aainstitutetext: Centre for Particle Theory, Department of Mathematical
Sciences,
Durham University, South Road, Durham DH1 3LE, UKbbinstitutetext: National
Institute of Science Education and Research Bhubaneshwar,
P.O. Jatni, Khurdha, 752050, Odisha, INDIAccinstitutetext: Homi Bhabha
National Institute, Training School Complex,
Anushakti Nagar, Mumbai 400094, INDIA

# Classifying three-character RCFTs with Wronskian Index equalling
$\mathbf{0}$ or $\mathbf{2}$

Arpit Das b,c Chethan N. Gowdigere b,c and Jagannath Santara
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In the modular linear differential equation (MLDE) approach to classifying
rational conformal field theories (RCFTs) both the MLDE and the RCFT are
identified by a pair of non-negative integers [n,l]. $\mathbf{n}$ is the
number of characters of the RCFT as well as the order of the MLDE that the
characters solve and $\mathbf{l}$, the Wronskian index, is associated to the
structure of the zeroes of the Wronskian of the characters. In this paper, we
study [3,0] and [3,2] MLDEs in order to classify the corresponding CFTs. We
reduce the problem to a “finite” problem: to classify CFTs with central charge
$0<c\leq 96$, we need to perform $6,720$ computations for the former and
$20,160$ for the latter. Each computation involves (i) first finding a
simultaneous solution to a pair of Diophantine equations and (ii) computing
Fourier coefficients to a high order and checking for positivity.

In the [3,0] case, for $0<c\leq 96$, we obtain many character-like solutions:
two infinite classes and a discrete set of $303$. After accounting for various
categories of known solutions, including Virasoro minimal models, WZW CFTs,
Franc-Mason vertex operator algebras and Gaberdiel-Hampapura-Mukhi novel coset
CFTs, we seem to have seven hitherto unknown character-like solutions which
could potentially give new CFTs. We also classify [3,2] CFTs for $0<c\leq 96$:
each CFT in this case is obtained by adjoining a constant character to a [2,0]
CFT, whose classification was achieved by Mathur-Mukhi-Sen three decades ago.

##  1 Introduction

Two dimensional conformal field theory (CFT) is a subject of primary
importance relevant to many areas in physics and mathematics;
DiFrancesco:1997nk ; Moore:1989vd ; Fuchs:2009iz ; Gaberdiel:1999mc  comprise
a partial list of references. In physics it is important for string theory
both in perturbation theory, initially and later for various aspects of non-
perturbative string theory also. Two dimensional CFTs describe two dimensional
critical systems at RG fixed points. They have contributed to many areas of
mathematics including representation theory, infinite dimensional algebras,
theory of modular forms, etc. An important subclass of two dimensional
conformal field theories is that of rational conformal field theories (RCFTs).
In these theories the central charge $c$ and the conformal dimensions of the
primary fields, the $h$s, are all rational numbers Anderson:1987ge . An
infinite number of RCFTs are known to exist. For example the infinite series
of Virasoro minimal models Belavin:1984vu  (see also DiFrancesco:1997nk ), and
the infinite set of Wess-Zumino-Witten conformal field theories (WZW CFTs)
Witten:1983ar . Even though an infinite number of RCFTs are known, there is no
known classification of them.

One classification scheme for RCFTs is based on the structure of their torus
partition functions. For a RCFT, the torus partition function can be expressed
as a sum of holomorphically factorised terms. The holomorphic factors are the
characters of the RCFT and the number of characters, denoted by $n$, is an
important detail of a RCFT for this classification scheme. Another important
detail of a RCFT relevant for this classification scheme arises as follows.
The $n$ characters are the linearly independent solutions of a single $n$-th
order ordinary differential equation on the moduli space of a torus, a modular
linear differential equation (MLDE) Eguchi:1987qd ; Mathur:1988gt ;
Anderson:1987ge . One then considers the Wronskian of these solutions which is
a modular function of weight $n(n-1)$. The Wronskian index of the RCFT,
denoted by $l$, is a number that takes non-negative integer values and is
associated with the structure of zeroes of the Wronskian and can be expressed
purely in terms of the RCFT data (the central charge, the number of characters
and the conformal dimensions of the primary fields). In this classification
scheme, a RCFT is identified in terms of these two numbers $\mathbf{[n,l]}$,
the number of characters and the Wronskian index.

Remarkably, and this is the primary reason this classification scheme exists,
the form of the MLDE gets fixed by this same pair $\mathbf{[n,l]}$. Actually
the coefficient functions of the MLDE are fixed by $\mathbf{[n,l]}$, apart
from some undetermined parameters. Thus two RCFTs with the same
$\mathbf{[n,l]}$ values are both solutions to the same MLDE but for a
different set of parameters in the MLDE. One thus sets up a MLDE for a given
number of characters and a given Wronskian index and studies all character-
like solutions (that is solutions with non-negative integer coefficients in
their $q$-expansion) and obtains all RCFTs with that number of characters and
that Wronskian index and thus classifies them. This is the MLDE approach to
the classification of RCFTs, first carried out in Mathur:1988na , where a
$\mathbf{[2,0]}$ MLDE was studied and all $\mathbf{[2,0]}$ RCFTs were
classified. The program has been implemented for $\mathbf{[2,2]}$ first in
Naculich:1988xv  and then in Hampapura:2015cea ; Gaberdiel:2016zke ;
Hampapura:2016mmz ; then partially for three-character RCFTs in Mathur:1988gt
; Mukhi:2020gnj ; Tener:2016lcn ; kaneko4 ; CM:2011 ; PSEZ:2016 . The MLDE
approach to RCFTs has been covered in both the physics Hampapura:2015cea ;
Gaberdiel:2016zke ; Mukhi:2020sxt ; Bantay:2010uy ; Tener:2016lcn ;
Harvey:2018rdc ; Harvey:2019qzs ; Bae:2017kcl ; Bae:2020xzl ; Kaidi:2020ecu ;
Kaidi:2021ent ; Bae:2021mej  and the maths literature kaneko5 ; kaneko1 ;
kaneko2 ; kaneko3 ; Gannon:2013jua ; arike1 ; franc1 ; kaneko4 ; mason1 . A
status report on the program can be found in Mukhi:2019xjy . An extension of
this program that includes fermionic RCFTs and up to three characters can be
found in Bae:2020xzl ; Bae:2021mej .

In this paper, we take on the task of classifying three-character CFTs. We do
restrict ourselves to the cases of $l=0$ and $l=2$, i.e. $\mathbf{[3,0]}$ and
$\mathbf{[3,2]}$ CFTs. The study of three-character CFTs are at the least one
level harder technically than two-character CFTs. One aspect of this can be
seen from the form of the MLDEs involved. In a MLDE (8), the coefficient
functions $\phi_{r}$ which are modular w.r.t the full modular group are fixed
by the Wronskian indiex $l$; the functional form is completely fixed leaving
only a few arbitrary parameters. For two character MLDEs studied so far in the
literature, viz. $\mathbf{[2,0]}$, $\mathbf{[2,2]}$ and $\mathbf{[2,4]}$, the
MLDE has only one arbitrary parameter (the $\mathbf{[2,4]}$ MLDE does have two
parameters to begin with but one of them gets fixed by other considerations
Chandra:2018pjq ). On the other hand both the $\mathbf{[3,0]}$ and
$\mathbf{[3,2]}$ MLDEs have two arbitrary parameters. Solving the MLDE means
also finding the value of the parameters for which solutions exist. Hence
obtaining two-character RCFTs involves scanning the set of possibilities for
one parameter which is the infinite set of all rational numbers. While solving
for three-character RCFTs (for $\mathbf{l}\leq 2$), one would need to scan
over the infinite set of _pairs_ of rational numbers.

There is another sense in which the problem of classifying three-character
CFTs is harder. There already exists an infinite number of such CFTs. In a
recent work Das:2020wsi , with the intention of correctly placing the known
classes of CFTs within the present classification scheme, we computed the
$\mathbf{[n,l]}$ values for an infinite number of CFTs. This allows us to
obtain a partial classification of all CFTs with a given $\mathbf{[n,l]}$. We
call this a partial classification because it partially answers the question
“what is the classification for all CFTs with this given $\mathbf{[n,l]}$?”
The partial classification for $\mathbf{[3,0]}$ CFTs from that paper is given
here (3.1.1). As one can see, there are two infinite classes (second and
fourth lines), $7$ irreducible CFTs and $8$ tensor product CFTs. Thus, even
before we start the problem, we are told that there will be an infinite number
of solutions, at least. This should be contrasted with what happens for
$\mathbf{[2,0]}$ CFTs. The partial classification from Das:2020wsi  reveals
that there are a finite number of such CFTs viz. $8$ (and it eventually turns
out that the full classification Mathur:1988na  also has only these $8$ CFTs
and nothing else). Here in this paper, at the end of the day, we do reproduce
the partial classification of Das:2020wsi  for $\mathbf{[3,0]}$ CFTs and find
two other infinite classes of solutions as well as a discrete set of CFTs and
we may have a few ($7$) potentially new CFTs. The subsequent case of
$\mathbf{[3,2]}$ CFTs is intriguing in it’s own way. There is not a single
such CFT among the infinite number of CFTs that we studied in Das:2020wsi .
Here in this paper, we do manage to classify all $\mathbf{[3,2]}$ CFTs up to
central charge $96$ and we find exactly nine new CFTs. But in a sense, they
are not as interesting since it seems that they are related to the nine
$\mathbf{[2,0]}$ admissible character solutions and it is not clear if they
should be considered as new CFTs. In any case, we have achieved
classification.

In this paper, we obtain solutions to the MLDEs by directly solving them; the
MLDE is a homogenous linear differential equation and one can solve it with
the Frobenius method. Although we do give a way of thinking about the MLDE as
an inhomogenous MLDE (one order lower), which turns out to be useful to
understand features of some of the solutions, our main method is the direct
Frobenius one. One directly plugs in the $q$-expansion of the characters:
$\chi_{i}(q)=q^{\alpha_{i}}\sum\limits_{n=0}^{\infty}f^{(i)}_{n}q^{n},~{}f^{(i)}_{0}\neq
0$ into the MLDE and solve it order by order. At the lowest order, one gets a
$\mathbf{n}$th order polynomial equation, the indicial equation, whose roots
are the indices $\alpha_{i}$. Higher orders give expressions for the
$f^{(i)}_{n}$’s which are rational functions of the parameters of the MLDE and
the index $\alpha_{i}$. Then one focusses on the identity character whose
index $\alpha_{0}$ (for unitary conformal field theories) is related to the
central charge ($\alpha_{0}=-\frac{c}{24}$) and each of it’s Fourier
coefficients are (non-negative) integers. There is a lot of purchase that
comes out of this integrality. Typically one imposes the fact that $m_{n}$’s
(the coefficients of $q^{n}$ in the identity character) are integers (just for
a few $n$’s) which constrains the possibilities for $\alpha_{0}$. One kind of
equation that comes out of this analysis is a polynomial equation for
$\alpha_{0}$ with integral coefficients; the integral-root theorem then tells
us the the kind of rational values that $\alpha_{0}$ and hence $c$ can take.
For example in the $\mathbf{[2,0]}$ and the $\mathbf{[2,2]}$ cases
Hampapura:2015cea , the integrality of $m_{1}$ implies that for
$c=\frac{p}{q}$ with $p$ and $q$ coprime, $q$ can be either $1$ or $5$ (we
give more details for $\mathbf{[2,0]}$ in section 2.2 below). This means that
$5c$ is always an integer; hence for a fixed range of central charge, say
$0<c\leq 1$, there are now only $5$ possibilities viz,
$\frac{1}{5},\frac{2}{5}\ldots\frac{5}{5}$. Prior to this stage of the
analysis, we had an infinite number of possibilities for $c$, even for a
finite range. Hence, the problem from here on, we say, has become a “finite”
problem. For the $\mathbf{[3,0]}$ and the $\mathbf{[3,2]}$ cases, we have to
impose integrality of both $m_{1}$ and $m_{2}$ to get a polynomial equation
for $\alpha_{0}$ with integral coefficients. It then turns out that, for the
$\mathbf{[3,0]}$ case, $70c$ is an integer, the problem becomes “finite” and
for $0<c\leq 1$ there are $70$ possibilities
$\frac{1}{70},\frac{2}{70}\ldots\frac{69}{70},\frac{70}{70}$. In this paper,
we study CFTs with $c\leq 96$ and hence consider $96\times 70=6720$
possibilities. For the $\mathbf{[3,2]}$ case, the corresponding number is
$96\times 210=20160$111It turns out that for the $\mathbf{[2,0]}$ case there
is a further constraint for the central charge viz. $c<10$, which makes even
the range of the central charge finite. This makes the number of possibilities
actually finite: $49$. There is a similar constraint in the $\mathbf{[2,2]}$
case Hampapura:2015cea  that makes the range of the central charge itself
finite and again results in a finite number of possibilities for $c$. For
three-character CFTs at least for the $\mathbf{[3,0]}$ case, we cannot expect
such a finiteness for the range of $c$ simply because even the known CFTs in
this class have an infinite range of central charge (3.1.1).. This strategy of
making the problem “finite” is a very important first step in our procedure to
classify three-character CFTs. Further steps are described in the subsequent
sections; they involve, for each of these finite possibilities, finding
simultaneous solutions to a pair of Diophantine equations, computing higher
order Fourier coefficients and checking for positivity.

_Disclaimer_ : In this paper, when we say, we solved a certain MLDE and
obtained a CFT, what we really would have done is, we solved that MLDE and
obtained a character-like solution a.k.a admissible character solution i.e. a
solution whose Fourier coefficients are non-negative. Strictly speaking,, we
should say we solved the MLDE and obtained an admissible character or
character-like solution. In some cases, the character-like solution does
constitute the characters of a known CFT and hence our words are justified; in
other cases, our words are not justified, especially in situations where the
solution is a hitherto unknown one there is more work to be done, before we
declare it a CFT.

This paper is organised as follows. In section 2 we give technical details
pertaining to MLDEs and the calculations needed to achieve classification of
RCFTs. Section 3 contains the classification of three-character RCFTs. We
first study the $\mathbf{[3,0]}$ case in 3.1: the classification of up to
central charge $96$ results in a large number of solutions, which are divided
into five mutually exclusive and exhaustive categories, which we describe
successively in 3.1.1, 3.1.2, 3.1.3, 3.1.4 and 3.1.5. We then study the
$\mathbf{[3,2]}$ case in 3.2. In section 4, we collect some observations on
the various new solutions of section 3.1, that seem to suggest that there are
novel coset dual pairs of CFTs among them. We conclude in section 5 where we
consolidate our results and give future directions. The appendix A tabulates
all the character-like solutions of both $\mathbf{[3,0]}$ and $\mathbf{[3,2]}$
MLDEs.

##  2 MLDE approach to RCFTs

###  2.1 Modular Linear Differential Equations

The characters of a RCFT are a set of $n$ functions, $\chi_{i}(\tau)$, on the
moduli space of the torus such that the torus partition function of the RCFT
can be written as:

$Z(\tau,\bar{\tau})=\sum_{i,j=0}^{n-1}M_{ij}\chi_{i}(\tau)\chi_{j}(\bar{\tau})$
(1)

The characters have a $q$-expansion:

$\chi_{i}(\tau)=\sum_{n=0}^{\infty}f_{n}^{(i)}q^{\alpha_{i}+n},\qquad
i=0,1,\cdots,n-1$ (2)

where $q=e^{2\pi i\tau}$ and $f_{n}^{(i)}$ are the Fourier coefficients of the
$i$th character. Here $\alpha_{i}$ are the exponents,

$\alpha_{i}=h_{i}-\frac{c}{24}$ (3)

where $h_{i}$ are the conformal dimensions of the primaries and $c$ is the
central charge. Note that, a single character can correspond to multiple
primaries. The modular invariance of the torus partition function,

$Z(\gamma\tau,\gamma\bar{\tau})=Z(\tau,\bar{\tau}),\quad\gamma=\begin{pmatrix}a~{}&b\\\
c~{}&d\end{pmatrix}\in\mathbf{SL(2,Z)}$ (4)

implies that the characters transform as vector-valued modular forms:

$\chi_{i}(\gamma\tau)=\sum_{k}V_{ik}(\gamma)\chi_{k}(\tau)$ (5)

where $V_{ik}$ are unitary matrices in the $n$-dimensional representation of
$\mathbf{SL(2,Z)}$.

The characters of a RCFT are solutions to an ordinary differential equation.
The appropriate notion of derivative is that of the Serre-Ramanujan
derivative. We denote the upper half plane by $\mathbb{H}$,

$\displaystyle\mathbb{H}=\\{\tau\in\mathbb{C}\ |\ \text{Im}(\tau)>0\\}.$ (6)

The Serre-Ramanujan derivative operator (see, for example, section 2.8 of
kilford  and exercise 5.1.8 of rammurthy1 ) is defined as follows:

$\displaystyle\mathcal{D}_{k}\,=12\,q\frac{\partial}{\partial q}-k\cdot E_{2}$
(7)

where $E_{2}(\tau)$ is the weight $2$ Eisenstein series. The operator
$\mathcal{D}_{k}$ is a linear operator which maps weight $k$ modular objects
to weight $k+2$ modular objects and satisfies the Leibniz rule. This is the
differential operator with which one sets up modular linear differential
equations.

With this we can write the most general $n^{th}$ order MLDE Mathur:1988na :

$\displaystyle\mathcal{D}^{n}\chi_{i}+\sum_{r=0}^{n-1}\phi_{r}(\tau)\mathcal{D}^{r}\chi_{i}=0,$
(8)

where the solutions $\chi_{i}$ are the characters of a $n$-character RCFT. In
the above, the coefficient functions $\phi_{r}(\tau)$ are modular w.r.t the
full modular group $\mathbf{SL(2,Z)}$ (even if the characters $\chi_{i}(\tau)$
themselves need not be, being modular w.r.t some congruence subgroup of
$\mathbf{SL(2,Z)}$) and are given by rational functions of the Eisenstein
series $E_{4}(\tau)$ and $E_{6}(\tau)$. Just as for differential equations
with ordinary derivatives, the co-efficient functions can be expressed in
terms of the Wronskians of the solutions. Define,

$\displaystyle
W_{r}=\left(\begin{array}[]{cccccc}\chi_{1}&\chi_{2}&\cdots&\cdots&\cdots&\chi_{n}\\\
\mathcal{D}\chi_{1}&\mathcal{D}\chi_{2}&\cdots&\cdots&\cdots&\mathcal{D}\chi_{n}\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
\mathcal{D}^{r-1}\chi_{1}&\mathcal{D}^{r-1}\chi_{2}&\cdots&\cdots&\cdots&\mathcal{D}^{r-1}\chi_{n}\\\
\mathcal{D}^{r+1}\chi_{1}&\mathcal{D}^{r+1}\chi_{2}&\cdots&\cdots&\cdots&\mathcal{D}^{r+1}\chi_{n}\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
\mathcal{D}^{n}\chi_{1}&\mathcal{D}^{n}\chi_{2}&\cdots&\cdots&\cdots&\mathcal{D}^{n}\chi_{n}\\\
\end{array}\right),$ (16)

the Wronskian, $W$, of (16) is given by $W\equiv W_{n}$. The coefficient
functions of (8) and the Wronskians (16) are related by:

$\displaystyle\phi_{r}=(-1)^{n-r}\frac{W_{r}}{W}.$ (17)

Now let us analyse the weights of various objects in (8). $\chi_{i}$s being
characters of a RCFT are weight $0$ modular functions. The first term,
$\mathcal{D}^{n}\chi_{i}$, is of weight $2n$ as each operation of
$\mathcal{D}$ increases the weight by $2$. Similarly $\mathcal{D}^{r}\chi_{i}$
is of weight $2r$ and hence $\phi_{r}(\tau)$ are modular functions of weight
$2(n-r)$. Note that since $W$ can have zeros in $\mathbb{H}$, so $\phi_{r}$s
are not modular forms but rather are modular functions222Modular forms are
holomorphic in $\mathbb{H}$ but modular functions, in general, are meromorphic
in $\mathbb{H}$.. Also, note that $W$ is a modular function of weight
$n(n-1)$.

####  2.1.1 The Valence Formula and the Wronskian Index

For a modular function $f$ of weight $k$, $\nu_{\tau_{0}}(f)$ denotes the
order of pole/zero of $f$ at $\tau=\tau_{0}$. Let us now look at the valence
formula (see, for example, section 3.1 of kilford ) which states that,

$\displaystyle\nu_{\iota\infty}(f)+\frac{1}{2}\nu_{\iota}(f)+\frac{1}{3}\nu_{\omega}(f)+\sum^{{}^{\prime}}_{\tau\neq\iota,\omega;\
\tau\in\mathcal{F}}\nu_{\tau}(f)=\frac{k}{12},$ (18)

where $\omega$ is one of the cube root of unity, $\mathcal{F}$ is the
Fundamental Domain and the ′ above the summation means that the sum excludes
$\tau\in\mathbb{H}$ which have $\text{Re}(\tau)=\frac{1}{2}$ or, which have
both $|\tau|=1$ and $\text{Re}(\tau)>0$.

Now, at $\tau\rightarrow\iota\infty$, $\chi_{i}\sim q^{\alpha_{i}}$ and hence,
$W\sim q^{\sum_{i=0}^{n-1}\alpha_{i}}$. Hence, $W$ has a
$-\sum_{i=0}^{n-1}\alpha_{i}$ order pole at $\iota\infty$. The last three
terms on the left hand side of (18) together can be written as $\frac{l}{6}$
where $l$ is a non-negative integer. Hence, the valence formula now reads

$\displaystyle\sum_{i=1}^{n-1}\alpha_{i}+\frac{l}{6}=\frac{n(n-1)}{12}.$ (19)

Now using (3), we have

$\frac{l}{6}=\frac{n(n-1)}{12}+\frac{n\,c}{24}-\sum_{i}h_{i}.$ (20)

$l$, which is some information about the zeroes of the Wronskian (at the cusp
points and the interior of the fundamental domain), is the Wronskian index of
the RCFT. This Wronskian index $l$, which is expressed purely in terms of the
RCFT data (as seen in (20)), proves to be important for the classification of
RCFTs.

###  2.2 $\mathbf{[2,0]}$ MLDE

In this subsection, we will illustrate the theory with the example of
$\mathbf{[2,0]}$ that first appeared in the classic paper Mathur:1988na . What
we do in the following is slightly different from Mathur:1988na , but it has
the advantage that it can be generalised to the three-character story in
section 3. Also thus we will recall the classification of $\mathbf{[2,0]}$
CFTs which show up in multiple ways in the classification of both
$\mathbf{[3,0]}$ and $\mathbf{[3,2]}$ CFTs that we take up in section 3.

Any $\mathbf{2}$-character MLDE has the form:

$\displaystyle\mathcal{D}^{2}\chi_{i}+\phi_{1}(\tau)~{}\mathcal{D}\chi_{i}+\phi_{0}(\tau)~{}\chi_{i}=0$
(21)

with $\phi_{1}(\tau)$ modular of weight $2$ and $\phi_{0}(\tau)$ modular of
weight $4$. From (17),
$\phi_{1}(\tau)=-\frac{W_{1}}{W},~{}\phi_{0}(\tau)=\frac{W_{0}}{W}$. For
vanishing Wronskian index $\mathbf{l}=0$, $W$ has no zeroes and
$\phi_{1}(\tau),\phi_{0}(\tau)$ are $\mathbf{SL(2,Z)}$ modular forms (i.e.
holomorphic) of weights $2$ and $4$ respectively. Hence they are respectively
$0$ and $E_{4}$. Thus we see how the $\mathbf{[n,l]}$ values determine the
MLDE in the $\mathbf{[2,0]}$ case to be

$\displaystyle\mathcal{D}^{2}\chi_{i}+\mu_{1,0}~{}E_{4}(\tau)~{}\chi_{i}=0$
(22)

where $\mu_{1,0}$ is the one single parameter of this MLDE; we use a notation
$\mu_{a,b}$ to be the constant parameter that multiplies the monomial
$E_{4}(\tau)^{a}\,E_{6}(\tau)^{b}$. Now to obtain $\mathbf{[2,0]}$ admissible
character solutions and CFTs, we plug in (2) into (22); at the leading order
we obtain the indicial equation

$\displaystyle\alpha^{2}-\frac{\alpha}{6}+\frac{\mu_{1,0}}{144}=0.$ (23)

We evaluate the above equation for the identity character
$\alpha_{0}=-\frac{c}{24}$ and use it to express the parameter in terms of
$\alpha_{0}$

$\displaystyle\mu_{1,0}=-144\,\alpha_{0}^{2}+24\alpha_{0}.$ (24)

Having thus converted the parameters into expressions pertaining to the
identity character, we now get to the next to leading order of (22),

$\displaystyle\frac{f_{1}}{f_{0}}=\frac{-576\,\alpha-240\mu_{1,0}}{144\alpha^{2}-264\alpha+\mu_{1,0}+120},$
(25)

which when evaluated for the identity character and further using (24), we
obtain

$\displaystyle m_{1}=\frac{1440\alpha_{0}^{2}-264\alpha_{0}}{5+12\alpha_{0}}.$
(26)

Here, $m_{1}$ is the Fourier coefficient (of $q^{1}$) of the identity
character and is a non-negative integer. Define

$N=-120\alpha_{0}$ (27)

and (26) becomes

$\displaystyle N^{2}+N(m_{1}+22)-50\,m_{1}=0$ (28)

a quadratic equation for $N$ with integer coefficients and by the integral
root theorem we can conclude that $N$ is an integer. $-120\alpha_{0}$ is an
integer means that $5c$ is an integer. This is the stage at which the problem
becomes “finite”: for the finite range $0<c\leq 1$, there are only five
possible values viz.
$\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},\frac{5}{5}$. Then we go back
to (26) and rewrite in terms of $c$ to get $m_{1}=\frac{5c^{2}+22c}{10-c}$ and
since the left hand side is positive as well as the numerator of the right
hand side, the denominator needs to be positive which implies $c<10$. Hence we
find that the allowed values of the central charge are the $49$ values
$\frac{1}{5},\frac{2}{5},\ldots\frac{48}{5},\frac{49}{5}$ or $1\leq N\leq 49$.

Now we compute higher Fourier coefficients. The Frobenius method gives the
Fourier coefficients of a character as rational functions of the index and the
parameter, which on using (24) and the fact that $\mathbf{l}=0$, become
rational functions of the central charge. We then just compute for each of the
$49$ cases and eliminate those cases for which the co-efficients are negative.
It turns out that there are $10$ admissible character solutions, given in
table 1.

Table 1: $\mathbf{[2,0]}$ $N$ | $c$ | Theory
---|---|---
2 | $\frac{2}{5}$ | $\mathcal{M}(5,2)$
5 | 1 | $(\mathbf{\hat{A}_{1}})_{1}$
10 | 2 | $(\mathbf{\hat{A}_{2}})_{1}$
14 | $\frac{14}{5}$ | $(\mathbf{\hat{G}_{2}})_{1}$
20 | 4 | $(\mathbf{\hat{D}_{4}})_{1}$
26 | $\frac{26}{5}$ | $(\mathbf{\hat{F}_{4}})_{1}$
30 | 6 | $(\mathbf{\hat{E}_{6}})_{1}$
35 | 7 | $(\mathbf{\hat{E}_{7}})_{1}$
38 | $\frac{38}{5}$ | $\mathbf{E_{7\frac{1}{2}}}$
40 | 8 | $(\mathbf{\hat{E}_{8}})_{1}$

Of these, the first eight admissible character solutions correspond to CFTs.
The first entry needs an explanation. What one obtains is two indices
$-\frac{1}{60}$ and $\frac{11}{60}$. The way the discussion has been set up,
we equate the smaller index to be $-\frac{c}{24}$ and the bigger one to be
$h-\frac{c}{24}$. This gives us a unitary CFT interpretation since we obtain
positive $c$ and $h$ ($\frac{2}{5}$ and $\frac{1}{5}$ respectively). But in
reality $\mathcal{M}(5,2)$ is a non-unitary CFT; this is also consistent with
the obtained indices. If make the opposite identification, i.e. equate the
bigger index $\frac{11}{60}$ to be $-\frac{c}{24}$ and the smaller index
$-\frac{1}{60}$ to be $h-\frac{c}{24}$ we would arrive at the correct numbers
for $\mathcal{M}(5,2)$ viz. $c=-\frac{22}{5}$ and $h=-\frac{1}{5}$. From only
the MLDE solution, one cannot decide if it is a unitary CFT or not. Here in
table 1, it turns out that only the first entry is a non-unitary CFT and the
other CFT entries (two to eight) are all unitary CFTs. In section 3 we have
many solutions that do not seem to be associated with any known CFTs; in all
such cases, we tabulate the data as if they are unitary CFTs (just as in table
1), but one should keep open the possibility that they could well be non-
unitary CFTs (as it happens for the first entry of table 1).

The ninth entry corresponding to $N=38$ is not associated with any CFT but it
is a genuine admissible character and seems to be associated with some other
mathematical structure (see JL:2006 ).

The last admissible character solution (for $N=40$) is strictly not a two-
character CFT. $(\mathbf{\hat{E}_{8}})_{1}$ is actually a one-character CFT;
it is a $\mathbf{[1,2]}$ CFT. The single character of this CFT turns out to be
the identity character of the above $N=40$ solution. The other character of
the solution is an “unstable” character. An unstable character is a character
with positive rational coefficients, but there is no positive integer
available, which when multiplied makes all the Fourier coefficients integers.
The positive integer needed keeps increasing as we increase the order to which
we compute. This is in contrast to usual characters, which in this context may
be termed “stable” characters, where usually after a certain order of
computation one finds a positive integer that when multiplied makes all
Fourier coefficients integers not only up to that order but also to all
subsequent larger orders of computation. This is the first instance when a CFT
with a lower number of characters shows up as a solution to a MLDE with a
bigger number of characters along with unstable characters. In our study of
solutions to $\mathbf{[3,0]}$ MLDEs, we will find in 3.1.2 many two and one
character CFTs that appear as solutions accompanied with one or two unstable
characters.

The analysis that is presented here in this section (2.2) to derive the famous
MMS classification table 1 is not the same as the one that appears in the
classic paper Mathur:1988na . They arrive at a finite number of central
charges ($22$) by requiring that the discriminant of (28) be a perfect square;
they then compute the higher Fourier coefficients for these cases and impose
positivity to obtain the classification in table 1. The analysis presented
here has the advantage that a slight generalisation works for the three-
character MLDEs that we study later in this paper.

###  2.3 Inhomogeneous MLDEs

The MLDEs that appear in the classification scheme being studied here (8) and
that originated with Mathur:1988na  are homogenous differential equations. In
this subsection, we develop inhomogeneous modular linear differential
equations. We will show that for every $\mathbf{[n,l]}$ MLDE there is an
associated inhomogeneous MLDE of order $\mathbf{n-1}$ which is equivalent to
it. That is, the solutions to the homogenous MLDE also solve the associated
inhomogeneous MLDE. This point of view for MLDE solutions helps explain some
of the solutions that we obtain in section 3.1.3 while solving
$\mathbf{[3,0]}$ homogenous MLDEs.

The starting point is Abel’s identity abel  variously referred to as Abel’s
formula and Abel’s differential equation identity. Abel’s identity allows the
computation of the Wronskian of the solutions of a differential equation
without having to solve the differential equation. The identity gives a first
order differential equation for the Wronskian; the coefficient functions for
this first order differential equation are given by the coefficient functions
of the differential equation under study. This theory, given by Abel for
differential equations on the complex plane, generalises to the setting of
modular differential equations with Serre-Ramanujan derivatives. First, from
(16) we obtain $\mathcal{D}W=W_{n-1}$, which when combined with (17) gives the
first order differential equation for the Wronskian:

$\mathcal{D}W+\phi_{n-1}\,W=0.$ (29)

We can solve this differential equation and obtain the Wronskians (see table 2
below for the general case and table 3 for the Wronskians of $\mathbf{[2,0]}$
RCFTs). We thus have the situation where we first have only the Wronskian
index, some partial information of the Wronskian, which determines the
coefficient function $\phi_{n-1}$ of the MLDE and which in turn via (29)
determines the Wronskian completely. For a $\mathbf{[n,0]}$ MLDE, $\phi_{n-1}$
vanishes and hence it’s Wronskian is a constant w.r.t the Serre-Ramanujan
derivative. Serre-Ramanujan constants w.r.t the $\mathcal{D}_{k}$ operator can
be determined by solving using a Frobenius like method:

$\displaystyle\mathcal{D}_{k}\,f=0\qquad\Longrightarrow\qquad
f(q)=f_{0}~{}\eta(q)^{2k}$ (30)

and are appropriate powers of the Dedekind-eta function, so as to have weight
$k$. We give the Wronskians for a few MLDEs in table 2, which includes all the
particular MLDEs we study in this paper. The $A_{1}$ that appears in the last
column is the single integration constant resulting from solving the first
order differential equation (29). $A_{1}$ can take different values for
different CFTs; we illustrate this in table 3 where we give the $A_{1}$ value
for the MMS $\mathbf{[2,0]}$ CFTs.

Table 2: Wronskians MLDE | $\phi_{n-1}$ | $W$
---|---|---
$\mathbf{[n,0]}$ | $0$ | $A_{1}~{}\eta(q)^{2n(n-1)}$
$\mathbf{[n,2]}$ | $\frac{4E_{6}}{E_{4}}$ | $A_{1}~{}\eta(q)^{2n(n-1)}\left[\frac{E_{4}}{\eta(q)^{8}}\right]$
$\mathbf{[n,3]}$ | $\frac{6E_{4}^{2}}{E_{6}}$ | $A_{1}~{}\eta(q)^{2n(n-1)}\left[\frac{E_{6}}{\eta(q)^{12}}\right]$
$\mathbf{[n,4]}$ | $\frac{8E_{6}}{E_{4}}$ | $A_{1}~{}\eta(q)^{2n(n-1)}\left[\frac{E_{4}}{\eta(q)^{8}}\right]^{2}$
Table 3: $\mathbf{[2,0]}$ Wronskians $c$ | Theory | Wronskian
---|---|---
$\frac{2}{5}$ | $\mathcal{M}(5,2)$ | $-\frac{1}{5}\,\eta(q)^{4}$
1 | $(\mathbf{\hat{A}_{1}})_{1}$ | $\frac{1}{2}\,\eta(q)^{4}$
2 | $(\mathbf{\hat{A}_{2}})_{1}$ | $\eta(q)^{4}$
$\frac{14}{5}$ | $(\mathbf{\hat{G}_{2}})_{1}$ | $\frac{14}{5}\,\eta(q)^{4}$
4 | $(\mathbf{\hat{D}_{4}})_{1}$ | $4\,\eta(q)^{4}$
$\frac{26}{5}$ | $(\mathbf{\hat{F}_{4}})_{1}$ | $\frac{78}{5}\,\eta(q)^{4}$
6 | $(\mathbf{\hat{E}_{6}})_{1}$ | $18\,\eta(q)^{4}$
7 | $(\mathbf{\hat{E}_{7}})_{1}$ | $42\,\eta(q)^{4}$

Solving (29) is effectively one integration. And we are left with
$\mathbf{n-1}$ integrations more to be performed. These are done by solving
the following $\mathbf{n-1}$-order MLDE, which is obtained as follows.
Consider the formula for the Wronskian; we will illustrate this for
$\mathbf{[3,0]}$:

$\displaystyle
A_{1}\eta^{12}=\mathcal{D}^{2}\chi_{3}\left[\chi_{1}\,\mathcal{D}\chi_{2}-\chi_{2}\,\mathcal{D}\chi_{1}\right]+\mathcal{D}\chi_{3}\left[\chi_{2}\,\mathcal{D}^{2}\chi_{1}-\chi_{1}\,\mathcal{D}^{2}\chi_{2}\right]+\chi_{3}\left[\mathcal{D}\chi_{1}\mathcal{D}^{2}\chi_{2}-\mathcal{D}\chi_{2}\,\mathcal{D}^{2}\chi_{1}\right]$

and rewrite it as

$\displaystyle\mathcal{D}^{2}\chi_{1}+\mathcal{D}\chi_{1}\left[\frac{\chi_{3}\mathcal{D}^{2}\chi_{2}-\chi_{2}\mathcal{D}^{2}\chi_{3}}{\chi_{2}\mathcal{D}\chi_{3}-\chi_{3}\mathcal{D}\chi_{2}}\right]+\chi_{1}\left[\frac{\mathcal{D}\chi_{2}\mathcal{D}^{2}\chi_{3}-\mathcal{D}\chi_{3}\mathcal{D}^{2}\chi_{2}}{\chi_{2}\mathcal{D}\chi_{3}-\chi_{3}\mathcal{D}\chi_{2}}\right]=\frac{A_{1}\eta^{12}}{\chi_{2}\mathcal{D}\chi_{3}-\chi_{3}\mathcal{D}\chi_{2}}$
(32)

In general, we obtain an inhomogeneous MLDE of order $\mathbf{n-1}$, the
inhomogeneous term given by the Wronskian from (29). The Wronskian is a known
function at this stage and appears in (32), if in addition, we consider
$\chi_{1}$ and $\chi_{2}$ as given, we can read (32) as a differential
equation for $\chi_{3}$. We then solve this first by obtaining a particular
solution. This can be done again with a Frobenius-like procedure. Let us
denote the particular solution by $A_{1}P(q)$. The general solution to (32) is
then given by a linear combination of the particular solution and the
homogenous solutions. The homogenous solutions of (32) are nothing but
$\chi_{2}$ and $\chi_{3}$. Hence the general solution is

$\displaystyle\chi_{1}(q)=A_{1}\,P(q)+A_{2}\,\chi_{2}(q)+A_{3}\,\chi_{3}(q).$
(33)

Let us denote the Fourier expansion of the particular solution and the Fourier
expansions of the other characters by

$\displaystyle
A_{1}P(q)=q^{\frac{1}{2}-\alpha_{2}-\alpha_{3}}\sum\limits_{n=0}^{\infty}g^{(1)}_{n}q^{n},\quad\chi_{2}(q)=q^{\alpha_{2}}\sum\limits_{n=0}^{\infty}g^{(2)}_{n}q^{n},\quad\chi_{3}(q)=q^{\alpha_{3}}\sum\limits_{n=0}^{\infty}g^{(3)}_{n}q^{n}.$
(34)

Then we have

$\displaystyle\chi_{1}(q)=q^{\frac{1}{2}-\alpha_{2}-\alpha_{3}}\sum\limits_{n=0}^{\infty}\left[g^{(1)}_{n}+A_{2}\,q^{-\frac{1}{2}+2\alpha_{2}+\alpha_{3}}\,g^{(2)}_{n}+A_{3}\,q^{-\frac{1}{2}+\alpha_{2}+2\alpha_{3}}\,g^{(3)}_{n}\right]q^{n}.$
(35)

If the values of $\alpha_{2}$ and $\alpha_{3}$ for which character-like
solutions exist are such that $-\frac{1}{2}+2\alpha_{2}+\alpha_{3}$ and
$-\frac{1}{2}+\alpha_{2}+2\alpha_{3}$ are not non-negative integers, then to
get character-like solution for $\chi_{1}$, we have to set $A_{2}$ and $A_{3}$
to be zero. This is what happens in most examples: the first character is
nothing but the particular solution of the inhomogeneous MLDE. But one can
imagine the following situation.

(i) If $-\frac{1}{2}+2\alpha_{2}+\alpha_{3}$ is a non-negative integer, then
$A_{2}$ isn’t required to vanish. $A_{2}$ can take any positive integral value
and we would have a good character solution for $\chi_{1}$.

$\displaystyle-\frac{1}{2}+2\alpha_{2}+\alpha_{3}\in\mathbf{Z_{\geq 0}},\qquad
A_{2}\in\mathbf{Z_{\geq 0}}$
$\displaystyle\chi_{1}(q)=q^{\frac{1}{2}-\alpha_{2}-\alpha_{3}}\sum\limits_{n=0}^{\infty}\left[g^{(1)}_{n}+A_{2}\,q^{-\frac{1}{2}+2\alpha_{2}+\alpha_{3}}\,g^{(2)}_{n}\right]q^{n}.$
(36)

We thus have an infinite number of admissible character solutions,
parametrised by $A_{2}$, in (2.3). All members of this infinite family have
the same indices and hence the same $c$, $h_{1}$, $h_{2}$ and also they have
the same Wronskian. But they are different solutions (they differ in the first
character).

(ii) If $-\frac{1}{2}+\alpha_{2}+2\alpha_{3}$ is a non-negative integer, then
$A_{3}$ isn’t required to vanish. $A_{3}$ can take any positive integral value
and we would have a good character solution for $\chi_{1}$.

$\displaystyle-\frac{1}{2}+\alpha_{2}+2\alpha_{3}\in\mathbf{Z_{\geq 0}},\qquad
A_{3}\in\mathbf{Z_{\geq 0}}$
$\displaystyle\chi_{1}(q)=q^{\frac{1}{2}-\alpha_{2}-\alpha_{3}}\sum\limits_{n=0}^{\infty}\left[g^{(1)}_{n}+A_{3}\,q^{-\frac{1}{2}+\alpha_{2}+2\alpha_{3}}\,g^{(3)}_{n}\right]q^{n}.$
(37)

We thus have an infinite number of admissible character solutions,
parametrised by $A_{3}$, in (2.3). All members of this infinite family have
the same indices and hence the same $c$, $h_{1}$, $h_{2}$ and also they have
the same Wronskian. But they are different solutions (they differ in the first
character).

In our study of character-like solutions to $\mathbf{[3,0]}$ MLDEs we will
encounter two infinite families of CFTs (in section 3.1.3) each family having
the same $c$, $h_{1}$, $h_{2}$ values, one following (2.3) and another (2.3).

##  3 A Classification of three-character CFTs

In this section, we study three character CFTs. First we study three character
CFTs with a vanishing Wronskian index i.e. $\mathbf{[3,0]}$ CFTs and
subsequently we study three character CFTs with a Wronskian index equalling
two i.e. $\mathbf{[3,2]}$ CFTs. We have been made aware by the recent
concurrent work Kaidi:2021ent  that for three character MLDEs new admissible
character solutions are expected when the Wronskian index is a multiple of
$3$.

###  3.1 Classifying $\mathbf{[3,0]}$ CFTs

Any $\mathbf{3}$-character MLDE has the form:

$\displaystyle\mathcal{D}^{3}\chi_{i}+\phi_{2}(\tau)~{}\mathcal{D}^{2}\chi_{i}+\phi_{1}(\tau)~{}\mathcal{D}\chi_{i}+\phi_{0}(\tau)~{}\chi_{i}=0$
(38)

with $\phi_{2}(\tau)$, $\phi_{1}(\tau)$ and $\phi_{0}(\tau)$ modular of weight
$2$, $4$ and $6$ respectively. From (17),
$\phi_{1}(\tau)=-\frac{W_{1}}{W},~{}\phi_{0}(\tau)=\frac{W_{0}}{W}$. For
vanishing Wronskian index $\mathbf{l}=0$, $W$ has no zeroes and
$\phi_{2}(\tau)$, $\phi_{1}(\tau)$, and $\phi_{0}(\tau)$ are
$\mathbf{SL(2,Z)}$ modular forms (i.e. holomorphic) of weights $2$, $4$ and
$6$ respectively. Hence they are respectively $0$, $E_{4}$ and $E_{6}$. Thus
we see how the $\mathbf{[n,l]}$ values determine the MLDE in the
$\mathbf{[3,0]}$ case to be

$\displaystyle\mathcal{D}^{3}\chi_{i}+\mu_{1,0}\,E_{4}\,\mathcal{D}\chi_{i}+\mu_{0,1}\,E_{6}\,\chi_{i}=0,$
(39)

This is a two parameter MLDE. Let us denote the three characters by
$\chi_{0}(q),\chi_{1}(q),\chi_{2}(q)$ where $\chi_{0}(q)$ is the identity
character. The characters have the $q$-expansion
$\chi_{i}(q)=q^{\alpha_{i}}\sum\limits_{n=0}^{\infty}f^{(i)}_{n}q^{n}$
$~{}f^{(i)}_{0}\neq 0$ and each $f^{(i)}_{n}$ a positive integer. $\alpha_{i}$
is referred to as the index of the character. The three indices are named such
that $\alpha_{0}<\alpha_{1}<\alpha_{2}$. The three rational numbers that
define a three-character RCFT viz. the central charge $c$, the conformal
dimensions of the non-identity characters $h_{1}$ and $h_{2}$ with
$h_{1}<h_{2}$, are taken to be related to the indices :
$\alpha_{0}=-\frac{c}{24},\alpha_{1}=h_{1}-\frac{c}{24},\alpha_{2}=h_{2}-\frac{c}{24}$.333In
the following, we will be searching for solutions with a negative
$\alpha_{0}$. This means that the assignment
$\alpha_{0}=-\frac{c}{24},\alpha_{1}=h_{1}-\frac{c}{24},\alpha_{2}=h_{2}-\frac{c}{24}$
together with $\alpha_{0}<\alpha_{1}<\alpha_{2}$ implies that we are searching
for solutions that give unitary CFTs. It may seem that we will miss out on
non-unitary CFTs. But it turns out non-unitary CFTs also show up as solutions
within this scheme. We can illustrate this with the example of the non-unitary
Virasoro minimal model $\mathcal{M}(7,2)$ for which we have
$c=-\frac{68}{7},h_{1}=-\frac{2}{7},h_{2}=-\frac{3}{7}$ which corresponds to
indices
$\alpha_{0}=\frac{17}{42},\alpha_{1}=\frac{5}{42},\alpha_{2}=-\frac{1}{42}$
which are not ordered according to $\alpha_{0}<\alpha_{1}<\alpha_{2}$. But if
we rename the indices
$\overline{\alpha_{0}}=-\frac{1}{42},\overline{\alpha_{1}}=\frac{5}{42},\overline{\alpha_{2}}=\frac{17}{42}$
so that they are correctly ordered
$\overline{\alpha_{0}}<\overline{\alpha_{1}}<\overline{\alpha_{2}}$ and define
central charges and conformal dimensions
$\overline{\alpha_{0}}=-\frac{\overline{c}}{24},\overline{\alpha_{1}}=\overline{h_{1}}-\frac{\overline{c}}{24},\overline{\alpha_{2}}=\overline{h_{2}}-\frac{\overline{c}}{24}$
we get
$\overline{c}=\frac{4}{7},\overline{h_{1}}=\frac{1}{7},\overline{h_{2}}=\frac{2}{7}$
which looks like a unitary CFT. And this is how the non-unitary CFT
$\mathcal{M}(7,2)$ shows up within our scheme of computation (see the second
row of table 11). It turns out Castro-Alvaredo:2017udm  that the central
charge $\overline{c}$ has a physical meaning as the central charge that
controls renormalisation group flows between non-unitary CFTs (it is called
$\mathbf{c_{eff}}$ in Castro-Alvaredo:2017udm ). Another non-unitary CFT that
shows up in our tables is $\mathcal{M}(5,2)^{\otimes 2}$ which has
$\mathbf{c_{eff}}=\frac{4}{5}$ (see third row of table 11). Thus among the
many solutions that we obtain in this paper which are completely unknown or
not yet fully characterised as CFTs, there could be non-unitary CFTs also. And
we would have tabulated their $\mathbf{c_{eff}}$.

A linear differential equation, when solved by a Frobenius-like method,
determines the ratios $\frac{f^{(i)}_{n}}{f^{(i)}_{0}}$. For the identity
character, $f^{(0)}_{0}=1$; in the following we will write $f^{(0)}_{n}\equiv
m_{n},n\geq 1$ and all the $m_{n}$’s are non-negative integers. For the other
characters, $i=1,2$, $f^{(i)}_{0}$ need not be $1$ and and hence in our search
for character-like solutions we will require that the ratios determined by the
differential equation are all positive rational numbers such that upon
multiplying by a suitable positive integer all of them become positive
integers. This step depends on the order $n$ up to which one solves the
differential equation. We will denote this positive integer by $D_{1}$ and
$D_{2}$ for the characters $\chi_{1}$ and $\chi_{2}$ respectively.
$f^{(1)}_{0}$ and $f^{(2)}_{0}$ cannot be determined by the method pursued
here which is the Frobenius-like method for solving linear differential
equations; they can be any multiple of $D_{1}$ and $D_{2}$ respectively.

At the leading order in the $q$-expansion, the differential equation (39)
gives the following indicial equation

$\displaystyle\alpha^{3}-\frac{\alpha^{2}}{2}+\left(\frac{\mu_{1,0}}{144}+\frac{1}{18}\right)\alpha+\frac{\mu_{0,1}}{1728}=0,$
(40)

whose solutions are the indices of the characters:
$\alpha_{0},\alpha_{1},\alpha_{2}$. At the next to leading order in the
$q$-expansion, the differential equation (39) gives the following equation

$\displaystyle\frac{f_{1}}{f_{0}}=\frac{42\mu_{0,1}-240\mu_{1,0}\alpha-192\alpha-1728\alpha^{2}}{80+\mu_{1,0}+288\alpha+432\alpha^{2}}.$
(41)

We now evaluate the two equations (39) and (40) for the identity character and
interpret these equations as two equations for the two parameters which give
the solution

$\displaystyle\mu_{0,1}$ $\displaystyle=$
$\displaystyle-1728\alpha_{0}^{3}+864\alpha_{0}^{2}-12\mu_{1,0}\alpha_{0}-96\alpha_{0}$
$\displaystyle\mu_{1,0}$ $\displaystyle=$
$\displaystyle-\frac{72576\,\alpha_{0}^{3}+(432\,m_{1}-34560)\alpha_{0}^{2}+(288\,m_{1}+4224)\alpha_{0}+80\,m_{1}}{m_{1}+744\,\alpha_{0}}.$
(42)

We have thus determined the two parameters of the MLDE in terms of objects
associated with the identity character viz. $\alpha_{0}$ and $m_{1}$. We now
consider the differential equation (39) at the next order in the
$q$-expansion, which when evaluated for the identity character, is an equation
for $m_{2}$. After using (3.1) we obtain a polynomial equation in the
variables $\alpha_{0}$, $m_{1}$ and $m_{2}$. We observe that if we define a
new variable

$N\equiv-1680\,\alpha_{0}$ (43)

the polynomial equation is the following one with indeterminates $N$, $m_{1}$
and $m_{2}$.

$\displaystyle
N^{4}+(m_{1}^{2}+93\,m_{1}-2\,m_{2}+955)N^{3}+(-2380\,m_{1}^{2}+28770\,m_{1}+7700\,m_{2}+167160)N^{2}$
$\displaystyle+(1372000\,m_{1}^{2}+10760400\,m_{1}-7330400\,m_{2}-9800\,m_{1}m_{2})N$
$\displaystyle+13720000\,m_{1}m_{2}-329280000\,m_{1}^{2}=0.$ (44)

If we consider (44) as a polynomial equation to determine $N$, which at this
stage is a rational number, we find that the coefficients are all integers.
Given that the coefficient of $N^{4}$ is $1$, which is the reason why we
choose the definition (43) in the first place, we can use the integer-root-
theorem to conclude that $N$ has to be an integer. This is a important turning
point in our analysis. $N$ which was just rational is now restricted to be an
integer. This means that the central charge, which is given by
$c=\frac{N}{70}$ is a rational number whose denominator (not necessarily in
the $\frac{p}{q}$ form with $p,q$ coprime) is always $70$. This also means
that the central charge when expressed in it’s irreducible form $\frac{p}{q}$
with $p,q$ coprime, $q$ can take values $1,2,5,7,10,14,35$ and $70$. More
significantly, our problem has become a “finite” problem in the sense that if
one is searching for CFTs in a certain range of central charge, say $0<c\leq
1$, one only needs to do $70$ computations for
$c=\frac{1}{70},\frac{2}{70},\frac{3}{70},\ldots\frac{69}{70},\frac{70}{70}.$
Furthermore, (44) is a equation with all indeterminates being positive
integers, hence a Diophantine equation. We pick a value of $N$, or
equivalently $c$, and look for positive integer solutions to (44). The
considerations that will give us the rest of the procedure for solving (39)
are as follows.

The two parameters of the MLDE given in (3.1), can now ((43)) can be thought
of as functions of $N$ and $m_{1}$. The indicial equation (40) becomes a cubic
equation whose coefficients are functions of $N$ and $m_{1}$ only. Furthermore
we readily know one of the roots viz. $\alpha_{0}$ or more pertinently
$-\frac{N}{1680}$. Thus the cubic indicial equation factorises into linear and
quadratic equations:

$\displaystyle(1680\alpha+N)\left[(8749440N-19756800m_{1})\alpha^{2}+(-5208N^{2}+11760\,m_{1}N-4374720N+9878400\,m_{1})\alpha\right.$
$\displaystyle\left.+\,N^{3}+14\,m_{1}N^{2}+924\,N^{2}-29400\,m_{1}N+141120\,N+9878400\,m_{1}\right]=0.$
(45)

The roots of the quadratic equation determine the other two indices
$\alpha_{1}$ and $\alpha_{2}$ and eventually $h_{1}$ and $h_{2}$. For rational
roots the discriminant has to be the square of a rational number, but since
the discriminant is an integer (on account of all the coefficients being
integers), it needs to be a perfect square to be able to result in rational
roots. If we denote the discriminant by $k^{2}$ with $k$ chosen to be
positive, we have

$\displaystyle
3457440000\,m_{1}^{2}-1657376000\,m_{1}N-8232000\,m_{1}^{2}N+55899200N^{2}$
$\displaystyle+3528000\,m_{1}N^{2}+4900\,m_{1}^{2}N^{2}+52080\,N^{3}-2100\,m_{1}N^{3}-31N^{4}=k^{2}.$
(46)

We note that (46) is an equation where all the indeterminates viz. $N$,
$m_{1}$ and $k$ are positive integers and hence a Diophantine equation. The
roots are then given by

$\displaystyle\frac{58800\,m_{1}-26040\,N+70\,m_{1}\,N-31\,N^{2}\pm
3\,k}{3360(70\,m_{1}-31\,N)}$ (47)

with $\alpha_{1}$ the smaller root and the $\alpha_{2}$ the bigger one.

The procedure444Most of the key equations (44), (3.1), (46) for the solution
of the $\mathbf{[3,0]}$ MLDE can already be found in an appendix of the early
paper on this subject Mathur:1988gt . We have been able to use that theory to
come up with this procedure that involves a finite number of computations for
a finite range of central charge. to obtain character like solutions to (39)
can now be stated.

(i) We solve the two Diophantine equations (44) and (46) simultaneously. We
fix a value of $N$ and then obtain solutions to (44). Each such solution is a
set of positive integers for $N$, $m_{1}$ and $m_{2}$. We take the $m_{1}$ and
$N$ values of each such solution and see if we can find positive integer
values for $k$ that satisfy (46). At the end of this procedure, we will have a
set of four positive integers, for $N$, $m_{1}$, $m_{2}$ and $k$ which
simultaneously solve (44) and (46). For example, for $0<N\leq 70$, we obtain
six simultaneous solutions to (44) and (46) given in Table 4.

Table 4: Simultaneous solutions to Diophantine equations for $0<N\leq 70$ $N$ | $m_{1}$ | $m_{2}$ | $k$
---|---|---|---
28 | 1 | 1 | 67032
35 | 0 | 1 | 265825
40 | 1 | 2 | 187200
56 | 2 | 3 | 178752
70 | 1 | 4 | 441000
70 | 3 | 4 | 137200

(ii) At this stage, we have the three indices: $\alpha_{0}$, $\alpha_{1}$ and
$\alpha_{2}$ from equations (43) and (47) which further means that we have the
$c$, $h_{1}$ and $h_{2}$ of the putative CFTs. But we have imposed positivity
on the Fourier coefficients of the character only upto order $q^{2}$ and only
for the identity character. Hence, now in this next stage of the procedure, we
impose positivity. That is we determine the Fourier coefficients of all the
characters upto some high order (for us $q^{2000}$), which are afterall
functions of the index of the character and the parameters in the MLDE which
in turn are functions of the known $\alpha_{0}$ and $m_{1}$ (3.1) and check if
they are positive. We reject solutions to the Diophantine equations which do
not survive the positivity constraints. We keep all the other solutions and in
the process also compute $D_{1}$ and $D_{2}$, the positive integers that when
multiplied to $\chi_{1}$ and $\chi_{2}$, makes all their Fourier coefficients
to be positive integers as well. For example, the $N=28$ solution in the first
row of Table 4 turns out to have negative Fourier coefficients for the
identity character and hence is rejected.

We implement the just delineated procedure for $N\leq 6720$ or in other words
for $c\leq 96$. Our results are tabulated in appendix A. Each row gives
details of a character-like solution to the $\mathbf{[3,0]}$ MLDE (39). At
this initial stage of analysis, we think of them as character-like solutions
and not yet CFTs. Every row consists, in the first four columns, of the
(simultaneous) solution to the Diophantine equations (44), (46). In the next
two columns, we give the $D_{1}$ and $D_{2}$ values (not for solutions that
will eventually turn out to be well-known CFTs, for example the ones that
occur in the partial classification of Das:2020wsi ). The next three columns,
that is the seventh, eighth and the ninth columns, consists of the $c$,
$h_{1}$ and $h_{2}$ values and the reader who is only interested in the
results of this paper can focus on just these columns (with the reminder of
the unitary CFT interpretation see footnote 3). The last column consists of an
assignment of the solution to one of five mutually exclusive and exhaustive
categories: $\mathbf{I},$ $\mathbf{II},$ $\mathbf{III},$ $\mathbf{IV}$ and
$\mathbf{V}$, which we describe below. Our computations for $N\leq 6720$ is
given in the eight tables 11 \- 18 of appendix A. These eight tables together
comprise $227$ rows of character-like solutions. Solutions with central
charges between two consecutive integers are grouped together and are given
between two horizontal double lines. If we were to tabulate all our
computations, we would have had $307$ rows. We find a pattern for the last
$80$ rows (and beyond) and hence don’t put it here. Each row, given in
appendix A, represents one character-like solution, except four rows, which
each represent multiple solutions and together comprise two infinite classes
of solutions. Of the $227$ rows of solutions, the split into the five
categories $\mathbf{I},$ $\mathbf{II},$ $\mathbf{III},$ $\mathbf{IV}$ and
$\mathbf{V}$ are $122,22,61,15,$ and $7$ respectively. If we had given all the
$307$ rows, the split into the five categories $\mathbf{I},$ $\mathbf{II},$
$\mathbf{III},$ $\mathbf{IV}$ and $\mathbf{V}$ would have been $202,22,61,15,$
and $7$ respectively.

####  3.1.1 Category $\mathbf{I}$

The first category of solutions are the known RCFTs with three characters and
vanishing Wronskian index. Recently, a computation of the $\mathbf{[n,l]}$
values for a large class of known RCFTs was made and tabulated in Das:2020wsi
. The results of that study can be viewed as a partial classification of RCFTs
for a given $\mathbf{[n,l]}$; the complete classification would come from a
study of all solutions to the associated $\mathbf{[n,l]}$ MLDE. We will recall
the partial classification for $\mathbf{[3,0]}$ CFTs from that paper (after
correcting a minor typo) here:

$\displaystyle(\mathbf{\hat{A}_{1}})_{2},\qquad(\mathbf{\hat{A}_{3}})_{1},\qquad(\mathbf{\hat{A}_{4}})_{1},$
$\displaystyle(\mathbf{\hat{B}_{r}})_{1}\quad\text{for all}\quad r\geq 3,$
$\displaystyle(\mathbf{\hat{C}_{2}})_{1},$
$\displaystyle(\mathbf{\hat{D}_{r}})_{1}\quad\text{for all}\quad r\geq 5,$
$\displaystyle(\mathbf{\hat{E}_{8}})_{2},$
$\displaystyle\mathcal{M}(7,2),\qquad\mathcal{M}(4,3),$
$\displaystyle\mathcal{M}(5,2)\otimes\mathcal{M}(5,2),\qquad(\mathbf{\hat{A}_{1}})_{1}\otimes(\mathbf{\hat{A}_{1}})_{1},\qquad(\mathbf{\hat{A}_{2}})_{1}\otimes(\mathbf{\hat{A}_{2}})_{1},\qquad(\mathbf{\hat{D}_{4}})_{1}\otimes(\mathbf{\hat{D}_{4}})_{1},$
$\displaystyle(\mathbf{\hat{E}_{6}})_{1}\otimes(\mathbf{\hat{E}_{6}})_{1},\qquad(\mathbf{\hat{E}_{7}})_{1}\otimes(\mathbf{\hat{E}_{7}})_{1},\qquad(\mathbf{\hat{F}_{4}})_{1}\otimes(\mathbf{\hat{F}_{4}})_{1},\qquad(\mathbf{\hat{G}_{2}})_{1}\otimes(\mathbf{\hat{G}_{2}})_{1}.$
(48)

In the tables 11 \- 18 of appendix A, we have $122$ solutions in category
$\mathbf{I}$. We will collect them into table 5 and table 6 here. The $N$ and
$m_{1}$ values of any row uniquely identifies that row. The reader can
correlate a row of table 5 or table 6 with a (unique) row from tables 11 \- 18
of appendix A by comparing the $N$ and $m_{1}$ values.

Table 5: Category I : irreducible

$N$ | $m_{1}$ | Theory
---|---|---
35 | 0 | $\mathcal{M}(4,3)$
40 | 1 | $\mathcal{M}(7,2)$
70 | 1 | $(\mathbf{\hat{so}(2)})_{1}$
105 | 3 | $(\mathbf{\hat{A}_{1}})_{2}$
175 | 10 | $(\mathbf{\hat{C}_{2}})_{1}$
210 | 15 | $(\mathbf{\hat{A}_{3}})_{1}$
245 | 21 | $(\mathbf{\hat{B}_{3}})_{1}$
280 | 24 | $(\mathbf{\hat{A}_{4}})_{1}$
1085 | 248 | $(\mathbf{\hat{E}_{8}})_{2}$
70r+35 | $\frac{2r(2r+1)}{2}$ | $(\mathbf{\hat{B}_{r}})_{1}$
70(r+1) | $\frac{2(r+1)(2r+1)}{2}$ | $(\mathbf{\hat{D}_{r+1}})_{1}$

  * •

where $4\leq r\leq 55$

Table 5 contains $113$ of the $122$ solutions of category $\mathbf{I}$. Each
of them corresponds to an irreducible three-character CFT, that is, it is not
a tensor-product of theories with fewer characters. We can locate them in the
partial classification (3.1.1) in the first five rows. The third row of table
5 is not there in (3.1.1). This is a known CFT, corresponding to two decoupled
copies of the Ising model. It is also a WZW CFT, $(\mathbf{\hat{so}(2)})_{1}$,
and can perhaps be included in the $(\mathbf{\hat{D}_{r}})_{1}$ series.

We performed computations up to $N\leq 6720$ or for $c\leq 96$. But we are
reporting only computations up to $N\leq 3920$ or for $c\leq 56$, see table
17. In the range $56<c\leq 96$, we have $80$ admissible character solutions
and all of them are in category $\mathbf{I}$. They are
$(\mathbf{\hat{B}_{r}})_{1}$ for $56\leq r\leq 95$ snd
$(\mathbf{\hat{D}_{r}})_{1}$ for $57\leq r\leq 96$. It is tempting to
conjecture that for all $c>96$ also there are only category $\mathbf{I}$
solutions (the $\mathbf{B}$ and $\mathbf{D}$ series WZW CFTs ) and attempt to
find a proof for the same.

Table 6: Category I : tensor S. No. | $N$ | $m_{1}$ | Theory
---|---|---|---
1. | 56 | 2 | $\mathcal{M}(5,2)^{\otimes 2}$
2. | 140 | 6 | $(\mathbf{\hat{A}_{1}})_{1}^{\otimes 2}$
3. | 280 | 16 | $(\mathbf{\hat{A}_{2}})_{1}^{\otimes 2}$
4. | 392 | 28 | $(\mathbf{\hat{G}_{2}})_{1}^{\otimes 2}$
5. | 560 | 56 | $(\mathbf{\hat{D}_{4}})_{1}^{\otimes 2}$
6. | 728 | 104 | $(\mathbf{\hat{F}_{4}})_{1}^{\otimes 2}$
7. | 840 | 156 | $(\mathbf{\hat{E}_{6}})_{1}^{\otimes 2}$
8. | 980 | 266 | $(\mathbf{\hat{E}_{7}})_{1}^{\otimes 2}$
9. | 1064 | 380 | $\mathbf{E_{7\frac{1}{2}}}^{\otimes 2}$

Table 6 contains the other $9$ of the $122$ solutions of category
$\mathbf{I}$. Each of them corresponds to the tensor product of two copies of
a $\mathbf{[2,0]}$ character-like solution (see table 1). Eight of these can
be located in the partial classification of CFTs (3.1.1) in the last two rows.
The ninth $\mathbf{E_{7\frac{1}{2}}}^{\otimes 2}$ is not a CFT, but very much
an admissible character solution JL:2006 .

Of all the solutions in category $\mathbf{I}$, two of them are non-unitary
CFTs, the second and first entries of tables 5 and 6 respectively. See
footnote 3.

####  3.1.2 Category $\mathbf{II}$

Even at the outset, it should already be stated that all the solutions in
category $\mathbf{II}$, except possibly two, are not to be considered as
admissible-character solutions that will give consistent CFTs. This is because
while all the three characters satisfy the positivity constraints, there are
one or more characters that are “unstable”. An unstable character is a
character with positive rational coefficients, but there is no positive
integer available, which when multiplied makes all the Fourier coefficients
integers. The positive integer needed keeps increasing as we increase the
order to which we compute. This is in contrast to usual characters, which in
this context may be termed “stable” characters, where usually after a certain
order of computation one finds a positive integer that when multiplied makes
all Fourier oefficients integers not only up to that order but also to all
subsequent larger orders of computation.

In the original MMS classification for $\mathbf{[2,0]}$ CFTs Mathur:1988na ,
given here in table 1, of the ten solutions that pass the positivity criteria,
only nine of them are admissible character solutions with only “stable”
characters. Eight of them (the first eight in table 1) are genuine CFTs and
one of them (the ninth entry of table 1), is related to another mathematical
structure JL:2006 . But that paper does include a solution (the last entry of
table 1), one of whose characters is an “unstable” character and the other
“stable” character is the character of the $(\mathbf{\hat{E}_{8}})_{1}$ CFT
which is a $\mathbf{[1,2]}$ CFT. Our category $\mathbf{II}$ solutions are
being reported here, keeping in line with this aspect of original MMS
classification table.

It is an interesting question to ask: which two-character CFTs along with one
“unstable” character are solutions to $\mathbf{[3,0]}$ MLDEs? There is also
the related question: which one-character CFTs along with two “unstable”
characters are solutions to $\mathbf{[3,0]}$ MLDEs? The table 7 answers these
questions.

The solutions of category $\mathbf{II}$ are related to $\mathbf{[n,l]}$ CFTs
or $\mathbf{[n,l]}$ admissible character solutions with $\mathbf{n}\leq 2$ and
$\mathbf{l}\leq 4$. Two solutions viz. S. No. 9 and 10 of table 7, have two
“unstable” characters and the stable character is that of a $\mathbf{[1,2]}$
and $\mathbf{[1,4]}$ CFT, respectively. $18$ solutions in table 7 have one
“unstable” character with the “stable” characters that of known two-character
CFTs. The $\mathbf{[2,0]}$ CFT $\mathcal{M}(5,2)$ is not there in table 7. It
turns out that there is a solution to the $\mathbf{[3,0]}$ MLDE (39), two of
whose linearly independent solutions are the characters of $\mathcal{M}(5,2)$
and the other is a solution with negative Fourier coefficients at low orders
(reminiscent of quasi-characters Mukhi:2020gnj ). This occurs for the
$N=28,m_{1}=1,m_{2}=1,k=67032$ solution to the Diophantine equations. But we
do not report this here in the tables.

The last two solutions left to discuss are curious ones: S. No.4 and S. No.15
of table 7. In each case all characters are “stable” characters. For S. No.4,
two of the characters are the characters of the $(\mathbf{\hat{D}_{4}})_{1}$
CFT and the third character is the non-identity character of the
$(\mathbf{\hat{D}_{4}})_{1}$ CFT. Thus this solution has the curious feature
that two of the three characters are identical. This means that Wronskian
vanishes, which does not contradict the fact that all $\mathbf{[3,0]}$
solutions should have $A_{1}\eta^{12}$ as the Wronskian (see section 2.3), if
$A_{1}=0$ (which is also a possibility). It is not clear to us if we should
consider this as a new $\mathbf{[3,0]}$ admissible character or just derived
from an old $\mathbf{[2,0]}$ solution. The solution in S. No.15 is similar,
except that the two-character CFT is a $\mathbf{[2,2]}$ CFT that appeared in
Hampapura:2015cea .

Table 7: Category II S.No. | $N$ | $m_{1}$ | Identification
---|---|---|---
1. | 70 | 3 | (2,0): $(\mathbf{\hat{A}_{1}})_{1}$
2. | 140 | 8 | (2,0): $(\mathbf{\hat{A}_{2}})_{1}$
3. | 196 | 14 | (2,0): $(\mathbf{\hat{G}_{2}})_{1}$
4. | 280 | 28 | (2,0): $(\mathbf{\hat{D}_{4}})_{1}$
5. | 364 | 52 | (2,0): $(\mathbf{\hat{F}_{4}})_{1}$
6. | 420 | 78 | (2,0): $(\mathbf{\hat{E}_{6}})_{1}$
7. | 490 | 133 | (2,0): $(\mathbf{\hat{E}_{7}})_{1}$
8. | 532 | 190 | (2,0): $\mathbf{E_{7\frac{1}{2}}}$
9. | 560 | 248 | (1,2): $(\mathbf{\hat{E}_{8}})_{1}$
10. | 1120 | 496 | (1,4): $(\mathbf{\hat{E}_{8}})_{1}^{\otimes 2}$
11. | 1148 | 410 | (2,2): CFT of Hampapura:2015cea
12 | 1190 | 323 | (2,2): GHM-dual of $(\mathbf{\hat{E}_{7}})_{1}$
13. | 1260 | 234 | (2,2): GHM-dual of $(\mathbf{\hat{E}_{6}})_{1}$
14. | 1316 | 188 | (2,2): GHM-dual of $(\mathbf{\hat{F}_{4}})_{1}$
15. | 1400 | 140 | (2,2): GHM-dual of $(\mathbf{\hat{D}_{4}})_{1}$
16. | 1484 | 106 | (2,2): GHM-dual of $(\mathbf{\hat{G}_{2}})_{1}$
17. | 1540 | 88 | (2,2): GHM-dual of $(\mathbf{\hat{A}_{2}})_{1}$
18. | 1610 | 69 | (2,2): GHM-dual of $(\mathbf{\hat{A}_{1}})_{1}$
19. | 1652 | 59 | (2,2): CFT of Hampapura:2015cea
20. | 2268 | 4 | (2,4) CFT of Chandra:2018pjq
21. | 2310 | 3 | (2,4) CFT of Chandra:2018pjq
22. | 2380 | 1 | (2,4) CFT of Chandra:2018pjq

####  3.1.3 Category $\mathbf{III}$

In their paper FM:2019 , C. Franc and G. Mason studied certain kinds of vertex
operator algebras of rank $3$, which seem to be related to $\mathbf{[3,0]}$
CFTs. The similarity is that they also study (certain kinds of) character-like
solutions to the $\mathbf{[3,0]}$ MLDE (39) that we are studying in this
paper. We have rediscovered all these Franc-Mason solutions here in our work.
Some of the Franc-Mason solutions pertain to well-known (even prior to FM:2019
) CFTs. There are Franc-Mason solutions that occur within our category
$\mathbf{I}$, category $\mathbf{II}$ as well as category $\mathbf{IV}$ (see
below). The rest of the Franc-Mason solutions find their place in the present
category $\mathbf{III}$, There are $61$ rows in the eight tables 11-18 of
appendix A. $57$ of these rows correspond to $57$ CFTs; the rest four rows
each correspond to multiple CFTs which we discuss below.

Now we discuss the first of the left over four rows of category
$\mathbf{III}$. This is in table 12 and occurs at
$N=560,m_{1}=d_{1},m_{2}=d_{2},k=k_{1}$, where $d_{1}$, $d_{2}$, $k_{1}$ are
integers: $0\leq d_{1}\leq 247$, $d_{2}=156+16d_{1}$ and
$k_{1}=4860800-19600d_{1}$. Note that these are $248$ different CFTs all with
the same central charge and even same scaling dimensions. The second of the
left over rows of category $\mathbf{III}$ is in table 12 and occurs at
$N=560,m_{1}=e_{1},m_{2}=e_{2},k=k_{2}$ where $e_{1}$, $e_{2}$, $k_{2}$ are
integers: $e_{1}\geq 249$, $e_{2}=156+16e_{1}$ and
$k_{2}=-4860800+19600e_{1}$. Note that these are an infinite class of CFTs all
with the same central charge and even same scaling dimensions. These two
classes of CFTs comprise the first infinite class of Franc-Mason solutions
FM:2019  ($(x=0,y=-1/2)$). We can understand these infinite solutions all with
the same central charge and scaling dimensions from an aspect of MLDE
solutions that we derived in section 2.3, equation (2.3): translating to the
notation of that equation, we have
$\alpha_{1}=-\frac{1}{3},\alpha_{2}=\frac{1}{6},\alpha_{3}=\frac{2}{3}\Longrightarrow-\frac{1}{2}+\alpha_{2}+2\alpha_{3}=1$.

Now we discuss the third of the left over four rows of category
$\mathbf{III}$. This is in table 13 and occurs at
$N=1120,m_{1}=d_{1},m_{2}=d_{2},k=k_{1}$, where $d_{1}$, $d_{2}$, $k_{1}$ are
integers:$0\leq d_{1}\leq 495$, $d_{2}=2296+136d_{1}$ and
$k_{1}=9721600-19600d_{1}$. Note that these are $496$ different CFTs all with
the same central charge and even same scaling dimensions. The fourth of the
left over rows of category $\mathbf{III}$ is in table 13 and occurs at
$N=1120,m_{1}=e_{1},m_{2}=e_{2},k=k_{2}$ where $e_{1}$, $e_{2}$, $k_{2}$ are
integers: $e_{1}\geq 497$, $e_{2}=2296+136e_{1}$ and
$k_{2}=-9721600+19600e_{1}$. Note that these are an infinite class of CFTs all
with the same central charge and even same scaling dimensions. These two
classes of CFTs comprise the second infinite class of Franc-Mason solutions
FM:2019 ($(x=1,y=0)$). We can understand these infinite solutions all with the
same central charge and scaling dimensions from an aspect of MLDE solutions
that we derived in section 2.3, equation (2.3): translating to the notation of
that equation, we have
$\alpha_{1}=-\frac{2}{3},\alpha_{2}=\frac{1}{3},\alpha_{3}=\frac{5}{6}\Longrightarrow-\frac{1}{2}+2\alpha_{2}+\alpha_{3}=1$.

####  3.1.4 Category $\mathbf{IV}$

The paper Gaberdiel:2016zke  gives a new kind of coset construction for CFTs,
which is sometimes referred to as a “novel coset construction” (to perhaps
distinguish it from WZW cosets). In this construction one starts with a
meromorphic CFT ($\mathcal{H}$) and quotients it by an affine sub-theory
($\mathcal{D}$), also referred to as the denominator theory and produces the
coset CFT ($\mathcal{C}$). Schematically one writes
$\mathcal{C}=\mathcal{H}/\mathcal{D}$ and refers to the $\mathcal{C}$-theory
as the novel coset dual of the $\mathcal{D}$-theory. We will refer to these
dual theories as “GHM-duals” in this paper.

We have $15$ solutions in category $\mathbf{IV}$ in the tables 11-18 of
appendix A. Each of these are GHM-dual CFTs. All of them belong to only one
particular case of GHM-duality among the many that are studied in
Gaberdiel:2016zke . This is the case where both the denominator CFT
$\mathcal{D}$ as well as the GHM-dual CFT $\mathcal{C}$ are $\mathbf{[3,0]}$
CFTs. It is convenient and suggestive to use the notation
$\tilde{\mathcal{D}}$ for $\mathcal{C}$. The central charges and the scaling
dimensions of the two CFTs, in this particular case of GHM-duality, are
related as follows:

$\displaystyle c^{\mathcal{D}}+c^{\tilde{\mathcal{D}}}=24,\qquad
h_{1}^{\mathcal{D}}+h_{1}^{\tilde{\mathcal{D}}}=2,\qquad
h_{2}^{\mathcal{D}}+h_{2}^{\tilde{\mathcal{D}}}=2.$ (49)

We collect all the solutions in category $\mathbf{IV}$ here in table 8. Each
of the GHM-dual CFTs are duals of category $\mathbf{I}$ CFTs. From our
computations, and from (49) which comprise necessary conditions, we can only
identify potential GHM-dual CFTs. Whether it is actually a GHM-dual CFT
requires more analysis involving meromorphic CFTs Schellekens:1993 ,
Chandra:2018ezv ; which is beyond the scope of the present work. But all the
CFTs in our table 8 are actual GHM-dual CFTs and the meromorphic CFT
associated to each and other details can be found in table 2 of
Gaberdiel:2016zke .

Table 8: Category IV S.No. | $N$ | $m_{1}$ | Identification
---|---|---|---
1. | 1050 | 255 | GHM-dual of $(\mathbf{\hat{D}_{9}})_{1}$
2. | 1190 | 221 | GHM-dual of $(\mathbf{\hat{D}_{7}})_{1}$
3. | 1225 | 210 | GHM-dual of $(\mathbf{\hat{B}_{6}})_{1}$
4. | 1260 | 198 | GHM-dual of $(\mathbf{\hat{D}_{6}})_{1}$
5. | 1295 | 185 | GHM-dual of $(\mathbf{\hat{B}_{5}})_{1}$
6. | 1330 | 171 | GHM-dual of $(\mathbf{\hat{D}_{5}})_{1}$
7. | 1365 | 156 | GHM-dual of $(\mathbf{\hat{B}_{4}})_{1}$
8. | 1400 | 120 | GHM-dual of $(\mathbf{\hat{A}_{4}})_{1}$
9. | 1435 | 123 | GHM-dual of $(\mathbf{\hat{B}_{3}})_{1}$
10. | 1470 | 105 | GHM-dual of $(\mathbf{\hat{A}_{3}})_{1}$
11. | 1505 | 86 | GHM-dual of $(\mathbf{\hat{C}_{2}})_{1}$
12. | 1575 | 45 | GHM-dual of $(\mathbf{\hat{A}_{1}})_{2}$
13 | 1645 | 0 | GHM-dual of $\mathcal{M}(4,3)$
14. | 2296 | 0 | GHM-dual of $\mathcal{M}(5,2)^{\otimes 2}$
15. | 2360 | 0 | GHM-dual of $\mathcal{M}(7,2)$

####  3.1.5 Category $\mathbf{V}$

In the last category, from the tables 11-18 we have seven rows corresponding
to seven admissible character solutions, which we have gathered in table 9.
These are new solutions. They do not seem to fall in any of the known classes
of solutions. At this stage of the analysis, one can only claim to have
obtained new character-like solutions, also referred to as admissible
character solutions. They are not yet CFTs; to establish that we need more
work wp . We can only say that we potentially have _seven new $\mathbf{[3,0]}$
CFTs_.

Table 9: Category V S.No. | $N$ | $m_{1}$ | $c$ | $h_{1}$ | $h_{2}$
---|---|---|---|---|---
1. | 840 | 318 | 12 | 1/3 | 5/3
2. | 1400 | 80 | 20 | 4/3 | 5/3
3. | 1400 | 728 | 20 | 1/3 | 8/3
4. | 1400 | 890 | 20 | 2/3 | 7/3
5. | 1960 | 1948 | 28 | 2/3 | 10/3
6. | 2520 | 3384 | 36 | 2/3 | 13/3
7. | 3080 | 3146 | 44 | 1/3 | 17/3

Here, we present the Fourier expansion (a few terms) of the seven potentially
new $\mathbf{[3,0]}$ RCFTs.

  * 1\. c = 12

    1. 1.

$\chi_{0}(q)=q^{-\frac{1}{2}}\left(1+318q+8514q^{2}+126862q^{3}+1269771q^{4}+9695754q^{5}+\mathcal{O}(q^{6})\right)$

    2. 2.

$\chi_{\frac{1}{3}}(q)=q^{-\frac{1}{6}}\left(1+178q+4634q^{2}+61924q^{3}+566277q^{4}+4041864q^{5}+\mathcal{O}(q^{6})\right)$

    3. 3.

$\chi_{\frac{5}{3}}(q)=q^{\frac{7}{6}}\left(1+23q+272q^{2}+2286q^{3}+15318q^{4}+87091q^{5}+\mathcal{O}(q^{6})\right)$

  * 2\. c = 20

    1. 1.

$\chi_{0}(q)=q^{-\frac{5}{6}}\left(1+80q+46790q^{2}+2654800q^{3}+68308625q^{4}+1122213936q^{5}+\mathcal{O}(q^{6})\right)$

    2. 2.

$\chi_{\frac{4}{3}}(q)=q^{\frac{1}{2}}\left(5+840q+34398q^{2}+744040q^{3}+10930320q^{4}+122840280q^{5}+\mathcal{O}(q^{6})\right)$

    3. 3.

$\chi_{\frac{5}{3}}(q)=q^{\frac{5}{6}}\left(4+355q+11240q^{2}+209810q^{3}+2791180q^{4}+29171789q^{5}+\mathcal{O}(q^{6})\right)$

  * 3\. c = 20

    1. 1.

$\chi_{0}(q)=q^{-\frac{5}{6}}\left(1+728q+106406q^{2}+3894424q^{3}+82707185q^{4}+1242579288q^{5}+\mathcal{O}(q^{6})\right)$

    2. 2.

$\chi_{\frac{1}{3}}(q)=q^{-\frac{1}{2}}\left(2+717q+83124q^{2}+3031214q^{3}+62776974q^{4}+904657032q^{5}+\mathcal{O}(q^{6})\right)$

    3. 3.

$\chi_{\frac{8}{3}}(q)=q^{\frac{11}{6}}\left(1+40q+806q^{2}+11076q^{3}+117599q^{4}+1031680q^{5}+\mathcal{O}(q^{6})\right)$

  * 4\. c = 20

    1. 1.

$\chi_{0}(q)=q^{-\frac{5}{6}}\left(1+890q+55700q^{2}+2695300q^{3}+68460905q^{4}+1122686166q^{5}+\mathcal{O}(q^{6})\right)$

    2. 2.

$\chi_{\frac{2}{3}}(q)=q^{-\frac{1}{6}}\left(5+728q+58000q^{2}+1822700q^{3}+33995325q^{4}+452188800q^{5}+\mathcal{O}(q^{6})\right)$

    3. 3.

$\chi_{\frac{7}{3}}(q)=q^{\frac{3}{2}}\left(10+423q+9180q^{2}+134925q^{3}+1516500q^{4}+13956435q^{5}+\mathcal{O}(q^{6})\right)$

  * 5\. c = 28

    1. 1.

$\chi_{0}(q)=q^{-\frac{7}{6}}\left(1+1948q+424724q^{2}+23967552q^{3}+1047902951q^{4}+31801518220q^{5}+\mathcal{O}(q^{6})\right)$

    2. 2.

$\chi_{\frac{2}{3}}(q)=q^{-\frac{1}{2}}\left(25+9084q+862632q^{2}+59416816q^{3}+2267443743q^{4}+55824667884q^{5}+\mathcal{O}(q^{6})\right)$

    3. 3.

$\chi_{\frac{10}{3}}(q)=q^{\frac{13}{6}}\left(11+628q+18439q^{2}+368372q^{3}+5618761q^{4}+69789060q^{5}+\mathcal{O}(q^{6})\right)$

  * 6\. c = 36

    1. 1.

$\chi_{0}(q)=q^{-\frac{3}{2}}\left(1+3384q+1337850q^{2}+186743064q^{3}+10611868761q^{4}+442989392472q^{5}+\mathcal{O}(q^{6})\right)$

    2. 2.

$\chi_{\frac{2}{3}}(q)=q^{-\frac{5}{6}}\left(2+1294q+243835q^{2}+19695524q^{3}+1210616824q^{4}+50488594386q^{5}+\mathcal{O}(q^{6})\right)$

    3. 3.

$\chi_{\frac{13}{3}}(q)=q^{\frac{17}{6}}\left(4+289q+10688q^{2}+268714q^{3}+5154084q^{4}+80349657q^{5}+\mathcal{O}(q^{6})\right)$

  * 7\. c = 44

    1. 1.

$\chi_{0}(q)=q^{-\frac{11}{6}}\left(1+3146q+1906130q^{2}+467324061q^{3}+61502739069q^{4}+5175295282540q^{5}+\mathcal{O}(q^{6})\right)$

    2. 2.

$\chi_{\frac{1}{3}}(q)=q^{-\frac{3}{2}}\left(13+20178q+8638218q^{2}+1682129028q^{3}+187246128501q^{4}+\mathcal{O}(q^{5})\right)$

    3. 3.

$\chi_{\frac{17}{3}}(q)=q^{\frac{23}{6}}\left(19+1702q+76646q^{2}+2317292q^{3}+52998283q^{4}+979386560^{5}+\mathcal{O}(q^{6})\right)$

We leave the further study of the above new admissible character solutions in
category $\mathbf{V}$ as well as the ones in category $\mathbf{III}$ for the
near future wp , especially the question as to whether they constitute
consistent CFTs. We have been informed that some work in that direction has
been done in Kaidi:2021ent .

###  3.2 Classifying $\mathbf{[3,2]}$ CFTs

We now study three character rational conformal field theories with a
Wronskian index given by $2$. The characters of every $\mathbf{[3,2]}$ CFT is
a solution to the $\mathbf{[3,2]}$ MLDE :

$\displaystyle\mathcal{D}^{3}\chi_{i}+\mu_{-1,1}\,\frac{E_{6}}{E_{4}}\,\mathcal{D}^{2}\chi_{i}+\mu_{1,0}\,E_{4}\,\mathcal{D}\chi_{i}+\mu_{0,1}\,E_{6}\,\chi_{i}=0,$
(50)

At this stage there are three parameters in the MLDE, but one of them gets
fixed: $\mu_{-1,1}=4$ and we are in the same situation as the $\mathbf{[3,0]}$
case i.e. two free parameters to scan. The analysis and the strategy follows
the $\mathbf{[3,0]}$ case and we will be brief here, highlighting only the
contrasts. At the leading order in the $q$-expansion, the differential
equation (50) gives the following indicial equation:

$\displaystyle
1728\,\alpha^{3}-288\,\alpha^{2}+12\,\mu_{1,0}\,\alpha+\mu_{0,1}=0$ (51)

Note that the sum of the indices is $\frac{1}{6}$ which is what is dictated by
the Wronskian index being two from (20) and which is a direct consequence of
choosing $\mu_{-1,1}=4$. At the next to leading order in the $q$-expansion,
the differential equation (50) gives the following equation

$\displaystyle\frac{f_{1}}{f_{0}}=-\frac{2\left(17280\alpha^{3}-19872\alpha^{2}+48\alpha(5\mu_{1,0}+66)-11\mu_{0,1}\right)}{432\alpha^{2}+384\alpha+\mu_{1,0}+120}$
(52)

We again now evaluate the two equations (51) and (52) for the identity
character and interpret these equations as two equations for the two
parameters which give the solution

$\displaystyle\mu_{0,1}$ $\displaystyle=$
$\displaystyle\frac{288\alpha_{0}(\alpha_{0}+1)\left(-1440\alpha_{0}^{2}+264\alpha_{0}+12\alpha_{0}m_{1}+5m_{1}\right)}{744\alpha_{0}+m_{1}}$
$\displaystyle\mu_{1,0}$ $\displaystyle=$
$\displaystyle-\frac{24\left(3024\,\alpha_{0}^{3}+6(3m_{1}-320)\alpha_{0}^{2}+8(2m_{1}+33)\alpha_{0}+5m_{1}\right)}{744\alpha_{0}+m_{1}}.$
(53)

We again have thus determined the two parameters of the MLDE in terms of
objects associated with the identity character viz. $\alpha_{0}$ and $m_{1}$.
We now consider the differential equation (50) at the next order in the
$q$-expansion, which when evaluated for the identity character, is an equation
for $m_{2}$. After using (3.2) we obtain a polynomial equation in the
variables $\alpha_{0}$, $m_{1}$ and $m_{2}$. We observe that if we define a
new variable

$\displaystyle N=-5040\,\alpha_{0}$ (54)

the polynomial equation is the following one with indeterminates $N$, $m_{1}$
and $m_{2}$.

$\displaystyle
N^{4}+\left(3m_{1}^{2}+279\,m_{1}-6\,m_{2}-75695\right)N^{3}-\left(26460\,m_{1}^{2}+2397150\,m_{1}-79380\,m_{2}+71137080\right)N^{2}$
$\displaystyle+\left(74440800\,m_{1}^{2}-264600\,m_{1}m_{2}+7888784400\,m_{1}-236728800\,m_{2}\right)N$
$\displaystyle-\left(87127488000\,m_{1}^{2}-1259496000\,m_{1}m_{2}\right)=0$
(55)

We can use the integer-root-theorem to conclude that $N$ has to be an integer.
This is again an important turning point in our analysis. $N$ which was just
rational is now restricted to be an integer. This means that the central
charge, which is given by $c=\frac{N}{210}$ is a rational number whose
denominator (not necessarily in the $\frac{p}{q}$ form with $p,q$ coprime) is
always $210$. Our problem has again become a “finite” problem in the sense
that if one is searching for CFTs in a certain range of central charge, say
$0<c\leq 1$, one only needs to do $210$ computations for
$c=\frac{1}{210},\frac{2}{210},\frac{3}{210},\ldots\frac{209}{210},\frac{210}{210}.$
Equation (3.2) is one of the Diophantine equations. Again we use (3.2) and
(54) in the indicial equation, which then factorises into linear and quadratic
equations:

$\displaystyle\left(\alpha+\frac{N}{5040}\right)\left((78744960\,N-533433600\,m_{1})\alpha^{2}+(105840\,Nm_{1}+88905600\,m_{1}-15624\,N^{2}-13124160\,N)\alpha\right.$
$\displaystyle\left.+42\,N^{2}m_{1}-299880\,Nm_{1}+444528000\,m_{1}+N^{3}-4116\,N^{2}-4656960\,N\right)=0$
(56)

The discriminant of the quadratic equation gives the second Diophantine
equation:

$\displaystyle-31\,N^{4}-140(45m_{1}-5332)\,N^{3}+4900(146320+7152\,m_{1}+9\,m_{1}^{2})\,N^{2}-8232000\,m_{1}(33\,m_{1}+8092)\,N$
$\displaystyle+418350240000\,m_{1}^{2}=k^{2}.$ (57)

The other two indices are given by

$\displaystyle\frac{26040\,N+31\,N^{2}-176400\,m_{1}-210\,Nm_{1}\pm
3\,k}{10080(31\,N-210\,m_{1})}.$ (58)

We follow the same procedure as for the $\mathbf{[3,0]}$ case. First we find
positive integer solutions to the two Diophantine equations (3.2) and (57). We
do this for $N\leq 20160$ to investigate central charge in the range $0<c\leq
96$. For every solution, we then compute further Fourier coefficients via the
Frobenius-like method and reject all solutions which don’t satisfy positivity
criteria necessary for character-like solutions.

Our results are tabulated in table 19 of appendix A. We find that there are
exactly nine character-like solutions. Each of these solutions has the
following feature. The character $\chi_{1}(q)$ is a constant character. How
this comes about is that $h_{1}=\frac{c}{24}$ so that $\alpha_{1}=0$ and
furthermore all Fourier coefficients of this character vanish except the first
one i.e $f_{0}^{(1)}\neq 0,f_{n}^{(1)}=0,n\neq 0$. Thus
$\chi_{1}(q)=f_{0}^{(1)}$ is a constant and we call this a constant character.
Every solution in table 19 is thus comprised of the characters of a
$\mathbf{[2,0]}$ CFT or more generally of an admissible character solution of
a $\mathbf{[2,0]}$ MLDE adjoined with a constant character,

We can show using elementary MLDE theory that this is an example of a general
phenomenon. One can always adjoin a constant character to a $\mathbf{[n,l}]$
CFT to obtain a $\mathbf{[n+1,n+l}]$ CFT. What we are seeing in our results is
for $\mathbf{n}=2$ and $\mathbf{l}=0$.

Our computations seem to suggest the following classification for
$\mathbf{[3,2]}$ CFTs: _The only admissible character solutions are the ones
obtained by adjoining a constant character to a $\mathbf{[2,0]}$ admissible
character solution._ Recent concurrent work Kaidi:2021ent  suggests that for
three character MLDEs new admissible character solutions are expected only
when the Wronskian index is a multiple of $3$. This is in agreement with our
classification above, which does not contain any new admissible character
solutions.

##  4 Potential GHM-dual pairs of $\mathbf{[3,0]}$ CFTs

Among the five categories of $\mathbf{[3,0]}$ solutions given in section 3.1,
we will consider categories $\mathbf{I}$, $\mathbf{II}$ and $\mathbf{IV}$ as
well-understood CFTs. Solutions in category $\mathbf{V}$ are new and need to
be studied more, as well as solutions in category $\mathbf{III}$, the Franc-
Mason vertex operator algebras FM:2019 . In this section, we will tabulate
some observations about solutions in these two categories and hope to present
a more complete study in the future wp .

In the novel coset construction of Gaberdiel-Hampapura-Mukhi Gaberdiel:2016zke
, the central charges and the scaling dimensions of the denominator
$\mathbf{[3,0]}$ CFT and the coset $\mathbf{[3,0]}$ CFT are related by (49).
We will define a pair of $\mathbf{[3,0]}$ CFTs $(T,\tilde{T})$ to be
potentially GHM-dual, or that the theories $(T,\tilde{T})$ constitute a
potential GHM-dual pair, if their central charges and scaling dimensions
follow:

$\displaystyle c+\tilde{c}=24,\qquad h_{1}+\tilde{h}_{1}=2,\qquad
h_{2}+\tilde{h}_{2}=2.$ (59)

We have been able to identify $27$ potential GHM-dual pairs of CFTs, which we
have tabulated in table 10. There are two kinds of these pairs of solutions.
The first kind ($17$ pairs) is one in which one of the CFTs of the pair is a
known WZW CFT and the other is a unknown CFT from category $\mathbf{III}$ or
category $\mathbf{V}$. This kind of a situation is closer to the GHM
construction where one of the CFTs (the denominator CFT) is a WZW CFT. The
second kind ($10$ pairs) is one where both the CFTs of the pair is an unknown
CFT from the category $\mathbf{III}$. This seems very interesting. We hope to
investigate and report on these potential GHM-dual pairs of CFTs in future
work wp .

Table 10: Potential GHM-dual Pairs of $\mathbf{[3,0]}$ CFTs $N$ | $m_{1}$ | $c$ | $h_{1}$ | $h_{2}$ | $\tilde{N}$ | $\tilde{m}_{1}$ | $\tilde{c}$ | $\tilde{h}_{1}$ | $\tilde{h}_{2}$ | GHM-dual pair
---|---|---|---|---|---|---|---|---|---|---
70 | 1 | 1 | 1/2 | 1/8 | 1610 | 23 | 23 | 3/2 | 15/8 |  $(\mathbf{\hat{so}(2)})_{1}$ $\leftrightarrow$ $\mathbf{III}$
120 | 6 | 12/7 | 2/7 | 3/7 | 1560 | 78 | 156/7 | 12/7 | 11/7 |  $\mathbf{III}$ $\leftrightarrow$ $\mathbf{III}$
140 | 6 | 2 | 1/2 | 1/4 | 1540 | 66 | 22 | 3/2 | 7/4 |  $(\mathbf{\hat{A}_{1}})_{1}^{\otimes 2}$ $\leftrightarrow$ $\mathbf{III}$
168 | 3 | 12/5 | 1/5 | 3/5 | 1512 | 27 | 108/5 | 9/5 | 7/5 |  $\mathbf{III}$ $\leftrightarrow$ $\mathbf{III}$
280 | 16 | 4 | 2/3 | 1/3 | 1400 | 80 | 20 | 4/3 | 5/3 |  $(\mathbf{\hat{A}_{2}})_{1}^{\otimes 2}$ $\leftrightarrow$ $\mathbf{V}$
392 | 28 | 28/5 | 4/5 | 2/5 | 1288 | 92 | 92/5 | 6/5 | 8/5 |  $(\mathbf{\hat{G}_{2}})_{1}^{\otimes 2}$ $\leftrightarrow$ $\mathbf{III}$
440 | 88 | 44/7 | 4/7 | 5/7 | 1240 | 248 | 124/7 | 10/7 | 9/7 |  $\mathbf{III}$ $\leftrightarrow$ $\mathbf{III}$
504 | 144 | 36/5 | 3/5 | 4/5 | 1176 | 336 | 84/5 | 7/5 | 6/5 |  $\mathbf{III}$ $\leftrightarrow$ $\mathbf{III}$
520 | 156 | 52/7 | 4/7 | 6/7 | 1160 | 348 | 116/7 | 10/7 | 8/7 |  $\mathbf{III}$ $\leftrightarrow$ $\mathbf{III}$
525 | 105 | 15/2 | 1/2 | 15/16 | 1155 | 231 | 33/2 | 3/2 | 17/16 |  $(\mathbf{\hat{B}_{7}})_{1}$ $\leftrightarrow$ $\mathbf{III}$
560 | $d_{1}$ | 8 | 1/2 | 1 | 1120 | $\tilde{d_{1}}$ | 16 | 3/2 | 1 |  $\mathbf{III}$ $\leftrightarrow$ $\mathbf{III}$
595 | 255 | 17/2 | 1/16 | 3/2 | 1085 | 465 | 31/2 | 31/16 | 1/2 |  $\mathbf{III}$ $\leftrightarrow$ $(\mathbf{\hat{B}_{15}})_{1}$
600 | 210 | 60/7 | 3/7 | 8/7 | 1080 | 378 | 108/7 | 11/7 | 6/7 |  $\mathbf{III}$ $\leftrightarrow$ $\mathbf{III}$
616 | 220 | 44/5 | 2/5 | 6/5 | 1064 | 380 | 76/5 | 8/5 | 4/5 |  $\mathbf{III}$ $\leftrightarrow$ $\mathbf{III}$
616 | 253 | 44/5 | 1/5 | 7/5 | 1064 | 437 | 76/5 | 9/5 | 3/5 |  $\mathbf{III}$ $\leftrightarrow$ $\mathbf{III}$
630 | 261 | 9 | 1/8 | 3/2 | 1050 | 435 | 15 | 15/8 | 1/2 |  $\mathbf{III}$ $\leftrightarrow$ $(\mathbf{\hat{D}_{15}})_{1}$
665 | 171 | 19/2 | 1/2 | 19/16 | 1015 | 261 | 29/2 | 3/2 | 13/16 |  $(\mathbf{\hat{B}_{9}})_{1}$ $\leftrightarrow$ $\mathbf{III}$
665 | 266 | 19/2 | 3/16 | 3/2 | 1015 | 406 | 29/2 | 29/16 | 1/2 |  $\mathbf{III}$ $\leftrightarrow$ $(\mathbf{\hat{B}_{14}})_{1}$
680 | 221 | 68/7 | 3/7 | 9/7 | 1000 | 325 | 100/7 | 11/7 | 5/7 |  $\mathbf{III}$ $\leftrightarrow$ $\mathbf{III}$
700 | 270 | 10 | 1/4 | 3/2 | 980 | 378 | 14 | 7/4 | 1/2 |  $\mathbf{III}$ $\leftrightarrow$ $(\mathbf{\hat{D}_{14}})_{1}$
728 | 104 | 52/5 | 3/5 | 6/5 | 952 | 136 | 68/5 | 7/5 | 4/5 |  $(\mathbf{\hat{F}_{4}})_{1}^{\otimes 2}$ $\leftrightarrow$ $\mathbf{III}$
735 | 210 | 21/2 | 1/2 | 21/16 | 945 | 270 | 27/2 | 3/2 | 11/16 |  $(\mathbf{\hat{B}_{10}})_{1}$ $\leftrightarrow$ $\mathbf{III}$
735 | 273 | 21/2 | 5/16 | 3/2 | 945 | 351 | 27/2 | 27/16 | 1/2 |  $\mathbf{III}$ $\leftrightarrow$ $(\mathbf{\hat{B}_{13}})_{1}$
770 | 231 | 11 | 1/2 | 11/8 | 910 | 273 | 13 | 3/2 | 5/8 |  $(\mathbf{\hat{D}_{11}})_{1}$ $\leftrightarrow$ $\mathbf{III}$
770 | 275 | 11 | 3/8 | 3/2 | 910 | 325 | 13 | 13/8 | 1/2 |  $\mathbf{III}$ $\leftrightarrow$ $(\mathbf{\hat{D}_{13}})_{1}$
805 | 253 | 23/2 | 1/2 | 23/16 | 875 | 275 | 25/2 | 3/2 | 9/16 |  $(\mathbf{\hat{B}_{11}})_{1}$ $\leftrightarrow$ $\mathbf{III}$
805 | 276 | 23/2 | 7/16 | 3/2 | 875 | 300 | 25/2 | 25/16 | 1/2 |  $\mathbf{III}$ $\leftrightarrow$ $(\mathbf{\hat{B}_{12}})_{1}$

##  5 Conclusion and Future Directions

In this paper, we have directly solved two two-parameter MLDEs, the
$\mathbf{[3,0]}$ and $\mathbf{[3,2]}$ MLDEs and classified admissible
character solutions and potential $\mathbf{[3,0]}$ and $\mathbf{[3,2]}$ RCFTs
for the range of central charge within $0<c\leq 96$. We have divided the
$\mathbf{[3,0]}$ solutions into five mutually exclusive and exhaustive
categories namely $\mathbf{I,\ II,\ III,\ IV,\ V}$. Out of these five
categories, category $\mathbf{V}$ includes _seven_ admissible character
solutions and _seven potentially new RCFTs_. We will investigate, in future,
which of these seven admissible character solutions in category $\mathbf{V}$
are actually RCFTs (by checking if their fusion rule coefficients come out to
be positive using the Verlinde formula); some work in this direction has
appeared in Kaidi:2021ent . At this point, we conjecture that for $c>96$ we
would only get the WZW CFTs belonging to the $\mathbf{\hat{B}}_{r}$ and
$\mathbf{\hat{D}}_{r}$ series. We have also classified $\mathbf{[3,0]}$
admissible character solutions and have discovered that each of them is a
$\mathbf{[2,0]}$ admissible character solutions adjoined with a constant
character.

Furthermore, while performing the above classification, we also observe that
though we began to search for solutions of Diophantine equations coming from
$\mathbf{[3,0]}$ MLDE, we can obtain $\mathbf{[n,l]}$ solutions with
$n\in\\{1,2\\}$ and $l\in\\{0,2,4\\}$. So, such solutions can be realised as
solutions to $\mathbf{[3,0]}$ MLDE with the additional character(s) being
either unstable or have negative $q$-expansion coefficients. These
$\mathbf{[n,l]}$ solutions have been previously discussed in this MLDE
approach to classification of RCFTs in the context of $1$-character and
$2$-character RCFTs. To our surprise we find that all such $\mathbf{[n,l]}$
solutions could be realised as a solutions to the $\mathbf{[3,0]}$ MLDE. We
can, in future, investigate the case of $\mathbf{[2,6]}$ and $\mathbf{[2,8]}$
theories as solutions to some higher-order MLDE inspired by our above
observations.

In this paper, we have directly solved the MLDEs. In the process we have
uncovered many phenomena. (i) It is possible for an infinite number of CFTs to
all have the same central charge and scaling dimensions: sections 2.3 and
3.1.3 (ii) It is possible for a CFT to have two identical characters: section
3.1.2 (iii) A $\mathbf{[n,l]}$ CFT can occur as a solution to $\mathbf{[N,L]}$
CFT for $\mathbf{n}<\mathbf{N}$: section 3.1.2 (iv) One can adjoin a constant
character to a $\mathbf{[n,l]}$ CFT and obtain a $\mathbf{[n+1,n+l]}$ CFT:
section 3.2. We have seen all these phenomena in the context of
$\mathbf{[3,0]}$ and $\mathbf{[3,2]}$ CFTs and expect them to occur for higher
characters as well.

We have established a basic template on how to go about solving directly a
MLDE: expressing the parameters in terms of objects associated with the
identity character, seeking out Diophantine equations that exist because of
the integrality of the Fourier coefficients of the identity character, making
the problem “finite” etc. We hope to bring this template to bear upon other
$\mathbf{[n,l]}$ CFTs.

Our observations in section 4 suggest that there are more examples of the
novel coset CFTs of Gaberdiel-Hampapura-Mukhi, which we hope to understand wp
. Admissible character solutions in category $\mathbf{III}$ and $\mathbf{V}$
which are to be considered still unknown, which are also potentially GHM-dual
to known WZW CFTs, have better prospects of being consistent CFTs. We hope to
report on these ongoing studies.

Note Added: During the final stages of this work we became aware of concurrent
work on this subject. The first paper Kaidi:2021ent  appeared while the first
version of our paper was being prepared. The second paper Bae:2021mej
appeared very soon after the first version of our paper and before the present
second version of our paper. There is an overlap of our $\mathbf{[3,0]}$
results with (small) parts of both these papers but the methods seem to be
quite different.

Acknowledgments

AD would like to express his gratitude to the School of Physics, NISER,
Bhubaneswar for the hospitality and resources required to complete this
project. AD would like to thank Sigma Samhita for her immense help in the type
setting of the tables required for this work. AD would also like to thank
Jishu Das for useful discussions on modular forms. AD would also like to thank
Rinkesh Panigrahi for discussions on Python. CNG thanks Jaban Meher for
helpful discussions and Avani for lending him the use of her computer. JS is
indebted to Sumedha and Senthil K. Kumar for useful discussions.

##  Appendix A Tabulating the character-like solutions of the $\mathbf{[3,0]}$
and $\mathbf{[3,2]}$ MLDEs

Table 11: Solutions to the $\mathbf{[3,0]}$ MLDE with $\mathbf{0<c\leq 7}$ $N$ | $m_{1}$ | $m_{2}$ | $k$ | $D_{1}$ | $D_{2}$ | $c$ | $h_{1}$ | $h_{2}$ | Category
---|---|---|---|---|---|---|---|---|---
35 | 0 | 1 | 265825 | $\cdots$ | $\cdots$ | 1/2 | 1/16 | 1/2 | I
40 | 1 | 2 | 187200 | $\cdots$ | $\cdots$ | 4/7 | 1/7 | 3/7 | I
56 | 2 | 3 | 178752 | $\cdots$ | $\cdots$ | 4/5 | 1/5 | 2/5 | I
70 | 1 | 4 | 441000 | 1 | 1 | 1 | 1/8 | 1/2 | I
70 | 3 | 4 | 137200 | $\cdots$ | $\cdots$ | 1 | 1/4 | 3/8 | II
105 | 3 | 9 | 532875 | $\cdots$ | $\cdots$ | 3/2 | 3/16 | 1/2 | I
120 | 6 | 12 | 264000 | 3 | 2 | 12/7 | 2/7 | 3/7 | III
140 | 6 | 17 | 548800 | $\cdots$ | $\cdots$ | 2 | 1/4 | 1/2 | I
140 | 8 | 17 | 176400 | $\cdots$ | $\cdots$ | 2 | 1/3 | 5/12 | II
168 | 3 | 18 | 1119552 | 3 | 1 | 12/5 | 1/5 | 3/5 | III
175 | 10 | 30 | 496125 | $\cdots$ | $\cdots$ | 5/2 | 5/16 | 1/2 | I
196 | 14 | 42 | 142688 | $\cdots$ | $\cdots$ | 14/5 | 2/5 | 9/20 | II
210 | 15 | 51 | 382200 | $\cdots$ | $\cdots$ | 3 | 3/8 | 1/2 | I
245 | 21 | 84 | 214375 | $\cdots$ | $\cdots$ | 7/2 | 7/16 | 1/2 | I
280 | 16 | 98 | 1411200 | $\cdots$ | $\cdots$ | 4 | 1/3 | 2/3 | I
280 | 24 | 124 | 784000 | $\cdots$ | $\cdots$ | 4 | 2/5 | 3/5 | I
280 | 28 | 134 | 0 | $\cdots$ | $\cdots$ | 4 | 1/2 | 1/2 | II
315 | 36 | 207 | 253575 | $\cdots$ | $\cdots$ | 9/2 | 1/2 | 9/16 | I
350 | 45 | 310 | 539000 | $\cdots$ | $\cdots$ | 5 | 1/2 | 5/8 | I
364 | 52 | 377 | 214032 | $\cdots$ | $\cdots$ | 26/5 | 11/20 | 3/5 | II
385 | 55 | 451 | 848925 | $\cdots$ | $\cdots$ | 11/2 | 1/2 | 11/16 | I
392 | 28 | 280 | 2283008 | $\cdots$ | $\cdots$ | 28/5 | 2/5 | 4/5 | I
420 | 66 | 639 | 1176000 | $\cdots$ | $\cdots$ | 6 | 1/2 | 3/4 | I
420 | 78 | 729 | 352800 | $\cdots$ | $\cdots$ | 6 | 7/12 | 2/3 | II
440 | 88 | 902 | 598400 | 11 | 44 | 44/7 | 4/7 | 5/7 | III
455 | 78 | 884 | 1512875 | $\cdots$ | $\cdots$ | 13/2 | 1/2 | 13/16 | I
490 | 91 | 1197 | 1852200 | $\cdots$ | $\cdots$ | 7 | 1/2 | 7/8 | I
490 | 133 | 1673 | 411600 | $\cdots$ | $\cdots$ | 7 | 5/8 | 3/4 | II
Table 12: Solutions to the $\mathbf{[3,0]}$ MLDE with $\mathbf{7<c\leq 12}$ $N$ | $m_{1}$ | $m_{2}$ | $k$ | $D_{1}$ | $D_{2}$ | $c$ | $h_{1}$ | $h_{2}$ | Category
---|---|---|---|---|---|---|---|---|---
504 | 144 | 1926 | 620928 | 12 | 9 | 36/5 | 3/5 | 4/5 | III
520 | 156 | 2236 | 832000 | 13 | 78 | 52/7 | 4/7 | 6/7 | III
525 | 105 | 1590 | 2186625 | $\cdots$ | $\cdots$ | 15/2 | 1/2 | 15/16 | I
532 | 190 | 2831 | 268128 | $\cdots$ | $\cdots$ | 38/5 | 13/20 | 4/5 | II
560 | 56 | 1052 | 3763200 | $\cdots$ | $\cdots$ | 8 | 1/2 | 1 | I
560 | 120 | 2076 | 2508800 | $\cdots$ | $\cdots$ | 8 | 1/2 | 1 | I
560 | 248 | 4124 | 0 | $\cdots$ | $\cdots$ | 8 | 3/4 | 3/4 | II
560 | $d_{1}$ | $d_{2}$ | $k_{1}$ | 1 | 1 | 8 | 1/2 | 1 | III
560 | $e_{1}$ | $e_{2}$ | $k_{2}$ | 1 | 1 | 8 | 1/2 | 1 | III
595 | 136 | 2669 | 2811375 | $\cdots$ | $\cdots$ | 17/2 | 1/2 | 17/16 | I
595 | 255 | 4216 | 478975 | 17 | 221 | 17/2 | 1/16 | 3/2 | III
600 | 210 | 4050 | 1560000 | 10 | 285 | 60/7 | 3/7 | 8/7 | III
616 | 220 | 4433 | 1655808 | 11 | 11 | 44/5 | 2/5 | 6/5 | III
616 | 253 | 4642 | 931392 | 11 | 242 | 44/5 | 1/5 | 7/5 | III
630 | 153 | 3384 | 3087000 | $\cdots$ | $\cdots$ | 9 | 1/2 | 9/8 | I
630 | 261 | 4500 | 970200 | 9 | 57 | 9 | 1/8 | 3/2 | III
665 | 171 | 4237 | 3328325 | $\cdots$ | $\cdots$ | 19/2 | 1/2 | 19/16 | I
665 | 266 | 4997 | 1466325 | 19 | 703 | 19/2 | 3/16 | 3/2 | III
680 | 221 | 5474 | 2692800 | 17 | 782 | 68/7 | 3/7 | 9/7 | III
700 | 190 | 5245 | 3528000 | $\cdots$ | $\cdots$ | 10 | 1/2 | 5/4 | I
700 | 270 | 5725 | 1960000 | 5 | 15 | 10 | 1/4 | 3/2 | III
728 | 104 | 3458 | 5136768 | $\cdots$ | $\cdots$ | 52/5 | 3/5 | 6/5 | I
735 | 210 | 6426 | 3678675 | $\cdots$ | $\cdots$ | 21/2 | 1/2 | 21/16 | I
735 | 273 | 6699 | 2443875 | 21 | 1225 | 21/2 | 5/16 | 3/2 | III
770 | 231 | 7799 | 3773000 | $\cdots$ | $\cdots$ | 11 | 1/2 | 11/8 | I
770 | 275 | 7931 | 2910600 | 11 | 187 | 11 | 3/8 | 3/2 | III
805 | 253 | 9384 | 3803625 | $\cdots$ | $\cdots$ | 23/2 | 1/2 | 23/16 | I
805 | 276 | 9430 | 3352825 | 23 | 1771 | 23/2 | 7/16 | 3/2 | III
840 | 156 | 7542 | 5644800 | $\cdots$ | $\cdots$ | 12 | 2/3 | 4/3 | I
840 | 222 | 10050 | 4704000 | 1 | 51 | 12 | 3/5 | 7/5 | III
840 | 276 | 11202 | 3763200 | $\cdots$ | $\cdots$ | 12 | 1/2 | 3/2 | I
840 | 318 | 8514 | 2822400 | 1 | 1 | 12 | 1/3 | 5/3 | V
Table 13: Solutions to the $\mathbf{[3,0]}$ MLDE with $\mathbf{12<c\leq 17}$ $N$ | $m_{1}$ | $m_{2}$ | $k$ | $D_{1}$ | $D_{2}$ | $c$ | $h_{1}$ | $h_{2}$ | Category
---|---|---|---|---|---|---|---|---|---
875 | 275 | 13250 | 4134375 | 125 | 2325 | 25/2 | 9/16 | 3/2 | III
875 | 300 | 13275 | 3644375 | $\cdots$ | $\cdots$ | 25/2 | 1/2 | 25/16 | I
910 | 273 | 15574 | 4459000 | 13 | 325 | 13 | 5/8 | 3/2 | III
910 | 325 | 15626 | 3439800 | $\cdots$ | $\cdots$ | 13 | 1/2 | 13/8 | I
945 | 270 | 18171 | 4729725 | 27 | 2871 | 27/2 | 11/16 | 3/2 | III
945 | 351 | 18279 | 3142125 | $\cdots$ | $\cdots$ | 27/2 | 1/2 | 27/16 | I
952 | 136 | 10438 | 6717312 | 119 | 68 | 68/5 | 4/5 | 7/5 | III
952 | 374 | 11985 | 2612288 | 119 | 12138 | 68/5 | 2/5 | 9/5 | III
980 | 266 | 21035 | 4939200 | $\cdots$ | $\cdots$ | 14 | 3/4 | 3/2 | I
980 | 378 | 21259 | 2744000 | $\cdots$ | $\cdots$ | 14 | 1/2 | 7/4 | I
1000 | 325 | 26850 | 3960000 | 55 | 2925 | 100/7 | 5/7 | 11/7 | III
1000 | 380 | 26850 | 2816000 | 55 | 11495 | 100/7 | 4/7 | 12/7 | III
1015 | 261 | 24157 | 5080075 | 29 | 3393 | 29/2 | 13/16 | 3/2 | III
1015 | 406 | 24592 | 2238075 | $\cdots$ | $\cdots$ | 29/2 | 1/2 | 29/16 | I
1050 | 255 | 27525 | 5145000 | 15 | 455 | 15 | 7/8 | 3/2 | IV
1050 | 435 | 28305 | 1617000 | $\cdots$ | $\cdots$ | 15 | 1/2 | 15/8 | I
1064 | 380 | 41762 | 2860032 | 57 | 3249 | 76/5 | 4/5 | 8/5 | I
1064 | 437 | 41762 | 1608768 | 57 | 19 | 76/5 | 3/5 | 9/5 | III
1080 | 378 | 45009 | 2808000 | 117 | 3510 | 108/7 | 6/7 | 11/7 | III
1080 | 456 | 45126 | 1123200 | 39 | 20424 | 108/7 | 4/7 | 13/7 | III
1085 | 248 | 31124 | 5126625 | $\cdots$ | $\cdots$ | 31/2 | 15/16 | 3/2 | I
1085 | 465 | 32426 | 873425 | $\cdots$ | $\cdots$ | 31/2 | 1/2 | 31/16 | I
1120 | 496 | 36984 | 0 | $\cdots$ | $\cdots$ | 16 | 1/2 | 2 | I
1120 | 496 | 69752 | 0 | $\cdots$ | $\cdots$ | 16 | 5/4 | 5/4 | II
1120 | $d_{1}$ | $d_{2}$ | $k_{1}$ | 1 | 1 | 16 | 1 | 3/2 | III
1120 | $e_{1}$ | $e_{2}$ | $k_{2}$ | 1 | 1 | 16 | 1 | 3/2 | III
1148 | 410 | 64739 | 578592 | 902 | $\cdots$ | 82/5 | 6/5 | 27/20 | II
1155 | 231 | 38940 | 4810575 | 33 | 4301 | 33/2 | 17/16 | 3/2 | III
1155 | 528 | 42009 | 1010625 | $\cdots$ | $\cdots$ | 33/2 | 1/2 | 33/16 | I
1160 | 348 | 58174 | 1856000 | 725 | 1972 | 116/7 | 8/7 | 10/7 | III
1176 | 336 | 59850 | 1448832 | 154 | 1452 | 84/5 | 6/5 | 7/5 | III
1176 | 534 | 73842 | 1138368 | 33 | 55924 | 84/5 | 1/5 | 12/5 | III
1190 | 221 | 43112 | 4498200 | 17 | 561 | 17 | 9/8 | 3/2 | IV
1190 | 323 | 60860 | 999600 | 51 | $\cdots$ | 17 | 5/4 | 11/8 | II
1190 | 561 | 47532 | 2165800 | $\cdots$ | $\cdots$ | 17 | 1/2 | 17/8 | I
Table 14: Solutions to the $\mathbf{[3,0]}$ MLDE with $\mathbf{17<c\leq 21}$ $N$ | $m_{1}$ | $m_{2}$ | $k$ | $D_{1}$ | $D_{2}$ | $c$ | $h_{1}$ | $h_{2}$ | Category
---|---|---|---|---|---|---|---|---|---
1225 | 210 | 47425 | 4073125 | 35 | 4655 | 35/2 | 19/16 | 3/2 | IV
1225 | 595 | 53585 | 3472875 | $\cdots$ | $\cdots$ | 35/2 | 1/2 | 35/16 | I
1240 | 248 | 58212 | 1686400 | 2108 | 2108 | 124/7 | 9/7 | 10/7 | III
1260 | 198 | 51849 | 3528000 | 9 | 75 | 18 | 5/4 | 3/2 | IV
1260 | 234 | 59805 | 1058400 | 1 | $\cdots$ | 18 | 4/3 | 17/12 | II
1260 | 598 | 83049 | 3528000 | 25 | 221 | 18 | 1/4 | 5/2 | III
1260 | 630 | 60201 | 4939200 | $\cdots$ | $\cdots$ | 18 | 1/2 | 9/4 | I
1288 | 92 | 30774 | 7501312 | 1196 | 299 | 92/5 | 6/5 | 8/5 | III
1288 | 690 | 36754 | 7501312 | 299 | 178802 | 92/5 | 3/5 | 11/5 | III
1295 | 185 | 56351 | 2855475 | 37 | 4921 | 37/2 | 21/16 | 3/2 | IV
1295 | 666 | 67414 | 6572125 | $\cdots$ | $\cdots$ | 37/2 | 1/2 | 37/16 | I
1316 | 188 | 62087 | 773808 | 4794 | $\cdots$ | 94/5 | 7/5 | 29/20 | II
1330 | 171 | 60895 | 2048200 | 19 | 627 | 19 | 11/8 | 3/2 | IV
1330 | 703 | 75259 | 8379000 | $\cdots$ | $\cdots$ | 19 | 1/2 | 19/8 | I
1365 | 156 | 65442 | 1098825 | 39 | 5083 | 39/2 | 23/16 | 3/2 | IV
1365 | 741 | 83772 | 10367175 | $\cdots$ | $\cdots$ | 39/2 | 1/2 | 39/16 | I
1400 | 80 | 46790 | 7056000 | 5 | 4 | 20 | 4/3 | 5/3 | V
1400 | 120 | 62630 | 3920000 | 4 | 13 | 20 | 7/5 | 8/5 | IV
1400 | 140 | 69950 | 0 | 5 | 5 | 20 | 3/2 | 3/2 | II
1400 | 728 | 106406 | 9878400 | 2 | 1 | 20 | 1/3 | 8/3 | V
1400 | 780 | 92990 | 12544000 | $\cdots$ | $\cdots$ | 20 | 1/2 | 5/2 | I
1400 | 890 | 55700 | 17640000 | 5 | 10 | 20 | 2/3 | 7/3 | V
1435 | 123 | 74374 | 1255625 | 5125 | 41 | 41/2 | 3/2 | 25/16 | IV
1435 | 820 | 102951 | 14916825 | $\cdots$ | $\cdots$ | 41/2 | 1/2 | 41/16 | I
1470 | 105 | 78666 | 2675400 | 637 | 21 | 21 | 3/2 | 13/8 | IV
1470 | 861 | 113694 | 17493000 | $\cdots$ | $\cdots$ | 21 | 1/2 | 21/8 | I
Table 15: Solutions to the $\mathbf{[3,0]}$ MLDE with $\mathbf{21<c\leq 26}$ $N$ | $m_{1}$ | $m_{2}$ | $k$ | $D_{1}$ | $D_{2}$ | $c$ | $h_{1}$ | $h_{2}$ | Category
---|---|---|---|---|---|---|---|---|---
1484 | 106 | 84429 | 1080352 | $\cdots$ | 15847 | 106/5 | 31/20 | 8/5 | II
1505 | 86 | 82775 | 4266675 | 5031 | 43 | 43/2 | 3/2 | 27/16 | IV
1505 | 903 | 125259 | 20279875 | $\cdots$ | $\cdots$ | 43/2 | 1/2 | 43/16 | I
1512 | 27 | 46386 | 10075968 | 459 | 42483 | 108/5 | 7/5 | 9/5 | III
1512 | 860 | 133851 | 17912832 | 833 | 3015426 | 108/5 | 2/5 | 14/5 | III
1512 | 1404 | 53730 | 46061568 | 459 | 153 | 108/5 | 4/5 | 12/5 | III
1540 | 66 | 86647 | 6036800 | 77 | 11 | 22 | 3/2 | 7/4 | III
1540 | 88 | 99935 | 1940400 | $\cdots$ | 22 | 22 | 19/12 | 5/3 | II
1540 | 946 | 137687 | 23284800 | $\cdots$ | $\cdots$ | 22 | 1/2 | 11/4 | I
1540 | 1298 | 98967 | 42257600 | 77 | 847 | 22 | 3/4 | 5/2 | III
1560 | 78 | 104754 | 3432000 | 5070 | 27170 | 156/7 | 11/7 | 12/7 | III
1560 | 1248 | 120874 | 40560000 | 130 | 799500 | 156/7 | 5/7 | 18/7 | III
1575 | 45 | 90225 | 7993125 | 4785 | 45 | 45/2 | 3/2 | 29/16 | IV
1575 | 990 | 151020 | 26515125 | $\cdots$ | $\cdots$ | 45/2 | 1/2 | 45/16 | I
1575 | 1640 | 99795 | 62346375 | 1595 | 956449 | 45/2 | 13/16 | 5/2 | III
1610 | 23 | 93449 | 10143000 | 575 | 23 | 23 | 3/2 | 15/8 | III
1610 | 69 | 131905 | 3155600 | $\cdots$ | 253 | 23 | 13/8 | 7/4 | II
1610 | 1035 | 165301 | 29978200 | $\cdots$ | $\cdots$ | 23 | 1/2 | 23/8 | I
1610 | 2323 | 100349 | 102557000 | 575 | 32683 | 23 | 7/8 | 5/2 | III
1624 | 58 | 136706 | 5183808 | 4959 | 1102 | 116/5 | 8/5 | 9/5 | III
1624 | 1711 | 146624 | 69981408 | 1653 | 910803 | 116/5 | 4/5 | 13/5 | III
1640 | 41 | 141122 | 7675200 | 4797 | 50922 | 164/7 | 11/7 | 13/7 | III
1645 | 0 | 96256 | 12493775 | 4371 | 47 | 47/2 | 3/2 | 31/16 | IV
1645 | 1081 | 180574 | 33681375 | $\cdots$ | $\cdots$ | 47/2 | 1/2 | 47/16 | I
1645 | 4371 | 100627 | 223103125 | 4371 | 1135003 | 47/2 | 15/16 | 5/2 | III
1652 | 59 | 164315 | 3954888 | $\cdots$ | 32509 | 118/5 | 33/20 | 9/5 | II
1680 | 1128 | 196884 | 37632000 | $\cdots$ | $\cdots$ | 24 | 1/2 | 3 | I
1715 | 1176 | 214277 | 41837425 | $\cdots$ | $\cdots$ | 49/2 | 1/2 | 49/16 | I
1750 | 1225 | 232800 | 46305000 | $\cdots$ | $\cdots$ | 25 | 1/2 | 25/8 | I
1785 | 1275 | 252501 | 51042075 | $\cdots$ | $\cdots$ | 51/2 | 1/2 | 51/16 | I
1820 | 1118 | 258869 | 39748800 | 117 | 3315 | 26 | 1/4 | 7/2 | III
1820 | 1326 | 273429 | 56056000 | $\cdots$ | $\cdots$ | 26 | 1/2 | 13/4 | I
Table 16: Solutions to the $\mathbf{[3,0]}$ MLDE with $\mathbf{26<c\leq 36}$ $N$ | $m_{1}$ | $m_{2}$ | $k$ | $D_{1}$ | $D_{2}$ | $c$ | $h_{1}$ | $h_{2}$ | Category
---|---|---|---|---|---|---|---|---|---
1848 | 1536 | 305286 | 73137792 | 2392 | 47018049 | 132/5 | 3/5 | 16/5 | III
1855 | 1378 | 295634 | 61354125 | $\cdots$ | $\cdots$ | 53/2 | 1/2 | 53/16 | I
1890 | 1431 | 319167 | 66943800 | $\cdots$ | $\cdots$ | 27 | 1/2 | 27/8 | I
1925 | 1485 | 344080 | 72832375 | $\cdots$ | $\cdots$ | 55/2 | 1/2 | 55/16 | I
1960 | 1540 | 370426 | 79027200 | $\cdots$ | $\cdots$ | 28 | 1/2 | 7/2 | I
1960 | 1948 | 424724 | 112896000 | 25 | 11 | 28 | 2/3 | 10/3 | V
1995 | 1596 | 398259 | 85535625 | $\cdots$ | $\cdots$ | 57/2 | 1/2 | 57/16 | I
2030 | 1653 | 427634 | 92365000 | $\cdots$ | $\cdots$ | 29 | 1/2 | 29/8 | I
2065 | 1711 | 458607 | 99522675 | $\cdots$ | $\cdots$ | 59/2 | 1/2 | 59/16 | I
2100 | 1770 | 491235 | 107016000 | $\cdots$ | $\cdots$ | 30 | 1/2 | 15/4 | I
2100 | 2778 | 683715 | 199214400 | 539 | 14421 | 30 | 3/4 | 7/2 | III
2135 | 1830 | 525576 | 114852325 | $\cdots$ | $\cdots$ | 61/2 | 1/2 | 61/16 | I
2135 | 3599 | 888770 | 279546225 | 47763 | 264580485 | 61/2 | 13/16 | 7/2 | III
2170 | 1891 | 561689 | 123039000 | $\cdots$ | $\cdots$ | 31 | 1/2 | 31/8 | I
2170 | 5239 | 1296885 | 440206200 | 9269 | 2295147 | 31 | 7/8 | 7/2 | III
2184 | 3612 | 958572 | 290293248 | 14877 | 250774426 | 156/5 | 4/5 | 18/5 | III
2205 | 1953 | 599634 | 131583375 | $\cdots$ | $\cdots$ | 63/2 | 1/2 | 63/16 | I
2240 | 2016 | 639472 | 140492800 | $\cdots$ | $\cdots$ | 32 | 1/2 | 4 | I
2268 | 4 | 113253 | 5882352 | 310124 | $\cdots$ | 162/5 | 11/5 | 47/20 | II
2275 | 2080 | 681265 | 149774625 | $\cdots$ | $\cdots$ | 65/2 | 1/2 | 65/16 | I
2296 | 0 | 90118 | 7971712 | 10168 | 615164 | 164/5 | 11/5 | 12/5 | IV
2310 | 3 | 86004 | 4998000 | 1105 | $\cdots$ | 33 | 9/4 | 19/8 | II
2310 | 2145 | 725076 | 159436200 | $\cdots$ | $\cdots$ | 33 | 1/2 | 33/8 | I
2345 | 2211 | 770969 | 169484875 | $\cdots$ | $\cdots$ | 67/2 | 1/2 | 67/16 | I
2360 | 0 | 63366 | 5852800 | 715139 | 848656 | 236/7 | 16/7 | 17/7 | IV
2380 | 1 | 58997 | 3439800 | 13 | $\cdots$ | 34 | 7/3 | 29/12 | II
2380 | 2278 | 819009 | 179928000 | $\cdots$ | $\cdots$ | 34 | 1/2 | 17/4 | I
2415 | 2346 | 869262 | 190772925 | $\cdots$ | $\cdots$ | 69/2 | 1/2 | 69/16 | I
2450 | 2415 | 921795 | 202027000 | $\cdots$ | $\cdots$ | 35 | 1/2 | 35/8 | I
2485 | 2485 | 976676 | 213697575 | $\cdots$ | $\cdots$ | 71/2 | 1/2 | 71/16 | I
2520 | 2556 | 1033974 | 225792000 | $\cdots$ | $\cdots$ | 36 | 1/2 | 9/2 | I
2520 | 3384 | 1337850 | 325987200 | 2 | 4 | 36 | 2/3 | 13/3 | V
Table 17: Solutions to the $\mathbf{[3,0]}$ MLDE with $\mathbf{36<c\leq 51}$ $N$ | $m_{1}$ | $m_{2}$ | $k$ | $D_{1}$ | $D_{2}$ | $c$ | $h_{1}$ | $h_{2}$ | Category
---|---|---|---|---|---|---|---|---|---
2555 | 2628 | 1093759 | 238317625 | $\cdots$ | $\cdots$ | 73/2 | 1/2 | 73/16 | I
2590 | 2701 | 1156102 | 251281800 | $\cdots$ | $\cdots$ | 37 | 1/2 | 37/8 | I
2625 | 2775 | 1221075 | 264691875 | $\cdots$ | $\cdots$ | 75/2 | 1/2 | 75/16 | I
2660 | 2850 | 1288751 | 278555200 | $\cdots$ | $\cdots$ | 38 | 1/2 | 19/4 | I
2695 | 2926 | 1359204 | 292879125 | $\cdots$ | $\cdots$ | 77/2 | 1/2 | 77/16 | I
2730 | 3003 | 1432509 | 307671000 | $\cdots$ | $\cdots$ | 39 | 1/2 | 39/8 | I
2765 | 3081 | 1508742 | 322938175 | $\cdots$ | $\cdots$ | 79/2 | 1/2 | 79/16 | I
2800 | 3160 | 1587980 | 338688000 | $\cdots$ | $\cdots$ | 40 | 1/2 | 5 | I
2835 | 3240 | 1670301 | 354927825 | $\cdots$ | $\cdots$ | 81/2 | 1/2 | 81/16 | I
2870 | 3321 | 1755784 | 371665000 | $\cdots$ | $\cdots$ | 41 | 1/2 | 41/8 | I
2905 | 3403 | 1844509 | 388906875 | $\cdots$ | $\cdots$ | 83/2 | 1/2 | 83/16 | I
2940 | 3486 | 1936557 | 406660800 | $\cdots$ | $\cdots$ | 42 | 1/2 | 21/4 | I
2975 | 3570 | 2032010 | 424934125 | $\cdots$ | $\cdots$ | 85/2 | 1/2 | 85/16 | I
3010 | 3655 | 2130951 | 443734200 | $\cdots$ | $\cdots$ | 43 | 1/2 | 43/8 | I
3045 | 3741 | 2233464 | 463068375 | $\cdots$ | $\cdots$ | 87/2 | 1/2 | 87/16 | I
3080 | 3146 | 1906130 | 372556800 | 13 | 19 | 44 | 1/3 | 17/3 | V
3080 | 3828 | 2339634 | 482944000 | $\cdots$ | $\cdots$ | 44 | 1/2 | 11/2 | I
3115 | 3916 | 2449547 | 503368425 | $\cdots$ | $\cdots$ | 89/2 | 1/2 | 89/16 | I
3150 | 4005 | 2563290 | 524349000 | $\cdots$ | $\cdots$ | 45 | 1/2 | 45/8 | I
3185 | 4095 | 2680951 | 545893075 | $\cdots$ | $\cdots$ | 91/2 | 1/2 | 91/16 | I
3220 | 4186 | 2802619 | 568008000 | $\cdots$ | $\cdots$ | 46 | 1/2 | 23/4 | I
3255 | 4278 | 2928384 | 590701125 | $\cdots$ | $\cdots$ | 93/2 | 1/2 | 93/16 | I
3290 | 4371 | 3058337 | 613979800 | $\cdots$ | $\cdots$ | 47 | 1/2 | 47/8 | I
3325 | 4465 | 3192570 | 637851375 | $\cdots$ | $\cdots$ | 95/2 | 1/2 | 95/16 | I
3360 | 4560 | 3331176 | 662323200 | $\cdots$ | $\cdots$ | 48 | 1/2 | 6 | I
$3395$ | $4656$ | $3474249$ | $687402625$ | $\cdots$ | $\cdots$ | 97/2 | 1/2 | 97/16 | I
$3430$ | $4753$ | $3621884$ | $713097000$ | $\cdots$ | $\cdots$ | 49 | 1/2 | 49/8 | I
$3465$ | $4851$ | $3774177$ | $739413675$ | $\cdots$ | $\cdots$ | 99/2 | 1/2 | 99/16 | I
$3500$ | $4950$ | $3931225$ | $766360000$ | $\cdots$ | $\cdots$ | 50 | 1/2 | 25/4 | I
$3535$ | $5050$ | $4093126$ | $793943325$ | $\cdots$ | $\cdots$ | 101/2 | 1/2 | 101/16 | I
$3570$ | $5151$ | $4259979$ | $822171000$ | $\cdots$ | $\cdots$ | 51 | 1/2 | 51/8 | I
Table 18: Solutions to the $\mathbf{[3,0]}$ MLDE with $\mathbf{51<c\leq 56}$ $N$ | $m_{1}$ | $m_{2}$ | $k$ | $D_{1}$ | $D_{2}$ | $c$ | $h_{1}$ | $h_{2}$ | Category
---|---|---|---|---|---|---|---|---|---
$3605$ | $5253$ | $4431884$ | $851050375$ | $\cdots$ | $\cdots$ | 103/2 | 1/2 | 103/16 | I
$3640$ | $5356$ | $4608942$ | $880588800$ | $\cdots$ | $\cdots$ | 52 | 1/2 | 13/2 | I
$3675$ | $5460$ | $4791255$ | $910793625$ | $\cdots$ | $\cdots$ | 105/2 | 1/2 | 105/16 | I
$3710$ | $5565$ | $4978926$ | $941672200$ | $\cdots$ | $\cdots$ | 53 | 1/2 | 53/8 | I
$3745$ | $5671$ | $5172059$ | $973231875$ | $\cdots$ | $\cdots$ | 107/2 | 1/2 | 107/16 | I
$3780$ | $5778$ | $5370759$ | $1005480000$ | $\cdots$ | $\cdots$ | 54 | 1/2 | 27/4 | I
$3815$ | $5886$ | $5575132$ | $1038423925$ | $\cdots$ | $\cdots$ | 109/2 | 1/2 | 109/16 | I
$3850$ | $5995$ | $5785285$ | $1072071000$ | $\cdots$ | $\cdots$ | 55 | 1/2 | 55/8 | I
3864 | 13110 | 13725282 | 2591631168 | 12971091 | 4897835680923668 | 276/5 | 4/5 | 33/5 | III
$3885$ | $6105$ | $6001326$ | $1106428575$ | $\cdots$ | $\cdots$ | 111/2 | 1/2 | 111/16 | I
$3920$ | $6216$ | $6223364$ | $1141504000$ | $\cdots$ | $\cdots$ | 56 | 1/2 | 7 | I
Table 19: Solutions to the $\mathbf{[3,2]}$ MLDE with $\mathbf{0<c\leq 96}$ $N$ | $m_{1}$ | $m_{2}$ | $k$ | $D_{1}$ | $D_{2}$ | $c$ | $h_{1}$ | $h_{2}$ | CFT
---|---|---|---|---|---|---|---|---|---
$84$ | $1$ | $1$ | $737352$ | $\cdots$ | $\cdots$ | 2/5 | 1/60 | 1/5 | ${\bf\mathcal{M}(5,2)}\oplus{\rm constant}$
$210$ | $3$ | $4$ | $2058000$ | $\cdots$ | $\cdots$ | $1$ | 1/24 | 1/4 | $(\mathbf{\hat{A}_{1}})_{1}\oplus{\rm constant}$
$420$ | $8$ | $17$ | $4762800$ | $\cdots$ | $\cdots$ | $2$ | $1/12$ | ${1}/{3}$ | $(\mathbf{\hat{A}_{2}})_{1}\oplus{\rm constant}$
$588$ | $14$ | $42$ | $7277088$ | $\cdots$ | $\cdots$ | ${14}/{5}$ | ${7}/{60}$ | ${2}/{5}$ | $(\mathbf{\hat{G}_{2}})_{1}\oplus{\rm constant}$
$840$ | $28$ | $134$ | $11289600$ | $\cdots$ | $\cdots$ | $4$ | ${1}/{6}$ | ${1}/{2}$ | $(\mathbf{\hat{D}_{4}})_{1}\oplus{\rm constant}$
$1092$ | $52$ | $377$ | $14768208$ | $\cdots$ | $\cdots$ | ${26}/{5}$ | ${13}/{60}$ | ${3}/{5}$ | $(\mathbf{\hat{F}_{4}})_{1}\oplus{\rm constant}$
$1260$ | $78$ | $729$ | $15876000$ | $\cdots$ | $\cdots$ | $6$ | ${1}/{4}$ | ${2}/{3}$ | $(\mathbf{\hat{E}_{6}})_{1}\oplus{\rm constant}$
$1470$ | $133$ | $1673$ | $13582800$ | $\cdots$ | $\cdots$ | $7$ | ${7}/{24}$ | ${3}/{4}$ | $(\mathbf{\hat{E}_{7}})_{1}\oplus{\rm constant}$
$1596$ | $190$ | $2831$ | $7775712$ | $\cdots$ | $\cdots$ | ${38}/{5}$ | ${19}/{60}$ | ${4}/{5}$ | $\mathbf{E_{7\frac{1}{2}}}\oplus{\rm constant}$

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