# Results of (3,0) GHM Duals

Arpit Das  Chethan N. Gowdigere  and Sunil Mukhi

# New meromorphic CFTs from cosets

Arpit Das  Chethan N. Gowdigere  and Sunil Mukhi

###### Abstract

In recent years it has been understood that new rational CFTs can be
discovered by applying the coset construction to meromorphic CFTs. Here we
turn this approach around and show that the coset construction, together with
the classification of meromorphic CFT with $c\leq 24$, can be used to predict
the existence of new meromorphic CFTs with $c\geq 32$ whose Kac-Moody algebras
are non-simply-laced and/or at levels greater than 1. This implies they are
non-lattice theories. Using three-character coset relations, we propose 34
infinite series of meromorphic theories with arbitrarily large central charge,
as well as 46 theories at $c=32$ and $c=40$.

##  1 Introduction

A rational 2d conformal field theory (RCFT) is characterised by having a
finite set of holomorphic characters $\chi_{i}(\tau)$ and a partition function
of the form:

$\mathcal{Z}(\tau,{\bar{\tau}})=\sum_{i,j=0}^{n-1}M_{ij}\,{\overline{\chi}}_{i}({\bar{\tau}})\chi_{j}(\tau)$
(1.1)

where $n$ is the total number of linearly independent characters of some
chiral algebra.

A classification programme for rational conformal field theories in 2d was
initiated in [1, 2]. It is based on the fact that characters are vector-valued
forms that solve a modular linear differential equation (MLDE) whose order is
the number of characters. Such equations have finitely many parameters and
these can be varied to scan for solutions that satisfy the basic criteria to
be those of a conformal field theory. We will refer to such solutions as
“admissible characters”. From the study of MLDE it emerged that an important
classifier for RCFT with a given number of characters is an integer $\ell\geq
0,\ell\neq 1$ called the Wronskian index (for a detailed review, see [3]).
Admissible characters for bosonic CFTs have been constructed in [4, 5, 6, 7,
8, 9, 10, 11, 12] and for fermionic CFTs in [13, 14, 15].

Admissible characters do not, in general, correspond to any RCFT. For
meromorphic theories (those with a single character, $n=1$) this is
particularly obvious. At $c=24$ there is an infinite set of admissible
characters as we will see below, but only a finite subset corresponds to a CFT
as shown in the seminal work of Schellekens [16]. Moreover a given character
in this finite set sometimes corresponds to multiple different CFTs. That an
analogous phenomenon holds for theories with multiple characters was made
explicit in recent times [7, 8, 9] with the discovery of infinite families of
admissible characters depending on unbounded integers.

The classification of RCFTs for a fixed number of characters and Wronskian
index thus requires two steps: the classification of admissible characters,
and the sub-classification of those that describe genuine RCFT. Both steps are
easiest for small values of $n$ as well as the Wronskian index $\ell$, because
in this case the characters are completely determined by their critical
exponents [2, 5]. Given a set of admissible characters, a useful method to
find corresponding CFT comes from the coset construction of [17, 18] as
implemented for meromorphic numerator theories in [19]111See [20, 21] for
earlier discussions of meromorphic cosets.. This works as follows. We pick a
known meromorphic CFT $\cal H$ as well as a known RCFT ${\cal C}$ with $n$
characters and suitable exponents, such that the coset:

${\tilde{\cal C}}=\frac{\cal H}{\cal C}$ (1.2)

obtained by embedding the Kac-Moody algebra of ${\cal C}$ in that of ${\cal
H}$ corresponds to the given admissible characters. Whenever this can be done,
we may conclude that these characters describe the CFT ${\tilde{\cal C}}$.
This approach has been implemented in [19, 22, 23, 24] where ${\cal H}$ is a
known CFT at $c=8,16,24$ (these are completely classified) or a lattice theory
at $c=32$ (these are too many to classify but many lattice theories are known
[25, 26]). In particular, in [19] this procedure was first used to construct
the CFTs’ corresponding to admissible characters with $(n,\ell)=(2,2)$ that
had originally been found nearly three decades earlier [4] but had not been
previously identified with CFTs.

Let us briefly review some basic aspects of the meromorphic coset relation
(more details can be found in [19, 10]). The numerator theory ${\cal H}$ is
typically an extension of a non-simple Kac-Moody algebra $\oplus_{i}{\cal
G}_{r_{i},k_{i}}$ by higher-spin generators that organise a subset of Kac-
Moody characters into a single character. We denote these theories by ${\cal
E}_{1}[\oplus_{i}{\cal G}_{r_{i},k_{i}}]$ where $\cal E$ stands for
“extension” and the subscript 1 tells us that the extended theory has a single
character. There are broadly two types of meromorphic theories: those
corresponding to free bosons on an even, self-dual lattice, for which the Kac-
Moody algebra only contains simply-laced factors
($A_{r},D_{r},E_{6},E_{7},E_{8}$) at level 1, and the rest, which we call non-
lattice theories. These are characterised by the presence of non-simply-laced
factors ($B_{r},C_{r},F_{4},G_{2}$) and/or levels greater than 1. Some non-
lattice theories can be derived as orbifolds of lattice theories, while others
are more complicated to construct [27, 16, 28]. The denominator theories
${\cal C}$, at least in the original references [19, 29, 8], are taken to be
affine theories belonging to a current-algebra minimal series [30] or
occasionally the Virasoro minimal series ${\cal M}(p,q)$ [31]. The coset
theories ${\tilde{\cal C}}$ again typically have non-simple Kac-Moody algebras
that are extended by other chiral generators so that they have a smaller
number $n$ of characters than the affine theories for the same algebras.
Following the notation introduced above for meromorphic theories, we denote
these by ${\cal E}_{n}[\oplus_{i}{\cal G}_{r_{i},k_{i}}]$ where the subscript
$n$ denotes the number of characters.

If the characters of ${\cal C}$ are denoted $\chi_{i}(\tau)$, those of
${\tilde{\cal C}}$ are denoted ${\tilde{\chi}}_{i}$ and the single character
of $\cal H$ is denoted $\chi^{\cal H}$, then the coset relation is embodied in
a holomorphic bilinear relation:

$\sum_{i=0}^{n-1}d_{i}\,\chi_{i}(\tau)\,{\tilde{\chi}}_{i}(\tau)=\chi^{\cal
H}(\tau)$ (1.3)

where $d_{i}$ are positive integers and $d_{0}=1$.

When both ${\cal H}$ and $\cal C$ correspond to CFTs having a Sugawara stress
tensor in terms of Kac-Moody currents, then by embedding the currents of $\cal
C$ in those of ${\cal H}$ one defines the stress tensor of the coset theory
$\tilde{\cal C}$. At a physics level of rigour, we take this to be a proof
that ${\cal C}$ is a genuine CFT [19]. But it is also possible for Eq. (1.3)
to be satisfied when one or both of ${\cal C},{\tilde{\cal C}}$ does not have
any Kac-Moody currents. Also there are cases where none of ${\cal
C},{\tilde{\cal C}},{\cal H}$ has Kac-Moody currents (for example, ${\cal H}$
may correspond to the Monster module [32, 33]). In such cases the coset
construction of [19] is more tricky to apply, since without a Sugawara
construction we do not have an explicit expression for the stress tensor of
${\cal C}$. At the same time, the bilinear relation can certainly be verified
as easily as for the Sugawara case. So it is compelling to believe that, even
in the absence of Sugawara stress tensors, if the bilinear relation holds and
${\cal H}$ and ${\cal C}$ are both CFTs, then so is ${\tilde{\cal C}}$. One
such example [29] arises when ${\cal C}$ is the Ising model and ${\tilde{\cal
C}}$ is the Baby Monster CFT [34].

As described above, the coset relation Eq. (1.2) has been used to find
interesting theories ${\tilde{\cal C}}$ given meromorphic theories ${\cal H}$
and known denominator CFTs ${\cal C}$ to divide them by. As input, this method
has relied on known classifications of meromorphic theories at $c=8,16,24$
[27, 16] as well as special classes of lattices at $c=32$ [25, 26] to find new
$n$-character CFTs. However, if ${\cal C}$ and ${\tilde{\cal C}}$ are both
known to describe CFTs then this relation can be used in the other direction:
to argue that ${\cal H}$ also corresponds to a CFT. Thus, in principle new
meromorphic theories can be found in this way.

In the present work we propose a practical way to carry this out. The key idea
is to first generate an extension-type CFT ${\tilde{\cal C}}$ using the coset
construction, with ${\cal H}$ being a known meromorphic theory in the
Schellekens list [16] and ${\cal C}$ being a suitable affine theory embedded
in it. Next, we remove ${\cal C}$ from the story and find bilinear relations
where ${\tilde{\cal C}}$ is paired with the characters of another known CFT
${\cal C}^{\prime}$. For $n=3$, large numbers of such coset pairs were
generated in [10]. Since ${\cal C}^{\prime}$ is a different theory from ${\cal
C}$ (and not a tensor product of ${\cal C}$ with something else), the result
is a new meromorphic theory at a different central charge.

The affine algebra of the new meromorphic theory is the sum of algebras of
${\cal C}^{\prime}$ and ${\tilde{\cal C}}$, denoted
$\mathcal{E}_{1}[\mathcal{C}^{\prime}\oplus\tilde{\mathcal{C}}]$, of the
multi-character theory $\mathcal{C}^{\prime}\oplus\tilde{\mathcal{C}}$. This
extension arises from the bilinear relation satisfied by ${\cal C}^{\prime}$
and ${\tilde{\cal C}}$. In each case we explicitly find the character of
${\cal H}$. In most cases it will turn out that ${\cal
C}^{\prime}\oplus{\tilde{\cal C}}$ is non-simply-laced and/or has factors of
level $>1$, showing that it must be a non-lattice theory.

Using the above approach we predict the existence of 34 infinite families of
novel meromorphic theories with central charges $c^{\mathcal{H}}=8(m+3)$ where
$m$ is an integer $\geq 1$. Every such infinite family starts with a member at
$c^{\mathcal{H}}=24$ that can be found in Schellekens’ list [16]. We also
predict the existence of 46 novel meromorphic theories with central charges
$c^{\mathcal{H}}=32$ and $c^{\mathcal{H}}=40$. We now turn to a description of
the method and then present our results.

##  2 Background

We start by briefly summarising the admissible characters for meromorphic
theories at $c^{\mathcal{H}}=8N$ where $N$ is an integer $\geq 1$. From the
theory of modular forms (see for example [35]) we know that any polynomial
222Actually any rational function of $j(q)$ would be invariant, but we are
specialising to polynomials as we want these functions to be holomorphic in
$\mathbb{H}\setminus\\{i\infty\\}$. of the Klein $j$-invariant $j(q)$ is a
modular invariant function. The $j$-invariant has the expansion:

$j(q)=q^{-1}+744+196884q+\cdots$ (2.1)

If we are willing to accept modular invariance up to a phase then the
polynomial can be multiplied by $j(q)^{\frac{1}{3}}$ or $j(q)^{\frac{2}{3}}$.
Thus the most general modular invariant is of the form:

$\chi^{\cal
H}(\tau)=j^{\frac{N}{3}-s}(j^{s}+a_{1}\,j^{s-1}+a_{2}\,j^{s-2}+\ldots\ldots+a_{s}),$
(2.2)

where $s:=\left\lfloor\frac{N}{3}\right\rfloor$.

For $c^{\cal H}=24,32,40$, the corresponding characters are of the form:

$\displaystyle c^{\mathcal{H}}=24:\,\,\,\chi^{\cal H}(\tau)=j+{\cal N},$ (2.3)
$\displaystyle c^{\mathcal{H}}=32:\,\,\,\chi^{\cal
H}(\tau)=j^{\frac{1}{3}}(j+{\cal N})$ (2.4) $\displaystyle
c^{\mathcal{H}}=40:\,\,\,\chi^{\cal H}(\tau)=j^{\frac{2}{3}}(j+{\cal N}),$
(2.5)

where we have renamed $a_{1}$ as ${\cal N}$. In the $c^{\cal H}=24$ case we
see that all values of ${\cal N}\geq-744$ are admissible, but it is known that
only a finite subset correspond to CFTs. [16].

For $n$-character RCFT, the characters are vector-valued modular forms that
have the general form:

$\chi_{i}(q)=q^{-\frac{c}{24}}(a_{i,0}+a_{i,1}q+a_{i,2}q^{2}+\cdots),~{}~{}i=0,1,\cdots,n-1$
(2.6)

For the characters to be admissible, the coefficients $a_{i,r}$ must be non-
negative integers. Moreover we must have $a_{0,0}=1$ (non-degeneracy of the
identity character). We also define
$D_{i}=a_{i,0},m_{1}=a_{0,1},m_{2}=a_{0,2}$. These are, respectively, the
ground-state degeneracy of each of the generalised primaries, the number of
spin-1 currents and the number of spin-2 currents.

For $n=2$ there is an explicit and complete classification of all admissible
characters [8]. However for $n\geq 3$ the complete classification remains an
open problem. Nevertheless, admissible characters can be found by writing a
general Modular Linear Differential Equation of the form:

$\big{(}D^{n}+\phi_{2}(\tau)D^{n-1}+\cdots+\phi_{2n}(\tau)\big{)}\chi=0$ (2.7)

where $D$ is a covariant derivative on torus moduli space and
$\phi_{2j}(\tau)$ are meromorphic modular functions of weight $2j$. The
maximum number of poles of the $\phi_{2r}$ is called the Wronskian index
$\ell$. Consider the above equation for a fixed order $n$ and a fixed value of
$\ell$. In this case the set of $\phi_{2j}$ depends on a finite number of
parameters, and one scans the parameter space to find those values where the
coefficients $a_{i,r}$ satisfy the admissibility criteria. Of relevance in the
present work will be solutions for $n=3$ and $\ell=0$, in which case the
equation becomes:

$\big{(}D^{3}+\mu_{1}E_{4}(\tau)D+\mu_{2}E_{6}(\tau)\big{)}\chi=0$ (2.8)

While this equation was first studied long ago in [5], all its admissible
solutions with $c\leq 96$ have been classified only recently [10] and we will
make use of some of these below (all these relevant solutions are given in
table LABEL:t6 in Sec. B).

Next we summarise some results about the coset construction. In [19] the
following relation between the central charge $c$ and conformal dimensions
$h_{i}$ of the characters $\chi_{i}$, and the corresponding quantities
${\tilde{c}},{\tilde{h}}_{i}$ of the coset dual ${\tilde{\chi}}_{i}$ was
derived:

$\ell+{\tilde{\ell}}=n^{2}+\left(\frac{c+{\tilde{c}}}{4}-1\right)n-6\sum_{i=1}^{n-1}(h_{i}+{\tilde{h}}_{i})$
(2.9)

We will be interested in the case $n=3$. Also, note that $c+{\tilde{c}}$ must
be a multiple of 8, so we write it as $8N$ where $N$ is an integer $\geq 1$
333In [19] the convention was to write $c+{\tilde{c}}=24N$ where $N$ is a
multiple of $\frac{1}{3}$.. Also, since the right hand side of the bilinear
relation is a character with integer dimensions, we must have
$h_{i}+{\tilde{h}}_{i}=n_{i}$, an integer $\geq 1$, for each $i$. Thus, for
$n=3$ the relation can be written:

$\ell+{\tilde{\ell}}=6\Big{(}N+1-\sum_{i=1}^{2}n_{i}\Big{)}$ (2.10)

Let us comment on the significance of the integers $n_{i}$. These denote the
new chiral currents, of spin $n_{i}$, created by “fusing” the primaries of two
RCFTs ${\cal C},{\tilde{\cal C}}$ via a bilinear relation into a meromorphic
theory ${\cal H}$. Whenever each $n_{i}\geq 2$, we have a special situation
where the only new currents in ${\cal H}$, relative to ${\cal
C}\oplus{\tilde{\cal C}}$, have spin $\geq 2$. In such cases the Kac-Moody
algebra of ${\cal H}$ is necessarily the direct sum of that of ${\cal
C},{\tilde{\cal C}}$. On the other hand if any $n_{i}=1$, we have new Kac-
Moody currents in ${\cal H}$ that were not present in the coset pair. Viewed
from the converse perspective starting with the meromorphic theory, the first
case $n_{i}\geq 2$ arises when the denominator theory ${\cal C}$ is an affine
theory for one (or more) of the simple factors in ${\cal H}$. In that case the
coset procedure just deletes the factor(s) and the remaining ones make up the
Kac-Moody algebra of ${\tilde{\cal C}}$. But in the second case, the algebra
of ${\cal C}$ undergoes a non-trivial embedding in one of the simple factors
of ${\cal H}$. This “Higgses” that factor down to a subalgebra given by the
commutant, and some currents are lost in the process. Only the first case
$n_{i}\geq 2$ will be of relevance in the present work, while the other case
involving embeddings will be discussed in [24, 23].

From now on we focus on coset dual pairs of characters with $n=3$ and
$\ell,{\tilde{\ell}}=0,0$. From the above relation it follows that
$n_{1}+n_{2}=N+1$. While the classification of admissible $(3,0)$ characters
has not been completed, significant progress has been made in [5, 36, 37, 19,
10]. We will make use of the results in [10], which subsumes the output of
most of the previous works and provides a complete set of admissible three-
character sets with $c\leq 96$ 444In an upcoming work [24] we re-examine this
partial classification of admissible characters and attempt to identify which
of them can be shown to exist as CFTs as well as which ones can be shown not
to correspond to any CFT..

In the notation of [10], to which we refer the reader for more details, the
admissible character sets fall into five categories, labelled
$\mathbf{I},\mathbf{II},\cdots,\mathbf{V}$. Of these, all entries in
categories $\mathbf{I},\mathbf{II},\mathbf{IV}$ have already been identified
as CFTs in [10] 555with a few exceptions that correspond to generalisations of
CFTs such as Intermediate Vertex Operator Algebras (IVOA) [38].. Among these
are characters that were previously identified as CFTs in [19] and which will
play a role in the present work. We will label them GHMD where the subscript
$D$ denotes the dimension of their Kac-Moody algebra (see all GHM solutions
listed in table LABEL:t6 in Sec. B). We will also make use of characters of
type $\mathbf{III}$ and $\mathbf{V}$ which were not identified with CFTs in
[10]. These will be studied in complete detail in work to appear [24]. The
list of relevant ${\bf III}$ and ${\bf V}$ solutions can be found in table
LABEL:t6 of Appendix B.

##  3 Constructing new meromorphic CFTs

###  3.1 $(3,0)$ cosets from $c=24$ meromorphic theories

We start by using the coset construction with ${\cal H}$ being one of the
meromorphic theories in Schellekens’ list, to identify 22 sets of admissible
characters as CFTs. Table LABEL:t0 shows coset pairings of characters
$\chi_{i},{\tilde{\chi}}_{i}$ to make one of these meromorphic theories. The
purpose of this exercise is to identify the theories $\tilde{\mathcal{C}}$
that we will use later on.

The entries in the table are as follows. The first column is a serial number
labelling the 22 cases of interest. The next four columns tell us the
properties of an affine (or minimal) model ${\cal C}$ that we use as the
denominator in the coset relation. Respectively, they provide the central
charge, conformal dimensions, dimension $m_{1}$ of the Kac-Moody algebra, and
the Kac-Moody algebra itself. The next four columns provide the same
properties for a coset theory ${\tilde{\cal C}}$ that combines with ${\cal C}$
in a bilinear relation that has been verified. The last four columns tell us
the Kac-Moody algebra of the meromorphic Schellekens theory, the integers
$(d_{1},d_{2})$ appearing in the bilinear relation Eq. (1.3), the integer
${\cal N}$ that specifies the character via $\chi^{\cal H}(\tau)=j(\tau)+{\cal
N}$, and the serial number of the corresponding theory in the table of [16].
Note that the dimension of the Kac-Moody algebra of the meromorphic theory is
${\cal N}+744$.

As can be seen in the Table, rows $4-9,13-17,22$ are cases where
$\tilde{\mathcal{C}}$ is of GHM type [19]. Among the rest, rows
$1,2,10-12,18-21$ were missed in [19] for various reasons – for example that
reference did not consider coset duals where ${\cal C}$ is a tensor product of
two identical two-character theories. Note that despite this, ${\tilde{\cal
C}}$ is not a tensor product of simpler theories. Finally, row 3 is the Ising-
Baby Monster pairing implicit in [34] and discussed in the present context in
[29]. This is the only known case of a $c=24$ coset where one or both (in this
case, both) of the entries have no Kac-Moody algebra. The coset relation must
then be understood in the more general sense mentioned below Eq. (1.3), and
the corresponding meromorphic theory is the Monster CFT. We will soon see that
for $c\geq 32$, the Baby Monster theory features in coset pairings with affine
theories.

Table 1: Coset relations for $c^{\mathcal{H}}=24$ with $(n_{1},n_{2})=(2,2)$. We identify $\mathcal{M}(4,3)\cong B_{0,1}$, $A_{1,2}\cong B_{1,1}$, $C_{2,1}\cong B_{2,1}$ and BM $\equiv$ Baby Monster. We further identify $U(1)\cong D_{1,1}$, $A_{1,1}^{\oplus 2}\cong D_{2,1}$ and $A_{3,1}\cong D_{3,1}$. # | $c$ | $(h_{1},h_{2})$ | $m_{1}$ | $\mathcal{C}$ | $\tilde{c}$ | $(\tilde{h}_{1},\tilde{h}_{2})$ | $\tilde{m}_{1}$ | $\tilde{\mathcal{C}}$ | Affine | $(d_{1},d_{2})$ | $\mathcal{N}$ | Sch.
---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  | algebra |  |  | #
1. | $12$ | $(\frac{1}{2},\frac{3}{2})$ | $276$ | $D_{12,1}$ | $12$ | $(\frac{3}{2},\frac{1}{2})$ | $276$ | $D_{12,1}$ | $D_{12,1}^{\oplus 2}$ | $2$ | 552 | 66
2. | $12$ | $(\frac{2}{3},\frac{4}{3})$ | $156$ | $E_{6,1}^{\oplus 2}$ | $12$ | $(\frac{4}{3},\frac{2}{3})$ | $156$ | $E_{6,1}^{\oplus 2}$ | $E_{6,1}^{\oplus 4}$ | $8$ | 312 | 58
3. | $\frac{1}{2}$ | $(\frac{1}{2},\frac{1}{16})$ | $0$ | $\mathcal{M}(4,3)$ | $\frac{47}{2}$ | $(\frac{3}{2},\frac{31}{16})$ | $0$ | BM | — | $(1,1)$ | 0 | 0
4. | $\frac{3}{2}$ | $(\frac{1}{2},\frac{3}{16})$ | $3$ | $A_{1,2}$ | $\frac{45}{2}$ | $(\frac{3}{2},\frac{29}{16})$ | $45$ | $\text{GHM}_{45}$ | $A_{1,2}^{\oplus 16}$ | $(1,1024)$ | 48 | 5
|  |  |  |  |  |  |  |  | $A_{3,4}^{\oplus 3}A_{1,2}$ |  |  | 7
|  |  |  |  |  |  |  |  | $A_{5,6}C_{2,3}A_{1,2}$ |  |  | 8
|  |  |  |  |  |  |  |  | $D_{5,8}A_{1,2}$ |  |  | 10
5. | $\frac{5}{2}$ | $(\frac{1}{2},\frac{5}{16})$ | $10$ | $C_{2,1}$ | $\frac{43}{2}$ | $(\frac{3}{2},\frac{27}{16})$ | $86$ | $\text{GHM}_{86}$ | $D_{4,2}^{\oplus 2}C_{2,1}^{\oplus 4}$ | $(1,512)$ | 96 | 25
|  |  |  |  |  |  |  |  | $A_{5,2}^{\oplus 2}A_{2,1}^{\oplus 2}C_{2,1}$ |  |  | 26
|  |  |  |  |  |  |  |  | $E_{6,4}A_{2,1}C_{2,1}$ |  |  | 28
6. | $\frac{7}{2}$ | $(\frac{1}{2},\frac{7}{16})$ | $21$ | $B_{3,1}$ | $\frac{41}{2}$ | $(\frac{3}{2},\frac{25}{16})$ | $123$ | $\text{GHM}_{123}$ | $B_{3,1}D_{6,2}C_{4,1}B_{3,1}$ | $(1,256)$ | 144 | 39
|  |  |  |  |  |  |  |  | $B_{3,1}A_{9,2}A_{4,1}$ |  |  | 40
7. | $\frac{9}{2}$ | $(\frac{1}{2},\frac{9}{16})$ | $36$ | $B_{4,1}$ | $\frac{39}{2}$ | $(\frac{3}{2},\frac{23}{16})$ | $156$ | $\text{GHM}_{156}$ | $D_{8,2}B_{4,1}^{\oplus 2}$ | $(1,128)$ | 192 | 47
|  |  |  |  |  |  |  |  | $B_{4,1}C_{6,1}^{\oplus 2}$ |  |  | 48
8. | $\frac{11}{2}$ | $(\frac{1}{2},\frac{11}{16})$ | $55$ | $B_{5,1}$ | $\frac{37}{2}$ | $(\frac{3}{2},\frac{21}{16})$ | $185$ | $\text{GHM}_{185}$ | $B_{5,1}E_{7,2}F_{4,1}$ | $(1,1)$ | 240 | 53
9. | $\frac{13}{2}$ | $(\frac{1}{2},\frac{13}{16})$ | $78$ | $B_{6,1}$ | $\frac{35}{2}$ | $(\frac{3}{2},\frac{19}{16})$ | $210$ | $\text{GHM}_{210}$ | $B_{6,1}C_{10,1}$ | $(1,32)$ | 288 | 56
10. | $\frac{17}{2}$ | $(\frac{1}{2},\frac{17}{16})$ | $136$ | $B_{8,1}$ | $\frac{31}{2}$ | $(\frac{3}{2},\frac{15}{16})$ | $248$ | $E_{8,2}$ | $B_{8,1}E_{8,2}$ | $(1,1)$ | 384 | 62
11. | $1$ | $(\frac{1}{2},\frac{1}{8})$ | $1$ | $U(1)$ | $23$ | $(\frac{3}{2},\frac{15}{8})$ | $23$ | ${\bf III_{50}}$ | $U(1)^{\oplus 24}$ | $(8,4096)$ | 24 | 1
12. | $2$ | $(\frac{1}{2},\frac{1}{4})$ | $6$ | $A_{1,1}^{\oplus 2}$ | $22$ | $(\frac{3}{2},\frac{7}{4})$ | $66$ | ${\bf III_{45}}$ | $A_{1,1}^{\oplus 24}$ | $(64,4096)$ | 72 | 15
|  |  |  |  |  |  |  |  | $A_{3,2}^{\oplus 4}A_{1,1}^{\oplus 4}$ |  |  | 16
|  |  |  |  |  |  |  |  | $A_{5,3}D_{4,3}A_{1,1}^{\oplus 3}$ |  |  | 17
|  |  |  |  |  |  |  |  | $A_{7,4}A_{1,1}^{\oplus 3}$ |  |  | 18
|  |  |  |  |  |  |  |  | $D_{5,4}C_{3,2}A_{1,1}^{\oplus 2}$ |  |  | 19
|  |  |  |  |  |  |  |  | $D_{6,5}A_{1,1}^{\oplus 2}$ |  |  | 20
13. | $3$ | $(\frac{1}{2},\frac{3}{8})$ | $15$ | $A_{3,1}$ | $21$ | $(\frac{3}{2},\frac{13}{8})$ | $105$ | $\text{GHM}_{105}$ | $A_{3,1}^{\oplus 8}$ | $(8,1024)$ | 120 | 30
|  |  |  |  |  |  |  |  | $D_{5,2}^{\oplus 2}A_{3,1}^{\oplus 2}$ |  |  | 31
|  |  |  |  |  |  |  |  | $A_{7,2}C_{3,1}^{\oplus 2}A_{3,1}$ |  |  | 33
|  |  |  |  |  |  |  |  | $D_{7,3}G_{2,1}A_{3,1}$ |  |  | 34
|  |  |  |  |  |  |  |  | $C_{7,2}A_{3,1}$ |  |  | 35
14. | $5$ | $(\frac{1}{2},\frac{5}{8})$ | $45$ | $D_{5,1}$ | $19$ | $(\frac{3}{2},\frac{11}{8})$ | $171$ | $\text{GHM}_{171}$ | $D_{5,1}^{\oplus 2}\,A_{7,1}^{\oplus 2}$ | $(8,256)$ | 216 | 49
15. | $6$ | $(\frac{1}{2},\frac{3}{4})$ | $66$ | $D_{6,1}$ | $18$ | $(\frac{3}{2},\frac{5}{4})$ | $198$ | $\text{GHM}_{198}$ | $D_{6,1}^{\oplus 4}$ | $(64,256)$ | 264 | 54
|  |  |  |  |  |  |  |  | $D_{6,1}\,A_{9,1}^{\oplus 2}$ |  |  | 55
16. | $7$ | $(\frac{1}{2},\frac{7}{8})$ | $91$ | $D_{7,1}$ | $17$ | $(\frac{3}{2},\frac{9}{8})$ | $221$ | $\text{GHM}_{221}$ | $D_{7,1}\,A_{11,1}\,E_{6,1}$ | $(8,64)$ | 312 | 59
17. | $9$ | $(\frac{1}{2},\frac{9}{8})$ | $153$ | $D_{9,1}$ | $15$ | $(\frac{3}{2},\frac{7}{8})$ | $255$ | $\text{GHM}_{255}$ | $D_{9,1}\,A_{15,1}$ | $(8,16)$ | 408 | 63
18. | $10$ | $(\frac{1}{2},\frac{5}{4})$ | $190$ | $D_{10,1}$ | $14$ | $(\frac{3}{2},\frac{3}{4})$ | $266$ | $E_{7,1}^{\oplus 2}$ | $D_{10,1}E_{7,1}^{\oplus 2}$ | $(1,2)$ | 456 | 64
19. | $4$ | $(\frac{1}{3},\frac{2}{3})$ | $16$ | $A_{2,1}^{\oplus 2}$ | $20$ | $(\frac{5}{3},\frac{4}{3})$ | $80$ | ${\bf V_{39}}$ | $A_{2,1}^{\oplus 12}$ | $(8748,972)$ | 96 | 24
|  |  |  |  |  |  |  |  | $A_{5,2}^{\oplus 2}C_{2,1}A_{2,1}^{\oplus 2}$ |  |  | 26
|  |  |  |  |  |  |  |  | $A_{8,3}A_{2,1}^{\oplus 2}$ |  |  | 27
20. | $\frac{28}{5}$ | $(\frac{2}{5},\frac{4}{5})$ | $28$ | $G_{2,1}^{\oplus 2}$ | $\frac{92}{5}$ | $(\frac{8}{5},\frac{6}{5})$ | $92$ | ${\bf III_{37}}$ | $E_{6,3}G_{2,1}^{\oplus 3}$ | $(50,1)$ | 120 | 32
21. | $\frac{52}{5}$ | $(\frac{3}{5},\frac{6}{5})$ | $104$ | $F_{4,1}^{\oplus 2}$ | $\frac{68}{5}$ | $(\frac{7}{5},\frac{4}{5})$ | $136$ | ${\bf III_{22}}$ | $C_{8,1}F_{4,1}^{\oplus 2}$ | $(50,1)$ | 240 | 52
22. | $4$ | $(\frac{2}{5},\frac{3}{5})$ | $24$ | $A_{4,1}$ | $20$ | $(\frac{8}{5},\frac{7}{5})$ | $120$ | $\text{GHM}_{120}$ | $A_{4,1}^{\oplus 6}$ | $(1250,1250)$ | 144 | 37
|  |  |  |  |  |  |  |  | $A_{9,2}B_{3,1}A_{4,1}$ |  |  | 40

Table LABEL:t0 includes a few self-dual pairs. In these cases we have:
$\chi_{0}=\tilde{\chi}_{0}$, $\chi_{1}=\tilde{\chi}_{2}$ and
$\chi_{2}=\tilde{\chi}_{1}$. Hence Eq.(1.3) (for three characters) becomes:

$\displaystyle\chi_{0}^{\mathcal{H}}=\chi_{0}^{2}+d_{1}\,\chi_{1}\chi_{2}+d_{2}\,\chi_{2}\chi_{1}=\chi_{0}^{2}+(d_{1}+d_{2})\,\chi_{1}\chi_{2}=\chi_{0}^{2}+d_{3}\,\chi_{1}\chi_{2}.$

Hence, in the $(d_{1},d_{2})$ column we just have one entry
$d_{3}=d_{1}+d_{2}$ for self-dual pairs. We follow this convention for all
tables whenever there are self-dual pairs involved in a bilinear relation.

###  3.2 New meromorphic theories at $c\geq 32$: infinite families

The next step is to combine the theories labelled ${\tilde{\cal C}}$ in Table
LABEL:t0 with suitable infinite series of affine theories to make modular
invariants with central charge $\geq 32$. One set of results is exhibited in
Table LABEL:t1, where 34 infinite families of coset pairs are described. The
theories labelled ${\tilde{\cal C}}$ are all taken from Table LABEL:t0.
However in each case the corresponding theory labelled ${\cal C}$ in Table
LABEL:t0 has been replaced by a new affine theory ${\cal C}$ labelled by an
arbitrary integer parameter $m$. The associated character is exhibited in the
last column of the table (see below for an explanation of the notation). The
$m=0$ case is the original one in Table LABEL:t0. The integers $(n_{1},n_{2})$
are, in all cases, given by $(2,m+2)$, verifying Eq. (2.10) between a pair of
three-characters with vanishing Wronskian index.

In this table, both ${\cal C}$ and ${\tilde{\cal C}}$ correspond to known
theories. Hence, from the coset pairing we conclude that the modular invariant
obtained by combining them in a bilinear relation also corresponds to a
genuine CFT. This is a meromorphic CFT that, in most cases, has to be of non-
lattice type since it either involves non-simply-laced algebras or levels
greater than 1, or both.

Table 2: Coset relations for $c^{\mathcal{H}}=8(m+3)$ with $(n_{1},n_{2})=(2,m+2)$. # | $c$ | $(h_{1},h_{2})$ | $m_{1}$ | $\mathcal{C}$ | $\tilde{c}$ | $(\tilde{h}_{1},\tilde{h}_{2})$ | $\tilde{m}_{1}$ | $\tilde{\mathcal{C}}$ | Sch. #
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  | ($m=0$)
1. | $\frac{16m+1}{2}$ | $\left(\frac{1}{2},\frac{16m+1}{16}\right)$ | 128$m^{2}$ \+ 8$m$ | $B_{8m,1}$ | $\frac{47}{2}$ | $\left(\frac{3}{2},\frac{31}{16}\right)$ | $0$ | BM | 0
2. | $\frac{16m+3}{2}$ | $\left(\frac{1}{2},\frac{16m+3}{16}\right)$ | 128$m^{2}$ \+ 40$m$ \+ 3 | $B_{8m+1,1}$ | $\frac{45}{2}$ | $\left(\frac{3}{2},\frac{29}{16}\right)$ | $45$ | $\mathcal{E}_{3}[A_{1,2}^{\oplus 15}]$ | 5
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[A_{3,4}^{\oplus 3}]$ | 7
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[A_{5,6}C_{2,3}]$ | 8
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[D_{5,8}]$ | 10
3. | $\frac{16m+5}{2}$ | $\left(\frac{1}{2},\frac{16m+5}{16}\right)$ | 128$m^{2}$ \+ 72$m$ \+ 10 | $B_{8m+2,1}$ | $\frac{43}{2}$ | $\left(\frac{3}{2},\frac{27}{16}\right)$ | $45$ | $\mathcal{E}_{3}[D_{4,2}^{\oplus 2}C_{2,1}^{\oplus 3}]$ | 25
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[A_{5,2}^{\oplus 2}A_{2,1}^{\oplus 2}]$ | 26
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[E_{6,4}A_{2,1}]$ | 28
4. | $\frac{16m+7}{2}$ | $\left(\frac{1}{2},\frac{16m+7}{16}\right)$ | 128$m^{2}$ \+ 104$m$ \+ 21 | $B_{8m+3,1}$ | $\frac{41}{2}$ | $\left(\frac{3}{2},\frac{25}{16}\right)$ | $123$ | $\mathcal{E}_{3}[D_{6,2}C_{4,1}B_{3,1}]$ | 39
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[A_{9,2}A_{4,1}]$ | 40
5. | $\frac{16m+9}{2}$ | $\left(\frac{1}{2},\frac{16m+9}{16}\right)$ | 128$m^{2}$ \+ 136$m$ \+ 36 | $B_{8m+4,1}$ | $\frac{39}{2}$ | $\left(\frac{3}{2},\frac{23}{16}\right)$ | $156$ | $\mathcal{E}_{3}[D_{8,2}B_{4,1}]$ | 47
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[C_{6,1}^{\oplus 2}]$ | 48
6. | $\frac{16m+11}{2}$ | $\left(\frac{1}{2},\frac{16m+11}{16}\right)$ | 128$m^{2}$ \+ 168$m$ \+ 55 | $B_{8m+5,1}$ | $\frac{37}{2}$ | $\left(\frac{3}{2},\frac{21}{16}\right)$ | $185$ | $\mathcal{E}_{3}[E_{7,2}F_{4,1}]$ | 53
7. | $\frac{16m+13}{2}$ | $\left(\frac{1}{2},\frac{16m+13}{16}\right)$ | 128$m^{2}$ \+ 200$m$ \+ 78 | $B_{8m+6,1}$ | $\frac{35}{2}$ | $\left(\frac{3}{2},\frac{19}{16}\right)$ | $210$ | $\mathcal{E}_{3}[C_{10,1}]$ | 56
8. | $\frac{16m+17}{2}$ | $\left(\frac{1}{2},\frac{16m+17}{16}\right)$ | 128$m^{2}$ \+ 264$m$ \+ 136 | $B_{8m+8,1}$ | $\frac{31}{2}$ | $\left(\frac{3}{2},\frac{15}{16}\right)$ | $248$ | $E_{8,2}$ | 62
9. | $8m+1$ | $\left(\frac{1}{2},\frac{8m+1}{8}\right)$ | 128$m^{2}$ \+ 24$m$ \+ 1 | $D_{8m+1,1}$ | $23$ | $\left(\frac{3}{2},\frac{15}{8}\right)$ | $23$ | $\mathcal{E}_{3}[D_{1,1}^{\oplus 23}]$ | 1
10. | $8m+2$ | $\left(\frac{1}{2},\frac{8m+2}{8}\right)$ | 128$m^{2}$ \+ 56$m$ \+ 6 | $D_{8m+2,1}$ | $22$ | $\left(\frac{3}{2},\frac{7}{4}\right)$ | $66$ | $\mathcal{E}_{3}[A_{1,1}^{\oplus 22}]$ | 15
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[A_{3,2}^{\oplus 4}A_{1,1}^{\oplus 2}]$ | 16
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[A_{5,3}D_{4,3}A_{1,1}]$ | 17
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[A_{7,4}A_{1,1}]$ | 18
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[D_{5,4}C_{3,2}]$ | 19
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[D_{6,5}]$ | 20
11. | $8m+3$ | $\left(\frac{1}{2},\frac{8m+3}{8}\right)$ | 128$m^{2}$ \+ 88$m$ \+ 15 | $D_{8m+3,1}$ | $21$ | $\left(\frac{3}{2},\frac{13}{8}\right)$ | $105$ | $\mathcal{E}_{3}[A_{3,1}^{\oplus 7}]$ | 30
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[D_{5,2}^{\oplus 2}A_{3,1}]$ | 31
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[A_{7,2}C_{3,1}^{\oplus 2}]$ | 33
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[D_{7,3}G_{2,1}]$ | 34
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[C_{7,2}]$ | 35
12. | $8m+5$ | $\left(\frac{1}{2},\frac{8m+5}{8}\right)$ | 128$m^{2}$ \+ 152$m$ \+ 45 | $D_{8m+5,1}$ | $19$ | $\left(\frac{3}{2},\frac{11}{8}\right)$ | $171$ | $\mathcal{E}_{3}[D_{5,1}A_{7,1}^{\oplus 2}]$ | 49
13. | $8m+6$ | $\left(\frac{1}{2},\frac{8m+6}{8}\right)$ | 128$m^{2}$ \+ 184$m$ \+ 66 | $D_{8m+6,1}$ | $18$ | $\left(\frac{3}{2},\frac{5}{4}\right)$ | $198$ | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$ | 54
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[A_{9,1}^{2}]$ | 55
14. | $8m+7$ | $\left(\frac{1}{2},\frac{8m+7}{8}\right)$ | 128$m^{2}$ \+ 216$m$ \+ 91 | $D_{8m+7,1}$ | $17$ | $\left(\frac{3}{2},\frac{9}{8}\right)$ | $221$ | $\mathcal{E}_{3}[A_{11,1}E_{6,1}]$ | 59
15. | $8m+9$ | $\left(\frac{1}{2},\frac{8m+9}{8}\right)$ | 128$m^{2}$ \+ 280$m$ \+ 153 | $D_{8m+9,1}$ | $15$ | $\left(\frac{3}{2},\frac{7}{8}\right)$ | $255$ | $\mathcal{E}_{3}[A_{15,1}]$ | 63
16. | $8m+10$ | $\left(\frac{1}{2},\frac{8m+10}{8}\right)$ | 128$m^{2}$ \+ 312$m$ \+ 190 | $D_{8m+10,1}$ | $14$ | $\left(\frac{3}{2},\frac{3}{4}\right)$ | $266$ | $E_{7,1}^{\oplus 2}$ | 64
17. | $8m+12$ | $\left(\frac{1}{2},\frac{8m+12}{8}\right)$ | 128$m^{2}$ \+ 376$m$ \+ 276 | $D_{8m+12,1}$ | $12$ | $\left(\frac{3}{2},\frac{1}{2}\right)$ | $276$ | $D_{12,1}$ | 66

The new meromorphic theories predicted by the above coset relations are
summarised in Table LABEL:t4. The first 15 rows have $B_{r,1}$ factors and the
next 19 rows have $D_{r,1}$ factors. We provide the Kac-Moody algebra of which
the meromorphic theory is an extension (in the first row, the extension is of
a combination of a Kac-Moody algebra with the Baby Monster module). When the
integer $m$ is equal to 0 the theory is part of the list in [16] and we
provide the serial number of that list where this entry can be found. For all
$m\geq 1$ the theory is new, to our knowledge.

Finally, for each case we provide the linear combination of character
bilinears corresponding to the character of the extension. Here we have
exhibited the way in which the 9 characters of the theory ${\cal
C}\oplus{\tilde{\cal C}}$ are combined into a meromorphic extension ${\cal
E}_{1}[{\cal C}\oplus{\tilde{\cal C}}]$ (in special cases where ${\cal
C}={\tilde{\cal C}}$ we have 6 rather than 9 characters for ${\cal
C}\oplus{\tilde{\cal C}}$, but the idea is the same). The quantities
${\hat{\chi}}_{0},{\hat{\chi}}_{2},{\hat{\chi}}_{m+2}$, labelled by their
conformal dimensions, are respectively the bilinears
$\chi_{0}{\tilde{\chi}}_{0},\chi_{1}{\tilde{\chi}}_{1},\chi_{2}{\tilde{\chi}}_{2}$
of the characters of ${\cal C},{\tilde{\cal C}}$ (these however are labelled
serially as $0,1,2$ rather than by their conformal dimensions, the latter can
be read off from the table) 666Note that the ${\tilde{\chi}}_{i}$ are not
characters of the Kac-Moody algebra, but of its three-character extension..
The ${\hat{\chi}}$’s are three of the 9 (or 6) characters in ${\cal
C}\oplus{\tilde{\cal C}}$. In general the bilinear identity involves
coefficients $d_{1},d_{2}$ (recall that $d_{0}=1$). Thus the column specifies
${\hat{\chi}}_{0}+d_{1}{\hat{\chi}}_{1}+d_{2}{\hat{\chi}}_{2}$ which defines
the new meromorphic theory ${\cal E}_{1}[{\cal C}\oplus{\tilde{\cal C}}]$.

Table 3: The $34$ infinite series of new meromorphic CFTs. In each series, $m=0$ corresponds to a Schellekens theory, whose Schellekens number S# is given in the second last column. BM denotes the Baby Monster CFT. Each theory has central charge $8m+24$. # | $\mathcal{H}$ | S# | Modular | # | $\mathcal{H}$ | S# | Modular
---|---|---|---|---|---|---|---
|  |  | invariant |  |  |  | invariant
1. | $\mathcal{E}_{1}[B_{8m,1}\text{BM}]$ | $0$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,{\hat{\chi}}_{m+2}$ | 2. | $\mathcal{E}_{1}[B_{8m+1,1}A_{1,2}^{\oplus 15}]$ | $5$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,1024{\hat{\chi}}_{m+2}$
3. | $\mathcal{E}_{1}[B_{8m+1,1}A_{3,4}^{\oplus 3}]$ | $7$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,1024{\hat{\chi}}_{m+2}$ | 4. | $\mathcal{E}_{1}[B_{8m+1,1}A_{5,6}C_{2,3}]$ | $8$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,1024{\hat{\chi}}_{m+2}$
5. | $\mathcal{E}_{1}[B_{8m+1,1}D_{5,8}]$ | $10$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,1024{\hat{\chi}}_{m+2}$ | 6. | $\mathcal{E}_{1}[B_{8m+2,1}C_{2,1}^{\oplus 3}D_{4,2}^{\oplus 2}]$ | $25$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,512{\hat{\chi}}_{m+2}$
7. | $\mathcal{E}_{1}[B_{8m+2,1}A_{2,1}^{\oplus 2}A_{5,2}^{\oplus 2}]$ | $26$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,512{\hat{\chi}}_{m+2}$ | 8. | $\mathcal{E}_{1}[B_{8m+2,1}A_{2,1}E_{6,4}]$ | $28$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,512{\hat{\chi}}_{m+2}$
9. | $\mathcal{E}_{1}[B_{8m+3,1}B_{3,1}C_{4,1}D_{6,2}]$ | $39$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,256{\hat{\chi}}_{m+2}$ | 10. | $\mathcal{E}_{1}[B_{8m+3,1}A_{4,1}A_{9,2}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,256{\hat{\chi}}_{m+2}$
11. | $\mathcal{E}_{1}[B_{8m+4,1}B_{4,1}D_{8,2}]$ | $47$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,128{\hat{\chi}}_{m+2}$ | 12. | $\mathcal{E}_{1}[B_{8m+4,1}\,C_{6,1}^{\oplus 2}]$ | $48$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,128{\hat{\chi}}_{m+2}$
13. | $\mathcal{E}_{1}[B_{8m+5,1}\,E_{7,2}\,F_{4,1}]$ | $53$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+{\hat{\chi}}_{m+2}$ | 14. | $\mathcal{E}_{1}[B_{8m+6,1}\,C_{10,1}]$ | $56$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+32\,{\hat{\chi}}_{m+2}$
15. | $\mathcal{E}_{1}[B_{8m+8,1}\,E_{8,2}]$ | $62$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+{\hat{\chi}}_{m+2}$ |  |  |  |
16. | $\mathcal{E}_{1}[D_{8m+1,1}D_{1,1}^{\oplus 23}]$ | $1$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+4096\,{\hat{\chi}}_{m+2}$ | 17. | $\mathcal{E}_{1}[D_{8m+2,1}A_{1,1}^{\oplus 22}]$ | $15$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+4096\,{\hat{\chi}}_{m+2}$
18. | $\mathcal{E}_{1}[D_{8m+2,1}A_{1,1}^{\oplus 2}A_{3,2}^{\oplus 4}]$ | $16$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+4096\,{\hat{\chi}}_{m+2}$ | 19. | $\mathcal{E}_{1}[D_{8m+2,1}A_{1,1}A_{5,3}D_{4,3}]$ | $17$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+4096\,{\hat{\chi}}_{m+2}$
20. | $\mathcal{E}_{1}[D_{8m+2,1}A_{1,1}A_{7,4}]$ | $18$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+4096\,{\hat{\chi}}_{m+2}$ | 21. | $\mathcal{E}_{1}[D_{8m+2,1}C_{3,2}D_{5,4}]$ | $19$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+4096\,{\hat{\chi}}_{m+2}$
22. | $\mathcal{E}_{1}[D_{8m+2,1}D_{6,5}]$ | $20$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+4096\,{\hat{\chi}}_{m+2}$ | 23. | $\mathcal{E}_{1}[D_{8m+3,1}A_{3,1}^{\oplus 7}]$ | $30$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+1024\,{\hat{\chi}}_{m+2}$
24. | $\mathcal{E}_{1}[D_{8m+3,1}A_{3,1}D_{5,2}^{\oplus 2}]$ | $31$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+1024\,{\hat{\chi}}_{m+2}$ | 25. | $\mathcal{E}_{1}[D_{8m+3,1}A_{7,2}C_{3,1}^{\oplus 2}]$ | $33$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+1024\,{\hat{\chi}}_{m+2}$
26. | $\mathcal{E}_{1}[D_{8m+3,1}\,D_{7,3}G_{2,1}]$ | $34$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+1024\,{\hat{\chi}}_{m+2}$ | 27. | $\mathcal{E}_{1}[D_{8m+3,1}C_{7,2}]$ | $35$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+1024\,{\hat{\chi}}_{m+2}$
28. | $\mathcal{E}_{1}[D_{8m+5,1}A_{7,1}^{\oplus 2}D_{5,1}]$ | $49$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+256\,{\hat{\chi}}_{m+2}$ | 29. | $\mathcal{E}_{1}[D_{8m+6,1}D_{6,1}^{\oplus 3}]$ | $54$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+256\,{\hat{\chi}}_{m+2}$
30. | $\mathcal{E}_{1}[D_{8m+6,1}A_{9,1}^{\oplus 2}]$ | $55$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+256\,{\hat{\chi}}_{m+2}$ | 31. | $\mathcal{E}_{1}[D_{8m+7,1}A_{11,1}E_{6,1}]$ | $59$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+64\,{\hat{\chi}}_{m+2}$
32. | $\mathcal{E}_{1}[D_{8m+9,1}A_{15,1}]$ | $63$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+16\,{\hat{\chi}}_{m+2}$ | 33. | $\mathcal{E}_{1}[D_{8m+10,1}E_{7,1}^{\oplus 2}]$ | $64$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+2\,{\hat{\chi}}_{m+2}$
34. | $\mathcal{E}_{1}[D_{8m+12,1}D_{12,1}]$ | $66$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+{\hat{\chi}}_{m+2}$ |  |  |  |

We now work out an example in detail. This is a typical case in this table and
should clarify the procedure that has been used for all cases. We pick Row 6
of Table LABEL:t1, involving the coset pairing of $B_{8m+5,1}$ with ${\cal
E}_{3}[E_{7,2}\oplus F_{4,1}]$. The resulting meromorphic theory is in row 13
of Table LABEL:t4. Notice that $B_{r,1}$ is a three-character affine theory
for all $r$ (the same is true for the $D_{r,1}$ series, see Appendix A for
details). We now claim that the two factors pair to a meromorphic character
with $c^{\mathcal{H}}=8(m+3)$ and $(n_{1},n_{2})=(2,m+2)$ and that the
bilinear identity is:

$\displaystyle\chi_{0}^{\mathcal{H}}$
$\displaystyle=\chi_{0}\tilde{\chi}_{0}+\chi_{\frac{1}{2}}\tilde{\chi}_{\frac{3}{2}}+\chi_{\frac{16m+11}{16}}\tilde{\chi}_{\frac{21}{16}}={\hat{\chi}}_{0}+{\hat{\chi}}_{2}+{\hat{\chi}}_{m+2}$
(3.1)

The notation has been explained above. It should be kept in mind that for
$m=0$ the last two terms in the last expression remain distinct although both
would be denoted by ${\hat{\chi}}_{2}$, one comes from the bilinear in
$h=\frac{1}{2},{\tilde{h}}=\frac{3}{2}$ while the other comes from the
bilinear in $h=\frac{11}{16},{\tilde{h}}=\frac{21}{16}$.777However for self-
dual pairs the two ${\hat{\chi}}_{2}$’s are indeed same at $m=0$. However for
$m\geq 1$, the case of interest here, there is no ambiguity in the notation.

Now we explain how the bilinear identity is proved for all $m$, with the
$m$-independent coefficients $(d_{1},d_{2})=(1,1)$ in this family. To start
with, we have explicitly verified the bilinear relation, to order $q^{2000}$
in the $q$-series, for $32\leq c^{\cal H}\leq 72$, which corresponds to $1\leq
m\leq 6$ by comparing the series expansion on both sides. We find
$(d_{1},d_{2})=(1,1)$ in all these cases, and the modular invariant on the RHS
of the bilinear relation to be:

$\begin{split}&c^{\mathcal{H}}=24:\,\,\,\chi^{\cal H}(\tau)=j-504,\\\
&c^{\mathcal{H}}=32:\,\,\,\chi^{\cal
H}(\tau)=j^{\frac{4}{3}}-456\,j^{\frac{1}{3}},\\\
&c^{\mathcal{H}}=40:\,\,\,\chi^{\cal
H}(\tau)=j^{\frac{5}{3}}-152\,j^{\frac{2}{3}},\\\
&c^{\mathcal{H}}=48:\,\,\,\chi^{\cal H}(\tau)=j^{2}+408\,j-129024,\\\
&c^{\mathcal{H}}=56:\,\,\,\chi^{\cal
H}(\tau)=j^{\frac{7}{3}}+1224\,j^{\frac{4}{3}}-374784\,j^{\frac{1}{3}},\\\
&c^{\mathcal{H}}=64:\,\,\,\chi^{\cal
H}(\tau)=j^{\frac{8}{3}}+2296\,j^{\frac{5}{3}}-659456\,j^{\frac{2}{3}},\\\
&c^{\mathcal{H}}=72:\,\,\,\chi^{\cal
H}(\tau)=j^{3}+3624\,j^{2}-839680\,j-33030144.\end{split}$ (3.2)

This led us to conjecture that $(d_{1},d_{2})=(1,1)$ independent of $m$, and
then immediately to a proof of the conjecture.

For the proof we switch to the notation of Eq.(2.2) and find a formula for the
coefficients in this example for all $m$. For this we first write:

$\displaystyle{\hat{\chi}}_{0}+d_{1}\,{\hat{\chi}}_{2}+d_{2}\,{\hat{\chi}}_{m+2}=q^{-\frac{c^{\mathcal{H}}}{24}}\left(\text{an
integral power series of q}\right)$ (3.3)

After cancelling the fractional power of $q$ – if any – from both sides of the
above equation, we see that:

$\displaystyle a_{r}(m)=$ $\displaystyle\,\text{coefficient of}\ q^{r}\
\text{in LHS of Eq.(\ref{b8m5_ghm185_9char00})}$ $\displaystyle-$
$\displaystyle\ \text{coefficient of}\ q^{r}\ \text{in}\
\left(j^{\frac{m+3}{3}}+a_{1}\,j^{\frac{m}{3}}+\ldots+a_{r-1}j^{\frac{m+3}{3}-(r-1)}\right).$
(3.4)

For $r=1$ we can write more explicitly:

$\displaystyle a_{1}(m)=$ $\displaystyle\ \text{coefficient of}\ q\ \text{in
LHS of Eq.(\ref{b8m5_ghm185_9char00})}\ -\ \text{coefficient of}\ q\
\text{in}\ j^{\frac{m+3}{3}}$ $\displaystyle=$ $\displaystyle\
128\,m^{2}+168\,m+240-248(m+3).$ (3.5)

Note that though $a_{1}(m)$ has been derived from Eq.(3.3), it only comes from
comparing $\mathcal{O}(q)$ coefficient on both sides of Eq.(3.3) and hence the
formula for $a_{1}(m)$ is independent of $d_{1}$ and $d_{2}$. This is because
on the LHS of Eq.(3.3) $d_{1}$ appears at $\mathcal{O}(q^{2})$ and $d_{2}$
appears at $\mathcal{O}(q^{m+2})$.

The characters $\chi_{i}$ of the $B$-series affine theory at level 1 are known
to be given by Jacobi $\theta$-constants. Using this representation one can
show that $m_{2}(m)$ is quartic in $m$ (recall that $m_{2}$ is the second-
level degeneracy for the identity character). This in turn would imply that
the coefficient of $q^{2}$ in $\chi^{\cal H}$ would be quartic in $m$ and
hence $a_{2}$ would be quartic in $m$. Using this, we now prove that
$d_{1}(m)$ is independent of $m$.

We first employ Mathematica to solve (for $32\leq c^{\cal H}\leq 72$ as in
Eq.(3.2) above):

$\displaystyle(m,\text{coefficient of}\ q^{2}\ \text{in}\ \chi^{\cal H})=$
$\displaystyle[(1,272432),(2,560268),(3,1121832),(4,2159876),$
$\displaystyle(5,3942688),(6,6804092)].$ (3.6)

which returns:

$\displaystyle\text{coefficient of}\ q^{2}\ \text{in }\chi^{\cal
H}=121108+\frac{337268}{3}\,m+\frac{89056}{3}\,m^{2}+\frac{19456}{3}\,m^{3}+\frac{8192}{3}\,m^{4}.$
(3.7)

From the above we can get a general formula for $a_{2}$ using Eq.(2.2),

$\displaystyle a_{2}(m)=$ $\displaystyle
121108+\frac{337268}{3}\,m+\frac{89056}{3}\,m^{2}+\frac{19456}{3}\,m^{3}+\frac{8192}{3}\,m^{4}$
$\displaystyle-4124(m+3)-30752(m+2)(m+3)-248\,m\,a_{1}.$ (3.8)

Comparing the $\mathcal{O}(q^{2})$ term on both sides of Eq.(1.3)), we find:

$\displaystyle\tilde{m}_{2}+\tilde{m}_{1}m_{1}(m)+m_{2}(m)+d_{1}(m)\,D_{1}(m)\tilde{D}_{1}$
$\displaystyle=\,a_{2}(m)+248\,m\,a_{1}(m)+4124(m+3)+30752(m+2)(m+3).$ (3.9)

Now from row 6 of Table LABEL:t1 we read off:

$\begin{split}&{\tilde{m}}_{1}=185\\\ &m_{1}(m)=128m^{2}+168m+55,\end{split}$
(3.10)

while from the $\theta$-constant representation of the characters of $B_{r,1}$
we get:

$\begin{split}D_{1}(r)&=2r+1=16m+11\\\
m_{2}(r)&=1+\frac{25}{6}r+\frac{23}{6}r^{2}-\frac{2}{3}r^{3}+\frac{2}{3}r^{4},\end{split}$
(3.11)

where $r=8m+5$ for the present example.

Finally, solving the MLDE for the ($m$-independent) $\tilde{\cal C}$ theory in
row 6 of Table LABEL:t1 gives:

$\begin{split}{\tilde{m}_{2}}&=56351,\qquad{\tilde{D}}_{1}=4921.\end{split}$
(3.12)

Inserting Eqs.(3.10), (3.11), (3.12) in Eq. (3.9), we get:

$\displaystyle 56351+$ $\displaystyle
185(128\,m^{2}+168\,m+55)+\left(1+\frac{25}{6}(8m+5)+\frac{23}{6}(8m+5)^{2}\right.$
$\displaystyle\left.-\frac{2}{3}(8m+5)^{3}+\frac{2}{3}(8m+5)^{4}\right)+d_{1}(m)4921(16m+11)$
$\displaystyle=$
$\displaystyle\,a_{2}(m)+248\,m\,a_{1}(m)+4124(m+3)+30752(m+2)(m+3)$ (3.13)

from which, using Eq. (3.8) and Eq. (3.5), we find that $d_{1}(m)=1$ for all
$m$. Similarly one can argue that $d_{2}=1$ $\forall\,m\geq 1$.

The corresponding results for the remaining infinite families can be explained
in the same way. In all cases the coefficients $d_{1},d_{2}$ in the bilinear
relation for all $m$ are the same as those obtained for $m=0$, i.e. the
$c^{\cal H}=24$ case. Thus we have verified (still to order $q^{2000}$ in the
$q$-series), that this new theory satisfies a bilinear relation with
${\tilde{\cal C}}$ forming a modular invariant with $c^{\cal H}=8(m+3)$ for
all $m$.

Now let us give below, explicitly, the $3$-character extension of the twelve
character theory $E_{7,2}F_{4,1}$,

$\displaystyle\tilde{\chi}_{0}=\chi^{E}_{0}\chi^{F}_{0}+\chi^{E}_{\frac{7}{5}}\chi^{F}_{\frac{3}{5}}$
$\displaystyle\tilde{\chi}_{\frac{3}{2}}=\chi^{E}_{\frac{3}{2}}\chi^{F}_{0}+\chi^{E}_{\frac{9}{10}}\chi^{F}_{\frac{3}{5}}$
$\displaystyle\tilde{\chi}_{\frac{21}{16}}=\chi^{E}_{\frac{21}{16}}\chi^{F}_{0}+\chi^{E}_{\frac{57}{80}}\chi^{F}_{\frac{3}{5}}$
(3.14)

where $\chi^{E}_{i}$’s represent the six characters of $E_{7,2}$ and
$\chi^{F}_{i}$s represent the two characters of $F_{4,1}$. The above
expressions are obtained by comparing the characters of the extension (found
from the MLDE approach) with the leading behaviour of the characters
$\chi^{E},\chi^{F}$ which is given by the dimensions of integrable
representations.

Using the above result, now we can explicitly write the $1$-character
extension of the $36$-character theory $B_{8m+5,1}E_{7,2}F_{4,1}$:

$\displaystyle\chi^{\mathcal{H}}$
$\displaystyle={\hat{\chi}}_{0}+{\hat{\chi}}_{2}+{\hat{\chi}}_{m+2}=\chi_{0}\tilde{\chi}_{0}+\chi_{\frac{1}{2}}\tilde{\chi}_{\frac{3}{2}}+\chi_{\frac{16m+11}{16}}\tilde{\chi}_{\frac{21}{16}}$
$\displaystyle=j^{1/3}\left(j+128\,m^{2}+168\,m+240-248(m+3)\right),$
$\displaystyle=\chi_{0}\chi^{E}_{0}\chi^{F}_{0}+\chi_{0}\chi^{E}_{\frac{7}{5}}\chi^{F}_{\frac{3}{5}}+\chi_{\frac{1}{2}}\chi^{E}_{\frac{3}{2}}\chi^{F}_{0}+\chi_{\frac{1}{2}}\chi^{E}_{\frac{9}{10}}\chi^{F}_{\frac{3}{5}}+\chi_{\frac{16m+11}{16}}\chi^{E}_{\frac{21}{16}}\chi^{F}_{0}+\chi_{\frac{16m+11}{16}}\chi^{E}_{\frac{57}{80}}\chi^{F}_{\frac{3}{5}},$
(3.15)

where ${\hat{\chi}}_{i}$s represent the nine characters of
$B_{8m+5,1}\oplus\mathcal{E}_{3}[E_{7,2}F_{4,1}]$, $\chi_{i}$’s represent the
three characters of $B_{8m+5,1}$, $\tilde{\chi}_{i}$’s represent the three
characters of $\mathcal{E}_{3}[E_{7,2}F_{4,1}]$.

###  3.3 New meromorphic theories at $c=32,40$ : finite families

In this section we exhibit a finite set of novel meromorphic theories with
central charges $c^{\mathcal{H}}=32$ and $c^{\mathcal{H}}=40$ only. As in the
previous cases, the bilinear relations for these examples have also been
verified to order $q^{2000}$. Tables LABEL:t2 and LABEL:t3 exhibit the coset
pairings which correspond respectively to $c^{\mathcal{H}}=32$ with
$(n_{1},n_{2})=(2,3)$, and $c^{\mathcal{H}}=40$ with $(n_{1},n_{2})=(3,3)$.
There exists no new family for $c^{\mathcal{H}}=40$ with
$(n_{1},n_{2})=(2,4)$. At $c^{\mathcal{H}}=32$ we find that there are 7
additional meromorphic theories. Out of these, 3 are novel non-lattice
meromorphic theories (second and third line of row 1, and row 2). The
remaining 4 are lattice meromorphic theories whose lattices can be found in
[39]. Similarly, at $c^{\mathcal{H}}=40$, there are 39 additional meromorphic
theories, of which 32 are novel non-lattice theories and the remaining 7 are
lattice theories.

Table 4: Coset relations for $c^{\mathcal{H}}=32$ with $(n_{1},n_{2})=(2,3)$. # | $c$ | $(h_{1},h_{2})$ | $m_{1}$ | $\mathcal{C}$ | $\tilde{c}$ | $(\tilde{h}_{1},\tilde{h}_{2})$ | $\tilde{m}_{1}$ | $\tilde{\mathcal{C}}$
---|---|---|---|---|---|---|---|---
1. | $12$ | $\left(\frac{2}{3},\frac{4}{3}\right)$ | 156 | $E_{6,1}^{\oplus 2}$ | $20$ | $\left(\frac{4}{3},\frac{5}{3}\right)$ | $80$ | $\mathcal{E}_{3}[A_{2,1}^{\oplus 10}]$
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[A_{5,2}^{\oplus 2}C_{2,1}]$
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[A_{8,3}]$
2. | $\frac{68}{5}$ | $\left(\frac{4}{5},\frac{7}{5}\right)$ | 136 | $\mathcal{E}_{3}[C_{8,1}]$ | $\frac{92}{5}$ | $\left(\frac{6}{5},\frac{8}{5}\right)$ | $92$ | $\mathcal{E}_{3}[E_{6,3}G_{2,1}]$
3. | $14$ | $\left(\frac{3}{4},\frac{3}{2}\right)$ | 266 | $E_{7,1}^{\oplus 2}$ | $18$ | $\left(\frac{5}{4},\frac{3}{2}\right)$ | $198$ | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$
|  |  |  |  |  |  |  | $\mathcal{E}_{3}[A_{9,1}^{\oplus 2}]$
4. | $15$ | $\left(\frac{7}{8},\frac{3}{2}\right)$ | 255 | $\mathcal{E}_{3}[A_{15,1}]$ | $17$ | $\left(\frac{9}{8},\frac{3}{2}\right)$ | $221$ | $\mathcal{E}_{3}[A_{11,1}E_{6,1}]$

Table 5: Coset relations for $c^{\mathcal{H}}=40$ with $(n_{1},n_{2})=(3,3)$. # | $c$ | $(h_{1},h_{2})$ | $m_{1}$ | $\mathcal{C}$ | $\tilde{c}$ | $(\tilde{h}_{1},\tilde{h}_{2})$ | $\tilde{m}_{1}$ | $\tilde{\mathcal{C}}$
---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |
1. | $20$ | $\left(\frac{1}{2},\frac{5}{2}\right)$ | 780 | $D_{20,1}$ | $20$ | $\left(\frac{5}{2},\frac{1}{2}\right)$ | $780$ | $D_{20,1}$
2. | $20$ | $\left(\frac{4}{3},\frac{5}{3}\right)$ | 80 | $\mathcal{E}_{3}[A_{2,1}^{\oplus 10}]$ | $20$ | $\left(\frac{5}{3},\frac{4}{3}\right)$ | $80$ | $\mathcal{E}_{3}[A_{2,1}^{\oplus 10}]$
|  |  |  | $\mathcal{E}_{3}[A_{2,1}^{\oplus 10}]$ |  |  |  | $\mathcal{E}_{3}[A_{5,2}^{\oplus 2}C_{2,1}]$
|  |  |  | $\mathcal{E}_{3}[A_{2,1}^{\oplus 10}]$ |  |  |  | $\mathcal{E}_{3}[A_{8,3}]$
|  |  |  | $\mathcal{E}_{3}[A_{5,2}^{\oplus 2}C_{2,1}]$ |  |  |  | $\mathcal{E}_{3}[A_{5,2}^{\oplus 2}C_{2,1}]$
|  |  |  | $\mathcal{E}_{3}[A_{5,2}^{\oplus 2}C_{2,1}]$ |  |  |  | $\mathcal{E}_{3}[A_{8,3}]$
|  |  |  | $\mathcal{E}_{3}[A_{8,3}]$ |  |  |  | $\mathcal{E}_{3}[A_{8,3}]$
3. | $20$ | $\left(\frac{7}{5},\frac{8}{5}\right)$ | 120 | $\mathcal{E}_{3}[A_{4,1}^{\oplus 5}]$ | $20$ | $\left(\frac{8}{5},\frac{7}{5}\right)$ | $120$ | $\mathcal{E}_{3}[A_{4,1}^{\oplus 5}]$
|  |  |  | $\mathcal{E}_{3}[A_{4,1}^{\oplus 5}]$ |  |  |  | $\mathcal{E}_{3}[A_{9,2}B_{3,1}]$
|  |  |  | $\mathcal{E}_{3}[A_{9,2}B_{3,1}]$ |  |  |  | $\mathcal{E}_{3}[A_{9,2}B_{3,1}]$
4. | $17$ | $\left(\frac{3}{2},\frac{9}{8}\right)$ | 221 | $\mathcal{E}_{3}[A_{11,1}E_{6,1}]$ | $23$ | $\left(\frac{3}{2},\frac{15}{8}\right)$ | $23$ | $\mathcal{E}_{3}[D_{1,1}^{\oplus 23}]$
5. | $\frac{35}{2}$ | $\left(\frac{3}{2},\frac{19}{16}\right)$ | 210 | $\mathcal{E}_{3}[C_{10,1}]$ | $\frac{45}{2}$ | $\left(\frac{3}{2},\frac{29}{16}\right)$ | $45$ | $\mathcal{E}_{3}[A_{1,2}^{\oplus 15}]$
|  |  |  | $\mathcal{E}_{3}[C_{10,1}]$ |  |  |  | $\mathcal{E}_{3}[A_{3,4}^{\oplus 3}]$
|  |  |  | $\mathcal{E}_{3}[C_{10,1}]$ |  |  |  | $\mathcal{E}_{3}[A_{5,6}C_{2,3}]$
|  |  |  | $\mathcal{E}_{3}[C_{10,1}]$ |  |  |  | $\mathcal{E}_{3}[D_{5,8}]$
6. | $18$ | $\left(\frac{5}{4},\frac{3}{2}\right)$ | 198 | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$ | $22$ | $\left(\frac{7}{4},\frac{3}{2}\right)$ | $66$ | $\mathcal{E}_{3}[A_{1,1}^{\oplus 22}]$
|  |  |  | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$ |  |  |  | $\mathcal{E}_{3}[A_{3,2}^{\oplus 4}A_{1,1}^{\oplus 2}]$
|  |  |  | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$ |  |  |  | $\mathcal{E}_{3}[A_{5,3}D_{4,3}A_{1,1}]$
|  |  |  | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$ |  |  |  | $\mathcal{E}_{3}[A_{7,4}A_{1,1}]$
|  |  |  | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$ |  |  |  | $\mathcal{E}_{3}[D_{5,4}C_{3,2}]$
|  |  |  | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$ |  |  |  | $\mathcal{E}_{3}[D_{6,5}]$
|  |  |  | $\mathcal{E}_{3}[A_{9,1}^{\oplus 2}]$ |  |  |  | $\mathcal{E}_{3}[A_{1,1}^{\oplus 22}]$
|  |  |  | $\mathcal{E}_{3}[A_{9,1}^{\oplus 2}]$ |  |  |  | $\mathcal{E}_{3}[A_{3,2}^{\oplus 4}A_{1,1}^{\oplus 2}]$
|  |  |  | $\mathcal{E}_{3}[A_{9,1}^{\oplus 2}]$ |  |  |  | $\mathcal{E}_{3}[A_{5,3}D_{4,3}A_{1,1}]$
|  |  |  | $\mathcal{E}_{3}[A_{9,1}^{\oplus 2}]$ |  |  |  | $\mathcal{E}_{3}[A_{7,4}A_{1,1}]$
|  |  |  | $\mathcal{E}_{3}[A_{9,1}^{\oplus 2}]$ |  |  |  | $\mathcal{E}_{3}[D_{5,4}C_{3,2}]$
|  |  |  | $\mathcal{E}_{3}[A_{9,1}^{\oplus 2}]$ |  |  |  | $\mathcal{E}_{3}[D_{6,5}]$
7. | $\frac{37}{2}$ | $\left(\frac{3}{2},\frac{21}{16}\right)$ | 185 | $\mathcal{E}_{3}[E_{7,2}F_{4,1}]$ | $\frac{43}{2}$ | $\left(\frac{3}{2},\frac{27}{16}\right)$ | $86$ | $\mathcal{E}_{3}[D_{4,2}^{\oplus 2}C_{2,1}^{\oplus 3}]$
|  |  |  | $\mathcal{E}_{3}[E_{7,2}F_{4,1}]$ |  |  |  | $\mathcal{E}_{3}[A_{5,2}^{\oplus 2}A_{2,1}^{\oplus 2}]$
|  |  |  | $\mathcal{E}_{3}[E_{7,2}F_{4,1}]$ |  |  |  | $\mathcal{E}_{3}[E_{6,4}A_{2,1}]$
8. | $19$ | $\left(\frac{3}{2},\frac{11}{8}\right)$ | 171 | $\mathcal{E}_{3}[D_{5,1}A_{7,1}^{\oplus 2}]$ | $21$ | $\left(\frac{3}{2},\frac{13}{8}\right)$ | $105$ | $\mathcal{E}_{3}[A_{3,1}^{\oplus 7}]$
|  |  |  | $\mathcal{E}_{3}[D_{5,1}A_{7,1}^{\oplus 2}]$ |  |  |  | $\mathcal{E}_{3}[D_{5,2}^{\oplus 2}A_{3,1}]$
|  |  |  | $\mathcal{E}_{3}[D_{5,1}A_{7,1}^{\oplus 2}]$ |  |  |  | $\mathcal{E}_{3}[A_{7,2}C_{3,1}^{\oplus 2}]$
|  |  |  | $\mathcal{E}_{3}[D_{5,1}A_{7,1}^{\oplus 2}]$ |  |  |  | $\mathcal{E}_{3}[D_{7,3}G_{2,1}]$
|  |  |  | $\mathcal{E}_{3}[D_{5,1}A_{7,1}^{\oplus 2}]$ |  |  |  | $\mathcal{E}_{3}[C_{7,2}]$
9. | $\frac{39}{2}$ | $\left(\frac{3}{2},\frac{23}{16}\right)$ | 156 | $\mathcal{E}_{3}[D_{8,2}B_{4,1}]$ | $\frac{41}{2}$ | $\left(\frac{3}{2},\frac{25}{16}\right)$ | $123$ | $\mathcal{E}_{3}[D_{6,2}C_{4,1}B_{3,1}]$
|  |  |  | $\mathcal{E}_{3}[D_{8,2}B_{4,1}]$ |  |  |  | $\mathcal{E}_{3}[A_{9,2}A_{4,1}]$
|  |  |  | $\mathcal{E}_{3}[C_{6,1}^{\oplus 2}]$ |  |  |  | $\mathcal{E}_{3}[D_{6,2}C_{4,1}B_{3,1}]$
|  |  |  | $\mathcal{E}_{3}[C_{6,1}^{\oplus 2}]$ |  |  |  | $\mathcal{E}_{3}[A_{9,2}A_{4,1}]$

The new meromorphic theories predicted by the above coset relations are
summarised in Table LABEL:t5. The format for the columns of this table is
similar to that of Table LABEL:t4. The first 7 entries are theories at $c=32$
and the next 39 are theories at $c=40$.

Table 6: The finite set of 46 new meromorphic CFTs at central charges $32$ and $40$. # | $\mathcal{H}$ | $c^{\mathcal{H}}$ | $\chi^{\mathcal{H}}$ | # | $\mathcal{H}$ | $c^{\mathcal{H}}$ | $\chi^{\mathcal{H}}$
---|---|---|---|---|---|---|---
1. | $\mathcal{E}_{1}[A_{2,1}^{\oplus 10}E_{6,1}^{\oplus 2}]$ | $32$ | ${\hat{\chi}}_{0}+972{\hat{\chi}}_{2}+2^{2}\cdot 3^{7}{\hat{\chi}}_{3}$ | 2. | $\mathcal{E}_{1}[A_{5,2}^{\oplus 2}C_{2,1}E_{6,1}^{\oplus 2}]$ | $32$ | ${\hat{\chi}}_{0}+972{\hat{\chi}}_{2}+2^{2}\cdot 3^{7}{\hat{\chi}}_{3}$
3. | $\mathcal{E}_{1}[A_{8,3}E_{6,1}^{\oplus 2}]$ | $32$ | ${\hat{\chi}}_{0}+972{\hat{\chi}}_{2}+2^{2}\cdot 3^{7}{\hat{\chi}}_{3}$ | 4. | $\mathcal{E}_{1}[C_{8,1}E_{6,3}G_{2,1}]$ | $32$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+1250{\hat{\chi}}_{3}$
5. | $\mathcal{E}_{1}[D_{6,1}^{\oplus 3}E_{7,1}^{\oplus 2}]$ | $32$ | ${\hat{\chi}}_{0}+256{\hat{\chi}}_{2}+64{\hat{\chi}}_{3}$ | 6. | $\mathcal{E}_{1}[A_{9,1}^{\oplus 2}E_{7,1}^{\oplus 2}]$ | $32$ | ${\hat{\chi}}_{0}+256{\hat{\chi}}_{2}+64{\hat{\chi}}_{3}$
7. | $\mathcal{E}_{1}[A_{11,1}A_{15,1}E_{6,1}]$ | $32$ | ${\hat{\chi}}_{0}+512{\hat{\chi}}_{2}+64{\hat{\chi}}_{3}$ |  |  |  |
8. | $\mathcal{E}_{1}[D_{20,1}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2{\hat{\chi}}_{3}$ | 9. | $\mathcal{E}_{1}[A_{2,1}^{\oplus 20}]$ | $40$ | ${\hat{\chi}}_{0}+2^{3}\cdot 3^{12}{\hat{\chi}}_{3}$
10. | $\mathcal{E}_{1}[A_{2,1}^{\oplus 10}A_{5,2}^{\oplus 2}C_{2,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{3}\cdot 3^{12}{\hat{\chi}}_{3}$ | 11. | $\mathcal{E}_{1}[A_{2,1}^{\oplus 10}A_{8,3}]$ | $40$ | ${\hat{\chi}}_{0}+2^{3}\cdot 3^{12}{\hat{\chi}}_{3}$
12. | $\mathcal{E}_{1}[A_{5,2}^{\oplus 4}C_{2,1}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2^{3}\cdot 3^{12}{\hat{\chi}}_{3}$ | 13. | $\mathcal{E}_{1}[A_{5,2}^{\oplus 2}A_{8,3}C_{2,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{3}\cdot 3^{12}{\hat{\chi}}_{3}$
14. | $\mathcal{E}_{1}[A_{8,3}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2^{3}\cdot 3^{12}{\hat{\chi}}_{3}$ | 15. | $\mathcal{E}_{1}[A_{4,1}^{\oplus 10}]$ | $40$ | ${\hat{\chi}}_{0}+2^{2}\cdot 5^{8}{\hat{\chi}}_{3}$
16. | $\mathcal{E}_{1}[A_{9,2}^{\oplus 2}B_{3,1}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2^{2}\cdot 5^{8}{\hat{\chi}}_{3}$ | 17. | $\mathcal{E}_{1}[A_{4,1}^{\oplus 5}A_{9,2}\,B_{3,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{2}\cdot 5^{8}{\hat{\chi}}_{3}$
18. | $\mathcal{E}_{1}[A_{11,1}D_{1,1}^{\oplus 23}E_{6,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{6}{\hat{\chi}}_{3}+2^{17}{\hat{\chi}}_{3}$ | 19. | $\mathcal{E}_{1}[A_{1,2}^{\oplus 15}C_{10,1}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$
20. | $\mathcal{E}_{1}[A_{3,4}^{\oplus 3}C_{10,1}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$ | 21. | $\mathcal{E}_{1}[A_{5,6}C_{2,3}C_{10,1}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$
22. | $\mathcal{E}_{1}[C_{10,1}D_{5,8}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$ | 23. | $\mathcal{E}_{1}[A_{1,1}^{\oplus 22}D_{6,1}^{\oplus 3}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$
24. | $\mathcal{E}_{1}[A_{1,1}^{\oplus 2}A_{3,2}^{\oplus 4}D_{6,1}^{\oplus 3}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$ | 25. | $\mathcal{E}_{1}[A_{1,1}A_{5,3}D_{4,3}D_{6,1}^{\oplus 3}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$
26. | $\mathcal{E}_{1}[A_{1,1}A_{7,4}D_{6,1}^{\oplus 3}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$ | 27. | $\mathcal{E}_{1}[C_{3,2}D_{5,4}D_{6,1}^{\oplus 3}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$
28. | $\mathcal{E}_{1}[D_{6,1}^{\oplus 3}D_{6,5}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$ | 29. | $\mathcal{E}_{1}[A_{1,1}^{\oplus 22}A_{9,1}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$
30. | $\mathcal{E}_{1}[A_{1,1}^{\oplus 2}A_{3,2}^{\oplus 4}A_{9,1}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$ | 31. | $\mathcal{E}_{1}[A_{1,1}A_{5,3}A_{9,1}^{\oplus 2}D_{4,3}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$
32. | $\mathcal{E}_{1}[A_{1,1}A_{7,4}A_{9,1}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$ | 33. | $\mathcal{E}_{1}[A_{9,1}^{\oplus 2}C_{3,2}D_{5,4}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$
34. | $\mathcal{E}_{1}[A_{9,1}^{\oplus 2}D_{6,5}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$ | 35. | $\mathcal{E}_{1}[C_{2,1}^{\oplus 3}D_{4,2}^{\oplus 2}E_{7,2}F_{4,1}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$
36. | $\mathcal{E}_{1}[A_{2,1}^{\oplus 2}A_{5,2}^{\oplus 2}E_{7,2}F_{4,1}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$ | 37. | $\mathcal{E}_{1}[A_{2,1}E_{6,4}E_{7,2}F_{4,1}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$
38. | $\mathcal{E}_{1}[A_{3,1}^{\oplus 7}A_{7,1}^{\oplus 2}D_{5,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{6}{\hat{\chi}}_{3}+2^{17}{\hat{\chi}}_{3}$ | 39. | $\mathcal{E}_{1}[A_{3,1}A_{7,1}^{\oplus 2}D_{5,1}D_{5,2}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2^{6}{\hat{\chi}}_{3}+2^{17}{\hat{\chi}}_{3}$
40. | $\mathcal{E}_{1}[A_{7,1}^{\oplus 2}A_{7,2}C_{3,1}^{\oplus 2}D_{5,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{6}{\hat{\chi}}_{3}+2^{17}{\hat{\chi}}_{3}$ | 41. | $\mathcal{E}_{1}[A_{7,1}^{\oplus 2}D_{5,1}D_{7,3}G_{2,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{6}{\hat{\chi}}_{3}+2^{17}{\hat{\chi}}_{3}$
42. | $\mathcal{E}_{1}[A_{7,1}^{\oplus 2}C_{7,2}D_{5,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{6}{\hat{\chi}}_{3}+2^{17}{\hat{\chi}}_{3}$ | 43. | $\mathcal{E}_{1}[B_{3,1}B_{4,1}C_{4,1}D_{6,2}D_{8,2}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$
44. | $\mathcal{E}_{1}[A_{4,1}A_{9,2}B_{4,1}D_{8,2}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$ | 45. | $\mathcal{E}_{1}[B_{3,1}C_{4,1}C_{6,1}^{\oplus 2}D_{6,2}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$
46. | $\mathcal{E}_{1}[A_{4,1}A_{9,2}C_{6,1}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$ |  |  |  |

##  4 Conclusions

The meromorphic theories we have identified in the present work have been
summarised in Tables LABEL:t4 and LABEL:t5. As mentioned in the Introduction,
the proposals in this work are made at a physics level of rigour and
considerable evidence provided. It should be possible in future to convert
these to a set of mathematically rigorous statements and proofs. We have
obtained infinitely many theories that can be thought of as generalisations of
34 non-lattice theories in Ref. [16] to arbitrary central charge $c=8N$. The
fact that infinitely many generalisations exist is ultimately due to
properties of the $B_{r,1}$ and $D_{r,1}$ affine theories, which have three
characters for all $r$ and whose modular transformation matrix is periodic in
$r$.

It is interesting to see the Baby Monster module make an appearance in this
discussion, in row 1 of Table LABEL:t4. This appears to illustrate a general
phenomenon: modules with and without Kac-Moody algebras appear on a similar
footing in general meromorphic theories (and presumably in more general
RCFTs). This is in contrast to the $c=24$ case where there is one entry (the
Monster CFT) that is the extension of modules without a Kac-Moody algebra,
namely ${\cal E}_{1}[{\cal M}(4,3)\,BM]$, while the rest are extensions of
only Kac-Moody algebras. Such a perspective may be helpful to derive at least
partial classifications for meromorphic theories at $c=32$, even though a
complete classification is understood to be virtually impossible due to the
enormous number (of order $10^{9}$) of possible theories.

The present work makes no claim to being complete. The goal has only been to
provide several examples of an interesting phenomenon: the prediction of
meromorphic theories at higher $c$ starting from coset pairings for
meromorphic theories at lower $c$. The present results already suggest many
new possibilities, for example we may conjecture that starting at every
$c^{\cal H}=8(2p+3)$ (where $p\in\mathbb{N}\cup\\{0\\}$) there is an infinite
series labelled by another parameter $m$ with $(n_{1},n_{2})=(p+2,p+m+2)$.
This series is the extension $\mathcal{E}_{1}[D_{8p+12,1}D_{8p+8m+12,1}]$. The
special case for $p=0$ and arbitrary $m$ corresponds to row 17 of table
LABEL:t1 which describes a series of meromorphic theories starting at $c^{\cal
H}=24$ given by $\mathcal{E}_{1}[D_{12,1}D_{8m+12,1}]$. Also the special case
$p=1$ and $m=0$ can be found in row 1 of table LABEL:t3 which describes a
meromorphic theory at $c^{\cal H}=40$ given by,
$\mathcal{E}_{1}[D_{20,1}D_{20,1}]$. Our conjecture subsumes these examples
and suggests infinite generalisations thereof. More generally, we have not yet
considered cases with multiple $B_{r,1}$ or $D_{r,1}$ factors. We hope to
return to all these and many more cases in the future [24].

The recursive nature of the process described here is a potentially very
useful feature. If a meromorphic theory is first discovered by any method at
any central charge $c=8N$, one can attempt to construct its generalisations at
higher values of $c$ using our approach.

Our results provide more evidence for the close relationship between general
RCFTs and meromorphic CFT. It is not true, as was implicitly thought earlier
(and is still occasionally claimed in the literature), that meromorphic
theories constitute some sort of “exotic” outliers in the space of all RCFTs.
Rather, the most general RCFTs are the ones with extensions of the usual
Virasoro and Kac-Moody algebras. The most familiar RCFTs, such as minimal
Virasoro and affine theories, are merely unextended special cases of the RCFT
landscape which is mostly populated by extensions. Meromorphic theories are
such extensions, and are intimately linked to $n$-character RCFT for $n>1$
through coset relations.

## Acknowledgements

AD and CNG would like to thank Jagannath Santara for collaboration on the
papers [3] and [10] and for very helpful discussions. AD would like to thank
Daniele Dorigoni, Nabil Iqbal and Madalena Lemos for useful discussions on Lie
algebras. He would also like to thank Sigma Samhita for her immense help in
the type setting of the tables required for this work. He would also like to
express his gratitude to Gabriel Arenas-Henriquez and Jose Cerqueira-sa for
helpful discussions on Mathematica. SM would like to thank Brandon Rayhaun for
very helpful discussions, and Keshav Dasgupta and Alex Maloney for hospitality
at McGill University, Montreal where part of this work was done. He is also
grateful for support from a grant by Precision Wires India Ltd. for String
Theory and Quantum Gravity research at IISER Pune.

##  Appendix A Some general features of $B_{r,1}$ and $D_{r,1}$ affine
theories

  * •

For $B_{r,1}$ we have the three characters as follows,

    * –

$\chi_{0}$: Identity (adj. irrep): $[0,\cdots,0]$ (which contains information
about the dimension of Lie algebra (= coefficient of $\mathcal{O}(q)$ =
$2r^{2}+r$) and hence is the adjoint irrep of $SO(2r+1)$).

    * –

$\chi_{\frac{1}{2}}$: Fundamental irrep: $\sigma_{1}$ (whose dimension (=
coefficient of $\mathcal{O}(q^{0})$ term = $2r+1$) matches with that of the
fundamental irrep of $SO(2r+1)$).

    * –

$\chi_{\frac{2r+1}{16}}$: Spinor irrep: $\sigma_{r}$ (whose dimension (=
coefficient of $\mathcal{O}(q^{0})$ term = $2^{r}$) matches with that of the
spinor irrep of $SO(2r+1)$)888Dimension of spinorial representation of
$SO(n\,\text{odd})=2^{\frac{n-1}{2}}$.

  * •

For $D_{r,1}$ we have the three characters as follows,

    * –

$\chi_{0}$: Identity (adj. irrep): $[0,\cdots,0]$ (which contains information
about the dimension of Lie algebra (= coefficient of $\mathcal{O}(q)$ =
$2r^{2}-r$) and hence is the adjoint irrep of $SO(2r)$)

    * –

$\chi_{\frac{1}{2}}$: Fundamental irrep: $\sigma_{1}$ (whose dimension (=
coefficient of $\mathcal{O}(q^{0})$ term = $2r$) matches with that of the
fundamental irrep of $SO(2r)$)

    * –

$\chi_{\frac{r}{8}}$: Spinor irrep: $\sigma_{r}\equiv\sigma_{r-1}$ (whose
dimension (= coefficient of $\mathcal{O}(q^{0})$ term = $2^{r-1}$) matches
with that of the spinor irrep of $SO(2r)$)999Dimension of spinorial
representation of $SO(n\,\text{even})=2^{\frac{n}{2}-1}$. In this case (even
dimension) we have chiral spinor irreps ($\sigma_{r}\equiv\sigma_{r-1}$) but
at the level of characters they will be treated on equal footing.

Now note that we have an infinite family whenever we have a factor of
$B_{r_{1},1}$ or $D_{r_{1},1}$ in a theory and the idea is to generate an
infinite set by increasing $r_{1}\to r_{1}+8k$ (with $k\in\mathbb{Z}$). Now,
we can explicitly compute and check the $(d_{1},d_{2})$ values for upto
$c^{\mathcal{H}}=40$ and we observe that we get the same $(d_{1},d_{2})$ for a
given coset pair forming a particular infinite family at every stage of
$c^{\cal H}$. This is because the only thing that differs in the bilinear
relation of one stage to another for a given infinite family is the rank (in
steps of 8) of $B_{r_{1},1}$ or $D_{r_{1},1}$ factors. However, increasing
$r_{1}\to r_{1}+8$ doesn’t change the number of irreps of $B_{r_{1},1}$ or
$D_{r_{1},1}$. Hence, we observe again that the values of $(d_{1},d_{2})$ at
one value of $c^{\cal H}$ will be the same for all other values of $c^{\cal
H}$.

The above discussion shows a very crucial point about 3-characters. As we
discussed above we were able to generate infinite families of meromorphic
theories because each of these theories had a $B_{r_{1},1}$ or $D_{r_{1},1}$
as their factors and as we know from [3], every $B_{r_{1},1}$ or $D_{r_{1},1}$
has 3 characters (except $D_{4,1}$ which has 2 due to triality). Thus, we
could generalise the Schellenkens coset pairs which always resulted in a
9-character theory and from Schellekens we know that the non-$B_{r_{1},1}$ or
non-$D_{r_{1},1}$ factors always admit a unique $3$-character extension.

##  Appendix B Relevant admissible character solutions

In this section we present (for completeness) the relevant admissible
character solutions to the $(3,0)$ MLDE and GHM solutions from [10] and [19]
respectively.

Table 7: Some admissible character solutions to the $(3,0)$ MLDE and GHM solutions. # | $c$ | $h_{1}$ | $h_{2}$ | $m_{1}$ | $D_{1}$ | $D_{2}$ | # | $c$ | $h_{1}$ | $h_{2}$ | $m_{1}$ | $D_{1}$ | $D_{2}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
${\bf III_{22}}$ | $\frac{68}{5}$ | $\frac{4}{5}$ | $\frac{7}{5}$ | $136$ | $119$ | $68$ | ${\bf III_{37}}$ | $\frac{92}{5}$ | $\frac{6}{5}$ | $\frac{8}{5}$ | $92$ | $1196$ | $299$
${\bf V_{39}}$ | $20$ | $\frac{4}{3}$ | $\frac{5}{3}$ | $80$ | $5$ | $4$ | ${\bf III_{45}}$ | $22$ | $\frac{3}{2}$ | $\frac{7}{4}$ | $66$ | $77$ | $11$
${\bf III_{50}}$ | $23$ | $\frac{3}{2}$ | $\frac{15}{8}$ | $23$ | $575$ | $23$ |  |  |  |  |  |  |
$\text{GHM}_{45}$ | $\frac{45}{2}$ | $\frac{3}{2}$ | $\frac{29}{16}$ | $45$ | $4785$ | $45$ | $\text{GHM}_{86}$ | $\frac{43}{2}$ | $\frac{3}{2}$ | $\frac{27}{16}$ | $86$ | $5031$ | $43$
$\text{GHM}_{105}$ | $21$ | $\frac{3}{2}$ | $\frac{13}{8}$ | $105$ | $637$ | $21$ | $\text{GHM}_{120}$ | $20$ | $\frac{7}{5}$ | $\frac{8}{5}$ | $120$ | $4$ | $13$
$\text{GHM}_{123}$ | $\frac{41}{2}$ | $\frac{3}{2}$ | $\frac{25}{16}$ | $123$ | $5125$ | $41$ | $\text{GHM}_{156}$ | $\frac{39}{2}$ | $\frac{3}{2}$ | $\frac{23}{16}$ | $156$ | $5083$ | $39$
$\text{GHM}_{171}$ | $19$ | $\frac{3}{2}$ | $\frac{11}{8}$ | $171$ | $627$ | $19$ | $\text{GHM}_{185}$ | $\frac{37}{2}$ | $\frac{3}{2}$ | $\frac{21}{16}$ | $185$ | $4921$ | $2368$
$\text{GHM}_{198}$ | $18$ | $\frac{3}{2}$ | $\frac{5}{4}$ | $198$ | $75$ | $9$ | $\text{GHM}_{210}$ | $\frac{35}{2}$ | $\frac{3}{2}$ | $\frac{19}{16}$ | $210$ | $4655$ | $35$
$\text{GHM}_{221}$ | $17$ | $\frac{3}{2}$ | $\frac{9}{8}$ | $221$ | $561$ | $17$ | $\text{GHM}_{255}$ | $15$ | $\frac{3}{2}$ | $\frac{7}{8}$ | $255$ | $455$ | $15$

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