$su(2)$ representation theory (and Awata-Yamada fusion algebra). This subsector has degeneracy $u$
	which can be factored out to identify a vacuum and although it does not include a quasi-character, its $S$-matrix elements with the quasi-character given by labels $(0,\lceil u/p\rceil)$ are non-vanishing \eqref{0c-reduced}. We have expectedly not found an
	evidence of the existence of a $k=0$ subsector\footnote{Note that the sectors $(n,k=0)$ and $(n,k\neq0)$ generally have different degeneracies and we do not expect a mixing. Again, the theories with degenerate eigenvalues of modular $T$-transformation have more options, but we restrict ourselves to the non-degenerate cases.}. Hence if a subsector without the quasi-character exists, it must be in the $(n,k\neq0)$ subsector which must include a candidate vacuum.

	We have explored a few possibilities of salvaging theories where the
	minimum of the conformal dimensions is equal to the conformal dimension
	of the primary associated with the quasi-character.  We now propose that the
	characters associated with these theories do not allow an
	RCFT description.  This proposal rules out a large number of theories.  A subset
	of theories which can be ruled out in this fashion can be located
	by equating the value of $(n,k)$ for which the conformal dimension is
	the minimum, with the labels of the quasi-character
	$(n=0,k=\lceil\frac{u}{p}\rceil)$. We have found the set of levels
	\begin{equation}
		(p,u)=(p, Np-1) \ , N \in \mathbb{N} \ , p\geq3 \ ,
	\end{equation}
	which satisfies the proposed elimination condition. It is worth
	reiterating that this criterion is based on the assumption that a subsector excluding the quasi-character, which is the vacuum of the unitarised theory, and including a candidate vacuum is in general not possible and the eigenvalues of $T$-transformation are non-degenerate. We hope to provide a refined criterion and more details elsewhere.

	\begin{comment}
		\begin{align}
			h(0,0)=0, h(1,0)=9/16, h(2,0)=3/2, h(0,1)=-1/6, h(1,1)=-5/48, h(2,1)=1/3.
		\end{align}

		The $S$-matrix in the ordered basis of the table is given below. Note that as before this is not the unitary $S$-matrix.
		\begin{align}
			S=
			\begin{pmatrix}
				1/2\sqrt{3} & -1/\sqrt{6} & 1/2\sqrt{3} & -1/2\sqrt{3} & 1/\sqrt{6} & -1/2\sqrt{3}\\
				-1/\sqrt{6} & 0 & 1/\sqrt{6} & -1/\sqrt{6} & 0 & 1/\sqrt{6}\\
				1/2\sqrt{3} & 1/\sqrt{6} & 1/2\sqrt{3} & -1/2\sqrt{3} & -1/\sqrt{6} & -1/2\sqrt{3}\\
				-1/\sqrt{3} & -\sqrt{2}/\sqrt{3} & -1/\sqrt{3} & -1/2\sqrt{3} & -1/\sqrt{6} & -1/2\sqrt{3}\\
				\sqrt{2}/\sqrt{3} & 0 & -\sqrt{2}/3 & -1/\sqrt{6} & 0 & 1/\sqrt{6}\\
				-1/\sqrt{3} & \sqrt{2}/3 & -1/\sqrt{3} & -1/2\sqrt{3} & 1/\sqrt{6} & -1/2\sqrt{3}
			\end{pmatrix}
		\end{align}
		For this case, $\chi_{j(0,1)}$ is the quasi-character. As is clear from the \textit{$S$-matrix}, there is no sub-sector, or none of the entries in the fourth column are $0$. This is also obvious from both of the examples as this also happens to be the state with the lowest conformal dimension and we know that the Perron-Frobenius vector has components which are either all positive or negative but not zero (we write both positive or negative for our analysis since there are always two characters which are negative of each other). For completeness, we mention that there exists no possibility of building a $D$ or $E$ type modular invariant since the dimensions are not separated by an integer (same modulo $1$).
	\end{comment}

	\section{Conclusion and Discussion}\label{conclusions}

	In our sojourn into the territory of Ka\v c-Wakimoto characters for
	the admissible levels of $\widehat{su}(2)$, we found that not all of the
	character vectors correspond to physical character vectors.  In most
	cases, the representation theory contains a quasi-character.  However,
	we still found a countably infinite set of cases, which is still a
	small set relative to the cases that involve quasi-character vectors,
	which can potentially be interpreted as character vectors of some
	RCFT.  The latter set of characters can be classified into three
	subsets, one at an isolated point $(p=3,u=4)$, and a couple of
	infinite series: the threshold levels series with $(p=2,u=2N+1)$, and
	another series with $(p=2N+3,u=2)$, where $N\in \mathbb{N}$.  For the
	threshold cases and the isolated levels, we have shown the positivity
	and integrality of the Fourier coefficients by writing these
	characters as sums of integrable highest weight characters of the WZW
	models.  The characters at threshold levels are the sums of characters
	of $\widehat{su}(2)_{4N}$, reaffirming an old
	observation \cite{Mukhi:1989bp}.  The characters of the isolated case
	can be decomposed into two independent modular invariant sectors, one
	related to $\widehat{so}(5)_1$ and the other to $\widehat{su}(2)_1$.  The
	$u=2$ series, with half odd-integer levels also seems to possess proper characters which we have
	checked to a reasonably large power of $q$-expansion. We find them to
	be related to the difference of characters of the $\widehat{su}(2)_{4N}$
	theory.  The partition functions either have fractional coefficients
	or have a non-unique vacuum state, strongly indicating that these
	theories are logarithmic CFTs. This curious blend for positive definite integer coefficients in the
	$q$-expansions of the characters on one hand and their relation to the
	difference of characters of $\widehat{su}(2)_{4N+4}$ makes this infinite
	series of theories quite intriguing.  We hope to develop a better
	understanding of this relation, but it is beyond the scope of this
	manuscript.  To make it more curious, the non-unitary theory does not
	possess a unique vacuum and its unitarised partner does not have well-defined
	fusion rules, again indicating that $u=2$ theories are not strictly
	RCFTs.  Nevertheless, the possibility of these half of odd-integer level
	theories being logarithmic CFTs seems promising and we hope to report
	on it in another paper.  Given that the characters of the fractional level
	theories belonging to these two infinite series are certain linear
	combinations of the characters of $\widehat{su}(2)$ at integrable levels,
	it implies that these CFTs do not add new characters to the
	pool of admissible characters.  This seems to be in resonance with
	an older observation \cite{Mukhi:1989bp}.

	We have proposed an elimination criterion which does not allow $(p\in prime, u\in prime)$, which contains at least one
	quasi-character, a modular invariant subsector excluding the quasi-character.  We have also proposed another elimination criterion for the theories with levels parametrised by $(p,Np-1)$, with $p\geq3$, which exhibit a quasi-character corresponding to the vacuum module of the unitarised theory. However, we do not have proof of the latter assertion.  It would be interesting to
	find a counter-example and an associated RCFT.  This may shed light on
	how the admissible characters combine to reproduce the
	representation theory at fractional levels. Our results put severe constraints on possible RCFT realisation of the characters at
	admissible levels. There are, however, arguments suggesting that the highest weight primaries need other modules for closure of the
	fusion algebra \cite{Gaberdiel:2001ny, Lesage:2002ch,
		Ridout:2008nh, Ridout:2010jk, Creutzig:2013hma, Creutzig:2013yca}.

	We hasten to point out that the existence of quasi-characters is not
	necessarily a lost cause.  As has been shown for
	two \cite{Chandra:2018pjq} and three \cite{Mukhi:2020gnj} character
	theories, admissible characters can be obtained from the
	quasi-characters either by adding two quasi-characters with proper
	coefficients or by multiplication of an appropriate power of the
	$j$-function.  We hope to revisit these issues soon.

	\subsection*{Acknowledgement}

We are immensely thankful to  Dileep P. Jatkar for his valuable suggestions, enlightening discussions, and patiently correcting the draft. I would also like to thank Subramanya Hegde for illuminating discussions, Kajal Singh for help with the draft and Md. Abhishek for comments on the draft.

redpajama_set_name: RedPajamaArXiv


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