Jan 2021 (v2.1)

arXiv:2011.07922 [hep-th]

Mathieu Moonshine and Siegel Modular Forms
Suresh Govindarajan† and Sutapa Samanta∗

†Department of Physics, Indian Institute of Technology Madras , Chennai 600036
INDIA

∗School of Physical Sciences, Indian Association for the Cultivation of
Science, Jadavpur, Kolkata 700032, INDIA

Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
A second-quantized version of Mathieu moonshine leads to product formulae for
functions that are potentially genus-two Siegel Modular Forms analogous to the
Igusa Cusp Form. The modularity of these functions do not follow in an obvious
manner. For some conjugacy classes, but not all, they match known modular
forms. In this paper, we express the product formulae for all conjugacy
classes of $M_{24}$ in terms of products of standard modular forms. This
provides a new proof of their modularity.

##  1 Introduction

Modular forms are important ingredients in string theory as well as in
mathematical physics. The generating function of a $\frac{1}{2}$-BPS state of
$\mathcal{N}=4$ superstring theory obtained by compactifying the heterotic
string on a six-torus is a modular form. Using dualities, the $\mathcal{N}=4$
superstring theory in four dimensions can also be obtained by compactifying
type II string theory on $K3\times T^{2}$. The generating function of the
degeneracy of electrically charged $\tfrac{1}{2}$-BPS state is given by the
twenty fourth power of the Dedekind $\eta$-function[1]. When the six-torus is
twined by a symmetry element, then the generating function is given by a
multiplicative $\eta$ product for a cycle shape that is associated with a
conjugacy class of $M_{24}$[2]. Given a cycle shape $\rho$, the $\eta$-product
associated with the cycle shape are defined by the following map:

$\rho=1^{a_{1}}2^{a_{2}}\cdots
n^{a_{n}}\quad\longrightarrow\quad\eta_{\rho}(\tau)=\eta(\tau)^{a_{1}}\eta(2\tau)^{a_{2}}\cdots\eta(n\tau)^{a_{n}}\
.$

A cycle shape is called balanced if there exists a positive integer $M$ such
that

$\rho=1^{a_{1}}2^{a_{2}}\cdots
n^{a_{n}}=\left(\tfrac{M}{1}\right)^{a_{1}}\left(\tfrac{M}{2}\right)^{a_{2}}\cdots\left(\tfrac{M}{n}\right)^{a_{n}}.$

Conway and Norton [3] observed that all the conjugacy classes of the Mathieu
group $M_{24}$ are given by a balanced cycle shape. Thus there is a connection
between the generating function of the $\frac{1}{2}$ BPS states and the
conjugacy classes of the Mathieu group. This is known as Mathieu moonshine [4,
5, 2].

Another Mathieu moonshine was discovered by Eguchi, Ooguri, and Tachikawa [6]
that relates the conjugacy class $1a$ of $M_{24}$ with the elliptic genus of
$K3$. This elliptic genus is a weight zero and index one Jacobi form. Later,
the Jacobi forms for all other conjugacy classes are constructed [7, 8, 9].
Gannon proved the existence of a class function that evaluates to all these
Jacobi forms[10]. We list the cycle shapes and the conjugacy classes of
$M_{24}$ in Appendix C.

Now, if we consider the $\frac{1}{4}$-BPS states of the same $\mathcal{N}=4$
superstring theory as mentioned earlier, the generating functions of the
degeneracy are genus-two Siegel Modular Forms in some cases[2]. In [2], the
authors proposed the following map relating cycle shapes to Siegel Modular
Forms:

$\rho=1^{a_{1}}2^{a_{2}}\cdots
n^{a_{n}}\quad\longrightarrow\quad\Phi_{k}^{\rho}(\mathbf{Z})\ ,$

where $\Phi_{k}^{\rho}(\mathbf{Z})$ is a Siegel Modular Form with weight
$k=(-2+\sum_{i}a_{i})$ of a suitable subgroup of $Sp(4,\mathbb{Z})$. For
conjugacy class $1a$, this is the well-known weight ten Igusa cusp form[1].
The two Mathieu moonshines were combined in a second-quantized version leading
to class function that implied a product formula for
$\Phi_{k}^{\rho}(\mathbf{Z})$ for all conjugacy classes for $M_{24}$ [11]. The
proof that these product formulae are indeed Siegel Modular Forms for all
conjugacy classes first appeared in [12]. It was also proven for a sub-class
that appear in the context of $L_{2}(11)$ moonshine in [13]. The square-root
of some of these Siegel Modular Forms also appear in the denominator formula
of Borcherds-Kac-Moody Lie super-algebra [2, 14, 15, 13, 16].

In this paper, we provide a new and distinct proof that the
$\Phi_{k}^{\rho}(\mathbf{Z})$ are Siegel Modular Forms. The various
connections are described in Fig. 1.

$\rho\in
M_{24}$$Z^{\rho}(\tau,z)$$\phi_{k,1}^{\rho}(\tau,z)$$\Phi_{k}^{\rho}(\mathbf{Z})$multiplicative
liftzeroth FJ Coeff. Figure 1: Moonshine for $M_{24}$.
$\phi_{k,1}^{\rho}(\tau,z)$ is constructed from the multiplicative
$\eta$-products and $Z^{\rho}(\tau,z)$ is the twined elliptic genus.
$\Phi_{k}^{\rho}(\mathbf{Z})$ is the genus-two Siegel Modular Form.

The main result of this paper is the following proposition that we prove.

######  Proposition 2.2.

For $g\in M_{24}$, let $\rho_{m}=[g^{m}]$, $\rho=\rho_{1}$,
$Z_{0,1}^{\rho_{m}}(\tau,z)$ be the Jacobi form of weight zero and index one
corresponding to $g^{m}$ and $\Phi_{k}^{\rho}(\mathbf{Z})$ be the modular form
defined by the multiplicative lift given in Eq. (2.6). Then

$\Big{(}\Phi_{k}^{\rho}(\mathbf{Z})\Big{)}^{p_{\rho}}=\prod_{m|p_{\rho}}\left(B_{\frac{p_{\rho}}{m}Z^{\rho_{m}}}(m\,\mathbf{Z})\right)\
,$

where $p_{\rho}$ is the length of the shortest cycle in the cycle shape for
the conjugacy class $\rho$ and $B_{\psi}(\mathbf{Z})$ is the Siegel Modular
Form given by Theorem 2.1.

Remark: Our proof differs from the one by Persson-Volpato[12] in that we show
that the product formula for the conjugacy class of an order $N$ element of
$M_{24}$ correspond to Siegel Modular Forms at level $N$ of $Sp(4,\mathbb{Z})$
with character corresponding to a $p_{\rho}$-th root of unity, where
$p_{\rho}$ is as defined above. In [12], it was shown that these products
correspond to subgroup at higher level $p_{\rho}N$.111We thank Daniel Persson
and Roberto Volpato for a useful email correspondence. Thus our result
complements theirs. Our proof also explicitly connects to Borcherds products
that appear in number theory. In the process we obtain a simple looking
formula that holds for all conjugacy classes.

After the introductory section, in Section 2, we discuss the ideas leading to
Proposition 2.2 and a sketch of its proof using two illustrative examples.
Section 3 summarises our results and concluding remarks. Appendix A provides
the definitions and examples of modular forms and Jacobi forms. Appendix B
lists the Jacobi forms for all conjugacy classes of $M_{24}$. A fairly long
appendix D provides the details of the proof of Proposition 2.2.

##  2 Construction of Siegel Modular Forms for $M_{24}$

In this section, we construct genus two Siegel Modular Forms for all the
conjugacy classes of $M_{24}$. We know that there’s a weight zero and index
one Jacobi form (see Appendix C) [6, 7, 8, 9] for every conjugacy class of
$M_{24}$. We apply the multiplicative lift to them to construct the Siegel
Modular Forms. This multiplicative lift naturally leads to a product formula.
We prove the modularity of the construction in two ways – (i) one is to
construct the sum side using the additive lift and showing that this is equal
to the product formula, (ii) the other is to compare the product formula with
the Borcherds products. The first method works in most cases, while the second
method works for all the cases.

###  2.1 The multiplicative lift

The Mathieu moonshine leads to a Siegel Modular Form which is denoted by
$\Phi_{k}^{\rho}(\mathbf{Z})$. The formula for $\Phi_{k}^{\rho}(\mathbf{Z})$
is given by [11]

$\displaystyle\Phi_{k}^{\rho}=s\,\phi^{\rho}_{k,1}\times\exp\left(-\sum_{m=1}^{\infty}s^{m}V_{m}\cdot
Z_{0,1}^{\rho}(\tau,z)\right)\ ,$ (2.1)

where $V_{m}$ is the generalized Hecke-like operator. This operator acts on a
weight zero and index one Jacobi form as follows

$\displaystyle V_{m}\cdot
Z_{0,1}^{\rho}(\tau,z)\equiv\frac{1}{m}\sum_{ad=m}\sum_{b=0}^{d-1}Z_{0,1}^{\rho_{a}}\left(\frac{a\tau+b}{d},az\right)\
.$ (2.2)

where $\rho$ denotes the conjugacy class of the element $g$ of order $N$. Then
$\rho_{a}$ denotes the conjugacy class of the element $g^{a}$. We set
$\rho_{0}=1^{24}$. The Fourier-Jacobi coefficients $f_{a}(n,\ell)$ of
$Z^{\rho_{a}}_{0,1}(\tau,z)$ is defined as follows:

$\displaystyle
Z_{0,1}^{\rho_{a}}(\tau,z)=\sum_{n=0}^{\infty}\sum_{\ell\in\mathbb{Z}}f_{a}(n,\ell)q^{n}r^{\ell}\
.$ (2.3)

Let us now define $c^{\alpha}(n,\ell)$ via following discrete Fourier
transform.

$\displaystyle
f_{a}(n,\ell)=\sum_{\alpha=0}^{N-1}\left(\omega_{N}^{a}\right)^{\alpha}c^{\alpha}(n,\ell)\
,$ (2.4)

where $\omega_{N}=\exp(2\pi i/N)$. The inverse Fourier transform of Eq. (2.4)
is given by

$\displaystyle c^{\alpha}=\frac{1}{N}\sum_{a=0}^{N-1}\exp(-2\pi
ia\alpha/N)f_{a}\ .$ (2.5)

Using (2.2), we obtain a product formula for (2.1) as follows:

$\displaystyle\Phi_{k}^{\rho}(\mathbf{Z})=s\,\phi_{k,1}^{\rho}\times\prod_{m=1}^{\infty}\prod_{\alpha=0}^{N-1}\prod_{n=0}^{\infty}\prod_{\begin{subarray}{c}\ell\in\mathbb{Z}\\\
4nm-\ell^{2}\geq
0\end{subarray}}\left(1-\omega_{n}^{\alpha}q^{n}r^{\ell}s^{m}\right)^{c^{\alpha}(nm,\ell)}\
,$ (2.6)

where the zeroth Fourier-Jacobi term, i.e., the coefficient of $s^{1}$ in the
Fourier expansion is given by

$\phi_{k,1}^{\rho}(\tau,z)=\frac{\vartheta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\
\eta_{\rho}(\tau)\ .$

Remark: Let $p_{\rho}$ (resp. $N_{\rho}$) be the length of the smallest (resp.
largest) cycle in $\rho$. From the result of Cheng and Duncan[17], one has
that $\eta_{\rho}(\tau)^{p_{\rho}}$ (and hence
$[\phi_{k,1}^{\rho}(\tau,z)]^{p_{\rho}}$) is a modular (Jacobi) form for the
subgroup $\Gamma_{0}(N_{\rho})$ of $SL(2,\mathbb{Z})$. We thus anticipate that
$[\Phi_{k}^{\rho}(\mathbf{Z})]^{p_{\rho}}$ will be a Siegel Modular Form for a
level $N_{\rho}$ subgroup of $Sp(4,\mathbb{Z})$.

For the cases when the order of $g(\in M_{24})$ is prime (i.e.,
$N=2,3,5,7,11,23$), one has the conjugacy class $g^{a}$ for $a\neq 0$ mod $N$
is the same as that of $g$. Thus, one has $\rho_{a}=\rho$ for $a\neq 0$ mod
$N$. Now using Eq. (2.5) we get

$c^{0}=\frac{1}{N}(f_{0}+(N-1)f_{1})\hskip 8.5359pt\text{and}\hskip
8.5359ptc^{1}=\frac{1}{N}(f_{0}-f_{1})\ .$

For these cases, on using the product representation for the theta and
$\eta$-functions that appear in $\phi^{\rho}_{k,1}(\tau,z)$ and using
$\prod_{n=0}^{N-1}(1-\omega^{n}x)=(1-x^{N})$, the formula given in Eq. (2.6)
simplifies to

$\displaystyle\Phi_{k}^{\rho}$
$\displaystyle=qrs\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{c^{0}(nm,\ell)-c^{1}(nm,\ell)}\Big{(}1-q^{nN}r^{\ell
N}s^{mN}\Big{)}^{c^{1}(nm,\ell)}\ ,$ (2.7)
$\displaystyle=qrs\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-q^{nN}r^{\ell
N}s^{mN}\Big{)}^{\frac{f_{0}(nm,\ell)-f_{1}(nm,\ell)}{N}}\ ,$ (2.8)

where $(n,\ell,m)>0$ implies $n>0,$ or $n=0$ and $m>0$, or $n=m=0$ and
$\ell<0$. We will treat the cases of composite $N$ separately.

###  2.2 The additive lift

The another construction of $\Phi^{\rho}(\mathbf{Z})$ is via additive lift.
The additive seed is the weak Jacobi form of weight $k$ and index $t$ for the
sub-group $\Gamma_{0}(Nq,q)$. This index $t$ can be integral or half integral.
Let us consider a Jacobi form of weight $k$ and index $t$,
$\phi_{k,t}(\tau,z)$. Then for $k>0$, the additive lift is given by

$\displaystyle\Phi^{\rho}(\mathbf{Z})=\sum_{m=1\mod
q}s^{mt}\phi^{\rho}|_{k,t}T_{-}(m)(\tau,z)\ ,$ (2.9)

where $T_{-}(m)$ is the Hecke-like operator introduced by Cléry and Gritsenko
[18] and is given by

$\displaystyle T_{-}(m):=\sum_{\begin{subarray}{c}ad=m\\\ (a,N)=1\\\
a>0\end{subarray}}\Gamma_{1}(Nq,q)\,\sigma_{a}\begin{pmatrix}a&bq\\\
0&d\end{pmatrix}\ ,$ (2.10)

where $\sigma_{a}=\left(\begin{smallmatrix}a^{-1}&0\\\
0&a\end{smallmatrix}\right)\mod Nq$ and $(m,q)=1$. For all $M_{24}$ conjugacy
classes ($\rho$), we have $t=1$ and $q=1$.

###  2.3 Modularity by comparing with the additive side

The additive seed for these cases are Jacobi forms of weight $k$ and index
one,

$\displaystyle\phi_{k,1}^{\rho}(\tau,z)=\frac{\vartheta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\
\eta_{\rho}(\tau)=\sum_{n,\ell}a(n,\ell)q^{n}r^{\ell}\ .$ (2.11)

A necessary but not sufficient condition for the compatibility of the additive
lift with the multiplicative lift is:

$\displaystyle\left.\left[\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta_{\rho}(\tau)\right]\right|_{k,1}T_{-}(2)(\tau,z)=-Z^{\rho}(\tau,z)\left[\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta_{\rho}(\tau)\right]\
.$ (2.12)

This is the coefficient of $s$ in both the lifts, i.e., the ones given in Eq.
(2.1) and Eq. (2.9). This condition holds for all the cycle shapes of $M_{24}$
except for the following eight cycle shapes where we observe that [19]

$\displaystyle-\frac{T_{2}\phi^{1^{2}11^{2}}(\tau,z)}{\phi^{1^{2}11^{2}}(\tau,z)}-Z^{1^{2}11^{2}}(\tau,z)=\frac{11}{2}\phi^{1^{2}11^{2}}(\tau,z)\
,$
$\displaystyle-\frac{T_{2}\phi^{1^{1}2^{1}7^{1}14^{1}}(\tau,z)}{\phi^{1^{1}2^{1}7^{1}14^{1}}(\tau,z)}-Z^{1^{1}2^{1}7^{1}14^{1}}(\tau,z)=7\phi^{1^{1}2^{1}7^{1}14^{1}}(\tau,z)\
,$
$\displaystyle-\frac{T_{2}\phi^{1^{1}3^{1}5^{1}15^{1}}(\tau,z)}{\phi^{1^{1}3^{1}5^{1}15^{1}}(\tau,z)}-Z^{1^{1}3^{1}5^{1}15^{1}}(\tau,z)=\frac{15}{2}\phi^{1^{1}3^{1}5^{1}15^{1}}(\tau,z)\
,$ (2.13)
$\displaystyle-\frac{T_{2}\phi^{2^{2}10^{2}}(\tau,z)}{\phi^{2^{2}10^{2}}(\tau,z)}-Z^{2^{2}10^{2}}(\tau,z)=10\phi^{2^{2}10^{2}}(\tau,z)\
,$
$\displaystyle-\frac{T_{2}\phi^{2^{1}4^{1}6^{1}12^{1}}(\tau,z)}{\phi^{2^{1}4^{1}6^{1}12^{1}}(\tau,z)}-Z^{2^{1}4^{1}6^{1}12^{1}}(\tau,z)=12\phi^{2^{1}4^{1}6^{1}12^{1}}(\tau,z)\
,$

and

$\displaystyle-\frac{T_{2}\phi^{4^{6}}(\tau,z)}{\phi^{4^{6}}(\tau,z)}-Z^{4^{6}}(\tau,z)=16\frac{\eta(2\tau)^{4}\eta(8\tau)^{4}}{\eta(4\tau)^{4}}\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\
,$
$\displaystyle-\frac{T_{2}\phi^{3^{8}}(\tau,z)}{\phi^{3^{8}}(\tau,z)}-Z^{3^{8}}(\tau,z)=18\frac{\eta(\tau)^{3}\eta(9\tau)^{3}}{\eta(3\tau)^{2}}\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\
,$ (2.14)
$\displaystyle-\frac{T_{2}\phi^{6^{4}}(\tau,z)}{\phi^{6^{4}}(\tau,z)}-Z^{6^{4}}(\tau,z)=2\left(\eta_{6^{4}}(\tau)+\frac{\eta(\tau)^{3}\eta(9\tau)^{3}}{\eta(3\tau)^{2}}+6\frac{\eta(2\tau)^{3}\eta(18\tau)^{3}}{\eta(6\tau)^{2}}\right.$
$\displaystyle\hskip
170.71652pt\left.+\frac{\eta(4\tau)^{3}\eta(36\tau)^{3}}{\eta(12\tau)^{2}}+\cdots\right)\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\
,$

where $T_{2}\phi^{\rho}$ is short for $\phi^{\rho}|_{k,1}T_{-}(2)$. All the
above Jacobi forms except the ones correspond to $3^{8}$ and $4^{6}$ are of
weight $k=0$ and we have naïvely applied a formula which assumes $k>0$.

Even for the cases for which the compatibility condition Eq. (2.12) holds, we
need to prove that all other terms also match. We do not pursue this method
here. We prove the modularity using another method which we discuss next.

###  2.4 Modularity by comparing with a Borcherds formula

We begin with a theorem due to Cléry-Gritsenko (see also [20, 21, 22]) that
leads to a Borcherds product formula for a meromorphic Siegel Modular Form
starting from a nearly holomorphic Jacobi form of weight zero and index $t$.

######  Theorem 2.1 (Cléry-Gritsenko[18]).

Let $\psi$ be a nearly holomorphic Jacobi form of weight $0$ and index $t$ of
$\Gamma_{0}(N)$. Assume that for all cusps of $\Gamma_{0}(N)$ one has
$\frac{h_{e}}{N_{e}}c_{f/e}(n,\ell)\in\mathbb{Z}$ if $4nt-\ell^{2}\leq 0$.
Then the product

$B_{\psi}(\mathbf{Z})=q^{A}r^{B}s^{C}\prod_{f/e\in\mathcal{P}}\prod_{\begin{subarray}{c}n,\ell,m\in\mathbb{Z}\\\
(n,\ell,m)>0\end{subarray}}\Big{(}1-(q^{n}r^{\ell}s^{tm})^{N_{e}}\Big{)}^{\frac{h_{e}}{N_{e}}c_{f/e}(nm,\ell)}\
,$

with

$A=\frac{1}{24}\sum_{\begin{subarray}{c}f/e\in\mathcal{P}\\\
\ell\in\mathbb{Z}\end{subarray}}h_{e}\,c_{f/e}(0,\ell),\
B=\frac{1}{2}\sum_{\begin{subarray}{c}f/e\in\mathcal{P}\\\
\ell\in\mathbb{Z}_{>0}\end{subarray}}\ell h_{e}\,c_{f/e}(0,\ell),\
C=\frac{1}{4}\sum_{\begin{subarray}{c}f/e\in\mathcal{P}\\\
\ell\in\mathbb{Z}\end{subarray}}\ell^{2}h_{e}\,c_{f/e}(0,\ell)\ ,$

defines a meromorphic Siegel Modular Form of weight

$k=\frac{1}{2}\sum_{\begin{subarray}{c}f/e\in\mathcal{P}\\\
\ell\in\mathbb{Z}\end{subarray}}\frac{h_{e}}{N_{e}}c_{f/e}(0,0)$

with respect to $\mathbf{\Gamma}_{t}(N)^{+}$ possibly with character. The
character is determined by the zeroth Fourier-Jacobi coefficient of
$B_{\psi}(\mathbf{Z})$, i.e., the coefficient of $s^{C}$ in the Fourier
expansion, which is a Jacobi form of weight $k$ and index $C$ of the Jacobi
subgroup of $\mathbf{\Gamma}_{t}(N)^{+}$.

Here $\mathcal{P}$ are the set of cusps for $\Gamma_{0}(N)$, $h_{e}$ is the
width of the cusp at $f/e\in\mathcal{P}$ and $N_{e}=N/e$. For any congruence
subgroup $\Gamma$, the cusps are defined as the $\Gamma$-equivalent classes of
$\mathbb{Q}\cup\\{\infty\\}$ where $\mathbb{Q}$ denotes the set of rational
numbers. The width of the cusp $\alpha\in\mathcal{P}$ is the minimal $h_{e}$
such that $(\begin{smallmatrix}1&h_{e}\\\
0&1\end{smallmatrix})\in\gamma^{-1}\Gamma\gamma$, where $\gamma\in
SL(2,\mathbb{Z})$ such that $\gamma(\infty)=\alpha$.

Remark: As discussed in [13], the condition on $c_{f/e}(n,\ell)$ for
$4nt-\ell^{2}\leq 0$ that $\frac{h_{e}}{N_{e}}c_{f/e}(n,\ell)\in\mathbb{Z}$
will be relaxed as follows. For all cusps that have identical values of
$(N_{e},h_{e})$, we require that the sum of
$\frac{h_{e}}{N_{e}}c_{f/e}(n,\ell)$ (with $4nt-\ell^{2}\leq 0$) for all such
cusps be integral. This ensures that $B_{\psi}$ has only zeros or poles at all
divisors – this is the reason for the condition in the theorem.

Define the projection[23], $\pi_{FE}$, as follows

$\pi_{FE}\left(\phi|M_{f/e}\right)(\tau,z):=\frac{1}{h_{e}}\sum_{b=0}^{h_{e}-1}\phi|M_{f/e}(\tau+b,z)\
,$ (2.15)

where $M_{f/e}=\left(\begin{smallmatrix}f&*\\\ e&*\end{smallmatrix}\right)\in
SL(2,\mathbb{Z})$ maps the cusp at $i\infty$ to $f/e$. It is the Fourier
coefficients of the projected Jacobi form defined above that appears in the
product formula.

Raum proved the modularity of the products such as those given in Eq. (2.6) by
considering products of rescaled Borcherds products[23]. His results were
however restricted to only conjugacy classes associated with elements of
$M_{24}$ that were primes or powers of primes. Further, there were some
computational errors in his work that we fix as well. We not only extend his
results but also provide a systematic method of obtaining the precise rescaled
Borcherds products that are needed for all conjugacy classes. For all type-I
conjugacy classes of $M_{24}$, we find that the product formula is equivalent
to a single Borcherds formula. More generally, we find that the following
holds.

######  Proposition 2.2.

For $g\in M_{24}$, let $\rho_{m}=[g^{m}]$, $\rho=\rho_{1}$,
$Z_{0,1}^{\rho_{m}}(\tau,z)$ be the Jacobi form of weight zero and index one
corresponding to $g^{m}$ and $\Phi_{k}^{\rho}(\mathbf{Z})$ be the modular form
defined by the multiplicative lift given in Eq. (2.6). Then

$\Big{(}\Phi_{k}^{\rho}(\mathbf{Z})\Big{)}^{p_{\rho}}=\prod_{m|p_{\rho}}\left(B_{\frac{p_{\rho}}{m}Z^{\rho_{m}}}(m\,\mathbf{Z})\right)\
,$

where $p_{\rho}$ is the length of the shortest cycle in the cycle shape for
the conjugacy class $\rho$ and $B_{\psi}(\mathbf{Z})$ is the Siegel Modular
Form given by Theorem 2.1.

###### Proof.

As the proof is mostly done by exhaustion, we illustrate this theorem for two
examples, one with $p_{\rho}=1$ when $N$ is prime and one with $p_{\rho}\neq
1$ here. All other conjugacy classes are worked out in the Appendix E.

$\boxed{\mathbf{p_{\rho}=1}}$

For all type-I conjugacy classes of $M_{24}$ i.e with $p_{\rho}=1$, we see
that the multiplicative lift can be expressed in terms single Borcherds
product. It is very easy to see for all the conjugacy classes with
$N=1,2,3,5,7,11,23$ where $N$ is the order of the element $g\in M_{24}$. For
prime $N$, there are only two cusps, one is at $i\infty$ (which is
$\Gamma_{0}(N)$ equivalent to $1/N$) and another is at $0/1$. To prove the
equality of multiplicative lift and the Borcherds product, we need to show
that

$\displaystyle\pi_{FE}(Z^{\rho})$ $\displaystyle=Z^{\rho}\ ,$ (2.16)
$\displaystyle\pi_{FE}(Z^{\rho}|S)$
$\displaystyle=\frac{1}{N}\left(Z^{1^{24}}-Z^{\rho}\right)\ .$ (2.17)

The first equation holds trivially since $Z^{\rho}$ has only integral powers
of $q$ in its Fourier-Jacobi expansion. The second part follows from a
calculation.

$\pi_{FE}(Z^{\rho}|S)(\tau,z)=\frac{2}{N+1}\phi_{0,1}(\tau,z)+\pi_{FE}(\alpha^{(N)}|S)(\tau)\phi_{-2,1}(\tau,z)\
,$

where $\alpha^{(N)}(\tau)=\frac{2N}{N+1}E_{2}^{(N)}(\tau)$ for $N=2,3,5,7$.
For $N=11$ and $23$, one has

$\displaystyle\alpha^{(11)}(\tau)$
$\displaystyle=\frac{11}{6}E_{2}^{(11)}(\tau)-\frac{22}{5}\eta_{1^{2}11^{2}}(\tau)\
,$ $\displaystyle\alpha^{(23)}(\tau)$
$\displaystyle=\tfrac{23}{12}E_{2}^{(23)}(\tau)-\tfrac{23}{22}f_{23,1}(\tau)-\tfrac{161}{22}f_{23,2}(\tau)\
,$

where $f_{23,a}(\tau)$ are defined in Appendix B.1. For $N=2,3,5,7$, computing
$\pi_{FE}\left(\alpha^{(N)}|S\right)(\tau)$, we obtain

$\pi_{FE}(\alpha^{(N)}|S)(\tau)=-\tfrac{2}{N(N+1)}\sum_{b=0}^{N-1}E_{2}^{(N)}(\tfrac{\tau+b}{N})=-\tfrac{2}{(N+1)}E_{2}^{(N)}\big{|}_{2}U_{N}=-\tfrac{2}{(N+1)}E_{2}^{(N)}(\tau)\
,$

where $U_{N}$ is the Hecke operator for $\Gamma_{0}(N)$ (defined by Atkin and
Lehner [24]) and $E_{2}^{(N)}$ is its eigenform with eigenvalue $+1$. Thus, we
get

$\displaystyle\pi_{FE}(Z^{\rho}|S)(\tau,z)$
$\displaystyle=\frac{2}{N+1}\phi_{0,1}(\tau,z)-\frac{1}{N}\alpha^{(N)}(\tau)\phi_{-2,1}(\tau,z)$
$\displaystyle=\frac{2}{N}\phi_{0,1}(\tau,z)-\frac{1}{N}Z^{\rho}(\tau,z)$
$\displaystyle=\frac{1}{N}\left(Z^{1^{24}}-Z^{\rho}\right)(\tau,z)\ ,$

which establishes Eq. (2.17) for prime $N=2,3,5,7$. A similar computation
holds for $N=11$ and $23$. For $N=11$, one can show that

$\pi_{FE}(\alpha^{(11)}|S)(\tau)=-(1/11)\alpha^{(11)}(\tau)\ ,$

and for $N=23$, one can show that

$\pi_{FE}(\alpha^{(23)}|S)(\tau)=-(1/23)\alpha^{(23)}(\tau)\ .$

Thus Eq. (2.17) holds for $N=11$ and $23$ as well.

$\boxed{\mathbf{p_{\rho}\neq 1}}$

Let us consider the cycle shape $12^{2}$. The multiplicative lift leads to the
product formula given by (data about the cusps are given in E.2.8)

$\Phi^{12^{2}}_{-1}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{2}\Big{)}^{x_{1}}\Big{(}1-(q^{n}r^{\ell}s^{m})^{3}\Big{)}^{x_{2}}\\\
\Big{(}1-(q^{n}r^{\ell}s^{m})^{4}\Big{)}^{x_{3}}\Big{(}1-(q^{n}r^{\ell}s^{m})^{6}\Big{)}^{x_{4}}\Big{(}1-(q^{n}r^{\ell}s^{m})^{12}\Big{)}^{x_{5}}\
,$ (2.18)

where

$\displaystyle x_{1}=\frac{1}{2}(f_{2}(nm,\ell)-f_{1}(nm,\ell)),\hskip
14.22636ptx_{2}=\frac{1}{3}(f_{3}(nm,\ell)-f_{1}(nm,\ell)),$ $\displaystyle
x_{3}=\frac{1}{4}(-f_{2}(nm,\ell)+f_{4}(nm,\ell)),$ $\displaystyle
x_{4}=\frac{1}{3}(f_{1}(nm,\ell)-f_{2}(nm,\ell)-f_{3}(nm,\ell)+f_{6}(nm,\ell)),$
$\displaystyle
x_{5}=\frac{1}{12}(f_{0}(nm,\ell)+f_{2}(nm,\ell)-f_{4}(nm,\ell)-f_{6}(nm,\ell))\
.$

Now to see if this can be written as the Borcherds product as stated in
Theorem 2.1 we need to compute the coefficients at different cusps. We need to
combine cusps with same $(h_{e},N_{e})$ value and one can show that the only
non-zero contributions are the following:

$\displaystyle\pi_{FE}\Big{(}Z^{12^{2}}|M_{\frac{1}{72}}\Big{)}$
$\displaystyle=-Z^{12^{2}},$
$\displaystyle\pi_{FE}\Big{(}Z^{12^{2}}|M_{\frac{5}{48}}+Z^{12^{2}}|M_{\frac{1}{48}}\Big{)}$
$\displaystyle=-Z^{12^{2}},$
$\displaystyle\pi_{FE}\Big{(}Z^{12^{2}}|M_{\frac{5}{24}}+Z^{12^{2}}|M_{\frac{1}{24}}\Big{)}$
$\displaystyle=Z^{12^{2}}\ .$

Using these and the product formula for Borcherds product given by Cléry-
Gritsenko, we get

$B_{Z^{12^{2}}}(\mathbf{Z})=\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{2}\Big{)}^{-\frac{1}{2}f_{1}(nm,\ell)}\\\
\Big{(}1-(q^{n}r^{\ell}s^{m})^{3}\Big{)}^{-\frac{1}{3}f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{6}\Big{)}^{\frac{1}{6}f_{1}(nm,\ell)}\
.$ (2.19)

This does not give all the terms that appear in the product form given in Eq.
(2.18). The missing terms can be accounted for by additional terms leading to

$\displaystyle\Big{(}\Phi_{-1}^{12^{2}}(\mathbf{Z})\Big{)}^{12}=B_{12Z^{12^{2}}}(\mathbf{Z})B_{6Z^{6^{4}}}(2\mathbf{Z})B_{4Z^{4^{6}}}(3\mathbf{Z})B_{3Z^{3^{8}}}(4\mathbf{Z})B_{2Z^{2^{12}}}(6\mathbf{Z})B_{Z^{1^{24}}}(12\mathbf{Z})\
.$ (2.20)

$\Big{(}\Phi_{-1}^{12^{2}}(\mathbf{Z})\Big{)}^{12}$ is a meromorphic Siegel
Modular Form of weight $(-12)$ at level 24. The zeroth Fourier-Jacobi term is
given by the additive seed
$\phi_{-1,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta(12\tau)^{2}$.
∎

##  3 Conclusions

We proved that there is a genus-two Siegel Modular Form (of a level $N$
subgroup of $Sp(2,\mathbb{Z})$) for every conjugacy class of $M_{24}$. Each of
these arise on evaluating the class function for $M_{24}$ in Eq. (2.1) on the
conjugacy class. This was achieved by expressing a product formula implied by
Mathieu moonshine as rescaled product of the Borcherds products for all
conjugacy classes. This makes their modularity manifest. This completes the
proof that was initiated by Raum[23] as well as resolving puzzles raised in
[11]. Our result also corrects Raum’s incorrect results for the cycle shapes
$2^{12},3^{8},2^{4}4^{4},4^{6}$, and $1^{2}2^{1}4^{1}8^{2}$.

Generalized Mathieu moonshine deals with pairs of commuting elements of
$(g,h)\in M_{24}$. To each pair, we have an eta product as well as a Jacobi
form[25]. Again, one has a product formula implied by second-quantized
moonshine[11, 12]. Their modularity has been proved in [12]. Our methods can
be used to provide explicit formulae similar to the ones obtained in this
paper.

Umbral moonshine is a generalization of Mathieu moonshine[26, 27]. These
connect finite groups, vector valued mock modular forms to the Niemeier
lattices in 24 dimensions. In a recent paper [16], we have shown that there
exist Siegel Modular Forms for the $A$-series of umbral moonshine. It is of
interest if such Siegel Modular Forms appear for all other cases.

It has been shown that there is no point in the space of CFTs associated with
the K3 sigma model where $M_{24}$ appears as a symmetry[28]. The best known
example obtains $\mathbb{Z}^{8}_{2}$ : $\mathbb{M}_{20}$ symmetry[29]. There
has been an interesting attempt to combine symmetries at different points in
the CFT moduli space and obtain all of $M_{24}$[30]. It remains to be seen if
there is a natural construction of a $M_{24}$ modules that leads to the class
functions associated with the Jacobi and Siegel Modular Forms.

Acknowledgments: We thank Mohammed Shabbir and S. Viwanath for useful
discussions. SS was supported by the Ramanujan Fellowship of Ayan Mukhopadhyay
(IIT Madras) when some of this work was carried.

##  Appendix A The modular group and congruence subgroups

Modular group:  It is a group of all fractional linear transformations on the
complex upper half plane, i.e, the transformations
$z\mapsto\frac{az+b}{cz+d}$, such that $a,b,c,$ and $d$ are all integers and
obey $ad-bc=1$.

These transformations are isomorphic to the group
$PSL(2,\mathbb{Z})=SL(2,\mathbb{Z})/\\{\pm\mathbf{1}\\}$. The projective
special linear group which we denote by $\Gamma$, is defined as follows:

$\Gamma=\Big{\\{}\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)|a,b,c,d\in\mathbb{Z},ad-bc=1\Big{\\}}\ .$

The complex upper half plane is defined as

$\mathbb{H}_{1}=\\{z\in\mathbb{C};\text{Im }(z)>0\\}\ .$

The modular group is generated by following two transformation:

$\displaystyle T=\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)\,\text{ such that }T\cdot z=z+1$ $\displaystyle
S=\left(\begin{smallmatrix}1&-1\\\ 1&0\end{smallmatrix}\right)\,\text{ such
that }S\cdot z=-\frac{1}{z}$

Now we define various subgroups of $SL(2,\mathbb{Z})$ which will be useful for
later discussions. These subgroups are obtained by imposing certain constrains
on the matrix elements and are called congruence subgroups. For an integer
$N>1$, the principal congruence subgroup of level $N$ is defined as follows:

$\Gamma(N)=\Big{\\{}\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in
SL(2,\mathbb{Z})|\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\equiv\left(\begin{smallmatrix}1&0\\\
0&1\end{smallmatrix}\right)\mod N\Big{\\}}\ .$

Another subgroup known as Hecke congruence subgroup of level $N$ denoted by
$\Gamma_{0}(N)$, is defined as

$\Gamma_{0}(N)=\Big{\\{}\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in
SL(2,\mathbb{Z})|\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\equiv\left(\begin{smallmatrix}*&*\\\
0&*\end{smallmatrix}\right)\mod N\Big{\\}}\ .$

Symplectic group:  The symplectic group, $Sp(2,\mathbb{Q})$ is a set of
$4\times 4$ matrices written in terms of four $2\times 2$ matrices $A,B,C,D$
(with $A,B,C,D$ taking values in $\mathbb{Q}$) as
$M=\left(\begin{smallmatrix}A&B\\\ C&D\end{smallmatrix}\right)$ satisfying
$AB^{T}=BA^{T}$, $CD^{T}=DC^{T}$ and $AD^{T}-BC^{T}=I$. This group naturally
acts on the Siegel upper half space $\mathbb{H}_{2}$, as

$\displaystyle\mathbf{Z}=\left(\begin{smallmatrix}\tau&z\\\
z&\tau^{\prime}\end{smallmatrix}\right)\longmapsto
M\cdot\mathbf{Z}\equiv(A\mathbf{Z}+B)(C\mathbf{Z}+D)^{-1}\ .$ (A.1)

The Siegel upper half space is defined as follows:

$\displaystyle\mathbb{H}_{2}=\Big{\\{}\mathbf{Z}=\left(\begin{smallmatrix}\tau&z\\\
z&\tau^{\prime}\end{smallmatrix}\right)\in M_{2}(\mathbb{C}),\
\text{Im}(\mathbf{Z})>0\Big{\\}}\ .$ (A.2)

The paramodular group at level $N$, $\mathbf{\Gamma}_{t}(N)$, is a subgroup of
$Sp(2,\mathbb{Q})$ is defined as follows:

$\displaystyle\mathbf{\Gamma}_{t}(N)=\left\\{\left(\begin{smallmatrix}*&*t&*&*\\\
*&*&*&*t^{-1}\\\ N&*Nt&*&*\\\ *Nt&*Nt&*t&*\end{smallmatrix}\right)\in
Sp(2,\mathbb{Q}),\text{ with }N,t\in\mathbb{Z}_{>0}\text{ and
}*\in\mathbb{Z}\right\\}\ .$ (A.3)

The $t=1,N=1$ case of the general paramodular group $\mathbf{\Gamma}_{t}(N)$
is the usual symplectic group over integers, i.e.,
$\mathbf{\Gamma}_{1}=Sp(2,\mathbb{Z})$. The embedding of
$\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma_{0}(N)$ in $\Gamma_{t}(N)$ is given by

$\widetilde{\gamma}=\widetilde{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}}\equiv\begin{pmatrix}a&0&b&0\\\ 0&1&0&0\\\ c&0&d&0\\\
0&0&0&1\end{pmatrix}\ ,\ c=0\mod N\ .$ (A.4)

The normal double extension of $\mathbf{\Gamma}_{t}(N)$ in $Sp(2,\mathbb{R})$
is defined as
$\mathbf{\Gamma}_{t}(N)^{+}=\mathbf{\Gamma}_{t}(N)\cup\mathbf{\Gamma}_{t}(N)V_{t}$
where

$\displaystyle V_{t}=\frac{1}{\sqrt{t}}\left(\begin{smallmatrix}0&t&0&0\\\
1&0&0&0\\\ 0&0&0&1\\\ 0&0&t&0\end{smallmatrix}\right)\in Sp(2,\mathbb{R})\ .$
(A.5)

As shown by Cléry and Gritsenko [31], this $\mathbf{\Gamma}_{t}(N)^{+}$ is
generated by $V_{t}$ and by its parabolic subgroup

$\displaystyle\mathbf{\Gamma}_{t}^{\infty}(N)=\left\\{\pm\left(\begin{smallmatrix}*&0&*&*\\\
*&1&*&*t^{-1}\\\ N&0&*&*\\\ 0&0&0&1\end{smallmatrix}\right)\in\Gamma_{t}(N),\
\forall*\in\mathbb{Z}\right\\}\ .$ (A.6)

##  Appendix B Modular forms and Jacobi forms

Modular form: A modular form of weight $k$, with $k$ being integer or half
integer, and character $v$ with respect to any of the subgroups of the modular
group $\Gamma$ is a holomorphic function
$f:\mathbb{H}_{1}\rightarrow\mathbb{C}$ satisfying,

$(f|_{k}M)(z)=v(M)f(z)\ ,$

where $M=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in\Gamma$
and $z\in\mathbb{H}_{1}$ and the slash operation is given by

$(f|_{k}M)(z)=(c\tau+d)^{-k}f(M\cdot z)\ .$

Jacobi form: A holomorphic function
$\phi_{k,m}(\tau,z):\mathbb{H}_{1}\times\mathbb{C}\rightarrow\mathbb{C}$ is
called a Jacobi form of weight $k$ and index $m$ if the function

$\widetilde{\phi}_{k}(\mathbf{Z})=\exp(2\pi
im\tau^{\prime})\phi_{k,m}(\tau,z)\ ,$

on $\mathbb{H}_{2}$ such that $\widetilde{\phi}_{k}(\mathbf{Z})$ is a modular
form of weight $k$ with respect to the Jacobi group
$\mathbf{\Gamma}^{J}(N)\subseteq Sp(2,\mathbb{Z})$ with character $v$, i.e.,
it satisfies

$\widetilde{\phi}|_{k}\ M(\mathbf{Z})=v(M)\
\widetilde{\phi}_{k}(\mathbf{Z})\hskip 28.45274pt\forall M\in\Gamma^{J}(N)\ ,$
(B.1)

and it is holomorphic at all cusps, i.e., let $\gamma\in SL(2,\mathbb{Z})$ and
$\tilde{\gamma}$, its embedding in $Sp(2,\mathbb{Z})$ as given in Eq. (A.4),
then it has a Fourier expansion

$\widetilde{\phi}|_{k}\tilde{\gamma}(\mathbf{Z})=s^{m}\
\sum_{\begin{subarray}{c}n,\ell\\\ 4nm-\ell^{2}\geq 0\end{subarray}}\
c_{\gamma}(n,\ell)\ q^{n}r^{\ell}$ (B.2)

where $q=e^{2\pi i\tau}$, $r=e^{2\pi iz}$ and $s=e^{2\pi i\tau^{\prime}}$ ,
$n,\ell\in\mathbb{Q}$.

Siegel Modular Form: A Siegel Modular Form of weight $k$ for the subgroup
$\mathbf{\Gamma}_{t}(N)$ with a character $v$ is a holomorphic function
$F:\mathbb{H}_{2}\rightarrow\mathbb{C}$ which satisfies

$\displaystyle(F|_{k}M)(\mathbf{Z})=v(M)F(\mathbf{Z})\ ,$ (B.3)

for all $M=\left(\begin{smallmatrix}A&B\\\
C&D\end{smallmatrix}\right)\in\mathbf{\Gamma}_{t}(N)$.

Here $|_{k}$ is the standard slash operator on the space of functions on
$\mathbb{H}_{2}:$

$\displaystyle(F|_{k}M)(\mathbf{Z}):=\det(C\mathbf{Z}+D)^{-k}F(M\cdot\mathbf{Z})\
.$ (B.4)

###  B.1 Examples of Modular and Jacobi forms

We discuss some examples of modular and Jacobi forms that enter our
discussions. The Dedekind eta function is defined by

$\eta(\tau)=q^{1/24}\prod_{m=1}^{\infty}(1-q^{m})\ .$

Its modular properties are given in Eq. (D.1). $E_{2}^{*}(\tau)$ is the weight
two non-holomorphic modular form of $SL(2,\mathbb{Z})$. It is given by

$E_{2}^{*}(\tau)=1-24\ \sum_{n=1}^{\infty}\sigma_{1}(n)\ q^{n}\ -\frac{3}{\pi\
\textrm{Im}\tau}\ ,$ (B.5)

where $\sigma_{\ell}(n)=\sum_{1\leq d|n}d^{\ell}$. Then for $N>1$,

$E_{2}^{(N)}(\tau):=\frac{1}{N-1}\Big{(}NE_{2}^{*}(N\tau)-E_{2}^{*}(\tau)\Big{)}$
(B.6)

is a weight two holomorphic modular form of $\Gamma_{0}(N)$ with constant
coefficient equal to $1$. For $\Gamma_{0}(23)$, we need the following weight-
two modular forms that we denote by $f_{23,1}(\tau)$ and $f_{23,2}(\tau)$. The
first few terms in their $q$-series are[8]

$\displaystyle f_{23,1}(\tau)$
$\displaystyle=2q-q^{2}-q^{4}-2q^{5}-5q^{6}+2q^{7}+4q^{9}+6q^{10}-6q^{11}+\cdots\
,$ $\displaystyle f_{23,2}(\tau)$
$\displaystyle=q^{2}-2q^{3}-q^{4}+2q^{5}+q^{6}+2q^{7}-2q^{8}-2q^{10}-2q^{11}+\cdots\
.$

For $a.b\in(0,1)\mod 2$, the genus-one theta functions are defined by

$\theta\left[\genfrac{}{}{0.0pt}{}{a}{b}\right]\left(\tau,z\right)=\sum_{l\in\mathbb{Z}}q^{\frac{1}{2}(l+\frac{a}{2})^{2}}\
r^{(l+\frac{a}{2})}\ e^{i\pi lb}\ ,$ (B.7)

with $r=\exp(2\pi iz)$. Let
$\vartheta_{1}\left(\tau,z\right)\equiv\theta\left[\genfrac{}{}{0.0pt}{}{1}{1}\right]\left(\tau,z\right)$,
$\vartheta_{2}\left(\tau,z\right)\equiv\theta\left[\genfrac{}{}{0.0pt}{}{1}{0}\right]\left(\tau,z\right)$,
$\vartheta_{3}\left(\tau,z\right)\equiv\theta\left[\genfrac{}{}{0.0pt}{}{0}{0}\right]\left(\tau,z\right)$
and
$\vartheta_{4}\left(\tau,z\right)\equiv\theta\left[\genfrac{}{}{0.0pt}{}{0}{1}\right]\left(\tau,z\right)$.
Using these, we can construct two index one Jacobi forms:

$\displaystyle\phi_{0,1}(\tau,z)$
$\displaystyle=4\sum_{a=3}^{4}\left[\frac{\vartheta_{a}(\tau,z)}{\vartheta_{a}(\tau,0)}\right]^{2}\
,$ (B.8) $\displaystyle\phi_{-2,1}(\tau,z)$
$\displaystyle=\frac{\vartheta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\ .$ (B.9)

Any Jacobi form, $Z_{0,1}(\tau,z)$, of weight zero and index one of
$\Gamma_{0}(N)$ ($N>1$) can be written as follows[22]

$Z_{0,1}(\tau,z)=A\ \phi_{0,1}(\tau,z)+B(\tau)\ \phi_{-2,1}(\tau,z)\ ,$

where $A$ is a constant and $B(\tau)$ is a weight two modular form of
$\Gamma_{0}(N)$.

##  Appendix C List of $M_{24}$ Jacobi forms

There are 26 conjugacy classes of $M_{24}$. The ATLAS nomenclature of the
conjugacy classes are as follows:

$\displaystyle\text{Type I :
}1a,2a,3a,5a,4b,7a,7b,8a,6a,11a,15a,15b,14a,14b,23a,23b,$
$\displaystyle\text{Type II : }12b,6b,4c,3b,2b,10a,21a,21b,4a,12a.$

The cycle shapes associated with type I conjugacy classes have at least one
one-cycle.

Conj. class | $\rho$ |  $Z^{\rho}(\tau,z)$
---|---|---
1a | $1^{24}$ |  2 $\phi_{0,1}(\tau,z)$—
2a | $1^{8}2^{8}$ |  $\frac{2}{3}\phi_{0,1}(\tau,z)+\frac{4}{3}E_{2}^{(2)}(\tau)\phi_{-2,1}(\tau,z)$
3a | $1^{6}3^{6}$ |  $\tfrac{1}{2}\phi_{0,1}(\tau,z)+\tfrac{3}{2}E^{(3)}_{2}(\tau)\ \phi_{-2,1}(\tau,z)$
5a | $1^{4}5^{4}$ |  $\tfrac{1}{3}\phi_{0,1}(\tau,z)+\tfrac{5}{3}E^{(5)}_{2}(\tau)\ \phi_{-2,1}(\tau,z)$
4b | $1^{4}2^{2}4^{4}$ |  $\tfrac{1}{3}\phi_{0,1}(\tau,z)+(-\tfrac{1}{3}E^{(2)}_{2}(\tau)+2E^{(4)}_{2}(\tau))\ \phi_{-2,1}(\tau,z)$
7a/b | $1^{3}7^{3}$ |  $\frac{1}{4}\phi_{0,1}(\tau,z)+\frac{7}{4}E_{2}^{(7)}(\tau)\phi_{-2,1}(\tau,z)$
8a | $1^{2}2^{1}4^{1}8^{2}$ |  $\tfrac{1}{6}\phi_{0,1}(\tau,z)+(-\tfrac{1}{2}E^{(4)}_{2}(\tau)+\tfrac{7}{3}E^{(8)}_{2}(\tau))\ \phi_{-2,1}(\tau,z)$
6a | $1^{2}2^{2}3^{2}6^{2}$ |  $\tfrac{1}{6}\phi_{0,1}(\tau,z)+(-\tfrac{1}{6}E^{(2)}_{2}(\tau)-\tfrac{1}{2}E^{(3)}_{2}(\tau)+\tfrac{5}{2}E^{(6)}_{2}(\tau))\ \phi_{-2,1}(\tau,z)$
11a | $1^{2}11^{2}$ |  $\tfrac{1}{6}\phi_{0,1}(\tau,z)+(\tfrac{11}{6}E^{(11)}_{2}(\tau)-\tfrac{22}{5}\eta_{1^{2}11^{2}}(\tau))\ \phi_{-2,1}(\tau,z)$
15a/b | $1^{1}3^{1}5^{1}15^{1}$ |  $\tfrac{1}{12}\phi_{0,1}(\tau,z)+(-\tfrac{1}{16}E^{(3)}_{2}(\tau)-\tfrac{5}{24}E_{2}^{(5)}(\tau)\newline +\tfrac{35}{16}E_{2}^{(15)}(\tau)-\frac{15}{4}\eta_{1^{1}3^{1}5^{1}15^{1}}(\tau))\ \phi_{-2,1}(\tau,z)$
14a/b | $1^{1}2^{1}7^{1}14^{1}$ |  $\tfrac{1}{12}\phi_{0,1}(\tau,z)+(-\tfrac{1}{36}E^{(2)}_{2}(\tau)-\tfrac{7}{12}E_{2}^{(7)}(\tau)\newline +\tfrac{91}{36}E_{2}^{(14)}(\tau)-\frac{14}{3}\eta_{1^{1}2^{1}7^{1}14^{1}}(\tau))\ \phi_{-2,1}(\tau,z)$
23a/b | $1^{1}23^{1}$ |  $\tfrac{1}{12}\phi_{0,1}(\tau,z)+(\tfrac{23}{12}E^{(23)}_{2}(\tau)-\tfrac{23}{22}f_{23,1}(\tau)\newline -\tfrac{161}{22}f_{23,2}(\tau))\ \phi_{-2,1}(\tau,z)$
12b | $12^{2}$ |  $2\,\eta_{1^{4}2^{-1}4^{1}6^{1}12^{-1}}(\tau)\ \phi_{-2,1}(\tau,z)$
6b | $6^{4}$ |  $2\,\eta_{1^{2}2^{2}3^{2}6^{-2}}(\tau)\ \phi_{-2,1}(\tau,z)$
4c | $4^{6}$ |  $2\,\eta_{1^{4}2^{2}4^{-2}}(\tau)\ \phi_{-2,1}(\tau,z)$
3b | $3^{8}$ |  $2\,\eta_{1^{6}3^{-2}}(\tau)\ \phi_{-2,1}(\tau,z)$
2b | $2^{12}$ |  $2\,\eta_{1^{8}2^{-4}}(\tau)\ \phi_{-2,1}(\tau,z)$
10a | $2^{2}10^{2}$ |  $2\,\eta_{1^{3}2^{1}5^{1}10^{-1}}(\tau)\ \phi_{-2,1}(\tau,z)$
21a/b | $3^{1}21^{1}$ |  $\left(\frac{7}{3}\eta_{1^{3}3^{-1}7^{3}21^{-1}}(\tau)-\frac{1}{3}\eta_{1^{6}3^{-2}}(\tau)\right)\ \phi_{-2,1}(\tau,z)$
4a | $2^{4}4^{4}$ |  $2\,\eta_{2^{8}4^{-4}}(\tau)\ \phi_{-2,1}(\tau,z)$
12a | $2^{1}4^{1}6^{1}12^{1}$ |  2$\,\eta_{1^{3}2^{-1}3^{-1}4^{2}6^{3}12^{-2}}(\tau)\ \phi_{-2,1}(\tau,z)$
Table 1: The Jacobi forms for all conjugacy classes of $M_{24}$.

##  Appendix D Rules for computing cusps

The $S$ and $T$-transformation of $\eta$-function with simple arguments are as
follows (see Chapter 2 of [32])

$\displaystyle\eta(N\tau)|_{S}=\tfrac{1}{\sqrt{N}}e^{-i\pi/4}\eta\left(\tfrac{\tau}{N}\right),$
$\displaystyle\left.\eta\left(\frac{\tau}{N}\right)\right|_{S}=\sqrt{N}e^{-i\pi/4}\eta(N\tau),$
(D.1) $\displaystyle\eta(\tau+a)=e^{i\pi a/12}\eta(\tau)\ .$

For more complicated arguments of $\eta$-function, let
$\psi_{j}:=\eta\left(\frac{\tau+j}{N}+j\right)$. Consider the following cases

  1. 1.

If $(j,N)=1$, then choose a $j^{\prime}$ such that $jj^{\prime}=N+1$ we have,

$\displaystyle\psi_{j}|_{S}(\tau)=\text{sign}(-j^{\prime})e^{-3\pi
i/4}\psi_{-j^{\prime}}(\tau)\ .$ (D.2)

  2. 2.

$(j,N)\neq 1$, but a simple change in variable of $\tau$ leads to the
condition $(j,N)=1$. In that case, we make the necessary change in the
variable. Then we choose a $j^{\prime}$ such that $jj^{\prime}=N+1$. An
example of such case is the following.

$\displaystyle\left.\eta\left(\frac{\tau-3}{6}\right)\right|_{S}=\tau^{-1/2}\eta\left(\frac{-\frac{1}{\tau}-3}{6}\right)=\tau^{-1/2}\eta\left(\frac{-\frac{1}{3\tau}-1}{2}\right)=\tau^{-1/2}\eta\left(\frac{-1-\tilde{\tau}}{2\tilde{\tau}}\right)\
,$

where $\tilde{\tau}=3\tau$. Now let us consider
$\psi^{(2)}_{-1}(\tilde{\tau})=\eta\left(\frac{\tilde{\tau}-1}{2}-1\right)$
and do the $S$-transformation.

$\left.\eta\left(\frac{\tilde{\tau}-1}{2}-1\right)\right|_{S}=e^{-i\pi/12}\left.\eta\left(\frac{\tilde{\tau}-1}{2}\right)\right|_{S}=e^{-i\pi/12}\tilde{\tau}^{-1/2}\eta\left(\frac{-\frac{1}{\tilde{\tau}}-1}{2}\right)\\\
=e^{-i\pi/12}\tilde{\tau}^{-1/2}\eta\left(\frac{-1-\tilde{\tau}}{2\tilde{\tau}}\right)\
.$

Again using Eq. (D.2) we get

$\displaystyle\left.\eta\left(\frac{\tau-1}{2}-1\right)\right|_{S}=e^{-3\pi
i/4}\eta\left(\frac{\tau+3}{2}+3\right)\ .$

Comparing these two and putting $\tilde{\tau}=3\tau$ back we get

$\displaystyle\left.\eta\left(\frac{\tau-3}{6}\right)\right|_{S}=\sqrt{3}e^{-i\pi/3}\eta\left(\frac{3\tau+1}{2}\right)\
.$

  3. 3.

$(j,N)\neq 1$, and there exists no change of variable in $\tau$ that can lead
us to the condition $(j,N)=1$. In that case, we need to use a more general
formula. Given $\psi_{j}(\tau)$, choose $j^{\prime}$ such that $jj^{\prime}=1$
mod $N$ and

$G=\begin{pmatrix}j&\frac{jj^{\prime}-1}{N}\\\
N&j^{\prime}\end{pmatrix}\in\Gamma_{0}(N)\ .$

Let $G=ST^{a}ST^{b}ST^{c}S$, then

$\psi_{j}\Big{|}_{S}(\tau)=e^{-\pi i}e^{(j+j^{\prime}+a+b+c)\pi
i/12}\,i^{J}\,\psi_{-j^{\prime}}(\tau)\ ,$

where $J$ is the number of changes of sign in the sequence $1,c,bc-1,abc-
a-c=N$.

##  Appendix E Proving Proposition 2.2

###  E.1  Type-I conjugacy classes

####  E.1.1 Cycle shape $\mathbf{1^{4}\cdot 2^{2}\cdot 4^{4}}$

The twining genera $Z^{\rho_{a}}(\tau,z),\,a=0,\ldots 3$, which contribute to
the multiplicative lifts are the one corresponding to conjugacy class $4b$ and
its various cusps.

$\displaystyle Z^{\rho_{0}}(\tau,z)=Z^{1^{24}}(\tau,z),$ $\displaystyle
Z^{\rho}(\tau,z)=Z^{\rho_{3}}(\tau,z)=Z^{1^{4}2^{2}4^{4}}(\tau,z),$
$\displaystyle Z^{\rho_{2}}(\tau,z)=Z^{1^{8}2^{8}}(\tau,z)\ .$

All these Jacobi forms are listed in table 1. The multiplicative seed is
$Z^{1^{4}2^{2}4^{4}}(\tau,z)$, and this is a Jacobi form of $\Gamma_{0}(4)$.
The Borcherds lift (multiplicative lift), given by Eq. (2.6) leads to the
following product formula:

$\Phi_{3}^{1^{4}2^{2}4^{4}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{c^{0}(nm,\ell)-c^{2}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{2}\right)^{c^{2}(nm,\ell)-c^{1}(nm,\ell)}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{4}\right)^{c^{1}(nm,\ell)}\ .$ (E.1)

Then using Eq. (2.5) we get,

$\displaystyle c^{0}=\frac{1}{4}(f_{0}+2f_{1}+f_{2}),\hskip
5.69046ptc^{1}=\frac{1}{4}(f_{0}-f_{2}),\hskip
5.69046ptc^{2}=\frac{1}{4}(f_{0}-2f_{1}+f_{2})\ .$ (E.2)

Using these we can rewrite the product formula (E.1) as

$\Phi_{3}^{1^{4}2^{2}4^{4}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{f_{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{2}\right)^{\frac{1}{2}(f_{2}(nm,\ell)-f_{1}(nm,\ell))}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{4}\right)^{\frac{1}{4}(f_{0}(nm,\ell)-f_{2}(nm,\ell))}\
.$ (E.3)

To see if it can be written as the Borcherds product as stated in theorem 2.1,
we need to compute the Fourier coefficients at different cusps. The cusps and
other data for $\Gamma_{0}(4)$ are as follows:222Link to compute cusps of
congruence subgroups $\Gamma_{0}(N)$: http://magma.maths.usyd.edu.au/calc/ and
the code is
$>$ G := CongruenceSubgroup(0,N);
$>$ Cusps(G);
$>$ Widths(G);

$f/e$ | $i\infty$ | $1/2$ | $0/1$
---|---|---|---
$h_{e}$ | 1 | 1 | 4
$N_{e}$ | 1 | 2 | 4

Using $M_{i\infty}=1,M_{0}=S$, and $M_{1/p}=-ST^{-p}S$ and using the
transformation rules under $S$ and $T$ as listed above in Section D, we can
show that

$\displaystyle\pi_{FE}\Big{(}Z^{1^{4}2^{2}4^{4}}|M_{\frac{1}{2}}\Big{)}=Z^{1^{8}2^{8}}-Z^{1^{4}2^{2}4^{4}},$
$\displaystyle\pi_{FE}\Big{(}Z^{1^{4}2^{2}4^{4}}|M_{\frac{0}{1}}\Big{)}=\frac{1}{4}\left(Z^{1^{24}}-Z^{1^{8}2^{8}}\right)\
.$

The Fourier expansion of the cusp about zero does not have integral power in
$q$ and hence does not contribute to the product formula. Using these
identities, we see that the Borcherds product as given by Clery-Gritsenko in
2.1 takes the form same as Eq. (E.3), i.e.,

$\displaystyle
B_{Z^{1^{4}2^{2}4^{4}}}(\mathbf{Z})=\Phi_{3}^{1^{4}2^{2}4^{4}}(\mathbf{Z})\ .$
(E.4)

The zeroth coefficient of the Siegel Modular Form, i.e., the $m=0$ term is
given by the additive seed
$\phi_{3,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta(\tau)^{4}\eta(2\tau)^{2}\eta(4\tau)^{4}$.
Hence $\Phi_{3}^{1^{4}2^{2}4^{4}}(\mathbf{Z})$ is a modular form of weight 3
at level 4.

####  E.1.2 Cycle shape $\mathbf{1^{2}\cdot 2^{2}\cdot 3^{2}\cdot 6^{2}}$

This is an element of order $6$. The twining genera that contribute to the
multiplicative lifts are the following:

$\displaystyle Z^{\rho_{0}}(\tau,z)=Z^{1^{24}}(\tau,z),$ $\displaystyle
Z^{\rho}(\tau,z)=Z^{\rho_{5}}(\tau,z)=Z^{1^{2}2^{2}3^{2}6^{2}}(\tau,z),$
$\displaystyle
Z^{\rho_{2}}(\tau,z)=Z^{\rho_{4}}(\tau,z)=Z^{1^{6}3^{6}}(\tau,z),$
$\displaystyle Z^{\rho_{3}}(\tau,z)=Z^{1^{8}2^{8}}(\tau,z)\ .$

The multiplicative seed $Z^{1^{2}2^{2}3^{2}6^{2}}(\tau,z)$ is a Jacobi form of
$\Gamma_{0}(6)$. The cusps and other data for $\Gamma_{0}(6)$ are the
following:

$f/e$ | $i\infty$ | $1/2$ | $1/3$ | $0/1$
---|---|---|---|---
$h_{e}$ | 1 | 3 | 2 | 6
$N_{e}$ | 1 | 3 | 2 | 6

The multiplicative lift given by Eq. (2.6) leads to the product formula,

$\Phi_{2}^{1^{2}2^{2}3^{2}6^{2}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{c^{0}(nm,\ell)+c^{1}(nm,\ell)-c^{2}(nm,\ell)-c^{3}(nm,\ell)}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{2}\right)^{c^{3}(nm,\ell)-c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{3}\right)^{c^{2}(nm,\ell)-c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{6}\right)^{c^{1}(nm,\ell)}\
.$ (E.5)

Then using Eq. (2.5) we express the $c^{\alpha}$ in terms of the Fourier-
Jacobi coefficients of $Z^{\rho_{s}}$ as follows:

$\displaystyle c^{0}=\frac{1}{6}(f_{0}+2f_{1}+2f_{2}+f_{3}),\hskip
5.69046ptc^{1}=\frac{1}{6}(f_{0}+f_{1}-f_{2}-f_{3}),$ $\displaystyle
c^{2}=\frac{1}{6}(f_{0}-f_{1}-f_{2}+f_{3}),\hskip
5.69046ptc^{3}=\frac{1}{6}(f_{0}-2f_{1}+2f_{2}-f_{3})\ .$ (E.6)

Then the above product formula becomes:

$\Phi_{2}^{1^{2}2^{2}3^{2}6^{2}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{f_{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{2}\right)^{\frac{1}{2}(f_{2}(nm,\ell)-f_{1}(nm,\ell))}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{3}\right)^{\frac{1}{3}(f_{3}(nm,\ell)-f_{1}(nm,\ell))}\left(1-(q^{n}r^{\ell}s^{m})^{6}\right)^{\frac{1}{6}(f_{0}(nm,\ell)+f_{1}(nm,\ell)-f_{2}(nm,\ell)-f_{3}(nm,\ell))}\
.$ (E.7)

Now we need to check if it can be written as the Borcherds product as stated
in theorem 2.1. For that we need to compute the Fourier-Jacobi coefficients at
cusps $i\infty,1/2,1/3$ and $0/1$. Straightforward computation shows that

$\displaystyle\pi_{FE}\Big{(}Z^{1^{2}2^{2}3^{2}6^{2}}|M_{\frac{1}{2}}\Big{)}=\frac{1}{3}\left(Z^{1^{8}2^{8}}-Z^{1^{2}2^{2}3^{2}6^{2}}\right),$
$\displaystyle\pi_{FE}\Big{(}Z^{1^{2}2^{2}3^{2}6^{2}}|M_{\frac{1}{3}}\Big{)}=\frac{1}{2}\left(Z^{1^{6}3^{6}}-Z^{1^{2}2^{2}3^{2}6^{2}}\right),$
$\displaystyle\pi_{FE}\Big{(}Z^{1^{2}2^{2}3^{2}6^{2}}|M_{\frac{0}{1}}\Big{)}=\frac{1}{6}\left(Z^{1^{24}}+Z^{1^{2}2^{2}3^{2}6^{2}}-Z^{1^{6}3^{6}}-Z^{1^{8}2^{8}}\right)\
.$

Using these and using the product formula for Borcherds product given by
Cléry-Gritsenko, we get expression for
$B_{Z}^{1^{2}2^{2}3^{2}6^{2}}(\mathbf{Z})$ which is same as Eq. (E.7), i.e.,

$\displaystyle
B_{Z^{1^{2}2^{2}3^{2}6^{2}}}(\mathbf{Z})=\Phi_{2}^{1^{2}2^{2}3^{2}6^{2}}(\mathbf{Z})\
.$ (E.8)

Hence $\Phi_{2}^{1^{2}2^{2}3^{2}6^{2}}(\mathbf{Z})$ is a modular form of
weight 2 and level 6. The zeroth coefficient of this Siegel Modular Form,
i.e., the $m=0$ term is given by the additive seed
$\phi_{2,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta(\tau)^{2}\eta(2\tau)^{2}\eta(3\tau)^{2}\eta(6\tau)^{2}$.

####  E.1.3 Cycle shape $\mathbf{1^{2}\cdot 2^{1}\cdot 4^{1}\cdot 8^{2}}$

This is a $M_{24}$ element of order $8$ and the conjugacy class is $8a$. The
twining genera that contribute to the multiplicative lift are the following:

$\displaystyle Z^{\rho_{0}}(\tau,z)=Z^{1^{24}}(\tau,z),$ $\displaystyle
Z^{\rho_{1}}(\tau,z)=Z^{\rho_{3}}(\tau,z)=Z^{\rho_{5}}(\tau,z)=Z^{\rho_{7}}(\tau,z)=Z^{1^{2}2^{1}4^{1}8^{2}}(\tau,z),$
$\displaystyle
Z^{\rho_{2}}(\tau,z)=Z^{\rho_{6}}(\tau,z)=Z^{1^{4}2^{2}4^{4}}(\tau,z),$
$\displaystyle Z^{\rho_{4}}(\tau,z)=Z^{1^{8}2^{8}}(\tau,z)\ .$

The multiplicative seed is given by $Z^{1^{2}2^{1}4^{1}8^{2}}$ and this is a
Jacobi form of $\Gamma_{0}(8)$. The cusps at level $8$ are the following:

$f/e$ | $i\infty$ | $1/2$ | $1/4$ | $0/1$
---|---|---|---|---
$h_{e}$ | 1 | 2 | 1 | 8
$N_{e}$ | 1 | 4 | 2 | 8

The Borcherds lift given by Eq. (2.6) leads to the following product formula:

$\Phi_{1}^{1^{2}2^{1}4^{1}8^{2}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{c^{0}(nm,\ell)-c^{4}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{2}\right)^{c^{4}(nm,\ell)-c^{2}(nm,\ell)}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{4}\right)^{c^{2}(nm,\ell)-c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{8}\right)^{c^{1}(nm,\ell)}\
.$ (E.9)

Then using Eq. (2.5), i.e.,

$\displaystyle c^{0}=\frac{1}{8}(f_{0}+4f_{1}+2f_{2}+f_{4}),\hskip
5.69046ptc^{1}=\frac{1}{8}(f_{0}-f_{4}),$ $\displaystyle
c^{2}=\frac{1}{8}(f_{0}-2f_{2}+f_{4}),\hskip
5.69046ptc^{4}=\frac{1}{8}(f_{0}-4f_{1}+2f_{2}+f_{4})\ ,$ (E.10)

we rewrite the product formula in terms of the Fourier-Jacobi coefficients
$f_{a}(n,\ell)$ as

$\Phi_{1}^{1^{2}2^{1}4^{1}8^{2}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{f_{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{2}\right)^{\frac{1}{2}(f_{2}(nm,\ell)-f_{1}(nm,\ell))}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{4}\right)^{\frac{1}{4}(f_{4}(nm,\ell)-f_{2}(nm,\ell))}\left(1-(q^{n}r^{\ell}s^{m})^{8}\right)^{\frac{1}{8}\left(f_{0}(nm,\ell)-f_{4}(nm,\ell)\right)}\
.$ (E.11)

To check if it can be written as the Borcherds product of Cléry-Gritsenko 2.1,
we need to compute the coefficients at cusps about $i\infty,1/2,1/4$ and
$0/1$. Computing $Z^{1^{2}2^{1}4^{1}8^{2}}$ at these cusps one can show that

$\displaystyle\pi_{FE}\Big{(}Z^{1^{2}2^{1}4^{1}8^{2}}|M_{\frac{1}{2}}\Big{)}=\frac{1}{2}\left(Z^{1^{8}2^{8}}-Z^{1^{4}2^{2}4^{4}}\right),$
$\displaystyle\pi_{FE}\Big{(}Z^{1^{2}2^{1}4^{1}8^{2}}|M_{\frac{1}{4}}\Big{)}=\left(Z^{1^{4}2^{2}4^{4}}-Z^{1^{2}2^{1}4^{1}8^{2}}\right),$
$\displaystyle\pi_{FE}\Big{(}Z^{1^{2}2^{1}4^{1}8^{2}}|M_{\frac{0}{1}}\Big{)}=\frac{1}{8}\left(Z^{1^{24}}-Z^{1^{8}2^{8}}\right)\
.$

Using these and using the product formula for Borcherds product given by
Cléry-Gritsenko, we get expression for
$B_{Z^{1^{2}2^{1}4^{1}8^{2}}}(\mathbf{Z})$ same as Eq. (E.11), i.e.,

$\displaystyle
B_{Z^{1^{2}2^{1}4^{1}8^{2}}}(\mathbf{Z})=\Phi_{1}^{1^{2}2^{1}4^{1}8^{2}}(\mathbf{Z})\
.$ (E.12)

Hence $\Phi_{1}^{1^{2}2^{1}4^{1}8^{2}}(\mathbf{Z})$ is a meromorphic Siegel
Modular Form of weight 1 at level 8. The zeroth coefficient of this Siegel
Modular Form, i.e., the $m=0$ term is given by the additive seed
$\phi_{1,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta(\tau)^{2}\eta(2\tau)\eta(4\tau)\eta(8\tau)^{2}$.

####  E.1.4 Cycle shape $\mathbf{1\cdot 2\cdot 7\cdot 14}$

This is an element of order $14$. The following twining genera that contribute
to the multiplicative lift :

$\displaystyle Z^{\rho_{0}}(\tau,z)=Z^{1^{24}}(\tau,z),$ $\displaystyle
Z^{\rho}(\tau,z)=Z^{\rho_{3}}(\tau,z)=Z^{\rho_{5}}(\tau,z)=Z^{\rho_{9}}(\tau,z)=Z^{\rho_{11}}(\tau,z)=Z^{\rho_{13}}(\tau,z)=Z^{1^{1}2^{1}7^{1}14^{1}}(\tau,z),$
$\displaystyle
Z^{\rho_{2}}(\tau,z)=Z^{\rho_{4}}(\tau,z)=Z^{\rho_{6}}(\tau,z)=Z^{\rho_{8}}(\tau,z)=Z^{\rho_{10}}(\tau,z)=Z^{\rho_{12}}(\tau,z)=Z^{1^{3}7^{3}}(\tau,z),$
$\displaystyle Z^{\rho_{7}}(\tau,z)=Z^{1^{8}2^{8}}(\tau,z)\ .$

The multiplicative seed for this case, i.e.,
$Z^{1^{1}2^{1}7^{1}14^{1}}(\tau,z)$ is a Jacobi form of $\Gamma_{0}(14)$. The
cusps and other data at level $14$ are the following:

$f/e$ | $i\infty$ | $1/2$ | $1/7$ | $0/1$
---|---|---|---|---
$h_{e}$ | 1 | 7 | 2 | 14
$N_{e}$ | 1 | 7 | 2 | 14

The Borcherds lift given by Eq. (2.6) leads to the following product formula:

$\Phi_{0}^{1^{1}2^{1}7^{1}14^{1}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{c^{0}(nm,\ell)+c^{1}(nm,\ell)-c^{2}(nm,\ell)-c^{7}(nm,\ell)}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{2}\right)^{c^{7}(nm,\ell)-c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{7}\right)^{c^{2}(nm,\ell)-c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{14}\right)^{c^{1}(nm,\ell)}\
.$ (E.13)

Again we rewrite $c^{\alpha}(n,\ell)$ in terms of $f_{a}(n,\ell)$ using Eq.
(2.5),

$\displaystyle c^{0}=\frac{1}{14}(f_{0}+6f_{1}+6f_{2}+f_{7}),\hskip
5.69046ptc^{1}=\frac{1}{14}(f_{0}+f_{1}-f_{2}-f_{7}),$ $\displaystyle
c^{2}=\frac{1}{14}(f_{0}-f_{1}-f_{2}+f_{7}),\hskip
5.69046ptc^{7}=\frac{1}{14}(f_{0}-6f_{1}+6f_{2}-f_{7})\ .$ (E.14)

Then the above product formula becomes

$\Phi_{0}^{1^{1}2^{1}7^{1}14^{1}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{f_{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{2}\right)^{\frac{1}{2}(f_{2}(nm,\ell)-f_{1}(nm,\ell))}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{7}\right)^{\frac{1}{7}(f_{7}(nm,\ell)-f_{1}(nm,\ell))}\left(1-(q^{n}r^{\ell}s^{m})^{14}\right)^{\frac{1}{14}\left(f_{0}(nm,\ell)+f_{1}(nm,\ell)-f_{2}(nm,\ell)-f_{7}(nm,\ell)\right)}\
.$ (E.15)

To see if it can be written as the Borcherds product as stated in theorem 2.1
we need to compute the Fourier-Jacobi coefficients of the multiplicative seed
at cusps about $i\infty,1/2,1/7$ and $0/1$. Computing the cusps, one can show
that

$\displaystyle\pi_{FE}\Big{(}Z^{1^{1}2^{1}7^{1}14^{1}}|M_{\frac{1}{2}}\Big{)}=\frac{1}{2}\left(Z^{1^{3}7^{3}}-Z^{1^{1}2^{1}7^{1}14^{1}}\right),$
$\displaystyle\pi_{FE}\Big{(}Z^{1^{1}2^{1}7^{1}14^{1}}|M_{\frac{1}{7}}\Big{)}=\frac{1}{7}\left(Z^{1^{8}2^{8}}-Z^{1^{1}2^{1}7^{1}14^{1}}\right),$
$\displaystyle\pi_{FE}\Big{(}Z^{1^{1}2^{1}7^{1}14^{1}}|M_{\frac{0}{1}}\Big{)}=\frac{1}{14}\left(Z^{1^{24}}+Z^{1^{1}2^{1}7^{1}14^{1}}-Z^{1^{8}2^{8}}-Z^{1^{3}7^{3}}\right)\
.$

Then the Borcherds product formula given by Cléry-Gritsenko leads to product
formula Eq. (E.15), i.e.,

$\displaystyle
B_{Z^{1^{1}2^{1}7^{1}14^{1}}}(\mathbf{Z})=\Phi_{0}^{1^{1}2^{1}7^{1}14^{1}}(\mathbf{Z})\
.$ (E.16)

Hence $\Phi_{0}^{1^{1}2^{1}7^{1}14^{1}}(\mathbf{Z})$ is a modular form of
weight zero at level 14. The zeroth coefficient of the Siegel Modular Form,
i.e., the $m=0$ term is given by the additive seed
$\phi_{0,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta(\tau)\eta(2\tau)\eta(7\tau)\eta(14\tau)$.

####  E.1.5 Cycle shape $\mathbf{1\cdot 3\cdot 5\cdot 15}$

This is an $M_{24}$ element of order $15$. The twining genera that contribute
to the multiplicative lift are the following:

$\displaystyle Z^{\rho_{0}}(\tau,z)=Z^{1^{24}}(\tau,z),$ $\displaystyle
Z^{\rho}(\tau,z)=Z^{\rho_{2}}(\tau,z)=Z^{\rho_{4}}(\tau,z)=Z^{\rho_{7}}(\tau,z),$
$\displaystyle\hskip
14.22636pt=Z^{\rho_{8}}(\tau,z)=Z^{\rho_{11}}(\tau,z)=Z^{\rho_{13}}(\tau,z)=Z^{\rho_{14}}(\tau,z)=Z^{1^{1}3^{1}5^{1}15^{1}}(\tau,z),$
$\displaystyle
Z^{\rho_{3}}(\tau,z)=Z^{\rho_{6}}(\tau,z)=Z^{\rho_{9}}(\tau,z)=Z^{\rho_{12}}(\tau,z)=Z^{1^{4}5^{4}}(\tau,z),$
$\displaystyle
Z^{\rho_{5}}(\tau,z)=Z^{\rho_{10}}(\tau,z)=Z^{1^{6}3^{6}}(\tau,z)\ .$

The multiplicative seed is a Jacobi form of $\Gamma_{0}(15)$. The cusps and
other data for $\Gamma_{0}(15)$ are the following:

$f/e$ | $i\infty$ | $1/3$ | $1/5$ | $0/1$
---|---|---|---|---
$h_{e}$ | 1 | 5 | 3 | 15
$N_{e}$ | 1 | 5 | 3 | 15

The Borcherds lift given by Eq. (2.6) leads to the following product formula:

$\Phi_{0}^{1^{1}3^{1}5^{1}15^{1}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{c^{0}(nm,\ell)+c^{1}(nm,\ell)-c^{3}(nm,\ell)-c^{5}(nm,\ell)}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{3}\right)^{c^{5}(nm,\ell)-c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{5}\right)^{c^{3}(nm,\ell)-c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{15}\right)^{c^{1}(nm,\ell)}\
.$ (E.17)

Then using Eq. (2.5), i.e.,

$\displaystyle c^{0}=\frac{1}{15}(f_{0}+8f_{1}+4f_{3}+2f_{5}),\hskip
5.69046ptc^{1}=\frac{1}{15}(f_{0}+f_{1}-f_{3}-f_{5}),$ $\displaystyle
c^{3}=\frac{1}{15}(f_{0}-2f_{1}-f_{3}+2f_{5}),\hskip
5.69046ptc^{5}=\frac{1}{15}(f_{0}-4f_{1}+4f_{3}-f_{5})\ ,$ (E.18)

we can rewrite the above product formula as

$\Phi_{0}^{1^{1}3^{1}5^{1}15^{1}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{f_{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{3}\right)^{\frac{1}{3}(f_{3}(nm,\ell)-f_{1}(nm,\ell))}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{5}\right)^{\frac{1}{5}(f_{5}(nm,\ell)-f_{1}(nm,\ell))}\left(1-(q^{n}r^{\ell}s^{m})^{15}\right)^{\frac{1}{15}\left(f_{0}(nm,\ell)+f_{1}(nm,\ell)-f_{3}(nm,\ell)-f_{5}(nm,\ell)\right)}\
.$ (E.19)

To see if it can be written as the Borcherds product as stated in theorem 2.1
we need to compute the Fourier coefficients at cusps about $i\infty,1/3,1/5$
and $1/15$. The cusp about zero does not have any integer power in $q$ and
hence does not contribute. Computing the seed, i.e.,
$Z^{1^{1}3^{1}5^{1}15^{1}}(\tau,z)$ at these cusps one can show that

$\displaystyle\pi_{FE}\Big{(}Z^{1^{1}3^{1}5^{1}15^{1}}|M_{\frac{1}{3}}\Big{)}=\frac{1}{3}\left(Z^{1^{4}5^{4}}-Z^{1^{1}3^{1}5^{1}15^{1}}\right),$
$\displaystyle\pi_{FE}\Big{(}Z^{1^{1}3^{1}5^{1}15^{1}}|M_{\frac{1}{5}}\Big{)}=\frac{1}{5}\left(Z^{1^{6}3^{6}}-Z^{1^{1}3^{1}5^{1}15^{1}}\right),$
$\displaystyle\pi_{FE}\Big{(}Z^{1^{1}3^{1}5^{1}15^{1}}|M_{\frac{0}{1}}\Big{)}=\frac{1}{15}\left(Z^{1^{24}}+Z^{1^{1}3^{1}5^{1}15^{1}}-Z^{1^{6}3^{6}}-Z^{1^{4}5^{4}}\right)\
.$

Using these and using the product formula for Borcherds product given by
Cléry-Gritsenko, we get expression for
$B_{Z^{1^{1}3^{1}5^{1}15^{1}}}(\mathbf{Z})$ which is identical to Eq. (E.19),
i.e.,

$\displaystyle
B_{Z^{1^{1}3^{1}5^{1}15^{1}}}(\mathbf{Z})=\Phi_{0}^{1^{1}3^{1}5^{1}15^{1}}(\mathbf{Z})\
.$ (E.20)

Hence $\Phi_{0}^{1^{1}3^{1}5^{1}15^{1}}(\mathbf{Z})$ is a meromorphic Siegel
Modular Form of weight zero at level 15. The zeroth coefficient of the Siegel
Modular Form, i.e., the $m=0$ term is given by the additive seed
$\phi_{0,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta(\tau)\eta(3\tau)\eta(5\tau)\eta(15\tau)$.

So we see that for all the type-I conjugacy classes of $M_{24}$, the length of
the lowest cycle in the balanced cycle-shape is one and all the product
formula obtained via the multiplicative lift matches with the Borcherds
product formula given by Cléry-Gritsenko.

###  E.2 Type-II conjugacy classes

Now we consider the cases for which there is no cycle shape of length one. The
corresponding conjugacy classes are called type-II conjugacy classes.

####  E.2.1 Cycle shape $\mathbf{2^{12}}$

This is an element of order $2$. There are only two twining genera which
contribute to the multiplicative lift. They are

$\displaystyle Z^{\rho_{0}}(\tau,z)=Z^{1^{24}}(\tau,z)\ ,\hskip
14.22636ptZ^{\rho}(\tau,z)=Z^{2^{12}}(\tau,z)\ .$

The multiplicative seed corresponding to this conjugacy class, i.e.,
$Z^{2^{12}}(\tau,z)$ is a Jacobi form of $\Gamma_{0}(4)$. The cusps and other
data of $\Gamma_{0}(4)$ are the following:

$f/e$ | $i\infty$ | $1/2$ | $0/1$
---|---|---|---
$h_{e}$ | 1 | 1 | 4
$N_{e}$ | 1 | 2 | 4

The Borcherds lift given by Eq. (2.6) leads to the following product formula:

$\Phi_{4}^{2^{12}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{c^{0}(nm,\ell)-c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{2}\right)^{c^{1}(nm,\ell)}\
.$ (E.21)

Then using Eq. (2.5), i.e.,

$\displaystyle c^{0}=\frac{1}{2}(f_{0}+f_{1}),\hskip
5.69046ptc^{1}=\frac{1}{2}(f_{0}-f_{1})\ ,$ (E.22)

we can rewrite the above product formula as

$\displaystyle\Phi_{4}^{2^{12}}=qrs\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{2}\Big{)}^{\frac{1}{2}(f_{0}(nm.\ell)-f_{1}(nm,\ell))}\
.$ (E.23)

To see if it can be written as the Borcherds product as stated in theorem 2.1,
we need to compute the coefficients at the cusps $i\infty$ and $1/2$. The
other cusps $0/1$ does not contribute as there are no integer power in $q$ in
the Fourier expansion. Then we get

$\displaystyle\pi_{FE}\Big{(}Z^{2^{12}}|M_{\frac{1}{2}}\Big{)}=-Z^{2^{12}}\ .$

Using this and the product formula for Borcherds product given by Cléry-
Gritsenko, we get

$B_{Z^{2^{12}}}(\mathbf{Z})=\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{2}\Big{)}^{-\frac{1}{2}f_{1}(nm,\ell)}\
.$ (E.24)

This does not give all the terms that appear in the product form obtained from
the Borcherds lift given in Eq. (E.23). The missing terms can be accounted for
by additional terms leading to

$\displaystyle\Big{(}\Phi_{4}^{2^{12}}(\mathbf{Z})\Big{)}^{2}=B_{2Z^{2^{12}}}(\mathbf{Z})B_{Z^{1^{24}}}(\mathbf{2Z})\
.$ (E.25)

Therefore $\Big{(}\Phi_{4}^{2^{12}}(\mathbf{Z})\Big{)}^{2}$ is a meromorphic
Siegel Modular Form of weight $8$ at level $4$. The zeroth coefficient of the
Siegel Modular Form, i.e., the $m=0$ term is given by the additive seed
$\phi_{4,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta(2\tau)^{12}$.

####  E.2.2 Cycle shape $\mathbf{2^{4}\cdot 4^{4}}$

This $M_{24}$ element is of order $4$. The twining genera that contribute to
the multiplicative lifts are,

$\displaystyle Z^{\rho_{0}}(\tau,z)=Z^{1^{24}}(\tau,z)\ ,$ $\displaystyle
Z^{\rho}(\tau,z)=Z^{\rho_{3}}(\tau,z)=Z^{2^{4}4^{4}}(\tau,z)\ ,$
$\displaystyle Z^{\rho_{2}}(\tau,z)=Z^{1^{8}2^{8}}(\tau,z)\ .$

The multiplicative seed, i.e., $Z^{2^{4}4^{4}}(\tau,z)$ is a Jacobi form of
$\Gamma_{0}(8)$. The cusps and other data at level $8$ are the following:

$f/e$ | $i\infty$ | $1/2$ | $1/4$ | $0/1$
---|---|---|---|---
$h_{e}$ | 1 | 2 | 1 | 8
$N_{e}$ | 1 | 4 | 2 | 8

Following the computation similar to the case of cycle shape
$1^{4}2^{2}4^{4}$, the Borcherds lift leads to the following product formula:

$\Phi_{2}^{2^{4}4^{4}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{f_{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{2}\right)^{\frac{1}{2}(f_{2}(nm,\ell)-f_{1}(nm,\ell))}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{4}\right)^{\frac{1}{4}(f_{0}(nm,\ell)-f_{2}(nm,\ell))}\
.$ (E.26)

To see if it can be written as the Borcherds product as stated in theorem 2.1
we need to compute the Fourier-Jacobi coefficients of the multiplicative seed
at the cusps about $i\infty$ and $1/4$ as the Jacobi forms have integer
coefficients for these cusps.

$\displaystyle\pi_{FE}\Big{(}Z^{2^{4}4^{4}}|M_{\frac{1}{4}}\Big{)}=-Z^{2^{4}4^{4}}\
.$

Using this and the product formula for Borcherds product given by Cléry-
Gritsenko, we get

$B_{Z^{2^{4}4^{4}}}(\mathbf{Z})=\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{2}\Big{)}^{-\frac{1}{2}f_{1}(nm,\ell)}\
.$ (E.27)

This product formula does not give all the terms that appear in the product
form obtained from the Borcherds lift given in Eq. (E.26). The missing terms
can be accounted for by additional terms leading to

$\displaystyle\Big{(}\Phi_{2}^{2^{4}4^{4}}(\mathbf{Z})\Big{)}^{2}=B_{2Z^{2^{4}4^{4}}}(\mathbf{Z})B_{Z^{1^{8}2^{8}}}(2\mathbf{Z})\
.$ (E.28)

Therefore $\Big{(}\Phi_{2}^{2^{4}4^{4}}(\mathbf{Z})\Big{)}^{2}$ is a
meromorphic Siegel Modular Form of weight $4$ at level 8. The zeroth
coefficient of the Siegel Modular Form, i.e., the $m=0$ term is given by the
additive seed
$\phi_{2,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta(2\tau)^{4}\eta(4\tau)^{4}$.

####  E.2.3 Cycle shape $\mathbf{3^{8}}$

This is an element of order $3$. The twining genera which contribute to the
multiplicative lifts are

$\displaystyle Z^{\rho_{0}}(\tau,z)=Z^{1^{24}}(\tau,z),$ $\displaystyle
Z^{\rho}(\tau,z)=Z^{\rho_{2}}(\tau,z)=Z^{3^{8}}(\tau,z)\ .$

The corresponding multiplicative seed $Z^{3^{8}}(\tau,z)$ is a Jacobi form of
$\Gamma_{0}(9)$. The cusps and other data for $\Gamma_{0}(9)$ are the
following:

$f/e$ | $i\infty$ | $1/2$ | $1/3$ | $0/1$
---|---|---|---|---
$h_{e}$ | 1 | 1 | 1 | 9
$N_{e}$ | 1 | 3 | 3 | 9

Cusps about $2/3$ can be computed by $M_{\frac{2}{3}}=ST^{-1}ST^{2}S$. The
cusp about $0$ does not have integer power in $q$ and hence does not
contribute. The Borcherds lift given leads to the product formula given by

$\displaystyle\Phi_{2}^{3^{8}}=qrs\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{3}\Big{)}^{\frac{1}{3}(f_{0}(nm.\ell)-f_{1}(nm,\ell)}\
.$ (E.29)

To see if it can be written as the Borcherds product as stated in theorem 2.1
we need to compute the coefficients at the cusps about $i\infty$, $1/3$ and
$2/3$. The contributions from the cusps at $1/3$ and $2/3$ have cube roots of
unity. However, the contributions of the two cusps add in the product formula
to give integral coefficients. One gets

$\displaystyle\pi_{FE}\Big{(}Z^{3^{8}}|M_{\frac{1}{3}}+Z^{3^{8}}|M_{\frac{2}{3}}\Big{)}=-Z^{3^{8}}\
.$

Using this and the product formula for Borcherds product given by Cléry-
Gritsenko, we get

$B_{Z^{3^{8}}}(\mathbf{Z})=\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{3}\Big{)}^{-\frac{1}{3}f_{1}(nm,\ell)}\
.$ (E.30)

This does not give all the terms that appear in the product form obtained from
the Borcherds lift given in Eq. (E.29). The missing terms can be accounted for
by additional terms leading to

$\displaystyle\Big{(}\Phi_{2}^{3^{8}}(\mathbf{Z})\Big{)}^{3}=B_{3Z^{4^{6}}}(\mathbf{Z})B_{Z^{1^{24}}}(\mathbf{3Z})\
.$ (E.31)

Therefore $\Big{(}\Phi_{2}^{3^{8}}(\mathbf{Z})\Big{)}^{3}$ is a meromorphic
Siegel Modular Form of weight $6$ at level $9$. The zeroth coefficient of the
Siegel Modular Form, i.e., the $m=0$ term is given by the additive seed
$\phi_{2,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta(3\tau)^{8}$.

####  E.2.4 Cycle shape $\mathbf{4^{6}}$

This is an element of order $4$. The twining genera which contribute to the
Siegel Modular Forms are

$\displaystyle Z^{\rho_{0}}(\tau,z)=Z^{1^{24}}(\tau,z),$ $\displaystyle
Z^{\rho}(\tau,z)=Z^{\rho_{3}}(\tau,z)=Z^{4^{6}}(\tau,z),$ $\displaystyle
Z^{\rho_{2}}(\tau,z)=Z^{2^{12}}(\tau,z)\ .$

The multiplicative seed $Z^{4^{6}}(\tau,z)$ is a Jacobi form of
$\Gamma_{0}(16)$. The cusps and other data at level $16$ are the following:

$f/e$ | $i\infty$ | $1/2$ | $1/4$ | 3/4 | 1/8 | $0/1$
---|---|---|---|---|---|---
$h_{e}$ | 1 | 4 | 1 | 1 | 1 | 16
$N_{e}$ | 1 | 8 | 4 | 4 | 2 | 16

Cusp about $3/4$ can be computed via following transformation:

$M_{\frac{3}{4}}=ST^{-1}ST^{2}ST^{-1}S\ .$

Following the computation similar to the case of cycle shape
$1^{4}2^{2}4^{4}$, the Borcherds lift leads to the following product formula:

$\Phi_{1}^{4^{6}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{f_{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{2}\right)^{\frac{1}{2}(f_{2}(nm,\ell)-f_{1}(nm,\ell))}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{4}\right)^{\frac{1}{4}(f_{0}(nm,\ell)-f_{2}(nm,\ell))}\
.$ (E.32)

To see if it can be written as the Borcherds product as stated in theorem 2.1,
we need to compute the coefficients at cusps about $i\infty$, ($1/4,3/4$) and
$1/8$. We need to pair up cusps with identical $(h_{e},N_{e})$. Computing
these cusps, we get

$\Big{(}Z^{4^{6}}|M_{\frac{1}{4}}\Big{)}=-iZ^{4^{6}},\text{ and
}\Big{(}Z^{4^{6}}|M_{\frac{3}{4}}\Big{)}=iZ^{4^{6}}\ .$

Then one obtains

$\displaystyle\pi_{FE}\Big{(}Z^{4^{6}}|M_{\frac{1}{4}}+Z^{4^{6}}|M_{\frac{3}{4}}\Big{)}=0,$
$\displaystyle\pi_{FE}\Big{(}Z^{4^{6}}|M_{\frac{1}{8}}\Big{)}=-Z^{4^{6}}\ .$

Using these and the product formula for Borcherds product given by Cléry-
Gritsenko, we get

$B_{Z^{4^{6}}}(\mathbf{Z})=\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{2}\Big{)}^{-\frac{1}{2}f_{1}(nm,\ell)}\
.$ (E.33)

This does not give all the terms that appear in the product form obtained from
the Borcherds lift given in Eq. (E.32). The missing terms can be accounted for
by additional terms leading to

$\displaystyle\Big{(}\Phi_{1}^{4^{6}}(\mathbf{Z})\Big{)}^{4}=B_{4Z^{4^{6}}}(\mathbf{Z})B_{2Z^{2^{12}}}(2\mathbf{Z})B_{Z^{1^{24}}}(\mathbf{4Z})\
.$ (E.34)

Therefore $\Big{(}\Phi_{1}^{4^{6}}(\mathbf{Z})\Big{)}^{4}$ is a modular form
of weight $4$ at level $16$. The zeroth coefficient of the Siegel Modular
Form, i.e., the $m=0$ term is given by the additive seed
$\phi_{1,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta(4\tau)^{6}$.

####  E.2.5 Cycle shape $\mathbf{6^{4}}$

This is an element of order $6$ corresponding to the conjugacy class $6b$. The
twining genera which contribute to the multiplicative lift are

$\displaystyle Z^{\rho_{0}}(\tau,z)=Z^{1^{24}}(\tau,z),$ $\displaystyle
Z^{\rho}(\tau,z)=Z^{\rho_{5}}(\tau,z)=Z^{6^{4}}(\tau,z),$ $\displaystyle
Z^{\rho_{2}}(\tau,z)=Z^{\rho_{4}}(\tau,z)=Z^{3^{8}}(\tau,z),$ $\displaystyle
Z^{\rho_{3}}(\tau,z)=Z^{2^{12}}(\tau,z)\ .$

The corresponding multiplicative seed, $Z^{6^{4}}(\tau,z)$ is a Jacobi form of
$\Gamma_{0}(36)$. The cusps and other data at level $36$ are the following:

c | $i\infty$ | 0 | $1/2$ | $1/3$ | $2/3$ | $1/4$ | $1/6$ | $5/6$ | $1/9$ | $1/12$ | $5/12$ | $1/18$
---|---|---|---|---|---|---|---|---|---|---|---|---
$h_{e}$ | 1 | 36 | 9 | 4 | 4 | 9 | 1 | 1 | 4 | 1 | 1 | 1
$N_{e}$ | 1 | 36 | 18 | 12 | 12 | 9 | 6 | 6 | 4 | 3 | 3 | 2

The cusps about $5/6$ and $5/12$ are respectively given by
$M_{\frac{5}{6}}=-ST^{-1}ST^{5}S$ and
$M_{\frac{5}{12}}=ST^{-2}ST^{2}ST^{-2}S$. Following the computation similar to
the cycle shape $1^{2}2^{2}3^{2}6^{2}$, the multiplicative lift leads to the
product formula given by

$\Phi_{0}^{6^{4}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{f_{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{2}\right)^{\frac{1}{2}(f_{2}(nm,\ell)-f_{1}(nm,\ell))}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{3}\right)^{\frac{1}{3}(f_{3}(nm,\ell)-f_{1}(nm,\ell))}\left(1-(q^{n}r^{\ell}s^{m})^{6}\right)^{\frac{1}{6}(f_{0}(nm,\ell)+f_{1}(nm,\ell)-f_{2}(nm,\ell)-f_{3}(nm,\ell))}\
.$ (E.35)

Now to see if can be written as the Borcherds product as stated in theorem 2.1
we need to compute the coefficients at cusps about $i\infty$, $(1/6,5/6)$,
$(1/12,5/12)$ and $1/18$. We need to pair up cusps with identical values of
$h_{e}$ and $N_{e}$ to get integral coefficients. Then from straightforward
calculations one obtains

$\displaystyle\pi_{FE}\Big{(}Z^{6^{4}}|M_{\frac{1}{6}}+Z^{6^{4}}|M_{\frac{5}{6}}\Big{)}$
$\displaystyle=Z^{6^{4}},$
$\displaystyle\pi_{FE}\Big{(}Z^{6^{4}}|M_{\frac{1}{12}}+Z^{6^{4}}|M_{\frac{5}{12}}\Big{)}$
$\displaystyle=-Z^{6^{4}},$
$\displaystyle\pi_{FE}\Big{(}Z^{6^{4}}|M_{\frac{1}{18}}\Big{)}$
$\displaystyle=-Z^{6^{4}}\ .$

Using these and the product formula for Borcherds product given by Cléry-
Gritsenko, we get

$B_{Z^{6^{4}}}(\mathbf{Z})=\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{2}\Big{)}^{-\frac{1}{2}f_{1}(nm,\ell)}\\\
\Big{(}1-(q^{n}r^{\ell}s^{m})^{3}\Big{)}^{-\frac{1}{3}f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{6}\Big{)}^{\frac{1}{6}f_{1}(nm,\ell)}\
.$ (E.36)

This does not give all the terms that appear in the product form given in Eq.
(E.35). The missing terms can be accounted for by additional terms leading to

$\displaystyle\Big{(}\Phi_{0}^{6^{4}}(\mathbf{Z})\Big{)}^{6}=B_{6Z^{6^{4}}}(\mathbf{Z})B_{3Z^{3^{8}}}(2\mathbf{Z})B_{2Z^{2^{12}}}(3\mathbf{Z})B_{Z^{1^{24}}}(6\mathbf{Z})\
.$ (E.37)

$\Big{(}\Phi_{0}^{6^{4}}(\mathbf{Z})\Big{)}^{6}$ is a meromorphic Siegel
Modular Form of weight zero at level 36. The zeroth coefficient of the Siegel
Modular Form, i.e., the $m=0$ term is given by the additive seed
$\phi_{0,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta(6\tau)^{4}$.

####  E.2.6 Cycle shape $\mathbf{2^{2}\cdot 10^{2}}$

This is an element of order $10$. The twining genera which contribute to the
multiplicative lift are

$\displaystyle Z^{\rho_{0}}(\tau,z)=Z^{1^{24}}(\tau,z),$ $\displaystyle
Z^{\rho}(\tau,z)=Z^{\rho_{3}}(\tau,z)=Z^{\rho_{7}}(\tau,z)=Z^{\rho_{9}}(\tau,z)=Z^{2^{2}10^{2}}(\tau,z),$
$\displaystyle
Z^{\rho_{2}}(\tau,z)=Z^{\rho_{4}}(\tau,z)=Z^{\rho_{6}}(\tau,z)=Z^{\rho_{8}}(\tau,z)=Z^{1^{4}5^{4}}(\tau,z),$
$\displaystyle Z^{\rho_{5}}(\tau,z)=Z^{2^{12}}(\tau,z)\ .$

The multiplicative seed, $Z^{2^{2}10^{2}}(\tau,z)$ is a Jacobi form of
$\Gamma_{0}(20)$. The cusps at level $20$ are the following:

c | $i\infty$ | $1/2$ | $1/4$ | $1/5$ | $1/10$ | 0/1
---|---|---|---|---|---|---
$h_{e}$ | 1 | 5 | 5 | 4 | 1 | 20
$N_{e}$ | 1 | 10 | 5 | 4 | 2 | 20

The Borcherds lift given by Eq. (2.6) leads to the following product formula:

$\Phi_{0}^{2^{2}10^{2}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{c^{0}(nm,\ell)+c^{1}(nm,\ell)-c^{2}(nm,\ell)-c^{5}(nm,\ell)}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{2}\right)^{c^{5}(nm,\ell)-c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{5}\right)^{c^{2}(nm,\ell)-c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{10}\right)^{c^{1}(nm,\ell)}\
.$ (E.38)

Then using Eq. (2.5), i.e.,

$\displaystyle c^{0}=\frac{1}{10}(f_{0}+4f_{1}+4f_{2}+f_{5}),\hskip
5.69046ptc^{1}=\frac{1}{10}(f_{0}+f_{1}-f_{2}-f_{5})\ ,$ (E.39) $\displaystyle
c^{2}=\frac{1}{10}(f_{0}-f_{1}-f_{2}+f_{5}),\hskip
5.69046ptc^{5}=\frac{1}{10}(f_{0}-4f_{1}+4f_{2}-f_{5})\ ,$ (E.40)

we can rewrite the above product formula as

$\Phi_{0}^{2^{2}10^{2}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{2}\Big{)}^{\frac{1}{2}(f_{2}(nm,\ell)-f_{1}(nm,\ell))}\\\
\Big{(}1-(q^{n}r^{\ell}s^{m})^{5}\Big{)}^{\frac{1}{5}(f_{5}(nm,\ell)-f_{1}(nm,\ell))}\Big{(}1-(q^{n}r^{\ell}s^{m})^{10}\Big{)}^{\frac{1}{10}(f_{0}(nm,\ell)+f_{1}(nm,\ell)-f_{2}(nm,\ell)-f_{5}(nm,\ell))}\
.$ (E.41)

Now to see if can be written as the Borcherds product as stated in theorem 2.1
we need to compute the coefficients at the cusps about $i\infty$, $1/2$, $1/4$
and $1/10$ as other cusps do not have integral power in $q$. Then from
straightforward calculations one obtains

$\displaystyle\pi_{FE}\Big{(}Z^{2^{2}10^{2}}|M_{\frac{1}{2}}\Big{)}$
$\displaystyle=\frac{1}{5}\left(Z^{2^{2}10^{2}}-Z^{2^{12}}\right),$
$\displaystyle\pi_{FE}\Big{(}Z^{2^{2}10^{2}}|M_{\frac{1}{4}}\Big{)}$
$\displaystyle=-\frac{1}{5}\left(Z^{2^{2}10^{2}}-Z^{2^{12}}\right),$
$\displaystyle\pi_{FE}\Big{(}Z^{2^{2}10^{2}}|M_{\frac{1}{10}}\Big{)}$
$\displaystyle=-Z^{2^{2}10^{2}}\ .$

Using these and the product formula for Borcherds product given by Cléry-
Gritsenko, we get

$B_{Z^{2^{2}10^{2}}}(\mathbf{Z})=\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{2}\Big{)}^{-\frac{1}{2}f_{1}(nm,\ell)}\\\
\Big{(}1-(q^{n}r^{\ell}s^{m})^{5}\Big{)}^{\frac{1}{5}(f_{5}(nm,\ell)-f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{10}\Big{)}^{\frac{1}{10}(f_{1}(nm,\ell)-f_{5}(nm,\ell))}\
.$ (E.42)

This does not give all the terms that appear in the product form given in Eq.
(E.41). The missing terms can be accounted for by additional terms leading to

$\displaystyle\Big{(}\Phi_{0}^{2^{2}10^{2}}(\mathbf{Z})\Big{)}^{2}=B_{2Z^{2^{2}10^{2}}}(\mathbf{Z})B_{Z^{1^{4}5^{4}}}(2\mathbf{Z})\
.$ (E.43)

$\Big{(}\Phi_{0}^{2^{2}10^{2}}(\mathbf{Z})\Big{)}^{2}$ is a meromorphic Siegel
Modular Form of weight zero at level 20. The zeroth coefficient of the Siegel
Modular Form, i.e., the $m=1$ term is given by the additive seed
$\phi_{0,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta(2\tau)^{2}\eta(10\tau)^{2}$.

####  E.2.7 Cycle shape $\mathbf{2\cdot 4\cdot 6\cdot 12}$

This is an element of order $12$. The twining genera which contribute to the
multiplicative lift are

$\displaystyle Z^{\rho_{0}}(\tau,z)=Z^{1^{24}}(\tau,z),$ $\displaystyle
Z^{\rho}(\tau,z)=Z^{\rho_{5}}(\tau,z)=Z^{\rho_{7}}(\tau,z)=Z^{\rho_{11}}(\tau,z)=Z^{2^{1}4^{1}6^{1}12^{1}}(\tau,z),$
$\displaystyle
Z^{\rho_{2}}(\tau,z)=Z^{\rho_{10}}(\tau,z)=Z^{1^{2}2^{2}3^{2}6^{2}}(\tau,z),$
$\displaystyle
Z^{\rho_{3}}(\tau,z)=Z^{\rho_{9}}(\tau,z)=Z^{2^{4}4^{4}}(\tau,z),$
$\displaystyle Z^{\rho_{4}}(\tau,z)=Z^{\rho_{8}}(\tau,z)=Z^{1^{6}3^{6}},$
$\displaystyle Z^{\rho_{6}}(\tau,z)=Z^{1^{8}2^{8}}\ .$

The corresponding multiplicative seed, $Z^{2^{1}4^{1}6^{1}12^{1}}(\tau,z)$ is
a Jacobi form of $\Gamma_{0}(24)$. The cusps at level $24$ are the following:

c | $i\infty$ | 0 | $1/2$ | $1/3$ | $1/4$ | $1/6$ | $1/8$ | $1/12$
---|---|---|---|---|---|---|---|---
$h_{e}$ | 1 | 24 | 6 | 8 | 3 | 2 | 3 | 1
$N_{e}$ | 1 | 24 | 12 | 8 | 6 | 4 | 3 | 2

The Borcherds lift given by Eq. (2.6) leads to the following product formula:

$\Phi_{0}^{2^{1}4^{1}6^{1}12^{1}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{c^{0}(nm,\ell)+c^{2}(nm,\ell)-c^{4}(nm,\ell)-c^{6}(nm,\ell)}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{2}\right)^{c^{6}(nm,\ell)-c^{2}(nm,\ell)-c^{3}(nm,\ell)+c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{3}\right)^{c^{4}(nm,\ell)-c^{2}(nm,\ell)}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{4}\right)^{c^{3}(nm,\ell)-c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{6}\right)^{c^{2}(nm,\ell)-c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{12}\right)^{c^{1}(nm,\ell)}\
.$ (E.44)

Then using Eq. (2.5), i.e.,

$\displaystyle
c^{0}=\frac{1}{12}(f_{0}+4f_{1}+2f_{2}+2f_{3}+2f_{4}+f_{6}),\hskip
5.69046ptc^{1}=\frac{1}{12}(f_{0}+f_{2}-f_{4}-f_{6})\ ,$ $\displaystyle
c^{2}=\frac{1}{12}(f_{0}+2f_{1}-f_{2}-2f_{3}-f_{4}+f_{6}),\hskip
5.69046ptc^{3}=\frac{1}{12}(f_{0}-2f_{2}+2f_{4}-f_{6})\ ,$ $\displaystyle
c^{6}=\frac{1}{12}(f_{0}-4f_{1}+2f_{2}-2f_{3}+2f_{4}+f_{6}),\hskip
5.69046ptc^{4}=\frac{1}{12}(f_{0}-2f_{1}-f_{2}+2f_{3}-f_{4}+f_{6})\ ,$

we can rewrite the above product formula as

$\Phi^{2^{1}4^{1}6^{1}12^{1}}_{0}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{2}\Big{)}^{\frac{1}{2}(f_{2}(nm,\ell)-f_{1}(nm,\ell))}\\\
\Big{(}1-(q^{n}r^{\ell}s^{m})^{3}\Big{)}^{\frac{1}{3}(f_{3}(nm,\ell)-f_{1}(nm,\ell))}\Big{(}1-(q^{n}r^{\ell}s^{m})^{4}\Big{)}^{\frac{1}{4}(-f_{2}(nm,\ell)+f_{4}(nm,\ell))}\\\
\Big{(}1-(q^{n}r^{\ell}s^{m})^{6}\Big{)}^{\frac{1}{3}(f_{1}(nm,\ell)-f_{2}(nm,\ell)-f_{3}(nm,\ell)+f_{6}(nm,\ell))}\\\
\Big{(}1-(q^{n}r^{\ell}s^{m})^{12}\Big{)}^{\frac{1}{12}(f_{0}(nm,\ell)+f_{2}(nm,\ell)-f_{4}(nm,\ell)-f_{6}(nm,\ell))}\
.$ (E.45)

Now to see if can be written as the Borcherds product as stated in theorem 2.1
we need to compute the coefficients at different cusps. Only the cusps about
$1/4,1/8$ and $1/12$ has integer power of $q$ in the Fourier expansion. The
other cusps do not contribute. Then from straightforward calculations one
obtains

$\displaystyle\pi_{FE}\Big{(}Z^{2^{1}4^{1}6^{1}12^{1}}|M_{\frac{1}{4}}\Big{)}$
$\displaystyle=\frac{1}{3}\left(Z^{2^{1}4^{1}6^{1}12^{1}}-Z^{2^{4}4^{4}}\right),$
$\displaystyle\pi_{FE}\Big{(}Z^{2^{1}4^{1}6^{1}12^{1}}|M_{\frac{1}{8}}\Big{)}$
$\displaystyle=-\frac{1}{3}\left(Z^{2^{1}4^{1}6^{1}12^{1}}-Z^{2^{4}4^{4}}\right),$
$\displaystyle\pi_{FE}\Big{(}Z^{2^{1}4^{1}6^{1}12^{1}}|M_{\frac{1}{12}}\Big{)}$
$\displaystyle=-Z^{2^{1}4^{1}6^{1}12^{1}}\ .$

Using these and the product formula for Borcherds product given by Cléry-
Gritsenko, we get

$B_{Z^{2^{1}4^{1}6^{1}12^{1}}}(\mathbf{Z})=\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{2}\Big{)}^{-\frac{1}{2}f_{1}(nm,\ell)}\\\
\Big{(}1-(q^{n}r^{\ell}s^{m})^{3}\Big{)}^{-\frac{1}{3}(f_{1}(nm,\ell)-f_{3}(nm,\ell))}\Big{(}1-(q^{n}r^{\ell}s^{m})^{6}\Big{)}^{\frac{1}{6}(f_{1}(nm,\ell)-f_{3}(nm,\ell))}\
.$ (E.46)

This does not give all the terms that appear in the product form given in Eq.
(E.45). The missing terms can be accounted for by additional terms leading to

$\displaystyle\Big{(}\Phi_{0}^{2^{1}4^{1}6^{1}12^{1}}(\mathbf{Z})\Big{)}^{2}=B_{2Z^{2^{1}4^{1}6^{1}12^{1}}}(\mathbf{Z})B_{Z^{1^{2}2^{2}3^{2}6^{2}}}(2\mathbf{Z})\
.$ (E.47)

$\Big{(}\Phi_{0}^{2^{1}4^{1}6^{1}12^{1}}(\mathbf{Z})\Big{)}^{2}$ is a
meromorphic Siegel Modular Form of weight zero at level 24. The zeroth
coefficient of the Siegel Modular Form, i.e., the $m=0$ term is given by the
additive seed
$\phi_{0,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta(2\tau)\eta(4\tau)\eta(6\tau)\eta(12\tau)$.

####  E.2.8 Cycle shape $\mathbf{12^{2}}$

This is an element of order $12$. The twining genera which contribute to the
multiplicative lift are

$\displaystyle Z^{\rho_{0}}(\tau,z)=Z^{1^{24}}(\tau,z),$ $\displaystyle
Z^{\rho}(\tau,z)=Z^{\rho_{5}}(\tau,z)=Z^{\rho_{7}}(\tau,z)=Z^{\rho_{11}}(\tau,z)=Z^{12^{2}}(\tau,z),$
$\displaystyle Z^{\rho_{2}}(\tau,z)=Z^{\rho_{10}}(\tau,z)=Z^{6^{4}}(\tau,z),$
$\displaystyle Z^{\rho_{3}}(\tau,z)=Z^{\rho_{9}}(\tau,z)=Z^{4^{6}}(\tau,z),$
$\displaystyle Z^{\rho_{4}}(\tau,z)=Z^{\rho_{8}}(\tau,z)=Z^{3^{8}},$
$\displaystyle Z^{\rho_{6}}(\tau,z)=Z^{2^{12}}\ .$

The corresponding multiplicative seed, $Z^{12^{2}}(\tau,z)$ is a Jacobi form
of $\Gamma_{0}(144)$. The cusps and other data at level $144$ are the
following:

c | $i\infty$ | 0 | $1/2$ | $1/3$ | 2/3 | $1/4$ | 3/4 | $1/6$ | 5/6 | $1/8$ | 1/9 | $1/12$ | 5/12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$h_{e}$ | 1 | 144 | 36 | 16 | 16 | 9 | 9 | 4 | 4 | 9 | 16 | 1 | 1
$N_{e}$ | 1 | 144 | 72 | 48 | 48 | 36 | 36 | 24 | 24 | 18 | 16 | 12 | 12
c | $7/12$ | $11/12$ | 1/16 | $1/18$ | 1/24 | $5/24$ | 1/36 | $7/36$ | $1/48$ | $5/48$ | 1/72
---|---|---|---|---|---|---|---|---|---|---|---
$h_{e}$ | 1 | 1 | 9 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$N_{e}$ | 12 | 12 | 9 | 8 | 6 | 6 | 4 | 4 | 3 | 3 | 2

####  E.2.9 Cycle shape $\mathbf{3\cdot 21}$

This is an element of order $21$. The twining genera that contribute to the
multiplicative lift are

$\displaystyle Z^{\rho_{0}}(\tau,z)=Z^{1^{24}}(\tau,z),$ $\displaystyle
Z^{\rho}(\tau,z)=Z^{\rho_{2}}(\tau,z)=Z^{\rho_{4}}(\tau,z)=Z^{\rho_{5}}(\tau,z)=Z^{\rho_{8}}(\tau,z)=Z^{\rho_{10}}(\tau,z)=Z^{\rho_{11}}(\tau,z)$
$\displaystyle\hskip
28.45274pt=Z^{\rho_{13}}(\tau,z)=Z^{\rho_{16}}(\tau,z)=Z^{\rho_{17}}(\tau,z)=Z^{\rho_{19}}(\tau,z)=Z^{\rho_{20}}(\tau,z)=Z^{3^{1}21^{1}}(\tau,z),$
$\displaystyle
Z^{\rho_{3}}(\tau,z)=Z^{\rho_{6}}(\tau,z)=Z^{\rho_{9}}(\tau,z)=Z^{\rho_{12}}(\tau,z)=Z^{\rho_{15}}(\tau,z)=Z^{\rho_{18}}(\tau,z)=Z^{1^{3}7^{3}}(\tau,z),$
$\displaystyle Z^{\rho_{7}}(\tau,z)=Z^{\rho_{14}}(\tau,z))=Z^{3^{8}}(\tau,z)\
.$

In this case the multiplicative seed, $Z^{3^{1}21^{1}}(\tau,z)$ is a Jacobi
form of $\Gamma_{0}(63)$. The cusps and other data for $\Gamma_{0}(63)$ are
the following:

c | $i\infty$ | $1/3$ | $2/3$ | $1/7$ | $1/9$ | 1/21 | 2/21 | 0/1
---|---|---|---|---|---|---|---|---
$h_{e}$ | 1 | 7 | 7 | 9 | 7 | 1 | 1 | 63
$N_{e}$ | 1 | 21 | 21 | 9 | 7 | 3 | 3 | 63

The Borcherds lift given by Eq. (2.6) leads to the following product formula:

$\Phi_{-1}^{3^{1}21^{1}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\left(1-q^{n}r^{\ell}s^{m}\right)^{c^{0}(nm,\ell)+c^{1}(nm,\ell)-c^{3}(nm,\ell)-c^{7}(nm,\ell)}\\\
\left(1-(q^{n}r^{\ell}s^{m})^{3}\right)^{c^{7}(nm,\ell)-c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{7}\right)^{c^{3}(nm,\ell)-c^{1}(nm,\ell)}\left(1-(q^{n}r^{\ell}s^{m})^{21}\right)^{c^{1}(nm,\ell)}\
.$ (E.48)

Then using Eq. (2.5), i.e.,

$\displaystyle c^{0}=\frac{1}{21}(f_{0}+12f_{1}+6f_{3}+2f_{7}),\hskip
5.69046ptc^{1}=\frac{1}{21}(f_{0}+f_{1}-f_{3}-f_{7})\ ,$ $\displaystyle
c^{3}=\frac{1}{21}(f_{0}-2f_{1}-f_{3}+2f_{7}),\hskip
5.69046ptc^{7}=\frac{1}{21}(f_{0}-6f_{1}+6f_{3}-f_{7})\ ,$

we can rewrite the above product formula as

$\Phi_{-1}^{3^{1}21^{1}}(\mathbf{Z})=qrs\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{3}\Big{)}^{\frac{1}{3}(f_{3}(nm,\ell)-f_{1}(nm,\ell))}\\\
\Big{(}1-(q^{n}r^{\ell}s^{m})^{7}\Big{)}^{\frac{1}{7}(f_{7}(nm,\ell)-f_{1}(nm,\ell))}\Big{(}1-(q^{n}r^{\ell}s^{m})^{21}\Big{)}^{\frac{1}{21}(f_{0}(nm,\ell)+f_{1}(nm,\ell)-f_{3}(nm,\ell)-f_{7}(nm,\ell))}\
.$ (E.49)

Now to see if can be written as the Borcherds product as stated in theorem 2.1
we need to compute the coefficients at different cusps. We need to consider
the contributions from the cusps at $i\infty$, $(1/3,2/3)$, $1/9$, and
$(1/21,2/21)$ as other cusps do not have integral power in $q$. Here again, we
need to pair up cusps with identical $h_{e}$ and $N_{e}$ to obtain integral
coefficients. Then from straightforward calculations one obtains

$\displaystyle\pi_{FE}\Big{(}Z^{3^{1}21^{1}}|M_{\frac{1}{3}}+Z^{3^{1}21^{1}}|M_{\frac{2}{3}}\Big{)}$
$\displaystyle=\frac{1}{7}(Z^{3^{1}21^{1}}-Z^{3^{8}}),$
$\displaystyle\pi_{FE}\Big{(}Z^{3^{1}21^{1}}|M_{\frac{1}{9}}\Big{)}$
$\displaystyle=-\frac{1}{7}(Z^{3^{1}21^{1}}-Z^{3^{8}}),$
$\displaystyle\pi_{FE}\Big{(}Z^{3^{1}21^{1}}|M_{\frac{1}{21}}+Z^{3^{1}21^{1}}|M_{\frac{2}{21}}\Big{)}$
$\displaystyle=-Z^{3^{1}21^{1}}\ .$

Using these and the product formula for Borcherds product given by Cléry-
Gritsenko, we get

$B_{Z^{3^{1}21^{1}}}(\mathbf{Z})=\prod_{(n,\ell,m)>0}\Big{(}1-q^{n}r^{\ell}s^{m}\Big{)}^{f_{1}(nm,\ell)}\Big{(}1-(q^{n}r^{\ell}s^{m})^{2}\Big{)}^{-\frac{1}{3}f_{1}(nm,\ell)}\\\
\Big{(}1-(q^{n}r^{\ell}s^{m})^{7}\Big{)}^{\frac{1}{7}(f_{7}(nm,\ell)-f_{1}(nm,\ell))}\Big{(}1-(q^{n}r^{\ell}s^{m})^{21}\Big{)}^{\frac{1}{21}(f_{1}(nm,\ell)-f_{7}(nm,\ell))}\
.$ (E.50)

This does not give all the terms that appear in the product form given in Eq.
(E.49). The missing terms can be accounted for by additional terms leading to

$\displaystyle\Big{(}\Phi_{-1}^{3^{1}21^{1}}(\mathbf{Z})\Big{)}^{3}=B_{3Z^{3^{1}21^{1}}}(\mathbf{Z})B_{Z^{1^{3}7^{3}}}(3\mathbf{Z})\
.$ (E.51)

$\Big{(}\Phi_{-1}^{3^{1}21^{1}}(\mathbf{Z})\Big{)}^{3}$ is a meromorphic
Siegel Modular Form of weight $(-3)$ at level 63. The zeroth coefficient of
the Siegel Modular Form, i.e., the $m=0$ term is given by the additive seed
$\phi_{-1,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\eta(3\tau)\eta(21\tau)$.

Unlike the type-I case, none of the conjugacy class has a cycle shape with
cycle length one. In all these cases, the multiplicative lift can be expressed
as a product of multiple rescaled Borcherds products. This proves that the
multiplicative lift can be used to construct a genus-two Siegel Modular Form
for all the conjugacy classes of $M_{24}$, including those for which the
additive construction is not known.

## References

  * [1] R. Dijkgraaf, E. P. Verlinde, and H. L. Verlinde, “Counting dyons in $\mathcal{N}=4$ string theory,” Nucl. Phys. B484 (1997) 543–561, arXiv:hep-th/9607026 [hep-th].
  * [2] S. Govindarajan and K. Gopala Krishna, “Generalized Kac-Moody Algebras from CHL dyons,” JHEP 04 (2009) 032, arXiv:0807.4451 [hep-th].
  * [3] J. H. Conway and S. P. Norton, “Monstrous moonshine,” Bulletin of the London Mathematical Society 11 no. 3, 308–339.
  * [4] D. Dummit, H. Kisilevsky, and J. McKay, “Multiplicative products of $\eta$-functions,” Contemp. Math. 045 (1985) 89–98.
  * [5] G. Mason, “$M_{24}$ and certain automorphic forms,” Contemp. Math. 45 (1985) 89–98.
  * [6] T. Eguchi, H. Ooguri, and Y. Tachikawa, “Notes on the K3 Surface and the Mathieu group $M_{24}$,” Exper. Math. 20 (2011) 91–96, arXiv:1004.0956 [hep-th].
  * [7] M. C. N. Cheng, “K3 Surfaces, $\mathcal{N}=4$ Dyons, and the Mathieu Group M24,” Commun. Num. Theor. Phys. 4 (2010) 623–658, arXiv:1005.5415 [hep-th].
  * [8] M. R. Gaberdiel, S. Hohenegger, and R. Volpato, “Mathieu twining characters for K3,” JHEP 09 (2010) 058, arXiv:1006.0221 [hep-th].
  * [9] T. Eguchi and K. Hikami, “Note on twisted elliptic genus of $K3$ surface,” Phys. Lett. B694 (2011) 446–455, arXiv:1008.4924 [hep-th].
  * [10] T. Gannon, “Much ado about Mathieu,” Adv. Math. 301 (2016) 322–358, arXiv:1211.5531 [math.RT].
  * [11] S. Govindarajan, “Unravelling Mathieu Moonshine,” Nucl. Phys. B864 (2012) 823–839, arXiv:1106.5715 [hep-th].
  * [12] D. Persson and R. Volpato, “Second Quantized Mathieu Moonshine,” Commun. Num. Theor. Phys. 08 (2014) 403–509, arXiv:1312.0622 [hep-th].
  * [13] S. Govindarajan and S. Samanta, “Two moonshines for $L_{2}(11)$ but none for $M_{12}$,” Nucl. Phys. B939 (2019) 566–598, arXiv:1804.06677 [hep-th].
  * [14] S. Govindarajan and K. Gopala Krishna, “BKM Lie superalgebras from dyon spectra in Z(N) CHL orbifolds for composite N,” JHEP 05 (2010) 014, arXiv:0907.1410 [hep-th].
  * [15] S. Govindarajan, “BKM Lie superalgebras from counting twisted CHL dyons,” JHEP 05 (2011) 089, arXiv:1006.3472 [hep-th].
  * [16] S. Govindarajan and S. Samanta, “BKM Lie superalgebras from counting twisted CHL dyons – II,” Nucl. Phys. B 948 (2019) 114770, arXiv:1905.06083 [hep-th].
  * [17] M. C. N. Cheng and J. F. R. Duncan, “On Rademacher Sums, the Largest Mathieu Group, and the Holographic Modularity of Moonshine,” Commun. Num. Theor. Phys. 6 (2012) 697–758, arXiv:1110.3859 [math.RT].
  * [18] V. Gritsenko and F. Clery, “The Siegel modular forms of genus 2 with the simplest divisor,” arXiv:0812.3962 [math.NT].
  * [19] T. Eguchi and K. Hikami, “Twisted Elliptic Genus for K3 and Borcherds Product,” Lett. Math. Phys. 102 (2012) 203–222, arXiv:1112.5928 [hep-th].
  * [20] V. A. Gritsenko and V. V. Nikulin, “Automorphic Forms and Lorentzian Kac–Moody Algebras. Part I,” arXiv:alg-geom/9610022 [math.AG].
  * [21] V. A. Gritsenko and V. V. Nikulin, “Automorphic forms and Lorentzian Kac-Moody algebras. Part 2,” arXiv:alg-geom/9611028 [alg-geom].
  * [22] H. Aoki and T. Ibukiyama, “Simple graded rings of siegel modular forms, differential operators and borcherds products,” International Journal of Mathematics 16 (03, 2005) 249–279.
  * [23] M. Raum, “$M_{24}$-twisted product expansions are Siegel modular forms,” Commun. Number Theory Phys. 7 no. 3, (2013) 469–495. https://doi.org/10.4310/CNTP.2013.v7.n3.a3.
  * [24] A. O. L. Atkin and J. Lehner, “Hecke operators on $\gamma_{0}(m)$,” Math. Ann. 185 (1970) 134–160.
  * [25] M. R. Gaberdiel, D. Persson, H. Ronellenfitsch, and R. Volpato, “Generalized Mathieu Moonshine,” Commun. Num. Theor Phys. 07 (2013) 145–223, arXiv:1211.7074 [hep-th].
  * [26] M. C. N. Cheng, J. F. R. Duncan, and J. A. Harvey, “Umbral Moonshine,” Commun. Num. Theor. Phys. 08 (2014) 101–242, arXiv:1204.2779 [math.RT].
  * [27] M. C. N. Cheng, J. F. R. Duncan, and J. A. Harvey, “Umbral Moonshine and the Niemeier Lattices,” arXiv:1307.5793 [math.RT].
  * [28] M. R. Gaberdiel, S. Hohenegger, and R. Volpato, “Symmetries of K3 sigma models,” Commun. Num. Theor. Phys. 6 (2012) 1–50, arXiv:1106.4315 [hep-th].
  * [29] M. R. Gaberdiel, A. Taormina, R. Volpato, and K. Wendland, “A K3 sigma model with $\mathbb{Z}^{8}_{2}$ : $\mathbb{M}_{20}$ symmetry,” JHEP 02 (2014) 022, arXiv:1309.4127 [hep-th].
  * [30] A. Taormina and K. Wendland, “The overarching finite symmetry group of Kummer surfaces in the Mathieu group $M_{24}$,” JHEP 08 (2013) 125, arXiv:1107.3834 [hep-th].
  * [31] F. Cléry and V. Gritsenko, “Siegel modular forms of genus 2 with the simplest divisor,” Proceedings of the London Mathematical Society 102 no. 6, (2011) 1024–1052.
  * [32] P. Niemann, “Some Generalized Kac-Moody Algebras With Known Root Multiplicities,” arXiv:math/0001029 [math.QA].