EPHOU-24-005

Moduli stabilization in finite modular symmetric models

Yoshihiko<EMAIL_ADDRESS>Komei<EMAIL_ADDRESS>Testutaro
<EMAIL_ADDRESS>
Tatsuo<EMAIL_ADDRESS>and Kaito
<EMAIL_ADDRESS>
1Department of Physics, University of Wisconsin-Madison, Madison, WI 53706,
USA
2Department of Physics, Keio University, Yokohama 223-8533, Japan
3Department of Physics, Hokkaido University, Sapporo 060-0810, Japan

###### Contents

  1. 1 Introduction
  2. 2 Modular symmetry and modular forms
    1. 2.1 Modular symmetry
      1. 2.1.1 Modular symmetry and modular groups
      2. 2.1.2 Fundamental domains and fixed points
      3. 2.1.3 Matter fields and modular forms
    2. 2.2 Expansion around fixed points
      1. 2.2.1 Expansion around $\tau=\mathrm{i}$
      2. 2.2.2 Expansion around $\tau=\omega$
      3. 2.2.3 Expansion around $\tau=\mathrm{i}\infty$
  3. 3 Model in supergravity
    1. 3.1 Scalar potential
    2. 3.2 Moduli stabilization and overview of our models
      1. 3.2.1 Single stabilizer case
      2. 3.2.2 Double stabilizer case
  4. 4 Single modular form potential
  5. 5 Double modular forms potential
    1. 5.1 Brief summary
    2. 5.2 Vacuum of $V_{6+4}$
    3. 5.3 Vacuum of $V_{6+4^{\prime}}$
    4. 5.4 Vacuum of $V_{4+4^{\prime}}$
    5. 5.5 $X$ masses
  6. 6 Conclusion
  7. A $A_{4}$ modular modular forms and their derivatives
    1. A.1 Modular form around $\tau=\mathrm{i}$
    2. A.2 Modular form around $\tau=\omega$
    3. A.3 $q$ expansion of modular form and its derivative
  8. B Details of potential coefficients
    1. B.1 Potential expansion around $\tau=\mathrm{i}$
      1. B.1.1 Potential coefficients
      2. B.1.2 Coefficients of potential derivative
    2. B.2 Potential expansion around $\tau=\omega$
  9. C Explicit form of Kähler metric
    1. C.1 Single stabilizer case
    2. C.2 Double stabilizer case
  10. D Hessian analysis
    1. D.1 $V_{6+4}$
    2. D.2 $V_{6+4^{\prime}}$
    3. D.3 $V_{4+4^{\prime}}$

##  1 Introduction

A modular symmetry is well-motivated from the geometrical symmetry of the
higher dimensional theories such as the superstring theory. For example, the
modulus parameter $\tau$ is the complex structure modulus when we consider a
torus or its orbifold compactification. Then $\tau$ is called modulus field
which is a dynamical degree of freedom in the four-dimensional effective field
theory.

The moduli fields play important roles both in particles physics and
cosmology. These days, the flavor symmetry based on the modular group [1] has
been attracting much attention. In the context of flavor physics, the three
generations of quarks and leptons transform non-trivially under the modular
flavor symmetry. In addition, Yukawa couplings are assumed to be modular forms
which are holomorphic functions of a modulus $\tau$ and also transform non-
trivially as similar to the matter fields. As discussed in Ref. [2], we note
that the (in)homogeneous finite modular group $\Gamma_{N}$ with the level
$N\leq 5$ is isomorphic to the well-known permutation groups, such as $S_{3}$,
$A_{4}$, $S_{4}$, and $A_{5}$. The flavor physics based on these new flavor
structures has been discussed actively in the literature [1, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12].

The scalar potential with a modular symmetry can be generally written with
modular forms, whereas such a potential has been studied in the context of the
inflation models [13, 14, 15, 16, 17, 18, 19]. Independently of the inflation
model, the potential typically has the following form

$\displaystyle
V\sim\sum_{i=1}c_{i}(2\mathop{\mathrm{Im}}\tau)^{k_{i}}|Y_{i}(\tau)|^{2},$
(1.1)

where $c_{i}$ is a constant, $Y_{i}(\tau)$ denotes a modular form, $k_{i}$ is
a weight of $Y_{i}(\tau)$, and $i$ labels each modular form contribution. When
the potential contains only a single modular form $Y(\tau)$, i.e.
$V=(2\mathop{\mathrm{Im}}\tau)^{k}|Y(\tau)|^{2},$ the vacuum is obviously
given by a zero point of the modular form $Y(\tau_{*})=0$ if $Y=0$ exists.
Here, $\tau_{*}$ is a value at $Y=0$. If there exist multiple contributions of
the modular forms in the scalar potential as in Eq. (1.1), competition among
$Y_{i}$ gives a vacuum away from $\tau_{*}$. Such a new vacuum can realize a
flavor hierarchy with a large vacuum expectation value (VEV) of $\tau$. On top
of that, if a CP-breaking vacuum could be realized, the matter-anti-matter
asymmetry could be explained.

In this work, we study vacua of scalar potential consisting of one or two
modular forms as in Eq. (1.1) with the modular symmetry and (approximate)
$R$-symmetry. Models with two modular forms help to understand how a stable
vacuum is generated and whether or not the CP symmetry is broken. For the
simplicity, we focus on the $A_{4}$ modular symmetry, which is a well-studied
non-Abelian discrete flavor symmetry, and the scalar potential written with
its singlet modular forms, $Y^{(4)}_{\bm{1}^{(\prime)}}$ and
$Y^{(6)}_{\bm{1}}$.

This paper is organized as follows. In Sec. 2, we give a brief review of
modular symmetry and modular forms. We also discuss the systematic expansion
of the modular forms at three fixed points of the modular symmetry. Our model
is introduced in Sec. 3. We show the details of the Kähler potential and
superpotential, and give overview of our models. In Sec. 4, we show the
profiles of potential with a single modular form and find that vacuum exists
at the fixed point. Sec. 5 shows how the stable vacuum at the fixed point
shifts by an additional modular form contribution in numerical and analytical
perturbative ways. Sec. 6 is devoted to our conclusion. The details of the
modular forms and formulae used in our analysis are summarized in appendices.

In the following parts, we will set the reduced Planck scale $M_{P}\approx
2.4\times 10^{18}~{}\mathrm{GeV}$ to be unity otherwise stated.

##  2 Modular symmetry and modular forms

In this section, we give a brief review of the modular symmetry and the
modular forms. We also discuss the expansion of the modular forms around their
fixed points.

###  2.1 Modular symmetry

####  2.1.1 Modular symmetry and modular groups

The homogeneous modular group $\Gamma\coloneqq\mathrm{SL}(2,\mathbb{Z})$ is
defined by

$\displaystyle\Gamma=\left\\{\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\bigg{|}\
a,b,c,d\in\mathbb{Z},\quad ad-bc=1\right\\},$ (2.1)

which is generated by

$\displaystyle S=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\qquad
T=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix},\qquad R=\begin{pmatrix}-1&0\\\
0&-1\end{pmatrix}.$ (2.2)

These generators satisfy the following relations:

$\displaystyle S^{2}=R,\qquad(ST)^{3}=R^{2}=S^{4}=\bm{1},\qquad TR=RT.$ (2.3)

Under this $\mathrm{SL}(2,\mathbb{Z})$ transformation, the modulus $\tau$
transforms as

$\displaystyle\gamma:\tau\mapsto\frac{a\tau+b}{c\tau+d},\qquad\gamma\in\mathrm{SL}(2,\mathbb{Z}).$
(2.4)

In particular, the generators, $S$ and $T$, act $\tau$ as

$\displaystyle S:\tau\mapsto-\frac{1}{\tau},\qquad T:\tau\mapsto\tau+1,$ (2.5)

and $\tau$ is invariant under $R=-\bm{1}$. This is called the modular
transformation and we introduce the modular group by
$\bar{\Gamma}=\mathrm{PSL}(2,\mathbb{Z})=\mathrm{SL}(2,\mathbb{Z})/\mathbb{Z}^{R}_{2}$,
where $\mathbb{Z}^{R}_{2}$ is generated by $R$.

The congruence subgroup of the level $N$, denoted by $\Gamma(N)$, is defined
by

$\displaystyle\Gamma(N)\coloneqq\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z}),\ \begin{pmatrix}a&b\\\
c&d\end{pmatrix}\equiv\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\mod N\right\\}.$
(2.6)

The quotients $\Gamma_{N}\coloneqq\bar{\Gamma}/\Gamma(N)$ for $N=2,3,4,$ and
$5$ are isomorphic to $S_{3}$, $A_{4}$, $S_{4}$, and $A_{5}$, respectively
[2]. In these quotients, $T$ also satisfies $T^{N}=\bm{1}$ with $N\leq 5$111If
$N>5$, additional relations are required in order to close the algebra [2].,
which generates an additional $\mathbb{Z}^{T}_{N}$ symmetry.

####  2.1.2 Fundamental domains and fixed points

Figure 1:  The fundamental domain of the $\mathrm{SL}(2,\mathbb{Z})$ for the
modulus $\tau$. The dots denote the three fixed points of the modular
transformations.

By the actions of the generators $S$ and $T$, the modulus $\tau$ takes its
value on the fundamental domain $\mathcal{F}$, which is defined by

$\displaystyle\mathcal{F}=\left\\{\tau\in\mathbb{C},\quad\mathop{\mathrm{Im}}\tau>0,\quad|\tau|>1,\quad|\mathop{\mathrm{Re}}\tau|<1/2,\quad\mathop{\mathrm{Re}}\tau=-1/2\right\\},$
(2.7)

where we decompose $\tau$ as
$\tau=\mathop{\mathrm{Re}}\tau+\mathrm{i}\mathop{\mathrm{Im}}\tau$. This is
the region shown in Fig. 1. In this region, it is found that we have three
fixed points for the modular transformation $\Gamma=\mathrm{SL}(2,\mathbb{Z})$

$\displaystyle\tau=\mathrm{i},\quad\omega,\quad\mathrm{i}\infty,$ (2.8)

which are denoted by the dots in Fig. 1. Here
$\omega\coloneqq\mathrm{e}^{2\pi\mathrm{i}/3}$ being the cubic root of unity.
Once $\tau$ takes a value, the modular symmetry is spontaneously broken but we
obtain the following residual symmetries corresponding to the fixed points:

  * •

$\tau=\mathrm{i}$: This is the fixed point of the $S$ transformation and the
residual symmetry is $\mathbb{Z}^{S}_{2}$.

  * •

$\tau=\omega$: This is the fixed point of the $ST$ transformation and the
residual symmetry is $\mathbb{Z}^{ST}_{3}$.

  * •

$\tau=\mathrm{i}\infty$: This is the fixed point of the $T$ transformation and
the residual symmetry is $\mathbb{Z}^{T}_{N}$.

These residual symmetries come from Eq. (2.3) and $T^{N}=\bm{1}$.

####  2.1.3 Matter fields and modular forms

Hereafter, we focus on the supergravity (SUGRA) formulation [20, 21, 22].
Under the modular transformation $\gamma_{N}\in\Gamma_{N}$, a matter
(super)field $X$ with the modular weight $k_{X}$ transforms as

$\displaystyle\gamma_{N}:X\mapsto(c\tau+d)^{k_{X}}\rho_{X}(\gamma)X,$ (2.9)

where $\rho_{X}(\gamma)$ denotes the representation matrix determined by the
representation of $X$ for $\Gamma_{N}$. A modular form $Y(\tau)$ depending on
$\tau$ similarly transforms under the modular transformation:

$\displaystyle\gamma_{N}:Y\mapsto(c\tau+d)^{k_{Y}}\rho_{Y}(\gamma)Y,$ (2.10)

where $k_{Y}$ and $\rho_{Y}(\gamma)$ are the modular weight and representation
matrix, respectively. The modular weights $k_{X}$ and $k_{Y}$ are taken as
even for simplicity.222 The weights can be taken also as odd, and then a
modular group contains the non-trivial element of $R$.  The matter Kähler
potential is assumed to be given by

$\displaystyle\mathcal{K}_{X}=\frac{|X|^{2}}{(-\mathrm{i}\tau+\mathrm{i}\bar{\tau})^{-k_{X}}}=\frac{|X|^{2}}{(2\mathop{\mathrm{Im}}\tau)^{-k_{X}}},$
(2.11)

which is invariant under the transformations (2.4) and (2.9).

The Kähler potential for the modulus field $\tau$ typically has the following
form

$\displaystyle\mathcal{K}_{\tau}=-h\log(-\mathrm{i}\tau+\mathrm{i}\bar{\tau}).$
(2.12)

Here, $h$ is a dimensionless constant, which is related to the choice of the
compact space in the higher dimensional theory. As shown in the following
section, $h$ is irrelevant as long as we focus on the scalar potential
generated by a coupling to matter field $X$, and hence we continue our
analysis without specifying the value of $h$.

In this paper, we focus on the case of $\Gamma_{3}\simeq A_{4}$ $(N=3)$ and
its modular forms for concreteness of the model building.

###  2.2 Expansion around fixed points

Since we focus on the potential vacuum close to the fixed points via
perturbation, we discuss techniques for expanding modular forms around each
fixed point, in particular for $\tau=\mathrm{i}$ and $\omega$. The
$q$-expansion of the modular form is useful for the fixed point
$\tau=\mathrm{i}\infty$, where $q=e^{2\pi\mathrm{i}\tau}$.

####  2.2.1 Expansion around $\tau=\mathrm{i}$

$\tau=i$ is a fixed point associated with the $S$ transformation

$\displaystyle S=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\qquad S^{2}=\bm{1},$
(2.13)

and the residual symmetry is $\mathbb{Z}_{2}^{S}$. Under the $S$
transformation, the modular form $Y^{(k)}$ with the weight $k$ transforms as

$\displaystyle
Y^{(k)}\mapsto(-\tau)^{k}\rho_{Y}(S)Y^{(k)}\quad\text{for}\quad\tau\mapsto-\frac{1}{\tau},$
(2.14)

where $\rho_{Y}$ is a representation matrix of $Y^{(k)}$. It is now convenient
to introduce

$\displaystyle s\coloneqq\frac{\tau-\mathrm{i}}{\tau+\mathrm{i}},$ (2.15)

and to consider the modular forms as functions of $s$ [23, 24, 25, 26]. This
parameter transforms as

$\displaystyle S:s\mapsto-s,$ (2.16)

under the $S$ transformation. The modular form $Y^{(k)}$ here transforms as

$\displaystyle
S:Y^{(k)}(s)\mapsto\biggl{(}\frac{1+s}{1-s}\biggr{)}^{k}\mathrm{i}^{-k}\rho_{Y}(S)Y^{(k)}(s).$
(2.17)

Let us consider the singlets of $A_{4}$ as a concrete example. The
representation matrix becomes $\rho_{Y}(S)=1$, then

$\displaystyle
Y^{(k)}(-s)=\biggl{(}\frac{1+s}{1-s}\biggr{)}^{k}\mathrm{i}^{-k}Y^{(k)}(s).$
(2.18)

If we define

$\displaystyle\tilde{Y}^{(k)}(s)\coloneqq(1-s)^{-k}Y^{(k)}(s),$ (2.19)

Eq. (2.18) is rewritten as the following simple form:

$\displaystyle\tilde{Y}^{(k)}(-s)=\mathrm{i}^{-k}\tilde{Y}^{(k)}(s).$ (2.20)

Considering $l$-th derivative and taking $s\to 0$ of this equation, we obtain
the useful relation [26]

$\displaystyle(\mathrm{i}^{2l}-\mathrm{i}^{-k})\frac{\mathrm{d}^{l}\tilde{Y}^{(k)}}{\mathrm{d}s^{l}}\bigg{|}_{s\to
0}=0.$ (2.21)

We note that if the derivative of $Y^{(k)}$ has a non-vanishing value, the
order of the derivatives $l$ and the modular weight $k$ need to satisfy
$\mathrm{i}^{2l}=\mathrm{i}^{-k}$. This tells us $Y^{(6)}_{\bm{1}}(\tau=i)=0$,
for instance. With this relation and the chain rule

$\displaystyle\frac{\mathrm{d}s}{\mathrm{d}\tau}=\frac{(1-s)^{2}}{2\mathrm{i}},$
(2.22)

we can derive the systematic expansion series of the singlet modular form
around $\tau=\mathrm{i}$. For $k=4$ and $6$, we can find

$\displaystyle Y^{(4)}_{\bm{r}}(\tau)$
$\displaystyle=\tilde{Y}^{(4)}_{\bm{r}}(0)+2\mathrm{i}\tilde{Y}^{(4)}_{\bm{r}}(0)(\tau-\mathrm{i})+\frac{1}{2}\biggl{[}-5\tilde{Y}^{(4)}_{\bm{r}}-\frac{1}{4}\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}(0)\biggr{]}(\tau-\mathrm{i})^{2}$
$\displaystyle\quad-\frac{\mathrm{i}}{3!}\biggl{[}15\tilde{Y}^{(4)}_{\bm{r}}(0)+\frac{9}{4}\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}(0)\biggr{]}(\tau-\mathrm{i})^{3}$
$\displaystyle\quad+\frac{1}{4!}\biggl{[}\frac{105}{2}\tilde{Y}^{(4)}_{\bm{r}}(0)+\frac{63}{4}\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}(0)+\frac{1}{16}\frac{\mathrm{d}^{4}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{4}}(0)\biggr{]}(\tau-\mathrm{i})^{4}+\mathcal{O}((\tau-\mathrm{i})^{5}),$
(2.23) $\displaystyle Y^{(6)}_{\bm{1}}(\tau)$
$\displaystyle=-\frac{\mathrm{i}}{2}\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}(0)(\tau-\mathrm{i})+\frac{7}{4}\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{r}}}{\mathrm{d}s}(0)(\tau-\mathrm{i})^{2}$
$\displaystyle\quad+\frac{\mathrm{i}}{3!}\biggl{[}21\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}(0)+\frac{1}{8}\frac{\mathrm{d}^{3}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s^{3}}(0)\biggr{]}(\tau-\mathrm{i})^{3}$
$\displaystyle\quad+\frac{1}{4!}\biggl{[}-126\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}(0)-\frac{9}{4}\frac{\mathrm{d}^{3}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s^{3}}(0)\biggr{]}(\tau-\mathrm{i})^{4}+\mathcal{O}((\tau-\mathrm{i})^{5}),$
(2.24)

where $\bm{r}=\bm{1},~{}\bm{1}^{\prime}$ for $k=4$. Note that the modular
forms in the left-handed side are functions of $\tau$ and that with a tilde in
the right-handed side is a function of $s$. See also App. A.1 for the details.

####  2.2.2 Expansion around $\tau=\omega$

$\tau=\omega$ is a fixed point associated with the $ST$ transformation

$\displaystyle ST=\begin{pmatrix}0&1\\\
-1&-1\end{pmatrix},\qquad(ST)^{3}=\bm{1},$ (2.25)

and the residual symmetry is $\mathbb{Z}_{3}^{ST}$. Under this transformation,
a modular form $Y^{(k)}$ with the weight $k$ transforms as

$\displaystyle Y^{(k)}\mapsto(-\tau-1)^{k}\rho_{Y}(ST)Y^{(k)}$ (2.26)

and this becomes the following form if $\tau$ is set to be $\omega$:

$\displaystyle Y^{(k)}(ST\omega)=\omega^{2k}\rho_{Y}(ST)Y^{(k)}(\omega).$
(2.27)

In a similar way to that in the expansion around $\tau=\mathrm{i}$, it is
convenient to introduce

$\displaystyle u\coloneqq\frac{\tau-\omega}{\tau-\omega^{2}},$ (2.28)

and consider the modular forms as the functions of $u$ [23, 24, 27, 26]. This
parameter transforms as $u\mapsto\omega^{2}u$. Eq. (2.27) is rewritten as

$\displaystyle
Y^{(k)}(\omega^{2}u)=\biggl{(}\frac{1-\omega^{2}u}{1-u}\biggr{)}^{k}\omega^{2k}\rho_{Y}(ST)Y^{(k)}(u).$
(2.29)

If we introduce

$\displaystyle\hat{Y}^{(k)}\coloneqq(1-u)^{-k}Y^{(k)},$ (2.30)

Eq. (2.29) becomes

$\displaystyle\hat{Y}^{(k)}(\omega^{2}u)=\omega^{2k}\rho_{Y}(ST)\hat{Y}^{(k)}=\omega^{-k}\rho_{Y}(ST)\hat{Y}^{(k)}(u).$
(2.31)

From this equation, the $ST$ transformations for $k=4,\ 6$ are given by

$\displaystyle\hat{Y}^{(4)}_{\bm{1}}(\omega^{2}u)$
$\displaystyle=\omega^{-4}\hat{Y}^{(4)}_{\bm{1}}(u)=\omega^{2}\hat{Y}^{(4)}_{\bm{1}}(u),$
(2.32) $\displaystyle\hat{Y}^{(4)}_{\bm{1}^{\prime}}(\omega^{2}u)$
$\displaystyle=\omega^{-3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}(u)=\hat{Y}^{(4)}_{\bm{1}^{\prime}}(u),$
(2.33) $\displaystyle\hat{Y}^{(6)}_{\bm{1}}(\omega^{2}u)$
$\displaystyle=\omega^{-6}\hat{Y}^{(6)}_{\bm{1}}(u)=\hat{Y}^{(6)}_{\bm{1}}(u),$
(2.34)

where we use $\rho_{\bm{1}}(ST)=1$ and $\rho_{\bm{1}^{\prime}}(ST)=\omega$.
Considering the $l$-th derivatives of these relations and taking $u\to 0$, we
obtain the following useful relation like Eq. (2.21)

$\displaystyle(\omega^{2l}-\omega^{q_{\bm{r}}-k})\frac{\mathrm{d}^{l}\hat{Y}^{(k)}_{\bm{r}}}{\mathrm{d}u^{l}}\bigg{|}_{u\to
0}=0,\qquad q_{\bm{r}}=\begin{cases}0&\bm{r}=\bm{1}\\\
1&\bm{r}=\bm{1}^{\prime}\end{cases}.$ (2.35)

It is noted that if $2l\equiv q_{\bm{r}}-k\mod 3$, the derivative of the
modular form can take a non-zero value. Using the chain rule

$\displaystyle\frac{\mathrm{d}u}{\mathrm{d}\tau}=\frac{(1-u)^{2}}{\sqrt{3}\mathrm{i}},$
(2.36)

we can get the following systematic expansions:

$\displaystyle Y^{(4)}_{\bm{1}}(\tau)$
$\displaystyle=-\frac{\mathrm{i}}{\sqrt{3}}\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}(0)(\tau-\omega)+\frac{5}{3}\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}(0)(\tau-\omega)^{2}$
$\displaystyle\quad+\frac{\mathrm{i}}{3!}\biggl{[}\frac{30}{\sqrt{3}}\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}(0)\biggr{]}(\tau-\omega)^{3}$
$\displaystyle\quad+\frac{1}{4!}\biggl{[}-\frac{280}{3}\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}(0)+\frac{1}{9}\frac{\mathrm{d}^{4}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u^{4}}(0)\biggr{]}(\tau-\omega)^{4}+\mathcal{O}((\tau-\omega)^{5}),$
(2.37) $\displaystyle Y^{(4)}_{\bm{1}^{\prime}}(\tau)$
$\displaystyle=\hat{Y}^{(4)}_{\bm{1}^{\prime}}(0)+\frac{4\mathrm{i}}{\sqrt{3}}\hat{Y}^{(4)}_{\bm{1}^{\prime}}(0)(\tau-\omega)-\frac{10}{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}(0)(\tau-\omega)^{2}$
$\displaystyle\quad+\frac{\mathrm{i}}{3!}\biggl{[}-\frac{40}{\sqrt{3}}\hat{Y}^{(4)}_{\bm{1}^{\prime}}(0)+\frac{1}{3\sqrt{3}}\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}(0)\biggr{]}(\tau-\omega)^{3}$
$\displaystyle\quad+\frac{1}{4!}\biggl{[}\frac{280}{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}(0)-\frac{28}{9}\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}(0)\biggr{]}(\tau-\omega)^{4}+\mathcal{O}((\tau-\omega)^{5}),$
(2.38) $\displaystyle Y^{(6)}_{\bm{1}}(\tau)$
$\displaystyle=\hat{Y}^{(6)}_{\bm{1}}(0)+2\sqrt{3}\mathrm{i}\hat{Y}^{(6)}_{\bm{1}}(0)(\tau-\omega)-7\hat{Y}^{(6)}_{\bm{1}}(\tau-\omega)^{2}$
$\displaystyle\quad+\frac{\mathrm{i}}{3!}\biggl{[}-\frac{112}{\sqrt{3}}\hat{Y}^{(6)}_{\bm{1}}(0)+\frac{1}{3\sqrt{3}}\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}(0)\biggr{]}(\tau-\omega)^{3}$
$\displaystyle\quad+\frac{1}{4!}\biggl{[}336\hat{Y}^{(6)}_{\bm{1}}(0)-4\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}(0)\biggr{]}(\tau-\omega)^{4}+\mathcal{O}((\tau-\omega)^{5}).$
(2.39)

The modular form in the left-handed side is a function of $\tau$ and that with
a hat in the right-handed side is a function of $u$. Note that
$Y^{(4)}_{\bm{1}}(\tau=\omega)=0$, for instance. See also App. A.1 for the
details of the derivation.

####  2.2.3 Expansion around $\tau=\mathrm{i}\infty$

At $\tau=\mathrm{i}\infty$, so-called $q$-expnasion works well, where
$q\coloneqq\exp(2\pi\mathrm{i}\tau)$ [23, 25, 26, 24]. The leading order
expansions of the trivial and non-trivial singlets with the modular weights 4
and 6 are given by

$\displaystyle Y^{(4)}_{\bm{1}}$ $\displaystyle\approx 1+240q,$ (2.40)
$\displaystyle Y^{(4)}_{\bm{1}^{\prime}}$ $\displaystyle\approx-12q^{1/3},$
(2.41) $\displaystyle Y^{(6)}_{\bm{1}}$ $\displaystyle\approx 1-504q.$ (2.42)

Note that $Y^{(4)}_{\bm{1}^{\prime}}(\tau=\mathrm{i}\infty)=0$ since $q\to 0$
in the large $\mathop{\mathrm{Im}}\tau$ limit. See also App. A.3 for the
higher order terms and their derivatives.

##  3 Model in supergravity

###  3.1 Scalar potential

In this section, we introduce the model and its scalar potential. We focus on
the moduli potential of the F-term SUGRA potential generated by a coupling to
matter field $X$. The F-term potential in SUGRA is given by [21, 22]

$\displaystyle
V=\mathrm{e}^{\mathcal{K}}\Bigl{[}\mathcal{K}^{i\bar{j}}D_{i}W\overline{D_{j}W}-3|W|^{2}\Bigr{]}$
(3.1)

where $\mathcal{K}_{i\bar{j}}$ is the total Kähler metric and
$\mathcal{K}^{i\bar{j}}$ denotes its inverse. $D_{i}$ acts on the
superpotential as $D_{i}W=\partial_{i}W+(\partial_{i}\mathcal{K})W$ with the
Kähler potential $\mathcal{K}$. The indices $i,j$ run all the superfield
components. The total Kähler potential is assumed to be

$\displaystyle\mathcal{K}=-h\log(-\mathrm{i}\tau+\mathrm{i}\bar{\tau})+\sum_{a}^{N_{a}}\biggl{[}\frac{|X_{a}|^{2}}{(-\mathrm{i}\tau+\mathrm{i}\bar{\tau})^{-k_{X_{a}}}}-\frac{|X_{a}|^{4}}{4\Lambda_{X_{a}}^{2}(-\mathrm{i}\tau+\mathrm{i}\bar{\tau})^{-2k_{X_{a}}}}\biggr{]}$
(3.2)

where the index $a$ runs all the matter fields $X_{a}$, $k_{X_{a}}$ is the
modular weight for $X_{a}$, $N_{a}$ is the total number of $X_{a}$, and we
choose $h=2$ in numerical analyses of this paper. $|X_{a}|^{4}$ terms are
introduced to stabilizer $X_{a}$ and their $\tau$-dependent coefficients are
chosen such that the action is modular invariant. $\Lambda_{X_{a}}$ is an
energy scale suppressing the higher order terms. We consider the following
superpotential

$\displaystyle W=\Lambda^{2}\sum_{a}^{N_{a}}c_{a}Y_{a}^{(k_{Y_{a}})}X_{a},$
(3.3)

where $Y_{a}^{(k_{Y_{a}})}$ is a modular form with the weight $k_{Y_{a}}$ and
$c_{a}$ is a numerical coefficient. $\Lambda$ is a typical energy scale of the
superpotential and we set $\Lambda=1$ for the simplicity hereafter. These
$X_{a}$’s are so-called stabilizer fields [28, 29, 30, 31, 32], which are
matter fields introduced to generate a scalar potential for a modulus field
[33]. Here, we assume that the representation of $X_{a}$ is a singlet such
that the superpotential is modular invariant. This system has an $R$-symmetry
under the phase transformation:

$\displaystyle\mathrm{U}(1)_{R}:X_{a}\mapsto\exp(2\mathrm{i}\alpha)X_{a},$
(3.4)

where $\alpha$ is a constant transformation parameter. This symmetry controls
the scalar potential as shown in the following sections. The modular weights
$k_{X_{a}}$ and $h$ satisfy the following relation

$\displaystyle-h=k_{Y_{a}}+k_{X_{a}}$ (3.5)

for each label $a$ [34, 18]. If $X_{a}$ are much smaller than the Planck
scale, i.e. $|X_{a}|^{2}\ll 1$, the scalar potential (3.1) is evaluated as

$\displaystyle
V=\sum_{a}^{N_{a}}(-\mathrm{i}\tau+\mathrm{i}\bar{\tau})^{k_{Y_{a}}}|c_{a}Y^{(k_{Y_{a}})}|^{2}+{\cal
O}(|X|^{2}),$ (3.6)

where we use (3.5). Hereafter, we study moduli stabilization including $X_{a}$
with $N_{a}=1$ and $N_{a}=2$ in Sec. 4 and Sec. 5, respectively.

###  3.2 Moduli stabilization and overview of our models

Before focusing on analysis of the $\tau$ vacuum, let us consider the
properties of the stabilizer fields in Eqs. (3.2) and (3.3). We give also
overviews of our models.

####  3.2.1 Single stabilizer case

First, we start the discussion with a single stabilizer case. The Kähler
potential and superpotential are given by

$\displaystyle\mathcal{K}$
$\displaystyle=Z_{X}|X|^{2}-\frac{Z_{X}^{2}|X|^{4}}{4\Lambda_{X}^{2}}-h\log(-\mathrm{i}\tau+\mathrm{i}\bar{\tau}),\qquad
Z_{X}=\frac{1}{(-\mathrm{i}\tau+\mathrm{i}\bar{\tau})^{-k_{X}}},$ (3.7)
$\displaystyle W$ $\displaystyle=Y(\tau)X.$ (3.8)

If we focus on the $X$-dependence in the scalar potential, it is expanded as

$\displaystyle V$
$\displaystyle=(2\mathop{\mathrm{Im}}\tau)^{-(h+k_{X})}|Y(\tau)|^{2}$
$\displaystyle\quad+(2\mathop{\mathrm{Im}}\tau)^{-h}\biggl{[}\frac{|Y|^{2}}{\Lambda^{2}_{X}}+\frac{|(h+k_{X})Y(\tau)-2\mathrm{i}\mathop{\mathrm{Im}}\tau\partial_{\tau}Y(\tau)|^{2}}{h}\biggr{]}|X|^{2}+\mathcal{O}(|X|^{4})$
(3.9) $\displaystyle\eqqcolon V_{0}+m_{X}^{2}|X|^{2}+\mathcal{O}(|X|^{4}).$
(3.10)

$m_{X}^{2}$ is positive definite if the modulus is stabilized such that the
conditions $Y\neq 0$ or $\partial_{\tau}Y\neq 0$ is satisfied. Then we obtain
$\braket{X}=0$ and this is the case in this paper, except for
$Y=Y^{(4)}_{{\bm{1}}^{\prime}}$.

The potential $V_{0}\coloneqq(2\mathop{\mathrm{Im}}\tau)^{-(h+k_{X})}|Y|^{2}$
gives de Sitter (dS) vacuum or supersymmetric one depending on
$\braket{Y(\tau)}$:

$\displaystyle\begin{cases}\displaystyle\braket{V_{0}}>0\ \text{for}\
\braket{Y}\neq 0\quad\Rightarrow\quad\text{dS vacuum}\\\ \\\
\displaystyle\braket{V_{0}}=0\ \text{for}\
\braket{Y}=0\quad\Rightarrow\quad\text{supersymmetric Minkowski
vacuum}\end{cases}.$ (3.11)

The modulus $\tau$ develops the VEV against the zero point value of $Y$
because of $V_{0}\propto|Y|^{2}$. Then $Y=0$ and $\partial_{\tau}Y\neq 0$ are
realized at the vacuum. We note that $\braket{W}=0$ is always satisfied for
$\braket{X}=0$ and F-term is given by $F_{X}\sim\partial_{X}W\sim Y=0$. Then
we obtain supersymmetric Minkowski vacuum with $\braket{Y}=0$ in a single $X$
model of this paper.

We note that the canonically normalized mass squared is given by
$m_{X}^{2}/\mathcal{K}_{X\bar{X}}=m_{X}^{2}/Z_{X}$. $\mathcal{K}_{X\bar{X}}$
is the coefficient of the kinetic term of $X$ and the explicit form of the
Kähler metric is shown in App. C.

##### Comment on the $R$-symmetry violation

Let us consider adding a $\tau$-dependent term $W_{0}(\tau)$ to the
superpotenial as

$\displaystyle W=Y(\tau)X+W_{0}(\tau),$ (3.12)

where $W_{0}(\tau)$ depends only the modulus $\tau$ [35, 36, 37, 38, 39,
33].333See also Ref.[40]. This $W_{0}$ will generally exist in the quantum
gravity since it violates the global $R$-symmetry, and it induces an
additional contribution to the scalar potential. Then, the potential is
roughly given by

$\displaystyle
V\sim\bigl{[}V_{0}-3|W_{0}|^{2}\bigr{]}+m_{X}^{2}|X|^{2}-3W_{0}\bar{Y}\bar{X}-3\overline{W_{0}}YX+\cdots,$
(3.13)

and $\braket{X}$ becomes as follows depending on $\braket{Y}$:

$\displaystyle\braket{X}\sim\begin{cases}\displaystyle\frac{\Lambda_{X}^{2}}{\braket{Y}}W_{0}&\text{for
}\braket{Y}\neq 0,\\\ \\\ \displaystyle 0&\text{for }\braket{Y}=0.\end{cases}$
(3.14)

where $m_{X}^{2}\sim|Y|^{2}/\Lambda_{X}^{2}$ is used for $Y\neq 0$. Note that
$\partial_{\tau}Y$ contributes to $m_{X}^{2}$. If $\braket{Y}=0$ or
$|\braket{Y}|\gg|W_{0}|$, the VEV of $X$ remains tiny as discussed above. A
shift of $\tau$ VEV will be also small and then smaller cosmological constant
$\braket{V}\sim\braket{V_{0}-3}{W_{0}}{{}^{2}}$ is obtained.

On the other hand, for the case of $|Y|\sim|W_{0}|$, it will be necessary to
study the detail of a modulus stabilization including the $W_{0}$ because we
roughly evaluate

$\displaystyle D_{X}W\sim Y\sim W_{0},\quad D_{\tau}W\sim D_{\tau}W_{0}\sim
W_{0},\qquad W=YX+W_{0}\sim W_{0}.$ (3.15)

Then the magnitude of the F-term scalar potential is characterized by $W_{0}$
[35, 36, 37, 38, 39, 33]. However, a concrete study is beyond our scope in
this paper.

####  3.2.2 Double stabilizer case

Next, let us discuss the model with two matter fields, say $X$ and $x$. In the
model, the Kähler potential and superpotential are given by

$\displaystyle\mathcal{K}$
$\displaystyle=Z_{X}|X|^{2}-\frac{Z_{X}^{2}|X|^{4}}{4\Lambda_{X}^{2}}+Z_{x}|x|^{2}-\frac{Z_{x}|x|^{4}}{4\Lambda_{x}^{2}}-h\log(-\mathrm{i}\tau+\mathrm{i}\bar{\tau}),$
(3.16) $\displaystyle Z_{X}$
$\displaystyle=\frac{1}{(-\mathrm{i}\tau+\mathrm{i}\bar{\tau})^{-k_{X}}},\qquad
Z_{x}=\frac{1}{(-\mathrm{i}\tau+\mathrm{i}\bar{\tau})^{-k_{x}}},$ (3.17)
$\displaystyle W$ $\displaystyle=Y(\tau)X+y(\tau)x,$ (3.18)

where an additional stabilizer field is denoted by $x$. The mixing between $X$
and $x$ vanishes due to the (approximate) $\mathrm{U}(1)_{R}$ symmetry. By Eq.
(3.1), we obtain

$\displaystyle V$
$\displaystyle=(2\mathop{\mathrm{Im}}\tau)^{-(h+k_{X})}|Y(\tau)|^{2}+(2\mathop{\mathrm{Im}}\tau)^{-(h+k_{x})}|y(\tau)|^{2}$
$\displaystyle\quad+(2\mathop{\mathrm{Im}}\tau)^{-h}\biggl{[}\frac{|Y(\tau)|^{2}}{\Lambda_{X}^{2}}+|y(\tau)|^{2}(2\mathop{\mathrm{Im}}\tau)^{-k_{x}+k_{X}}+\frac{|(h+k_{X})Y(\tau)-2\mathrm{i}\mathop{\mathrm{Im}}\tau\partial_{\tau}Y(\tau)|^{2}}{h}\biggr{]}|X|^{2}$
$\displaystyle\quad+(2\mathop{\mathrm{Im}}\tau)^{-h}\biggl{[}\frac{|y(\tau)|^{2}}{\Lambda_{x}^{2}}+|Y(\tau)|^{2}(2\mathrm{i}\mathop{\mathrm{Im}}\tau)^{-k_{X}+k_{x}}+\frac{|(h+k_{x})y(\tau)-2\mathrm{i}\mathop{\mathrm{Im}}\tau\partial_{\tau}y(\tau)|^{2}}{h}\biggr{]}|x|^{2}$
$\displaystyle\quad+\frac{(2\mathop{\mathrm{Im}}\tau)^{-h}}{h}\biggl{[}\Bigl{\\{}\bigl{(}(h+k_{X})Y(\tau)-2\mathrm{i}\mathop{\mathrm{Im}}\tau\partial_{\tau}Y\bigr{)}\bigl{(}(h+k_{x})\overline{y(\tau)}+2\mathrm{i}\mathop{\mathrm{Im}}\tau\overline{\partial_{\tau}y(\tau)}\bigr{)}$
$\displaystyle\qquad-
hY(\tau)\overline{y(\tau)}\Bigr{\\}}X\bar{x}+\mathrm{c.c.}\biggr{]}+\cdots$
(3.19) $\displaystyle\eqqcolon
V_{0}+m_{X}^{2}|X|^{2}+m_{x}^{2}|x|^{2}+(M^{2}X\bar{x}+\mathrm{c.c.})+\cdots,$
(3.20)

where we expand the scalar potential to the second order of stabilizers. The
ellipsis denotes the higher order terms in $X$ and $x$. The quadratic term in
this potential is expressed as

$\displaystyle
V\ni\begin{pmatrix}\bar{X}&\bar{x}\end{pmatrix}\begin{pmatrix}m_{X}^{2}&\overline{M^{2}}\\\
M^{2}&m_{x}^{2}\end{pmatrix}\begin{pmatrix}X\\\ x\end{pmatrix}.$ (3.21)

We note that this simple form results from the $\mathrm{U}(1)_{R}$ symmetry.
Then we obtain the following two mass eigenvalues from this mass squared
matrix:

$\displaystyle
m_{\pm}^{2}=\frac{1}{2}\left[\frac{m_{X}^{2}}{\mathcal{K}_{X\bar{X}}}+\frac{m_{x}^{2}}{\mathcal{K}_{x\bar{x}}}\pm\sqrt{4\frac{|M|^{4}}{\mathcal{K}_{X\bar{X}}\mathcal{K}_{x\bar{x}}}+\left(\frac{m_{X}^{2}}{\mathcal{K}_{X\bar{X}}}-\frac{m_{x}^{2}}{\mathcal{K}_{x\bar{x}}}\right)^{2}}\right].$
(3.22)

Here we normalize the mass term of Eq. (3.21) with the Kähler metric so that
the stabilizer fields have canonically normalized kinetic terms. For the
details of the Kähler metric, see App. C. If the stabilizer fields are
stabilized at $\braket{X}=\braket{x}=0$, the Kähler metric reduces to
$\mathcal{K}_{X\bar{X}}=Z_{X}$ and $\mathcal{K}_{x\bar{x}}=Z_{x}$, and the
mass eigenvalues become

$\displaystyle
m_{\pm}^{2}=\frac{1}{2}\left[\frac{m_{X}^{2}}{Z_{X}}+\frac{m_{x}^{2}}{Z_{x}}\pm\sqrt{4\frac{|M|^{4}}{Z_{X}Z_{x}}+\left(\frac{m_{X}^{2}}{Z_{X}}-\frac{m_{x}^{2}}{Z_{x}}\right)^{2}}\right].$
(3.23)

Depending on the VEVs of the modular forms, the following three ways can be
considered in general:

  1. 1.

$\braket{Y}\neq 0,~{}\braket{y}\neq 0$.
The mass squared parameters in Eq. (3.21) satisfy

$\displaystyle m_{X}^{2}>m_{x}^{2}\gtrsim|M|^{2}$ (3.24)

for $\Lambda_{X}\sim\Lambda_{x}\lesssim 1$, and the mass eigenvalues become
approximately

$\displaystyle
m_{+}^{2}\approx\frac{m_{X}^{2}}{\mathcal{K}_{X\bar{X}}}+\frac{|M|^{4}/\mathcal{K}_{X\bar{X}}\mathcal{K}_{x\bar{x}}}{m_{X}^{2}/\mathcal{K}_{X\bar{X}}-m_{x}^{2}/\mathcal{K}_{x\bar{x}}},\qquad
m_{-}^{2}\approx\frac{m_{x}^{2}}{\mathcal{K}_{x\bar{x}}}-\frac{|M|^{4}/\mathcal{K}_{X\bar{X}}\mathcal{K}_{x\bar{x}}}{m_{X}^{2}/\mathcal{K}_{X\bar{X}}-m_{x}^{2}/\mathcal{K}_{x\bar{x}}},$
(3.25)

which leads to $m_{\pm}^{2}\geq 0$. From this, the vacuum is characterized by
$\braket{X}=\braket{x}=0$. The vacuum is dS because of
$V\sim|Y|^{2}+|y|^{2}\neq 0$.

  2. 2.

$\braket{Y}\neq 0,~{}\braket{y}=0$.
The mass squared parameters have the following relation:

$\displaystyle m_{X}^{2}\gtrsim m_{x}^{2}\sim|M|^{2}$ (3.26)

for $\Lambda_{X}\sim\Lambda_{x}\lesssim 1$. Even in this case, the lighter
mass squared eigenvalue are positive definite;

$\displaystyle
m_{-}^{2}\approx\frac{m_{x}^{2}}{\mathcal{K}_{x\bar{x}}}-\frac{|M|^{4}}{m_{X}^{2}\mathcal{K}_{x\bar{x}}}\approx\frac{m_{x}^{2}}{\mathcal{K}_{x\bar{x}}}=\frac{m_{x}^{2}}{Z_{x}}\geq
0,$ (3.27)

where we assume $\braket{X}=\braket{x}=0$ here. The vacuum is dS because of
$V\sim|Y|^{2}\neq 0$.

  3. 3.

$\braket{Y}=\braket{y}=0$.
If the derivatives of the modular forms do not vanish, we obtain non-vanishing
mass squared terms for the stabilizer fields and a vanishing cosmological
constant $V=0$.

The cases 1 and 2 are realized in the following concrete models and 3 is not
the case in this paper. dS or Minkowski vacua are obtained if we have multi
modular form contribution.

Similarly to the single $X$ case, $W_{0}(\tau)$ which violates $R$-symmetry
explicitly can be added to the superpotential. It is expected that the vacuum
energy can be changed as discussed already, but a concrete study is beyond our
scope in this paper.

##  4 Single modular form potential

(a) $V^{(6)}_{\bm{1}}$

(b) $V^{(4)}_{\bm{1}}$

(c) $V^{(4)}_{\bm{1}^{\prime}}$

Figure 2:  3D plots of the single modular form potential (4.3). These
potentials have their global vacua at $\tau=\mathrm{i},~{}\omega,$ and
$\mathrm{i}\infty$ on the fundamental domain, respectively. The points are
fixed points of the modular transformation.

First, let us start the vacuum analysis of the single modular form potential.
In this case, the superpotential has a simple form

$\displaystyle W^{(k_{Y})}_{\bm{r}}=Y^{(k_{Y})}_{\bm{r}}X_{-h-k_{Y}},$ (4.1)

and Kähler metric is given by Eq. (C.2). $k_{Y}$ and $\bm{r}$ denote the
modular weight and representation of the singlet modular form
$Y^{(k_{Y})}_{\bm{r}}$, respectively. As examples we will consider cases with
using

$\displaystyle
Y^{(k_{Y})}_{\bm{r}}=Y^{(4)}_{\bm{1}},~{}Y^{(4)}_{\bm{1}^{\prime}},~{}\text{or}~{}Y^{(6)}_{\bm{1}}.$
(4.2)

Here, $Y^{(k_{Y})}_{\bm{r}}=0$ at $\tau=\omega,~{}\mathrm{i}\infty,$ or
$\mathrm{i}$ respectively as discussed in Sec. 2.2, which are also fixed
points of the modular transformation. $X$ has the modular weight $-h-k_{Y}$
from Eq. (3.5). A representation of the $X$ is a trivial singlet when $X$ is
coupled to $Y^{(4)}_{\bm{1}}$ or $Y^{(6)}_{\bm{1}}$, whereas it is a non-
trivial singlet $\bm{1}^{\prime\prime}$ when $X$ is coupled to
$Y^{(4)}_{\bm{1}^{\prime}}$. The scalar potential is given by

$\displaystyle
V^{(k_{Y})}_{\bm{r}}\coloneqq(2\mathop{\mathrm{Im}}\tau)^{k_{Y}}|Y^{(k_{Y})}_{\bm{r}}|^{2}.$
(4.3)

Thus the minimum of the scalar potential exists at
$\tau=\omega,~{}\mathrm{i}\infty,$ or $\mathrm{i}$ on the fundamental domain
when we take
$Y^{(k_{Y})}_{\bm{r}}=Y^{(4)}_{\bm{1}},~{}Y^{(4)}_{\bm{1}^{\prime}}{\rm
or}~{}Y^{(6)}_{\bm{1}}$, respectively.

$\tau$ and $X$ are stabilized in the models with the potential constructed by
$Y^{(4)}_{\bm{1}}$ or $Y^{(6)}_{\bm{1}}$. Recovering the overall scale of
$\Lambda$ and the Planck scale, we obtain the modulus mass squared
$m_{\tau}^{2}\sim\partial_{\tau}\partial_{\bar{\tau}}V/\mathcal{K}_{\tau\bar{\tau}}$,
where $\mathcal{K}_{\tau\bar{\tau}}=hM_{P}^{2}/(2{\rm Im}\tau)^{2}$, at the
respective supersymmetric Minkowski vacuum444 The $M_{p}$-dependence in the
modulus mass will not be shown explicitly in the following sections. :

$\displaystyle
m_{\tau}\sim\frac{\Lambda^{2}}{M_{P}}|\partial_{\tau}Y^{(k)}_{\bm{1}}(\langle\tau\rangle)|\sim
10\frac{\Lambda^{2}}{M_{P}}.$ (4.4)

Here, $\langle\tau\rangle=\omega$ and $\mathrm{i}$ for $k=4$ and $6$,
respectively. On the other hand, the potential constructed by
$Y^{(4)}_{\bm{1}^{\prime}}$ is a kind of runaway potential because
$V^{(4)}_{\bm{1}^{\prime}}\sim|q|^{2/3}$ in the $q$-expansion as shown in Eq.
(2.41). Then, there exists no vacuum at a finite value of $\tau$ on the
fundamental domain and hence $X$ is not be stabilized owing to the vanishing
mass squared at $\tau=\mathrm{i}\infty$.

The 3D plots of the scalar potentials $V^{(6)}_{\bm{1}}$, $V^{(4)}_{\bm{1}}$,
and $V^{(4)}_{\bm{1}^{\prime}}$ are shown in Fig. 2. It is found that the
potentials (a) and (b) have the minimum at $\tau=\mathrm{i}$ and
$\tau=\omega$, respectively. The potential (c) with
$Y^{(4)}_{\bm{1}^{\prime}}$ is runaway as mentioned above.

The modulus VEVs at the fixed points are useful for particle models since the
models with a finite modular symmetry for a flavor hierarchy often require the
VEVs nearby the fixed points [23, 41, 24, 27, 42, 43, 44, 45, 46, 47, 48, 49].
On the other hand, a small deviation of the VEV from the fixed point is
necessary for a realistic flavor hierarchy, and hence we will consider the
deformation of the potential by adding an another modular form contribution.

##  5 Double modular forms potential

We discuss modulus stabilization with two modular forms in the potential. The
superpotential in each model is given by

$\displaystyle W_{6+4}$ $\displaystyle\coloneqq
Y^{(6)}_{\bm{1}}X_{-h-6}+cY^{(4)}_{\bm{1}}X_{-h-4},$ $\displaystyle
W_{6+4^{\prime}}$ $\displaystyle\coloneqq
Y^{(6)}_{\bm{1}}X_{-h-6}+cY^{(4)}_{\bm{1}^{\prime}}X_{-h-4}^{\prime},$ (5.1)
$\displaystyle W_{4+4^{\prime}}$ $\displaystyle\coloneqq
Y^{(4)}_{\bm{1}}X_{-h-4}+cY^{(4)}_{\bm{1}^{\prime}}X_{-h-4}^{\prime},$

where $c$ is a dimensionless parameter controlling an additional modular form
contribution. The index of $X$ denotes the modular weight of the stabilizer
field. $X_{-h-4}$ and $X_{-h-6}$ are trivial singlets whereas
$X_{-h-4}^{\prime}$ is $\bm{1}^{\prime\prime}$. Here, suppose that $X$’s are
stabilized at the origin, and we will confirm it numerically in Sec 5.5. Then,
the scalar potential is given by the following form as discussed in Eq.
(3.19):

$\displaystyle V_{6+4}\coloneqq V^{(6)}_{\bm{1}}+c^{2}V^{(4)}_{\bm{1}},\quad
V_{6+4^{\prime}}\coloneqq
V^{(6)}_{\bm{1}}+c^{2}V^{(4)}_{\bm{1}^{\prime}},\quad
V_{4+4^{\prime}}\coloneqq V^{(4)}_{\bm{1}}+c^{2}V^{(4)}_{\bm{1}^{\prime}},$
(5.2)

where $V^{(k_{Y})}_{\bm{r}}$ is a single modular form potential defined by Eq.
(4.3). We also use the notation $V_{r}\ (r=6+4,\ 6+4^{\prime},\
4+4^{\prime})$.

In the rest part of this section, we show 3D plots of these scalar potentials
and their vacuum shifts from a fixed point. To this end, we consider a
perturbation of a $c$-dependent term to a fixed point to exhibit instabilities
via results developed in Sec. 2.2.

###  5.1 Brief summary

| $c\lesssim 2$ | $c\gtrsim 2$ |
---|---|---|---
Vacuum | $\braket{\tau}=\mathrm{i}$ | $\braket{\tau}=\omega$ | dS
(a) $V_{6+4}$

| $c\lesssim 5$ | $c\gtrsim 5$ | large $c$ |
---|---|---|---|---
Vacuum | $\braket{\tau}=\mathrm{i}$ | $\braket{\tau}=\mathrm{i}(1+y(c))$ | $\braket{\tau}=\mathrm{i}y(c)$ | dS
(b) $V_{6+4^{\prime}}$

| $c\lesssim 2$ | $c\gtrsim 2$ |
---|---|---|---
Vacuum | $\braket{\tau}=\omega$ | $\braket{\tau}=-1/2+\mathrm{i}y(c)$ | dS
(c) $V_{4+4^{\prime}}$

Table 1:  A brief summary of the vacua and $c$ values corresponding to each
vacuum. $\braket{\tau}=\mathrm{i}$ and $\omega$ are the fixed points and give
CP-conserving vacua. Real numbers of $y(c)$’s in the table (b) and (c) are
fluctuations of $\mathop{\mathrm{Im}}(\tau)$ around fixed points. $y(c)$ have
non-trivial $c$ dependence. When $c\gtrsim 5$ in (b), $y(c)<1$ is given by the
stationary condition of Eq. (5.23). For a larger $c$ in (b), $y(c)>1$ is the
solution of the stationary condition which is approximated to that of Eq.
(5.29). A large $y(c)>1$ in the table (c) for $c\gtrsim 2$ is approximately
given by the solution of Eq. (5.42).

Before moving to a concrete analysis, let us summarize our results here. Tab.
1 summarizes where the vacua of the potentials exist for each value of $c$ as
it varies. The potential minimum is shifted from the vacuum obtained only with
one modular form to a new vacuum due to a $c$-dependent contribution of the
additional modular form. A non-trivial vacuum away from the fixed points is
realized for the scalar potential with $Y^{(4)}_{\bm{1}^{\prime}}$ for a large
$c$. After all, the vacua obtained are CP-conserving one with the positive
cosmological constant. The $X$ fields have positive mass squared at these
vacua as shown in Sec 5.5.

###  5.2 Vacuum of $V_{6+4}$

(a) $c=1$

(b) $c=2.5$

(c) $c=5$

Figure 3:  The 3D plots of $V_{6+4}$ for $c=1,\ 2.5,$ and $5$, respectively.
The potential minimum is at $\tau=\mathrm{i}$ for a small $c$, while it moves
to $\tau=\omega$ as $c$ increases. This is because $V_{6+4}\sim
V^{(4)}_{\bm{1}}$ for a large $c$.

Let us start our discussion with focusing on the vacuum of $V_{6+4}$. For the
details, see Apps. B and D. Fig. 3 shows that the minimum of $V_{6+4}$ shifts
from $\tau=\mathrm{i}$ to $\tau=\omega$ as $c$ increases. A critical change
occurs when $c\gtrsim 2$ as discussed below. In the following part, we study
how the potential minimum changes in a perturbative way around the fixed
points, and it turns out that the vacua at the fixed points are stable against
a perturbation, depending on $c$.

##### Expansion around $\tau=\mathrm{i}$.

Let us expand the scalar potential $V_{6+4}$ by using
$\tau\eqqcolon\mathrm{i}+x+\mathrm{i}y$:

$\displaystyle V_{6+4}(x,y)$
$\displaystyle\approx\bigl{[}2.84\,y^{2}-2.84\,y^{3}+8.44\,y^{4}$
$\displaystyle\quad+\left(2.84-2.84\,y^{2}-7.09\,y^{4}\right)x^{2}-15.5\,x^{4}\bigr{]}\times
10^{3}$
$\displaystyle\quad+c^{2}\bigl{[}33.9+237\,y^{2}-237\,y^{3}+622\,y^{4}$
$\displaystyle\quad+\left(-305+846\,y-1.10\times
10^{3}\,y^{2}\right)\,x^{2}+1.20\times 10^{3}\,x^{4}\bigr{]}.$ (5.3)

Here, we expand the potential to the fourth order in fluctuations to see the
$c$-dependence of the VEV of $\tau$. The first derivatives of $V_{6+4}$ are
given by

$\displaystyle\frac{\partial V_{6+4}}{\partial x}$
$\displaystyle=4\mathcal{X}_{4}(c)\Bigl{(}x^{2}+\frac{1}{2}\frac{\mathcal{X}_{2}(c,y)}{\mathcal{X}_{4}(c)}\Bigr{)}x,$
(5.4) $\displaystyle\frac{\partial V_{6+4}}{\partial y}$
$\displaystyle=\mathcal{Y}_{1}(c,x)+\mathcal{Y}_{2}(c,x)y+\mathcal{Y}_{3}(c)y^{2}+\mathcal{Y}_{4}(c)y^{3}.$
(5.5)

The coefficients $\mathcal{X}_{2,4}$ and $\mathcal{Y}_{1,2,3,4}$ are
summarized in Tabs. 6 and 7 in App. B. For a small $c$ and $y\ll 1$, the
signatures of $\mathcal{X}_{2}$ and $\mathcal{X}_{4}$ become

$\displaystyle\mathcal{X}_{2}\approx
823.23-2755.7y-14367y^{2}>0,\qquad\mathcal{X}_{4}\approx(1.12c^{2}-15.5)\times
10^{3}<0.$ (5.6)

Then, $V_{6+4}$ has the minimum at $x=0$. At $x=0$, we obtain
$\mathcal{Y}_{1}=0$ and the stationary condition of Eq. (5.5) reduces to

$\displaystyle\frac{\partial V_{6+4}}{\partial
y}\bigg{|}_{x=0}=\bigl{[}\mathcal{Y}_{4}(c)y^{2}+\mathcal{Y}_{3}(c)y+\mathcal{Y}_{2}(x=0,c)\bigr{]}y=0.$
(5.7)

We find that the discriminant of $y$’s second order equation in the bracket of
Eq. (5.7) is always negative

$\displaystyle\mathcal{Y}_{3}^{2}(c)-4\mathcal{Y}_{2}(x=0,c)\mathcal{Y}_{4}(c)<0$
(5.8)

and hence the stationary point is given by $y=0$. Thus the fixed point
$\tau=\mathrm{i}$ remains vacuum for a small $c$. At $\tau=\mathrm{i}$ vacuum,
$Y^{(6)}_{\bm{1}}=0$ and $Y^{(4)}_{\bm{1}}\neq 0$ and hence
$\braket{V_{6+4}}\sim c^{2}\braket{}{Y^{(4)}_{\bm{1}}}{{}^{2}}>0$.

The threshold value of $c$ triggering the vacuum shift from $\tau=\mathrm{i}$
to $\tau=\omega$ is discussed with the expansion around $\tau=\omega$ as
below.

Figure 4: The $c$-dependence of the Hessian matrix at the $\tau=\mathrm{i}$
vacuum. $H_{11}$ becomes negative when $c\gtrsim 3$ and $H_{22}$ monotonically
increases.

In order to see the stability of the vacuum at $\tau=\mathrm{i}$, let us
introduce the Hessian matrix:

$\displaystyle\begin{pmatrix}H_{11}(x,y;c)&H_{12}(c,x,y)\\\
H_{21}(x,y;c)&H_{22}(c,x,y)\end{pmatrix}$
$\displaystyle\coloneqq\begin{pmatrix}\partial^{2}V/\partial
x^{2}&\partial^{2}V/\partial x\partial y\\\ \partial^{2}V/\partial y\partial
x&\partial^{2}V/\partial y^{2}\end{pmatrix}$ (5.9)
$\displaystyle=\begin{pmatrix}\partial^{2}V/\partial x^{2}&0\\\
0&\partial^{2}V/\partial y^{2}\end{pmatrix}.$ (5.10)

In the last equality, we used the fact that the off-diagonal components vanish
when CP is conserved at the vacuum. Here, CP transformation is given by

$\displaystyle\mathrm{CP}:\tau\mapsto-\bar{\tau}.$ (5.11)

Thus the vacuum at $\tau=\mathrm{i}$ preserves CP. The $c$-dependence of the
Hessian matrix at $\tau=\mathrm{i}$ is shown in Fig. 4. Recovering the overall
scale of $\Lambda$, we obtain the modulus mass at $\tau=\mathrm{i}$

$\displaystyle
m_{\tau}\sim\Lambda^{2}|\partial_{\tau}Y^{(6)}_{\bm{1}}(\tau=i)|\sim
10\Lambda^{2}.$ (5.12)

Note that $H_{11}$ becomes tachyonic when $c\gtrsim 3$. Then, the vacuum at
$\tau=\mathrm{i}$ becomes unstable since $Y^{(4)}_{\bm{1}}\neq 0$ at
$\tau=\mathrm{i}$ and $V_{6+4}$ is approximately given by
$c^{2}|Y^{(4)}_{\bm{1}}|^{2}$ which costs energy for such a large $c$.

##### Expansion around $\omega$.

By using Eqs. (2.37), (2.39), and $\tau\eqqcolon\omega+x+\mathrm{i}y$, the
potential can be expanded as

$\displaystyle V_{6+4}$ $\displaystyle\approx\left(224-448y^{2}+7.51\times
10^{3}y^{3}-1.22\times 10^{4}y^{4}\right)$
$\displaystyle\quad+\left(-448-2.05\times 10^{4}y+7.32\times
10^{4}y^{2}\right)x^{2}-1.16\times 10^{4}x^{4}$
$\displaystyle\quad+c^{2}\Bigl{[}328y^{2}-379y^{3}-109y^{4}+\left(328-379y-656y^{2}\right)\,x^{2}-546x^{4}\Bigr{]}.$
(5.13)

The first derivatives of the potential are given by

$\displaystyle\frac{\partial V_{6+4}}{\partial x}\approx
2\big{\\{}-449+328c^{2}\big{\\}}x,\qquad\frac{\partial V_{6+4}}{\partial
y}\approx 2\bigl{\\{}-449+328c^{2}\bigr{\\}}y.$ (5.14)

For $c\gtrsim 1$, the second term in the bracket of two $\partial V$’s becomes
dominant and $\tau=\omega$ becomes a local minimum:
$\partial^{2}V_{6+4}/\partial x^{2}=\partial^{2}V_{6+4}/\partial y^{2}>0$. For
$c\gtrsim 2$, we numerically find $V_{6+4}(\tau=\mathrm{i})\sim
c^{2}|Y^{(4)}_{\bm{1}}(\tau=\mathrm{i})|^{2}\geq
V_{6+4}(\tau=\omega)\sim|Y^{(6)}_{\bm{1}}(\tau=\omega)|^{2}$, and hence
$\tau=\omega$ becomes the global minimum. Recovering the overall scale of
$\Lambda$, we obtain the modulus mass at $\tau=\omega$

$\displaystyle m_{\tau}\sim
c\Lambda^{2}|\partial_{\tau}Y^{(4)}_{\bm{1}}(\tau=\omega)|\sim
10c\Lambda^{2}.$ (5.15)

Note that $\tau=\omega$ is the CP-conserving minimum with $T$ symmetry of
$\tau\equiv\tau-1$:

$\displaystyle\mathrm{CP}:~{}\tau=\omega=-\frac{1}{2}+\mathrm{i}\frac{\sqrt{3}}{2}\mapsto\frac{1}{2}+\mathrm{i}\frac{\sqrt{3}}{2}\equiv-\frac{1}{2}+\mathrm{i}\frac{\sqrt{3}}{2}=\omega.$
(5.16)

At $\tau=\omega$ vacuum, $Y^{(6)}_{\bm{1}}\neq 0$ and $Y^{(4)}_{\bm{1}}=0$ and
hence $\braket{V_{6+4}}\sim\braket{}{Y^{(6)}_{\bm{1}}}{{}^{2}}>0$.

###  5.3 Vacuum of $V_{6+4^{\prime}}$

(a) $c=1$

(b) $c=6$

(c) $c=100$

Figure 5:  The 3D plots of $V_{6+4^{\prime}}$ for $c=1,\ 6$, and $100$,
respectively. $V_{6+4^{\prime}}$ can have a vacuum at
$\braket{\tau}\to\mathrm{i}\infty$ as $c$ increases. For $c=\mathcal{O}(100)$,
$V_{6+4^{\prime}}$ has the minimum at $\mathop{\mathrm{Im}}\tau\sim 3$.

Next, we consider the vacuum of the potential $V_{6+4^{\prime}}$. Fig. 5 shows
the potential can shift away from the fixed point $\tau=\mathrm{i}$ to
$\mathop{\mathrm{Im}}\tau\gg 1$ as $c$ increases. In the following part, let
us show this change of $\braket{\tau}$ in Fig. 5 by perturbative analysis.

##### Expansion around $\tau=\mathrm{i}$.

As in the case of $V_{6+4}$, we expand the potential to the fourth order in
fluctuations

$\displaystyle V_{6+4^{\prime}}(x,y)$
$\displaystyle\approx\bigl{[}2.84\,y^{2}-2.84\,y^{3}+8.44\,y^{4}$
$\displaystyle\quad+\left(2.84-2.84\,y-7.09\,y^{2}\right)x^{2}-15.5\,x^{4}\bigr{]}\times
10^{3}$
$\displaystyle\quad+c^{2}\bigl{[}33.9-88.0\,y^{2}+88.0\,y^{3}+11.6\,y^{4}$
$\displaystyle\quad+\left(20.2+128\,y+360\,y^{2}\right)\,x^{2}-62.7\,x^{4}\bigr{]},$
(5.17)

where we use $\tau\eqqcolon\mathrm{i}+x+\mathrm{i}y$. The first derivatives of
$V_{6+4^{\prime}}$ are similarly given by Eqs. (5.4) and (5.5), and see Tabs.
6 and 7 in App. B for the details of $\mathcal{X}$ and $\mathcal{Y}$. For
$y\ll 1$, we focus on a small $c$ and then find

$\displaystyle\mathcal{X}_{2}$
$\displaystyle=2836.9-2836.9y-7092.2y^{2}+c^{2}(20.224+128.49y+360.09y^{2})>0,$
(5.18) $\displaystyle\mathcal{X}_{4}$ $\displaystyle=-62.735c^{2}-15532<0.$
(5.19)

Then $\partial V_{6+4^{\prime}}/\partial x=0$ is satisfied by $x=0,\
\pm\sqrt{-\mathcal{X}_{2}/(2\mathcal{X}_{4})}$, and the $x=0$ is the minimum.

Figure 6:  (Left) The behavior of the non-zero solution in Eq. (5.23) as the
function of $c$. The vacuum shifts in the $\mathop{\mathrm{Im}}\tau$ direction
as $c$ increases. (Right) The behavior of the solution $y$ as the function of
$c$, which is approximately given by Eq. (5.29). It is found that $y\sim 2.5$
is realized with about $c\sim 100$ [44, 45].

Let us consider the stationary condition of the imaginary direction:

$\displaystyle\frac{\partial V_{6+4^{\prime}}}{\partial
y}\bigg{|}_{x=0}=\bigl{(}\mathcal{Y}_{4}(c)y^{2}+\mathcal{Y}_{3}(c)y+\mathcal{Y}_{2}(x=0,c)\bigr{)}y=0.$
(5.20)

Whether the solution of $y\neq 0$ exist or not depends on the discriminant of
the equation in the bracket of (5.20). It is found that numerically

$\displaystyle\mathcal{Y}_{3}^{2}-4\mathcal{Y}_{2}\mathcal{Y}_{4}>0\quad\text{when}\
c\gtrsim 5$ (5.21)

and hence the stationary points of $V_{6+4^{\prime}}$ are

$\displaystyle y(c)=\begin{cases}\displaystyle 0&c\lesssim 5\\\
\displaystyle\frac{-\mathcal{Y}_{3}+\sqrt{\mathcal{Y}_{3}^{2}-4\mathcal{Y}_{2}\mathcal{Y}_{4}}}{2\mathcal{Y}_{4}},~{}0&c\gtrsim
5\end{cases},$ (5.22)

where we drop
$\displaystyle\frac{-\mathcal{Y}_{3}-\sqrt{\mathcal{Y}_{3}^{2}-4\mathcal{Y}_{2}\mathcal{Y}_{4}}}{2\mathcal{Y}_{4}}$
which is located outside the fundamental domain. For $c\gtrsim 5$, $y=0$ is
the local maximum. The explicit form of the non-zero solution is given by

$\displaystyle\frac{-\mathcal{Y}_{3}+\sqrt{\mathcal{Y}_{3}^{2}-4\mathcal{Y}_{2}\mathcal{Y}_{4}}}{2\mathcal{Y}_{4}}$
$\displaystyle=-2.8413+\frac{2155.3+4.7426\sqrt{-3569.4+93.7802c^{2}+0.52736c^{4}}}{726.34+c^{2}}.$
(5.23)

The left panel of Fig. 6 shows the behavior of non-zero $y(c)$ as a function
of $c$. The VEV of $y$ stays zero for $c\leq 5.68$, and monotonically
increases when $c\gtrsim 5$.

The $c$-dependence of the Hessian diagonal components (5.9) is shown in Fig.
7. Note that off the diagonal components of the Hessian are vanishing at the
CP-conserving vacuum at $\tau=i(1+y(c))$. Recovering the overall scale of
$\Lambda$, we obtain the modulus mass at $\langle\tau\rangle=i(1+y(c))$

$\displaystyle
m_{\tau}\sim\Lambda^{2}|\partial_{\tau}Y^{(6)}_{\bm{1}}(\langle\tau\rangle)|\sim
10\Lambda^{2}.$ (5.24)

The mass difference between $x$ and $y$ gets bigger as $c$ increases, and the
reason is shown later at the end of this section. In the left panel the
Hessian is evaluated at $x=y=0$ for $c\lesssim 5.68$, whereas it is evaluated
at $\displaystyle
x=0,~{}y(c)=\frac{-\mathcal{Y}_{3}+\sqrt{\mathcal{Y}_{3}^{2}-4\mathcal{Y}_{2}\mathcal{Y}_{4}}}{2\mathcal{Y}_{4}}$
for $c\gtrsim 5.68$. Then, the masses of the modulus fields become
monotonically heavy as $c$ is increasing.

Figure 7:  (Left) The $c$-dependence of the Hessian diagonal components.
(Right) The values of the Hessian matrix evaluated at $\tau=\mathrm{i}y(c)$.
See the text for the details.

##### Expansion around $\tau=\mathrm{i}\infty$.

The contribution from the runaway potential
$V^{(4)}_{\bm{1}^{\prime}}\sim|Y^{(4)}_{\bm{1}^{\prime}}|^{2}$ is dominant in
$V_{6+4^{\prime}}$ for a large $c$, and the vacuum exists at a large
$\mathop{\mathrm{Im}}\tau$. For a large $\mathop{\mathrm{Im}}\tau$ VEV, the
scalar potential is given by following form with a $q$-expansion

$\displaystyle V_{6+4^{\prime}}$
$\displaystyle=(2y)^{6}|1-504q|^{2}+c^{2}(2y)^{4}|-12q^{1/3}|^{2}+\mathcal{O}(q^{3})$
$\displaystyle=(2y)^{6}(1-1008\mathrm{e}^{-2\pi y}\cos 2\pi
x+504^{2}\mathrm{e}^{-4\pi y})+c^{2}(2y)^{4}144\mathrm{e}^{-4\pi
y/3}+\mathcal{O}(q^{3}).$ (5.25)

Here, we used $q=\exp(2\pi\mathrm{i}\tau)=\exp[2\pi(\mathrm{i}x-y)]$ and $y\gg
1$. The first derivative of $V_{6+4^{\prime}}$ with respect to $x$ is

$\displaystyle\frac{\partial V_{6+4^{\prime}}}{\partial x}\propto+\sin 2\pi
x.$ (5.26)

Thus the minimum is located at $x=0$ independently of $c$. Hence the vacuum is
still CP symmetric for a large $c$. The potential gradient along the
$y$-direction at $x=0$ is

$\displaystyle\frac{\partial V_{6+4^{\prime}}}{\partial y}\bigg{|}_{x=0}$
$\displaystyle\approx 384y^{3}\Bigl{[}\mathrm{e}^{-4\pi y}(\mathrm{e}^{2\pi
y}-504)(336\pi y+\mathrm{e}^{2\pi y}-504)y^{2}-8c^{2}\mathrm{e}^{-4\pi
y/3}(\pi y-3)\Bigr{]}.$ (5.27)

In the $y\gg 1$ limit, the stationary condition at the leading order in $q$
reduces to

$\displaystyle\frac{\partial V_{6+4^{\prime}}}{\partial y}\bigg{|}_{x=0}\sim
384y^{3}\bigl{[}-8c^{2}\mathrm{e}^{-4\pi y/3}(\pi y-3)+y^{2}\bigr{]}=0.$
(5.28)

Equivalently, the above condition is rewritten as

$\displaystyle y+\frac{3}{4\pi}\log\frac{y^{2}}{y-3/\pi}=\frac{3}{2\pi}\log
c+\frac{3}{4\pi}\log 8\pi.$ (5.29)

The right panel in Fig. 6 shows the solution $y(c)$ of this equation as the
function of $c$. $y(c)$ turns out to increase as $c$ does. In order to see the
stability around the stationary point at $\tau=\mathrm{i}y(c)$, the second
derivative of the potential is evaluated as

$\displaystyle\frac{\partial^{2}V_{6+4^{\prime}}}{\partial
y^{2}}\bigg{|}_{x=0,~{}y=y(c)}$ $\displaystyle=384\mathrm{e}^{-4\pi
y}y^{4}\Bigl{[}-8c^{2}\mathrm{e}^{8\pi y/3}(\pi y-3)+(\mathrm{e}^{2\pi
y}-504)(336\pi y+\mathrm{e}^{2\pi y}-504)y^{2}\Bigr{]}\bigg{|}_{y=y(c)}$
$\displaystyle\sim 128y(c)^{4}\frac{4\pi^{2}y(c)^{2}-9\pi y(c)-18}{\pi
y(c)-3}.$ (5.30)

The $y(c)$-dependence of the diagonal components of the Hessian are shown in
the right panel in Fig. 7, while the off-diagonal components are vanishing.
The mass of the $\mathop{\mathrm{Im}}\tau$ gets heavier as $y(c)$ (and $c$)
increase. This is because the mass is proportional to Eq. (5.30). On the other
hand, the mass of $\mathop{\mathrm{Re}}\tau$ gets lighter as $y(c)$ increase.
This is seen from $V_{6+4^{\prime}}\sim-\mathrm{e}^{-2\pi y}\cos 2\pi x$.
Hence, recovering the overall scale of $\Lambda$, we roughly find

$\displaystyle m_{x}\sim 10^{2}y(c)^{4}e^{-\pi y(c)}\Lambda^{2},\qquad
m_{y}\sim 10^{2}y(c)^{7/2}\Lambda^{2},$ (5.31)

where we have canonically normalized the masses by multiplying $y(c)^{2}$ to
the second derivatives of $V_{6+4^{\prime}}$. The explicit form of the Hessian
matrix component is shown in Eq. (D.10).

In this model, the vacuum exists at a point apart from any fixed points. Thus
$\braket{V_{6+4^{\prime}}}\sim\braket{}{Y_{\bm{1}}^{(6)}}{{}^{2}+|Y_{{\bm{1}}^{\prime}}^{(4)}|^{2}}>0$.

###  5.4 Vacuum of $V_{4+4^{\prime}}$

(a) $c=1$

(b) $c=2$

(c) $c=3$

Figure 8:  The 3D plots of $V_{4+4^{\prime}}$ for $c=1,\ 2,$ and $3$,
respectively.

Similarly to $V_{6+4^{\prime}}$, the potential $V_{4+4^{\prime}}$ can have the
minimum at $\tau=-1/2+\mathrm{i}\mathop{\mathrm{Im}}\tau$, where
$\mathop{\mathrm{Im}}\tau\geq\sqrt{3}/2$, depending on $c$ as shown in Fig. 8.
We show $c$-dependence of the modulus VEV as in the previous sections.

##### Expansion around $\tau=\omega$.

Eqs. (2.37) and (2.39) give us the following potential expansion:

$\begin{split}V_{4+4^{\prime}}(x,y)=&~{}328y^{2}-379y^{3}-109y^{4}+\left(328-379y-656y^{2}\right)\,x^{2}-546x^{4}\\\
&+c^{2}\left\\{36.9-49.2y^{2}-351y^{3}+485y^{4}+\left(-49.2+940y+3.04\times
10^{3}y^{2}\right)x^{2}+551x^{4}\right\\},\end{split}$ (5.32)

where $\tau\eqqcolon\omega+x+\mathrm{i}y$. The first derivatives of
$V_{4+4^{\prime}}$ are given by

$\displaystyle\frac{\partial V_{4+4^{\prime}}}{\partial x}$
$\displaystyle=2(327.6-49.16c^{2})x,$ (5.33) $\displaystyle\frac{\partial
V_{4+4^{\prime}}}{\partial y}$ $\displaystyle=2(327.6-49.16c^{2})y.$ (5.34)

Here, the only leading contribution is considered. For a small $c\lesssim 2$,
the minimum is located at $x=y=0$: the second derivatives of
$V_{4+4^{\prime}}$ are given by

$\displaystyle\frac{\partial^{2}V_{4+4^{\prime}}}{\partial
x^{2}}\bigg{|}_{x=y=0}=\frac{\partial^{2}V_{4+4^{\prime}}}{\partial
y^{2}}\bigg{|}_{x=y=0}=2(328-49.2c^{2}),\qquad\frac{\partial^{2}V_{4+4^{\prime}}}{\partial
x\partial y}=0.$ (5.35)

Thus, recovering the overall scale of $\Lambda$, we obtain the modulus mass at
$\tau=\omega$

$\displaystyle
m_{\tau}\sim\Lambda^{2}|\partial_{\tau}Y^{(4)}_{\bm{1}}(\tau=\omega)|\sim
10\Lambda^{2}.$ (5.36)

When $c$ gets larger than $2$ and hence the contribution from
$Y^{(4)}_{\bm{1}^{\prime}}$ becomes dominant in $V_{4+4^{\prime}}$, there
appear tachyonic modes at $x=y=0$. The instability indicates that the vacuum
can be shifted from $\tau=\omega=-1/2+\mathrm{i}\sqrt{3}/2$ to a larger
$\mathop{\mathrm{Im}}\tau$.

##### Expansion around $\tau=-1/2+\mathrm{i}\infty$.

When a $c$ is large, we use $q$ expansion and the modular forms are expressed
as

$\displaystyle Y^{(4)}_{\bm{1}}\approx 1+240q,\qquad
Y^{(4)}_{\bm{1}^{\prime}}\approx-12q^{1/3},$ (5.37)

for $|q|\ll 1$, where $q=e^{2\pi\mathrm{i}\tau}$. We parameterize
$\tau=x+\mathrm{i}y~{}(y\gg 1)$ and the potential is given by

$\displaystyle V_{4+4^{\prime}}$
$\displaystyle\approx(2y)^{4}\left(|1+240q|^{2}+c^{2}\left|-12q^{1/3}\right|^{2}\right)$
$\displaystyle=16y^{4}\Bigl{[}1+57600\mathrm{e}^{-4\pi y}+480\mathrm{e}^{-2\pi
y}\cos(2\pi x)+144c^{2}\mathrm{e}^{-4\pi y/3}\Bigr{]}.$ (5.38)

We note that the signature of $\cos(2\pi x)$ is opposite to that in Eq.
(5.25). Due to this cosine function, this potential has the minimum at
$x=-1/2$:

$\displaystyle\frac{\partial V_{4+4^{\prime}}}{\partial x}\propto-\sin(2\pi
x),\qquad\frac{\partial^{2}V_{4+4^{\prime}}}{\partial x^{2}}\propto-\cos(2\pi
x).$ (5.39)

Hence, CP is preserved at the vacuum with the T-symmetry like in a case of
$\tau=\omega$. The potential gradient with respect to $y$ is

$\displaystyle\frac{\partial V_{4+4^{\prime}}}{\partial y}\bigg{|}_{x=-1/2}$
$\displaystyle=64y^{3}\Bigl{[}(1-240\mathrm{e}^{-2\pi
y})(1-240\mathrm{e}^{-2\pi y}+240\pi\mathrm{e}^{-2\pi
y}y)-48c^{2}\mathrm{e}^{-4\pi y/3}(\pi y-3)\Bigr{]}.$ (5.40)

For $y\gg 1$, the stationary condition approximately reduces to

$\displaystyle\frac{\partial V_{4+4^{\prime}}}{\partial
y}\bigg{|}_{x=-1/2}\approx 64y^{3}\Bigl{[}1-48c^{2}\mathrm{e}^{-4\pi y/3}(\pi
y-3)\Bigr{]}=0.$ (5.41)

Equivalently, the above condition reads

$\displaystyle 1-48c^{2}\mathrm{e}^{-4\pi y/3}(\pi y-3)=0.$ (5.42)

The solution $y(c)$ of this Eq. (5.42) as a function of the $c$ is shown in
the left panel in Fig. 9. The $y(c)$ gets larger as $c$ increases. The second
derivative of $V_{4+4^{\prime}}$ is evaluated as

$\displaystyle\frac{\partial^{2}V_{4+4^{\prime}}}{\partial
y^{2}}\bigg{|}_{x=-1/2,y=y(c)}$ $\displaystyle=64\mathrm{e}^{-4\pi
y}y^{2}\Bigl{[}16c^{2}\mathrm{e}^{8\pi y/3}(4\pi^{2}y^{2}-24\pi
y+27)-480\mathrm{e}^{2\pi y}(\pi^{2}y^{2}-4\pi y+3)$
$\displaystyle\qquad\qquad\qquad+57600(4\pi^{2}y^{2}-8\pi
y+3)+3\mathrm{e}^{4\pi y}\Bigr{]}\bigg{|}_{y=y(c)}$ $\displaystyle\approx
64y(c)^{2}\Bigl{[}16c^{2}\mathrm{e}^{-4\pi y(c)/3}(4\pi^{2}y(c)^{2}-24\pi
y(c)+27)+3\Bigr{]}.$ (5.43)

In the last equality, we picked up just leading terms in $|q|\ll 1$. Putting
the solution of Eq. (5.42) into this equation, we obtain the following simple
form:

$\displaystyle\frac{\partial^{2}V_{4+4^{\prime}}}{\partial
y^{2}}\bigg{|}_{\text{solution of
Eq.~{}\eqref{eq:minimum-44p}}}=\frac{64\pi}{3}\frac{y(c)^{3}(4\pi
y(c)-15)}{\pi y(c)-3},$ (5.44)

and the signature of this equation is positive if $y(c)\geq 15/4\pi$ for
$c\gtrsim 2$. The $y(c)$ is the minimum of $V_{4+4^{\prime}}$ for a large $y$
and thus $x=-1/2$ and $y>15/4\pi$ is the minimum for $c\gtrsim 2$.

Figure 9:  (Left) The behavior of the solution $y$ as the function of $c$
which is approximately given by Eq. (5.42). (Right) The diagonal components of
the Hessian matrix at $\tau=\mathrm{i}y$, where $y$ is approximately given by
the solution of Eq. (5.42).

We give some comments on the Hessian analysis. We note that the off-diagonal
component with $x=-1/2$ vanishes and the $y$-dependence of the diagonal
components are shown in the right panel in Fig. 9. This figure shows
numerically that the vacuum at $\langle\tau\rangle=-1/2+\mathrm{i}y(c)$ is
stable. Hence, recovering the overall scale of $\Lambda$, we roughly find

$\displaystyle m_{x}\sim 10^{2}y(c)^{3}e^{-\pi y(c)}\Lambda^{2},\qquad
m_{y}\sim 10^{2}y(c)^{5/2}\Lambda^{2},$ (5.45)

where we have canonically normalized the masses by multiplying $y(c)^{2}$ to
the second derivatives of $V_{6+4^{\prime}}$ as in the previous section.

In this model, there exists the vacuum at a point apart from any fixed points.
Thus
$\braket{V_{4+4^{\prime}}}\sim\braket{}{Y_{\bm{1}}^{(4)}}{{}^{2}+|Y_{{\bm{1}}^{\prime}}^{(4)}|^{2}}>0$.

###  5.5 $X$ masses

We numerically show the mass eigenvalues of $X$ fields, $m_{\pm}$ given in Eq.
(3.23). With recovered overall scale of $\Lambda$, the explicit forms read

$\displaystyle m^{2}_{X}$
$\displaystyle=\Lambda^{4}(2\mathop{\mathrm{Im}}\tau)^{-h}\biggl{[}\frac{|Y(\tau)|^{2}}{\Lambda_{X}^{2}}+c^{2}|y(\tau)|^{2}(2\mathrm{i}\mathop{\mathrm{Im}}\tau)^{k_{X}-k_{x}}+\frac{|(h+k_{X})Y(\tau)-2\mathrm{i}\mathop{\mathrm{Im}}\tau\partial_{\tau}Y(\tau)|^{2}}{h}\biggr{]},$
(5.46) $\displaystyle m^{2}_{x}$
$\displaystyle=\Lambda^{4}(2\mathop{\mathrm{Im}}\tau)^{-h}\biggl{[}c^{2}\frac{|y(\tau)|^{2}}{\Lambda_{x}^{2}}+|Y(\tau)|^{2}(2\mathrm{i}\mathop{\mathrm{Im}}\tau)^{-k_{X}+k_{x}}+c^{2}\frac{|(h+k_{x})y(\tau)-2\mathrm{i}\mathop{\mathrm{Im}}\tau\partial_{\tau}y(\tau)|^{2}}{h}\biggr{]},$
(5.47) $\displaystyle M^{2}$
$\displaystyle=c\Lambda^{4}\frac{(2\mathop{\mathrm{Im}}\tau)^{-h}}{h}$
$\displaystyle\quad\times\biggl{[}\Bigl{\\{}\bigl{(}(h+k_{X})Y(\tau)-2\mathrm{i}\mathop{\mathrm{Im}}\tau\partial_{\tau}Y\bigr{)}\bigl{(}(h+k_{x})\overline{y(\tau)}+2\mathrm{i}\mathop{\mathrm{Im}}\tau\overline{\partial_{\tau}y(\tau)}\bigr{)}-hY(\tau)\overline{y(\tau)}\Bigr{\\}}\biggr{]},$
(5.48)

where $Y$ and $y$ are corresponding to $Y^{(4)}_{\bm{1}}$,
$Y^{(4)}_{\bm{1}^{\prime}}$, or $Y^{(6)}_{\bm{1}}$, depending on the models.
Note that $y$ differs by $c$ from that exhibited in the Sec. 3.2.2. The mass
eigenvalues can be roughly estimated as

$\displaystyle
m_{\pm}^{2}\sim\Lambda^{4}\times\biggl{(}\frac{|Y(\tau)|^{2}}{\Lambda_{X}^{2}}+|Y(\tau)+\partial_{\tau}Y(\tau)|^{2}\biggr{)}\sim(10\Lambda^{2})^{2},$
(5.49)

where $Y(\tau)\sim cy(\tau)$ and $\Lambda_{X}\sim\Lambda_{x}\sim 1$ are
assumed. Then masses of $X$-fields can be comparable to that of $\tau$.

In the plots of Figs. 10, 11, and 12, we take $h=2$ and
$\Lambda_{X}=\Lambda_{x}=1$ for simplicity. All the parameters of physical
mass squared turn out to be positive definite. Hence, the vacua studied in the
previous sections are stable.

Figure 10:  (Left) Eigenvalues of stabilizers’ masses for $V_{6+4}$ model at
$\tau=\mathrm{i}$ for a small $c$. (Right) The same figure at $\tau=\omega$
for $c\gtrsim 2$. Note that $m_{-}^{2}$ is independent of $c$ owing to the
nature of modular forms at $\tau=\omega$.

Figure 11:  (Left) Eigenvalues of stabilizers’ masses for $V_{6+4^{\prime}}$
model at $\tau=\mathrm{i}$ for $c\lesssim 5.68$, whereas the same figure at
$\tau=\mathrm{i}y$ for $c\gtrsim 5.68$, where $y$ is given by Eq. (5.23).
(Right) The same figure at $\tau=\mathrm{i}y$, where $y>1$ is approximately
given by the vacuum solution of Eq. (5.29).

Figure 12:  (Left) Eigenvalues of stabilizers’ masses for $V_{4+4^{\prime}}$
model at $\tau=\omega$. (Right) The same figure at the vacuum of
$\tau=-1/2+\mathrm{i}y$, where $y$ is approximately given by the solution of
Eq. (5.42).

##  6 Conclusion

In this paper, we study the moduli stabilization in the modular symmetric
SUGRA with (approximate) $\mathrm{U}(1)_{R}$ symmetry, in which the
superpotential consists of multiple modular form coupled to the matter
(stabilizer) fields. First, we study some properties of stabilizer fields and
show that the vacuum becomes dS or Minkowski if we have a multiple modular
forms in the superpotential. Our models focuses on the cases with one or two
modular forms coupled to the stabilizer fields. For the single stabilizer
case, we show that there exists vacuum at the zero point of the modular form,
$\tau=\mathrm{i},\ \omega$ and $\mathrm{i}\infty$, depending on the weight and
representation of the modular form. The zero points are also regarded as the
fixed point of the modular symmetry on the fundamental domain. For two
stabilizer fields coupled to modular forms, the vacuum can deviate from the
fixed points when non-trivial singlet modular form is added to the potential.
We obtain dS vacua owing to the non-vanishing modular form contribution. We
used perturbations of the modulus against the fixed points to study these
shifts of vacua analytically, and confirmed the resultant VEV of modulus also
numerically. As a consequence, there can exist the vacuum at
$\mathop{\mathrm{Im}}\tau\gg 1$, and all the vacua obtained in the paper are
CP-conserving.

In the string theory, there will exist many moduli other than $\tau$ which is
focused on in this paper. If the potential scale $\Lambda$ which is associated
also with modulus- and $X$-masses is much larger than scales relevant to
stabilizatoin terms of such moduli, our stabilization mechanism works well.
This ie because such moduli will be then much lighter than $\tau$ and a shift
of $\tau$ VEV which can be caused by light moduli will be very small [50].

Several issues which we did not discussed in this paper will be left. A global
$\mathrm{U}(1)_{R}$ symmetry will be absent in the quantum gravity since it is
a global symmetry, and is explicitly violated by $W_{0}(\tau)$ in the
superpotential. In such models, CP violation can be also obtained at the
vacuum owing to a competition between $W_{0}(\tau)$ and $Y^{\prime}(\tau)X$ in
the superpotential [33], where $Y^{\prime}(\tau)$ is a modular function. As an
another approach to CP violation, there may exist (explicit) violation of the
modular symmetry on top of $\mathrm{U}(1)_{R}$ violation, which is motivated
in an UV completed model. In such models, CP violation is caused at the vacuum
by an interference between the violation term and $Y(\tau)X$ in the
superpotential [51], where $Y(\tau)$ is a modular form. In this paper we
assume the Kähler potential given by Eq. (3.2), but we can also consider the
following additional term to the Kähler potential [52]:

$\displaystyle\delta\mathcal{K}=\sum_{a}\frac{|Y^{(k_{a})}_{a}X_{a}|^{2}}{(-\mathrm{i}\tau+\mathrm{i}\bar{\tau})^{-h}},$
(6.1)

which is introduced to realize a consistent inflationary scenario in Ref.
[18]. This term can have a large contribution in a large
$\mathop{\mathrm{Im}}\tau$ region. A general study of the moduli stabilization
including $W_{0}(\tau)$ and additional corrections can be essential in the
quantum gravity.

## Acknowledgments

The work of Y.A. is supported by JSPS Overseas Research Fellowships. This work
is supported in part by he Grant-in-Aid for Scientific Research from the
Ministry of Education, Science, Sports and Culture (MEXT), Japan No.
JP22K03601 (T.H.), JP23K03375 (T.K.) and JP24KJ0249 (K.N.).

##  Appendix A $A_{4}$ modular modular forms and their derivatives

| $\mathrm{i}\infty$ | $\mathrm{i}$ | $\omega$
---|---|---|---
$Y^{(4)}_{\bm{1}}$ | $1$ | $1.45576$ | $0$
$Y^{(4)}_{\bm{1}^{\prime}}$ | $0$ | $-1.45576$ | $-1.01247+1.75367\mathrm{i}$
$Y^{(6)}_{\bm{1}}$ | $1$ | $0$ | $2.88154$
Table 2:  The values of the modular forms at fixed points.

In this appendix, we summarize the features of the modular forms
$Y^{(4)}_{\bm{1}}$, $Y^{(4)}_{\bm{1}^{\prime}}$, and $Y^{(6)}_{\bm{1}}$ which
we use in the main part of this paper. The modular forms values at the fixed
points $\tau=\mathrm{i},\ \omega,\ \mathrm{i}\infty$ are listed in Tab. 2.

###  A.1 Modular form around $\tau=\mathrm{i}$

As discussed in Sec. 2.2.1, the $l$-th derivative of the modular form with the
weight $k$ satisfies

$\displaystyle(\mathrm{i}^{2l}-\mathrm{i}^{-k})\frac{\mathrm{d}^{l}\tilde{Y}^{(k)}_{\bm{r}}}{\mathrm{d}s^{l}}\bigg{|}_{s\to
0}=0,$ (A.1)

where $\tilde{Y}^{(k)}_{\bm{r}}(s)=(1-s)^{k}Y^{(k)}_{\bm{r}}(\tau)$ and
$s=(\tau-\mathrm{i})/(\tau+\mathrm{i})$. The modular form is expanded as

$\displaystyle
Y^{(k)}_{\bm{r}}=\sum_{l=0}\frac{1}{l!}\frac{\mathrm{d}^{l}Y^{(k)}_{\bm{r}}}{\mathrm{d}\tau^{l}}(\mathrm{i})(\tau-\mathrm{i})^{l},$
(A.2)

and the derivatives are related to $\tilde{Y}^{(k)}_{\bm{r}}(s)$ and its
derivatives as

$\displaystyle\frac{\mathrm{d}Y^{(k)}_{\bm{r}}}{\mathrm{d}\tau}$
$\displaystyle=\frac{\mathrm{i}(1-s)^{1+k}}{2}\biggl{[}k\tilde{Y}^{(k)}_{\bm{r}}-(1-s)\frac{\mathrm{d}\tilde{Y}^{(k)}_{\bm{r}}}{\mathrm{d}s}\biggr{]},$
(A.3) $\displaystyle\frac{\mathrm{d}^{2}Y^{(k)}_{\bm{r}}}{\mathrm{d}\tau^{2}}$
$\displaystyle=-\frac{(1-s)^{2+k}}{4}\biggl{[}k(1+k)\tilde{Y}^{(k)}_{\bm{r}}-(1-s)\Bigl{(}2(1+k)\frac{\mathrm{d}\tilde{Y}^{(k)}_{\bm{r}}}{\mathrm{d}s}-(1-s)\frac{\mathrm{d}^{2}\tilde{Y}^{(k)}_{\bm{r}}}{\mathrm{d}s^{2}}\Bigr{)}\biggr{]},$
(A.4)
$\displaystyle\frac{\mathrm{d}^{3}\tilde{Y}^{(k)}_{\bm{r}}}{\mathrm{d}\tau^{3}}$
$\displaystyle=-\frac{\mathrm{i}(1-s)^{3+k}}{8}\biggl{[}k(2+3k+k^{2})\tilde{Y}^{(k)}_{\bm{r}}-(1-s)\Bigl{\\{}3(2+3k+k^{2})\frac{\mathrm{d}\tilde{Y}^{(k)}_{\bm{r}}}{\mathrm{d}s}$
$\displaystyle\qquad-(1-s)\Bigl{(}3(2+k)\frac{\mathrm{d}^{2}\tilde{Y}^{(k)}_{\bm{r}}}{\mathrm{d}s^{2}}-(1-s)\frac{\mathrm{d}^{3}\tilde{Y}^{(k)}_{\bm{r}}}{\mathrm{d}s^{3}}\Bigr{)}\Bigr{\\}}\biggr{]},$
(A.5) $\displaystyle\frac{\mathrm{d}^{4}Y^{(k)}_{\bm{r}}}{\mathrm{d}\tau^{4}}$
$\displaystyle=\frac{(1-s)^{4+k}}{16}\biggl{[}k(6+11k+6k^{2}+k^{3})\tilde{Y}^{(k)}_{\bm{r}}-(1-s)\Bigl{\\{}4(6+11k+6k^{2}+k^{3})\frac{\mathrm{d}\tilde{Y}^{(k)}_{\bm{r}}}{\mathrm{d}s}$
$\displaystyle\qquad-(1-s)\Bigl{(}6(6+5k+k^{2})\frac{\mathrm{d}^{2}\tilde{Y}^{(k)}_{\bm{r}}}{\mathrm{d}s^{2}}-(1-s)\Bigl{(}4(3+k)\frac{\mathrm{d}^{3}\tilde{Y}^{(k)}_{\bm{r}}}{\mathrm{d}s^{3}}-(1-s)\frac{\mathrm{d}^{4}\tilde{Y}^{(k)}_{\bm{r}}}{\mathrm{d}s^{4}}\Bigr{)}\Bigr{)}\Bigr{\\}}\biggr{]}.$
(A.6)

In the left-hand side, $\tilde{Y}^{(k)}_{\bm{r}}$ is a function of $s$. We
note that CP transformation acts on the modular form as [53]

$\displaystyle\mathrm{CP}:Y^{(k)}_{\bm{1}^{(\prime)}}(\tau)\mapsto
Y^{(k)}_{\bm{1}^{(\prime)}}(-\bar{\tau})=\overline{Y^{(k)}_{\bm{1}^{(\prime)}}}(\tau),$
(A.7)

which implies that the modular forms take real values at $\tau=\mathrm{i}$.
From these equations, we obtain the following expansions of the weight $4$ and
$6$ modular forms around the fix point $\tau=\mathrm{i}$.

##### Weight 4.

$\displaystyle Y^{(4)}_{\bm{r}}$
$\displaystyle=\tilde{Y}^{(4)}_{\bm{r}}(0)+2\mathrm{i}\tilde{Y}^{(4)}_{\bm{r}}(0)(\tau-\mathrm{i})+\frac{1}{2}\biggl{[}-5\tilde{Y}^{(4)}_{\bm{r}}-\frac{1}{4}\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}(0)\biggr{]}(\tau-\mathrm{i})^{2}$
$\displaystyle\quad-\frac{\mathrm{i}}{3!}\biggl{[}15\tilde{Y}^{(4)}_{\bm{r}}(0)+\frac{9}{4}\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}(0)\biggr{]}(\tau-\mathrm{i})^{3}$
$\displaystyle\quad+\frac{1}{4!}\biggl{[}\frac{105}{2}\tilde{Y}^{(4)}_{\bm{r}}(0)+\frac{63}{4}\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}(0)+\frac{1}{16}\frac{\mathrm{d}^{4}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{4}}(0)\biggr{]}(\tau-\mathrm{i})^{4}+\mathcal{O}((\tau-\mathrm{i})^{5}),$
(A.8)

with $\bm{r}=\bm{1},\ \bm{1}^{\prime}$.

##### Weight 6.

$\displaystyle Y^{(6)}_{\bm{1}}$
$\displaystyle=-\frac{\mathrm{i}}{2}\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}(0)(\tau-\mathrm{i})+\frac{7}{4}\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{r}}}{\mathrm{d}s}(0)(\tau-\mathrm{i})^{2}$
$\displaystyle\quad+\frac{\mathrm{i}}{3!}\biggl{[}21\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}(0)+\frac{1}{8}\frac{\mathrm{d}^{3}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s^{3}}(0)\biggr{]}(\tau-\mathrm{i})^{3}$
$\displaystyle\quad+\frac{1}{4!}\biggl{[}-126\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}(0)-\frac{9}{4}\frac{\mathrm{d}^{3}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s^{3}}(0)\biggr{]}(\tau-\mathrm{i})^{4}+\mathcal{O}((\tau-\mathrm{i})^{5}).$
(A.9)

###  A.2 Modular form around $\tau=\omega$

As shown in Sec. 2.2.2, the $l$-th derivative of the modular form is
characterized by

$\displaystyle(\omega^{2l}-\omega^{q_{\bm{r}}-k})\frac{\mathrm{d}^{l}\hat{Y}^{(k)}_{\bm{r}}}{\mathrm{d}u^{l}}=0,$
(A.10)

where we introduce $u=(\tau-\omega)/(\tau-\omega^{2})$ and
$Y^{(k)}_{\bm{r}}=(1-u)^{k}\hat{Y}^{(k)}_{\bm{r}}$. $q_{\bm{r}}$ takes $0$ and
$1$ for the singlet $\bm{1}$ and the non-trivial singlet $\bm{1}^{\prime}$,
respectively. The modular form is expanded as

$\displaystyle
Y^{(k)}_{\bm{r}}=\sum_{l=0}\frac{1}{l!}\frac{\mathrm{d}^{l}Y^{(k)}_{\bm{r}}}{\mathrm{d}\tau^{l}}(\omega)(\tau-\omega)^{l},$
(A.11)

and the derivatives are given by

$\displaystyle\frac{\mathrm{d}Y^{(k)}_{\bm{r}}}{\mathrm{d}\tau}$
$\displaystyle=\frac{\mathrm{i}(1-u)^{1+k}}{\sqrt{3}}\biggl{[}k\hat{Y}^{(k)}_{\bm{r}}-(1-u)\frac{\mathrm{d}\hat{Y}^{(k)}_{\bm{r}}}{\mathrm{d}u}\biggr{]},$
(A.12)
$\displaystyle\frac{\mathrm{d}^{2}Y^{(k)}_{\bm{r}}}{\mathrm{d}\tau^{2}}$
$\displaystyle=-\frac{(1-u)^{2+k}}{3}\biggl{[}k(1+k)\hat{Y}^{(k)}_{\bm{r}}-(1-u)\Bigl{(}2(1+k)\frac{\mathrm{d}\hat{Y}^{(k)}_{\bm{r}}}{\mathrm{d}u}-(1-u)\frac{\mathrm{d}^{2}\hat{Y}^{(k)}_{\bm{r}}}{\mathrm{d}u^{2}}\Bigr{)}\biggr{]},$
(A.13)
$\displaystyle\frac{\mathrm{d}^{3}Y^{(k)}_{\bm{r}}}{\mathrm{d}\tau^{3}}$
$\displaystyle=-\frac{\mathrm{i}(1-u)^{3+k}}{3\sqrt{3}}\biggl{[}k(2+3k+k^{2})\hat{Y}^{(k)}_{\bm{r}}-(1-u)\Bigl{\\{}3(2+3k+k^{2})\frac{\mathrm{d}\hat{Y}^{(k)}_{\bm{r}}}{\mathrm{d}u}$
$\displaystyle\qquad-(1-u)\Bigl{(}3(2+k)\frac{\mathrm{d}^{2}\hat{Y}^{(k)}_{\bm{r}}}{\mathrm{d}u^{2}}-(1-u)\frac{\mathrm{d}^{3}\hat{Y}^{(k)}_{\bm{r}}}{\mathrm{d}u^{3}}\Bigr{)}\Bigr{\\}}\biggr{]}$
(A.14)
$\displaystyle\frac{\mathrm{d}^{4}Y^{(k)}_{\bm{r}}}{\mathrm{d}\tau^{4}}$
$\displaystyle=\frac{(1-u)^{4+k}}{9}\biggl{[}k(6+11k+6k^{2}+k^{3})\hat{Y}^{(k)}_{\bm{r}}-(1-u)\Bigl{\\{}4(6+11k+6k^{2}+k^{3})\frac{\mathrm{d}\hat{Y}^{(k)}_{\bm{r}}}{\mathrm{d}u}$
$\displaystyle\qquad-(1-u)\Bigl{(}6(6+5k+k^{2})\frac{\mathrm{d}^{2}\hat{Y}^{(k)}_{\bm{r}}}{\mathrm{d}u^{2}}-(1-u)\Bigl{(}4(3+k)\frac{\mathrm{d}^{3}\hat{Y}^{(k)}_{\bm{r}}}{\mathrm{d}u^{3}}-(1-u)\frac{\mathrm{d}^{4}\hat{Y}^{(k)}_{\bm{r}}}{\mathrm{d}u^{4}}\Bigr{)}\Bigr{)}\Bigr{\\}}\biggr{]}.$
(A.15)

$\hat{Y}^{(k)}_{\bm{r}}$ in the left-hand side is a function of $u$. From
these, we obtain the following expansions of the modular forms.

##### Weight 4 trivial singlet.

$\displaystyle Y^{(4)}_{\bm{1}}$
$\displaystyle=-\frac{\mathrm{i}}{\sqrt{3}}\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}(0)(\tau-\omega)+\frac{5}{3}\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}(0)(\tau-\omega)^{2}$
$\displaystyle\quad+\frac{\mathrm{i}}{3!}\biggl{[}\frac{30}{\sqrt{3}}\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}(0)\biggr{]}(\tau-\omega)^{3}$
$\displaystyle\quad+\frac{1}{4!}\biggl{[}-\frac{280}{3}\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}(0)+\frac{1}{9}\frac{\mathrm{d}^{4}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u^{4}}(0)\biggr{]}(\tau-\omega)^{4}+\mathcal{O}((\tau-\omega)^{5}).$
(A.16)

##### Weight 4 non-trivial singlet.

$\displaystyle Y^{(4)}_{\bm{1}^{\prime}}$
$\displaystyle=\hat{Y}^{(4)}_{\bm{1}^{\prime}}(0)+\frac{4\mathrm{i}}{\sqrt{3}}\hat{Y}^{(4)}_{\bm{1}^{\prime}}(0)(\tau-\omega)-\frac{10}{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}(0)(\tau-\omega)^{2}$
$\displaystyle\quad+\frac{\mathrm{i}}{3!}\biggl{[}-\frac{40}{\sqrt{3}}\hat{Y}^{(4)}_{\bm{1}^{\prime}}(0)+\frac{1}{3\sqrt{3}}\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}(0)\biggr{]}(\tau-\omega)^{3}$
$\displaystyle\quad+\frac{1}{4!}\biggl{[}\frac{280}{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}(0)-\frac{28}{9}\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}(0)\biggr{]}(\tau-\omega)^{4}+\mathcal{O}((\tau-\omega)^{5}).$
(A.17)

##### Weight 6 trivial singlet.

$\displaystyle Y^{(6)}_{\bm{1}}$
$\displaystyle=\hat{Y}^{(6)}_{\bm{1}}(0)+2\sqrt{3}\mathrm{i}\hat{Y}^{(6)}_{\bm{1}}(0)(\tau-\omega)-7\hat{Y}^{(6)}_{\bm{1}}(\tau-\omega)^{2}$
$\displaystyle\quad+\frac{\mathrm{i}}{3!}\biggl{[}-\frac{112}{\sqrt{3}}\hat{Y}^{(6)}_{\bm{1}}(0)+\frac{1}{3\sqrt{3}}\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}(0)\biggr{]}(\tau-\omega)^{3}$
$\displaystyle\quad+\frac{1}{4!}\biggl{[}336\hat{Y}^{(6)}_{\bm{1}}(0)-4\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}(0)\biggr{]}(\tau-\omega)^{4}+\mathcal{O}((\tau-\omega)^{5}).$
(A.18)

###  A.3 $q$ expansion of modular form and its derivative

In the $\mathop{\mathrm{Im}}\tau\gg 1$ region, the modular forms are expaded
by $q=\exp(2\pi\mathrm{i}\tau)$ [23, 25, 26, 24] as

$\displaystyle Y^{(4)}_{\bm{1}}(\tau)$
$\displaystyle=1+240q+2160q^{2}+6720q^{3}+\mathcal{O}(q^{4}),$ (A.19)
$\displaystyle Y^{(4)}_{\bm{1}^{\prime}}(\tau)$
$\displaystyle=q^{1/3}(-12q+96q^{2}-230q^{3}+\mathcal{O}(q^{4})),$ (A.20)
$\displaystyle Y^{(6)}_{\bm{1}}(\tau)$
$\displaystyle=1-504q-16632q^{2}-122976q^{3}+\mathcal{O}(q^{4}),$ (A.21)

which is called as the $q$-expansion. The derivatives are systematically given
by these formulae and listed below.

##### Weight 4 trivial singlet.

$\displaystyle\frac{\mathrm{d}Y^{(4)}_{\bm{1}}}{\mathrm{d}\tau}$
$\displaystyle=2\pi\mathrm{i}(240q+4320q^{2}+20160q^{3}+\mathcal{O}(q^{4})),$
(A.22)
$\displaystyle\frac{\mathrm{d}^{2}Y^{(4)}_{\bm{1}}}{\mathrm{d}\tau^{2}}$
$\displaystyle=-960\pi^{2}q-34560\pi^{2}q^{2}-40320\pi^{i}q^{3}+\mathcal{O}(q^{4}),$
(A.23)
$\displaystyle\frac{\mathrm{d}^{3}Y^{(4)}_{\bm{1}}}{\mathrm{d}\tau^{3}}$
$\displaystyle=-1920\mathrm{i}\pi^{3}q-138240\mathrm{i}\pi^{3}q^{2}-1451520\mathrm{i}\pi^{3}q^{3}+\mathcal{O}(q^{4}),$
(A.24)
$\displaystyle\frac{\mathrm{d}^{4}Y^{(4)}_{\bm{1}}}{\mathrm{d}\tau^{4}}$
$\displaystyle=3840\pi^{4}q+552960\pi^{4}q^{2}+8709120\pi^{4}q^{3}+\mathcal{O}(q^{4}).$
(A.25)

##### Weight 4 non-trivial singlet.

$\displaystyle\frac{\mathrm{d}Y^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}\tau}$
$\displaystyle=-2\pi\mathrm{i}q^{1/3}(4+128q+560q^{2}+\mathcal{O}(q^{3})),$
(A.26)
$\displaystyle\frac{\mathrm{d}^{2}Y^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}\tau^{2}}$
$\displaystyle=\frac{\pi^{2}q^{1/3}}{3}(16-2048q+15680q^{2}+\mathcal{O}(q^{3})),$
(A.27)
$\displaystyle\frac{\mathrm{d}^{3}Y^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}\tau^{3}}$
$\displaystyle=\frac{\mathrm{i}\pi^{3}q^{1/3}}{9}(32-16384q+219520q^{2}+\mathcal{O}(q^{3})),$
(A.28)
$\displaystyle\frac{\mathrm{d}^{4}Y^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}\tau^{4}}$
$\displaystyle=-\frac{\pi^{4}q^{1/3}}{27}(-64+131072q-3073280q^{2}+\mathcal{O}(q^{3})).$
(A.29)

##### Wight 6 trivial singlet.

$\displaystyle\frac{\mathrm{d}Y^{(6)}_{\bm{1}}}{\mathrm{d}\tau}$
$\displaystyle=-2\pi\mathrm{i}(504q+332604q^{2}+368928q^{3}+\mathcal{O}(q^{4})),$
(A.30)
$\displaystyle\frac{\mathrm{d}^{2}Y^{(6)}_{\bm{1}}}{\mathrm{d}\tau^{2}}$
$\displaystyle=2016\pi^{2}q+266112\pi^{2}q^{2}+4427136\pi^{2}q^{3}+\mathcal{O}(q^{4}),$
(A.31)
$\displaystyle\frac{\mathrm{d}^{3}Y^{(6)}_{\bm{1}}}{\mathrm{d}\tau^{3}}$
$\displaystyle=4032\pi^{3}\mathrm{i}q+106448\pi^{3}\mathrm{i}q^{2}+26562816\pi^{3}\mathrm{i}q^{3}+\mathcal{O}(q^{4}),$
(A.32)
$\displaystyle\frac{\mathrm{d}^{4}Y^{(6)}_{\bm{1}}}{\mathrm{d}\tau^{4}}$
$\displaystyle=-8064\pi^{4}q-4257792\pi^{4}q^{2}-159376896\pi^{4}q^{3}+\mathcal{O}(q^{4}).$
(A.33)

##  Appendix B Details of potential coefficients

In this appendix, we discuss the perturbative expansion of a single modular
form potential:

$\displaystyle
V^{(k)}_{\bm{r}}=(2\mathop{\mathrm{Im}}\tau)^{k}|Y^{(k)}_{\bm{r}}|^{2},$ (B.1)

where $k$ is the modular weight of the modular form $Y^{(k)}_{\bm{r}}$ and
$\bm{r}$ denotes the representaiton. We show the fourth order perturbation
around the fix points $\tau=\mathrm{i},\ \omega$.

###  B.1 Potential expansion around $\tau=\mathrm{i}$

####  B.1.1 Potential coefficients

We introduce the perturbation coefficient by

$\displaystyle V^{(k)}_{\bm{r}}=\sum_{i,j}C^{(k)}_{\bm{r}}[i:j]x^{i}y^{j},$
(B.2)

where the modulus is expanded as $\tau=\mathrm{i}+x+\mathrm{i}y$.

##### Weight 4 trivial and non-trivial singlets.

For the weight 4 singlets, the coefficients (B.2) are given by

$\displaystyle C^{(4)}_{\bm{r}}[0:0]$
$\displaystyle=16|\tilde{Y}^{(4)}_{\bm{r}}|^{2},$ (B.3a) $\displaystyle
C^{(4)}_{\bm{r}}[0:2]$
$\displaystyle=4\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\tilde{Y}^{(4)}_{\bm{r}}-16|\tilde{Y}^{(4)}_{\bm{r}}|^{2},$
(B.3b) $\displaystyle C^{(4)}_{\bm{r}}[0:3]$
$\displaystyle=-4\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\tilde{Y}^{(4)}_{\bm{r}}+16|\tilde{Y}^{(4)}_{\bm{r}}|^{2},$
(B.3c) $\displaystyle C^{(4)}_{\bm{r}}[0:4]$
$\displaystyle=-\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\tilde{Y}^{(4)}_{\bm{r}}+\frac{1}{12}\frac{\mathrm{d}^{4}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{4}}\tilde{Y}^{(4)}_{\bm{r}}+\frac{1}{4}\left|\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\right|^{2}-6|\tilde{Y}^{(4)}_{\bm{r}}|^{2},$
(B.3d) $\displaystyle C^{(4)}_{\bm{r}}[1:1]$
$\displaystyle=4\mathrm{i}\biggl{[}\overline{\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}}\tilde{Y}^{(4)}_{\bm{r}}-\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\overline{\tilde{Y}^{(4)}_{\bm{r}}}\biggr{]}=0,$
(B.3e) $\displaystyle C^{(4)}_{\bm{r}}[1:2]$
$\displaystyle=6\mathrm{i}\biggl{[}-\overline{\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}}\tilde{Y}^{(4)}_{\bm{r}}+\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\overline{\tilde{Y}^{(4)}_{\bm{r}}}\biggr{]}=0,$
(B.3f) $\displaystyle C^{(4)}_{\bm{r}}[1:3]$
$\displaystyle=\frac{\mathrm{i}}{6}\biggl{[}12\overline{\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}}\tilde{Y}^{(4)}_{\bm{r}}-12\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\overline{\tilde{Y}^{(4)}_{\bm{r}}}+\overline{\frac{\mathrm{d}^{4}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{4}}}\tilde{Y}^{(4)}_{\bm{r}}-\frac{\mathrm{d}^{4}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{4}}\overline{\tilde{Y}^{(4)}_{\bm{r}}}\biggr{]}=0,$
(B.3g) $\displaystyle C^{(4)}_{\bm{r}}[2:0]$
$\displaystyle=-4\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\tilde{Y}^{(4)}_{\bm{r}}+16|\tilde{Y}^{(4)}_{\bm{r}}|^{2},$
(B.3h) $\displaystyle C^{(4)}_{\bm{r}}[2:1]$
$\displaystyle=12\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\tilde{Y}^{(4)}_{\bm{r}}+16|\tilde{Y}^{(4)}_{\bm{r}}|^{2},$
(B.3i) $\displaystyle C^{(4)}_{\bm{r}}[2:2]$
$\displaystyle=\frac{1}{2}\left|\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\right|^{2}-18\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\tilde{Y}^{(4)}_{\bm{r}}-\frac{1}{2}\frac{\mathrm{d}^{4}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{4}}\tilde{Y}^{(4)}_{\bm{r}}+4|\tilde{Y}^{(4)}_{\bm{r}}|^{2}$
(B.3j) $\displaystyle C^{(4)}_{\bm{r}}[3:0]$
$\displaystyle=2\mathrm{i}\biggl{[}\overline{\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}}\tilde{Y}^{(4)}_{\bm{r}}-\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\overline{\tilde{Y}^{(4)}_{\bm{r}}}\biggr{]}=0$
(B.3k) $\displaystyle C^{(4)}_{\bm{r}}[3:1]$
$\displaystyle=\frac{\mathrm{i}}{6}\biggl{[}-60\overline{\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}}\tilde{Y}^{(4)}_{\bm{r}}-\overline{\frac{\mathrm{d}^{4}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{4}}}\tilde{Y}^{(4)}_{\bm{r}}+60\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\overline{\tilde{Y}^{(4)}_{\bm{r}}}+\frac{\mathrm{d}^{4}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{4}}\overline{\tilde{Y}^{(4)}_{\bm{r}}}\biggr{]}=0$
(B.3l) $\displaystyle C^{(4)}_{\bm{r}}[4:0]$
$\displaystyle=7\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\tilde{Y}^{(4)}_{\bm{r}}+\frac{1}{12}\frac{\mathrm{d}^{4}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{4}}\tilde{Y}^{(4)}_{\bm{r}}+\frac{1}{4}\left|\frac{\mathrm{d}^{2}\tilde{Y}^{(4)}_{\bm{r}}}{\mathrm{d}s^{2}}\right|^{2}+10|\tilde{Y}^{(4)}_{\bm{r}}|^{2}$
(B.3m) $\displaystyle C^{(4)}_{\bm{r}}[0:1]$
$\displaystyle=C^{(4)}_{\bm{r}}[1:0]=0,$ (B.3n)

where $\bm{r}=\bm{1},\ \bm{1}^{\prime}$. The numerical values of these
coefficients are summarized in Tabs. 3 and 4. The higher order fluctuation
than the fourth-order are noted as – in these tables and the same notation is
used in the following parts. The explicit forms of the potential expansion
become follows:

$\displaystyle V^{(4)}_{\bm{1}}(\mathrm{i}+x+\mathrm{i}y)$
$\displaystyle\approx 33.9+237\,y^{2}-237\,y^{3}+622\,y^{4}$
$\displaystyle\quad+\left(-305+846\,y-1.10\times
10^{3}\,y^{2}\right)\,x^{2}+1.20\times 10^{3}\,x^{4},$ (B.4) $\displaystyle
V^{(4)}_{\bm{1}^{\prime}}(\mathrm{i}+x+\mathrm{i}y)$ $\displaystyle\approx
33.9-88.0\,y^{2}+88.0\,y^{3}+11.6\,y^{4}$
$\displaystyle\quad+\left(20.2+128\,y+360\,y^{2}\right)\,x^{2}-62.7\,x^{4}.$
(B.5)

$i\backslash j$ | $0$ | $1$ | $2$ | $3$ | $4$
---|---|---|---|---|---
$0$ | $33.908$ | $0$ | $236.75$ | $-236.75$ | $621.76$
$1$ | $0$ | $0$ | $0$ | $0$ | –
$2$ | $-304.57$ | $845.88$ | $-1101.5$ | – | –
$3$ | $0$ | $0$ | – | – | –
$4$ | $1197.0$ | – | – | – | –
Table 3: Numerical values of $C^{(4)}_{\bm{1}}[i:j]$. $i\backslash j$ | $0$ | $1$ | $2$ | $3$ | $4$
---|---|---|---|---|---
$0$ | $33.908$ | $0$ | $-88.039$ | $88.039$ | $11.619$
$1$ | $0$ | $0$ | $0$ | $0$ | –
$2$ | $20.224$ | $128.49$ | $360.09$ | – | –
$3$ | $0$ | $0$ | – | – | –
$4$ | $-62.735$ | – | – | – | –
Table 4: Numerical values of $C^{(4)}_{\bm{1}^{\prime}}[i:j]$.

##### Weight 6 singlet.

For the weight 6 singlet modular form, the coefficients are

$\displaystyle C^{(6)}_{\bm{1}}[0:2]$
$\displaystyle=16\left|\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}\right|^{2},$
(B.6a) $\displaystyle C^{(6)}_{\bm{1}}[0:3]$
$\displaystyle=-16\left|\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}\right|^{2},$
(B.6b) $\displaystyle C^{(6)}_{\bm{1}}[0:4]$
$\displaystyle=\frac{4}{3}\frac{\mathrm{d}^{3}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s^{3}}\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}-12\left|\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}\right|^{2},$
(B.6c) $\displaystyle C^{(6)}_{\bm{1}}[1:3]$
$\displaystyle=\frac{4\mathrm{i}}{3}\biggl{[}\overline{\frac{\mathrm{d}^{3}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s^{3}}}\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}-\frac{\mathrm{d}^{3}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s^{3}}\overline{\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}}\biggr{]}=0,$
(B.6d) $\displaystyle C^{(6)}_{\bm{1}}[2:0]$
$\displaystyle=16\left|\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}\right|^{2},$
(B.6e) $\displaystyle C^{(6)}_{\bm{1}}[2:1]$
$\displaystyle=-16\left|\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}\right|^{2},$
(B.6f) $\displaystyle C^{(6)}_{\bm{1}}[2:2]$
$\displaystyle=-40\left|\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}\right|^{2},$
(B.6g) $\displaystyle C^{(6)}_{\bm{1}}[3:1]$
$\displaystyle=\frac{4\mathrm{i}}{3}\biggl{[}\overline{\frac{\mathrm{d}^{3}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s^{3}}}\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}-\frac{\mathrm{d}^{3}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s^{3}}\overline{\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}}\biggr{]}=0,$
(B.6h) $\displaystyle C^{(6)}_{\bm{1}}[4:0]$
$\displaystyle=-\frac{4}{3}\frac{\mathrm{d}^{3}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s^{3}}\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}-28\left|\frac{\mathrm{d}\tilde{Y}^{(6)}_{\bm{1}}}{\mathrm{d}s}\right|^{2},$
(B.6i) $\displaystyle C^{(6)}_{\bm{1}}[0:0]$
$\displaystyle=C^{(6)}_{\bm{1}}[0:1]=C^{(6)}_{\bm{1}}[1:0]=C^{(6)}_{\bm{1}}[1:1]=C^{(6)}_{\bm{1}}[1:2]=C^{(6)}_{\bm{1}}[3:0]=0$
(B.6j)

and their numerical values are listed in Tab. 5. From these, the explicit
expansion form of $V^{(6)}_{\bm{1}}$ is given by

$\displaystyle V^{(6)}_{\bm{1}}(\mathrm{i}+x+\mathrm{i}y)\times
10^{-3}\approx$ $\displaystyle~{}2.84y^{2}-2.84y^{3}+8.44y^{4}$
$\displaystyle+\left(2.84-2.84y-7.09y^{2}\right)x^{2}-15.5x^{4}.$ (B.7)

$i\backslash j$ | $0$ | $1$ | $2$ | $3$ | $4$
---|---|---|---|---|---
$0$ | $0$ | $0$ | $2836.9$ | $-2836.9$ | $8439.7$
$1$ | $0$ | $0$ | $0$ | $0$ | –
$2$ | $2836.9$ | $-2836.9$ | $-7092.2$ | – | –
$3$ | $0$ | $0$ | – | – | –
$4$ | $-15532$ | – | – | – | –
Table 5: Numerical values of $C^{(6)}_{\bm{1}}[i:j]$.

####  B.1.2 Coefficients of potential derivative

In Sec. 5.2, we introduced the following notation for the potential
derivatives around $\tau=\mathrm{i}$:

$\displaystyle\frac{\partial V_{r}}{\partial x}$
$\displaystyle=4\mathcal{X}_{4}(c)\biggl{[}x^{2}+\frac{1}{2}\frac{\mathcal{X}_{2}(c,y)}{\mathcal{X}_{4}(c)}\biggr{]}x,$
(B.8) $\displaystyle\frac{\partial V_{r}}{\partial y}$
$\displaystyle=\mathcal{Y}_{1}(c,x)+\mathcal{Y}_{2}(c,x)y+\mathcal{Y}_{3}(c)y^{3}+\mathcal{Y}_{4}(c)y^{3},$
(B.9)

where the label runs $r=6+4,\ 6+4^{\prime}$. These have following forms with
the coefficients (B.3) and (B.6)

$\displaystyle\mathcal{X}_{2}(c,y)$
$\displaystyle=C^{(6)}_{\bm{1}}[2:0]+c^{2}\bigl{(}C^{(4)}_{\bm{r}}[2:0]+C^{(4)}_{\bm{r}}[2:1]y+C^{(4)}_{\bm{r}}[2:2]y^{2}\bigr{)},$
(B.10) $\displaystyle\mathcal{X}_{4}(c)$
$\displaystyle=C^{(6)}_{\bm{1}}[4:0]+c^{2}C^{(4)}_{\bm{r}}[4:0],$ (B.11)
$\displaystyle\mathcal{Y}_{1}(c,x)$
$\displaystyle=\bigl{(}C^{(6)}_{\bm{1}}[2:1]+c^{2}C^{(4)}_{\bm{r}}[2:1]\bigr{)}x^{2},$
(B.12) $\displaystyle\mathcal{Y}_{2}(c,x)$
$\displaystyle=2\Bigl{[}C^{(6)}_{\bm{1}}[0:2]+C^{(6)}_{\bm{1}}[2:2]x^{2}+c^{2}\bigl{(}C^{(4)}_{\bm{r}}[0:1]+C^{(4)}_{\bm{r}}[2:2]x^{2}\bigr{)}\Bigr{]},$
(B.13) $\displaystyle\mathcal{Y}_{3}(c)$
$\displaystyle=3\bigl{(}C^{(6)}_{\bm{1}}[0:3]+c^{2}C^{(4)}_{\bm{r}}[0:3]\bigr{)},$
(B.14) $\displaystyle\mathcal{Y}_{4}(c)$
$\displaystyle=4\bigl{(}C^{(6)}_{\bm{1}}[0:4]+c^{2}C^{(4)}_{\bm{r}}[0:4]\bigr{)}.$
(B.15)

Here, we assume the expansion of fluctuations up to the fourth order as
discussed in the previous appendix, then $\mathcal{X}_{4}$, $\mathcal{Y}_{3}$,
and $\mathcal{Y}_{4}$ do not have the $x$ and $y$ dependence. The explicit
forms are summarized Tabs. 6 and 7.

| $V_{6+4}$ | $V_{6+4^{\prime}}$
---|---|---
$\mathcal{X}_{2}$ | $\begin{matrix}2836.9-2836.9y-7092.2y^{2}~{}~{}~{}~{}\\\ \qquad-c^{2}(304.57-845.88y+1101.5y^{2})\end{matrix}$ | $\begin{matrix}2836.9-2836.9y-7092.2y^{2}~{}~{}~{}~{}\\\ \qquad+c^{2}(20.224+128.49y+360.09y^{2})\end{matrix}$
$\mathcal{X}_{4}$ | $-15532+1197.0c^{2}$ | $-15532-62.735c^{2}$
Table 6: The concrete forms of the coefficients in Eqs. (B.10) and (B.11). | $V_{6+4}$ | $V_{6+4^{\prime}}$
---|---|---
$\mathcal{Y}_{1}$ | $(-2836.9+845.88c^{2})x^{2}$ | $(-2836.9+128.49c^{2})x^{2}$
$\mathcal{Y}_{2}$ | $\begin{matrix}2(2836.9-7092.2x^{2})~{}~{}~{}~{}\\\ \qquad+2c^{2}\left(236.75-1101.5x^{2}\right)\end{matrix}$ | $\begin{matrix}2(2836.9-7092.2x^{2})~{}~{}~{}~{}\\\ \qquad-2c^{2}\left(88.039-360.09x^{2}\right)\end{matrix}$
$\mathcal{Y}_{3}$ | $3\left(2836.9+236.75c^{2}\right)$ | $3\left(2836.9-88.039c^{2}\right)$
$\mathcal{Y}_{4}$ | $4\left(8439.7+621.76c^{2}\right)$ | $4\left(8439.7+11.619c^{2}\right)$
Table 7: The concrete forms of the coefficients in Eqs. (B.12)–(B.15).

###  B.2 Potential expansion around $\tau=\omega$

As similar to Eq. (B.2), we introduce the expansion coefficients around
$\tau=\omega$ as

$\displaystyle V^{(k)}_{\bm{r}}=\sum_{i,j}D^{(k)}_{\bm{r}}[i:j]x^{i}y^{j}$
(B.16)

where we introduce the fluctuation by $\tau=\omega+x+\mathrm{i}y$. The
concrete relations between these coefficients and the modular forms are given
as follows.

##### Weight 4 trival singlet.

For the weight 4 trivial singlet, the coefficients are

$\displaystyle D^{(4)}_{\bm{1}}[0:2]$
$\displaystyle=3\left|\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}\right|^{2},$
(B.17a) $\displaystyle D^{(4)}_{\bm{1}}[0:3]$
$\displaystyle=-2\sqrt{3}\left|\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}\right|^{2},$
(B.17b) $\displaystyle D^{(4)}_{\bm{1}}[0:4]$
$\displaystyle=-\left|\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}\right|^{2},$
(B.17c) $\displaystyle D^{(4)}_{\bm{1}}[2:0]$
$\displaystyle=3\left|\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}\right|^{2},$
(B.17d) $\displaystyle D^{(4)}_{\bm{1}}[2:1]$
$\displaystyle=-2\sqrt{3}\left|\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}\right|^{2},$
(B.17e) $\displaystyle D^{(4)}_{\bm{1}}[2:2]$
$\displaystyle=-6\left|\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}\right|^{2},$
(B.17f) $\displaystyle D^{(4)}_{\bm{1}}[4:0]$
$\displaystyle=-5\left|\frac{\mathrm{d}\hat{Y}^{(4)}_{\bm{1}}}{\mathrm{d}u}\right|^{2}$
(B.17g) $\displaystyle D^{(4)}_{\bm{1}}[0:0]$
$\displaystyle=D^{(4)}_{\bm{1}}[0:1]=D^{(4)}_{\bm{1}}[1:0]=D^{(4)}_{\bm{1}}[1:1]=D^{(4)}_{\bm{1}}[1:2]=D^{(4)}_{\bm{1}}[1:3]$
$\displaystyle=D^{(4)}_{\bm{1}}[3:0]=D^{(4)}_{\bm{1}}[3:1]=0,$ (B.17h)

and their numerical values are listed in Tab. 8. The potential expansion is
expressed as

$\displaystyle V^{(4)}_{\bm{1}}(\omega+x+\mathrm{i}y)\approx$ $\displaystyle
328y^{2}-379y^{3}-109y^{4}+\left(328-379y-656y^{2}\right)\,x^{2}-546x^{4}.$
(B.18)

$i\backslash j$ | $0$ | $1$ | $2$ | $3$ | $4$
---|---|---|---|---|---
$0$ | $0$ | $0$ | $327.80$ | $-378.51$ | $-109.27$
$1$ | $0$ | $0$ | $0$ | $0$ | –
$2$ | $327.80$ | $-378.51$ | $-655.60$ | – | –
$3$ | $0$ | $0$ | – | – | –
$4$ | $-546.33$ | – | – | – | –
Table 8:  Numerical values of $D^{(4)}_{\bm{1}}[i:j]$.

##### Weight 4 non-trivial singlet.

For the weight 4 non-trivial singlet, the expansion coefficients are given by

$\displaystyle D^{(4)}_{\bm{1}^{\prime}}[0:0]$
$\displaystyle=9|\hat{Y}^{(4)}_{\bm{1}^{\prime}}|^{2},$ (B.19a) $\displaystyle
D^{(4)}_{\bm{1}^{\prime}}[0:2]$
$\displaystyle=-12|\hat{Y}^{(4)}_{\bm{1}^{\prime}}|^{2}$ (B.19b)
$\displaystyle D^{(4)}_{\bm{1}^{\prime}}[0:3]$
$\displaystyle=\frac{1}{2\sqrt{3}}\biggl{[}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}}\hat{Y}^{(4)}_{\bm{1}^{\prime}}+\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(4)}_{\bm{1}^{\prime}}}+48|\hat{Y}^{(4)}_{\bm{1}^{\prime}}|^{2}\biggr{]}$
(B.19c) $\displaystyle D^{(4)}_{\bm{1}^{\prime}}[0:4]$
$\displaystyle=-\frac{1}{2}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}}\hat{Y}^{(4)}_{\bm{1}^{\prime}}-\frac{1}{2}\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(4)}_{\bm{1}^{\prime}}}-6|\hat{Y}^{(4)}_{\bm{1}^{\prime}}|^{2},$
(B.19d) $\displaystyle D^{(4)}_{\bm{1}^{\prime}}[1:2]$
$\displaystyle=\frac{\sqrt{3}\mathrm{i}}{2}\biggl{[}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}}\hat{Y}^{(4)}_{\bm{1}^{\prime}}-\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(4)}_{\bm{1}^{\prime}}}\biggr{]}$
(B.19e) $\displaystyle D^{(4)}_{\bm{1}^{\prime}}[1:3]$
$\displaystyle=2\mathrm{i}\biggl{[}-\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}}\hat{Y}^{(4)}_{\bm{1}^{\prime}}+\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(4)}_{\bm{1}^{\prime}}}\biggr{]}$
(B.19f) $\displaystyle D^{(4)}_{\bm{1}^{\prime}}[2:0]$
$\displaystyle=-12|\hat{Y}^{(4)}_{\bm{1}^{\prime}}|^{2}$ (B.19g)
$\displaystyle D^{(4)}_{\bm{1}^{\prime}}[2:1]$
$\displaystyle=-\frac{\sqrt{3}}{2}\biggl{[}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}}\hat{Y}^{(4)}_{\bm{1}^{\prime}}+\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(4)}_{\bm{1}^{\prime}}}-16|\hat{Y}^{(4)}_{\bm{1}^{\prime}}|^{2}\biggr{]}$
(B.19h) $\displaystyle D^{(4)}_{\bm{1}^{\prime}}[2:2]$
$\displaystyle=3\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}}\hat{Y}^{(4)}_{\bm{1}^{\prime}}+3\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(4)}_{\bm{1}^{\prime}}}+4|\hat{Y}^{(4)}_{\bm{1}^{\prime}}|^{2}$
(B.19i) $\displaystyle D^{(4)}_{\bm{1}^{\prime}}[3:0]$
$\displaystyle=\frac{\mathrm{i}}{2\sqrt{3}}\biggl{[}-\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}}\hat{Y}^{(4)}_{\bm{1}^{\prime}}+\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(4)}_{\bm{1}^{\prime}}}\biggr{]}$
(B.19j) $\displaystyle D^{(4)}_{\bm{1}^{\prime}}[3:1]$
$\displaystyle=2\mathrm{i}\biggl{[}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}}\hat{Y}^{(4)}_{\bm{1}^{\prime}}-\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(4)}_{\bm{1}^{\prime}}}\biggr{]}$
(B.19k) $\displaystyle D^{(4)}_{\bm{1}^{\prime}}[4:0]$
$\displaystyle=-\frac{1}{2}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}}\hat{Y}^{(4)}_{\bm{1}^{\prime}}-\frac{1}{2}\frac{\mathrm{d}^{3}\hat{Y}^{(4)}_{\bm{1}^{\prime}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(4)}_{\bm{1}^{\prime}}}+10|\hat{Y}^{(4)}_{\bm{1}^{\prime}}|^{2},$
(B.19l) $\displaystyle D^{(4)}_{\bm{1}^{\prime}}[0:1]$
$\displaystyle=D^{(4)}_{\bm{1}^{\prime}}[1:0]=D^{(4)}_{\bm{1}^{\prime}}[1:1]=0,$
(B.19m)

and the numerical values are summarized in Tab. 9.

$\displaystyle V^{(4)}_{\bm{1}^{\prime}}(\omega+x+\mathrm{i}y)$
$\displaystyle\approx 36.9-49.2y^{2}-351y^{3}+485y^{4}$
$\displaystyle\quad+(-49.2+940y+3.04\times 10^{3}y^{2})x^{2}+551x^{4}$ (B.20)

$i\backslash j$ | $0$ | $1$ | $2$ | $3$ | $4$
---|---|---|---|---|---
$0$ | $36.904$ | $0$ | $-49.206$ | $-351.30$ | $485.45$
$1$ | $0$ | $0$ | $0$ | $0$ | –
$2$ | $-49.206$ | $940.26$ | $3043.9$ | – | –
$3$ | $0$ | $0$ | – | – | –
$4$ | $551.06$ | – | – | – | –
Table 9:  Numerical values of $D^{(4)}_{\bm{1}^{\prime}}[i:j]$.

##### Weight 6 trivial singlet.

The expansion coefficients for the weight 6 trivial singlet become

$\displaystyle D^{(6)}_{\bm{1}}[0:0]$
$\displaystyle=27|\hat{Y}^{(6)}_{\bm{1}}|^{2},$ (B.21a) $\displaystyle
D^{(6)}_{\bm{1}}[0:2]$ $\displaystyle=-54|\hat{Y}^{(6)}_{\bm{1}}|^{2},$
(B.21b) $\displaystyle D^{(6)}_{\bm{1}}[0:3]$
$\displaystyle=\frac{\sqrt{3}}{2}\biggl{[}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}}\hat{Y}^{(6)}_{\bm{1}}+\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(6)}_{\bm{1}}}\biggr{]}+36\sqrt{3}|\hat{Y}^{(6)}_{\bm{1}}|^{2},$
(B.21c) $\displaystyle D^{(6)}_{\bm{1}}[0:4]$
$\displaystyle=-\frac{3}{2}\biggl{[}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}}\hat{Y}^{(6)}_{\bm{1}}+\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(6)}_{\bm{1}}}\biggr{]}-9|\hat{Y}^{(6)}_{\bm{1}}|^{2},$
(B.21d) $\displaystyle D^{(6)}_{\bm{1}}[1:2]$
$\displaystyle=\frac{3\sqrt{3}\mathrm{i}}{2}\biggl{[}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}}\hat{Y}^{(6)}_{\bm{1}}-\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(6)}_{\bm{1}}}\biggr{]},$
(B.21e) $\displaystyle D^{(6)}_{\bm{1}}[1:3]$
$\displaystyle=-6\mathrm{i}\biggl{[}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}}\hat{Y}^{(6)}_{\bm{1}}-\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(6)}_{\bm{1}}}\biggr{]},$
(B.21f) $\displaystyle D^{(6)}_{\bm{1}}[2:0]$
$\displaystyle=-54|\hat{Y}^{(6)}_{\bm{1}}|^{2},$ (B.21g) $\displaystyle
D^{(6)}_{\bm{1}}[2:1]$
$\displaystyle=-\frac{3\sqrt{3}}{2}\biggl{[}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}}\hat{Y}^{(6)}_{\bm{1}}+\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(6)}_{\bm{1}}}\biggr{]}+36\sqrt{3}|\hat{Y}^{(6)}_{\bm{1}}|^{2},$
(B.21h) $\displaystyle D^{(6)}_{\bm{1}}[2:2]$
$\displaystyle=9\biggl{[}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}}\hat{Y}^{(6)}_{\bm{1}}+\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(6)}_{\bm{1}}}\biggr{]}+6|\hat{Y}^{(6)}_{\bm{1}}|^{2},$
(B.21i) $\displaystyle D^{(6)}_{\bm{1}}[3:0]$
$\displaystyle=-\frac{\sqrt{3}\mathrm{i}}{2}\biggl{[}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}}\hat{Y}^{(6)}_{\bm{1}}-\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(6)}_{\bm{1}}}\biggr{]},$
(B.21j) $\displaystyle D^{(6)}_{\bm{1}}[3:1]$
$\displaystyle=6\mathrm{i}\biggl{[}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}}\hat{Y}^{(6)}_{\bm{1}}-\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(6)}_{\bm{1}}}\biggr{]},$
(B.21k) $\displaystyle D^{(6)}_{\bm{1}}[4:0]$
$\displaystyle=-\frac{3}{2}\biggl{[}\overline{\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}}\hat{Y}^{(6)}_{\bm{1}}+\frac{\mathrm{d}^{3}\hat{Y}^{(6)}_{\bm{1}}}{\mathrm{d}u^{3}}\overline{\hat{Y}^{(6)}_{\bm{1}}}\biggr{]}+63|\hat{Y}^{(6)}_{\bm{1}}|^{2},$
(B.21l) $\displaystyle D^{(6)}_{\bm{1}}[0:1]$
$\displaystyle=D^{(6)}_{\bm{1}}[1:0]=D^{(6)}_{\bm{1}}[1:1]=0,$ (B.21m)

and their numerical values are listed in Tab. 10. We expand the potential as

$\displaystyle V^{(6)}_{\bm{1}}(\omega+x+\mathrm{i}y)$
$\displaystyle=224-448y^{2}+7510y^{3}-12200y^{4}$
$\displaystyle\quad+\left(-449-20500y+73200y^{2}\right)x^{2}-11600x^{4}.$
(B.22)

$i\backslash j$ | $0$ | $1$ | $2$ | $3$ | $4$
---|---|---|---|---|---
$0$ | $224.19$ | $0$ | $-448.38$ | $7514.8$ | $-12194$
$1$ | $0$ | $0$ | $0$ | $0$ | –
$2$ | $-448.38$ | $-20473$ | $73164$ | – | –
$3$ | $0$ | $0$ | – | – | –
$4$ | $-11596$ | – | – | – | –
Table 10:  Numerical values of $D^{(6)}_{\bm{1}}[i:j]$.

##  Appendix C Explicit form of Kähler metric

###  C.1 Single stabilizer case

When the number of the stabilizer field is unity, we assume the following
Kähler potential

$\displaystyle\mathcal{K}=Z_{X}|X|^{2}-\frac{Z_{X}^{2}|X|^{4}}{4\Lambda_{X}^{2}}-h\log(-\mathrm{i}\tau+\mathrm{i}\bar{\tau}),\qquad
Z_{X}=(-\mathrm{i}\tau+\mathrm{i}\bar{\tau})^{k_{X}}.$ (C.1)

The Kähler metric is introduced by

$\displaystyle\mathcal{K}_{i\bar{j}}$
$\displaystyle=\begin{pmatrix}\mathcal{K}_{\tau\bar{\tau}}&\mathcal{K}_{\tau\bar{X}}\\\
\mathcal{K}_{X\bar{\tau}}&\mathcal{K}_{X\bar{X}}\end{pmatrix},$ (C.2)

and the explicit forms of these components are given by

$\displaystyle\mathcal{K}_{\tau\bar{\tau}}$
$\displaystyle=\frac{2h+2(-1+k_{X})k_{X}(2\mathop{\mathrm{Im}}\tau)^{k_{X}}|X|^{2}-k_{X}(-1+2k_{X})k_{X}(2\mathop{\mathrm{Im}}\tau)^{2k_{X}}|X|^{4}/\Lambda_{X}^{2}}{8(\mathop{\mathrm{Im}}\tau)^{2}},$
(C.3a) $\displaystyle\mathcal{K}_{\tau\bar{X}}$
$\displaystyle=-\mathrm{i}k_{X}(2\mathop{\mathrm{Im}}\tau)^{-1+k_{X}}X(1-(2\mathop{\mathrm{Im}}\tau)^{k_{X}}|X|^{2}/\Lambda_{X}^{2},$
(C.3b) $\displaystyle\mathcal{K}_{X\bar{\tau}}$
$\displaystyle=\mathrm{i}k_{X}(2\mathop{\mathrm{Im}}\tau)^{-1+k_{X}}\bar{X}(1-(2\mathop{\mathrm{Im}}\tau)^{k_{X}}|X|^{2}/\Lambda_{X}^{2},$
(C.3c) $\displaystyle\mathcal{K}_{X\bar{X}}$
$\displaystyle=(2\mathop{\mathrm{Im}}\tau)^{k_{X}}(1-(2\mathop{\mathrm{Im}}\tau)^{k_{X}}|X|^{2}/\Lambda_{X}^{2}).$
(C.3d)

If the VEV of $X$ is $\braket{X}=0$, $\mathcal{K}_{X\bar{X}}$ is reduced to
$Z_{X}$.

###  C.2 Double stabilizer case

The Kähler potential is

$\displaystyle\mathcal{K}$
$\displaystyle=Z_{X}|X|^{2}-\frac{Z_{X}^{2}|X|^{4}}{4\Lambda_{X}^{2}}+Z_{x}|x|^{2}-\frac{Z_{x}|x|^{4}}{4\Lambda_{x}^{2}}-h\log(-\mathrm{i}\tau+\mathrm{i}\bar{\tau}),$
(C.4) $\displaystyle Z_{X}$
$\displaystyle=(-\mathrm{i}\tau+\mathrm{i}\bar{\tau})^{k_{X}},\qquad
Z_{x}=(-\mathrm{i}\tau+\mathrm{i}\bar{\tau})^{k_{x}}.$ (C.5)

We write the Kähler metric of this case as

$\displaystyle\mathcal{K}_{i\bar{j}}$
$\displaystyle=\begin{pmatrix}\mathcal{K}_{\tau\bar{\tau}}&\mathcal{K}_{\tau\bar{X}}&\mathcal{K}_{\tau\bar{x}}\\\
\mathcal{K}_{X\bar{\tau}}&\mathcal{K}_{X\bar{X}}&\mathcal{K}_{X\bar{x}}\\\
\mathcal{K}_{x\bar{\tau}}&\mathcal{K}_{x\bar{X}}&\mathcal{K}_{x\bar{x}}\end{pmatrix},$
(C.6)

and the components are

$\displaystyle\mathcal{K}_{\tau\bar{\tau}}$
$\displaystyle=\frac{1}{8(\mathop{\mathrm{Im}}\tau)^{2}}\biggl{[}2h+2(-1+k_{X})k_{X}(2\mathop{\mathrm{Im}}\tau)^{k_{X}}|X|^{2}-k_{X}(-1+2k_{X})k_{X}(2\mathop{\mathrm{Im}}\tau)^{2k_{X}}|X|^{4}/\Lambda_{X}^{2}$
$\displaystyle\qquad+2(-1+k_{x})k_{x}(2\mathop{\mathrm{Im}}\tau)^{k_{x}}|x|^{2}-k_{x}(-1+2k_{x})k_{x}(2\mathop{\mathrm{Im}}\tau)^{2k_{x}}|x|^{4}/\Lambda_{X}^{2}\biggr{]},$
(C.7) $\displaystyle\mathcal{K}_{\tau\bar{X}}$
$\displaystyle=-\mathrm{i}k_{X}(2\mathop{\mathrm{Im}}\tau)^{-1+k_{X}}X(1-(2\mathop{\mathrm{Im}}\tau)^{k_{X}}|X|^{2}/\Lambda_{X}^{2},$
(C.8) $\displaystyle\mathcal{K}_{\tau\bar{x}}$
$\displaystyle=-\mathrm{i}k_{x}(2\mathop{\mathrm{Im}}\tau)^{-1+k_{x}}x(1-(2\mathop{\mathrm{Im}}\tau)^{k_{x}}|x|^{2}/\Lambda_{x}^{2},$
(C.9) $\displaystyle\mathcal{K}_{X\bar{\tau}}$
$\displaystyle=\mathrm{i}k_{X}(2\mathop{\mathrm{Im}}\tau)^{-1+k_{X}}\bar{X}(1-(2\mathop{\mathrm{Im}}\tau)^{k_{X}}|X|^{2}/\Lambda_{X}^{2},$
(C.10) $\displaystyle\mathcal{K}_{X\bar{X}}$
$\displaystyle=(2\mathop{\mathrm{Im}}\tau)^{k_{X}}(1-(2\mathop{\mathrm{Im}}\tau)^{k_{X}}|X|^{2}/\Lambda_{X}^{2}),$
(C.11) $\displaystyle\mathcal{K}_{x\bar{\tau}}$
$\displaystyle=\mathrm{i}k_{x}(2\mathop{\mathrm{Im}}\tau)^{-1+k_{x}}\bar{x}(1-(2\mathop{\mathrm{Im}}\tau)^{k_{x}}|x|^{2}/\Lambda_{x}^{2},$
(C.12) $\displaystyle\mathcal{K}_{x\bar{x}}$
$\displaystyle=(2\mathop{\mathrm{Im}}\tau)^{k_{x}}(1-(2\mathop{\mathrm{Im}}\tau)^{k_{x}}|x|^{2}/\Lambda_{x}^{2}),$
(C.13) $\displaystyle\mathcal{K}_{X\bar{x}}$
$\displaystyle=\mathcal{K}_{x\bar{X}}=0.$ (C.14)

We note that $\mathcal{K}_{X\bar{x}}$ and $\mathcal{K}_{x\bar{X}}$ are
vanished because of $\mathrm{U}(1)_{R}$ symmetry. If
$\braket{X}=\braket{x}=0$, the diagonal components reduce as
$\mathcal{K}_{X\bar{X}}=Z_{X}$, $\mathcal{K}_{x\bar{x}}=Z_{x}$.

##  Appendix D Hessian analysis

In this appendix, we will summarize the components of the Hessian matrices in
the potential vacua for $V_{6+4}$, $V_{6+4^{\prime}}$, and $V_{4+4^{\prime}}$,
respectively. The definition is given by Eq. (5.9). Here we leave the
fluctuation dependence of $(x,y)$, which is introduced by
$\tau=\tau_{*}+x+\mathrm{i}y$, where $\tau_{*}$ denotes the modulus VEV.

###  D.1 $V_{6+4}$

  * •

$\tau=\mathrm{i}$

$\displaystyle H_{11}(x,y;c)$
$\displaystyle=10^{3}(-186x^{2}+2(2.84-2.84y-7.09y^{2}))$
$\displaystyle\qquad+c^{2}(14400x^{2}+2(-305+846y-1100y^{2})),$ (D.1)
$\displaystyle H_{12}(x,y;c)$
$\displaystyle=H_{21}(x,y;c)=2c^{2}x(846-2200y),$ (D.2) $\displaystyle
H_{22}(x,y;c)$ $\displaystyle=10^{3}(5.68-14.18x^{2}-17.04y+101.28y^{2})$
$\displaystyle\qquad+c^{2}(474-2200x^{2}-1422y+7464y^{2}).$ (D.3)

  * •

$\tau=\omega$

$\displaystyle H_{11}(x,y;c)$
$\displaystyle=-139200x^{2}+2(-449-20500y+73200y^{2})$
$\displaystyle\qquad+c^{2}(-6552x^{2}+2(328-379y-656y^{2})),$ (D.4)
$\displaystyle H_{12}(x,y;c)$
$\displaystyle=H_{21}(x,y;c)=2c^{2}x(-379-1310y),$ (D.5) $\displaystyle
H_{22}(x,y;c)$ $\displaystyle=-896+146400x^{2}+45060y-146400y^{2}$
$\displaystyle\qquad+c^{2}(656-1312x^{2}-2274y-1308y^{2}).$ (D.6)

###  D.2 $V_{6+4^{\prime}}$

  * •

$\tau=\mathrm{i}$

$\displaystyle H_{11}(x,y;c)$ $\displaystyle\sim 5.67\times
10^{3}+c^{2}(720y^{2}+257y+40.4)$ $\displaystyle\qquad-(1.86\times
10^{5}+753c^{2})x^{2},$ (D.7) $\displaystyle H_{12}(x,y;c)$
$\displaystyle=H_{21}(c,x,y)\sim 1.44\times 10^{3}c^{2}xy+257c^{2}x,$ (D.8)
$\displaystyle H_{22}(x,y;c)$ $\displaystyle\sim 5.67\times
10^{3}-176c^{2}-(1.42\times 10^{4}-720c^{2})x^{2}$
$\displaystyle\qquad+(1.70\times 10^{4}-528c^{2})y+(1.01\times
10^{5}+139c^{2})y^{2}.$ (D.9)

  * •

$\tau=\mathrm{i}\infty$

$\displaystyle H_{11}(x,y,;c)$ $\displaystyle=(2.5805\times
10^{5})\pi^{2}y^{6}\mathrm{e}^{-2\pi y}\cos(2\pi x),$ (D.10) $\displaystyle
H_{12}(x,y,;c)$ $\displaystyle=H_{21}(x,y,;c)=-(2.5805\times 10^{5})\pi
y^{5}(\pi y-3)\mathrm{e}^{-2\pi y}\sin(2\pi x),$ (D.11) $\displaystyle
H_{22}(x,y,;c)$ $\displaystyle=128y^{2}\bigg{\\{}3y^{2}\left(84672e^{-4\pi
y}\left(8\pi^{2}y^{2}-24\pi y+15\right)+5\right)$
$\displaystyle\qquad-1008e^{-2\pi y}y^{2}\left(15-12\pi
y+2\pi^{2}y^{2}\right)\cos(2\pi x)$
$\displaystyle\qquad+c^{2}8\mathrm{e}^{-\frac{4\pi
y}{3}}\left(4\pi^{2}y^{2}-24\pi y+27\right)\bigg{\\}}.$ (D.12)

###  D.3 $V_{4+4^{\prime}}$

  * •

$\tau=\omega$

$\displaystyle H_{11}(x,y;c)$ $\displaystyle=655.60-6556.0x^{2}$
$\displaystyle\qquad+c^{2}\left(6612.7x^{2}+6087.8y^{2}+1880.5y-98.411\right),$
(D.13) $\displaystyle H_{12}(x,y;c)$ $\displaystyle=2c^{2}(940.26+6087.8y)x,$
(D.14) $\displaystyle H_{22}(x,y;c)$
$\displaystyle=655.60+2271.0y-1311.2y^{2}-1311.2x^{2}$
$\displaystyle\qquad+c^{2}\left(-98.411+2107.8y+5825.4y^{2}+6087.8x^{2}\right).$
(D.15)

  * •

$\tau=-1/2+\mathrm{i}\infty$

$\displaystyle H_{11}(x,y;c)$
$\displaystyle=-30720\pi^{2}y^{4}\mathrm{e}^{-2\pi y}\cos(2\pi x),$ (D.16)
$\displaystyle H_{12}(x,y;c)$ $\displaystyle=H_{21}(x,y,;c)=30720\pi(\pi
y-2)y^{3}e^{-2\pi y}\sin(2\pi x),$ (D.17) $\displaystyle H_{22}(x,y;c)$
$\displaystyle=64y^{2}\bigg{\\{}3+480e^{-2\pi y}\left(3-4\pi
y+\pi^{2}y^{2}\right)\cos(2\pi x)$ $\displaystyle\qquad\qquad+57600e^{-4\pi
y}\left(3-8\pi y+4\pi^{2}y^{2}\right)$
$\displaystyle\qquad\qquad+16c^{2}e^{-\frac{4\pi y}{3}}\left(27-24\pi
y+4\pi^{2}y^{2}\right)\bigg{\\}}.$ (D.18)

## References

  * [1] F. Feruglio, Are neutrino masses modular forms?, 1706.08749 [hep-ph].
  * [2] R. de Adelhart Toorop, F. Feruglio, and C. Hagedorn, Finite Modular Groups and Lepton Mixing, Nucl. Phys. B 858 (2012) 437–467 [1112.1340 [hep-ph]].
  * [3] T. Kobayashi, K. Tanaka, and T. H. Tatsuishi, Neutrino mixing from finite modular groups, Phys. Rev. D 98 no. 1, (2018) 016004 [1803.10391 [hep-ph]].
  * [4] J. T. Penedo and S. T. Petcov, Lepton Masses and Mixing from Modular $S_{4}$ Symmetry, Nucl. Phys. B 939 (2019) 292–307 [1806.11040 [hep-ph]].
  * [5] P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov, Modular A5 symmetry for flavour model building, JHEP 04 (2019) 174 [1812.02158 [hep-ph]].
  * [6] G.-J. Ding, S. F. King, and X.-G. Liu, Neutrino mass and mixing with $A_{5}$ modular symmetry, Phys. Rev. D 100 no. 11, (2019) 115005 [1903.12588 [hep-ph]].
  * [7] X.-G. Liu and G.-J. Ding, Neutrino Masses and Mixing from Double Covering of Finite Modular Groups, JHEP 08 (2019) 134 [1907.01488 [hep-ph]].
  * [8] P. P. Novichkov, J. T. Penedo, and S. T. Petcov, Double cover of modular $S_{4}$ for flavour model building, Nucl. Phys. B 963 (2021) 115301 [2006.03058 [hep-ph]].
  * [9] X.-G. Liu, C.-Y. Yao, and G.-J. Ding, Modular invariant quark and lepton models in double covering of $S_{4}$ modular group, Phys. Rev. D 103 no. 5, (2021) 056013 [2006.10722 [hep-ph]].
  * [10] X.-G. Liu, C.-Y. Yao, B.-Y. Qu, and G.-J. Ding, Half-integral weight modular forms and application to neutrino mass models, Phys. Rev. D 102 no. 11, (2020) 115035 [2007.13706 [hep-ph]].
  * [11] T. Kobayashi and M. Tanimoto, Modular flavor symmetric models, 2307.03384 [hep-ph].
  * [12] G.-J. Ding and S. F. King, Neutrino Mass and Mixing with Modular Symmetry, 2311.09282 [hep-ph].
  * [13] T. Kobayashi, D. Nitta, and Y. Urakawa, Modular invariant inflation, JCAP 08 (2016) 014 [1604.02995 [hep-th]].
  * [14] T. Higaki and F. Takahashi, Elliptic inflation: interpolating from natural inflation to R2-inflation, JHEP 03 (2015) 129 [1501.02354 [hep-ph]].
  * [15] R. Schimmrigk, Modular Inflation Observables and $j$-Inflation Phenomenology, JHEP 09 (2017) 043 [1612.09559 [hep-th]].
  * [16] M. Lynker and R. Schimmrigk, Modular Inflation at Higher Level $N$, JCAP 06 (2019) 036 [1902.04625 [astro-ph.CO]].
  * [17] R. Schimmrigk, Large and small field inflation from hyperbolic sigma models, Phys. Rev. D 105 no. 6, (2022) 063541 [2108.05400 [hep-th]].
  * [18] Y. Abe, T. Higaki, F. Kaneko, T. Kobayashi, and H. Otsuka, Moduli inflation from modular flavor symmetries, JHEP 06 (2023) 187 [2303.02947 [hep-ph]].
  * [19] G.-J. Ding, S.-Y. Jiang, and W. Zhao, Modular Invariant Slow Roll Inflation, 2405.06497 [hep-ph].
  * [20] S. Ferrara, D. Lust, A. D. Shapere, and S. Theisen, Modular Invariance in Supersymmetric Field Theories, Phys. Lett. B 225 (1989) 363.
  * [21] J. Wess and J. Bagger, Supersymmetry and supergravity.  Princeton University Press, Princeton, NJ, USA, 1992.
  * [22] D. Z. Freedman and A. Van Proeyen, Supergravity.  Cambridge Univ. Press, Cambridge, UK, 5, 2012.
  * [23] H. Okada and M. Tanimoto, Modular invariant flavor model of $A_{4}$ and hierarchical structures at nearby fixed points, Phys. Rev. D 103 no. 1, (2021) 015005 [2009.14242 [hep-ph]].
  * [24] P. P. Novichkov, J. T. Penedo, and S. T. Petcov, Fermion mass hierarchies, large lepton mixing and residual modular symmetries, JHEP 04 (2021) 206 [2102.07488 [hep-ph]].
  * [25] F. Feruglio, Universal Predictions of Modular Invariant Flavor Models near the Self-Dual Point, Phys. Rev. Lett. 130 no. 10, (2023) 101801 [2211.00659 [hep-ph]].
  * [26] S. Kikuchi, T. Kobayashi, K. Nasu, and Y. Yamada, Moduli trapping mechanism in modular flavor symmetric models, JHEP 08 (2023) 081 [2307.13230 [hep-ph]].
  * [27] S. T. Petcov and M. Tanimoto, $A_{4}$ modular flavour model of quark mass hierarchies close to the fixed point $\tau=\omega$, Eur. Phys. J. C 83 no. 7, (2023) 579 [2212.13336 [hep-ph]].
  * [28] R. Kallosh and A. Linde, New models of chaotic inflation in supergravity, JCAP 11 (2010) 011 [1008.3375 [hep-th]].
  * [29] R. Kallosh, A. Linde, and T. Rube, General inflaton potentials in supergravity, Phys. Rev. D 83 (2011) 043507 [1011.5945 [hep-th]].
  * [30] R. Kallosh, A. Linde, and B. Vercnocke, Natural Inflation in Supergravity and Beyond, Phys. Rev. D 90 no. 4, (2014) 041303 [1404.6244 [hep-th]].
  * [31] S. Ferrara, R. Kallosh, and A. Linde, Cosmology with Nilpotent Superfields, JHEP 10 (2014) 143 [1408.4096 [hep-th]].
  * [32] T. Higaki and Y. Tatsuta, Inflation from periodic extra dimensions, JCAP 07 (2017) 011 [1611.00808 [hep-th]].
  * [33] V. Knapp-Perez, X.-G. Liu, H. P. Nilles, S. Ramos-Sanchez, and M. Ratz, Matter matters in moduli fixing and modular flavor symmetries, Phys. Lett. B 844 (2023) 138106 [2304.14437 [hep-th]].
  * [34] H. Ohki, S. Uemura, and R. Watanabe, Modular flavor symmetry on a magnetized torus, Phys. Rev. D 102 no. 8, (2020) 085008 [2003.04174 [hep-th]].
  * [35] O. Lebedev, H. P. Nilles, and M. Ratz, De Sitter vacua from matter superpotentials, Phys. Lett. B 636 (2006) 126–131 [hep-th/0603047].
  * [36] E. Dudas, C. Papineau, and S. Pokorski, Moduli stabilization and uplifting with dynamically generated F-terms, JHEP 02 (2007) 028 [hep-th/0610297].
  * [37] H. Abe, T. Higaki, T. Kobayashi, and Y. Omura, Moduli stabilization, F-term uplifting and soft supersymmetry breaking terms, Phys. Rev. D 75 (2007) 025019 [hep-th/0611024].
  * [38] R. Kallosh and A. D. Linde, O’KKLT, JHEP 02 (2007) 002 [hep-th/0611183].
  * [39] H. Abe, T. Higaki, and T. Kobayashi, More about F-term uplifting, Phys. Rev. D 76 (2007) 105003 [0707.2671 [hep-th]].
  * [40] S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, De Sitter vacua in string theory, Phys. Rev. D 68 (2003) 046005 [hep-th/0301240].
  * [41] F. Feruglio, V. Gherardi, A. Romanino, and A. Titov, Modular invariant dynamics and fermion mass hierarchies around $\tau=i$, JHEP 05 (2021) 242 [2101.08718 [hep-ph]].
  * [42] S. Kikuchi, T. Kobayashi, K. Nasu, S. Takada, and H. Uchida, Quark hierarchical structures in modular symmetric flavor models at level 6, Phys. Rev. D 107 no. 5, (2023) 055014 [2301.03737 [hep-ph]].
  * [43] Y. Abe, T. Higaki, J. Kawamura, and T. Kobayashi, Quark masses and CKM hierarchies from $S_{4}^{\prime}$ modular flavor symmetry, Eur. Phys. J. C 83 no. 12, (2023) 1140 [2301.07439 [hep-ph]].
  * [44] S. T. Petcov and M. Tanimoto, A4 modular flavour model of quark mass hierarchies close to the fixed point $\tau$ = i$\infty$, JHEP 08 (2023) 086 [2306.05730 [hep-ph]].
  * [45] Y. Abe, T. Higaki, J. Kawamura, and T. Kobayashi, Quark and lepton hierarchies from S4’ modular flavor symmetry, Phys. Lett. B 842 (2023) 137977 [2302.11183 [hep-ph]].
  * [46] Y. Abe, T. Higaki, J. Kawamura, and T. Kobayashi, Fermion hierarchies in SU(5) grand unification from ${\Gamma}_{6}^{\prime}$ modular flavor symmetry, JHEP 08 (2023) 097 [2307.01419 [hep-ph]].
  * [47] I. de Medeiros Varzielas, M. Levy, J. T. Penedo, and S. T. Petcov, Quarks at the modular S4 cusp, JHEP 09 (2023) 196 [2307.14410 [hep-ph]].
  * [48] S. Kikuchi, T. Kobayashi, K. Nasu, S. Takada, and H. Uchida, Sp(6, Z) modular symmetry in flavor structures: quark flavor models and Siegel modular forms for $\widetilde{\Delta}\left(96\right)$, JHEP 04 (2024) 045 [2310.17978 [hep-ph]].
  * [49] S. Kikuchi, T. Kobayashi, and K. Nasu, CP phase in modular flavor models and discrete Froggatt-Nielsen models, 2312.11809 [hep-ph].
  * [50] H. Abe, T. Higaki, and T. Kobayashi, Remark on integrating out heavy moduli in flux compactification, Phys. Rev. D 74 (2006) 045012 [hep-th/0606095].
  * [51] T. Higaki, T. Kobayashi, K. Nasu, and H. Otsuka, Moduli stabilization, spontaneous CP violation and modular symmetry, to appear .
  * [52] M.-C. Chen, S. Ramos-Sánchez, and M. Ratz, A note on the predictions of models with modular flavor symmetries, Phys. Lett. B 801 (2020) 135153 [1909.06910 [hep-ph]].
  * [53] P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov, Generalised CP Symmetry in Modular-Invariant Models of Flavour, JHEP 07 (2019) 165 [1905.11970 [hep-ph]].