the upper half-plane $\mathcal{H}$, and transforms as knopp2004vector ;
bantay2007vector ; marks2010structure ; gannon2014theory ;
franc2016hypergeometric ; franc2018structure

$Y(\gamma\tau)=(c\tau+d)^{k}\rho(\gamma)Y(\tau),~{}~{}~{}\gamma\in\Gamma\,,$
(180)

where $k$ is called the modular weight and $\rho(\gamma)$ is the irreducible
representation matrix of $SL(2,\mathbb{Z})$. Moreover, $Y(\tau)$ is generally
required to satisfy the so-called moderate growth condition, i.e., there
exists an integer $n$ such that for each components $Y_{j}(\tau)$ we have
$|Y_{j}(\tau)|<\texttt{Im}(\tau)^{n}$ for any fixed $\tau$ with
$\texttt{Im}(\tau)\gg 0$. In other word, $Y_{j}(\tau)$ is bounded as
$\texttt{Im}(\tau)\rightarrow+\infty$.

In the present work, we will focus on the case that $k$ is a non-negative
integer and $\rho$ has finite image. Finite image means that, although there
are infinitely many elements $\gamma$, there are only a finite number of
representation matrices $\rho(\gamma)$. Note that $\rho$ can be taken to be
unitary since the image of $\rho$ is finite, and it is also completely
reducible by Maschke’s theorem rao2006linear , i.e., it can be decomposed into
a direct sum of irreducible representations. Therefore, we can further
restrict $\rho$ to the unitary irreducible representations without loss of
generality. Also the representation matrix $\rho(\gamma)$ should have finite
order for any $\gamma\in SL(2,\mathbb{Z})$ so that one can always work in the
basis where $\rho(T)$ is diagonal, and the eigenvalues of $\rho(T)$ are roots
of unity. Finally, the finiteness of the image of $\rho$ implies that the
kernel $\ker(\rho)$ is a normal subgroup of $SL(2,\mathbb{Z})$ with finite
index.

In the original paradigm of modular flavor symmetry Feruglio:2017spp , the
modular form of level $N$ satisfying the condition in Eq. (100) is assumed,
$\ker(\rho)$ is chosen to be the principal subgroup $\Gamma(N)$, and the
representation matrices $\rho(\gamma)$ form finite modular group $\Gamma_{N}$
or $\Gamma^{\prime}_{N}$. In the framework of VVMF, $\Gamma(N)$ and the
corresponding modular forms are not required, and we start from the more
general VVMF obeying Eq. (180).

The theory of the VVMF has been developed by Mason et al
marks2009classification ; marks2010structure  in recent years, and the
explicit form of VVMF can be obtained by solving the so-called modular linear
differential equation Liu:2021gwa ; marks2009classification ;
marks2010structure . In the following, we report the results for VVMF of low
dimensions Liu:2021gwa .

  * •

one-dimensional irreducible VVMFs

It is known that the modular generators $S$ and $T$ the relations
$S^{4}=(ST)^{3}=1$. Consequently the $SL(2,\mathbb{Z})$ group has 12 one-
dimensional irreducible representations denoted as $\mathbf{1}_{p}$ with

$\mathbf{1}_{p}:~{}~{}~{}~{}~{}\rho_{\mathbf{1}_{p}}(S)=i^{p}\,,\qquad\rho_{\mathbf{1}_{p}}(T)=e^{\frac{i\pi}{6}p}\,,$
(181)

where $p=0,1,\dots,11$. The lowest weight of the VVMFs in the representation
$\mathbf{1}_{p}$ is equal to $p$ and accordingly the minimal weight VVMF $Y$
is found to take the following form

$Y=\eta^{2p}\,.$ (182)

We would like to mention that $\eta$ is the famous Dedekind-eta function
defined in Eq. (118). One can check that $\eta^{2p}(\tau)$ satisfies the
following transformation form

$\displaystyle\eta^{2p}(S\tau)=(-\tau)^{p}i^{p}\eta^{2p}(\tau)=(-\tau)^{p}\rho_{\mathbf{1}_{p}}(S)\eta^{2p}(\tau),$
$\displaystyle\eta^{2p}(T\tau)=e^{\frac{i\pi}{6}p}\eta^{2p}(\tau)=\rho_{\mathbf{1}_{p}}(T)\eta^{2p}(\tau)\,.$
(183)

Hence $\eta^{2p}(\tau)$ is indeed the weight $p$ one-dimensional VVMFs in the
singlet irrep $\mathbf{1}_{p}$. The free module theorem implies that all one-
dimensional VVMFs can be expressed as marks2010structure :

$\mathcal{M}(\mathbf{1}_{p})=\mathbb{C}[E_{4},E_{6}]\eta^{2p}\,,\qquad
p=0,1,\dots,11\,.$ (184)

Here $\mathbb{C}[E_{4},E_{6}]$ denotes the polynomial of the Eisenstein series
$E_{4}(\tau)$ and $E_{6}(\tau)$ cohen2017modular :

$\displaystyle
E_{4}(\tau)=1+240\sum_{n=1}^{\infty}\sigma_{3}(n)q^{n}=1+240q+2160q^{2}+6720q^{3}+17520q^{4}+\ldots\,,$
$\displaystyle
E_{6}(\tau)=1-504\sum_{n=1}^{\infty}\sigma_{5}(n)q^{n}=1-504q-16632q^{2}-122976q^{3}-532728q^{4}+\ldots\,,$
(185)

where $\sigma_{k}(n)=\sum_{d|n}d^{k}$ is the sum of the $k$th power of the
divisors of $n$ and $q=e^{2\pi i\tau}$. Notice that $E_{4}(\tau)$ and
$E_{6}(\tau)$ are scalar-valued modular forms of $SL(2,\mathbb{Z})$ with
modular weight 4 and 6 respectively.

  * •

two-dimensional irreducible VVMFs

All the 2-d irreducible unitary representations of $SL(2,\mathbb{Z})$ with
finite image have been fully obtained in Ref. mason20082 . It is known that
there are 54 inequivalent 2-d irreps and up to a possible similar
transformation the representation matrices are given by

$\displaystyle\rho(T)=\begin{pmatrix}e^{2\pi ir_{1}}&0\\\ 0&e^{2\pi
ir_{2}}\end{pmatrix},\qquad 0\leq r_{1},r_{2}<1\,,$
$\displaystyle\rho(S)=\frac{(\lambda_{1}\lambda_{2})^{2}}{\lambda_{2}-\lambda_{1}}\begin{pmatrix}1&\sqrt{-(\lambda_{1}\lambda_{2})^{5}(\lambda_{1}-\lambda_{2})^{2}-1}\\\
\sqrt{-(\lambda_{1}\lambda_{2})^{5}(\lambda_{1}-\lambda_{2})^{2}-1}&-1\end{pmatrix}\,,$
(186)

with $\lambda_{1,2}=e^{2\pi ir_{1,2}}$. We will denote the 2-d irreducible
representation (irrep) as $\mathbf{2}_{(r_{1},r_{2})}$, and the allowed values
of the unordered pair $(r_{1},r_{2})$ are given in Liu:2021gwa . We can read
out the exponent matrix $L$ as $L=\texttt{diag}(r_{1},r_{2})$ and the minimal
weight $k_{0}=6\text{Tr}(L)-1=6(r_{1}+r_{2})-1$. The corresponding minimal
weight VVMF in the doublet representation $\mathbf{2}_{(r_{1},r_{2})}$ is
found to be Liu:2021gwa :

$Y(\tau)=\begin{pmatrix}~{}~{}\eta^{12(r_{1}+r_{2})-2}K^{\frac{6(r_{1}-r_{2})+1}{12}}~{}{}_{2}F_{1}(\frac{6(r_{1}-r_{2})+1}{12},\frac{6(r_{1}-r_{2})+5}{12};r_{1}-r_{2}+1;K)\\\
C\eta^{12(r_{1}+r_{2})-2}K^{\frac{6(r_{2}-r_{1})+1}{12}}~{}{}_{2}F_{1}(\frac{6(r_{2}-r_{1})+1}{12},\frac{6(r_{2}-r_{1})+5}{12};r_{2}-r_{1}+1;K)\\\
\end{pmatrix}$ (187)

up to an overall irrelevant constant. The coefficient $C$ depends on the
explicit form of the representation matrix $\rho(S)$, and its value can be
fixed to satisfy the condition Eq. (180) of VVMF for $\gamma=S$. The function
$K(\tau)$ is defined as $K(\tau)=1728/j(\tau)$, where $j(\tau)$ is the modular
$j$-invariant cohen2017modular  with the $q$-expansion:

$j(\tau)=q^{-1}+744+196884q+21493760q^{2}+\dots\,.$ (188)

${}_{n}F_{n-1}$ is the generalized hypergeometric series and it is defined by
the formula

${}_{n}F_{n-1}(a_{1},\dots,a_{n};b_{1},\dots,b_{n-1};z)=\sum_{m\geq
0}^{\infty}\dfrac{\prod_{j=1}^{n}(a_{j})_{m}}{\prod_{k=1}^{n-1}(b_{k})_{m}}\dfrac{z^{m}}{m!}\,,$
(189)

and $(a)_{m}$ is the Pochhammer symbol defined by

$(a)_{m}=\begin{cases}1\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\quad\qquad
m=0\,,\\\ a(a+1)\dots(a+m-1)\,,\ \qquad m\geq 1\,.\end{cases}$ (190)

Hence we know the module of two-dimensional VVMFs is given by

$\mathcal{M}(\mathbf{2}_{(r_{1},r_{2})})=\mathbb{C}[E_{4},E_{6}]Y\oplus\mathbb{C}[E_{4},E_{6}]D_{k_{0}}Y\,,$
(191)

where $D_{k_{0}}$ is the modular differential operators acting on the VVMFs of
weight $k_{0}$ Bruinier2008The ,

$D_{k_{0}}\equiv\theta-\frac{k_{0}E_{2}}{12}\,,\quad\theta\equiv
q\frac{d}{dq}=\frac{1}{2\pi i}\frac{d}{d\tau},\quad k_{0}\in\mathbb{Z}\,,$
(192)

with $E_{2}(\tau)$ being the well-known quasi-modular Eisenstein series of
weight 2 cohen2017modular

$E_{2}(\tau)=1-24\sum_{n=1}^{\infty}\sigma_{1}(n)q^{n}=1-24q-72q^{2}-96q^{3}-168q^{4}-144q^{5}+\ldots\,.$
(193)

For example, the group $\Gamma_{2}\cong S_{3}$ has doublet irreducible
representation with Kobayashi:2018vbk

$\rho(S)=\frac{1}{2}\begin{pmatrix}-1~{}&~{}-\sqrt{3}\\\
-\sqrt{3}~{}&~{}1\end{pmatrix},~{}~{}~{}~{}\rho(T)=\begin{pmatrix}1~{}&~{}0\\\
0~{}&~{}-1\end{pmatrix}\,,$ (194)

which implies $r_{1}=0$ and $r_{2}=1/2$ and the weight
$k_{0}=6(r_{1}+r_{2})-1=2$. From the general expression of two-dimensional
VVMF in Eq. (187) and considering the $S$ transformation, we can fix the
parameter $C=\frac{1}{3}$ and

$Y_{\mathbf{2}_{(0,1/2)}}(\tau)=2\sqrt{3}\begin{pmatrix}~{}~{}\eta^{4}K^{-1/6}~{}{}_{2}F_{1}(-\frac{1}{6},\frac{1}{6};\frac{1}{2};K)\\\
\frac{1}{3}\eta^{4}K^{1/3}~{}{}_{2}F_{1}(\frac{1}{3},\frac{2}{3};\frac{3}{2};K)\\\
\end{pmatrix}\,,$ (195)

where an overall constant $2\sqrt{3}$ is multiplied. The $q$-expansion of
$Y_{\mathbf{2}_{(0,1/2)}}(\tau)$ is found to be

$Y_{\mathbf{2}_{(0,1/2)}}(\tau)=\begin{pmatrix}1+24q+24q^{2}+96q^{3}+24q^{4}+144q^{5}+\ldots\\\
8\sqrt{3}\,q^{1/2}\left(1+4q+6q^{2}+8q^{3}+13q^{4}+12q^{5}+\ldots\right)\end{pmatrix}\,.$
(196)

It agrees, up to an overall factor, with the $q$-expansion derived in
Kobayashi:2018vbk , where the $Y_{(0,1/2)}(\tau)$ is the weight 2 modular form
of level 2 and it is expressed in terms of the eta-function and its
derivative.

  * •

three-dimensional irreducible VVMFs

Similar to the 2-d case, the 3-d irreps of $SL(2,\mathbb{Z})$ are determined
by the three rational numbers $r_{1},r_{2}$ and $r_{3}$
tuba2001representations . In the $T$ diagonal basis, the 3-d representation
matrix $\rho(T)$ can be parameterized as

$\rho(T)=\begin{pmatrix}e^{2\pi ir_{1}}~{}&0~{}&0\\\ 0~{}&e^{2\pi
ir_{2}}~{}&0\\\ 0~{}&0~{}&e^{2\pi ir_{3}}\end{pmatrix}\,,\qquad 0\leq
r_{1},r_{2},r_{3}<1\,,$ (197)

where the parameters $r_{1,2,3}$ have to fulfill the following constraint:

$r_{1}+r_{2}+r_{3}\in\frac{1}{12}\mathbb{Z}\,.$ (198)

This 3-d irrep is denoted as $\mathbf{3}_{(r_{1},r_{2},r_{3})}$. The
exponential matrix is $L=\texttt{diag}(r_{1},r_{2},r_{3})$ and the minimal
weight $k_{0}=4\text{Tr}(L)-2=4(r_{1}+r_{2}+r_{3})-2$. The minimal weight VVMF
$Y$ in the triplet representation $\mathbf{3}_{(r_{1},r_{2},r_{3})}$ is
determined to be Liu:2021gwa

$Y(\tau)=\begin{pmatrix}~{}~{}\eta^{8(r_{1}+r_{2}+r_{3})-4}K^{\frac{a_{1}+1}{6}}~{}{}_{3}F_{2}(\frac{a_{1}+1}{6},\frac{a_{1}+3}{6},\frac{a_{1}+5}{6};r_{1}-r_{2}+1,r_{1}-r_{3}+1;K)\\\
C_{1}\eta^{8(r_{1}+r_{2}+r_{3})-4}K^{\frac{a_{2}+1}{6}}~{}{}_{3}F_{2}(\frac{a_{2}+1}{6},\frac{a_{2}+3}{6},\frac{a_{2}+5}{6};r_{2}-r_{3}+1,r_{2}-r_{1}+1;K)\\\
C_{2}\eta^{8(r_{1}+r_{2}+r_{3})-4}K^{\frac{a_{3}+1}{6}}~{}{}_{3}F_{2}(\frac{a_{3}+1}{6},\frac{a_{3}+3}{6},\frac{a_{3}+5}{6};r_{3}-r_{1}+1,r_{3}-r_{2}+1;K)\end{pmatrix}\,,$
(199)

with $a_{1}=4r_{1}-2r_{2}-2r_{3},a_{2}=4r_{2}-2r_{1}-2r_{3}$ and
$a_{3}=4r_{3}-2r_{1}-2r_{2}$. The constants $C_{1,2}$ are fixed by the
representation matrix of $\rho(S)$ in the $T$-diagonal basis. Hence the 3-d
VVMFs module is of the form

$\mathcal{M}(\mathbf{3}_{(r_{1},r_{2},r_{3})})=\mathbb{C}[E_{4},E_{6}]Y\oplus\mathbb{C}[E_{4},E_{6}]D_{k_{0}}Y\oplus\mathbb{C}[E_{4},E_{6}]D^{2}_{k_{0}}Y\,.$
(200)

It is known that there are three linearly independent weight 2 modular forms
of level 3 and they can be arranged into a $A_{4}$ triplet with the following
representation matrices for the modular generators,

$\displaystyle\rho(S)=\frac{1}{3}\begin{pmatrix}-1~{}&~{}2~{}&~{}2\\\
2~{}&~{}-1~{}&~{}2\\\
2~{}&~{}2~{}&~{}-1\end{pmatrix},~{}~{}~{}~{}\rho(T)=\begin{pmatrix}1~{}&~{}0~{}&~{}0\\\
0~{}&~{}\omega~{}&~{}0\\\
0~{}&~{}0~{}&~{}\omega^{2}\end{pmatrix},~{}~{}~{}\omega=e^{2\pi i/3}\,,$ (201)

which is given by Eq. (497). This implies $r_{1}=0$, $r_{2}=\frac{1}{3}$ and
$r_{3}=\frac{2}{3}$ and the modular weight $k_{0}=4(r_{1}+r_{2}+r_{3})-2=2$.
Taking into account the modular transformation $S$, we find
$C_{1}=-\frac{1}{2}$, $C_{2}=-\frac{1}{8}$, and the VVMF in the triplet
representation of $A_{4}$ is given by

$Y_{\mathbf{3}_{(0,1/3,2/3)}}(\tau)=2\sqrt{3}\begin{pmatrix}~{}~{}\eta^{4}K^{-1/6}~{}{}_{3}F_{2}(-\frac{1}{6},\frac{1}{6},\frac{1}{2};\frac{2}{3},\frac{1}{3};K)\\\
-\frac{1}{2}\eta^{4}K^{1/6}~{}{}_{3}F_{2}(\frac{1}{6},\frac{1}{2},\frac{5}{6};\frac{2}{3},\frac{4}{3};K)\\\
-\frac{1}{8}\eta^{4}K^{1/2}~{}{}_{3}F_{2}(\frac{1}{2},\frac{5}{6},\frac{7}{6};\frac{5}{3},\frac{4}{3};K)\end{pmatrix}\,,$
(202)

where the overall factor $2\sqrt{3}$ is included. We find that the
$q$-expansion of $Y_{\mathbf{3}_{(0,1/3,2/3)}}(\tau)$ reads as

$\displaystyle
Y_{\mathbf{3}_{(0,1/3,2/3)}}(\tau)=\begin{pmatrix}1+12q+36q^{2}+12q^{3}+84q^{4}+72q^{5}+\ldots\\\
-6q^{1/3}\left(1+7q+8q^{2}+18q^{3}+14q^{4}+31q^{5}+\ldots\right)\\\
-18q^{2/3}\left(1+2q+5q^{2}+4q^{3}+8q^{4}+6q^{5}+\ldots\right)\end{pmatrix}\,,$
(203)

which exactly coincides with the $q$-expansion derived in Feruglio:2017spp ,
where $Y_{\mathbf{3}_{(0,1/3,2/3)}}(\tau)$ is expressed in terms of the eta-
function and its derivative. We see that the theory of VVMF provides a
convenient way to obtain the explicit expression of modular forms which is the
crucial element of modular invariant model.

Finally we summarise some more general finite modular groups. Since the irrep
$\rho$ of $\Gamma=SL(2,\mathbb{Z})$ has finite image, the image
$\text{Im}(\rho)$ forms a discrete finite group and it can play the role of
discrete flavor symmetry. Due to a fundamental theorem of homomorphism we can
define $\text{Im}(\rho)$ as the quotient group
$\text{Im}(\rho)\cong\Gamma/\ker(\rho)$ where $\Gamma=SL(2,\mathbb{Z})$ and
$\ker(\rho)$ is a normal subgroup determined by the representation $\rho$. The
$\ker(\rho)$ consists of the inverse image of the identity elements.

In the paradigm of modular flavor symmetry Feruglio:2017spp , $\ker(\rho)$ was
only restricted to principle congruence subgroups $\Gamma(N)$. The image
$\text{Im}(\rho)$ is the homogeneous or inhomogeneous finite modular group
$\Gamma^{\prime}_{N}=\Gamma/\Gamma(N)$ or $\Gamma_{N}=\Gamma/\pm\Gamma(N)$ and
it is taken as the flavor symmetry to address the flavor structure of quarks
and leptons. From the view of VVMF, the kernel $\ker(\rho)$ can be a general
normal subgroup of $SL(2,\mathbb{Z})$ instead of $\Gamma(N)$, and accordingly
the finite modular groups are not necessarily $\Gamma_{N}$ or
$\Gamma^{\prime}_{N}$. Thus the framework of modular invariance can be
extended significantly by considering VVMFs, and we have more possible choices
for the finite modular groups to construct modular invariant models.

In table 7 we list the normal subgroups of $SL(2,\mathbb{Z})$ with index $\leq
72$ as well as the corresponding finite modular group, where we omit the few
cases for which the finite modular group is abelian. Besides the known
principal congruence subgroups $\Gamma(N)$ as well as the inhomogeneous and
homogeneous finite modular groups, we see other normal subgroups of
$SL(2,\mathbb{Z})$. Additional relators are elements of $\ker(\rho)$, when
they are added to the $SL(2,\mathbb{Z})$ presentation relations
$S^{4}=(ST)^{3}=1$, $S^{2}T=TS^{2}$, the finite modular groups
$\text{Im}(\rho)\cong\Gamma/\ker(\rho)$ are produced. Moreover, if the element
$S^{2}\in\ker(\rho)$, we have $\rho(S^{2})=\mathds{1}_{d}$. Then the
corresponding finite modular group has only even representations, and the
VVMFs in these irreps must be of even weights.

It is known that the number of VVMFs in a given irrep $\rho_{Y}$ generally
increases with the modular weight $k_{Y}$. However, the module
$\mathcal{M}(\rho_{Y})$ is finitely generated by $d=\dim\rho_{Y}$ independent
bases denoted by $\\{Y_{1},\dots,Y_{d}\\}$, thus $Y_{I_{1}...I_{n}}(\tau)$ can
be written uniformly in the form of
$Y^{(k_{Y})}_{\rho_{Y}}=\alpha_{1}Y_{1}+\dots+\alpha_{d}Y_{d}$ with
$\alpha_{i}\in\mathbb{C}[E_{4},E_{6}]$, note that $\alpha_{i}$ are polynomials
of $E_{4}$ and $E_{6}$. Some $\alpha_{i}$ for vanishing for small value of
modular weight $k_{Y}$, nevertheless there are at most $d$ independent
alignments $Y_{1}$, $Y_{2}$,…, $Y_{d}$. This implies that the superpotential
is strongly constrained by the modular symmetry, and it can only take a finite
number of possible forms for a given $\mathcal{G}_{f}$, although there are
infinite possible weight and representation assignments for the matter fields.

Normal subgroups $\ker(\rho)$ | Finite modular groups $\Gamma/\ker(\rho)\cong\text{Im}(\rho)$
---|---
Index | Label | Additional relations | Group structure |  GAP Id
6 | $\Gamma(2)$ | $T^{2}$ | $S_{3}$ | $[6,1]$
12 | $-$ | $S^{2}T^{2}$ | $Z_{3}\rtimes Z_{4}\cong 2D_{3}$ | $[12,1]$
$\pm\Gamma(3)$ | $S^{2},T^{3}$ | $A_{4}$ | $[12,3]$
18 | $-$ | $ST^{-2}ST^{2}$ | $S_{3}\times Z_{3}$ | $[18,3]$
24 | $\Gamma(3)$ | $T^{3}$ | $T^{\prime}$ | $[24,3]$
$-$ | $S^{2}T^{3}$
$\pm\Gamma(4)$ | $S^{2},T^{4}$ | $S_{4}$ | $[24,12]$
$-$ | $S^{2},(ST^{-1}ST)^{2}$ | $A_{4}\times Z_{2}$ | $[24,13]$
36 | $-$ | $S^{3}T^{-2}ST^{2}$ | $(Z_{3}\rtimes Z_{4})\times Z_{3}$ | $[36,6]$
42 | $-$ | $T^{6},(ST^{-1}S)^{2}TST^{-1}ST^{2}$ | $Z_{7}\rtimes Z_{6}$ | $[42,1]$
$-$ | $T^{6},ST^{-1}ST(ST^{-1}S)^{2}T^{2}$
48 | $-$ | $S^{2}T^{4}$ | $2O$ | $[48,28]$
$-$ | $T^{8},ST^{4}ST^{-4}$ | $GL(2,3)$ | $[48,29]$
$\Gamma(4)$ | $T^{4}$ | $A_{4}\rtimes Z_{4}\cong S^{\prime}_{4}$ | $[48,30]$
$-$ | $(ST^{-1}ST)^{2}$ | $A_{4}\times Z_{4}$ | $[48,31]$
$-$ | $S^{2}(ST^{-1}ST)^{2}$ | $T^{\prime}\times Z_{2}$ | $[48,32]$
$-$ | $T^{12},ST^{3}ST^{-3}$ | $((Z_{4}\times Z_{2})\rtimes Z_{2})\rtimes Z_{3}$ | $[48,33]$
54 | $-$ | $T^{6},(ST^{-1}ST)^{3}$ | $(Z_{3}\times Z_{3})\rtimes Z_{6}$ | $[54,5]$
60 | $\pm\Gamma(5)$ | $S^{2},T^{5}$ | $A_{5}$ | $[60,5]$
72 | $-$ | $T^{12},ST^{4}ST^{-4}$ | $S_{4}\times Z_{3}$ | $[72,42]$
$\pm\Gamma(6)$ | $S^{2},T^{6},(ST^{-1}STST^{-1}S)^{2}T^{2}$ | $A_{4}\times S_{3}$ | $[72,44]$
Table 7:  The normal subgroups of $SL(2,\mathbb{Z})$ with index less than $78$
and the corresponding finite modular groups. It is defined
$\pm\Gamma(N)=\left\\{\pm\gamma,\gamma\in\Gamma(N)\right\\}$. Note that
$2D_{3}$ is the binary dihedral group of order 12 and $2O$ is the binary
octahedral group which is the Schur cover of permutation group $S_{4}$ of type
“$-$”. Additional relators are some elements of $\ker(\rho)$, the quotient
group $\Gamma/\ker(\rho)$ is produced when the relations “Additional relators
=1” together with $S^{4}=(ST)^{3}=1$ and $S^{2}T=TS^{2}$ are imposed.

##  5 CP symmetry and modular invariance

The interplay between the generalized CP symmetry (gCP) and modular symmetry
is nontrivial. It is found that the CP transformation can be consistently
defined in modular invariant supersymmetric theories with a single modulus
Baur:2019kwi ; Baur:2019iai ; Novichkov:2019sqv . In particular, the modular
symmetry can constrain the CP transformation laws of the modulus $\tau$, of
chiral matter multiplets and of modular forms Novichkov:2019sqv , and several
modular invariant models where CP is spontaneously broken have been
constructed Novichkov:2019sqv ; Kobayashi:2019uyt ; Novichkov:2020eep ;
Liu:2020akv ; Yao:2020zml ; Yao:2020qyy ; Okada:2020brs . Analogously for
multidimensional moduli space, consistent CP definitions have been examined in
the context of symplectic modular invariant theories, where the relevant
flavour group is the Siegel modular group Ishiguro:2020nuf ; Baur:2020yjl ;
Ding:2021iqp . The CP-conserving vacua in Calabi-Yau compactifications are
discussed in Ref. Ishiguro:2020nuf . In this section we formulate the
conditions for generalised CP-symmetry consistent with modular invariance, and
derive the modular transformations of matter fields and modular forms, before
discussing CP fixed points and their implications for flavor theories.

###  5.1 Generalized CP symmetry compatible with modular invariance

In a theory invariant under $\Gamma=SL(2,\mathbb{Z})$, consistent CP
transformations correspond to outer automorphism $u(\gamma)$ of $\Gamma$
Holthausen:2012dk ; Chen:2014tpa :

$\mathcal{CP}~{}\gamma~{}\mathcal{CP}^{-1}=u(\gamma)~{}~{}~{}.$ (204)

Each outer automorphism $u(\gamma)$ of $\Gamma$ can be described by Reiner :

$u(\gamma)=\chi(\gamma)~{}U~{}\gamma~{}U^{-1}\,,~{}~{}~{}~{}\text{or}~{}~{}~{}~{}u(\gamma)=\chi(\gamma)\gamma\,,$
(205)

with

$U=\begin{pmatrix}1~{}&~{}0\\\ 0~{}&~{}-1\end{pmatrix}$ (206)

and the map $\chi(\gamma)$, called character of the modular group, is a
homomorphism of $\Gamma$ into $\\{+1,-1\\}$. From the relations
$S^{4}=(ST)^{3}=1$ satisfied by the modular generators $S$ and $T$, we see the
modular group has a trivial character with

$\chi_{1}(S)=\chi_{1}(T)=1$ (207)

and another nontrivial character with

$\chi_{2}(S)=\chi_{2}(T)=-1\,.$ (208)

Hence the modular group $SL(2,\mathbb{Z})$ has two independent outer
automorphisms $u_{1,2}=\chi_{1,2}(\gamma)~{}U~{}\gamma~{}U^{-1}$, satisfying
the following relations:

$u^{2}_{1}=u^{2}_{2}=(u_{1}u_{2})^{2}=\mathds{1}\,.$ (209)

Hence the outer automorphism group is isomorphic to a Klein group $K_{4}\cong
Z_{2}\times Z_{2}=\\{u_{1},u_{2},u_{3}=u_{1}u_{2},u_{4}=\mathds{1}\\}$161616We
denote by $\mathds{1}$ the identity element of the outer automorphism.. We can
derive the action of the outer automorphisms on the modular generators $S$ and
$T$ as follows Ding:2021iqp :

$\displaystyle~{}u_{1}(S)=S^{-1}\,,\qquad u_{1}(T)=T^{-1}\,,$
$\displaystyle~{}u_{2}(S)=-S^{-1}\,,\qquad u_{2}(T)=-T^{-1}\,,$
$\displaystyle~{}u_{3}(S)=-S\,,\qquad u_{3}(T)=-T\,,$
$\displaystyle~{}u_{4}(S)=S\,,\qquad u_{4}(T)=T\,.$ (210)

We assume that the CP transformation of modulus $\tau$ is linear and, for
convenience, we write:

$\tau\xrightarrow{{\cal CP}}({\cal CP})\tau\equiv\tau_{{\cal
CP}}=p_{1}~{}\tau^{*}+p_{2}\,,$ (211)

where $p_{1}$ and $p_{2}$ are parameters to be determined. The inverse CP
transformation reads:

$\displaystyle\tau^{*}\xrightarrow{{\cal CP}^{-1}}\tau^{*}_{{\cal
CP}^{-1}}=\frac{\tau-p_{2}}{p_{1}}\,.$ (212)

Enforcing the “consistency condition chain” of Eq. (204) on the modular
generators $S$ and $T$ for the automorphisms $u_{a}(\gamma)(a=1,...,4)$, we
can obtain:

$\displaystyle\frac{p_{2}\tau-p^{2}_{1}-p^{2}_{2}}{\tau-
p_{2}}=u_{a}(S)\tau\,,$ $\displaystyle\tau+p_{1}=u_{a}(T)\tau\,.$ (213)

The elements $u_{1}$ and $u_{2}$ have the same action on $\tau$ and similarly
for $u_{3,4}$. The above equation can be straightforwardly solved to obtain

$p_{1}=\left\\{\begin{array}[]{cc}-1,{}&~{}\text{for $u_{1,2}$}\\\
+1,{}&~{}\text{for $u_{3,4}$}\end{array}\right.\,,\qquad~{}~{}p_{2}=0\,.$
(214)

Thus we have $\tau_{{\cal CP}}=-\tau^{*}$ for the automorphisms $u_{1,2}$ and
$\tau_{{\cal CP}}=\tau^{*}$ for $u_{3,4}$. Since ${\tt Im}(\tau_{{\cal
CP}})>0$, the admissible CP transformation of the modulus $\tau$ is
Baur:2019kwi ; Baur:2019iai ; Novichkov:2019sqv ; Acharya:1995ag ; Dent:2001cc
; Giedt:2002ns

$\tau\xrightarrow{{\cal CP}}\tau_{{\cal CP}}=-\tau^{*}~{}\,.$ (215)

This represents the correct transformation law of the modulus for both
$u_{1,2}$ outer automorphisms and we should instead discard $u_{3,4}$. Notice
that the same gCP transformation of $\tau$ can be derived from the consistency
condition, as shown in the Appendix B. From Eq. (215), we see
$\tau^{*}_{\mathcal{CP}^{-1}}=-\tau$ and the action of ${\cal CP}$ on the
modulus is involutive with ${\cal CP}^{2}\tau=\tau$.

Now let us check explicitly that the gCP transformation of Eq. (215) is indeed
compatible with Eq. (204). Applying the consistency transformation chain
$\mathcal{CP}\rightarrow\gamma\rightarrow\mathcal{CP}^{-1}$ for the modulus
$\tau$, we have

$\tau\stackrel{{\scriptstyle\mathcal{CP}}}{{\longrightarrow}}-\tau^{\ast}\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}-\frac{a\tau^{\ast}+b}{c\tau^{\ast}+d}\xrightarrow{\mathcal{CP}^{-1}}=\frac{a\tau-b}{-c\tau+d}=\gamma^{\prime}\tau\,.$
(216)

Since $\gamma$ and $-\gamma$ give the same action on modulus $\tau$, we see
that the generalized CP transformation indeed corresponds to the automorphism
$u_{1}$ or $u_{2}$ of the modular group with

$u_{1,2}:\gamma\rightarrow\gamma^{\prime}=u_{1,2}(\gamma)=\chi_{1,2}(\gamma)~{}U~{}\gamma~{}U^{-1}=\chi_{1,2}(\gamma)\begin{pmatrix}a~{}&-b\\\
-c~{}&d\end{pmatrix},~{}~{}~{}\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma\,,$ (217)

where $\chi_{1}(\gamma)$ is the trivial character given in Eq. (207) and
$\chi_{2}(\gamma)$ is the nontrivial character in Eq. (208).

If we perform a modular transformation $\gamma$ and then the CP transformation
of Eq. (215), we get

$\tau\xrightarrow{{\gamma}}\frac{a\tau+b}{c\tau+d}\xrightarrow{{\cal
CP}}(\gamma\circ{\cal CP})\tau=\frac{-a\tau^{*}+b}{-c\tau^{*}+d}\,,$ (218)

which is another allowed CP transformation. Since the theory is invariant
under $\Gamma$, this choice should not be view as independent from (215),
which we take as representative element in the class (215) with $a=d=1$,
$b=c=0$. We can rewrite Eq. (218) as

$\tau\xrightarrow{{\gamma\circ{\cal
CP}}}\frac{a^{\prime}\tau^{*}+b^{\prime}}{c^{\prime}\tau^{*}+d^{\prime}}~{}$
(219)

with

$a^{\prime}=-a,~{}~{}~{}b^{\prime}=b,~{}~{}~{}c^{\prime}=-c,~{}~{}~{}d^{\prime}=d\,.$
(220)

One can check

$a^{\prime}d^{\prime}-b^{\prime}c^{\prime}=-ad+bc=-(ad-bc)=-1\,.$ (221)

Note that the gCP transformation on $\tau$ always corresponds to minus
determinant with $a^{\prime}d^{\prime}-b^{\prime}c^{\prime}=-1$.

By combining CP and the modular transformations we get the extended modular
group $\Gamma^{*}=GL(2,\mathbb{Z})$ and the full symmetry transformation of
the complex modulus is Novichkov:2019sqv

$\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma^{*}:\quad\begin{cases}\tau\to\cfrac{a\tau+b}{c\tau+d}&\quad\text{if}\quad
ad-bc=1\,,\\\ \tau\to\cfrac{a\tau^{*}+b}{c\tau^{*}+d}&\quad\text{if}\quad ad-
bc=-1\,.\end{cases}$ (222)

where $\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$. The first line of (222)
refers to a modular transformation, while the second line refers to a CP
transformation. For the case of the CP transformation in (215) the matrix
$\gamma$ is represented by $U$ in Eq. (206). Notice that the action of
$\gamma$ and $-\gamma$ on $\tau$ is the same and the full symmetry group
acting on moduli is isomorphic to $PGL(2,\mathbb{Z})\equiv
GL(2,\mathbb{Z})/\\{\pm\mathds{1}_{2}\\}$. Here $PGL(2,\mathbb{Z})$ is the
group of integral $2\times 2$ matrices with determinant $\pm 1$, with the
matrices $M$ and $-M$ being identified.

###  5.2 CP transformations of matter chiral multiplets $\varphi$

For a generic chiral supermultiplets $\varphi$, its transfromation properties
are characterized by $-k_{\varphi}$ and $\rho_{\mathbf{r}}$, where
$-k_{\varphi}$ is the modular weight of $\varphi$ and $\rho_{\mathbf{r}}$ is
an irreducible representation of the finite modular group $\Gamma_{N}$ or
$\Gamma^{\prime}_{N}$. Under a modular transformation $\gamma$, the multiplet
$\varphi(x)$ transforms according to

$\varphi(x)\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}(c\tau+d)^{-k_{\varphi}}\rho_{\mathbf{r}}(\gamma)\varphi(x)\,,~{}~{}~{}\gamma=\begin{pmatrix}a~{}&b\\\
c~{}&d\end{pmatrix}\in\Gamma\,.$ (223)

The action of generalized CP symmetry on the multiplet $\varphi$ is

$\varphi(x)\stackrel{{\scriptstyle\mathcal{CP}}}{{\longrightarrow}}X_{\mathbf{r}}\overline{\varphi}(x_{\mathcal{P}})\,,$
(224)

where $x_{\mathcal{P}}=(t,-\vec{x})$, a bar denotes the hermitian conjugate
superfield, and $X_{\mathbf{r}}$ is a unitary matrix that acts on flavor
space. If we perform a CP transformation, followed by a modular transformation
and subsequently an inverse CP transformation, we end up with

$\varphi(x)\xrightarrow{{\cal
CP}}X_{\mathbf{r}}\overline{\varphi}(x_{\mathcal{P}})\xrightarrow{\gamma}X_{\mathbf{r}}(c\tau^{*}+d)^{-k_{\varphi}}\rho^{*}_{\mathbf{r}}(\gamma)\overline{\varphi}(x_{\mathcal{P}})\xrightarrow{{\cal
CP}^{-1}}(c\tau^{*}_{\mathcal{CP}^{-1}}+d)^{-k_{\varphi}}X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(\gamma)X_{\mathbf{r}}^{-1}\varphi(x)\,,$
(225)

where $\tau_{\mathcal{CP}^{-1}}$ is the result of applying the inverse CP
transformation $\mathcal{CP}^{-1}$ to the modulus $\tau$. Since the theory is
assumed to be invariant under both the modular and CP transformations, the
resulting transformation should be amount to another modular transformation
$\gamma^{\prime}$, i.e.,

$(c\tau^{*}_{\mathcal{CP}^{-1}}+d)^{-k_{\varphi}}X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(\gamma)X_{\mathbf{r}}^{-1}\varphi(x)=(c_{\gamma^{\prime}}\tau+d_{\gamma^{\prime}})^{-k_{\varphi}}\rho_{\mathbf{r}}(\gamma^{\prime})\varphi(x)$
(226)

which implies

$X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(\gamma)X_{\mathbf{r}}^{-1}=\left(\frac{c_{\gamma^{\prime}}\tau+d_{\gamma^{\prime}}}{c\tau^{*}_{\mathcal{CP}^{-1}}+d}\right)^{-k_{\varphi}}\rho_{\mathbf{r}}(\gamma^{\prime})\,,$
(227)

where we have denoted

$\gamma^{\prime}=\begin{pmatrix}a_{\gamma^{\prime}}&b_{\gamma^{\prime}}\\\
c_{\gamma^{\prime}}&d_{\gamma^{\prime}}\end{pmatrix}\in\Gamma\,.$ (228)

From the transformation of the modulus $\tau$ in Eq. (216), we know the
element $\gamma^{\prime}=u_{1,2}(\gamma)$ given in Eq. (217), thus we can read
out

$c_{\gamma^{\prime}}=-\chi_{1,2}(\gamma)c,~{}~{}~{}d_{\gamma^{\prime}}=\chi_{1,2}(\gamma)d\,.$
(229)

Considering $\tau^{*}_{\mathcal{CP}^{-1}}=-\tau$, the consistency condition of
Eq. (227) for the modular symmetry and generalized CP symmetry becomes

$X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(\gamma)X^{-1}_{\mathbf{r}}=\chi_{1,2}^{-k_{\varphi}}(\gamma)\rho_{\mathbf{r}}(u_{1,2}(\gamma))\,.$
(230)

Since the modular group has two possible characters $\chi_{1,2}(\gamma)$ given
in Eqs. (207,208), two kinds of generalized CP transformations could be
consistently defined in the context of $SL(2,\mathbb{Z})$. The coressponding
consistency conditions are given by Novichkov:2019sqv ; Ding:2021iqp

$X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(S)X^{-1}_{\mathbf{r}}=\rho_{\mathbf{r}}(S^{-1}),~{}~{}~{}X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(T)X^{-1}_{\mathbf{r}}=\rho_{\mathbf{r}}(T^{-1})\,.$
(231)

for the gCP transformation corresponding to the automorphism $u_{1}$ with
trivial character $\chi_{1}$ in Eq. (210). A second gCP transfromation
corresponding to $u_{2}$ with nontrivial character $\chi_{2}$ could be
defined171717The consistency condition for the second gCP transformation reads
as:
$X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(S)X^{-1}_{\mathbf{r}}=\sigma\rho_{\mathbf{r}}(S^{-1}),~{}~{}~{}X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(S)X^{-1}_{\mathbf{r}}=\sigma\rho_{\mathbf{r}}(T^{-1}),~{}~{}~{}\sigma=(-1)^{-k_{\varphi}}\rho_{\mathbf{r}}(S^{2})\,.$
Obviously the second gCP transformation $X_{\mathbf{r}}$ depends on
$\rho_{\mathbf{r}}(S^{2})$ and the modualr weight $k_{\varphi}$, and it can be
defined if and only if the level $N$ is even, the dimension of representation
$\rho_{\mathbf{r}}$ is even, and the traces of $\rho_{\mathbf{r}}(T)$ and
$\rho_{\mathbf{r}}(S)$ are vanishing Novichkov:2020eep ., nevertheless it is
difficult to build phenomenologically viable models of fermion masses and
mixing exploiting this second gCP transformation Novichkov:2020eep . Hence we
shall be concerned with the gCP transformation defermined by Eq. (231) in this
work.

In the basis where both $\rho_{\mathbf{r}}(S)$ and $\rho_{\mathbf{r}}(T)$ are
symemtric matrices, one has
$\rho_{\mathbf{r}}(S^{-1})=\rho^{\dagger}_{\mathbf{r}}(S)=\rho^{*}_{\mathbf{r}}(S)$
and
$\rho_{\mathbf{r}}(T^{-1})=\rho^{\dagger}_{\mathbf{r}}(T)=\rho^{*}_{\mathbf{r}}(T)$
so that one can determine $X_{\mathbf{r}}=\mathds{1}_{\mathbf{r}}$ up to an
irrelevant phase, i.e., the gCP transformation has the canonical form. If all
the Clebsch-Gordon coeffcients of the finite modular group are real in the
above symemtric basis, the gCP symemtry would enforce all coupling constants
in the Lagrangian to be real. Thus the real part of modulus $\tau$ would be
the unique source of CP violation, and gCP symemtry can enhance the predictive
power of modular invariant models. For the finite modular groups $\Gamma_{N}$
and $\Gamma^{\prime}_{N}$ with level $N\leq 6$, it is known that the symmetric
basis can always be achieved.

###  5.3 CP transformations of modular form $Y_{\mathbf{r}}(\tau)$

Considering a multiplet $Y_{\mathbf{r}}(\tau)$ of modular forms of level $N$
and weight $k_{Y}$, it transforms in certain irreducible representation
$\rho_{\mathbf{r}}$ of the finite modular group $\Gamma_{N}$ or
$\Gamma^{\prime}_{N}$, i.e.

$Y_{\mathbf{r}}(\gamma\tau)=(c\tau+d)^{k_{Y}}\rho_{\mathbf{r}}(\gamma)Y_{\mathbf{r}}(\tau),~{}~{}~{}\gamma\in\Gamma\,.$
(232)

In general, there can be several linearly independent such multiplets,
particularly for higher weight corresponding large $k_{Y}$. We start by
examining the case where there is only one. Under gCP it transforms as:

$Y_{\mathbf{r}}(\tau)\xrightarrow{\cal CP}Y_{\mathbf{r}}(-\tau^{*})\,.$ (233)

We can prove that the modular multiplet $Y_{\mathbf{r}}(\tau)$ has the same
gCP transformation as that of matter fields in Eq. (224), i.e.,

$Y_{\mathbf{r}}(-\tau^{*})=X_{\mathbf{r}}Y_{\mathbf{r}}^{*}(\tau)\,.$ (234)

In order to show this, we define

$\widetilde{Y}_{\mathbf{r}}(\tau)=X_{\mathbf{r}}^{-1*}Y_{\mathbf{r}}^{*}(-\tau^{*})\,,$
(235)

where $X_{\mathbf{r}}$ is the solution of the consistency condition Eq. (231)
for the concerned automorphism $u_{1}$. The modular transfromation of
$\widetilde{Y}_{\mathbf{r}}(\tau)$ under a generic element $\gamma\in\Gamma$
is

$\displaystyle\widetilde{Y}_{\mathbf{r}}(\gamma(\tau))$ $\displaystyle=$
$\displaystyle X_{\mathbf{r}}^{-1*}Y_{\mathbf{r}}^{*}(-(\gamma(\tau))^{*})$
(236) $\displaystyle=$ $\displaystyle
X_{\mathbf{r}}^{-1*}Y_{\mathbf{r}}^{*}(u_{1}(\gamma)(-\tau^{*}))$
$\displaystyle=$ $\displaystyle
X_{\mathbf{r}}^{-1*}(c\tau+d)^{k_{Y}}\rho^{*}_{\mathbf{r}}(u_{1}(\gamma))~{}Y_{\mathbf{r}}^{*}(-\tau^{*})$
$\displaystyle=$
$\displaystyle(c\tau+d)^{k_{Y}}~{}\rho_{\mathbf{r}}(\gamma)~{}\widetilde{Y}_{\mathbf{r}}(\tau)\,,$

where the identities $-(\gamma(\tau))^{*}=u(\gamma)(-\tau^{*})$ and
$X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(\gamma)X^{-1}_{\mathbf{r}}=\rho_{\mathbf{r}}(u_{1}(\gamma))$
have been used181818From Eq. (216), we know
$\mathcal{CP}\circ\gamma\circ\mathcal{CP}^{-1}=u_{1}(\gamma)$ which gives
$\mathcal{CP}\circ\gamma=u_{1}(\gamma)\circ\mathcal{CP}$. Consequently we have
$\mathcal{CP}\circ\gamma(\tau)=u_{1}(\gamma)\circ\mathcal{CP}(\tau)$ which
leads to $-(\gamma(\tau))^{*}=u_{1}(\gamma)(-\tau^{*})$. Moreover, From Eq.
(230), we know
$X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(\gamma)X^{-1}_{\mathbf{r}}=\rho_{\mathbf{r}}(u_{1}(\gamma))$
which gives
$\rho^{*}_{\mathbf{r}}(\gamma)X^{-1}_{\mathbf{r}}=X^{-1}_{\mathbf{r}}\rho_{\mathbf{r}}(u_{1}(\gamma))$
and
$\rho_{\mathbf{r}}(\gamma)X^{-1*}_{\mathbf{r}}=X^{-1*}_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(u_{1}(\gamma))$..
Since $\widetilde{Y}_{\mathbf{r}}(\tau)$ and $Y_{\mathbf{r}}(\tau)$ transform
in the same way and, by assumption, there is only one linearly independent
such modular form, we conclude that they are proportional,
$\widetilde{Y}_{\mathbf{r}}(\tau)\propto Y_{\mathbf{r}}(\tau)$ which gives:

$Y_{\mathbf{r}}(-\tau^{*})=\lambda~{}X_{\mathbf{r}}Y^{*}_{\mathbf{r}}(\tau)\,.$
(237)

By performing two gCP transformation in succsssion, we obtain

$Y_{\mathbf{r}}(\tau)=|\lambda|^{2}~{}X_{\mathbf{r}}X_{\mathbf{r}}^{*}Y_{\mathbf{r}}(\tau)=|\lambda|^{2}~{}Y_{\mathbf{r}}(\tau)\,,$
(238)

where we have chosen the gCP transfroamtion $X_{\mathbf{r}}$ to be a constant
unitary symmetric matrix fulfilling $X_{\mathbf{r}}X_{\mathbf{r}}^{*}=1$ which
corresponds to $\mathcal{CP}^{2}=1$191919The action of ${\cal CP}$ in the
modulus, see Eq. (215), is involutive, that is ${\cal CP}^{2}\tau=\tau$. The
action is $\mathcal{CP}^{2}$ on the matter field is
$\varphi(x)\stackrel{{\scriptstyle\mathcal{CP}^{2}}}{{\longrightarrow}}X_{\mathbf{r}}X^{*}_{\mathbf{r}}\varphi(x)$.
From the consistency condtion in Eq. (224), we obtain:
$(c\tau+d)^{-k_{\varphi}}X_{\mathbf{r}}X_{\mathbf{r}}^{*}~{}\rho_{\mathbf{r}}(\gamma)~{}X^{-1*}_{\mathbf{r}}X_{\mathbf{r}}^{-1}\varphi=(c\tau+d)^{-k_{\varphi}}\rho_{\mathbf{r}}(\gamma)\varphi~{}~{}~{}.$
By the Schur’s Lemma, the product $X_{\mathbf{r}}X_{\mathbf{r}}^{*}$ is
proportional to the identity. It acts without conjugating the matter fields
and represents the element $\gamma_{{\cal CP}^{2}}$ or, more precisely, the
element of the finite group $\Gamma_{N}$ or $\Gamma^{\prime}_{N}$ that
corresponds to $\gamma_{{\cal CP}^{2}}$. As a consequence we have:
$X_{\mathbf{r}}X_{\mathbf{r}}^{*}=\mathds{1}_{\mathbf{r}}~{}~{}\text{or}~{}~{}X_{\mathbf{r}}X_{\mathbf{r}}^{*}=\rho_{\mathbf{r}}(S^{2})=\pm\mathds{1}_{\mathbf{r}}\,.$
Therefore we must have
$X_{\mathbf{r}}X_{\mathbf{r}}^{*}=\mathds{1}_{\mathbf{r}}$ for the
inhomegenous finite modular group $\Gamma_{N}$.. The non-vanishing constant
$\lambda$ can be absorbed by an appropriate choice of phase of the whole
multiplet $Y_{\mathbf{r}}(\tau)$ so that we have

$Y_{\mathbf{r}}(\tau)\xrightarrow{\cal
CP}Y_{\mathbf{r}}(-\tau^{*})=X_{\mathbf{r}}Y_{\mathbf{r}}^{*}(\tau)\,.$ (239)

Hence the modular multiplets have the same gCP transformation law as that of
matter fields.

If there are $s$ linearly independent multiplets $Y^{a}_{\mathbf{r}}(\tau)$
$(a=1,\ldots,s)$ transforming as in Eq. (232), Eq. (236) holds individually
for all
$\widetilde{Y}^{a}_{\mathbf{r}}(\tau)=X_{\mathbf{r}}^{-1*}Y^{a*}_{\mathbf{r}}(-\tau^{*})$
and we have:

$Y^{a}_{\mathbf{r}}(-\tau^{*})=\lambda^{a}_{~{}b}~{}X_{\mathbf{r}}Y^{b*}_{\mathbf{r}}(\tau)~{}~{}~{},$
(240)

where
$\lambda^{a}_{~{}c}\lambda^{*c}_{~{}~{}b}X_{\mathbf{r}}X^{*}_{\mathbf{r}}=\delta^{a}_{~{}b}\mathds{1}_{\mathbf{r}}$.
When $X_{\mathbf{r}}$ is involutive with
$X_{\mathbf{r}}X^{*}_{\mathbf{r}}=\mathds{1}_{\mathbf{r}}$ we have
$\lambda^{a}_{~{}c}\lambda^{*c}_{~{}~{}b}=\delta^{a}_{~{}b}$, it is always
possible to factorize the matrix $\lambda$ into $\lambda=\eta^{-1}\eta^{*}$
and we obtain202020If $(\mathds{1}+\lambda)$ is invertible, we take
$\eta=(\mathds{1}+\lambda)^{-1}$, then $\lambda\eta^{*-1}=\eta^{-1}$. If
$(\mathds{1}+\lambda)$ is not invertible, we can always find a complex number
$u$ with $|u|=1$ such that $-u^{2}$ is not an eigenvalue of $\lambda$. Hence,
$\lambda+u^{2}\mathds{1}$ is invertible. In this case, we take
$\eta=(u^{-1}\lambda+u\mathds{1})^{-1}$, then $\lambda\eta^{*-1}=\eta^{-1}$.
This construction was given by Prof. Marc van Leeuwen que:matrix_decom .:

$\eta^{a}_{~{}b}Y^{b}_{\mathbf{r}}(-\tau^{*})=X_{\mathbf{r}}[\eta^{a}_{~{}b}Y^{b}_{\mathbf{r}}]^{*}(\tau)~{}~{}~{}.$
(241)

We see that, by performing the change of basis
$Y^{a}_{\mathbf{r}}(\tau)\to\eta^{a}_{~{}b}Y^{b}_{\mathbf{r}}(\tau)$, Eq.
(236) holds independently for each multiplet $Y^{a}(\tau)$. In short, when the
action of gCP on the field space is involutive with
$X_{\mathbf{r}}X^{*}_{\mathbf{r}}=\mathds{1}_{r}$, one can properly choose the
basis of the modular forms such that matter fields and modular forms transform
in a similar way under gCP, i.e.,

$\varphi(x)\stackrel{{\scriptstyle\mathcal{CP}}}{{\longrightarrow}}X_{\mathbf{r}}\overline{\varphi}(x_{\mathcal{P}})\,,~{}~{}~{}Y^{a}(\tau)\stackrel{{\scriptstyle\mathcal{CP}}}{{\longrightarrow}}Y^{a}_{\mathbf{r}}(-\tau^{*})=X_{\mathbf{r}}Y^{b*}_{\mathbf{r}}(\tau)\,.$
(242)

In this case, it is convenient to move to the basis where
$X_{\mathbf{r}}=\mathds{1}_{\mathbf{r}}$212121The CP transfromation matrix
$X_{\mathbf{r}}$ would be a unitary and symmetric matrix if
$X_{\mathbf{r}}X^{*}_{\mathbf{r}}=\mathds{1}_{\mathbf{r}}$, and the
corresponding Takagi factorization is
$X_{\mathbf{r}}=\Omega_{\mathbf{r}}\Omega^{T}_{\mathbf{r}}$ where
$\Omega_{\mathbf{r}}$ is a unitary matrix. If we perform a change of basis
with a unitary matrix $\Omega_{\mathbf{r}}$ in the field space:
$\varphi^{\prime}=\Omega^{\dagger}_{\mathbf{r}}\varphi$, it is easy to see
that the gCP transformation of $\varphi^{\prime}$ is
$\varphi^{\prime}(x)\stackrel{{\scriptstyle\mathcal{CP}}}{{\longrightarrow}}\overline{\varphi^{\prime}}(x_{\mathcal{P}})$..
Then the consistency condition of Eq. (231) implies that the representation
matrices $\rho_{\mathbf{r}}(S)$ and $\rho_{\mathbf{r}}(T)$ of the modular
generators are symmetric. As we explained early, if both
$\rho_{\mathbf{r}}(S)$ and $\rho_{\mathbf{r}}(T)$ are symmetric and unitary
matrices, we would have $X_{\mathbf{r}}=\mathds{1}_{\mathbf{r}}$ which is
exactly the canonical CP transformation. If the Clebsh-Gordan coefficients are
all real in the adopted basis, the requirement of gCP invairance would
enforces all free parameters in the superpotential to be real.

###  5.4 CP fixed points

In a modular invariant theory with gCP symmetry, the vacuum expectation value
of the modulus $\tau$ would be the unique source breaking modular and gCP
symmetry. However, there exist certain values of $\tau$ which conserve CP. If
$\tau$ is left invariant by CP transformation of Eq. (215), i.e.

$\tau\xrightarrow{{\cal CP}}\tau_{{\cal CP}}=-\tau^{*}=\tau\,.$ (243)

We see $\tau$ lies on the imaginary axis, $\Re\tau=0$. The most general gCP
transfromation of $\tau$ is $\gamma\circ{\cal CP}$ given by Eq. (218), it is
the composition of the a generic modular transformation $\gamma$ and the CP
transformation of Eq. (215). Thus the most general value of $\tau$ preserving
the residual gCP symmetry $\gamma\circ{\cal CP}$ satisfies the following
condition

$(\gamma\circ{\cal CP})\tau=\tau,~{}~{}~{}\gamma=\begin{pmatrix}a~{}&~{}b\\\
c~{}&~{}d\end{pmatrix}\in\Gamma\,,$ (244)

which gives

$\frac{-a\tau^{*}+b}{-c\tau^{*}+d}=\tau\,,$ (245)

or equivalently

$\displaystyle
c\left(\Re\tau\right)^{2}-\left(a+d\right)\Re\tau+b+c\left(\Im\tau\right)^{2}$
$\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle\left(a-d\right)\Im\tau$
$\displaystyle=$ $\displaystyle 0\,.$ (246)

Thus we have

$\displaystyle a=d\,,$ (247)

and

$\displaystyle\Re\tau=\left\\{\begin{array}[]{cc}\dfrac{a}{c}\pm\frac{1}{c}\sqrt{1-c^{2}\left(\Im\tau\right)^{2}}\,,{}&~{}c\neq
0\\\ \dfrac{b}{2a}\,,{}&~{}c=0\end{array}\right.$ (250)

Without loss of generality, we firstly consider that $\tau$ lies in the
fundamental domain. Solving the constraints in Eqs. (247, 250) together with
the determinant condition $ad-bc=1$, we find the CP conserved values of $\tau$
are the imaginary axis and the boundary of the fundamental domain
$\mathcal{D}$, the corresponding residual CP symmetry is summarized in table
8. At the intersection points, the residual symmetry is enhanced. For
instance, the modulus $\tau=e^{2i\pi/3}$ is invariant under both CP
transformations $T^{-1}\circ\mathcal{CP}$ and $S\circ\mathcal{CP}$ so that it
preserves the modular symmetry $ST$ which is generated by the two residual CP
transfromations. Thus the full residual symmetry group of $\tau=e^{2i\pi/3}$
is $\left(Z^{ST}_{3}\rtimes Z^{S\circ\mathcal{CP}}_{2}\right)\times Z^{R}_{2}$
defined as

$\displaystyle\left(Z^{ST}_{3}\rtimes Z^{S\circ\mathcal{CP}}_{2}\right)\times
Z^{R}_{2}$ $\displaystyle=$
$\displaystyle\Big{\\{}ST,R,S\circ\mathcal{CP}|R^{2}=(S\circ\mathcal{CP})^{2}=(ST)^{3}=1,(ST)R=R(ST),$
(251)
$\displaystyle\hskip-14.45377pt(S\circ\mathcal{CP})R=R(S\circ\mathcal{CP}),(S\circ\mathcal{CP})ST(S\circ\mathcal{CP})^{-1}=(ST)^{-1}\Big{\\}}\,.$

Similarly the symmetric point $\tau=i$ is invariant under both modular
transfromation $S$ and CP transformation $\mathcal{CP}$, and consequently the
residual symmetry group is $Z^{S}_{4}\rtimes
Z^{\mathcal{CP}}_{2}=\left\\{S,\mathcal{CP}|S^{4}=1,\mathcal{CP}^{2}=1,\mathcal{CP}\,S\,\mathcal{CP}^{-1}=S^{-1}\right\\}$.
Moreover, the symmetric point $\tau=i\infty$ is invariant under $T$, $R$ and
$\mathcal{CP}$, and therefore it preserves the residual symmetry group

$\left(Z^{T}\rtimes\mathcal{CP}\right)\times
Z^{R}_{2}=\left\\{T,R,\mathcal{CP}|R^{2}=\mathcal{CP}^{2}=1,\mathcal{CP}\,T\,\mathcal{CP}^{-1}=T^{-1},\mathcal{CP}\,R=R\,\mathcal{CP},TR=RT\right\\}\,.$
(252)

Figure 9: The CP conserving lines in the fundamental domain $\mathcal{D}$ are
displayed in blue and the corresponding residual CP symmetries are labelled.
The intersection points $\tau=e^{\pi i/3},\,i,e^{2\pi i/3}$ labelled in red
enjoy two different residual CP symmetries so that a residual modular symmetry
can be generated. For instance, the self-dual point $\tau=i$ is invariant
under both CP transfromations $Z^{\mathcal{CP}}_{2}$ and
$Z^{S\circ\mathcal{CP}}_{2}$, consequently it is also invariant under the
modular generator $S$.

We plot the CP conserving points and the corresponding residual CP symmetries
in figure 9. Furthermore we find that all the CP conserving points in the
upper half complex plane are related to those in the fundamental domain by
some modular transformation, and the corresponding generalized CP stabilizer
given by

$\tau^{\prime}=\gamma_{1}\tau^{\mathcal{CP}}_{0}\,,~{}~{}~{}~{}\gamma^{\prime}\circ\mathcal{CP}=\gamma_{1}\gamma^{\mathcal{CP}}_{0}\circ\mathcal{CP}\,\gamma^{-1}_{1}\,,$
(253)

where $\gamma_{1}$ is an arbitrary modular transformation,
$\tau^{\mathcal{CP}}_{0}$ and $\gamma^{\mathcal{CP}}_{0}$ take the values
summarized in table 8.

$\gamma^{\mathcal{CP}}_{0}$ | $\tau^{\mathcal{CP}}_{0}$ | Residual subgroup
---|---|---
$\pm 1$ | $\left\\{\tau|\Re\tau=0,\Im\tau\geq 1\right\\}$ | $Z^{\mathcal{CP}}_{2}\times Z^{R}_{2}$
$\pm T$ | $\left\\{\tau|\Re\tau=1/2,\Im\tau\geq\sqrt{3}/2\right\\}$ | $Z^{T\circ\mathcal{CP}}_{2}\times Z^{R}_{2}$
$\pm T^{-1}$ | $\left\\{\tau|\Re\tau=-1/2,\Im\tau\geq\sqrt{3}/2\right\\}$ | $Z^{T^{-1}\circ\mathcal{CP}}_{2}\times Z^{R}_{2}$
$\pm S$ | $\left\\{\tau|(\Re\tau)^{2}+(\Im\tau)^{2}=1,|\Re\tau|\leq 1/2\right\\}$ | $Z^{S\circ\mathcal{CP}}_{2}\times Z^{R}_{2}$
$\pm 1,\pm S$ | $i$ | $Z^{S}_{4}\rtimes Z^{\mathcal{CP}}_{2}$
$\pm T,\pm S$ | $e^{i\pi/3}$ | $\left(Z^{TS}_{3}\rtimes Z^{S\circ\mathcal{CP}}_{2}\right)\times Z^{R}_{2}$
$\pm T^{-1},\pm S$ | $e^{2i\pi/3}$ | $\left(Z^{ST}_{3}\rtimes Z^{S\circ\mathcal{CP}}_{2}\right)\times Z^{R}_{2}$
Table 8: The CP conserving values of modulus $\tau$ in the fundamental domain
$\mathcal{D}$ and the corresponding residual symmetry group. The symbol
$\gamma^{CP}_{0}$ is defined by the condition
$(\gamma^{\mathcal{CP}}_{0}\circ{\cal
CP})\tau^{\mathcal{CP}}_{0}=\tau^{\mathcal{CP}}_{0}$ which is a special case
of Eq. (244) with $\tau^{\mathcal{CP}}_{0}$ in the fundamental domain. We
would like to mention that the definition of $\gamma\circ\mathcal{CP}$ is
given in Eq. (218). Notice that the modular generators $S$ and $T$ are given
in Eq. (77) and $R=S^{2}$.

###  5.5 Implications for flavor mixing at CP fixed points

Considering a point $\tau^{\mathcal{CP}}_{0}$ of the fundamental domain where
CP is conserved, then there is an element $\gamma^{\mathcal{CP}}_{0}$ of
$\Gamma$ such that $(\gamma^{\mathcal{CP}}_{0}\circ{\cal
CP})\tau^{\mathcal{CP}}_{0}=\tau^{\mathcal{CP}}_{0}$ which gives
$\gamma^{\mathcal{CP}}_{0}(-\tau^{\mathcal{CP}*}_{0})=\tau^{\mathcal{CP}}_{0}$.
Taking the imaginary part on both sides of this equation and using the
identities $\Im(-\tau^{*})=\Im\tau$ and
$\Im(\gamma\tau)=\dfrac{\Im\tau}{|c\tau+d|^{2}}$, we obtain the automorphy
factor

$|J(\gamma^{\mathcal{CP}}_{0},-\tau^{\mathcal{CP}*}_{0})|=1$ (254)

which implies that $J(\gamma^{\mathcal{CP}}_{0},-\tau^{\mathcal{CP}*}_{0})$ is
a pure phase. Note that the automorphy factor $J(\gamma,\tau)=c\tau+d$ is
defined in Eq. (103). We assume that the lepton sector is described by the
superpotential:

$\mathcal{W}=-E^{c}_{i}\mathcal{Y}^{e}_{ij}(\tau)L_{j}H_{d}-\frac{1}{2\Lambda}L_{i}\mathcal{Y}^{\nu}_{ij}(\tau)L_{j}H_{u}H_{u}\,,$
(255)

where the neutrino masses are described by the Weinberg operator. Under the
action of modular transfromation $\gamma$, the matter multiplets $\varphi$
$(\varphi=H_{u,d},E^{c},L)$ transform as:

$\varphi\xrightarrow{\gamma}J^{-k_{\varphi}}(\gamma,\tau)\rho_{\varphi}(\gamma)\varphi\,,~{}~{}~{}\varphi\xrightarrow{\mathcal{CP}}X_{\varphi}\overline{\varphi}(x_{\mathcal{P}})\,.$
(256)

The weights $k_{E^{c},L}$ carry a flavour index and $J^{k_{E^{c},L}}$ are
diagonal matrices in flavour space. The invariance of the $\mathcal{W}$ under
the modular transformation $\gamma$ implies:

$\displaystyle\mathcal{Y}^{e}(\gamma\tau)$
$\displaystyle=J^{k_{H_{d}}}\rho_{H_{d}}^{\dagger}(\gamma)~{}J^{k_{E^{c}}}\rho_{E^{c}}^{*}(\gamma)~{}\mathcal{Y}^{e}(\tau)~{}\rho_{L}^{\dagger}(\gamma)J^{k_{L}}~{}~{}~{}\,,$
$\displaystyle\mathcal{Y}^{\nu}(\gamma\tau)$
$\displaystyle=J^{2k_{H_{u}}}\rho_{H_{u}}^{2\dagger}(\gamma)~{}J^{k_{L}}\rho_{L}^{*}(\gamma)~{}\mathcal{Y}^{\nu}(\tau)~{}\rho_{L}^{\dagger}(\gamma)J^{k_{L}}~{}~{}~{}\,,$
(257)

On the other hand, the invariance of $\mathcal{W}$ under the CP transformation
$\mathcal{CP}$ requires:

$\displaystyle\mathcal{Y}^{e}(-\tau^{*})$
$\displaystyle=X_{H_{d}}^{\dagger}~{}X_{E^{c}}^{*}~{}\mathcal{Y}^{e*}(\tau)~{}X_{L}^{\dagger}\,,$
$\displaystyle\mathcal{Y}^{\nu}(-\tau^{*})$
$\displaystyle=X_{H_{u}}^{2\dagger}~{}X_{L}^{*}~{}\mathcal{Y}^{\nu*}(\tau)~{}X_{L}^{\dagger}~{}~{}~{}.$
(258)

From eqs. (257) and (258), at the point $\tau^{\mathcal{CP}}_{0}$ enjoying
residual CP symmetry we obtain:

$\displaystyle\mathcal{Y}^{e*}(\tau^{\mathcal{CP}}_{0})$
$\displaystyle=\Omega_{H_{d}}~{}\Omega_{E^{c}}^{T}~{}\mathcal{Y}^{e}(\tau^{\mathcal{CP}}_{0})~{}\Omega_{L}\,,$
$\displaystyle\mathcal{Y}^{\nu*}(\tau^{\mathcal{CP}}_{0})$
$\displaystyle=\Omega_{H_{u}}^{2}~{}\Omega_{L}^{T}~{}\mathcal{Y}^{\nu}(\tau^{\mathcal{CP}}_{0})~{}\Omega_{L}\,,$
(259)

where we have defined the unitary matrices:

$\Omega_{\varphi}=J^{-k_{\varphi}}(\gamma^{\mathcal{CP}}_{0},-\tau^{\mathcal{CP}*}_{0})\rho_{\varphi}(\gamma^{\mathcal{CP}}_{0})X_{\varphi}\,,\qquad\qquad(\varphi=H_{u,d},E^{c},L)\,.$
(260)

Notice that $\Omega_{\varphi}$ is the CP transfromation matrix of the matter
multiplet $\varphi$ for the residual gCP symmetry
$\gamma^{\mathcal{CP}}_{0}\circ{\cal CP}$. The charged lepton and neutrino
mass matrices are given by $M_{e}=\mathcal{Y}^{e}v_{d}$ and
$M_{\nu}=\mathcal{Y}^{\nu}v^{2}_{u}/\Lambda$. Thus at the point $\tau$
enjoying residual CP invariance we have

$\displaystyle\Omega_{L}^{\dagger}~{}M^{\dagger}_{e}(\tau^{\mathcal{CP}}_{0})M_{e}(\tau^{\mathcal{CP}}_{0})~{}\Omega_{L}=\left[M^{\dagger}_{e}(\tau^{\mathcal{CP}}_{0})M_{e}(\tau^{\mathcal{CP}}_{0})\right]^{*}\,,$
$\displaystyle\Omega_{L}^{\dagger}~{}M^{\dagger}_{\nu}(\tau^{\mathcal{CP}}_{0})M_{\nu}(\tau^{\mathcal{CP}}_{0})~{}\Omega_{L}=\left[M^{\dagger}_{\nu}(\tau^{\mathcal{CP}}_{0})M_{\nu}(\tau^{\mathcal{CP}}_{0})\right]^{*}\,.$
(261)

We see that the hermitian combination of the neutrino and charged lepton mass
matrices
$M^{\dagger}_{\nu}(\tau^{\mathcal{CP}}_{0})M_{\nu}(\tau^{\mathcal{CP}}_{0})$
and $M^{\dagger}_{e}(\tau^{\mathcal{CP}}_{0})M_{e}(\tau^{\mathcal{CP}}_{0})$
are invariant under a common transformation of the left-handed charged leptons
and the left-handed neutrinos, which represents a combination of CP and
modular transformations. The unitary matrix $\Omega_{L}$ should be symmetric
otherwise the neutrino and the charged lepton mass spectrum would be
constrained to be partially degenerate Feruglio:2012cw . Furthermore, one can
always go to the basis in which $\Omega_{L}=1$ by performing basis
transfromation. Then the conditions eq. (261) fulfilled at the CP fixed point
$\tau^{\mathcal{CP}}_{0}$ imply that both Dirac and Majorana CP phases are
trivial Feruglio:2012cw . Therefore values of moduli deviating from residual
CP symmetry fixed points (the fundamental domain boundary or the imaginary
axis) are required to accommodate the observed non-degenerate lepton masses
and a non-trivial Dirac CP phase.

##  6 Modular invariance with multiple moduli

More fundamental theory at high energy generally involve extra dimension. For
instance, it is known that superstring theory can only be consistently defined
in 10 dimensional spacetime, the extra 6 dimensions have to be compactified in
the form of a Calabi–Yau manifold. As a consequence, the low energy effective
theories at our 4D universe have multiple moduli which depend on the geometry
of extra 6 dimensions. In this section, we shall study the scenario of several
moduli $\tau_{1},\tau_{2},\ldots,\tau_{M}$ from the bottom-up approach,
starting with factorizable moduli, before discussing non-factorizable
generalisations based on symplectic modular invariance, Siegel modular groups
and automorphic forms.

###  6.1 Factorizable moduli

Firstly we consider the direct product of several modular transformation
groups $\Gamma^{1}\times\Gamma^{2}\times\ldots\times\Gamma^{M}$, where each
$\Gamma^{j}$ for $j=1,...,M$ is the $SL(2,\mathbb{Z})$ group. The modulus
field for each modular symmetry $\Gamma^{j}$ is denoted as $\tau_{j}$. As
shown in Eq. (75), any modular transformation $\gamma_{j}$ in $\Gamma^{j}$
acts on the modulus $\tau_{j}$ as follows deMedeirosVarzielas:2019cyj

$\displaystyle\gamma_{j}:\tau_{j}\to\gamma_{j}\tau_{j}=\frac{a_{j}\tau_{j}+b_{j}}{c_{j}\tau_{j}+d_{j}},~{}~{}~{}\gamma_{j}\in\Gamma^{j}\,,$
(262)

while other modulus $\tau_{i}$ is invariant under $\gamma_{j}$, i.e.,
$\gamma_{j}\tau_{i}=\tau_{i}$ for $i\neq j$. Analogous to Eq. (81), each
modular group $\Gamma^{j}$ has the principal congruence subgroup
$\Gamma^{j}(N_{j})$, and a series of finite modular groups
$\Gamma_{N_{j}}^{{}^{\prime}j}=\Gamma^{j}/\Gamma^{j}(N_{j})$ and
$\Gamma_{N_{j}}^{j}=\Gamma^{j}/\pm\Gamma^{j}(N_{j})$ can be defined. Notice
that $N_{i}$ is unnecessary to be identical to $N_{j}$ for $i\neq j$. Thus the
finite modular group is extended to the direct product of several finite
modular groups
$\Gamma_{N_{1}}^{1}\times\Gamma_{N_{2}}^{2}\times\cdots\times\Gamma_{N_{M}}^{M}$,
and any $\Gamma_{N_{j}}^{j}$ could be $\Gamma_{N_{j}}^{{}^{\prime}j}$.

The modular transformation of a generic chiral superfield $\Phi_{I}$ is
characterized by the modular weight $k_{I,j}$ and the unitary representation
$\rho_{\mathbf{r}_{I},j}$ of $\Gamma_{N_{j}}^{j}$ under each modular group
$\Gamma^{j}$, i.e.,

$\displaystyle\Phi_{I}\rightarrow\prod_{j=1,...,M}(c_{j}\tau_{j}+d_{j})^{-k_{I,j}}\bigotimes_{j=1,...,M}\rho_{\mathbf{r}_{I},j}(\gamma_{j})\Phi_{I}\,,$
(263)

where $\bigotimes$ denotes the outer product of the representation matrices
for $\rho_{\mathbf{r}_{I},1}$, $\rho_{\mathbf{r}_{I},2}$, …,
$\rho_{\mathbf{r}_{I},M}$. In the framework of the $\mathcal{N}=1$ global
supersymmetry, the most general action invariant under the extended modular
group can be written as

$\mathcal{S}=\int d^{4}xd^{2}\theta
d^{2}\bar{\theta}\;\mathcal{K}(\Phi_{I},\bar{\Phi}_{I};\tau_{1},...,\tau_{M},\overline{\tau}_{1},...,\overline{\tau}_{M})+\left[\int
d^{4}xd^{2}\theta\;\mathcal{W}(\Phi_{I},\tau_{1},...,\tau_{M})+\text{h.c.}\right]\,.$
(264)

Invariance of the action $\mathcal{S}$ under Eqs. (262, 263) requires the
invariance of the Kähler potential up to a Kähler transformation. Similar to
the case of single modulus, the requirement of invariance of the Kähler
potential can be easily satisfied and many terms are allowed. For
illustration, the minimal Kähler potential is taken as usual.

$\displaystyle\mathcal{K}(\Phi_{I},\overline{\Phi}_{I};\tau_{1},...,\tau_{M},\overline{\tau}_{1},...,\overline{\tau}_{M})$
$\displaystyle=$
$\displaystyle-\sum^{M}_{j=1}h_{j}\log(-i\tau_{j}+i\overline{\tau}_{j})$ (265)
$\displaystyle+$
$\displaystyle\sum_{I}\,\frac{\overline{\Phi}_{I}\Phi_{I}}{\displaystyle\prod_{j=1,...,M}(-i\tau_{j}+i\overline{\tau}_{j})^{k_{I,j}}}\,,$

where all $h_{j}$ are positive constants. Since each modular symmetry is
independent from each other, one modulus field getting a VEV leaves the rest
of the Kähler potential still satisfying the other modular symmetries. If all
modulus fields acquire VEVs, the above Kähler potential leads to kinetic terms
for the scalar components of the supermultiplets $\phi_{I}$ and the modulus
fields $\tau_{j}$ as follow

$\displaystyle\sum^{M}_{j=1}\frac{h_{j}}{\langle-i\tau_{j}+i\overline{\tau}_{j}\rangle^{2}}\partial_{\mu}\overline{\tau}_{j}\partial^{\mu}\tau_{j}+\sum_{I}\frac{\partial_{\mu}\overline{\Phi}_{I}\partial^{\mu}\Phi_{I}}{\displaystyle\prod_{j=1,...,M}\langle-i\tau_{j}+i\overline{\tau}_{j}\rangle^{k_{I,j}}}\,.$
(266)

The superpotential $\mathcal{W}(\Phi_{I};\tau_{1},...,\tau_{M})$ is in general
a holomorphic function of the moduli $\tau_{1},\ldots,\tau_{M}$ and
superfields $\phi_{I}$. Modular invariance imposes strong constraint on the
superpotential $\mathcal{W}$ which can be expanded in power series of
$\Phi_{I}$ as

$\displaystyle\mathcal{W}(\Phi_{I};\tau_{1},...,\tau_{M})=\sum_{n}\sum_{\\{I_{1},\cdots,I_{n}\\}}\left(Y_{I_{1}...I_{n}}\Phi_{I_{1}}\cdots\Phi_{I_{n}}\right)_{\mathbf{1}}\,.$
(267)

For each term to be modular invariant, the modular forms $Y_{I_{1}...I_{n}}$
should transform as

$\displaystyle Y_{I_{1}...I_{n}}(\tau_{1},...,\tau_{M})\to
Y_{I_{1}...I_{n}}(\gamma_{1}\tau_{1},...,\gamma_{M}\tau_{M})$
$\displaystyle\hskip
56.9055pt=\prod^{M}_{j=1}(c_{j}\tau_{j}+d_{j})^{k_{Y,j}}\bigotimes^{M}_{j=1}\rho_{\mathbf{r}_{Y},j}(\gamma_{j})Y_{I_{1}...I_{n}}(\tau_{1},...,\tau_{M})\,,$
(268)

satisfying the conditions

$\displaystyle k_{Y,j}=k_{I_{1},j}+\cdots k_{I_{n},j}\;,$
$\displaystyle\rho_{\mathbf{r}_{Y},j}\otimes\rho_{\mathbf{r}_{I_{1}},j}\otimes\ldots\otimes\rho_{\mathbf{r}_{I_{n}},j}\ni\mathbf{1}\,,$
(269)

for $j=1,...,M$.

###  6.2 Symplectic modular invariance

Siegel modular forms can be thought of as modular forms in more than one
variable, they generalize the usual single variable modular forms on
$SL(2,\mathbb{Z})$ in that group $SL(2,\mathbb{Z})$ is replaced by the Siegel
modular group $\Gamma_{g}=Sp(2g,\mathbb{Z})$ and the upper half plane is
replaced by the Siegel upper half plane $\mathcal{H}_{g}$, where the integer
$g\geq 1$ is called the degree or genus. The Siegel modular group
$\Gamma=Sp(2g,\mathbb{Z})$ arises as the duality group in string Calabi-Yau
compactifications Cecotti:1988qn ; Cecotti:1988ad ; Cecotti:1989kn ;
Dixon:1989fj ; Candelas:1990pi ; Strominger:1990pd ; Ferrara:1991uz ;
Font:1992uk ; Ishiguro:2020nuf ; Baur:2020yjl ; Nilles:2021glx ;
Ishiguro:2021ccl . Siegel modular forms are relevant in the context of string
one-loop corrections Mayr:1995rx ; Stieberger:1998yi .

The Siegel modular group $\Gamma=Sp(2g,\mathbb{Z})$ is the group of $2g\times
2g$ matrices with integer entries and satisfy the symplectic condition

$\gamma^{t}~{}J~{}\gamma=J,~{}~{}~{}J=\begin{pmatrix}0~{}&~{}\mathds{1}_{g}\\\
-\mathds{1}_{g}~{}&~{}0\end{pmatrix},~{}~{}~{}\gamma=\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\in\Gamma_{g}\,,$ (270)

where the superscript $t$ denotes the transpose, and all the four blocks $A$,
$B$, $C$, $D$ are $g$ dimensional matrices with integer elements. Thus blocks
$A$, $B$, $C$ and $D$ satisfy the conditions,

$A^{t}C=C^{t}A\,,\quad B^{t}D=D^{t}B\,,\quad A^{t}D-C^{t}B=\mathds{1}_{g}\,,$
(271)

or equivalently

$AB^{t}=BA^{t}\,,\quad CD^{t}=DC^{t}\,,\quad AD^{t}-BC^{t}=\mathds{1}_{g}\,.$
(272)

For the lowest genus $g=1$, the above symplectic condition is exactly the
condition of unit determinant of $\gamma$. Consequently the Siegel modular
group $\Gamma_{1}=Sp(2,\mathbb{Z})$ is exactly identical with the usual
modular group $SL(2,\mathbb{Z})$. For any element $\gamma\in\Gamma_{g}$, both
$\gamma^{t}$ and $\gamma^{-1}$ are also elements of $\Gamma_{g}$ with

$\gamma^{-1}=\begin{pmatrix}D^{t}~{}&~{}-B^{t}\\\
-C^{t}~{}&~{}A^{t}\end{pmatrix}\,.$ (273)

The single modulus $\tau$ in the upper half plane is generalized to the Siegel
upper half plane $\mathcal{H}_{g}$ which is represented by $g\times g$
symmetric complex matrices with positive definite imaginary part222222The
modulus $\tau$ can be parameterized as $\tau=X+iY$ where both $X$ and $Y$ are
real symmetric matrices with $Y\neq 0$, and all the eigenvalues of $Y$ are
positive. This is exactly the condition of $\texttt{Im}(\tau)>0$ in Eq.
(274).,

$\mathcal{H}_{g}=\left\\{\tau\in
GL(g,\mathbb{C})~{}\Big{|}~{}\tau^{t}=\tau,\quad\texttt{Im}(\tau)>0\right\\}\,.$
(274)

The action of $\gamma\in\Gamma_{g}$ on the Siegel upper half plane
$\mathcal{H}_{g}$ is defined as follow,

$\tau\to\gamma\tau=(A\tau+B)(C\tau+D)^{-1}\,.$ (275)

The Siegel modular group $\Gamma_{g}$ can be generated by the generators $S$
and $T_{i}$ with

$S=\begin{pmatrix}0~{}&~{}\mathds{1}_{g}\\\
-\mathds{1}_{g}~{}&~{}0\end{pmatrix}\,,\quad
T_{i}=\begin{pmatrix}\mathds{1}_{g}~{}&~{}B_{i}\\\
0~{}&~{}\mathds{1}_{g}\end{pmatrix}\,,$ (276)

where $\\{B_{i}\\}$ is a basis for the $g\times g$ integer symmetric matrices.
Notice that $S$ coincides with the invariant symplectic form $J$ satisfying
$S^{2}=-\mathds{1}_{2g}$. For the case of $g=2$, without loss of generality we
could choose

$B_{1}=\begin{pmatrix}1~{}&0\\\ 0~{}&0\\\
\end{pmatrix},~{}~{}~{}B_{2}=\begin{pmatrix}0~{}&0\\\ 0~{}&1\\\
\end{pmatrix},~{}~{}~{}B_{3}=\begin{pmatrix}0~{}&1\\\ 1~{}&0\\\
\end{pmatrix}\,.$ (277)

Under the action of generators $S$ and $T_{i}$, the modulus $\tau$ transforms
as:

$\tau\xrightarrow{S}-\tau^{-1},~{}~{}~{}\tau\xrightarrow{T_{i}}\tau+B_{i}\,.$
(278)

In a similarly way as the single modulus case, we can define the principal
congruence subgroups $\Gamma_{g}(n)$ of level $n$ as

$\Gamma_{g}(n)=\left\\{\gamma\in\Gamma_{g}~{}\Big{|}~{}\gamma\equiv\mathds{1}_{2g}\,~{}(\texttt{mod}~{}n)\right\\}\,,$
(279)

where $n$ is a generic positive integer, and $\Gamma_{g}(1)=\Gamma_{g}$. The
group $\Gamma_{g}(n)$ is a normal subgroup of $\Gamma_{g}$, and the quotient
group $\Gamma_{g,n}\equiv\Gamma_{g}/\Gamma_{g}(n)$, which is known as finite
Siegel modular group, has finite order Koecher1954Zur ; thesis_Fiorentino :

$|\Gamma_{g,n}|=n^{g(2g+1)}\prod_{p|n}\prod_{1\leq k\leq
g}(1-\dfrac{1}{p^{2k}})\,,$ (280)

where the product is over the prime divisors $p$ of $n$. For the simplest case
of $g=1$, we have

$\displaystyle|\Gamma_{1,n}|=n^{3}\prod_{p|n}(1-\dfrac{1}{p^{2}})\,.$ (281)

This is consistent with the dimension formula of $SL(2,\mathbb{Z}_{n})$
Gunning1962 ; Schoeneberg .

Modular group $SL(2,\mathds{Z})\equiv\Gamma$ |  Siegel modular group $Sp(2g,\mathds{Z})\equiv\Gamma_{g}$
---|---
$\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$ | $\gamma=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$
$a,b,c,d\in\mathds{Z}$ | $A,B,C,D\in GL(g,\mathds{Z})$
$ad-bc=1$ |  $\gamma^{t}J\gamma=J$, $J=\begin{pmatrix}0&\mathds{1}_{g}\\\ -\mathds{1}_{g}&0\end{pmatrix}$
$H_{1}=\left\\{\tau\in\mathds{C}|\text{Im}(\tau)>0\right\\}$ | $H_{g}=\left\\{\tau\in GL(g,\mathds{C})|\tau^{t}=\tau,\text{Im}(\tau)>0\right\\}$
$\tau\rightarrow\gamma\tau=\frac{a\tau+b}{c\tau+d}$ | $\tau\rightarrow\gamma\tau=\left(A\tau+B\right)\left(C\tau+D\right)^{-1}$
$\Gamma(N)=\left\\{\gamma\in SL(2,\mathds{Z})|\gamma=\mathds{1}_{2}\mod{N}\right\\}$ | $\Gamma_{g}(N)=\left\\{\gamma\in Sp(2g,\mathds{Z})|\gamma=\mathds{1}_{2g}\mod{N}\right\\}$
$\Gamma^{\prime}_{N}=SL(2,\mathds{Z})/\Gamma(N)$ | $\Gamma_{g,N}=Sp(2g,\mathds{Z})/\Gamma_{g}(N)$
$Y(\gamma\tau)=(c\tau+d)^{k}Y(\tau),~{}~{}\gamma\in\Gamma(N)$ | $Y(\gamma\tau)=\left[\text{det}(C\tau+D)\right]^{k}Y(\tau),~{}~{}\gamma\in\Gamma_{g}(N)$
Table 9: The comparison between the modular group $SL(2,\mathds{Z})$ and the
Siegel modular group $Sp(2g,\mathds{Z})$, where $g$ is a positive integer and
it corresponds to the genus. Notice that $Sp(2,\mathds{Z})$ is isomorphic to
$SL(2,\mathds{Z})$.

####  6.2.1 Siegel modular forms and gCP in symplectic modular invariant
theory

The Siegel modular forms $Y(\tau)$ of integer weight $k$ and level $n$ at
genus $g$ are holomorphic functions of the moduli $\tau$, and they fulfill the
following transformation properties under the principal congruence subgroups
$\Gamma_{g}(n)$:

$Y(\gamma\tau)=[\det(C\tau+D)]^{k}Y(\tau)\,,\quad\quad\gamma=\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\in\Gamma_{g}(n)\,.$ (282)

For lower levels $n=1,2$, by taking $\gamma=-\mathds{1}_{2g}$ in Eq. (282), we
can see that the Siegel modular forms at genus $g$ of weight $k$ vanish if
$kg$ is odd. The Siegel modular forms of given weight $k$, level $n$ and genus
$g$ expand a linear space $\mathcal{M}_{k}(\Gamma_{g}(n))$ of finite
dimensional Bruinier2008The  and there are no non-vanishing forms of negative
weight Bruinier2008The . The product of two Siegel modular forms of weight
$k$, $k^{\prime}$ at level $n$ and genus $g$ is a Siegel modular form of
weight $k+k^{\prime}$. The full set of Siegel modular forms with respect to
$\Gamma_{g}(n)$ form a positive graded ring
$\mathcal{M}(\Gamma_{g}(n))=\bigoplus_{k\geq
0}\mathcal{M}_{k}(\Gamma_{g}(n))$.

The Siegel modular forms of weight $k$, level $n$ and genus $g$ are invariant
up to the automorphy factor $[\det(C\tau+D)]^{k}$ under $\Gamma_{g}(n)$ but
they transform under the quotient group
$\Gamma_{g,n}\equiv\Gamma_{g}/\Gamma_{g}(n)$. Similar to the case $g=1$
Feruglio:2017spp ; Liu:2019khw , it is always possible to choose a basis
$\\{Y_{i}(\tau)\\}$ in the space $\mathcal{M}_{k}(\Gamma_{g}(n))$ such that
the action of $\Gamma_{g}$ on the elements of the basis is described by a
unitary representation $\rho_{\mathbf{r}}$ of the finite Siegel modular group
$\Gamma_{g,n}=\Gamma_{g}/\Gamma_{g}(n)$ Ding:2020zxw :

$Y_{i}(\gamma\tau)=[\det(C\tau+D)]^{k}\rho_{\mathbf{r}}(\gamma)_{ij}Y_{j}(\tau),~{}~{}~{}\quad\gamma=\begin{pmatrix}A~{}&B\\\
C~{}&D\end{pmatrix}\in\Gamma_{g}\,.$ (283)

At variance with Eq. (282), where only transformations of $\Gamma_{g}(n)$ were
considered, in the previous equation the full Siegel modular group
$\Gamma_{g}$ is acting. The comparison between the modular group
$SL(2,\mathds{Z})\equiv\Gamma$ and the Siegel modular group
$Sp(2g,\mathds{Z})\equiv\Gamma_{g}$ is summarized in table 9. One see that
$\Gamma_{g}$ is a natural generation of $\Gamma$ at high genus $g$, and the
matrix elements $a$, $b$, $c$, $d$ are replaced by $A$, $B$, $C$, $D$ which
are $g\times g$ integer matrices, and the automorphy factor $c\tau+d$ is
replaced by $\det(C\tau+D)$.

Then the modular invariant theory with single modulus sketched in section 3.4
can be straightforwardly generalized to the case with multiple moduli based on
the Siegel modular symmetry $\Gamma_{g}$. Analogously each terms of the
superpotential should be invariant under the finite Siegel modular group
$\Gamma_{g,n}$ and its modular weight should be zero. Moreover, similar to the
case of single modulus discussed in section 5.1, the consistent CP
transformation of the matrix $\tau$ is determined to be Ding:2021iqp

$\tau\xrightarrow{{\cal CP}}\tau_{{\cal CP}}=-\tau^{*}\,,$ (284)

up to a Siegel modular transformation. Combining the above CP and Siegel
modular transformations, we get the extended Siegel modular group
$\Gamma_{g}^{*}=GSp(2g,\mathbb{Z})$ and the full symmetry transformation of
the complex moduli is

$\displaystyle\tau\xrightarrow{\gamma}(A\tau+B)(C\tau+D)^{-1}\,,$
$\displaystyle\tau\xrightarrow{g{\cal
CP}}(-A\tau^{*}+B)(-C\tau^{*}+D)^{-1}\,,$ (285)

where $\gamma=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ is the Siegel modular
transformation satisfying $\gamma^{t}J\gamma=J$. The CP symmetry acts on a
generic matter chiral supermultiplet $\varphi$ as follows,

$\varphi(x)\stackrel{{\scriptstyle\mathcal{CP}}}{{\longrightarrow}}X_{\mathbf{r}}\overline{\varphi}(x_{\mathcal{P}})\,,$
(286)

where $X_{\mathbf{r}}$ is a unitary matrix in the flavor space. The
consistency between the CP symmetry and modular symmetry requires
$X_{\mathbf{r}}$ should fulfill the following conditions Ding:2021iqp ,

$X_{\mathbf{r}}~{}\rho_{\mathbf{r}}^{*}(S)~{}X_{\mathbf{r}}^{-1}=\rho_{\mathbf{r}}(S^{-1})\,,~{}~{}~{}X_{\mathbf{r}}~{}\rho_{\mathbf{r}}^{*}(T_{i})~{}X_{\mathbf{r}}^{-1}=\rho_{\mathbf{r}}(T_{i}^{-1})\,.$
(287)

In the same fashion as section 5.3, one can show that CP transformation of the
modular form multiplets is identical with that of single modulus case, i.e.,

$Y^{a}_{\mathbf{r}}(\tau)\stackrel{{\scriptstyle\mathcal{CP}}}{{\longrightarrow}}Y^{a}_{\mathbf{r}}(-\tau^{*})=\lambda^{a}_{~{}b}X_{\mathbf{r}}Y^{b*}_{\mathbf{r}}(\tau)\,,$
(288)

with
$\lambda^{a}_{~{}c}\lambda^{*c}_{~{}b}X_{\mathbf{r}}X^{*}_{\mathbf{r}}=\delta^{a}_{~{}b}\mathds{1}_{\mathbf{r}}$,
where $a,b$ label linearly independent modular form multiplets of the same
type. In the basis where the matrices $\rho_{\mathbf{r}}(S)$ and
$\rho_{\mathbf{r}}(T_{i})$ are symmetric and unitary, the gCP transformation
would coincide with the canonical CP $X_{\mathbf{r}}=\mathds{1}$ and one can
always work with modular forms $Y^{a}_{\mathbf{r}}(\tau)$ with
$\lambda^{a}_{~{}b}=\delta^{a}_{~{}b}$.

####  6.2.2 Invariant loci in moduli space and reduced finite Siegel modular
group

From Eq. (280), we see that the order of the finite Siegel modular group
$\Gamma_{g,n}$ increases quickly with the genus $g$ and level $n$. At the
genus $g=2$, we have Ding:2020zxw

$\Gamma_{2,2}\cong S_{6},~{}~{}~{}\Gamma_{2,3}\cong Sp(4,F_{3})\,,$ (289)

and

$|\Gamma_{2,2}|=720,~{}~{}~{}|\Gamma_{2,3}|=51840\,.$ (290)

Note that $S_{6}$ is the permutation group of six objects, and $Sp(4,F_{3})$
is isomorphic to the double covering of Burkhardt group. However, $S_{6}$ has
no three dimensional irreducible representations to which the three
generations of left-handed lepton fields are usually assigned232323The $S_{6}$
group has two one-dimensional, four five-dimensional, two nine-dimensional,
two ten-dimensional and one sixteen-dimensional irreducible representations..

Small finite Siegel modular groups can be obtained by restricting the theory
to a subregion $\Omega$ of the moduli space $\mathcal{H}_{g}$ Ding:2020zxw .
We define a region $\Omega$ whose points $\tau$ are individually left
invariant under the action of a subgroup $H$ of $\Gamma_{g}$, i.e.

$h~{}\tau=\tau\,,~{}~{}~{}h\in H,~{}~{}\tau\in\Omega.$ (291)

Because $\gamma\tau=-\gamma\tau$, we also consider the projective group
$\bar{H}=H/\\{\pm\mathds{1}_{2g}\\}$. Both $H$ and $\bar{H}$ are called
stabilizers. The group $N(H)$ that, as a whole, leaves the region $\Omega$
invariant includes the elements $\gamma$ of $\Gamma_{g}$ such that:

$\gamma\tau=\tau^{\prime}\,,\qquad\tau,\tau^{\prime}\in\Omega\,,$ (292)

which requires

$\gamma^{-1}H\gamma=H\,.$ (293)

Therefore $N(H)$ is the normalizer of $H$. Analogously we can define the
principal congruence subgroup of $N(H)$, denoted as $N(H,n)$:

$N(H,n)=\left\\{\hat{\gamma}\in
N(H)~{}\Big{|}~{}\hat{\gamma}=\mathds{1}_{2g}\,~{}(\texttt{mod}~{}n)\right\\}~{}~{}~{}.$
(294)

Obviously $N(H,n)$ is a subgroup of $\Gamma_{g}(n)$, and it is also a normal
subgroup of $N(H)$. The finite modular subgroup $N_{n}(H)$ corresponding to
the modular subgroup $N(H)$ is the quotient group

$N_{n}(H)=N(H)/N(H,n)\,,$ (295)

which is a subgroup of finite Siegel modular group $\Gamma_{g,n}$. In short,
we can consistently truncate the moduli space to the subspace $\Omega$, and
substitute $\Gamma_{g}$ and $\Gamma_{g,n}$ with $N(H)$ and $N_{n}(H)$,
respectively. As a result, in our supersymmetric action we can restrict the
moduli $\tau$ to the region $\Omega$, which supersedes the full moduli space
${\cal H}_{g}$, and replace the group $\Gamma_{g}$ with $N(H)$. An element
$\gamma$ of $N(H)$ induces the transformation laws

$\left\\{\begin{array}[]{l}\tau\to\gamma\tau=(A\tau+B)(C\tau+D)^{-1}\\\\[5.69046pt]
\varphi_{I}\to[\det(C\tau+D)]^{k_{I}}\rho_{\mathbf{r}_{I}}(\gamma)\varphi_{I}~{}~{}~{}.\end{array}\right.\qquad\quad\gamma=\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\in N(H)\,,$ (296)

where $\rho_{\mathbf{r}_{I}}(\gamma)$ is a unitary representation of a finite
group $N_{n}(H)$.

The genus 2 Siegel modular invariant theories is the simplest non trivial
generalization of modular invariant supersymmetric theories studied in
Ferrara:1989bc ; Ferrara:1989qb ; Feruglio:2017spp . The moduli space ${\cal
H}_{2}$ has complex dimension 3 and describes 3 moduli:

$\tau=\left(\begin{array}[]{cc}\tau_{1}&\tau_{3}\\\
\tau_{3}&\tau_{2}\end{array}\right)\,,\qquad\det(\texttt{Im}(\tau))>0\,,\qquad\text{Tr}(\texttt{Im}(\tau))>0\,.$
(297)

The invariant loci in Siegel upper half plane $\mathcal{H}_{2}$ have been
classified by Gottschling gottschling1961fixpunkte ;
gottschling1961fixpunktuntergruppen ; gottschling1967uniformisierbarkeit . It
is established that $\mathcal{H}_{2}$ has two inequivalent invariant loci of
complex dimension two,

$\begin{pmatrix}\tau_{1}&0\\\
0&\tau_{2}\end{pmatrix}\,,~{}~{}~{}\begin{pmatrix}\tau_{1}&\tau_{3}\\\
\tau_{3}&\tau_{1}\end{pmatrix}\,,$ (298)

five inequivalent invariant loci of complex dimension one,

$\displaystyle\begin{pmatrix}i&0\\\
0&\tau_{2}\end{pmatrix}\,,~{}~{}~{}\begin{pmatrix}\omega&0\\\
0&\tau_{2}\end{pmatrix}\,,~{}~{}~{}\begin{pmatrix}\tau_{1}&0\\\
0&\tau_{1}\end{pmatrix}\,,~{}~{}~{}\begin{pmatrix}\tau_{1}&1/2\\\
1/2&\tau_{1}\end{pmatrix}\,,~{}~{}~{}\begin{pmatrix}\tau_{1}&\tau_{1}/2\\\
\tau_{1}/2&\tau_{1}\end{pmatrix}\,,$ (299)

and six independent isolated fixed points,

$\displaystyle\begin{pmatrix}\eta&\frac{1}{2}(\eta-1)\\\
\frac{1}{2}(\eta-1)&\eta\end{pmatrix}\,,~{}~{}~{}\begin{pmatrix}i&0\\\
0&i\end{pmatrix}\,,~{}~{}~{}\begin{pmatrix}\omega&0\\\
0&\omega\end{pmatrix}\,,~{}~{}~{}\dfrac{i\sqrt{3}}{3}\begin{pmatrix}2&1\\\
1&2\end{pmatrix}\,,~{}~{}~{}\begin{pmatrix}\omega&0\\\ 0&i\end{pmatrix}\,,$
(300)

with $\zeta=e^{2\pi i/5},~{}\eta=\frac{1}{3}(1+i2\sqrt{2}),~{}\omega=e^{2\pi
i/3}$. The corresponding stabilizers $\bar{H}$, the normalizers $N(H)$ and the
finite modular group $N_{2}(H)$ for each locus are given in Ref. Ding:2020zxw
. It is remarkable that the restricted finite modular group $N_{2}(H)$ has
small order and it is promising as a flavor symmetry. For instance, $N_{2}(H)$
is isomorphic to $(S_{3}\times S_{3})\rtimes Z_{2}$ and $S_{4}\times Z_{2}$
respectively for the two-dimensional loci in Eq. (298). Fermion mass models
based on symplectic modular symmetry are given in Refs. Ding:2020zxw ;
Ding:2021iqp ; Kikuchi:2023dow .

###  6.3 Multiple moduli and automorphic forms

The modular flavor symmetry with single modulus can be naturally generalized
to more general supersymmetric modular invariant theories involving multiple
moduli, where a discrete subgroup $\Gamma$ of a non-compact Lie group $G$
plays the role of flavour symmetry and the symmetry breaking sector spans an
Hermitian Symmetric Space which is a coset space of the type
$\mathcal{M}=G/K$, here $K$ is a maximal compact subgroup of $G$ Ding:2020zxw
. Then the modular forms which are building blocks of Yukawa couplings, would
be replaced by the more general automorphic forms Ding:2020zxw .

It is notable that the well-known moduli space
$\mathcal{H}=\left\\{\tau\in\mathbb{C}|\texttt{Im}(\tau)>0\right\\}$ can be
equivalently defined as the quotient space $G/K$ Feruglio:2017spp , where
$G=SL(2,\mathbb{R})$ and $K=SO(2)$ is a maximal compact subgroup of $G$. To be
more specific, a generic element g of $SL(2,\mathbb{R})$ can be decomposed
into the product of two matrices:

$SL(2,\mathbb{R})\ni\textsf{g}=\begin{pmatrix}\sqrt{y}~{}&x/\sqrt{y}\\\
0~{}&1/\sqrt{y}\end{pmatrix}~{}\textsf{k}\,,~{}~{}y>0~{},\qquad\textsf{k}=\left(\begin{matrix}\cos\theta~{}&-\sin\theta\\\
\sin\theta~{}&\cos\theta\end{matrix}\right)\,,$ (301)

where k belongs to $K=SO(2)$, while the first factor in the above
decomposition of g is an element of the coset $SL(2,\mathbb{R})/SO(2)$. It is
obvious that $\tau_{0}=i$ is invariant under the action of $K=SO(2)$,

$\textsf{k}\cdot i=\begin{pmatrix}\cos\theta~{}&-\sin\theta\\\
\sin\theta~{}&\cos\theta\end{pmatrix}\cdot
i=\frac{\cos\theta\;i-\sin\theta}{\sin\theta\;i+\cos\theta}=i\,.$ (302)

Hence any modulus $\tau=x+iy$ $(y>0)$ can be reached from the fixed complex
number $\tau_{0}=i$ by a $SL(2,\mathbb{R})$ transformation g, i.e.,

$\textsf{g}~{}\tau_{0}=\left(\begin{array}[]{cc}\sqrt{y}{}&x/\sqrt{y}\\\
0{}&1/\sqrt{y}\end{array}\right)\cdot i=x+i~{}y=\tau\,.$ (303)

This explicitly shows the one-to-one correspondence between the elements of
${\cal H}$ and those of $SL(2,\mathbb{R})/SO(2)$.

In a similar fashion, the generic element $\tau$ of the multi-dimensional
moduli space $\mathcal{M}$ can be represented by the action of $g\in G$ on an
element $\tau_{0}$ left invariant by $K$:

$\tau=g~{}\tau_{0},~{}~{}~{}g\in G\,,$ (304)

with

$h~{}\tau_{0}=\tau_{0}\,,~{}~{}\text{for any}~{}~{}h\in K\,.$ (305)

In finite discrete subgroup $\Gamma$ of $G$ is chosen as the candidate flavor
group, and its action on the moduli $\tau$ is given by

$\tau\stackrel{{\scriptstyle\gamma}}{{\rightarrow}}\gamma\tau=\left(\gamma
g\right)\tau_{0},~{}~{}~{}\gamma\in\Gamma\,.$ (306)

Given a normal subgroup $\Gamma_{normal}$ of $\Gamma$ with finite index, we
define the finite group $\Gamma_{finite}=\Gamma/\Gamma_{normal}$. Being
finite, the finite group $\Gamma_{finite}$ admits unitary representations
$\rho(\gamma)$. A general transformation of the supermultiplets
$\varphi^{(I)}$ of each sector is characterized by the weight $k_{I}$ and the
representation $\rho^{(I)}$ of $\Gamma_{finite}$:

$\tau\stackrel{{\scriptstyle\gamma}}{{\rightarrow}}\gamma\tau,~{}~{}~{}\varphi^{(I)}\xrightarrow{\gamma}j(\gamma,\tau)^{k_{I}}~{}\rho^{(I)}(\gamma)\varphi^{(I)}~{},~{}~{}~{}~{}\gamma\in\Gamma~{}\,,$
(307)

where $j(\gamma,\tau)$ is the automorphy factor and it satisfies the so-called
cocycle condition

$j(\gamma_{1}\gamma_{2},\tau)=j(\gamma_{1},\gamma_{2}\tau)j(\gamma_{2},\tau)\,.$
(308)

In short, the modular invariance of single modulus can be naturally
generalized to the case with multiple moduli Feruglio:2022ivc ;
Feruglio:2022cgv :

$\displaystyle\begin{array}[]{clc}SL(2,\mathbb{R}){}&~{}\rightarrow{}&~{}\text{Lie
group}~{}~{}G\\\ SO(2){}&~{}\rightarrow{}&~{}\text{maximal compact
subgroup}~{}~{}K\subset G\\\
\mathcal{H}=SL(2,\mathbb{R})/SO(2){}&~{}\rightarrow{}&~{}\mathcal{M}=G/K\\\
SL(2,\mathbb{Z}){}&~{}\rightarrow{}&~{}\text{infinite discrete
subgroup}~{}~{}\Gamma\subset G\\\ \Gamma(N){}&~{}\rightarrow{}&~{}\text{normal
subgroup~{}~{}}\Gamma_{normal}\subset\Gamma\\\
\Gamma^{\prime}_{N}{}&~{}\rightarrow{}&~{}\Gamma_{finite}=\Gamma/\Gamma_{normal}\\\
c\tau+d{}&~{}\rightarrow{}&~{}\text{automorphy
factor}~{}~{}j(\gamma,\tau)\end{array}\,.$ (316)

In the rigid $\mathcal{N}=1$ supersymmetry, a candidate minimal Kähler
potential is given by:

$\mathcal{K}_{\tt min}=-h\log
Z(\tau,\bar{\tau})+\sum_{I}Z(\tau,\bar{\tau})^{k_{I}}|\varphi^{(I)}|^{2}\,,~{}~{}Z(\tau,\bar{\tau})\equiv[j^{\dagger}(\textsf{g},\tau_{0})j(\textsf{g},\tau_{0})]^{-1}\,,$
(317)

where $h$ is a real constant whose sign is chosen to guarantee local
positivity of the metric for the moduli $\tau$. By construction, the above
potential is invariant under $\Gamma$ up to a Kähler transformation for a
general choice of $G$, $K$, $\Gamma$, $\Gamma_{normal}$ and $j(g,\tau)$.
Nevertheless this is not the most general Kähler potential invariant under
$\Gamma$. There are other terms which are compatible with invariance under
$\Gamma$ in a pure bottom-up approach. Generally additional assumptions or
inputs from a top-down approach are needed in order to reduce the
arbitrariness of the predictions Nilles:2020nnc ; Nilles:2020kgo ;
Nilles:2020tdp ; Nilles:2020gvu .

The superpotential could be expanded in powers of the supermultiplets
$\varphi^{(I)}$ as follows:

$\mathcal{W}=\sum_{n}Y_{I_{1}...I_{n}}(\tau)~{}\varphi^{(I_{1})}...\varphi^{(I_{n})}\,.$
(318)

The $n$-th order term is invariant provided the functions
$Y_{I_{1}...I_{n}}(\tau)$ obey:

$Y_{I_{1}...I_{n}}(\gamma\tau)=j(\gamma,\tau)^{k_{Y}(n)}\rho^{(Y)}(\gamma)~{}Y_{I_{1}...I_{n}}(\tau)\,.$
(319)

Each term of the superpotential should be invariant under the action of
$\Gamma$ and its total weight should be vanishing such that $k_{Y}(n)$ and
$\rho^{(Y)}$ have to fulfill

$\displaystyle
k_{Y}(n)+k_{I_{1}}+....+k_{I_{n}}=0\,,~{}~{}~{}\rho^{(Y)}\times\rho^{{(I_{1})}}\times...\times\rho^{{(I_{n})}}\supset\mathbf{1}\,,$

where $\mathbf{1}$ denotes the invariant singlet of the finite group
$\Gamma_{finite}$. When we restrict to transformation $\gamma$ of the group
$\Gamma_{normal}$ in eq. (319), we obtain:

$Y_{I_{1}...I_{n}}(\gamma\tau)=j(\gamma,\tau)^{k_{Y}(n)}~{}Y_{I_{1}...I_{n}}(\tau)\,,\quad\gamma\in\Gamma_{normal}\,.$
(320)

Thus the function

$\Psi(g)=j(g,\tau_{0})^{-k_{Y}(n)}Y_{I_{1}...I_{n}}(g\tau_{0})$ (321)

is an automorphic form for $G$, $K$ and $\Gamma_{normal}$. The automorphic
form $\Psi(g)$ is a smooth complex function and it satisfies

$\displaystyle\Psi(\gamma\textsf{g})$ $\displaystyle=$
$\displaystyle\Psi(\textsf{g})\,,~{}\quad~{}\gamma\in\Gamma_{normal}~{}~{}\,,$
$\displaystyle\Psi(gh)$ $\displaystyle=$ $\displaystyle
j(h,\tau_{0})^{-1}~{}\Psi(\textsf{g})\,,\quad h\in K\,.$ (322)

Moreover, the automorphic form $\Psi(g)$ is required to be an eigenfunction of
all the Casimir operators of $G$ and it should satisfy moderate growth
condition Borel ; Borel2 ; klingen1990introductory . The symplectic modular
invariant theory in section 6.2 corresponds to the following choice

$G=Sp(2g,\mathbb{R})\,,~{}K=Sp(2g,\mathbb{R})\cap
O(2g,\mathbb{R})\,,~{}\tau_{0}=i\mathds{1}_{g},~{}\Gamma=Sp(2g,\mathbb{Z})\,,~{}\Gamma_{normal}=\Gamma_{g}(n)\,,$
(323)

and the automorphy factor $j(\gamma,\tau)=\det(C\tau+D)$. We can also choose
other non-compact Hermitian Symmetric Space as multi-dimensional moduli space
which are summarized in table 10, and the symplectic modular group is exactly
the type $\mathbf{III}_{g}$.

Type | Group $G$ | Compact subgroup $K$ | $\dim_{\mathbb{C}}G/K$
---|---|---|---
$\mathbf{I}_{m,n}$ | $U(m,n)$ | $U(m)\times U(n)$ | $mn$
$\mathbf{II}_{m}$ | $SO^{*}(2m)$ | $U(m)$ | $\frac{1}{2}m(m-1)$
$\mathbf{III}_{m}$ | $Sp(2m)$ | $U(m)$ | $\frac{1}{2}m(m+1)$
$\mathbf{IV}_{m}$ | $SO(m,2)$ | $SO(m)\times SO(2)$ | $m$
$\mathbf{V}$ | $E_{6,-14}$ | $SO(10)\times SO(2)$ | $16$
$\mathbf{VI}$ | $E_{7,-25}$ | $E_{6}\times U(1)$ | $27$
Table 10:  Irreducible hermitian symmetric manifolds of noncompact type and
their complex dimension Ding:2020zxw .

##  7 Fermion mass hierarchies and texture zeros from modular symmetry

The modular symmetry is quite predictive by minimizing the symmetry breaking
sector, nevertheless it does not yet provide a convincing explanation of the
charged lepton masses. Usually the mass hierarchy is achieved by hand by
introducing one parameter for each charged lepton species. The values of the
free parameters are adjusted to reproduce the charged lepton masses, which are
not predicted, but just fitted as in the Standard Model. In this section, we
shall review how the fermion mass hierarchies can be naturally reproduced in
the framework of modular symmetry.

###  7.1 Weighton mechanism

Froggatt-Nielsen (FN) mechanism is a well-known approach to understand the
quark and lepton mass hierarchies Froggatt:1978nt . The FN mechanism assumes
an additional $U(1)_{FN}$ flavor symmetry under which the quarks and leptons
carry various charges. The $U(1)_{FN}$ flavor symmetry is spontaneously broken
by the vacuum expectation value (VEV) of a new “flavon” field $\theta$, where
$\theta$ is a neutral scalar under the SM but carries one unit of $U(1)_{FN}$
charge. In the FN mechanism, the fermion Yukawa couplings (except the top
quark Yukawa coupling) are forbidden at the renormalisable level due to the
$U(1)_{FN}$ symmetry. The small effective Yukawa couplings then originate from
non-renormalisable contact operators where the FN charges of fermions are
compensated by powers of $\theta$, leading to suppression by powers of the
small ratio $\langle\theta\rangle/M_{FN}$, where $\langle\theta\rangle$ and
$M$ denote the VEV of $\theta$ and the cut-off scale of the contact
interaction respectively.

The fermion mass hierarchies can be naturally reproduced in the framework of
modular symmetry through the so called weighton mechanism King:2020qaj . The
mechanism is analogous to the FN mechanism, but without requiring any Abelian
symmetry to be introduced, nor any SM singlet flavon to break it. The modular
weights of fermion fields play the role of FN charges, and a SM singlet field
$\phi$ with $k_{\phi}=1$ (i.e. weight $-1$). The field $\phi$ plays the role
of a flavon and it is called a “weighton”. This mechanism can be illustrated
with the benchmark modular invariant model for leptons in Feruglio:2017spp .
The representation and weight assignments matter fields are assumed to be
Feruglio:2017spp :

$L\sim\left(\mathbf{3},1\right)\,,~{}~{}e^{c}\sim\left(\mathbf{1},1\right)\,,~{}~{}\mu^{c}\sim\left(\mathbf{1}^{\prime\prime},1\right)\,,~{}~{}\tau^{c}\sim\left(\mathbf{1}^{\prime},1\right)\,,~{}~{}N^{c}\sim\left(\mathbf{3},1\right)\,,~{}~{}H_{u,d}\sim\left(\mathbf{1},0\right)\,,$
(324)

where the first numbers in the parentheses denote the transformation under
$A_{4}$ modular symmetry and the second numbers are the modular weight. Thus
the superpotential for the charged lepton Yukawa couplings read as

$\displaystyle\mathcal{W}_{e}$ $\displaystyle=$ $\displaystyle\alpha
e^{c}(LY^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{d}+\beta\mu^{c}(LY^{(2)}_{\mathbf{3}})_{\mathbf{1}^{\prime}}H_{d}+\gamma\tau^{c}(LY^{(2)}_{\mathbf{3}})_{\mathbf{1}^{\prime\prime}}H_{d}$
(325) $\displaystyle=$ $\displaystyle\alpha
e^{c}(L_{1}Y_{1}+L_{2}Y_{3}+L_{3}Y_{2})H_{d}+\beta\mu^{c}(L_{3}Y_{3}+L_{1}Y_{2}+L_{2}Y_{1})H_{d}$
$\displaystyle~{}+\gamma\tau^{c}(L_{2}Y_{2}+L_{3}Y_{1}+L_{1}Y_{3})H_{d}\,,$

where $Y_{1,2,3}(\tau)$ are the three independent level 3 and weight 2 modular
forms given in Eq. (132). Consequently the charged lepton Yukawa coupling
matrix takes the following form:

$\mathcal{Y}_{e}=\begin{pmatrix}\alpha\,Y_{1}~{}&~{}\alpha\,Y_{3}~{}&~{}\alpha\,Y_{2}\\\
\beta\,Y_{2}~{}&~{}\beta\,Y_{1}~{}&~{}\beta\,Y_{3}\\\
\gamma\,Y_{3}~{}&~{}\gamma\,Y_{2}~{}&~{}\gamma\,Y_{1}\end{pmatrix}\,.$ (326)

For $\tau=i\infty$ the modular triplet has a simple vaccum alignment

$\left(Y_{1},Y_{2},Y_{3}\right)|_{\tau=i\infty}=\left(1,0,0\right)\,,$ (327)

which can be seen from Eq. (133). Hence the charged lepton Yukawa is diagonal
in the limit $\tau=i\infty$,

$\mathcal{Y}_{e}|_{\tau=i\infty}=\begin{pmatrix}\alpha~{}&~{}0~{}&~{}0\\\
0~{}&~{}\beta~{}&~{}0\\\ 0~{}&~{}0~{}&~{}\gamma\end{pmatrix}\,,$ (328)

which implies

$m_{e}:m_{\mu}:m_{\tau}=\alpha:\beta:\gamma\,.$ (329)

Hierarchial values of the three couplings $\alpha$, $\beta$, $\gamma$ are
necessary to accommodate the observed charged lepton masses. In detailed
numerical analysis, the best fit values of modulus and the free couplings are
found to be Ding:2019zxk

$\langle\tau\rangle=0.0386+2.23i\,,~{}~{}\beta/\alpha=207.908\,,~{}~{}\gamma/\alpha=3673.38\,,$
(330)

which is in agreement with the estimate of Eq. (329).

This model can be recast in natural form by introducing a single weighton
$\phi$ which is a SM and $A_{4}$ singlet with $k_{\phi}=1$. The representation
assignments of lepton fields are kept intact while their modular weights are
changed as follows King:2020qaj ,

$\displaystyle
L\sim\left(\mathbf{3},1\right)\,,~{}~{}e^{c}\sim\left(\mathbf{1},-3\right)\,,~{}~{}\mu^{c}\sim\left(\mathbf{1}^{\prime\prime},-1\right)\,,~{}~{}\tau^{c}\sim\left(\mathbf{1}^{\prime},0\right)\,,$
$\displaystyle
N^{c}\sim\left(\mathbf{3},1\right)\,,~{}~{}H_{u,d}\sim\left(\mathbf{1},0\right)\,,~{}~{}\phi\sim\left(\mathbf{1},1\right)\,.$
(331)

Then the superpotential for the charged lepton masses takes the form

$\displaystyle\mathcal{W}_{e}$ $\displaystyle=$
$\displaystyle\alpha_{e}\left(\frac{\phi}{\Lambda}\right)^{4}e^{c}(LY^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{d}+\beta_{e}\left(\frac{\phi}{\Lambda}\right)^{2}\mu^{c}(LY^{(2)}_{\mathbf{3}})_{\mathbf{1}^{\prime}}H_{d}+\gamma_{e}\frac{\phi}{\Lambda}\tau^{c}(LY^{(2)}_{\mathbf{3}})_{\mathbf{1}^{\prime\prime}}H_{d}\,,$
(332)

which gives rise to a charged lepton Yukawa coupling matrix similar to Eq.
(326), except that it involves powers of $\phi$ controlling the hierarchies,

$\mathcal{Y}_{e}=\begin{pmatrix}\alpha_{e}\tilde{\phi}^{4}\,Y_{1}~{}&~{}\alpha_{e}\tilde{\phi}^{4}\,Y_{3}~{}&~{}\alpha_{e}\tilde{\phi}^{4}\,Y_{2}\\\
\beta_{e}\tilde{\phi}^{2}Y_{2}~{}&~{}\beta_{e}\tilde{\phi}^{2}Y_{1}~{}&~{}\beta_{e}\tilde{\phi}^{2}Y_{3}\\\
\gamma_{e}\tilde{\phi}Y_{3}~{}&~{}\gamma_{e}\tilde{\phi}Y_{2}~{}&~{}\gamma_{e}\tilde{\phi}Y_{1}\end{pmatrix},~{}~{}~{}\tilde{\phi}\equiv\frac{\langle\phi\rangle}{\Lambda}\,,$
(333)

where $\langle\phi\rangle$ denote the VEV of the weighton $\phi$ and $\Lambda$
is a dimensionful cut-off flavor scale. Similar to the usual driving field
mechanism familiar from flavon models Altarelli:2005yx , the weighton may be
driven by a leading order superpotential term242424A $U(1)_{R}$ symmetry in
necessary in the driving field mechanism, the driving superfield $\chi$ carry
two unit $R$ charges, the weighton $\phi$ and Higgs superfields have $R=0$ and
the matter superfields have $R=1$.

$\mathcal{W}_{driv}=\chi\left(Y^{(4)}_{\mathbf{1}}\frac{\phi^{4}}{M_{fl}^{2}}-M^{2}\right),$
(334)

where $\chi$ is an $A_{4}$ singlet driving superfield with zero modular
weight, while $M$ is a free dimensionful mass scale. The F-flatness condition
$F_{\chi}=\frac{\partial W_{driv}}{\partial\chi}=0$ applied to Eq. (334) then
drives a weighton vev,
$\langle\phi\rangle=(M^{2}M^{2}_{fl}/Y^{(4)}_{\mathbf{1}})^{1/4}$. In the
limit $\tau\rightarrow i\infty$, the above Yukawa coupling matrix
$\mathcal{Y}_{e}$ is diagonal and the charged lepton mass ratios are

$m_{e}:m_{\mu}:m_{\tau}=\alpha_{e}\tilde{\phi}^{4}:\beta_{e}\tilde{\phi}^{2}:\gamma_{e}\tilde{\phi}\,.$
(335)

It is remarkable that the electron, muon and tau masses are suppressed by
$\tilde{\phi}^{4}$, $\tilde{\phi}^{2}$ and $\tilde{\phi}$ respectively.
Assuming order one coefficients
$\alpha_{e},\beta_{e},\gamma_{e}\sim\mathcal{O}(1)$, the charged lepton mass
hierarchies can be naturally produced for the small parameter
$\tilde{\phi}\approx 1/15$. Furthermore, there will be additional terms
corresponding to higher weight modular forms, $Y^{(4)}_{\mathbf{3}}$,
compensated by extra powers of weighton fields $\phi$, which will give
corrections to the charged lepton superpotential King:2020qaj ,

$\Delta\mathcal{W}_{e}=\alpha^{\prime}_{e}\left(\frac{\phi}{\Lambda}\right)^{6}e^{c}(LY^{(4)}_{\mathbf{3}})_{\mathbf{1}}H_{d}+\beta^{\prime}_{e}\mu^{c}\left(\frac{\phi}{\Lambda}\right)^{4}(LY^{(4)}_{\mathbf{3}})_{\mathbf{1}^{\prime}}H_{d}+\gamma^{\prime}_{e}\tau^{c}\left(\frac{\phi}{\Lambda}\right)^{3}(LY^{(4)}_{\mathbf{3}})_{\mathbf{1}^{\prime\prime}}H_{d}\,,$
(336)

which are suppressed by $\tilde{\phi}^{2}$ with respect to the leading
superpotential in Eq. (333). The weighton mechanism can also be applied to
explain the quark mass hierarchies King:2020qaj ; Kuranaga:2021ujd .

###  7.2 Mass hierarchies, large lepton mixing and residual modular symmetry

As shown in section 3.3, the modular group $SL(2,\mathds{Z})$ only has three
independent fixed points $\tau_{S}=i$, $\tau_{ST}=e^{2\pi i/3}=\omega$ and
$\tau_{T}=i\infty$ which are left invariant under the action of $S$, $ST$ and
$T$. Since certain residual symmetry listed in table 3 is preserved at the
fixed points, the modular forms has definite alignments at the modular fixed
points. For example, the weight 2 and level 3 modular forms $Y_{1,2,3}(\tau)$
in Eq. (133) in the basis of Eq. (131) have the following alignments at fixed
points:

$\displaystyle\left(Y_{1},Y_{2},Y_{3}\right)\Big{|}_{\tau=i\infty}$
$\displaystyle=$ $\displaystyle\left(1,0,0\right)\,,$
$\displaystyle\left(Y_{1},Y_{2},Y_{3}\right)\Big{|}_{\tau=i}$ $\displaystyle=$
$\displaystyle Y_{1}(i)\left(1,1-\sqrt{3},-2+\sqrt{3}\right)\,,$
$\displaystyle\left(Y_{1},Y_{2},Y_{3}\right)\Big{|}_{\tau=\omega}$
$\displaystyle=$ $\displaystyle
Y_{1}(\omega)\left(1,\omega,-\omega^{2}/2\right)\,,$ (337)

with $Y_{1}(i)\simeq 1.023$ and $Y_{1}(\omega)\simeq 0.949$. As a consequence,
the residual modular symmetry group would enforce some entries of the fermion
mass matrix to be vanishing in proper representation basis of the finite
modular group. As the value of the modulus $\tau$ moves away from the fixed
points, these entries will generically become non-zero. The magnitudes of such
residual-symmetry-breaking entries will be controlled by the size of the
departure of $\tau$ from the fixed points and by the field transformation
properties under the residual symmetry group Okada:2020ukr ; Feruglio:2021dte
; Novichkov:2021evw ; Feruglio:2022kea ; Feruglio:2023mii . This is shown in
what follows.

After electroweak symmetry breaking, the fermion mass term can be generally
written as252525For Majorana fermions, $\psi$ and $\psi^{c}$ are identical and
this implies $k^{c}=k$ and $\rho^{c}=\rho$.

$\mathcal{L}_{m}=\psi^{c}_{i}M_{ij}(\tau)\psi_{j}\,.$ (338)

The matter superfields $\psi^{(c)}_{i}$ can be arranged into a column vector
$\psi^{(c)}$ transforming under the finite modular symmetry

$\psi\xrightarrow{\gamma}\,(c\tau+d)^{-k}\rho(\gamma)\,\psi\,,~{}~{}~{}\psi^{c}\,\xrightarrow{\gamma}\,(c\tau+d)^{-k^{c}}\rho^{c}(\gamma)\,\psi^{c}\,,$
(339)

where $\rho(\gamma)$ and $\rho^{c}(\gamma)$ are representations of the finite
modular group $\Gamma_{N}$ or $\Gamma^{\prime}_{N}$. Hence each entry
$M(\tau)_{ij}$ is a modular form of level $N$ and weight $K\equiv k+k^{c}$,
and modular invariance requires the mass matrix $M(\tau)$ to transform as

$M(\tau)\,\xrightarrow{\gamma}\,M(\gamma\tau)=(c\tau+d)^{K}\rho^{c}(\gamma)^{*}M(\tau)\rho(\gamma)^{\dagger}\,.$
(340)

One can use this transformation rule to constrain the form of the mass matrix
$M(\tau)$ by taking $\tau$ to be close to the modular fixed points and setting
$\gamma$ to the residual symmetry generator.

  * •

$\tau\rightarrow i\infty$

In modular symmetry models with $\tau$ in the vicinity of $i\infty$, it is
convenient to choose the $T$-diagonal basis
$\rho^{(c)}(T)=\text{diag}(\rho^{(c)}_{i})$. The multiplication rule $T^{N}=1$
in Eq. (83, 85) implies generally $(\rho^{c}_{i}\rho_{j})^{*}=\zeta^{p_{ij}}$
with $\zeta=e^{2\pi i/N}$ and $0\leq p_{ij}<N$. Then Eq. (340) implies that
the entries $M_{ij}$ can be expanded as Novichkov:2021evw :

$M_{ij}(\tau)=a_{0}\,q^{p_{ij}}_{N}+a_{1}\,q^{p_{ij}+N}_{N}+a_{2}\,q^{p_{ij}+2N}+\ldots\,,~{}~{}~{}q_{N}\equiv\exp\left(2\pi
i\tau/N\right)~{}~{}q_{N}=e^{2\pi i\tau/N}$ (341)

in the vicinity of $i\infty$. Hence $M_{ij}$ is of order
$\mathcal{O}(\epsilon^{p_{ij}})$ with
$\epsilon=|q_{N}|=e^{-2\pi\texttt{Im}(\tau)/N}$.

  * •

$\tau\rightarrow\omega$

For the analysis of models where $\tau$ is in the vicinity of
$\tau_{ST}=\omega$, it is convenient to switch to the basis where the product
$ST$ is represented by a diagonal matrix, i.e.,
$\rho^{(c)}(ST)=\text{diag}(\rho^{(c)}_{i})$, where the diagonal elements are
power of $\omega$ because of $(ST)^{3}=1$. Depending on the transformation
matrices $\rho^{(c)}(ST)$ and the modular weights $k$ and $k^{c}$, it turns
out that the entries $M_{ij}\sim\mathcal{O}(\epsilon^{l_{ij}})$ for
$\omega^{K}\rho^{c}_{i}\rho_{j}=\omega^{l_{ij}}$ with $l_{ij}=0,1,2$, where
$\epsilon=\left|\frac{\tau-\omega}{\tau-\omega^{2}}\right|$ Novichkov:2021evw
.

  * •

$\tau\rightarrow i$

In this case, it is convenient to switch to the basis where the modular
generator $S$ is represented by a diagonal matrix with
$\rho^{(c)}(S)=\text{diag}(\rho^{(c)}_{i})$, where the diagonal elements are
powers of $i$ because of $S^{4}=1$. It is found that the entries
$M_{ij}\sim\mathcal{O}(\epsilon^{n_{ij}})$ for
$i^{K}\rho^{c}_{i}\rho_{j}=(-1)^{n_{ij}}$ with $n_{ij}=0,1$, where
$\epsilon=\left|\frac{\tau-i}{\tau+i}\right|$ Novichkov:2021evw .

In short, each entry $M_{ij}$ the fermion mass matrix is of order
$\mathcal{O}(\epsilon^{l})$, where $\epsilon$ parameterises the deviation of
$\tau$ from the fixed points, and the power $l$ can be extracted from products
of factors which correspond to representations of the residual symmetry group.
It is notable that the value of $l$ only depends on the representations of the
matter fields. The fermion mass matrices in the vicinity of modular fixed
points have been analyzed for the homogeneous finite modular groups
$\Gamma^{\prime}_{N}$ of levels $N\leq 5$, it is found that hierarchical
spectra can be obtained for a small list of representation pairs
Novichkov:2021evw , the most promising of which are collected in table 11.
Here the coupling constants in modular flavor models are assumed to be of the
same order of magnitude, and the modular forms should be properly normalized
Petcov:2023fwh . Furthermore, it is claimed that the modular invariant models
exhibit a universal behavior around the fixed points, and the scaling of
fermion masses and mixing parameters is independent of the details of the
theory Feruglio:2022kea ; Feruglio:2023mii .

$N$ |  $\Gamma^{\prime}_{N}\;(\text{or}~{}\Gamma_{N})$ |  Mass ratios |  Modulus $\tau$ |  Viable $\mathbf{r}\otimes\mathbf{r}^{c}$
---|---|---|---|---
2 |  $S_{3}$ |  $(1,\epsilon,\epsilon^{2})$ | $\tau\simeq\omega$ | $[\mathbf{2}\oplus\mathbf{1}^{(\prime)}]\otimes[\mathbf{1}\oplus\mathbf{1}^{(\prime)}\oplus\mathbf{1}^{\prime}]$
3 |  $\Gamma^{\prime}_{3}=T^{\prime}\;(~{}\text{or}~{}A_{4})$ |  $(1,\epsilon,\epsilon^{2})$ | $\tau\simeq\omega$ | $[\mathbf{1}_{a}\oplus\mathbf{1}_{a}\oplus\mathbf{1}_{a}^{\prime}]\otimes[\mathbf{1}_{b}\oplus\mathbf{1}_{b}\oplus\mathbf{1}_{b}^{\prime\prime}]$
$\tau\simeq i\infty$ | $[\mathbf{1}_{a}\oplus\mathbf{1}_{a}\oplus\mathbf{1}_{a}^{\prime}]\otimes[\mathbf{1}_{b}\oplus\mathbf{1}_{b}\oplus\mathbf{1}_{b}^{\prime\prime}]$
$\text{with}~{}\mathbf{1}_{a}\neq(\mathbf{1}_{b})^{*}$
4 |  $S_{4}^{\prime}\;(\;\text{or}~{}S_{4})$ |  $(1,\epsilon,\epsilon^{2})$ | $\tau\simeq\omega$ | $[\mathbf{3}_{a}\text{, or }\mathbf{2}\oplus\mathbf{1}^{(\prime)}\text{, or }\mathbf{\hat{2}}\oplus\mathbf{\hat{1}}^{(\prime)}]$
$\otimes[\mathbf{1}_{b}\oplus\mathbf{1}_{b}\oplus\mathbf{1}_{b}^{\prime}]$
$(1,\epsilon,\epsilon^{3})$ | $\tau\simeq i\infty$ |  $\mathbf{3}\hphantom{{}^{\prime}}\otimes[\mathbf{2}\oplus\mathbf{1}\text{, or }\mathbf{1}\oplus\mathbf{1}\oplus\mathbf{1}^{\prime}]$,
$\mathbf{3}^{\prime}\otimes[\mathbf{2}\oplus\mathbf{1}^{\prime}\text{, or
}\mathbf{1}\oplus\mathbf{1}^{\prime}\oplus\mathbf{1}^{\prime}],$
$\mathbf{\hat{3}}^{\prime}\otimes[\mathbf{\hat{2}}\oplus\mathbf{\hat{1}}\text{,
or }\mathbf{\hat{1}}\oplus\mathbf{\hat{1}}\oplus\mathbf{\hat{1}}^{\prime}],$
$\mathbf{\hat{3}}\hphantom{{}^{\prime}}\otimes[\mathbf{\hat{2}}\oplus\mathbf{\hat{1}^{\prime}}\text{,
or
}\mathbf{\hat{1}}\oplus\mathbf{\hat{1}}^{\prime}\oplus\mathbf{\hat{1}}^{\prime}]$
5 |  $A_{5}^{\prime}\;(\;\text{or}~{}A_{5})$ |  $(1,\epsilon,\epsilon^{4})$ | $\tau\simeq i\infty$ | $\mathbf{3}\otimes\mathbf{3}^{\prime}$
Table 11:  Hierarchical mass patterns which can be realized in the vicinity of
symmetric points Novichkov:2021evw . These patterns are unaffected by the
exchange $\mathbf{r}\leftrightarrow\mathbf{r}^{c}$ and may only be viable for
certain weights, where $\mathbf{r}$ and $\mathbf{r}^{c}$ are representation of
$\psi$ and $\psi^{c}$ respectively. Subscripts run over irreps of a certain
dimension, and $\mathbf{1}_{a}^{\prime\prime\prime}=\mathbf{1}_{a}$ for $N=3$,
while $\mathbf{1}_{a}^{\prime\prime}=\mathbf{1}_{a}$ for $N=4$. Primes in
parenthesis are uncorrelated. Notice the representations without a hat have a
direct correspondence with $\Gamma_{N}$ representations, while the hatted
representations are specific to $\Gamma^{\prime}_{N}$.

The three generations of left-handed lepton fields are usually assigned to a
triplet of flavor symmetry in order to produce the large lepton mixing angles.
In the symmetry limit with $\tau$ at the fixed points, the charged lepton and
neutrino sectors would share the same residual symmetry so that the lepton
mixing matrix would be block diagonal Ding:2019gof  if the lepton masses are
non-degenerate. Hence the triplet assignment of left-handed leptons is
disfavored in the vicinity of modular fixed points. There are only four cases
in which the deviation of $\tau$ from the fixed point can generate
hierarchical charged-lepton masses together with the viable lepton mixing
Reyimuaji:2018xvs ; Novichkov:2021evw ,

  1. (i)

$L\leadsto 1\oplus 1\oplus 1$, $E^{c}\leadsto 1\oplus r$, where $1$ is some
real singlet of the flavour symmetry, and $r$ is some (possibly reducible)
representation such that $r\not\supset 1$;

  2. (ii)

$L\leadsto\mathbf{1}\oplus\mathbf{1}\oplus\mathbf{1}^{*}$,
$E^{c}\leadsto\mathbf{1}^{*}\oplus r$, where $\mathbf{1}$ is some complex
singlet of the flavour symmetry, $\mathbf{1}^{*}$ is its conjugate, and $r$ is
some (possibly reducible) representation such that
$r\not\supset\mathbf{1},\mathbf{1}^{*}$;

  3. (iii)

all charged-lepton masses vanish in the symmetric limit, i.e. the
corresponding hierarchical pattern involves only positive powers of
$\epsilon$, e.g. $(\epsilon,\epsilon^{2},\epsilon^{3})$. The small tau lepton
mass in comparison to the electroweak scale could be explained in this case,
and the three left-handed leptons can be a triplet of the modular group;

  4. (iv)

all light neutrino masses vanish in the symmetric limit, i.e. $L$ decomposes
into three (possibly identical) complex singlets none of which are conjugated
to each other.

Motivated by the model independent and universal behaviors around the modular
fixed points, a concrete model with the $S^{\prime}_{4}$ modular symmetry has
been constructed, and the light neutrino masses are generated by the type-I
seesaw mechanism Novichkov:2021evw . The modular transformations of the lepton
fields are assumed to be

$L_{1},L_{2}\sim(\mathbf{\hat{1}},2),\quad
L_{3}\sim(\mathbf{\hat{1}^{\prime}},2),\quad
E^{c}\sim(\mathbf{\hat{3}},4),\quad N^{c}\sim(\mathbf{3}^{\prime},1)\,,$ (342)

where the first numbers in the parenthesis denote the transformations under
$S^{\prime}_{4}$ and the second numbers are the modular weights. The
corresponding superpotential for lepton mass reads Novichkov:2021evw :

$\displaystyle\mathcal{W}$
$\displaystyle=\left[\alpha_{1}\left(Y^{(4,6)}_{\mathbf{3^{\prime}},1}E^{c}L_{1}\right)_{\mathbf{1}}+\alpha_{2}\left(Y^{(4,6)}_{\mathbf{3^{\prime}},2}E^{c}L_{1}\right)_{\mathbf{1}}\right.$
(343)
$\displaystyle+\left.\alpha_{3}\left(Y^{(4,6)}_{\mathbf{3^{\prime}},1}E^{c}L_{2}\right)_{\mathbf{1}}+\alpha_{4}\left(Y^{(4,6)}_{\mathbf{3^{\prime}},2}E^{c}L_{2}\right)_{\mathbf{1}}+\alpha_{5}\left(Y^{(4,6)}_{\mathbf{3}}E^{c}L_{3}\right)_{\mathbf{1}}\right]H_{d}$
$\displaystyle+\left[g_{1}\left(Y^{(4,3)}_{\mathbf{\hat{3}}}N^{c}L_{1}\right)_{\mathbf{1}}+g_{2}\left(Y^{(4,3)}_{\mathbf{\hat{3}}}N^{c}L_{2}\right)_{\mathbf{1}}+g_{3}\left(Y^{(4,3)}_{\mathbf{\hat{3}^{\prime}}}N^{c}L_{3}\right)_{\mathbf{1}}\right]H_{u}$
$\displaystyle+\Lambda\left(Y^{(4,2)}_{\mathbf{2}}(N^{c})^{2}\right)_{\mathbf{1}}\,.$

Since $L_{1}$ and $L_{2}$ are indistinguishable, one of the constants
$\alpha_{i}$, with $i=1,\dots,4$, are effectively not an independent parameter
and can be set to zero by a suitable rotation without loss of generality. We
choose to set $\alpha_{2}=0$. All the coupling constants are enforced to be
real by the gCP symmetry, and consequently the model depends on 9 effective
parameters including $\text{Re}\,\tau$ and $\text{Im}\,\tau$. The best fit
value of the modulus is found to be $\tau\simeq-0.496+0.877i$ which is close
to the fixed point $\omega$, the charged lepton mass hierarchies are produced
without fine-tuning Novichkov:2021evw , and the leptonic Dirac CP violation
phase is predicted to be around $\pi$. See Novichkov:2021evw  for more
numerical results. The closeness of the modulus to the modular symmetry fixed
points can also help to explain the quark mass hierarchies Petcov:2023vws ;
deMedeirosVarzielas:2023crv . Furthermore, it is shown that the modular
symmetry models exhibit a universal behavior in the vicinity of fixed points,
independent of details of the models such as the modular weights and
representation assignments of matter field under the finite modular groups
Feruglio:2022koo ; Feruglio:2023mii .

###  7.3 Texture zeros and modular symmetry

Texture zero is an interesting attempt to understand the patterns of fermion
masses and flavor mixing, it assumes that some entries of the fermion mass
matrices are vanishing in order to reduce the number of free parameters
Fritzsch:1977za ; Weinberg:1977hb ; Wilczek:1977uh . A typical example is the
Fritzsch-type quark mass matricesFritzsch:1977vd ; Fritzsch:1979zq :

$M_{u}=\begin{pmatrix}0~{}&A_{u}~{}&0\\\ A^{*}_{u}~{}&0~{}&B_{u}\\\
0~{}&B^{*}_{u}~{}&C_{u}\end{pmatrix},~{}~{}~{}~{}M_{d}=\begin{pmatrix}0~{}&A_{d}~{}&0\\\
A^{*}_{d}~{}&0~{}&B_{d}\\\ 0~{}&B^{*}_{d}~{}&C_{d}\end{pmatrix}\,.$ (344)

It relates the Cabibbo angle $\theta_{\rm C}$ to the ratio between the down
and strange quark masses via

$\tan\theta_{\rm C}\approx\sqrt{\frac{m_{d}}{m_{s}}}\,,$ (345)

which is the Gatto-Sartori relation Gatto:1968ss . As regard the texture zero
of lepton sector, in the basis of diagonal charged lepton mass matrix, the
Majorana neutrino mass matrix can have at most two zero entries and only seven
out of the total fifteen two-zero textures are compatible with current
experimental data at $3\sigma$ level Frampton:2002yf ; Xing:2002ta ;
Fritzsch:2011qv ; Meloni:2014yea ,

$\displaystyle\mathbf{A}_{1}:\begin{pmatrix}0&0&\times\\\ 0&\times&\times\\\
\times&\times&\times\end{pmatrix},~{}~{}~{}\mathbf{A}_{2}:\begin{pmatrix}0&\times&0\\\
\times&\times&\times\\\
0&\times&\times\end{pmatrix},~{}~{}~{}\mathbf{B}_{1}:\begin{pmatrix}\times&\times&0\\\
\times&0&\times\\\
0&\times&\times\end{pmatrix},~{}~{}~{}\mathbf{B}_{2}:\begin{pmatrix}\times&0&\times\\\
0&\times&\times\\\ \times&\times&0\end{pmatrix}\,,$
$\displaystyle\mathbf{B}_{3}:\begin{pmatrix}\times&0&\times\\\ 0&0&\times\\\
\times&\times&\times\end{pmatrix},~{}~{}~{}\mathbf{B}_{4}:\begin{pmatrix}\times&\times&0\\\
\times&\times&\times\\\
0&\times&0\end{pmatrix},~{}~{}~{}\mathbf{C}:\begin{pmatrix}\times&\times&\times\\\
\times&0&\times\\\ \times&\times&0\end{pmatrix}\,,$ (346)

where $\times$ denotes an arbitrary complex number. The phenomenology of
texture zeros in both quark and lepton sectors have been widely studied in
literature, see Refs. Fritzsch:1999ee ; Gupta:2012fsl  for reviews.
Systematical and complete analyses of all possible texture zeros have been
carried out for both quark Ludl:2015lta  and lepton mass matrices Ludl:2014axa
. It is found that the predictivity of pure texture zero models is quite weak
and the predictive mass matrices need relations among the non-zero matrix
elements. The most straightforward way to impose vanishing Yukawa couplings is
by enforcing them with Abelian flavour symmetries Grimus:2004hf ;
GonzalezFelipe:2014zjk ; Camara:2020efq . In such case, the non-zero entries
of the fermion mass matrices are uncorrelated since the three generations of
matter fields are not linked by the Abelian symmetry group.

One can freely assign the modular weights of matter fields in the bottom-up
approach of modular flavor symmetry, thus the modular symmetry has the merits
of both abelian flavor symmetry and discrete non-abelian flavor symmetry.
Moreover, it is known that modular forms in certain representations of
$\Gamma^{\prime}_{N}$ (or $\Gamma_{N}$) are absent at lower weights. In
addition, the even weight modular forms at level $N$ are in the irreducible
representations of $\Gamma_{N}$ Feruglio:2017spp  while the odd weight modular
forms are in the irreducible representations specific to $\Gamma^{\prime}_{N}$
Liu:2019khw . The structure of modular forms makes the modular symmetry to be
a natural framework producing texture zeros of fermion mass matrices.
Moreover, the nonzero entries are related by modular symmetry so that modular
symmetry can considerably improve the predictive power of texture zero. The
possible texture zeros of quark and lepton mass matrices have been
systematically analyzed in the context of $T^{\prime}$ modular symmetry
Lu:2019vgm ; Ding:2022aoe . The possible texture zero patterns of the charged
fermions mass matrices and neutrino mass matrix in the $T^{\prime}$ modular
symmetry are summarized in table 12 and table 13 respectively.

It was shown that the two-zero textures of the Majorana neutrino mass matrix
can be reproduced from the $A_{4}$ modular group Zhang:2019ngf , where all
matter fields were assigned to $A_{4}$ singlets and thus correlations among
nonzero entries of lepton mass matrices are lost. Moreover, the vanishing
entries of fermion mass matrices can naturally appear Ding:2019gof  if the
modulus $\tau$ is at certain modular fixed point listed in table 3. The
nearest neighbor interaction form of the quark mass matrices can be obtained
at the fixed point $\tau=\omega$ in the $A_{4}$ modular flavor symmetry by
taking account multi-Higgs fields Kikuchi:2022svo .

$\mathcal{C}_{1}^{(1)}:\left(\begin{array}[]{ccc}0&\times&\times\\\
\times&\times&\times\\\ \times&\times&\times\end{array}\right)$
---
$\mathcal{C}_{2}^{(1)}:\left(\begin{array}[]{ccc}0&\times&\times\\\
\times&0&\times\\\
\times&\times&\times\end{array}\right)\,,~{}~{}\mathcal{C}_{2}^{(2)}:\left(\begin{array}[]{ccc}\times&\times&\times\\\
\times&\times&\times\\\
0&0&\times\end{array}\right)\,,~{}~{}\mathcal{C}_{2}^{(3)}:\left(\begin{array}[]{ccc}\times&\times&0\\\
\times&\times&0\\\ \times&\times&\times\end{array}\right)$
$\mathcal{C}_{3}^{(1)}:\left(\begin{array}[]{ccc}\times&\times&0\\\
\times&\times&\times\\\
0&0&\times\end{array}\right)\,,~{}~{}\mathcal{C}_{3}^{(2)}:\left(\begin{array}[]{ccc}\times&0&\times\\\
\times&0&\times\\\
0&\times&\times\end{array}\right)\,,~{}~{}\mathcal{C}_{3}^{(3)}:\left(\begin{array}[]{ccc}0&\times&\times\\\
\times&0&\times\\\ \times&\times&0\end{array}\right)$
$\mathcal{C}_{4}^{(1)}:\left(\begin{array}[]{ccc}\times&\times&0\\\
\times&\times&0\\\
0&0&\times\end{array}\right)\,,~{}~{}\mathcal{C}_{4}^{(2)}:\left(\begin{array}[]{ccc}0&\times&0\\\
\times&0&0\\\
\times&\times&\times\end{array}\right)\,,~{}~{}\mathcal{C}_{4}^{(3)}:\left(\begin{array}[]{ccc}0&\times&\times\\\
\times&0&\times\\\ 0&0&\times\end{array}\right)$
$\mathcal{C}_{6}^{(1)}:\left(\begin{array}[]{ccc}0&\times&0\\\ \times&0&0\\\
0&0&\times\end{array}\right)$
Table 12:  Texture-zero classification for the (rank-3) charged fermions mass
matrices which can be realised from $T^{\prime}$ modular symmetry, up to row
and column permutations. In the notation of Grimus et al Ludl:2014axa ,
$\mathcal{C}_{3}^{(1)}=3_{1}^{(l)}$, $\mathcal{C}_{3}^{(2)}=3_{2}^{(l)}$,
$\mathcal{C}_{3}^{(3)}=3_{3}^{(l)}$, $\mathcal{C}_{4}^{(1)}=4_{4}^{(l)}$,
$\mathcal{C}_{4}^{(2)}=4_{1}^{(l)}$, $\mathcal{C}_{4}^{(3)}=4_{2}^{(l)}$,
$\mathcal{C}_{6}^{(1)}=6_{1}^{(l)}$.
$\mathcal{W}_{1}^{(1)}:\left(\begin{array}[]{ccc}\times&\times&\times\\\
\times&\times&\times\\\
\times&\times&0\end{array}\right)\,,\quad\mathcal{W}_{1}^{(2)}:\left(\begin{array}[]{ccc}\times&0&\times\\\
0&\times&\times\\\ \times&\times&\times\end{array}\right)$
---
$\mathcal{W}_{2}^{(1)}:\left(\begin{array}[]{ccc}0&\times&\times\\\
\times&0&\times\\\
\times&\times&\times\end{array}\right)\,,\quad\mathcal{W}_{2}^{(2)}:\left(\begin{array}[]{ccc}\times&\times&0\\\
\times&\times&0\\\ 0&0&\times\end{array}\right)$
$\mathcal{W}_{3}^{(1)}:\left(\begin{array}[]{ccc}0&0&\times\\\ 0&0&\times\\\
\times&\times&\times\end{array}\right)\,,\quad\mathcal{W}_{3}^{(2)}:\left(\begin{array}[]{ccc}\times&\times&0\\\
\times&\times&0\\\ 0&0&0\end{array}\right)$
$\mathcal{W}_{4}^{(1)}:\left(\begin{array}[]{ccc}0&0&\times\\\ 0&0&\times\\\
\times&\times&0\end{array}\right)\,,\quad\mathcal{W}_{4}^{(2)}:\left(\begin{array}[]{ccc}0&\times&0\\\
\times&\times&0\\\
0&0&0\end{array}\right)\,,\quad\mathcal{W}_{4}^{(3)}\left(\begin{array}[]{ccc}\times&0&0\\\
0&\times&0\\\
0&0&0\end{array}\right)\,,\quad\mathcal{W}_{4}^{(4)}:\left(\begin{array}[]{ccc}\times&0&0\\\
0&0&\times\\\ 0&\times&0\end{array}\right)$
$\mathcal{W}_{5}^{(1)}:\left(\begin{array}[]{ccc}0&\times&0\\\ \times&0&0\\\
0&0&0\end{array}\right)$
Table 13: Texture-zero patterns for the Majorana neutrino mass matrix
$M_{\nu}$ which can be realised by $T^{\prime}$ modular symmetry, up to row
and column permutations. Notice that only $\mathcal{W}_{1}^{(1)}$ ,
$\mathcal{W}_{2}^{(2)}$ , $\mathcal{W}_{3}^{(1)}$ , $\mathcal{W}_{3}^{(2)}$ ,
$\mathcal{W}_{4}^{(1)}$ and $\mathcal{W}_{4}^{(4)}$ can be obtained if
neutrino masses are described by the Weinberg operator. On the other hand, all
the above textures except $\mathcal{W}_{4}^{(4)}$ can be achieved if neutrino
masses are generated via the minimal type I seesaw mechanism. The above
texture zeros of neutrino mass matrices have been discussed in Ludl:2014axa ,
see Ref. Ding:2022aoe  for the notation correspondence between ours and those
of Ludl:2014axa .

##  8 Examples of modular models

In this section we apply the formalism of the previous sections to some
examples of concrete models. We begin with some general model building
considerations and an overview of models in the current literature. We then
discuss a general class of models of leptons based on $A_{4}$ before turning
to minimal models of leptons (and quarks) based on $S^{\prime}_{4}$ with a
single modulus field. Finally we discuss a model of leptons based on three
$S_{4}$ groups with three moduli fields, the Littlest Modular Seesaw model, so
called because it is highly predictive.

###  8.1 General model building considerations and overview of models

In general, the building blocks of modular flavor symmetry models are modular
forms of level $N$ and matter supermultiplets $\varphi$ which transform in
representations $\rho_{\varphi}$ of a finite modular group $\Gamma_{N}$ (or
$\Gamma^{\prime}_{N}$) and carry certain modular weights $k_{\varphi}$. When
considering a particular model, one has to specify both $\rho_{\varphi}$ and
$k_{\varphi}$ of each matter field262626The representation $\rho_{\varphi}$
and weight $k_{\varphi}$ of matter fields can be freely assigned in bottom-up
models, while they could possibly be fixed in the top-down approach
Nilles:2020nnc ; Nilles:2020kgo ; Nilles:2020tdp ; Baur:2020jwc ;
Nilles:2020gvu ., then it is straightforward to write out the modular
invariant superpotential for fermion masses by using the modular forms of
level $N$ listed in Appendix C.

Many modular flavor symmetry models for lepton masses and mixing, based on
different choices of level, modular weights and representations of matter
fields have been proposed. Moreover, the modular symmetry has been also
extended to the quark sector to describe the flavor structure of both quarks
and leptons with a common modulus $\tau$. A summary of the modular invariant
models in the current literature is given in table 14.

In modular invariant models for leptons, the three generations of lepton
doublets are usually assigned to be an irreducible triplet of the finite
modular group $\Gamma_{N}$ or $\Gamma^{\prime}_{N}$ in order to minimize the
number of free parameters and the Kähler potential is assumed to be the
minimal form otherwise additional free parameters and sizable corrections
would be invoked Chen:2019ewa ; Lu:2019vgm . The superpotential of charged
leptons usually depends only 3 dimensionless parameters. The latter are
adjusted to reproduce the charged lepton masses. All the remaining physical
observables such as neutrino masses and lepton mixing angles and CP violation
phases are usualy determined by the VEV of modulus $\tau$ and additional two
(or three) free couplings. The gCP symmetry could enforce the couplings to be
real in certain representation basis and the predictive power of the model
would be improved further. Hence the modular invariant models could involve a
small number of free parameters, and they can predict not only the lepton
mixing but also lepton masses. Moreover, modular symmetry has been used to
address the strong CP problem Feruglio:2023uof ; Ahn:2023iqa , and it is also
exploited to new physics beyond SM Nomura:2019jxj ; Okada:2020oxh ;
Okada:2020dmb ; Zhang:2021olk ; Tanimoto:2021ehw ; Kobayashi:2021ajl ;
Kobayashi:2021uam ; Kobayashi:2021pav ; Behera:2022wco ; Kobayashi:2022jvy ;
Mishra:2023cjc ; Abe:2023ylh ; Nomura:2023kwz . The moduli in modular flavor
symmetric models may possibly play the role of inflaton Gunji:2022xig ;
Abe:2023ylh .

| $\Gamma_{N}/\Gamma^{\prime}_{N}$ | leptons alone | quarks alone | leptons $\&$ quarks
---|---|---|---|---
$N=2$ | $S_{3}$ | Kobayashi:2018vbk ; Okada:2019xqk ; Meloni:2023aru  | Kobayashi:2018wkl  | —
$N=3$ | $A_{4}$ | Feruglio:2017spp ; Kobayashi:2018vbk ; Kobayashi:2018scp ; Criado:2018thu ; Kobayashi:2018wkl ; Novichkov:2018yse ; Nomura:2019yft ; Ding:2019zxk ; Nomura:2019lnr ; Asaka:2019vev ; Ding:2019gof ; Zhang:2019ngf ; Nomura:2019xsb ; Kobayashi:2019gtp ; Wang:2019xbo ; Ding:2020yen ; Aoki:2020eqf ; Asaka:2020tmo ; Okada:2020brs ; Kashav:2021zir ; Okada:2021qdf ; deMedeirosVarzielas:2021pug ; Kang:2022psa ; Abbas:2022slb ; Devi:2023vpe ; CentellesChulia:2023zhu ; Nomura:2023usj ; Kumar:2023moh ; Kobayashi:2023qzt  | Okada:2018yrn ; Petcov:2023vws ; Petcov:2022fjf ; Kikuchi:2023jap  | Okada:2019uoy ; King:2020qaj ; Abbas:2020qzc ; Okada:2020rjb ; Yao:2020qyy ; Nomura:2021yjb
$T^{\prime}$ | Liu:2019khw ; Okada:2022kee ; Ding:2022aoe  | — | Lu:2019vgm ; Benes:2022bbg
$N=4$ | $S_{4}$ | Ding:2019gof ; Penedo:2018nmg ; Novichkov:2018ovf ; deMedeirosVarzielas:2019cyj ; Kobayashi:2019mna ; King:2019vhv ; Criado:2019tzk ; Wang:2019ovr ; Wang:2020dbp ; Nomura:2021ewm  | — | Qu:2021jdy
$S^{\prime}_{4}$ | Novichkov:2020eep ; Ding:2022nzn  | Abe:2023ilq  | Liu:2020akv ; Abe:2023qmr
$N=5$ | $A_{5}$ | Novichkov:2018nkm ; Ding:2019xna ; Criado:2019tzk ; deMedeirosVarzielas:2022ihu  | — | —
$A^{\prime}_{5}$ | Wang:2020lxk ; Wang:2021mkw ; Behera:2021eut ; Behera:2022wco  | — | Yao:2020zml
$N=6$ | $\Gamma_{6}\cong S_{3}\times A_{4}$ | — | Kikuchi:2023cap  | —
$\Gamma^{\prime}_{6}\cong S_{3}\times T^{\prime}$ | Li:2021buv  | — | —
$N=7$ | $\Gamma_{7}$ | Ding:2020msi  | — | —
$\Gamma^{\prime}_{7}$ | — | — | —

Table 14:  Summary of modular invariant models based on SM gauge group.

###  8.2 Overview of $A_{4}$ modular models

The study of modular symmetry initiated from the modular group $A_{4}$
Feruglio:2017spp , it is instructive to recapitulate the $A_{4}$ modular
invariant lepton model since it comprises all the essential features of the
modular symmetry models. The model is realized at level $N=3$, and light
neutrinos get their masses via the type I see-saw mechanism. Both the left-
handed lepton fields $L$ and the right-handed neutrino fields $N^{c}$ are
assumed to be triplet of $A_{4}$, and the three right-handed charged leptons
$e^{c}$, $\mu^{c}$ and $\tau^{c}$ are assigned to transform as $A_{4}$
singlets $\mathbf{1}$, $\mathbf{1}^{\prime\prime}$ and $\mathbf{1}^{\prime}$
respectively. All the lepton fields carry one unit of modular weight, and the
Higgs fields $H_{u,d}$ are invariant under $A_{4}$ with zero modular weight,
i.e.

$L\sim\left(\mathbf{3},1\right)\,,~{}~{}e^{c}\sim\left(\mathbf{1},1\right)\,,~{}~{}\mu^{c}\sim\left(\mathbf{1}^{\prime\prime},1\right)\,,~{}~{}\tau^{c}\sim\left(\mathbf{1}^{\prime},1\right)\,,~{}~{}N^{c}\sim\left(\mathbf{3},1\right)\,,~{}~{}H_{u,d}\sim\left(\mathbf{1},0\right)\,,$
(347)

where the first numbers in the parentheses denote the transformation under
$A_{4}$ modular symmetry and the second numbers are the modular weight. In a
standard notation, the superpotential for the charged lepton masses is given
by:

$\mathcal{W}_{e}=\alpha
e^{c}(LY^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{d}+\beta\mu^{c}(LY^{(2)}_{\mathbf{3}})_{\mathbf{1}^{\prime}}H_{d}+\gamma\tau^{c}(LY^{(2)}_{\mathbf{3}})_{\mathbf{1}^{\prime\prime}}H_{d}\,,$
(348)

where $Y^{(2)}_{\mathbf{3}}(\tau)=(Y_{1},Y_{2},Y_{3})^{T}$ denotes the
irreducible triplet of level-3 weight-2 modular forms in Eq. (132). The
couplings $\alpha$, $\beta$ and $\gamma$ can be made real through redefinition
of the fields $e^{c}$, $\mu^{c}$, $\tau^{c}$. Using the contraction rules of
two $A_{4}$ triplets in Eq. (574), we can read out the charged lepton mass
matrix as follow,

$M_{e}=\begin{pmatrix}\alpha Y_{1}(\tau)~{}&\alpha Y_{3}(\tau)~{}&\alpha
Y_{2}(\tau)\\\ \beta Y_{2}(\tau)~{}&\beta Y_{1}(\tau)~{}&\beta Y_{3}(\tau)\\\
\gamma Y_{3}(\tau)~{}&\gamma Y_{2}(\tau)~{}&\gamma
Y_{1}(\tau)\end{pmatrix}v_{d}\,.$ (349)

The superpotential relevant to neutrino masses is:

$\mathcal{W}_{\nu}=g_{1}((N^{c}\,L)_{\mathbf{3}_{S}}Y^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{u}+g_{2}((N^{c}\,L)_{\mathbf{3}_{A}}Y^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{u}+\frac{1}{2}\Lambda_{L}(\left(N^{c}N^{c}\right)_{\mathbf{3_{S}}}Y^{(2)}_{\mathbf{3}})_{\mathbf{1}}\,,$
(350)

where the phase of $g_{1}$ can be absorbed into the left-handed lepton field
$L$, while $g_{2}$ is complex. The Dirac neutrino mass matrix $m_{D}$ and
heavy Majorana neutrino mass matrix $m_{N}$ take the following form

$\displaystyle\qquad\quad~{}~{}~{}M_{N}=\begin{pmatrix}2Y_{1}(\tau)~{}&~{}-Y_{3}(\tau)~{}&~{}-Y_{2}(\tau)\\\
-Y_{3}(\tau)~{}&~{}2Y_{2}(\tau)~{}&~{}-Y_{1}(\tau)\\\
-Y_{2}(\tau)~{}&~{}-Y_{1}(\tau)~{}&~{}2Y_{3}(\tau)\end{pmatrix}\Lambda_{L}\,,$
$\displaystyle
M_{D}=\begin{pmatrix}2g_{1}Y_{1}(\tau)~{}&~{}(-g_{1}+g_{2})Y_{3}(\tau)~{}&~{}(-g_{1}-g_{2})Y_{2}(\tau)\\\
(-g_{1}-g_{2})Y_{3}(\tau)~{}&~{}2g_{1}Y_{2}(\tau)~{}&~{}(-g_{1}+g_{2})Y_{1}(\tau)\\\
(-g_{1}+g_{2})Y_{2}(\tau)~{}&~{}(-g_{1}-g_{2})Y_{1}(\tau)~{}&~{}2g_{1}Y_{3}(\tau)\end{pmatrix}v_{u}\,.$
(351)

The light neutrino mass matrix is $M_{\nu}=-M_{D}^{T}M_{N}^{-1}M_{D}$. Charged
lepton masses can be reproduced by adjusting the parameters $\alpha$, $\beta$
and $\gamma$, while neutrino masses and the lepton mass matrix depend also on
additional five parameters: one overall scale
$|g_{1}|^{2}v^{2}_{u}/\Lambda_{L}$, the complex combination $g_{2}/g_{1}$ and
the $\tau$ VEV. This model can accommodate both normal ordering and inverted
ordering neutrino mass spectrum. The best fit values of the free parameters as
well as the neutrino masses and mixing parameters at the best fit points are
shown in table 15.

Goodness of fit against NuFit 5.2 values without SK atmospheric data
---
NO | $\chi_{\text{min}}^{2}$ | ${\tt Re}(\tau)$ | ${\tt Im}(\tau)$ | $|g_{2}/g_{1}|$ | ${\tt arg}(g_{2}/g_{1})$ | $\frac{|g_{1}|^{2}v_{u}^{2}}{\Lambda_{L}}({\rm eV})$ | $\alpha v_{d}({\rm MeV})$ | $\beta/\alpha$ | $\gamma/\alpha$ |
$1.921$ | $0.049$ | $2.264$ | $1.109$ | $1.899$ | $0.049$ | $0.360$ | $210.767$ | $3581.720$ |
$\sin^{2}\theta_{12}$ | $\sin^{2}\theta_{13}$ | $\sin^{2}\theta_{23}$ | $\delta_{CP}/\pi$ | $\alpha_{21}/\pi$ | $\alpha_{31}/\pi$ | $m_{1}({\rm eV})$ | $m_{2}({\rm eV})$ | $m_{3}({\rm eV})$ | $|m_{\beta\beta}|({\rm eV})$
$0.303$ | $0.0220$ | $0.575$ | $1.416$ | $0.023$ | $1.014$ | $0.0985$ | $0.0988$ | $0.1105$ | $0.0985$
IO | $\chi_{\text{min}}^{2}$ | ${\tt Re}(\tau)$ | ${\tt Im}(\tau)$ | $|g_{2}/g_{1}|$ | ${\tt arg}(g_{2}/g_{1})$ | $\frac{|g_{1}|^{2}v_{u}^{2}}{\Lambda_{L}}({\rm eV})$ | $\alpha v_{d}({\rm MeV})$ | $\beta/\alpha$ | $\gamma/\alpha$ |
$1.129$ | $0.498$ | $1.141$ | $0.968$ | $0.022$ | $0.028$ | $68.517$ | $16.527$ | $4.826\times 10^{-3}$ |
$\sin^{2}\theta_{12}$ | $\sin^{2}\theta_{13}$ | $\sin^{2}\theta_{23}$ | $\delta_{CP}/\pi$ | $\alpha_{21}/\pi$ | $\alpha_{31}/\pi$ | $m_{1}({\rm eV})$ | $m_{2}({\rm eV})$ | $m_{3}({\rm eV})$ | $|m_{\beta\beta}|({\rm eV})$
$0.305$ | $0.0222$ | $0.575$ | $1.745$ | $0.728$ | $0.813$ | $0.0638$ | $0.0644$ | $0.0405$ | $0.0332$
Goodness of fit against NuFit 5.2 values with SK atmospheric data
NO | $\chi_{\text{min}}^{2}$ | ${\tt Re}(\tau)$ | ${\tt Im}(\tau)$ | $|g_{2}/g_{1}|$ | ${\tt arg}(g_{2}/g_{1})$ | $\frac{|g_{1}|^{2}v_{u}^{2}}{\Lambda_{L}}({\rm eV})$ | $\alpha v_{d}({\rm MeV})$ | $\beta/\alpha$ | $\gamma/\alpha$ |
$0.129$ | $0.284$ | $1.871$ | $1.123$ | $4.651$ | $0.032$ | $0.362$ | $3544.019$ | $209.654$ |
$\sin^{2}\theta_{12}$ | $\sin^{2}\theta_{13}$ | $\sin^{2}\theta_{23}$ | $\delta_{CP}/\pi$ | $\alpha_{21}/\pi$ | $\alpha_{31}/\pi$ | $m_{1}({\rm eV})$ | $m_{2}({\rm eV})$ | $m_{3}({\rm eV})$ | $|m_{\beta\beta}|({\rm eV})$
$0.303$ | $0.0222$ | $0.445$ | $1.298$ | $1.944$ | $0.933$ | $0.0635$ | $0.0641$ | $0.0809$ | $0.0629$
IO | $\chi_{\text{min}}^{2}$ | ${\tt Re}(\tau)$ | ${\tt Im}(\tau)$ | $|g_{2}/g_{1}|$ | ${\tt arg}(g_{2}/g_{1})$ | $\frac{|g_{1}|^{2}v_{u}^{2}}{\Lambda_{L}}({\rm eV})$ | $\alpha v_{d}({\rm MeV})$ | $\beta/\alpha$ | $\gamma/\alpha$ |
$2.793$ | $0.498$ | $1.139$ | $0.971$ | $1.018$ | $0.028$ | $68.431$ | $16.530$ | $4.826\times 10^{-3}$ |
$\sin^{2}\theta_{12}$ | $\sin^{2}\theta_{13}$ | $\sin^{2}\theta_{23}$ | $\delta_{CP}/\pi$ | $\alpha_{21}/\pi$ | $\alpha_{31}/\pi$ | $m_{1}({\rm eV})$ | $m_{2}({\rm eV})$ | $m_{3}({\rm eV})$ | $|m_{\beta\beta}|({\rm eV})$
$0.307$ | $0.0222$ | $0.564$ | $1.731$ | $0.728$ | $0.812$ | $0.0636$ | $0.0642$ | $0.0405$ | $0.0331$

Table 15:  The best fit values of free couplings and lepton masses and lepton
mixing parameters in the original $A_{4}$ modular model of Feruglio
Feruglio:2017spp , we use the latest global fit results of NuFIT v5.2
without/with SK atmospheric data Esteban:2020cvm .

In the above original $A_{4}$ modular model Feruglio:2017spp , the right-
handed lepton fields $e^{c}$, $\mu^{c}$ and $\tau^{c}$ have the same modular
weight and they transform as different $A_{4}$ singlets. In fact, we can
distinguish $e^{c}$, $\mu^{c}$ and $\tau^{c}$ through the combination of
modular weights and representation assignments under $A_{4}$. If two or all of
$e^{c}$, $\mu^{c}$ and $\tau^{c}$ are assigned to the same $A_{4}$ singlet
representation, they could be distinguished by different modular weights. The
left-handed lepton fields are assigned to be $A_{4}$ triplet in order to
produce the large lepton mixing angles. Guided by the principle of simplicity,
we use the lower weight modular forms of level 3 as much as possible. Then the
superpotential for charged lepton masses can take the following ten possible
forms Ding:2019zxk :

  * •

$(e^{c},\mu^{c},\tau^{c})\sim(\mathbf{1},\mathbf{1},\mathbf{1})$,
$(k_{e^{c}}+k_{L},k_{\mu^{c}}+k_{L},k_{\tau^{c}}+k_{L})=(2,4,6)$

The modular invariant superpotential for charged lepton Yukawa couplings is
given by

$\displaystyle\mathcal{W}_{e}$ $\displaystyle=$ $\displaystyle\alpha
e^{c}(LY^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{d}+\beta\mu^{c}(LY^{(4)}_{\mathbf{3}})_{\mathbf{1}}H_{d}+\gamma_{1}\tau^{c}(LY^{(6)}_{\mathbf{3}I})_{\mathbf{1}}H_{d}+\gamma_{2}\tau^{c}(LY^{(6)}_{\mathbf{3}II})_{\mathbf{1}}H_{d}$
(352) $\displaystyle=$ $\displaystyle\alpha
e^{c}\left(L_{1}Y_{1}+L_{2}Y_{3}+L_{3}Y_{2}\right)H_{d}$
$\displaystyle+\beta\mu^{c}\Big{[}L_{1}(Y^{2}_{1}-Y_{2}Y_{3})\,+\,L_{2}(Y^{2}_{2}-Y_{1}Y_{3})+L_{3}(Y^{2}_{3}-Y_{1}Y_{2})\Big{]}H_{d}$
$\displaystyle+\gamma_{1}\tau^{c}\Big{[}L_{1}(Y^{3}_{1}+2Y_{1}Y_{2}Y_{3})\,+\,L_{2}(Y^{2}_{1}Y_{3}+2Y^{2}_{3}Y_{2})\,+\,L_{3}(Y^{2}_{1}Y_{2}+2Y^{2}_{2}Y_{3})\Big{]}H_{d}$
$\displaystyle+\gamma_{2}\tau^{c}\Big{[}L_{1}(Y^{3}_{3}+2Y_{1}Y_{2}Y_{3})\,+\,L_{2}(Y^{2}_{3}Y_{2}+2Y^{2}_{2}Y_{1})\,+\,L_{3}(Y^{2}_{3}Y_{1}+2Y^{2}_{1}Y_{2})\Big{]}H_{d}\,,~{}~{}~{}~{}~{}~{}$

where $Y^{(4)}_{\mathbf{3}}$ and $Y^{(6)}_{\mathbf{3}I}$,
$Y^{(6)}_{\mathbf{3}II}$ are weight 4 and weight 6 modular forms at level 3
respectively, and they are given in Eqs. (135, 136).

  * •

$(e^{c},\mu^{c},\tau^{c})\sim(\mathbf{1}^{\prime},\mathbf{1}^{\prime},\mathbf{1}^{\prime})$,
$(k_{e^{c}}+k_{L},k_{\mu^{c}}+k_{L},k_{\tau^{c}}+k_{L})=(2,4,6)$

$\displaystyle\mathcal{W}_{e}$ $\displaystyle=$ $\displaystyle\alpha
e^{c}(LY^{(2)}_{\mathbf{3}})_{\mathbf{1}^{\prime\prime}}H_{d}+\beta\mu^{c}(LY^{(4)}_{\mathbf{3}})_{\mathbf{1}^{\prime\prime}}H_{d}+\gamma_{1}\tau^{c}(LY^{(6)}_{\mathbf{3}I})_{\mathbf{1}^{\prime\prime}}H_{d}+\gamma_{2}\tau^{c}(LY^{(6)}_{\mathbf{3}II})_{\mathbf{1}^{\prime\prime}}H_{d}$
(353) $\displaystyle=$ $\displaystyle\alpha
e^{c}(L_{1}Y_{3}+L_{2}Y_{2}+L_{3}Y_{1})H_{d}$
$\displaystyle+\beta\mu^{c}\Big{[}L_{1}(Y^{2}_{2}-Y_{1}Y_{3})+L_{2}(Y^{2}_{3}-Y_{1}Y_{2})+L_{3}(Y^{2}_{1}-Y_{2}Y_{3})\Big{]}H_{d}$
$\displaystyle+\gamma_{1}\tau^{c}\Big{[}L_{1}(Y^{2}_{1}Y_{3}+2Y^{2}_{3}Y_{2})+L_{2}(Y^{2}_{1}Y_{2}+2Y^{2}_{2}Y_{3})+L_{3}(Y^{3}_{1}+2Y_{1}Y_{2}Y_{3})\Big{]}H_{d}$
$\displaystyle+\gamma_{2}\tau^{c}\Big{[}L_{1}(Y^{2}_{3}Y_{2}+2Y^{2}_{2}Y_{1})+L_{2}(Y^{2}_{3}Y_{1}+2Y^{2}_{1}Y_{2})+L_{3}(Y^{3}_{3}+2Y_{1}Y_{2}Y_{3})\Big{]}H_{d}\,.~{}~{}~{}~{}~{}~{}$

  * •

$(e^{c},\mu^{c},\tau^{c})\sim(\mathbf{1}^{\prime\prime},\mathbf{1}^{\prime\prime},\mathbf{1}^{\prime\prime})$,
$(k_{e^{c}}+k_{L},k_{\mu^{c}}+k_{L},k_{\tau^{c}}+k_{L})=(2,4,6)$

$\displaystyle\mathcal{W}_{e}$ $\displaystyle=$ $\displaystyle\alpha