Modular flavour symmetry and orbifolds

Francisco J. de Anda‡***E-mail<EMAIL_ADDRESS>Stephen F. King⋄†††E-mail:
<EMAIL_ADDRESS>
‡ Tepatitlán’s Institute for Theoretical Studies, C.P. 47600, Jalisco, México,

Dual CP Institute of High Energy Physics, C.P. 28045, Colima, México.

⋄ School of Physics and Astronomy, University of Southampton,

SO17 1BJ Southampton, United Kingdom.

We develop a bottom-up approach to flavour models which combine modular
symmetry with orbifold constructions. We first consider a 6d orbifold
$\mathbb{T}^{2}/\mathbb{Z}_{N}$, with a single torus defined by one complex
coordinate $z$ and a single modulus field $\tau$, playing the role of a flavon
transforming under a finite modular symmetry. We then consider 10d orbifolds
with three factorizable tori, each defined by one complex coordinate $z_{i}$
and involving the three moduli fields $\tau_{1},\tau_{2},\tau_{3}$
transforming under three finite modular groups. Assuming supersymmetry,
consistent with the holomorphicity requirement, we consider all 10d orbifolds
of the form $(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{N}\times\mathbb{Z}_{M})$, and
list those which have fixed values of the moduli fields (up to an integer).
The key advantage of such 10d orbifold models over 4d models is that the
values of the moduli are not completely free but are constrained by geometry
and symmetry. To illustrate the approach we discuss a 10d modular seesaw model
with $S_{4}^{3}$ modular symmetry based on
$(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$ where
$\tau_{1}=i,\ \tau_{2}=i+2$ are constrained by the orbifold, while
$\tau_{3}=\omega$ is determined by imposing a further remnant $S_{4}$ flavour
symmetry, leading to a highly predictive example in the class CSD$(n)$ with
$n=1-\sqrt{6}$.

##  1 Introduction

The Standard Model (SM), despite its many successes, does not account for the
origin of neutrino mass nor the quark and lepton family replication, and gives
no insight into the fermion masses and mixing parameters. One approach is to
introduce a family symmetry which may be a finite discrete or continuous,
gauged or global, Abelian or non-Abelian. Large lepton mixing has motivated
studies of non-Abelian finite discrete groups such as $A_{4},S_{4},A_{5}$ (for
reviews see e.g. [1, 2]). However such family symmetries must eventually be
spontaneously broken by new Higgs fields called flavons, and it turns out that
the vacuum alignment of such flavon fields plays a crucial role determining
the physical predictions of such models.

Another interesting class of symmetries arise from the modular group
$SL(2,\mathbb{Z})$, which is the group of $2\times 2$ matrices with integer
elements, the kind you first learn about in high school with positive or
negative elements, but with unit determinant. Geometrically, such a group is
the symmetry of a torus, which essentially has a flat geometry in two
dimensions (when it is cut open) and the symmetry corresponds to the discrete
coordinate transformations which leave the torus invariant, in other words the
different choices of two dimensional lattice vectors describing the same
torus. The two dimensional space may be conveniently associated with the real
and imaginary directions of the complex plane, with the lattice vectors
becoming complex vectors in the Argand plane. The modular symmetry in the
upper half of the complex plane, $PSL(2,\mathbb{Z})$, has particularly nice
features which rely on holomorphicity, the lack of complex conjugation
symmetry, reminiscent of supersymmetry.

At first sight, modular symmetry does not look like a promising starting point
for a family symmetry, for one thing it is an infinite group, since there are
an infinite number of $2\times 2$ matrices with integer elements and unit
determinant. Secondly, it is not immediately obvious what a torus has got to
do with particle physics. With the advent of superstring theory and extra
dimensions, this second question may at least find an answer, since orbifold
compactifications of two extra dimensions are often done on a torus [3, 4],
and in superstring theory, the single lattice vector which describes the torus
(in the convention that the other lattice vector has unit length and lies
along the real axis) is promoted to the status of a field, called the modulus
field $\tau$, where its vacuum expectation value (VEV) fixes the geometry of
the torus. Moreover, it is possible to obtain a finite discrete group from the
infinite modular group as discussed below.

The infinite modular group has a series of infinite normal subgroups called
the principle congruence subgroups $\Gamma(N)$ of level $N$, whose elements
are equal to the $2\times 2$ unit matrix mod $N$ (where typically $N$ is an
integer called the level of the group). For a given choice of level $N>2$, the
quotient group $\Gamma_{N}=PSL(2,\mathbb{Z})/\Gamma(N)$ is finite and may be
identified with the groups $\Gamma_{N}=A_{4},S_{4},A_{5}$ for levels
$N=3,4,5$, which may be subsequently be used as a family symmetry [5]. Indeed
the only flavon present in such theories is the single modulus field $\tau$,
whose VEV fixes the value of Yukawa couplings which form representations of
$\Gamma_{N}$ and are modular forms, leading to very predictive theories
independent of flavons [5].

Following the above observations [5], there has been considerable activity in
applying modular symmetry to flavour models, and also in extending the
framework to more general settings, following the bottom-up approach (see [6]
for more details and extensive references). For example the modular $S_{4}$
group was studied in [7, 8, 9]. To enhance the predictivity of such models,
rather than considering the VEV of $\tau$ to be a free complex parameter, it
is interesting to consider fixed points or stabilizers which are special
values for the modulus field $\tau$ such as $\tau=i,\omega,i\infty$ where part
of the modular transformations are preserved. However such an approach with
one modulus ‡‡‡Recently it has been claimed that a single modulus at $\tau=i$
can provide a good phenomenological description of leptons, but this requires
that the neutrino mass matrix is infinite at the fixed point [10]. is rather
too restrictive and generally calls for additional moduli fields which can be
introduced in a straightforward way by considering additional modular groups,
with one modulus per modular group, as suggested in [11, 12, 13, 14, 15]. A
recent example of a model of this kind was based on three finite modular
groups $S_{4}^{3}$ broken to its diagonal subgroup $S_{4}$, with three moduli
fields in the low energy theory located at three different fixed points, for
example $\tau_{1}=i,\ \tau_{2}=i+2,\ \tau_{3}=\omega$, leading to a very
predictive and successful phenomenological description of the neutrino and
charged lepton masses and lepton mixing based on a version of the littlest
seesaw [16].

While there has been considerable effort devoted to studying modular symmetry
arising from orbifolds in top-down heterotic string constructions [17],
§§§Top-down approaches suggest that the finite modular symmetry will typically
be accompanied by a flavour symmetry leading to so called eclectic symmetry
[18, 19, 20, 21, 22]. there has been little work on bottom-up approaches which
combine orbifolds together with modular symmetry. In the bottom-up approach to
modular symmetry as applied to flavour models, orbifolds are usually not
considered at all. Instead the formalism of modular symmetry and modular forms
is adopted and flavour models then constructed, without any reference to the
underlying orbifold [5]. However there have been some bottom-up attempts to
relate modular symmetry to orbifold GUTs, such as the model based on
supersymmetric $SU(5)$ in 6d, where the two extra dimensions are compactified
on a $\mathbb{T}^{2}/\mathbb{Z}_{2}$, leading to a remnant $A_{4}$ with single
modulus field located at the fixed point $\tau=\omega$ of the orbifold [23].
In this model, there was also an $A_{4}$ flavour symmetry commuting with the
$A_{4}$ modular symmetry, which was a pre-curser to the eclectic flavour
symmetry approach [23].

In this paper we develop a bottom-up approach to flavour models which combines
modular symmetry with orbifold constructions. We shall consider orbifolds in
10d which can provide three modular groups and three moduli fields in the low
energy theory (below the compactification scales). We assume that the 6 extra
dimensions are factorisable into 3 tori, each defined by one complex
coordinate $z_{i}$. Assuming supersymmetry, consistent with the holomorphicity
requirement, we consider all the orbifolds of the form
$(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{N}\times\mathbb{Z}_{M})$, and list all the
available orbifolds, which have fixed values of the moduli fields (up to an
integer). The key advantage of such 10d orbifold models over 4d models is that
the values of the moduli are not completely free but are constrained geometry
and symmetry.

To illustrate the approach, we focus on the orbifold example
$(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$, and discuss in
detail the fixed points, with the choices $\tau_{1}=i,\ \tau_{2}=i+2$ being
constrained by the orbifold, while $\tau_{3}$ is unconstrained but may be
fixed by specifying a remnant symmetry. Motivated by model building
considerations we consider $\tau_{3}=\omega$ determined by imposing a remnant
$S_{4}$ flavour symmetry. We assume an $S_{4}^{3}$ modular symmetry,
associated with each of the three moduli. We show that such a model can
reproduce a minimal 4d modular seesaw model of leptons based on three finite
modular groups $S_{4}^{3}$ broken to the diagonal modular subgroup $S_{4}$. In
the 4d models the three moduli fields were simply assumed to lie at the fixed
points, $\tau_{1}=i,\ \tau_{2}=i+2,\ \tau_{3}=\omega$ [16], but in the 10d
model, these values are constrained by geometry and symmetry. The resulting
model is in the class CSD$(n)$ with $n=1-\sqrt{6}$, where the atmospheric
angle $\theta_{23}$ is restricted to the second octant.

The bottom-up approach to modular symmetry from orbifolds followed here can
readily be extended to Grand Unified Theories (GUTs), with up to three moduli
groups and moduli fields, including a remnant flavour symmetry, leading to a
bottom-up version of the ecletic flavour symmetry in orbifold GUTs as
anticipated in [23].

The layout of the remainder of the paper is as follows: In Sec. 2, the general
SUSY preserving orbifolding is presented and shown how it fixes the modulus.
This is shown for the case of 6 and 10 spacetime dimensions, as well as an
specific detailed example. In Sec. 3 we describe the basics of the modular
symmetry $S_{4}$, its corresponding modular forms at the fixed points as well
as how it can arise as a remnant symmetry in orbifolding. In Sec. 4 we present
a viable and predictive lepton model which uses $S_{4}^{3}$ modular symmetry
in a $(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$ orbifold.
Finally in Sec. 5 we present our conclusions.

##  2 Orbifolding

Modular symmetries have proved themselves very useful in model building. They
may provide predictive flavor structure specially for the lepton sector
without requiring the addition of extra fields nor complicated symmetry
breaking mechanisms. A model with modular symmetry requires to be built in 6
dimensions (at least) and start with $\mathcal{N}=1$ SUSY, as the modular
transformations are essentially transformations of the extra dimensional part
of the enhanced Poincaré symmetry coupled with a SUSY transformation on the
fields.

Most models assume a 6 dimensional spacetime with SUSY where the extra
dimensions are compactified as a torus (with twist angle $\tau$) and build a
model using the assumed modular symmetries. However assuming the extra
dimensions to be a torus can’t lead to a viable theory as the resulting model
after compactification would have no chirality and $\mathcal{N}=2$ SUSY. The
standard solution is to compactify the extra dimensions as an orbifold, which
we now present its basics.

###  2.1 The orbifold $\mathbb{T}^{2}/\mathbb{Z}_{N}$

The two extra dimensional coordinates can be treated as a single complex
coordinate $z=x_{5}+ix_{6}$. The torus compactification is done by identifying

$z\sim z+1,\ \ \ z\sim z+\tau,$ (1)

where $\tau$ is called the twist angle and, for now, it is an arbitrary
complex number. This identification restricts the range of the complex
coordinate. The $\\{1,\tau\\}$ are called the basis vectors which generate the
lattice of the extra dimensional plane and define the torus.

The torus by itself leads to a non chiral theory after compactification. The
solution is to assume orbifolding, which is equivalent to assume that the
extra dimensional part of the Poincaré group is not a full symmetry. This is
done by modding out a discrete subgroup of the extra dimensional Lorentz
group, which is called orbifolding. In 6 dimensions, the extra dimensional
part of the Lorentz group is

$SO(1,5)/SO(1,3)\simeq SO(2)\simeq U(1),$ (2)

which correspond to rotation in the 2 extra dimensions. One can mod out by any
discrete subgroup $F\in U(1)$, which can only be $F=\mathbb{Z}_{N}$, with $N$
an arbitrary integer (for now). It has to be a discrete group to avoid
reducing the dimensionality. The $\mathbb{Z}_{N}$ orbifolding is achieved by
the identification

$z\sim e^{2i\pi/N}z,$ (3)

which further restricts the range of the extra dimensional coordinates. The
orbifold has fixed points which allow boundary conditions that generate
chirality, may break the gauge symmetry and reduce the enhanced SUSY.
Therefore they may lead to a consistent model after compactification.

To avoid dimensional reduction and therefore for the orbifold to be
consistent, the orbifold action in Eq.3 must be equivalent to an integer
number of lattice transformations as in Eq. 3. In other words, there must
exist integer numbers $a,b\in\mathbb{Z}$ such that a solution exists for

$e^{2i\pi/N}z=z+a+b\tau.$ (4)

It is enough to find a solution for each of the basis vectors $\\{1,\tau\\}$,

$e^{2i\pi/N}=a+b\tau,\ \ \ e^{2i\pi/N}\tau=c+d\tau,$ (5)

where there must exist $a,b,c,d\in\mathbb{Z}$ that solve these equations. It
is clear that there is no solution for arbitrary $N$ and $\tau$. This
restricts the $N$ and $\tau$ to be one of

$\begin{split}N=2,&\ \ \ \tau=z\in\mathbb{C},\\\ N=3,&\ \ \ \tau=\omega,\\\
N=4,&\ \ \ \tau=i,\\\ N=6,&\ \ \ \tau=\\{\omega,\rho/\sqrt{3}\\},\\\
\end{split}$ (6)

where $\omega=e^{2i\pi/3}$ and $\rho=e^{i\pi/6}$ and all the solutions are
valid up to an integer.

Therefore working with an orbifold may fix $\tau$ geometrically, adding
predictivity, and solves the chirality problem therefore allowing a viable
model.

###  2.2 The orbifold
$(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{N}\times\mathbb{Z}_{M})$

Many models may require various independent modular symmetries or different
$\tau$ values to achieve a better fit. One such model is presented in Sec. 4.
As it needs 3 independent modular symmetries, we focus on 10 dimensional
spaces with $\mathcal{N}=1$ SUSY before and after compactification.

In the 10 dimensional case, one can orbifold by a discrete subgroup of the
extra dimensional part of the Lorentz group

$SO(1,9)/SO(1,3)\simeq SO(6)\simeq SU(4),$ (7)

which corresponds to rotations in the extra 6 dimensions. The former $SU(4)$
can be identified with the $SU(4)_{\mathcal{R}}$ of the enhanced
$\mathcal{N}=4$ SUSY. As we want to preserve simple SUSY after
compactification, the discrete orbifolding group must be $F\subset SU(3)$. As
it is rank 2, a general 10d SUSY preserving abelian factorisable orbifolding
is

$(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{N}\times\mathbb{Z}_{M})$ (8)

which can be compactified by the basis vectors

$z_{i}\sim z_{i}+1,\ \ \ z_{i}\sim z_{i}+\tau_{i},$ (9)

and the orbifolding defined by

$\begin{split}\theta_{N}:\
(x,z_{1},z_{2},z_{3})&\sim(x,\alpha_{N}z_{1},\beta_{N}z_{2},\gamma_{N}z_{3}),\\\
\theta_{M}:\
(x,z_{1},z_{2},z_{3})&\sim(x,\alpha_{M}z_{1},\beta_{M}z_{2},\gamma_{M}z_{3}),\end{split}$
(10)

where $\alpha_{N,M},\beta_{N,M},\gamma_{N,M}$ are Nth, Mth roots of unity.

The choice of the phases of the orbifolding are restricted by the preservation
of $\mathcal{N}=1$ SUSY. The $\tau_{i}$ must be fixed so that the lattice is
unchanged by the orbifold transformation. The $\tau_{i}$ are fixed, as they
must such that the orbifolding identification does not change the lattice and
therefore the torus remains unchanged. Therefore there must exist integers
$a,b,c,d$ such that

$\begin{split}(\delta,\delta\tau)&=(a+b\tau,c+d\tau),\end{split}$ (11)

for each corresponding $\delta=\alpha_{N,M},\beta_{N,M},\gamma_{N,M}$

These restrictions limit the available (SUSY preserving [24]) orbifolds to be
as in Table 1, which displays all the available orbifolds with some of the
$\tau_{i}$ fixed as shown (up to an integer), while the non-fixed values are
indicated by the complex number $z$.

$(N,M)$ | $(\alpha_{N},\beta_{N},\gamma_{N})$ | $(\alpha_{M},\beta_{M},\gamma_{M})$ | $(\tau_{1},\tau_{2},\tau_{3})$
---|---|---|---
$(3,1)$ | $(\omega,\omega,\omega)$ | $(1,1,1)$ | $(\omega,\omega,\omega)$
$(4,1)$ | $(i,i,-1)$ | $(1,1,1)$ | $(i,i,z)$
$(6,1)_{I}$ | $(-\omega^{2},-\omega^{2},\omega^{2})$ | $(1,1,1)$ | $(\\{\omega,\rho/\sqrt{3}\\},\\{\omega,\rho/\sqrt{3}\\},\omega)$
$(6,1)_{II}$ | $(-\omega^{2},\omega,-1)$ | $(1,1,1)$ | $(\\{\omega,\rho/\sqrt{3}\\},\omega,z)$
$(2,2)$ | $(1,-1,-1)$ | $(-1,1,-1)$ | $(z,z,z)$
$(4,2)$ | $(i,-i,1)$ | $(1,-1,-1)$ | $(i,i,z)$
$(6,2)_{I}$ | $(-\omega^{2},1,-\omega)$ | $(1,-1,-1)$ | $(\\{\omega,\rho/\sqrt{3}\\},z,\\{\omega,\rho/\sqrt{3}\\})$
$(6,2)_{II}$ | $(\omega^{2},-\omega^{2},-\omega^{2})$ | $(1,-1,-1)$ | $(\omega,\\{\omega,\rho/\sqrt{3}\\},\\{\omega,\rho/\sqrt{3}\\})$
$(3,3)$ | $(1,\omega,\omega^{2})$ | $(\omega,1,\omega^{2})$ | $(\omega,\omega,\omega)$
$(6,3)$ | $(-\omega^{2},1,-\omega)$ | $(1,\omega,\omega^{2})$ | $(\\{\omega,\rho/\sqrt{3}\\},\\{\omega,\rho/\sqrt{3}\\},\omega)$
$(4,4)$ | $(1,i,-i)$ | $(i,1,-i)$ | $(i,i,i)$
$(6,6)$ | $(1,-\omega^{2},-\omega)$ | $(-\omega^{2},1,-\omega)$ | $(\\{\omega,\rho/\sqrt{3}\\},\\{\omega,\rho/\sqrt{3}\\},\\{\omega,\rho/\sqrt{3}\\})$
Table 1:  Comprehensive list of 6d abelian factorisable and SUSY preserving
orbifolds $(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{N}\times\mathbb{Z}_{M})$, where
$\omega=e^{2i\pi/3}$ and $\rho=e^{i\pi/6}$, and the fixed points of $\tau_{i}$
are specified only up to an integer. For example, $\tau_{2}=i,i+1,i+2$ and so
on are all equivalent. The values of the complex numbers $z$ are not
restricted by the orbifold, but particular values of $z$ may be fixed by a
remnant global symmetry.

###  2.3 The orbifold
$(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$

In this subsection we discuss an example of an orbifold chosen from Table 1
corresponding to $(N,M)=(4,2)$ which leads to an interesting model ¶¶¶This
example is not unique, there are other choices which also lead to viable
models.. The full model based on the resulting orbifold
$(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$ will be presented
in Sec. 4. The model we have in mind is an extra dimensional version of a four
dimensional model based on three finite modular groups $S_{4}^{3}$ broken to a
diagonal subgroup $S_{4}$, with the three moduli fields in the low energy
theory located at three different fixed points, namely $\tau_{1}=i,\
\tau_{2}=i+2,\ \tau_{3}=\omega$. In a 4d framework, this was shown to lead to
a very predictive and successful phenomenological description of the neutrino
and charged lepton masses and lepton mixing based on a type of littlest seesaw
[16].

In the 10d framework considered here, the desired moduli fields $\tau_{i}$ for
such model are in principle consistent with the orbifold divisors
$\mathbb{Z}_{2}\times\mathbb{Z}_{2},\ \mathbb{Z}_{4},\
\mathbb{Z}_{4}\times\mathbb{Z}_{2}$. However
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ does not fix any of the $\tau_{i}$, so is
not so restrictive. The $\mathbb{Z}_{4}$ orbifold divisor fixes the $\tau_{i}$
as needed by the model, but does not have the necessary fixed branes to build
consistent interactions. We are then left with the only viable and predictive
choice being the orbifold divisor $\mathbb{Z}_{4}\times\mathbb{Z}_{2}$, which
can lead to the desired fixed points, as we discuss below.

We assume, then, a 10d spacetime where the 6 extra dimensions are factorisable
into 3 torii, each defined by one complex coordinate $z_{i}$ with $i=1,2,3,$
and compactified as in Eq. 9

$z_{i}\sim z_{i}+1,\ \ \ z_{i}\sim z_{i}+\tau_{i},$ (12)

The orbifold $(\mathbb{T}^{2})^{3}/\mathbb{Z}_{4}\times\mathbb{Z}_{2}$ as
defined by the orbifolding actions in Eq. 10, using Table 1 with $(N,M)=(4,2)$
then implies,

$\begin{split}\theta_{4}&:(x,z_{1},z_{2},z_{3})\sim(x,iz_{1},-iz_{2},z_{3}),\\\
\theta_{2}&:(x,z_{1},z_{2},z_{3})\sim(x,z_{1},-z_{2},-z_{3}).\end{split}$ (13)

In the orbifold approach, $(1,\tau_{i})$ define the twist and the basis
vectors of each torus. For the orbifold to be consistent, the orbifolding
actions $\theta_{2,4}$ must not change the lattice, i.e. its action over the
lattice basis vectors $(1,\tau_{i})$ must be a linear combination of the
original lattice vectors, with integer coefficients. Therefore there must
exist integers $a_{1,2,3},b_{1,2,3},c_{1,2,3},d_{1,2,3}\in\mathbb{Z}$ such
that, as in Eq. 11

$\begin{split}(i,i\tau_{1,2})&=(a_{1,2}+b_{1,2}\tau_{1,2},c_{1,2}+d\tau_{1,2}),\\\
(-1,-\tau_{3})&=(a_{3}+b_{3}\tau_{3},c_{3}+d\tau_{3}),\\\ \end{split}$ (14)

In the present example, solving Eq. 14 gives,

$\begin{split}\tau_{1,2}&=i+n_{1,2},\ \ \ |\ \ \ n_{1,2}\in\mathbb{Z},\\\
\tau_{3}&\in\mathbb{C}.\end{split}$ (15)

which corresponds to the result given in Table 1 with $(N,M)=(4,2)$. We
emphasise that the twists $\tau_{i}$ are fixed geometrically by the orbifold
actions. Therefore in the orbifold approach to modular symmetries, the moduli
fields are not a completely free choice, but are constrained as in Table 1.

Each orbifold action in Eq. 13, leaves some invariant subspaces which are
called fixed branes

$\begin{split}\theta_{4}&:\left(x,\left\\{0,\frac{i+1}{2}\right\\},\left\\{0,\frac{i+1}{2}\right\\},z_{3}\right),\\\
\theta^{2}_{4}&:\left(x,\left\\{0,\frac{1}{2},\frac{i}{2},\frac{i+1}{2}\right\\},\left\\{0,\frac{1}{2},\frac{i}{2},\frac{i+1}{2}\right\\},z_{3}\right),\\\
\theta_{2}&:\left(x,z_{1},\left\\{0,\frac{1}{2},\frac{i}{2},\frac{i+1}{2}\right\\},\left\\{0,\frac{1}{2},\frac{\tau_{3}}{2},\frac{\tau_{3}+1}{2}\right\\}\right),\\\
\theta_{2}\theta_{4}&:\left(x,\left\\{0,\frac{i+1}{2}\right\\},\left\\{0,\frac{i+1}{2}\right\\},\left\\{0,\frac{1}{2},\frac{\tau_{3}}{2},\frac{\tau_{3}+1}{2}\right\\}\right),\\\
\theta_{2}\theta_{4}^{2}&:\left(x,\left\\{0,\frac{1}{2},\frac{i}{2},\frac{i+1}{2}\right\\},z_{2},\left\\{0,\frac{1}{2},\frac{\tau_{3}}{2},\frac{\tau_{3}+1}{2}\right\\}\right).\end{split}$
(16)

When building a model, fields can be chosen to be located in any of the
previous branes or in the bulk.

We want a minimal model where all fields can behave as modular forms (with
different $\tau_{i}$ depending on their location) but can interact with each
other, we will only use the 6d branes

$\begin{split}\mathbb{T}^{2}_{A}&=(x,z_{1},0,0),\\\
\mathbb{T}^{2}_{B}&=(x,0,z_{2},0),\\\
\mathbb{T}^{2}_{C}&=(x,0,0,z_{3}),\end{split}$ (17)

where all of them touch at the origin brane, where all interactions happen.

From Eq. 13, we note that the $z_{1}$ only feels the $\theta_{4}$ action,
therefore the $\mathbb{T}^{2}_{A}$ is a $\mathbb{Z}_{4}$ orbifold. As the
action of $\theta_{2}$ on $z_{2}$ is also contained in $\theta_{4}$, the
$\mathbb{T}^{2}_{B}$ is also a $\mathbb{Z}_{4}$ orbifold. Finally the $z_{3}$
only feels the $\theta_{2}$ action, therefore the $\mathbb{T}^{2}_{C}$ is a
$\mathbb{Z}_{2}$ orbifold.

##  3 Modular $S_{4}$ symmetries in the orbifold approach

So far we have considered possible orbifolds in which the VEVs of the moduli
fields $\tau_{i}$ are fixed at least partially by the geometry. We now turn to
the modular symmetries of the fields $\tau_{i}$ which are broken by the VEVs
of the moduli fields $\tau_{i}$. In general such modular symmetries are
infinite but have a series of infinite normal subgroups called the principle
congruence subgroups $\Gamma(N)$ of level $N$, whose elements are equal to the
$2\times 2$ unit matrix mod $N$ (where typically $N$ is an integer called the
level of the group). For a given choice of level $N>2$, the quotient group
$\Gamma_{N}=PSL(2,\mathbb{Z})/\Gamma(N)$ is finite and may be identified with
the groups $\Gamma_{N}=A_{4},S_{4},A_{5}$ for levels $N=3,4,5$, which may be
subsequently be used as a family symmetry [5]. In this section we consider the
case $N=4$ which corresponds to modular $S_{4}$ symmetries.

With two extra dimensions the single complex modulus $\tau$ has an infinite
modular symmetry $\overline{\Gamma}=SL(2,\mathbb{Z})$ as follows. The modular
group $\overline{\Gamma}$ is the group of linear fraction transformations
which acts on the complex modulus $\tau$ in the upper half complex plane as
follow,

$\tau\rightarrow\gamma\tau=\frac{a\tau+b}{c\tau+d},~{}~{}\text{with}~{}~{}a,b,c,d\in\mathbb{Z},~{}~{}ad-
bc=1,~{}~{}~{}\mathop{\mathrm{Im}}\tau>0\,.$ (18)

The modular group $\overline{\Gamma}$ can be generated by two generators $S$
and $T$

$S:\tau\mapsto-\frac{1}{\tau},~{}~{}~{}~{}\quad T:\tau\mapsto\tau+1.$ (19)

From the infinite modular group the finite subgroup
$\Gamma_{N}=PSL(2,\mathbb{Z})/\Gamma(N)$ may be obtained. A crucial element of
the modular invariance approach is the modular form $f(\tau)$ of weight $k$
and level $N$. The modular form $f(\tau)$ is a holomorphic function of the
complex modulus $\tau$ and it is required to transform under the action of
$\overline{\Gamma}(N)$ as follows,

$f\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{k}f(\tau)$ (20)

The modular forms of level $N=4$ have been constructed in [7, 25] .

The associated finite modular group $\Gamma_{4}$ has two generators $S$ and
$T$ which fulfill the following rations

$S^{2}=(ST)^{3}=(TS)^{3}=T^{4}=1\,.$ (21)

The finite modular group $\Gamma_{4}$ is isomorphic to the permutation group
$S_{4}$ of four objects. In order to see the correlation between $S_{4}$ and
tri-bimaximal mixing and the connection to $S_{3}$, $A_{4}$ groups more
easily, it is convenient to generate the $S_{4}$ group in terms of three
generators $\hat{S}$, $\hat{T}$ and $\hat{U}$ with the multiplication rules
[26, 27],

$\displaystyle\hat{S}^{2}=\hat{T}^{3}=\hat{U}^{2}=(\hat{S}\hat{T})^{3}=(\hat{S}\hat{U})^{2}=(\hat{T}\hat{U})^{2}=(\hat{S}\hat{T}\hat{U})^{4}=1\,,$
(22)

where $\hat{S}$ and $\hat{T}$ alone generate the group $A_{4}$, while
$\hat{T}$ and $\hat{U}$ alone generate the group $S_{3}$. The generators $S$,
$T$ can be expressed in terms of $\hat{S}$, $\hat{T}$ and $\hat{U}$

$S=\hat{S}\hat{U},\,T=\hat{S}\hat{T}^{2}\hat{U},\,ST=\hat{T}$ (23)

or vice versa

$\hat{S}=(ST^{2})^{2},~{}~{}~{}\hat{T}=ST,~{}~{}~{}\hat{U}=T^{2}ST^{2}\,,$
(24)

with the explicit matrices being

$\hat{S}=\frac{1}{3}\left(\begin{array}[]{ccc}-1{}&2&~{}2\\\ 2{}&-1&2\\\
2{}&2&~{}-1\end{array}\right),\ \ \
\hat{T}=\left(\begin{array}[]{ccc}1{}&0&~{}0\\\ 0{}&\omega^{2}&0\\\
0{}&0&~{}\omega\end{array}\right),\ \ \
\hat{U}=\mp\left(\begin{array}[]{ccc}1{}&0&~{}0\\\ 0{}&0&1\\\
0{}&1&~{}0\end{array}\right),$ (25)

where the minus sign in $\hat{U}$ applies for the 3 representation while the
plus sign is for the $\textbf{3}^{\prime}$ representation.

| $S$ | $T$
---|---|---
$\mathbf{1}$, $\mathbf{1}^{\prime}$ | $\pm 1$ | $\pm 1$
$\mathbf{2}$ | $\left(\begin{array}[]{cc}0&~{}1\\\ 1&~{}0\end{array}\right)$ | $\left(\begin{array}[]{cc}0{}&~{}\omega^{2}\\\ \omega{}&~{}0\end{array}\right)$
$\mathbf{3}$, $\mathbf{3}^{\prime}$ | $\pm\frac{1}{3}\left(\begin{array}[]{ccc}1{}&-2&~{}-2\\\ -2{}&-2&~{}1\\\ -2{}&1&~{}-2\end{array}\right)$ | $\pm\frac{1}{3}\left(\begin{array}[]{ccc}1{}&-2\omega^{2}&~{}-2\omega\\\ -2{}&-2\omega^{2}&~{}\omega\\\ -2{}&\omega^{2}&~{}-2\omega\end{array}\right)$
Table 2: The representation matrices of the generators $S$ and $T$ in the five
irreducible representations of $S_{4}$, where $\omega=e^{2\pi
i/3}=-1/2+i\sqrt{3}/2$ is a cubic root of unity.

We assume $\mathcal{N}=1$ SUSY in 10d and this abelian orbifold preserves
$\mathcal{N}=1$ SUSY in 4d after compactification [24]. Therefore we can
assume 3 independent modular symmetry groups, each associated with a different
tori [11, 12, 13, 14]. We assume three discrete modular symmetries
$S_{4}^{A,B,C}$ associated to each complex coordinate $z_{1,2,3}$
correspondingly.

With the assumed $S_{4}$ modular symmetries, the corresponding moduli from
Eq.15, which have an arbitrary integer, now can only be

$n=0,1,2,3,$ (26)

where it is now limited to a choice of one in four.

###  3.1 Fixed points and $S_{4}$ modular forms

In most models using modular symmetries, the $\tau$ is a free parameter that
is minimized by a potential and treated as a VEV. A standard strategy to
increase the predictivity of the model is to restrict to fixed points which
are geometrically preferred. These point $\bar{\tau}$ are defined as the
points that are invariant under some element of the modular group $\gamma\in
S_{4}$ called the stabilizer.

In an orbifold, the $\tau$ is not a free parameter and it is fixed by the
geometry of the orbifold itself. However, there are a finite number of
choices, which allow specific modular forms which are listed in Table 3 [28].
All the presented $S_{4}$ modular forms are defined in the basis from Table 2.

$\tau$ |  $Y^{(2)}_{\mathbf{3}}(\tau)$, $Y^{(6)}_{\mathbf{3,I}}(\tau)$ |  $Y^{(4)}_{\mathbf{3}}(\tau)$, $Y^{(6)}_{\mathbf{3^{\prime}}}(\tau)$ |  $Y^{(4)}_{\mathbf{3^{\prime}}}(\tau)$, $Y^{(6)}_{\mathbf{3,II}}(\tau)$
---|---|---|---
$i$ | $(1,1+\sqrt{6},1-\sqrt{6})$ | $(1,-\frac{1}{2},-\frac{1}{2})$ | $(1,1-\sqrt{\frac{3}{2}},1+\sqrt{\frac{3}{2}})$
$i+1$ | $(1,-\frac{\omega}{3}(1+i\sqrt{2}),-\frac{\omega^{2}}{3}(1+i\sqrt{2}))$ | $(0,1,-\omega)$ | $(1,\frac{i\omega}{\sqrt{2}},\frac{i\omega^{2}}{\sqrt{2}})$
$i+2$ | $(1,\frac{1}{3}(-1+i\sqrt{2}),\frac{1}{3}(-1+i\sqrt{2}))$ | $(0,1,-1)$ | $(1,-\frac{i}{\sqrt{2}},-\frac{i}{\sqrt{2}})$
$i+3$ | $(1,\omega(1+\sqrt{6}),\omega(1-\sqrt{6}))$ | $(1,-\frac{\omega}{2},-\frac{\omega^{2}}{2})$ | $(1,\omega(1-\sqrt{\frac{3}{2})},\omega^{2}(1+\sqrt{\frac{3}{2}}))$
$\tau$ | $Y^{(2)}_{\mathbf{3}}(\tau)$ |  $Y^{(4)}_{\mathbf{3}}(\tau)$,$Y^{(4)}_{\mathbf{3^{\prime}}}(\tau)$ |  $Y^{(6)}_{\mathbf{3,II}}(\tau)$,$Y^{(6)}_{\mathbf{3^{\prime}}}(\tau)$ | $Y^{(6)}_{\mathbf{3,I}}(\tau)$
$\omega$ | $(0,1,0)$ | $(0,0,1)$ | $(1,0,0)$ | $(0,0,0)$
$\omega+1$ | $(1,1,-\frac{1}{2})$ | $(1,-\frac{1}{2},1)$ | $(1,-2,-2)$
$\omega+2$ | $(1,-\frac{\omega^{2}}{2},\omega)$ | $(1,\omega^{2},-\frac{\omega}{2})$ | $(1,-2\omega^{2},-2\omega)$
$\omega+3$ | $(1,\omega,-\frac{\omega^{2}}{2})$ | $(1,-\frac{\omega}{2},\omega^{2})$ | $(1,-2\omega,-2\omega^{2})$
$\rho/\sqrt{3}$ | $(1,-\frac{\omega}{2},\omega^{2})$ | $(1,\omega,-\frac{\omega^{2}}{2})$ | $(1,-2\omega,-2\omega^{2})$
$\rho/\sqrt{3}+1$ | $(0,0,1)$ | $(0,1,0)$ | $(1,0,0)$
$\rho/\sqrt{3}+2$ | $(1,-\frac{1}{2},1)$ | $(1,1,-\frac{1}{2})$ | $(1,-2,-2)$
$\rho/\sqrt{3}+3$ | $(1,\omega^{2},-\frac{\omega}{2})$ | $(1,-\frac{\omega^{2}}{2},\omega)$ | $(1,-2\omega^{2},-2\omega)$ |

Table 3:  The alignments of triplet modular forms
$Y_{\mathbf{3},\mathbf{3^{\prime}}}(\tau)$ of level 4 up to weight 6 with the
available fixed moduli in orbifolds. We have ignored the overall constant
appearing in each alignment.

In the $(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$ orbifold,
it will be assumed that

$\tau_{1}=i,\ \ \ \tau_{2}=i+2,$ (27)

which are particular cases of Eq. 15 which are phenomenologicaly preferred, as
described in the Sec. 4. However the choice of $\tau_{3}$ is undetermined by
the $(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$ orbifold, and
instead shall be fixed by assuming a remnant $S_{4}$ symmetry, as discussed in
the next subsection.

###  3.2 $S_{4}$ Remnant Symmetry

The orbifold $\mathbb{Z}_{2}$, associated with the third torus
$\mathbb{T}^{2}_{C}$, does not fix $\tau$. However, supposing that the twist
angle is $\tau=\omega=e^{2i\pi/3}$ would leave a remnant $S_{4}$ symmetry
(which is a subgroup of the extra dimensional Poincaré group) after
compactification [29, 30]. We shall assume that there is a remnant $S_{4}$
after compactification, therefore fixing uniquely

$\tau_{3}=\omega.$ (28)

We focus on the branes of the fixus torus $\mathbb{T}_{C}$ [30, 31, 32],

$\bar{z}=\\{0,\ 1/2,\ \omega/2,\ \omega^{2}/2\\},$ (29)

which are naturally invariant under the orbifold transformations

$T_{1}:\ \bar{z}\to\bar{z}+1,\ \ \ T_{2}:\ \bar{z}\to\bar{z}+\omega,\ \ \
Z:\bar{z}\to-\bar{z}.$ (30)

The set of branes is invariant under the permutation set of them. However not
all permutations are Poincaré transformations.

These fixed branes and are permuted by the Poincaré transformations

$S_{1}:\bar{z}\to\bar{z}+1/2,\ \ \ S_{2}:\bar{z}+\omega/2,\ \ \
R:\bar{z}\to\omega\bar{z},\ \ \ P:\bar{z}\to\bar{z}^{*},\ \ \
P^{\prime}:\bar{z}\to-\bar{z}^{*},$ (31)

which, after orbifolding, generate the remnant symmetry. We can write these
operations explicitly $S_{1}[(12)(34)],\ S_{2}[(13)(24)],\ R[(243)(1)],\
P[(34)(1)(2)],\ P^{\prime}[(34)(1)(2)].$ There are only 3 independent
transformations since $S_{2}=R^{2}\cdot S_{1}\cdot R,\ \ P=P^{\prime}$.

These symmetry transformations relate to the $S_{4}$ generators with
$\hat{S}=S_{1},\ \hat{T}=R,\ \hat{U}=P$ satisfying Eq. 22 which is the
presentation rules for the $S_{4}$ symmetry [1].

The $\mathbb{T}_{A,B}$ have only 2 branes from Eq. 16. Therefore its remnant
symmetry can only be $\mathbb{Z}_{2}$.

With the assumption of an $S_{4}$ remnant symmetry, the $\tau_{3}$ is fixed
geometrically to be equal to $\omega$ [23]. .

##  4  A realistic orbifold model

(a) The extra dimensional space for $\mathbb{T}^{2}_{A}$. The $\mathbb{Z}_{4}$
orbifolding identifies the four isosceles triangles labeled as $a$.

(c) The extra dimensional space for $\mathbb{T}^{2}_{C}$. I The
$\mathbb{Z}_{2}$ orbifolding identifies the two equilateral triangles labeled
as $e$.

(b) The extra dimensional space for $\mathbb{T}^{2}_{B}$. The $\mathbb{Z}_{4}$
orbifolding is done by rotating the space by $\pi/2$ and creating drawing the
lattice (dotted pink). One identifies the overlaps, which is this case are
four quadrilaterlas labeled as $b$, four isosceles triangles labeled as $c$
and four right angle triangles labeled as $d$.

Figure 1: Visualization of the extra dimensional space for each of the
fundamental tori $\mathbb{T}^{2}_{A,B,C}.$ Identifying together opposite sides
we obtain $\mathbb{T}^{2}$. The orbifolding is described in each subfigure.
The dots represent the fixed points.

We now turn to a concrete 10d bottom-up orbifold model with three factorizable
tori built from the fundamental space depicted geometrically in Fig. 1. The
10d model is compactified on an orbifold
$(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$ and we assume
three finite modular symmetries $S^{A,B,C}_{4}$. Furthermore there is a
remnant $S_{4}$ symmetry whose only role is to fix $\tau_{3}=\omega$ ∥∥∥As
discussed later, remnant $S_{4}$ symmetry may be further employed to control
the Kähler potential.. This uniquely fixes the moduli geometrically to be
$\tau_{1}=i,\ \tau_{2}=i+2,\ \tau_{3}=\omega,$ (up to a choice in four).

The field content which defines the model is given in Table 4.

Field | $S_{4}^{A}$ | $S_{4}^{B}$ | $S_{4}^{C}$ |  $2k_{A}$ |  $2k_{B}$ |  $2k_{C}$ | Loc
---|---|---|---|---|---|---|---
$L$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{3}$ | 0 | 0 | 0 | $\mathbb{T}^{2}_{C}$
$e^{c}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 0 |  $-6$ | $\mathbb{T}^{2}_{C}$
$\mu^{c}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 0 |  $-4$ | $\mathbb{T}^{2}_{C}$
$\tau^{c}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 0 |  $-2$ | $\mathbb{T}^{2}_{C}$
$N_{a}^{c}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | $-4$ | 0 | $\mathbb{T}^{2}_{B}$
$N_{s}^{c}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $-2$ | 0 | 0 | $\mathbb{T}^{2}_{A}$
$\Phi_{BC}$ | $\mathbf{1}$ | $\mathbf{3}$ | $\mathbf{3}$ | 0 | 0 | 0 | Bulk
$\Phi_{AC}$ | $\mathbf{3}$ | $\mathbf{1}$ | $\mathbf{3}$ | 0 | 0 | 0 | Bulk

Yuk/Mass | $S_{4}^{A}$ | $S_{4}^{B}$ | $S_{4}^{C}$ |  $2k_{A}$ |  $2k_{B}$ |  $2k_{C}$
---|---|---|---|---|---|---
$Y_{e}(\tau_{3})$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{3}$ | 0 | 0 | $6$
$Y_{\mu}(\tau_{3})$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{3}$ | 0 | 0 | $4$
$Y_{\tau}(\tau_{3})$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{3}$ | 0 | 0 | $2$
$Y_{a}(\tau_{2})$ | $\mathbf{1}$ | $\mathbf{3}$ | $\mathbf{1}$ | 0 | $4$ | 0
$Y_{s}(\tau_{1})$ | $\mathbf{3}$ | $\mathbf{1}$ | $\mathbf{1}$ | $2$ | 0 | 0
$M_{a}(\tau_{2})$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | $8$ | 0
$M_{s}(\tau_{1})$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $4$ | 0 | 0

Table 4: Transformation properties of fields and modular forms (Yuk/Mass)
under the modular symmetries $S^{A,B,C}_{4}$ with modular weights $k_{A,B,C}$.
The Higgs fields $H_{u,d}$ (not displayed) transform trivially under all the
modular $S_{4}$ symmetries. The leptons $L\sim(2,-1/2)$, and
$e^{c},\mu^{c},\tau^{c}\sim(1,1)$ have the usual SM $SU(2)_{L}\times U(1)_{Y}$
quantum numbers and the right-handed neutrinos $N^{c}_{a,s}$ are SM singlets.
The Higgs $\Phi$ which break the three modular symmetries to their diagonal
subgroup, live in the 10d bulk, while the leptons live in the 2d subspaces as
shown.

The resulting 4d Lagrangian is [16], ignoring the dimensionless coupling
coefficients,

$\displaystyle w_{\ell}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda}\left[L\Phi_{BC}Y_{a}N_{a}^{c}+L\Phi_{AC}Y_{s}N_{s}^{c}\right]H_{u}$
$\displaystyle+\left[LY_{e}e^{c}+LY_{\mu}\mu^{c}+LY_{\tau}\tau^{c}\right]H_{d}$
$\displaystyle+\frac{1}{2}M_{a}N_{a}^{c}N_{a}^{c}+\frac{1}{2}M_{s}N_{s}^{c}N_{s}^{c}\,.$

and the modular Yukawa forms are fixed by the moduli
$\tau_{1}=i,\tau_{2}=i+2,\tau_{3}=\omega$ resulting in the alignments, using
Tables 3 and 4, ignoring the overall constants,

$\begin{split}Y_{a}&=(0,1,-1)^{T},\\\ Y_{s}&=(1,1+\sqrt{6},1-\sqrt{6})^{T},\\\
Y_{\tau}&=(0,1,0)^{T},\\\ Y_{\mu}&=(0,0,1)^{T},\\\
Y_{e}&=(1,0,0)^{T}.\end{split}$ (33)

The $\Phi$ fields are assumed to obtain a diagonal VEV that breaks two modular
symmetries into the diagonal one [16].

Hence, the charged-lepton mass matrix is simply given by

$M_{l}=v_{d}\begin{pmatrix}\left(Y_{e}\right)_{1}&\left(Y_{\mu}\right)_{1}&\left(Y_{\tau}\right)_{1}\\\
\left(Y_{e}\right)_{3}&\left(Y_{\mu}\right)_{3}&\left(Y_{\tau}\right)_{3}\\\
\left(Y_{e}\right)_{2}&\left(Y_{\mu}\right)_{2}&\left(Y_{\tau}\right)_{2}\end{pmatrix},$
(34)

where $v_{d}$ stands for $\left<H_{d}\right>$, and we ignore the dimensionless
coupling coefficients.

Plugging in the specific shapes of the modular forms given in Eq. 33 we arrive
at a diagonal charged-lepton mass matrix for $\tau_{C}=\omega$, including the
dimensionless coupling coefficients:

$M_{l}=v_{d}\begin{pmatrix}y_{e}&0&0\\\ 0&y_{\mu}&0\\\
0&0&y_{\tau}\end{pmatrix}.$ (35)

The Dirac neutrino mass matrix is then given by:

$M_{D}=v_{u}\begin{pmatrix}\left(Y_{a}\right)_{1}&\left(Y_{s}\right)_{1}\\\
\left(Y_{a}\right)_{3}&\left(Y_{s}\right)_{3}\\\
\left(Y_{a}\right)_{2}&\left(Y_{s}\right)_{2}\end{pmatrix},$ (36)

where, as usual, $v_{u}$ denotes the $H_{u}$ VEV, and the $2\times 3$
structure comes from the CSD with just two RH neutrinos. We have ignored the
dimensionless coupling coefficients. Choosing specific stabilisers for the two
remaining moduli fields, we can achieve a CSD(3.45) structure with
$n=1-\sqrt{6}$:

$M_{D}=v_{u}\begin{pmatrix}0&b\\\ a&b\left(1+\sqrt{6}\right)\\\
-a&b\left(1-\sqrt{6}\right)\end{pmatrix}.$ (37)

The type-I seesaw mechanism will lead to an effective mass matrix for the
light neutrinos:

$m_{\nu}=M_{D}\cdot M_{R}^{-1}\cdot
M_{D}^{T}=v_{u}^{2}\begin{pmatrix}\dfrac{b^{2}}{M_{s}}&\dfrac{b^{2}n}{M_{s}}&\dfrac{b^{2}(2-n)}{M_{s}}\\\\[12.0pt]
.&\dfrac{a^{2}}{M_{a}}+\dfrac{b^{2}n^{2}}{M_{s}}&-\dfrac{a^{2}}{M_{a}}+\dfrac{b^{2}n(2-n)}{M_{s}}\\\\[12.0pt]
.&.&\dfrac{a^{2}}{M_{a}}+\dfrac{b^{2}(2-n)^{2}}{M_{s}}\end{pmatrix},$ (38)

where $n=1-\sqrt{6}\approx-1.45$. This can be rewritten in terms of 3
independent physical parameters

$m_{\nu}=m_{a}\left(\begin{matrix}0&0&0&\\\ 0&1&-1\\\
0&-1&1\end{matrix}\right)+m_{b}e^{i\eta}\left(\begin{matrix}1&n&2-n&\\\
n&n^{2}&n(2-n)\\\ 2-n&n(2-n)&(2-n)^{2}\end{matrix}\right),$ (39)

where

$m_{a}=\left|\frac{v_{u}^{2}a^{2}}{M_{a}}\right|,\ \ \
m_{b}=\left|\frac{v_{u}^{2}b^{2}}{M_{s}}\right|,\ \ \ \rho={\rm
Arg}\left(\frac{a^{2}}{b^{2}}\right).$ (40)

Therefore the model has only these 3 parameters for the whole neutrino sector.

###  4.1 Eclectic symmetry with the remnant $S_{4}$

As discussed in Sec. 3.2, the remnant symmetry $S_{4}$ acts on the branes and
in the previous model, it has been identified with the modular symmetry
$S_{4}^{C}$. Therefore we have built a model whose flavour structure is
completely defined by the modular symmetries $S_{4}^{A}\times S_{4}^{B}\times
S_{4}^{C}$. However, it is known that having purely modular symmetries to
define the flavour structure complicates the Kähler [33].

In our setup, the only modular multiplet is the lepton doublet $L$. In
general, the minimal Kähler potential for a superfield $L$ would be a single
term $\mathcal{K}=L\bar{L}$. However as $L$ is modular form, the Kähler
potential is enhanced to include terms

$\mathcal{K}=L\bar{L}+\sum_{k}a_{k}(Y_{k}\overline{Y_{k}})_{\textbf{1}}(L\overline{L})_{\textbf{1}}+\sum_{k}b_{k}(Y_{k}L)_{\textbf{1}}(\overline{Y_{k}L})_{\textbf{1}},$
(41)

where the sum is over all available modular forms and the fields inside a
parenthesis $()_{\textbf{1}}$ are contracted into a modular symmetry singlet
and the $a_{k},b_{k}$ are arbitrary dimensionless constants. The $b_{k}$ terms
appear only the case where $L$ is something larger than a singlet, like in our
model, which is a triplet. In that case the sum is done over the three
different modular forms from Table 3, depending on the chosen $\tau$.

The $a_{k}$ terms can be absorbed as an overall normalization of the field
$L$, therefore they are not relevant. The $b_{k}$ terms are absorbed by the
normalization of each component of the field $L$ and therefore introducing
parameters that change the flavour structure given from the superpotential.

In our model, the only nontrivial modular form is the lepton doublet $L$ which
will have 3 $b_{k}$ Kähler terms. These parameters will affect the charged
lepton mass matrix and can be reabsorbed in the definition of
$y_{e,\mu,\tau}$, therefore preserving the same flavour structure there.
However these parameters also affect the normalization of each left handed
neutrino independently, introducing these 3 extra free parameters to the
neutrino mass matrix and therefore reducing the predictiveness of the model.
In general, for all modular symmetry models, it is assumed that these
parameters are negligible, although they not necessarily need be.

One could avoid the presence of the unwanted terms by enhancing the modular
symmetry by adding a standard flavour symmetry, such that the undesired Kähler
terms are forbidden by the standard flavour symmetry [34]. Relating flavor
symmetries and modular symmetries are called eclectic symmetries [18, 19, 22].

As an alternative model to the one presented in the previous section, we could
use the remnant symmetry as a standard flavour symmetry, as described in Sec
3.2. This way the $S_{4}^{C}$ becomes standard flavour symmetry while
$S_{4}^{A,B}$ remain as modular symmetries, thus having a trivial (where they
all commute) eclectic symmetry. With this assumption the lepton doublet is no
longer a modular form and there are no $b_{k}$ Kähler terms. The model would
require an extra $\mathbb{Z}_{3}$ shaping symmetry which would differentiate
the three charged lepton singlets. Furthermore the modular forms
$Y_{e,\mu,\tau}$ would not be available and they would have to be replaced by
3 flavon $S_{4}$ triplets $\phi_{e,\mu,\tau}$ whose VEV has the same desired
alignments. This could be easily achieved through the orbifold boundary
conditions and a very simple alignment superpotential [29]. With these
changes, the flavour structure and all the phenomenological implications would
be exactly the same as the model described in the previous subsection. Thus
the same flavour structure CSD($1-\sqrt{6}$) can be achieved easily through
modular or eclectic $S_{4}^{3}$ symmetry.

###  4.2 Numerical Fit

|  | without SK atmospheric data | with SK atmospheric data
---|---|---|---
|  |  NuFit $\pm 1\sigma$ | Model |  NuFit $\pm 1\sigma$ | Model
| $\theta_{12}/^{\circ}$ | $33.41_{-0.72}^{+0.75}$ | 34.34 | $33.41_{-0.72}^{+0.75}$ | 34.30
| $\theta_{23}/^{\circ}$ | $49.1_{-1.3}^{+1.0}$ | 48.31 | $42.2_{-0.9}^{+1.1}$ | 46.98
| $\theta_{13}/^{\circ}$ | $8.54_{-0.12}^{+0.11}$ | 8.54 | $8.58_{-0.11}^{+0.11}$ | 8.75
| $\delta/^{\circ}$ | $197_{-25}^{+42}$ | 284 | $232_{-26}^{+36}$ | 278
| $\dfrac{\Delta m^{2}_{21}}{10^{-5}~{}\text{eV}^{2}}$ | $7.41_{-0.20}^{+0.21}$ | 7.42 | $7.41_{-0.20}^{+0.21}$ | 7.13
| $\dfrac{\Delta m^{2}_{3\ell}}{10^{-3}~{}\text{eV}^{2}}$ | $+2.511_{-0.021}^{+0.028}$ | 2.510 | $+2.507_{-0.027}^{+0.026}$ | 2.520
| $\dfrac{m_{a}}{10^{-3}~{}{\rm eV}}$ |  | $31.47$ |  | 30.50
| $\dfrac{m_{b}}{10^{-3}~{}{\rm eV}}$ |  | $2.28$ |  | 2.32
| $\eta/\pi$ |  | $1.24$ |  | $1.26$
| $\chi^{2}$ |  | $6.3$ |  | 26.61
Table 5: Normal Ordering NuFit 5.2 values [35, 36] for the neutrino
observables, and the best fit point from the model. The best fit is for NuFit
data without SK atmospheric data where the atmospheric angle $\theta_{23}$ is
in the second octant, as preferred by the model.

With the CSD($1-\sqrt{6}$) structure, we can achieve the fits shown in Table
5. Note that in both best fits, there is a unique physical phase $\eta\approx
5\pi/4$ which could point to a geometrical origin.

To quantify how good the fit is we use

$\chi^{2}=\sum_{i}\left(\frac{x_{i}^{\rm exp}-x_{i}^{\rm
model}}{\sigma_{i}^{\rm exp}}\right)^{2},$ (42)

where it is summed over all 6 experimental neutrino values
$(\theta_{12},\theta_{13},\theta_{23},\delta,\Delta m^{2}_{21},\Delta
m^{2}_{3\ell})$. The model fits these 6 observables plus the lightest neutrino
mass (which is zero), the Majorana phases (where one is unphysical) which
determine neutrinoless beta decay parameter which is just equal to the $(1,1)$
element of the neutrino mass matrix, $m_{ee}=m_{b}$ [16]. Note that the 6
experimentally constrained observables are being fit with only 3 real
parameters $(m_{a},m_{b},\eta)$, which is a non-trivial achievement. Overall
these 3 parameters are predicting 9 neutrino observables, which shows that the
model is highly predictive. In particular the model requires a normal neutrino
mass squared ordering with the lightest neutrino being massless, and predicts
the atmospheric angle to be in the second octant, $\theta_{23}\approx
48^{\circ}$, with close to maximal leptonic CP violation, $\delta\approx
280^{\circ}$.

##  5 Conclusions

In recent years modular symmetries have been applied to flavour models in
bottom-up approaches, where finite modular groups $\Gamma_{N}$ may result from
the quotient group of the modular symmetry by its principal congruence
subgroup of level $N$, where for example $N=4$ corresponds to $S_{4}$. In such
approaches the role of the flavon field is played by a complex modulus field
$\tau$ in orbifold models with two extra dimensions.

In this paper we have discussed modular symmetry models arising from bottom-up
orbifold constructions. The simplest example in 6d involves the orbifold
$\mathbb{T}^{2}/\mathbb{Z}_{N}$ with a single torus defined by one complex
coordinate $z$ and a single modulus field $\tau$, playing the role of a flavon
transforming under a finite modular symmetry. More generally we have
considered bottom-up orbifolds in 10d, where the 6 extra dimensions are
factorisable into 3 tori, each defined by one complex coordinate $z_{i}$ and
involving the three moduli fields $\tau_{1},\tau_{2},\tau_{3}$ transforming
under three independent finite modular groups. Assuming supersymmetry,
consistent with the holomorphicity requirement, we consider all the orbifolds
of the form $(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{N}\times\mathbb{Z}_{M})$, and
list all the available orbifolds, which have fixed values of the moduli fields
(up to an integer). The key advantage of such 10d orbifold models over 4d
models is that the values of the moduli are not completely free but are
constrained by geometry and symmetry.

To illustrate the approach we have shown how a recently proposed littlest
modular seesaw model with $S_{4}^{3}$ modular symmetry could result from such
an orbifold construction. We have shown how this model may arise from an
$(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$ orbifold with
$\tau_{1}=i,\ \tau_{2}=i+2$ being fixed by the geometry of the
$\mathbb{Z}_{4}$ orbifold, while $\tau_{3}=\omega$ is determined by imposing a
further remnant $S_{4}$ flavour symmetry, commuting with the three $S_{4}$
modular symmetries. The $\tau_{1}=i$ leads to an alignment
$(1,1+\sqrt{6},1-\sqrt{6})$ and CSD$(n)$ with $n=1-\sqrt{6}$, where the
atmospheric angle $\theta_{23}$ is restricted to lie in the second octant. The
$\chi^{2}$ fit shows that this is a highly predictive and successful
description of all neutrino phenomenology, with two real parameters describing
all neutrino mixing and mass ratios.

An alternative case $n=1+\sqrt{6}$, which prefers the atmospheric angle
$\theta_{23}$ to be in the first octant, which was possible in the 4d model
[16], is not allowed here since it corresponds to an alignment
$(1,1-\sqrt{6},1+\sqrt{6})$ and a stabilizer $\tau_{1}=(-8+i)/13$ which is not
achievable in the 10d orbifold model considered here. Whereas in the 4d model
[16], the fixed points were selected in an ad hoc way, in the 10d orbifold
model considered here the values of $\tau_{1,2}$ are fixed by the geometry to
be equal to $i$ (up to an integer).

In the modular symmetry model considered here, the remnant $S_{4}$ plays no
role apart from fixing $\tau_{3}=\omega$. However in an alternative orbifold
model, the combination of flavour symmetry and modular symmetry could be used
to control the corrections to the Kähler potential, as in top-down eclectic
flavour symmetry. This would reintroduce flavons, and lead to a more
complicated model which is beyond the scope of the main discussion here,
although we have briefly sketched the consequences.

Finally we note that the bottom-up approach to modular symmetry from orbifolds
followed here can readily be extended to GUTs, with up to three moduli groups
and moduli fields. One could similarly include a remnant flavour symmetry,
leading to a bottom-up version of the ecletic flavour symmetry in orbifold
GUTs.

## Acknowledgements

SFK acknowledges the STFC Consolidated Grant ST/L000296/1 and the European
Union’s Horizon 2020 Research and Innovation programme under Marie Sklodowska-
Curie grant agreement HIDDeN European ITN project (H2020-MSCA-
ITN-2019//860881-HIDDeN). SFK would also like to thank CERN for its
hospitality.

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