# Modular Family Symmetry in F-Theory GUTs from the Bottom-up

Vasileios Basiouris<EMAIL_ADDRESS>Physics Department, University of
Ioannina, 45110, Ioannina, Greece  Miguel Crispim Romão
<EMAIL_ADDRESS>Institute for Particle Physics Phenomenology,
Department of Physics, Durham University, Durham DH1 3LE, U.K.  Laboratório de
Instrumentação e Física Experimental de Partículas, Escola de Ciências,
Departamento de Física, Universidade do Minho, 4701-057 Braga, Portugal
School of Physics and Astronomy, University of Southampton,Southampton SO17
1BJ, United Kingdom  Stephen F. King<EMAIL_ADDRESS>School of Physics
and Astronomy, University of Southampton,Southampton SO17 1BJ, United Kingdom
George K. Leontaris<EMAIL_ADDRESS>Physics Department, University of Ioannina,
45110, Ioannina, Greece

###### Abstract

Finite modular family symmetry can emerge from top-down approaches based on
heterotic string theory or Type IIB string theory. We show that, in addition
to such approaches, it can also emerge from local F-Theory bottom-up
constructions. As a first example of the new approach, we have analysed in
detail a concrete F-Theory Fluxed $SU(5)$ Grand Unified Theory (GUT) with
modular $S_{4}$ family symmetry. The model fits the fermion mass and mixing
data very well and serves as a demonstration of the bottom-up approach to
modular family symmetry in F-Theory fluxed GUTs.

Report Number: IPPP/24/42

##  1 Introduction

The predictive power of discrete flavour symmetries in fermion mass
hierarchies has long been emphasised. In attempting to explain the fermion
mass hierarchies and mixings, various non-abelian discrete groups were
successfully applied as family symmetries to numerous extensions of the
standard model (see e.g. [1] and references therein).

Such family symmetry must be spontaneously broken by vacuum expectation values
(VEVs) of scalar flavon fields, leading to the vacuum alignment problem, which
leads to additional complications. Modular invariance was suggested to
overcome such complications [2]. In the simplest examples, the only flavon is
a complex modulus $\tau$ that transforms under the modular group
$SL(2,\mathbb{Z})$. Finite modular symmetries $\Gamma_{N}$ can emerge as the
quotient group of $SL(2,\mathbb{Z})$ with one of its principal congruence
subgroups $\Gamma(N)$ of level $N$. Yukawa couplings are represented by
modular forms which transform under the finite level $N$ quotient group,
$\Gamma_{N}$, that further restricts the possible mass textures and increases
the predictive power of the effective theory. Subsequently, such finite
modular family symmetry has been widely studied from the top-down and bottom-
up perspectives (see e.g. [3, 4] for recent reviews).

From the top-down perspective, there have been attempts to investigate such
symmetries in the context of heterotic superstring motivated models [5].
Indeed, upon orbifold compactification of the six extra dimensions, the
modular group and its congruence group emerge naturally in the four-
dimensional effective theory. The origin of the modular group, in particular,
is attributed to the specific geometry of the compactification manifold.

Another example is an effective supergravity model from Type IIB
compactification [6, 7]. Its massless spectrum contains several moduli which
correspond to deformations of the Calabi-Yau (CY) manifold. Among them, the
complex structure, the Kähler and the axio-dilaton, the latter being a complex
scalar combination $\tau$ where the real part is the RR axion $C_{0}$ and its
imaginary is the inverse of the string coupling $g_{s}^{-1}=e^{-\phi}$ where
$\phi$ is the dilaton field. Taking into account that the massless spectrum is
described by an associated supergravity and the fact that $g_{s}$ takes only
positive values, the field $\tau$ admits values subject to $SL(2,\mathbb{Z})$,
which is nothing else than S-duality.

Several quantities and parameters of the effective theory, such as the Kähler
potential, the superpotential, and Yukawa couplings, depend on the field
$\tau$ and have well-defined transformation properties under (some congruence
group of) $SL(2,\mathbb{Z})$. Analogously, in toroidal compactifications,
there are discrete symmetries, dubbed T-dualities, which relate Type IIB
theory with compactification radius $R$ to Type IIA associated with the
inverse radius $1/R$. Constraints similar to the above mentioned are also
expected to be imposed on the effective theory (see [8] and review [9]).

In this paper, we extend these ideas to the rich framework of local F-Theory,
which can be applied using bottom-up constructions. We discuss the explicit
constructions of Yukawa couplings in F-Theory as integrals of overlapping
wavefunctions around the intersection of matter curves, each a Riemannian
surface. In this construction, the wavefunctions are solutions to the
equations of motion of the compactified Yang-Mills (YM) theory on the
7-branes, which the solutions have an undetermined holomorphic dependency on
the coordinates of the matter curve. This observation leads us to conclude
that the structure of the Yukawa couplings will depend on the complex
structure moduli of the matter curves in such a way that if these curves
exhibit modular symmetries, then the Yukawa couplings can be modular forms
associated with these symmetries.

Using this observation, we argue that one may explicitly construct fluxed GUTs
with discrete modular symmetry, within a bottom-up approach where the fields
on chosen matter curves, subject to specified fluxes, transform as a
representation of a discrete modular family group and carry assigned modular
weights. Additionally, the Yukawa matrices, arising from the overlapping
wavefunction integrals at matter curve intersections, are promoted to modular
forms of appropriate weight. As a first example of the new approach, we have
analysed in detail a concrete F-Theory Fluxed $SU(5)$ GUT with modular $S_{4}$
family symmetry. We show that, although the model fits the data very well,
with the best point having vanishing $\chi^{2}$, it is over-parameterised and
therefore lacks predictive power. Nevertheless, it provides some insight into
fermion mass hierarchies and serves as a demonstration of the bottom-up
approach to modular family symmetry in F-Theory fluxed GUTs.

This work is organised as follows. In section 2 we revise modular symmetry
invariant field theories to set notation and definitions. In section 3 we
discuss the origin of discrete finite modular symmetries in F-Theory
compactifications in three stages: in section 3.1 we discuss the existence of
Type IIB vacua, which is a perturbative limit of F-Theory, with explicit
finite modular symmetry and how the modular symmetry of the background
geometry mixes with the axio-dilaton modular symmetry; in section 3.2 we
discuss how the Yukawa couplings in Type IIB and F-Theory depend on the axio-
dilaton and transform under its modular symmetry; then, in section 3.3 we
review how the Yukawa couplings are obtained in F-Theory and argue that they
inherit modular symmetry properties from the geometry of the matter curves
supporting the matter fields. Equipped with these arguments, in section 4 we
present an F-Theory fluxed GUT with modular symmetry, including a fit to
experimental data in section 4.1. In section 5 we conclude and provide an
outlook for our framework.

##  2 Modular Symmetry Invariant Supersymmetric Field Theories

In this section, we introduce modular forms of even weight, following [2] to
set the notation and definitions that we will be using for the rest of this
work. We first define the homogeneous modular group, $\Gamma$, is composed of
$2\times 2$ matrices such that

$\Gamma=SL(2,\mathbb{Z})=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\Big{|}\ a,b,c,d\in\mathbb{Z},\ ad-bc=1\right\\}\ .$ (1)

$\Gamma$ has two generators, $S$ and $T$, which respect

$\displaystyle S^{4}=(ST)^{3}=I,$ (2)

where $I$ is the identity element, and can be represented in matrix form as

$\displaystyle S=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\
T=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}\ .$ (3)

Another closely related group is the inhomogeneous modular group,
$\bar{\Gamma}$, which is the group of linear fraction transformations,
$\gamma$, acting on the upper half complex plane,
$\mathcal{H}=\\{\tau\in\mathbb{C},\ \Im(\tau)>0\\}$, as

$\displaystyle\tau\mapsto\gamma\tau=\frac{a\tau+b}{c\tau+d},\ \text{s.t.
}a,b,c,d\in\mathbb{Z},\ ad-bc=1\ .$ (4)

Since the elements $\pm\gamma$ produce the same linear fractional
transformation, we can see that this group is related to $\Gamma$ as

$\displaystyle\bar{\Gamma}=\Gamma/\\{\pm I\\}=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}/\\{\pm\mathbb{I}\\}\Big{|}\ a,b,c,d\in\mathbb{Z},\ ad-
bc=1\right\\},$ (5)

where $\mathbb{I}$ is the identity $2\times 2$ matrix, and we note that it is
isomorphic to the projective matrix group $PSL(2,\mathbb{Z})\cong
SL(2,\mathbb{Z})/\\{\pm I\\}$. $\bar{\Gamma}$ is also generated by $S$ and
$T$, but since $I$ and $-I$ are identified, we have $S^{2}=I$, and they act on
a complex number $\tau\in\mathcal{H}$ as

$\displaystyle S:\tau\mapsto-\frac{1}{\tau},\ T:\tau\mapsto\tau+1$ (6)

An important sequence of subgroups of $\Gamma$ are the principal congruence
subgroups of level $N$, $\Gamma(N)$, which are given by

$\displaystyle\Gamma(N)=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma\Big{|}\ \begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\mod N\right\\}\ ,$ (7)

which are infinite normal subgroups of $\Gamma$. Since the congruence of unity
are all the integers, we have $\Gamma(1)=\Gamma$. Furthermore, we define
$\bar{\Gamma}(N)=\Gamma(N)/\\{\pm I\\}$ for $N=1,2$, and since $-I$ is not an
element of $\Gamma(N)$ for $N>2$ we have $\bar{\Gamma}(N)=\Gamma(N)$ for
$N>2$.

The quotients $\Gamma_{N}=\bar{\Gamma}/\bar{\Gamma}(N)$ are called
inhomogeneous finite modular groups of level $N$. They are finite and
discrete, and are generated by $S$ and $T$ respecting

$\displaystyle S^{2}=(ST)^{3}=T^{N}=I\ .$ (8)

Inhomogeneous finite modular groups $\Gamma_{N}$ have been studied in great
detail in the literature [2, 10, 11, 12, 13], with $\Gamma_{2}\cong S_{3}$,
$\Gamma_{3}\cong A_{4}$, $\Gamma_{4}\cong S_{4}$, and $\Gamma_{5}\cong A_{5}$,
as well as higher levels [14, 15]. For a full, comprehensive, list see [4] and
references therein.

An important aspect of modular symmetries is the notion of modular forms that
are holomorphic functions of a complex parameter $\tau\in\mathcal{H}$, which
we will take to be the modulus of a compact space, with special transformation
properties. A modular form $f(\tau)$, of weight $k$ and level $N$ is a
holomorphic function of $\gamma\in\mathcal{H}$ that under the action of
$\gamma\in\bar{\Gamma}(N)$ transforms as

$\displaystyle f(\gamma\tau)=(c\tau+d)^{k}f(\tau),\
\forall\gamma\in\bar{\Gamma}(N)\ ,$ (9)

and it can been shown [16] that, for even $k$, these span a finite dimension
linear space, $\mathcal{M}_{k}(\bar{\Gamma}(N))$, and in which it is possible
to find a basis such that the modular forms furnish a representation of the
inhomogeneous finite modular group $\Gamma_{N}$

$\displaystyle f_{r}(\gamma\tau)=(c\tau+d)^{k}\rho_{r}(\gamma)f_{r}(\tau),\
\forall\gamma\in\bar{\Gamma}\ ,$ (10)

where $\gamma$ is a representative element of $\Gamma_{N}$, $\rho_{r}(\gamma)$
is the representation matrix of $\gamma$ in the representation $r$ of
$\Gamma_{N}$.

We will now use these to build supersymmetric theories with modular
invariance, in order to construct a framework for model building. We will
follow [2, 17] and provide the uniform notation and definitions that we will
use for the remainder of this work.

An $\mathcal{N}=1$ supersymmetric theory is defined by two functions: the
Kähler potential, $K(\Phi_{I},\bar{\Phi}_{I},\tau,\bar{\tau})$ \- a real
function of the chiral superfields $\\{\Phi_{I}\\}$ and the moduli
$\\{\tau\\}$ \- and the superpotential $W(\Phi,\tau)$ \- a holomorphic
function of the same superfields. We will consider the case where there is
only one modulus field.

The superpotential admits the generic expansion

$\displaystyle
W(\Phi_{I},\tau)=\sum_{n}Y_{I_{1},\dots,I_{n}}\Phi_{I_{1}}\dots\Phi_{I_{n}}\
,$ (11)

where the couplings $Y_{I_{1},\dots,I_{n}}\Phi_{I_{1}}$ are modular forms of
weight $k_{Y}$, and will then transform under the action of $\bar{\Gamma}$.
The chiral superfields, $\\{\Phi_{I}\\}$, will also transform non-trivially,
but are otherwise not modular forms. Under the action of modular group, we
have

$\displaystyle\tau\mapsto\gamma\tau=\frac{a\tau+b}{c\tau+d}$ (12)
$\displaystyle\Phi_{I}\mapsto(c\tau+d)^{-k_{I}}\rho_{I}(\gamma)\Phi_{I}$ (13)
$\displaystyle
Y_{I_{1},\dots,I_{n}}\mapsto(c\tau+d)^{k_{Y}}\rho_{Y}(\gamma)Y_{I_{1},\dots,I_{n}}$
(14)

where $\gamma$ is an element of $\bar{\Gamma}$, and $\rho_{I}(\gamma)$ and
$\rho_{Y}(\gamma)$ are representation matrices of $\gamma$ in the finite
modular subgroup, $\Gamma_{N}$, $-k_{I}$ and $k_{Y}$ are modular weights.
Therefore, superpotential level invariance is obtained when the sum of the
modular weights for each coupling vanish, and if there is a singlet in the
product of representations

$\displaystyle k_{Y}$ $\displaystyle=k_{I_{1}}+\dots k_{I_{n}}$ (15)
$\displaystyle\mathbf{1}$
$\displaystyle\subset\rho_{Y}\rho_{I_{1}}\dots\rho_{I_{n}}\ .$ (16)

When considering the full supergravity theory, the relevant defining quantity
which needs to be invariant is the Kähler function,
$G(\Phi_{I},\bar{\Phi}_{I},\tau,\bar{\tau})$, given by

$\displaystyle
G(\Phi_{I},\bar{\Phi}_{I},\tau,\bar{\tau})=K(\Phi_{I},\bar{\Phi}_{I},\tau,\bar{\tau})+\ln
W(\Phi_{I},\tau)+\ln\bar{W}(\bar{\Phi}_{I},\bar{\tau})\ .$ (17)

This allows for the sum of modular weights to be non-vanishing at the
superpotential level, as long as the leftover is absorbed by the Kähler
potential through a Kähler transformation. However, to do so one usually needs
an explicit form of the Kähler potential. For example, if one considers a
Kähler potential with the customary no-scale structure with a chiral
superfield expansion

$\displaystyle
K(\Phi_{I},\bar{\Phi}_{I},\tau,\bar{\tau})=-h\ln(-i(\tau-\bar{\tau}))+\sum_{I}(-i(\tau-\bar{\tau}))^{-k_{I}}|\Phi_{I}|^{2}\
,$ (18)

with $h$ a positive number, the superpotential now transforms as

$\displaystyle W(\Phi,\tau)\mapsto(c\tau+d)^{-h}W(\Phi,\tau)\ .$ (19)

This effectively amounts to change the condition for vanishing modular weights
to

$\displaystyle k_{Y}=k_{I_{1}}+\dots k_{I_{n}}-h\ ,$ (20)

whilst each superpotential coupling still needs to be an $\Gamma_{N}$
invariant singlet.

##  3 The Origin of Discrete Finite Modular Symmetry from Type IIB to F-Theory

We now discuss how the finite modular symmetry can arise in F-Theory
constructions. We first revisit Type IIB (the perturbative limit of F-Theory)
vacua with discrete finite modular symmetry and explicitly obtain an $S_{4}$
invariant vacuum. We then turn to F-Theory, which inherits the S-duality from
Type IIB, to identify the axio-dilaton modular symmetry, which endows the
matter Yukawa couplings with modular symmetry transformation properties.
Finally, we present our conjecture that F-Theory matter curves can carry a
geometric modular symmetry, which will manifest itself in the Yukawa
couplings, endowing F-Theory fluxed GUTs with a discrete modular family
symmetry.

###  3.1 Finite Modular Symmetry in Type IIB Orientifold Toroidal
Compactifications

We now discuss the origins of finite modular symmetries in Type IIB string
theory. To this effect, we will study, expanding on [6] Type IIB orientifold
compactifications, where one can stabilise the moduli in a vacuum that is
invariant to finite modular symmetries. The starting point is Type IIB, which
exhibits an explicit modular invariance for the axio-dilaton irrespective of
the details of the compact space. Upon choosing a factorisable toroidal
orientifold for the compactification, $T^{6}/\mathbb{Z}_{2}=(T_{1}^{2}\times
T_{2}^{2}\times T_{3}^{2})/\mathbb{Z}_{2}$ the theory will also manifest the
modular invariance associated with the complex structure moduli of each of the
tori, in other words we will have
$SL(2,\mathbb{Z})_{\tau}\otimes(\otimes_{i=1}^{3}SL(2,\mathbb{Z})_{i})$ before
the complex structure moduli are stabilised by Type IIB flux configurations.
Once the fluxes acquire nonvanishing VEVs, we will show that the supersymmetry
preserving vacuum transforms non-trivially under a congruence subgroup of
order $N$, $\bar{\Gamma}(N)$, of the original modular symmetries, therefore
breaking the preserved symmetry to $\Gamma_{N}$.

As mentioned, we start with the Type IIB string theory, which is characterised
by the strong-weak coupling duality (S-duality for short) which relates the
theory with string coupling $g_{s}$ to that with $g_{s}^{-1}$. S-duality is a
non-perturbative symmetry based on the $SL(2,\mathbb{Z})$ modular group and is
realised by the axio-dilaton modulus $\tau$ whose imaginary component is
identified with the inverse string coupling

$\tau=C_{0}+ie^{-\phi}\equiv C_{0}+i\frac{1}{g_{s}}\equiv C_{0}+is~{},$ (21)

where $\phi$ is the dilaton, and for convenience the definition $s=g_{s}^{-1}$
has been introduced.

The four-dimensional (4d) effective action of the string moduli is described
by the Kähler potential and the superpotential, both dependent on the complex
structure moduli. The Kähler potential is parametrised in terms of the moduli
and the axio-dilaton

$\displaystyle
K=-\ln(-i(\tau-\bar{\tau}))-2\ln(\mathcal{V})-2\ln\left(e^{-\frac{3}{2}\phi}\int
J\wedge J\wedge J\right)\ ,$ (22)

where $\phi$ is the dilaton, $\tau$ is the axio-dilaton defined in (21),
$\mathcal{V}$ is the volume of the compactified space, and $J$ its Kähler form
$dJ=0$ which depends on the complex coordinates $z^{i}$ and $g_{i\bar{j}}$ the
Kähler metric, and in its most general form is given by

$J=ig_{i\bar{j}}dz^{i}\wedge d\bar{z}^{j}~{}.$

The superpotential for the moduli fields, is given by the standard Gukov-Vafa-
Witten formula

$\displaystyle W$ $\displaystyle=\int G_{3}\wedge\Omega,$ (23)

were the three form flux $G_{3}$ and the holomorphic three form $\Omega$ are
given

$\displaystyle G_{3}$ $\displaystyle=F_{3}-\tau H_{3}$ (24)
$\displaystyle\Omega$ $\displaystyle=dz_{1}\wedge dz_{2}\wedge dz_{3}\ .$ (25)

Finally, for the toroidal case we define $z_{i}=x_{i}+\tau_{i}y_{i}$ so that,

$dz_{i}=dx_{i}+\tau^{i}dy^{i}~{},$

where ${z_{i}}$ corresponds to the three complex coordinates of the
compactified space, and $\tau^{i}$ are the complex structure moduli of the
orientifold. In general, the complex structure moduli form a matrix,
$\tau^{ij}$, parameterising the 3-cycles of the compactification, but here we
take it to be a diagonal matrix, as we will be considering factorisable
toroidal orientifolds.

Let now the basis for 3-forms be $(\alpha_{i},\beta^{j})$

$\displaystyle\alpha_{0}=dx^{1}\wedge dx^{2}\wedge
dx^{3},\;\;\alpha_{i}=\dfrac{1}{2}\epsilon_{ilm}\;dx^{l}\wedge dx^{m}\wedge
dy^{i}$ $\displaystyle\beta^{0}=dy^{1}\wedge dy^{2}\wedge
dy^{3},\;\;\beta^{i}=-\dfrac{1}{2}\epsilon_{ilm}dy^{l}\wedge dy^{m}\wedge
dx^{i}~{},$ (26)

where we notice that there is no sum in $i=1,2,3$, and we have

$\displaystyle\int\alpha_{i}\wedge\beta^{j}=\delta^{j}_{i}~{}\ ,$ (27)

where the integral is over the compact space.

The 3-form field strengths are expanded in terms of the basis as

$\displaystyle
F_{3}=m^{0}\alpha_{0}+m^{i}\alpha_{i}+n_{i}\beta^{i}+n_{0}\beta^{0}$
$\displaystyle
H_{3}=p^{0}\alpha_{0}+p^{i}\alpha_{i}+q_{i}\beta^{i}+q_{0}\beta^{0}\ ,$ (28)

where $m$, $n$, $p$, and $q$ are quantised flux components, and are therefore
integer valued. In fact, in the absence of exotic O3-planes, these are all
even integers. The 3-form fluxes induce a D3-brane charge which has to fulfill
a tadpole cancellation condition

$\displaystyle N_{D3}+\frac{1}{2}N_{\rm flux}=\frac{1}{4}N_{O3}\ ,$ (29)

where $N_{D3}$ is the number of D3-branes, $N_{O3}$ the number of O3-planes to
be set by the details of the compactified space, and

$\displaystyle N_{\rm flux}=\int H_{3}\wedge F_{3}\ ,$ (30)

which can be explicitly evaluated to obtain

$\displaystyle
p^{0}n_{0}-q_{0}m^{0}+\sum_{i}(p^{i}n_{i}-q_{i}m^{i})=2\left(\frac{1}{4}N_{O3}-N_{D3}\right)\
.$ (31)

The most general form of the superpotential for the moduli fields in the basis
presented above is then

$\displaystyle W$ $\displaystyle=(m^{0}-\tau
p^{0})(\Pi_{j}\tau_{j})-(m^{i}-\tau p^{i})(\Pi_{j\neq i}\tau_{j})-(n_{i}-\tau
q_{i})\tau^{i}-(n_{0}-\tau q_{0})$
$\displaystyle=\tau_{1}\tau_{3}\tau_{2}(m^{0}-p^{0}\tau)-\tau_{3}\tau_{2}(m^{1}-p^{1}\tau)-\tau_{1}\tau_{3}(m^{2}-p^{2}\tau)-\tau_{1}\tau_{2}(m^{3}-p^{3}\tau)-$
$\displaystyle-\tau_{1}(n_{1}-\tau q_{1})-\tau_{2}(n_{2}-\tau
q_{2})-\tau_{3}(n_{3}-\tau q_{3})-(n_{0}-\tau q_{0})\ .$ (32)

The non-vanishing flux components will fix the moduli along flat directions,
where the potential is minimised, $D_{\tau}W=D_{\tau_{k}}W=0$, without
breaking supersymmetry. Along these flat directions, the invariance of the
fluxes under the modular symmetries of the axio-dilaton, $\tau$, and the
complex structure moduli, $\tau_{i}$, will lead to vacua which are invariant
under a finite modular subgroup. To see this, we focus on factorisable
toroidal orientifold compactification $T^{6}/\mathbb{Z}_{2}=(T^{2}_{1}\times
T^{2}_{2}\times T^{2}_{3})/\mathbb{Z}_{2}$.

The Type IIB action and the superpotential, eq. 23, are invariant under the
axio-dilaton modular symmetry, $SL(2,\mathbb{Z})_{\tau}$, according to which
the axio-dilaton and the 3-forms $F_{3}$, $H_{3}$ transform as [18]

$\displaystyle\tau^{\prime}=R(\tau)=\dfrac{a\tau+b}{c\tau+d}\ $ (33)
$\displaystyle\begin{pmatrix}F_{3}^{\prime}\\\
H_{3}^{\prime}\end{pmatrix}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}F_{3}\\\ H_{3}\end{pmatrix},\;\;R\in
SL(2,\mathbb{Z})_{\tau}\ .$ (34)

Furthermore, since the compactified space is a factorised torus, we can
identity three complex moduli $\tau_{i}$, where each has its own modular
symmetry for vanishing fluxes. Each torus $T_{i}$, is defined as the quotient
of the complex plane $\mathbb{C}/\Lambda_{i}$, where $\Lambda_{i}$ is a
lattice spanned by the vectors
$\mathbf{e}_{i}=(e_{y_{i}},e_{x_{i}})^{T}=(\tau_{i},1)^{T}$. One can further
define $\xi_{i}=(y_{i},x_{i})^{T}$ where the coordinates
$x_{i}\in[0,1),\,y_{\in}[0,1]$ and $z_{i}$ are introduced according to [19]

$z_{i}=\xi_{i}^{T}\mathbf{e}_{i}\equiv(y_{i},x_{i})\left(\begin{array}[]{c}e_{y_{i}}\\\
e_{x_{i}}\end{array}\right)$ (35)

and for $(e_{y_{i}},e_{x_{i}})^{T}=(\tau_{i},1)^{T}$ in particular,

$z_{i}=(y_{i},x_{i})\left(\begin{array}[]{c}\tau_{i}\\\
1\end{array}\right)\equiv x_{i}+\tau_{i}y_{i}z\ .$ (36)

Under modular symmetry $SL(2,\mathbf{Z})$

$\displaystyle R_{i}=\begin{pmatrix}a_{i}&b_{i}\\\
c_{i}&d_{i}\end{pmatrix},\;R_{i}\in SL(2,\mathbf{Z})$ (37)

the vectors $\mathbf{e}_{i}$ transform according to

$\displaystyle\mathbf{e}^{\prime}_{i}=R_{i}\mathbf{e}_{i}~{}.$ (38)

Both vectors, $\mathbf{e}^{\prime}$ and $\mathbf{e}_{i}$, span the same
lattice, and, since $R_{i}R_{i}^{-1}=I$,

$z=(y_{i},x_{i})R_{i}R_{i}^{-1}\left(\begin{array}[]{c}e_{y_{i}}\\\
e_{x_{i}}\end{array}\right)=e^{\prime}_{x_{i}}(x_{i}^{\prime}+\tau_{i}^{\prime}{y^{\prime}_{i}}),$

where the modulus $\tau_{i}$ describes the shape of the torus transforms as

$\displaystyle\tau_{i}=\frac{e_{y_{i}}}{e_{x_{i}}}\mapsto\tau^{\prime}_{i}=\frac{e^{\prime}_{y_{i}}}{e^{\prime}_{x_{i}}}=\frac{a_{i}\tau_{i}+b_{i}}{c_{i}\tau_{i}+d_{i}}~{}.$
(39)

Thus, we recover the modular symmetry transformation presented in the previous
section. This transformation also affects the real coordinates

$\displaystyle z_{i}=(y_{i},x_{i})\mathbf{e}_{i}\mapsto
z^{\prime}_{i}=(y_{i}^{\prime},x_{i}^{\prime})\mathbf{e}_{i}^{\prime}=(y_{i}^{\prime},x_{i}^{\prime})R_{i}\mathbf{e}_{i}\
,$ (40)

therefore, modular invariance implies that $(y_{i},x_{i})$ transform under
$SL(2,\mathbb{Z})_{i}$ as

$\displaystyle\begin{pmatrix}y_{i}^{\prime}\\\
x_{i}^{\prime}\end{pmatrix}=(R_{i}^{-1})^{T}\begin{pmatrix}y_{i}\\\
x_{i}\end{pmatrix}\ .$ (41)

For the remaining of the analysis, it is useful to consider the transformation
properties of the 1-forms. Thus, for the torus $T_{i}^{2}$, according to the
above reasoning, we have [19]

$\omega=\omega_{l}d\xi_{i}^{l},\;d\xi_{i}^{k}=\begin{pmatrix}dy_{i}^{k}\\\
dx_{i}^{k}\end{pmatrix}~{},\;\omega_{k}^{\prime}=R_{kl}\omega_{l}\ .$ (42)

As can be readily checked, an immediate consequence of the above setup is that
the holomorphic 3-form $\Omega$, defined in eq. 25, transforms as

$\displaystyle\Omega\mapsto\frac{\Omega}{\Pi_{i=1}^{3}(c_{i}\tau_{i}+d_{i})}\
.$ (43)

Furthermore, for a factorisable orientifold, in the large volume limit, the
last term of the Kähler potential, eq. 22, takes the explicit form

$\displaystyle-2\ln\left(e^{-\frac{3}{2}\phi}\int J\wedge J\wedge
J\right)=-\ln\left(i(\tau_{1}-\bar{\tau}_{1})(\tau_{2}-\bar{\tau}_{2})(\tau_{3}-\bar{\tau}_{3})\right)\
,$ (44)

and under $\otimes_{i=1}^{3}SL(2,\mathbb{Z})_{i}$ transforms as

$\displaystyle-\ln\left(i(\tau_{1}-\bar{\tau}_{1})(\tau_{2}-\bar{\tau}_{2})(\tau_{3}-\bar{\tau}_{3})\right)\mapsto$
$\displaystyle-\ln\left(i(\tau_{1}-\bar{\tau}_{1})(\tau_{2}-\bar{\tau}_{2})(\tau_{3}-\bar{\tau}_{3})\right)$
$\displaystyle+\ln\left(\Pi_{i=1}^{3}|c_{i}\tau_{i}+d_{i}|^{2}\right)\ ,$ (45)

where we notice that the extra term cancels exactly the factor from eq. 43 in
the supergravity action, which implies that $G_{3}$ needs to be invariant
under the tori modular symmetries.

Therefore, under the axio-dilaton and the three tori modular symmetries, both
3-forms $H_{3}$, $F_{3}$, and the real coordinates pairs $(x_{i},y_{i})$ on
which the 3-form basis is defined transform non-trivially, while $G_{3}$
itself remains invariant under the tori modular symmetries. This will imprint
non-trivial constraints on the flux data. Furthermore, along flat directions
of the superpotential, the flux data allowed by modular invariance will fix
the moduli. To see this, we first introduce the following configuration for
the fluxes

$\displaystyle p_{3}=-fm^{0},\;q_{1}=fm^{2},\;q_{2}=fm^{1},\;q_{0}=fn_{3},$
(46)

where $f$ is an integer. For this set of fluxes, the superpotential is given
by

$\displaystyle W$
$\displaystyle=\left(f\tau-\tau_{3}\right)\left(\tau_{1}\left(m^{2}-m^{0}\tau_{2}\right)+m^{1}\tau_{2}+n_{3}\right)\
.$ (47)

Using the definition (42), the 3-forms can now be written as [19]

$\displaystyle F_{3}=A_{ij}d\xi_{1}^{i}\wedge d\xi_{2}^{j}\wedge dx_{3}\ ,$
(48) $\displaystyle H_{3}=B_{ij}d\xi_{1}^{i}\wedge d\xi_{2}^{j}\wedge dy_{3}\
,$ (49)

where $B_{ij}=-fA_{ij}$ with

$\displaystyle A=\left(\begin{array}[]{cc}-n_{3}&m^{1}\\\ m^{2}&m^{0}\\\
\end{array}\right)~{}.$ (52)

Under the transformation of the modular symmetries associated with the tori
$i=1,2$, i.e. $SL(2,\mathbb{Z})_{1}\times SL(2,\mathbb{Z})_{2}$, the 3-forms
$F_{3}$, $H_{3}$ transform as

$\displaystyle F_{3}$
$\displaystyle\mapsto(R_{1}^{-1}A(R_{2}^{-1})^{T})_{ij}d\xi_{1}^{i}\wedge
d\xi_{2}^{j}\wedge dx_{3}$ (53) $\displaystyle H_{3}$
$\displaystyle\mapsto(R_{1}^{-1}A(R_{2}^{-1})^{T})_{ij}d\xi_{1}^{i}\wedge
d\xi_{2}^{j}\wedge dx_{3}\ ,$ (54)

and in order for $G_{3}$ to remain invariant the following relation must hold
true

$R_{1}^{-1}A(R_{2}^{-1})^{T}=A~{}.$ (55)

This imposes non-trivial constraints on the values of the flux data. We now
consider the superpotential in eq. 47 and its flat supersymmetric directions,
$\partial_{\tau}W=\partial_{\tau_{i}}W=W=0$, which yield

$\displaystyle\tau_{3}$ $\displaystyle=f\tau$ (56) $\displaystyle\tau_{1}$
$\displaystyle=\frac{-n_{3}-m^{1}\tau_{2}}{m^{2}-m^{0}\tau_{2}}\ .$ (57)

From eq. 56, we see that for $f=1$ the axio-dilaton $\tau$ and the complex
structure $\tau_{3}$ are identified, $\tau=\tau_{3}$. This implies that the
diagonal $SL(2,\mathbb{Z})\subset SL(2,\mathbb{Z})_{\tau}\times
SL(2,\mathbb{Z})_{\tau_{3}}$ remains unbroken by the vacuum

$\displaystyle\tau^{\prime}_{3}=R_{3}(\tau_{3})=R(\tau)=\tau^{\prime}\ ,$ (58)

and therefore we have $R=R_{3}$, effectively connecting the axio-dilaton
modular symmetry with that of the torus $T^{2}_{3}$.

We now focus on the symmetries associated with the tori with labels $i=1,2$.
Following the above discussion, we first solve eq. 55 with respect to $R_{2}$

$\displaystyle R_{2}=A^{T}(R_{1}^{-1})^{T}(A^{-1})^{T}~{}.$ (59)

Next from eq. 52 we have

$\displaystyle A^{T}=\begin{pmatrix}-n_{3}&m^{2}\\\
m^{1}&m^{0}\end{pmatrix},\;\;(A^{-1})^{T}=\dfrac{1}{m^{1}m^{2}+m^{0}n_{3}}\begin{pmatrix}-m^{0}&m^{2}\\\
m^{1}&n_{3}\end{pmatrix}\ ,$ (60)

from which we finally get111Here we make use of $\det(R_{1})=1$ to simplify
the denominator in eq. 60 arising from $(R_{1})^{-1}$.

$\displaystyle
R_{2}=\begin{pmatrix}\dfrac{m^{1}m^{2}a_{1}+m^{0}m^{2}b_{1}+m^{1}n_{3}c_{1}+m^{0}n_{3}d_{1}}{m^{1}m^{2}+m^{0}n_{3}}&\dfrac{-(m^{2})^{2}b_{1}+(n_{3})^{2}c_{1}+m^{2}n_{3}(a_{1}-d_{1})}{m^{1}m^{2}+m^{0}n_{3}}\\\\[15.0pt]
\dfrac{(m^{0})^{2}b_{1}-(m^{1})^{2}c_{1}+m^{0}m^{1}(a_{1}-d_{1})}{m^{1}m^{2}+m^{0}n_{3}}&\dfrac{m^{0}n_{3}b_{1}-m^{0}m^{2}b_{1}-m^{1}n_{3}c_{1}+m^{1}m^{2}d_{1}}{m^{1}m^{2}+m^{0}n_{3}}\end{pmatrix}~{}.$
(61)

The above result generalises that of [6], which can be reproduced in the limit
$(n_{3},m^{0})\rightarrow 0$. Furthermore, one can show that the vacuum
direction set by eq. 57 is invariant under
$\tau_{1}\mapsto\tau_{1}^{\prime}=R_{1}(\tau_{1})$ and
$\tau_{2}\mapsto\tau_{2}^{\prime}=R_{2}(\tau_{2})$ with $R_{2}$ given by eq.
61. Being an element of $SL(2,\mathbb{Z})_{2}$, the entries of $R_{2}$ are
integers, and the determinant equals unity. This is not a trivial requirement,
as the entries are now parametrically defined by the entries of an $R_{1}$
element and flux data. However, we can find which congruence subgroup,
$\Gamma(N)$, of $SL(2,\mathbb{Z})_{2}$ the matrix $R_{2}$ belongs to. To do
this, we first consider the case where the following relations hold

$\displaystyle m^{1}=-2m^{0},\;m^{0}=n_{3},\;n_{3}=xm^{2}\ .$ (62)

With these, $R_{2}$ can be expressed as

$\displaystyle
R_{2}=\begin{pmatrix}\dfrac{x(b_{1}-2c_{1}+d_{1}x)-2a_{1}}{x^{2}-2}&\dfrac{-b_{1}+x(a_{1}+c_{1}x-d_{1})}{x^{2}-2}\\\
\dfrac{-4c_{1}+x(-2a_{1}+b_{1}x+2d_{1})}{x^{2}-2}&\dfrac{x(a_{1}x-b_{1}+2c_{1})-2d_{1}}{x^{2}-2}\end{pmatrix}\
.$ (63)

We can now find the explicit congruence subgroup of level $N$ to which $R_{2}$
corresponds to, once the fluxes are fixed. To do so, we first inspect the off-
diagonal terms in eq. 63. The requirement that $R_{2}\in SL(2,\mathbb{Z})_{2}$
readily suggests that $q_{1}$ is proportional to $x^{2}-2$ while $s_{1}$ is
proportional to $(x^{2}-2)/4$. Therefore, we can identify
$\Gamma\left(4/(x^{2}-2)^{2}\right)$ as the principal congruence subgroup of
$SL(2,\mathbb{Z})_{1}$ of level $N=4/(x^{2}-2)^{2}$. Since $N$ needs to be an
integer, it can take only two possible values

$\displaystyle N=\begin{cases}1&,\ x=-2,0,2\\\ 4&,\ x=-1,1\end{cases}~{}.$
(64)

We observe that the values $\ x=-2,0,2$ lead to $N=1$, i.e., a trivial finite
modular group, hence we focus on the second solution, $x^{2}=1$ with $N=4$. In
this case, eq. 63 takes the form

$\displaystyle
R_{2}=\begin{pmatrix}-x(b_{1}-2c_{1}+d_{1}x)+2a_{1}&b_{1}-x(a_{1}+c_{1}x-d_{1})\\\
4c_{1}-x(-2a_{1}+b_{1}x+2d_{1})&-x(a_{1}x-b_{1}+2c_{1})+2d_{1}\end{pmatrix}\
.$ (65)

Additionally, if $R_{1}\in\Gamma\left(4/(x^{2}-2)^{2}\right)$, we have

$\displaystyle b_{1}$ $\displaystyle=c_{1}=0\mod 4/(x^{2}-2)^{2}$ (66)
$\displaystyle a_{1}$ $\displaystyle=d_{1}=1\mod 4/(x^{2}-2)^{2}\ ,$ (67)

which leads to

$\displaystyle R_{2}\mod 4/(x^{2}-2)^{2}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\mod 4/(x^{2}-2)^{2}\ ,$ (68)

regardless of the sign of $x$. Therefore, we have encountered the principal
congruence subgroup of level $N=4/(x^{2}-2)^{2}=4$ of the homogeneous modular
groups associated with the moduli with labels $i=1,2$, which will lead to a
finite modular group $\Gamma_{4}\simeq\bar{\Gamma}/\bar{\Gamma}(4)\simeq
S_{4}$. An explicit choice of fluxes that produces a vacuum that breaks the
full modular group to $\Gamma_{4}$ is

$\displaystyle\\{m^{1}=-4,\,m^{2}=2,\,m^{0}=2,\,n_{3}=2\\}\ ,$ (69)

which produces the total flux, c.f. eq. 31, $N_{\rm flux}=8$. To check if this
is a valid Type IIB solution, we first notice that the factorisable toroidal
orientifold $T^{6}/\mathbb{Z}_{2}=(T^{2}_{1}\times T^{2}_{2}\times
T^{2}_{3})/\mathbb{Z}_{2}$ has 64 fixed points, each associated with an
O3-plane. To preserve $\mathcal{N}=1$ SUSY in 4d, there cannot be
anti-D3-branes, for which $N_{D3}\geq 0$. Therefore, $N_{\rm flux}=8$ is
consistent with the tadpole cancellation condition eq. 29

$\displaystyle N_{\rm
flux}=2\left(\frac{1}{4}N_{O3}-N_{D3}\right)=2(16-N_{D3})\leq 32\ .$ (70)

In summary, in this section, we have derived the supersymmetric conditions on
the fluxes of the moduli superpotential which predict an $S_{4}$ finite
modular group from Type IIB orientifold compactification. However, this result
pertains only to the Type IIB supergravity action and does not include matter
fields and their interactions. To address this, we now move towards F-Theory
constructions.

###  3.2  The axio-dilaton in Type IIB and F-Theory

The axio-dilaton is related to the string coupling $g_{s}$ as
$\tau=C_{0}+i/g_{s}$. In Type IIB (and its geometric counterpart, F-Theory
which inherits S-duality from Type IIB) the Yukawa coupling is expected to
depend on the string coupling $g_{s}$ as

$\lambda(g_{s},z_{i})=g_{s}^{\alpha}\lambda(z_{i})\ ,$ (71)

but it is also expected to depend on the complex structure moduli through
$z_{i}=x_{i}+\tau_{i}y_{i}$, and possible flux parameters. These moduli fields
will each transform under their respective $SL(2,\mathbb{Z})$ symmetries. In
this subsection we shall be concerned with the axio-dilaton which is common to
both Type IIB and F-Theory.

The Yukawa coupling $\lambda$ will then be transformed under the
$SL(2,\mathbb{Z})_{\tau}$, modular group associated with the axio-dilaton.
From eq. 21

$\tau-\bar{\tau}=2is\to{\rm Im\tau}\equiv
s=\frac{\tau-\bar{\tau}}{2i}=\frac{1}{g_{s}}\ ,$ (72)

and therefore

$\frac{1}{g_{s}}\equiv{\rm Im\tau}\;\to\;{\rm Im\tau^{\prime}}=\frac{ad-
bc}{|c\tau+d|^{2}}\,{\rm Im\tau}\;=\;\frac{1}{|c\tau+d|^{2}}\,{\rm
Im\tau}\;\equiv\frac{1}{|c\tau+d|^{2}}\frac{1}{g_{s}}~{},$ (73)

where the fact that $ad-bc=1$ has been utilised. Hence, for an arbitrary power
of $g_{s}$, we have

$\gamma g_{s}^{-\alpha}=\frac{g_{s}^{-\alpha}}{|c\tau+d|^{2\alpha}}\ .$ (74)

Focusing on $\alpha=1$, for the $T$ generator we have $a=b=1,c=0,d=1$, and
hence

$\displaystyle\tau\to\tau+1$ $\displaystyle:$ $\displaystyle C_{0}\to
C_{0}+1,\;s\to s\ .$ (75)

On the other hand, for the $S$ generator we take $a=0,b=1,c=-1,d=0$, and so
the denominator in eq. 74 becomes

$|c\tau+d|^{2}=\tau\bar{\tau}=C_{0}^{2}+s^{2}\ ,$ (76)

and therefore the transformation acquires a specific structure given only in
terms of the axion $C_{0}$ and the inverse string coupling $g_{s}^{-1}=s$

$\displaystyle\tau\to-\frac{1}{\tau}$ $\displaystyle:$ $\displaystyle
C_{0}\to-\frac{C_{0}}{C_{0}^{2}+s^{2}},\;s\to\frac{s}{C_{0}^{2}+s^{2}}\ .$
(77)

This case is known as strong-weak duality or S-duality since it transforms the
string coupling $g_{s}$ to its inverse $g_{s}^{-1}$.

Recall now that the axio-dilaton part of the tree-level Kähler potential is

$K=-\log(-i(\tau-\bar{\tau}))+\cdots=-\log\left(\frac{\tau-\bar{\tau}}{2i}\right)-\log(2)+\cdots=-\log(s)+\cdots,$
(78)

so that the $SL(2,\mathbb{Z})_{\tau}$ transformation implies that
$-\log(s)\to-\log\frac{s}{|c\tau+d|^{2}}$ and thus the exponential $e^{K}$
transforms as

$e^{K}\to|c\tau+d|^{2}e^{K}\ .$ (79)

On the other hand, the gravitino mass is

$m_{3/2}^{2}=e^{K}|W|^{2}\ ,$ (80)

and since it must stay invariant we must have

$W\to\frac{W}{c\tau+d}\;\Rightarrow\;|W|^{2}\to\frac{|W|^{2}}{|c\tau+d|^{2}}\
.$ (81)

It is now apparent that if S-duality is to be maintained by the perturbative
superpotential Yukawa couplings [20], then the fields must have transformation
properties with respect to it. A generic trilinear term with a tree-level
Yukawa coupling of MSSM fields has the form

$W\supset\lambda_{ij}(g_{s})f_{i}f_{j}h\ .$ (82)

In the simplest context, the Yukawa coupling could simply be taken
$\lambda_{ij}(g_{s})\to\lambda_{ij}(z_{k})g_{s}^{-1/2}$ where the parameters
$\lambda_{ij}(z_{k})$ may depend on other moduli fields. Then, the
$\tau\to-1/\tau$ transformation discussed above entails

$\lambda\propto{g_{s}^{-1/2}}\;\xrightarrow[]{{\tau\,\to\,-\frac{1}{\tau}}}\;\frac{g_{s}^{-1/2}}{|C_{0}^{2}+g_{s}^{-2}|^{1/2}}\
,$ (83)

which matches exactly the transformation property of the tree-level
superpotential $W$ w.r.t. axio-dilaton $\tau$.

In a more general context, as we will see, the dependence of the Yukawa
couplings on moduli fields is more involved and non-zero modular weights for
the matter fields $f_{i},f_{j},h$ should also be considered. Furthermore, the
Yukawa couplings, which are $3\times 3$ matrices in the flavour space, could
transform non-trivially under the congruence group left over from the
supersymmetric conditions imposed on fluxes of the moduli superpotential part.
The specific choice of fluxes of the previous section indicates that the
underlying flavour symmetry governing the Yukawa lagangian is
$\Gamma_{4}\simeq\Gamma/\bar{\Gamma}(4)\simeq S_{4}$ with Yukawa matrices
being certain modular forms which belong to specific representations of the
$S_{4}$ group. Additional restrictions are expected to be derived from the
geometric structure of the compactification manifold to further suggest a
specific implementation of the above scenario. In the following, we continue
with F-Theory, where some of these hints become more transparent.

###  3.3 Yukawa Couplings and Fermion Mass Matrices in F-Theory

We now turn our attention to the Yukawa couplings in F-Theory. Our starting
point is an effective F-Theory GUT model, which is derived from an ADE-type
singularity with the world-volume of a 7-brane that wraps the space
$R^{3,1}\times{S}$ with ${S}$ being a Kähler manifold of two complex
dimensions $z_{1},z_{2}$. At low energies, F-Theory is described by an eight-
dimensional YM theory on $R^{3,1}\times\cal{S}$ which must be topologically
twisted to preserve $N=1$ supersymmetry.

The compactification space is a fibred eight-dimensional space (CY fourfold
$CY_{4}$) where the fibre over the base $B_{3}=CY_{3}$ associated with the
six-dimensional compact space is described by a two-dimensional torus whose
modulus is the axio-dilaton $\tau=C_{0}+ie^{-\phi}=C_{0}+i/g_{s}$. Therefore,
the $SL(2,\mathbb{Z})_{\tau}$ S-duality describes the variation of the modulus
$\tau$ of the 2-torus over the compactification manifold. The geometric
configuration consists of 7-branes filling the Minkowski 4D-space while
wrapping a 4D ‘surface’ $S$ – associated with some GUT symmetry – which is a
complex Kähler manifold so that supersymmetry is preserved. The four-
dimensional effective F-Theory model arises upon compactification of the
eight-dimensional theory on $S$. The possible GUT groups, in particular, are
associated with specific types of geometric singularities where the modulus
$\tau$ acquires certain values. The massless fields of the low energy spectrum
reside on Riemann surfaces, called matter curves, formed by 7-branes
intersecting the GUT surface, while Yukawa couplings are formed at specific
points where triple intersections of matter curves occur. 222Equivalently, the
torus over $B_{3}$ can be described by the Weierstraśs equation
$y^{2}=x^{3}+f(z)x^{2}+g(z)$, where $z$ is a coordinate of the complex
projective space $CP^{1}$ (Riemann sphere). Then
$\tau=\tau(z)=C_{0}(z)+ie^{-\phi(z)}$ and singularities occur at
$\Delta(z_{i})=0$. The torus is associated with the invariant $j(\tau(z))$,
which, together with the vanishing of $\Delta$ determines
$\tau\sim\frac{1}{2\pi i}\log(z-z_{i})$. Hence, for given $z$, $\tau$ is
fixed. Also, going around the singularity, there is a shift to the real part
of the modulus $C_{0}\to C_{0}+1$ that corresponds to $\tau\to\tau+1$ of
$SL(2,\mathbb{Z})_{\tau}$.

Within this framework, the corresponding gauge theory is that of the eight-
dimensional $N=1$ supersymmetric YM theory with minimal field content. The
bosonic spectrum, in particular, includes the gauge field $A$ and a
holomorphic two-form scalar $\Phi$. Both fields are found in the adjoint
representation and descend from the decomposition of the 10-dimensional gauge
field ${\cal A}$. Since 7-branes are wrapped on a curved $R^{(3,1)}\times S$
space, unbroken ${\cal N}=1$ supersymmetry requires $\Phi$ to be a holomorphic
(2,0)-form as a result of the topological twisting [21]. 333Within such an
F-Theory framework it is well known that there are many complex structure
moduli, associated with the positions of the 7-branes. The positions of the
7-branes are determined by tuning the complex structure moduli and can produce
additional structure in the elliptic fibration [22].

The superpotential $W_{8d}$ of the eight-dimensional fields and an associated
D-term take the form

$W_{8d}=m_{*}^{4}\int_{S}{\rm Tr}(F\wedge\Phi),\;\;D=\int_{S}\omega\wedge
F+\frac{1}{2}[\Phi,\bar{\Phi}]~{},$ (84)

where $F=dA-iA\wedge A$ and
$\omega=ig/2(dz_{1}\wedge\bar{d}z_{1}+dz_{2}\wedge\bar{d}z_{2})$ is the Kähler
form on $S$.

The eight-dimensional fields can be organised as one ${\cal N}=1$ vector
multiplet, ${\bf V}$, and two ${\cal N}=1$ chiral supermultiplets, ${\bf
A}_{\bar{m}}$ and ${\bf\Phi}_{mn}$,

$\displaystyle{\bf V}$ $\displaystyle=(A_{\mu},\eta^{\alpha},{\cal D})$ (85)
$\displaystyle{\bf A}_{\bar{m}}$
$\displaystyle=(A_{\bar{m}},\psi_{\bar{m}}^{\alpha},{\cal G}_{\bar{m}}))$ (86)
$\displaystyle{\bf\Phi}_{mn}$
$\displaystyle=(\varphi_{mn},\chi_{mn}^{\alpha},{\cal H}_{mn})\ ,$ (87)

where ${\cal G}_{\bar{m}},{\cal H}_{\bar{m}\bar{n}}$ are $F$-term components,
and ${\cal D}$ represents the $D$-term, whilst,
$\eta^{\dot{\alpha}},\psi_{\bar{m}}^{\alpha},\chi_{mn}^{\dot{\alpha}}$ are the
fermionic components, which, in the twisted YM theory are associated with a
zero, one- and two-form respectively [23],

$\eta^{\dot{\alpha}},\;\psi^{\alpha}=\psi_{\bar{m}}^{\alpha}d\bar{z}^{\bar{m}},\;\chi^{\dot{\alpha}}=\bar{\chi}_{\bar{m}\bar{n}}^{\dot{\alpha}}d\bar{z}^{\bar{m}}\wedge
d\bar{z}^{n}~{}.$

The indices $m,n$ take the values 1,2, the complex scalars
$A_{\bar{m}},\varphi_{mn}$ have dimensions of mass $M$ and ${\cal G},{\cal
H},{\cal D}$ of squared mass $M^{2}$.

To preserve the supersymmetric vacuum, all variations of the eight-dimensional
fields must vanish. In the context of the four-dimensional theory, this
corresponds to imposing the F- and D-flantess of the superpotential.
Minimising the superpotential (84) and imposing D-flatness, one arrives at the
following equations

$\displaystyle\bar{\partial}_{A}\Phi$ $\displaystyle=0$ (88) $\displaystyle
F^{(2,0)}$ $\displaystyle=0$ (89) $\displaystyle\omega\wedge
F+\frac{1}{2}[\Phi^{\dagger},\Phi]$ $\displaystyle=0~{}.$ (90)

The above equations have long been derived in reference [21] and are the basic
ingredients for studying the properties of fields in generic 7-brane
configurations. Here, we are interested in solutions for massless fields
residing on 7-brane configurations. The equations can be solved by expanding
the fields $A,\Phi$ assuming linear fluctuations around the background:

$\displaystyle A_{\bar{m}}\to\langle A_{\bar{m}}\rangle+a_{\bar{m}},\;\;$
$\displaystyle\;\;\Phi\to\langle\Phi\rangle+\varphi~{},$ (91)

with the definitions

$\displaystyle
a=a_{\bar{z}_{1}}d{\bar{z}_{1}}+a_{\bar{z}_{2}}d{\bar{z}_{2}},\;\;$
$\displaystyle\;\;\varphi=\varphi_{\bar{z}_{1}\bar{z}_{2}}d{\bar{z}_{1}}\wedge
d{\bar{z}_{2}}\ .$ (92)

Then, keeping only linear terms regarding the fluctuations $\varphi,a$, in the
holomorphic gauge the EoM take the form

$\displaystyle\bar{\partial}_{\langle A\rangle}a$ $\displaystyle=0$ (93)
$\displaystyle\bar{\partial}_{\langle
A\rangle}\varphi-i[a,\langle\bar{\Phi}\rangle]$ $\displaystyle=0$ (94)
$\displaystyle\omega\wedge\partial_{\langle
A\rangle}a-\frac{1}{2}[\langle\bar{\Phi}\rangle,\varphi]$ $\displaystyle=0\ .$
(95)

Substituting the expansions of the fields into (84) it is found that the
holomorphic trilinear Yukawa coupling is written in terms of $\phi$ and $a$ as
follows

$W_{\rm Yuk}=-im_{*}^{4}\int_{S}\text{Tr}(\varphi\wedge a\wedge a)\ ,$ (96)

where $m_{*}$ is the scale associated with the supergravity limit of F-Theory.

The fluctuations $\varphi$ and $a$ can be determined by solving the equations
(93-95) for a variety of diagonal or non-diagonal backgrounds [24, 25], the
latter being known as T-branes [24]. They are associated with the zero-modes
residing on the matter curves and when three of the latter define triple
intersection a Yukawa coupling is formed. Depending on the details of the
model, it is often the case that multiple zero-modes are accommodated on the
same matter curve.

It can be shown that the general form of the solution for zero modes localised
on a specific matter curve, say $z_{2}$ , takes the generic form

$\varphi=R_{a}\chi_{a}=R_{a}f(z_{2})g(z_{1},\bar{z}_{1},q)e^{-\sqrt{M_{z_{1}}^{4}+m^{4}}z_{1}\bar{z}_{1}}e^{\pm
2M_{z_{2}}^{2}z_{2}\bar{z}_{2}}\ ,$ (97)

where $M_{z_{i}}$ appear when fluxes are also introduced 444For example, in a
$U(3)$ model the flux assumes the form $\langle
F\rangle=-(2i/3)M^{2}(\bar{z}_{1}\wedge dz_{1}-\bar{z}_{2}\wedge dz_{2}){\rm
diag}(1,-2,1)$. In a generic context, however, when non-Abelian T-branes are
considered, a non-primitive flux is required $\omega\wedge F\neq 0$ to satisfy
the D-term [25]. For a comprehensive presentation, see review [26].

It can be observed that locally the solution is described by a Gaussian
profile, with its peak along the matter curve and waning out along the
transverse direction $z_{1}$. The function $f(z_{2})$ is a holomorphic
function of $z_{2}$ left undetermined from the equations of motion and $R_{a}$
encodes the group structure [25] associated with the background. Analogous
solutions can be written for the other intersecting matter curves in the
vicinity of the triple intersection. The integration (96) 555In section 3 eq
(3.28) of [24]\- using the notion of twisted one-forms, the connection
$\psi=a+\hat{\kappa}\wedge\varphi,\,\hat{\kappa}=g^{1\bar{i}}g^{2\bar{j}}\bar{\Omega}_{\overline{ijz}}d\bar{z}$
is implemented - and the Yukawa coupling receives a symmetric form. See the
appendix for the relevant computation. is performed over the three overlapping
wavefunctions where all of them are peaked at the triple intersection and
since they are strongly localised, the integral can be restricted to a small
region near the intersection point. At every triple intersection the gauge
symmetry is enhanced and generically zero-mode states are assembled into
representations of the higher symmetry. At the same time, multiple states
accommodated on a certain matter curve may be organised into representations
of the underlying symmetry of the complex structure of the matter curve.

Furthermore, assuming for example toroidal compactifications, the function $f$
may depend explicitly on the complex structure moduli of the curve, and thus
it is conceivable that they may transform as modular forms, as we argue in the
next section.

We discuss now the overall dependence of the Yukawa coupings on the mass
scales of the theory, and their relation to the axio-dilaton modulus [27]. In
string frame, the overall scale $m_{*}$ in (96), is given by
$m_{*}^{8}=m_{s}^{8}g_{s}^{-2}$ [27], hence, the resulting dependence of the
Yukawa coupling on $g_{s}$ is (see details in section 4 of [27])

$\lambda\propto\frac{m_{*}^{4}}{m_{s}^{4}}=\frac{1}{g_{s}}\ .$ (98)

The string coupling is related to the GUT scale. Indeed, let ${\cal V}_{S}\sim
R_{S}^{4}\sim 1/m^{4}_{GUT}$ be the volume of the GUT surface and ${\cal
V}_{B}\sim R^{6}$ that of the base $B_{3}$ of the fibration. Compactification
to four dimensions implies $M_{Pl}^{2}\approx m_{*}^{8}{\cal V}_{B}$ while
from the kinetic term of the field strength it follows
$\alpha_{GUT}^{-1}\approx m_{*}^{4}{\cal V}_{S}$. Combining these relations we
obtain a rough estimate

$\frac{1}{\alpha_{GUT}}\sim{m_{*}}^{4}{\cal
V}_{S}=\frac{m_{*}^{4}}{m_{s}^{4}}\frac{m_{s}^{4}}{m_{GUT}^{4}}=\frac{m_{s}^{4}}{m_{\small
GUT}^{4}}\frac{1}{g_{s}}\ ,$ (99)

Taking into account the various normalisation effects, the dependence on
$g_{s}$ is more involved. One finds [25, 28]

$\lambda=C\,a_{GUT}^{3/4}\ ,$ (100)

where $C$ may depend on other moduli fields via the wavefunctions of the form
(97) involved in the triple intersections. This implies that S-duality
symmetry is preserved only if the undetermined parts of the wavefunctions
associated with the Yukawa coupling under consideration exhibit the
appropriate dependence on $g_{s}$.

##  4 Bottom-up Approach to F-Theory Fluxed GUTs with Finite Discrete Modular
Symmetry

Hitherto, we described a basic F-Theory approach to Yukawa couplings and
presented a generic solution for the EoM. From the above analysis we inferred
that the Yukawa coupling inherits group properties encoded in the matter
wavefunctions. The latter depend on the complex structure moduli through
holomorphic functions of the complex coordinates $z_{i}$ left unspecified by
the EoM. Nevertheless, from the preceding sections and more particularly from
Section 3.1, we know that the fluxed superpotential of the moduli fields is
subject to modular restrictions. Therefore, if the Yukawa sector for the
ordinary matter of the superpotential is required to retain the same modular
symmetry or a subgroup thereof, its origin is expected to come from the yet
unspecified part of that solution. We are then given the opportunity to
consider the wavefunctions transformed as modular forms. For example,

$f(\tau_{i})\to(c\tau_{i}+d)^{-k_{i}}f(\tau_{i})\ ,$ (101)

where $\tau_{i}$ is a complex structure modulus associated with the complex
coordinate $z_{i}$. Additionally, the holomorphic Yukawa coupling, being
formed at the intersection of three matter curves, would be naturally
transformed in a non-trivial representation of the congruence subgroup of the
modular group.

To illustrate the main idea of the bottom-up approach to F-Theory fluxed GUTs
with modular symmetry, we give a simple example of an $SU(5)$ GUT embedded in
$E_{6}$ which has been derived in an F-Theory framework [29]. The novel
feature of this example is the inclusion of an $S_{4}$ finite modular family
symmetry.

As has already been pointed out, in this context matter fields reside on
curves which are formed at the intersections of 7-branes with the GUT surface
$S$, itself wrapped with 7-branes. We consider a divisor with an $E_{6}$
geometric singularity which, according to the F-Theory prescription,
corresponds to an $E_{6}$ gauge symmetry of the effective theory. In the
present setup, there are three matter curves accommodating three
$27_{t^{\prime}_{i}}$ representations of $E_{6}$. These are distinguished from
each other by the weights $t^{\prime}_{i}$ of the $SU(3)$ Cartan sublagebra
$(t^{\prime}_{1}+t^{\prime}_{2}+t^{\prime}_{3}=0)$. We impose a ${\cal Z}_{2}$
monodromy $t_{1}^{\prime}\leftrightarrow t_{2}^{\prime}$, and hence only two
distinct matter curves remain, e.g. $\Sigma_{27_{t_{1,3}^{\prime}}}$, and use
$U(1)$ fluxes to reduce the gauge symmetry down to $SU(5)$. Alternatively, one
may derive this model starting from the maximum admissible (well behaved)
singularity that corresponds to a $E_{8}$ gauge symmetry subsequently
decomposed to

$E_{8}\supset SU(5)\times SU(5)_{\perp}\supset SU(5)\times
U(1)^{4}_{t_{i}},\;\;\sum_{i=1}^{5}t_{i}=0~{},$ (102)

where now $t_{i}$ correspond to the Cartan subalgebra of $SU(5)_{\perp}$. The
$E_{6}$ and $SU(5)\times SU(5)_{\perp}$ properties of the matter and Higgs
multiplets are given in Table 1. Due to the aforementioned restrictions on
$t_{i}^{\prime}$ and the monodromy imposed, the only allowed trilinear $E_{6}$
term in the superpotential is $W\supset
27_{t_{1}^{\prime}}27_{t_{1}^{\prime}}27_{t_{3}^{\prime}}$. We then assign the
fermion supermultiplets to $27_{t^{\prime}_{1}}$ and the Higgs fields to
$27_{t^{\prime}_{3}}$.

$E_{6}$ | $SO(10)$ | $SU(5)$ | Weight
---|---|---|---
$27_{t_{1}^{\prime}}$ | $16$ | $\overline{5}_{3}$ | $t_{1}+t_{5}$
$27_{t_{1}^{\prime}}$ | $16$ | $10_{M}$ | $t_{1}$
$27_{t_{1}^{\prime}}$ | $16$ | $\theta_{15}$ | $t_{1}-t_{5}$
$27_{t_{1}^{\prime}}$ | $10$ | $5_{1}$ | $-t_{1}-t_{3}$
$27_{t_{1}^{\prime}}$ | $10$ | $\overline{5}_{2}$ | $t_{1}+t_{4}$
$27_{t_{1}^{\prime}}$ | $1$ | $\theta_{14}$ | $t_{1}-t_{4}$
$27_{t_{3}^{\prime}}$ | $16$ | $\overline{5}_{5}$ | $t_{3}+t_{5}$
$27_{t_{3}^{\prime}}$ | $16$ | $10_{2}$ | $t_{3}$
$27_{t_{3}^{\prime}}$ | $16$ | $\theta_{35}$ | $t_{3}-t_{5}$
$27_{t_{3}^{\prime}}$ | $10$ | $5_{H_{u}}$ | $-2t_{1}$
$27_{t_{3}^{\prime}}$ | $10$ | $\overline{5}_{4}$ | $t_{3}+t_{4}$
$27_{t_{3}^{\prime}}$ | $1$ | $\theta_{34}$ | $t_{3}-t_{4}$
Table 1: $SO(10)$ and $SU(5)$ decompositions of $27\in E_{6}$. The $SU(5)$
indices in $5_{i},10_{j}$ representations designate their origin of the
corresponding matter curve ($\Sigma_{5_{i}}$ and $\Sigma_{10_{j}}$), and
$10_{M}$ accommodates ordinary matter fields.

We break the $SU(5)$ gauge symmetry by turning on a flux along $U(1)_{Y}\in
SU(5)$, which also splits the $10$ and $\bar{5}$ representations of $SU(5)$.
However, anomaly cancellation conditions impose constraints on the
multiplicities of the latter which are as follows:

$\displaystyle
M_{10_{M}}=M_{5_{1}}=-M_{5_{2}}=-M_{5_{3}},\;M_{10_{2}}=-M_{5_{4}}=-M_{5_{5}}=M_{5_{H_{u}}}.$
(103)

Furthermore, to eliminate extraneous and exotic matter derived from the
decomposition of the 78-dimensional representation, we impose the conditions

$\displaystyle M_{10_{3}}=M_{10_{4}}=M_{5_{6}}=N_{8}=N_{9}=0,$ (104)

These imply that [29]

$\displaystyle\tilde{N}\equiv N_{7}~{}.$ (105)

The SM zero mode states derived from the complete $27_{t^{\prime}_{i}}$
representations after various successive symmetry-breaking stages with the
$U(1)$ fluxes shown in the last column of Table 2. Their multiplicities are
expressed in terms of the flux integers which have remained undetermined by
the consistency conditions mentioned above.

$E_{6}$ $SO(10)$ $SU(5)$ Weight vector $N_{Y}$ $M_{U(1)}$ SM particle content
$27_{t_{1}^{\prime}}$ $16$ $\overline{5}_{3}$ $t_{1}+t_{5}$ $\tilde{N}$
$-M_{5_{3}}$ $-M_{5_{3}}d^{c}+(-M_{5_{3}}+\tilde{N})L$ $27_{t_{1}^{\prime}}$
$16$ $10_{M}$ $t_{1}$ $-\tilde{N}$ $-M_{5_{3}}$
$-M_{5_{3}}Q+(-M_{5_{3}}+\tilde{N})u^{c}+(-M_{5_{3}}-\tilde{N})e^{c}$
$27_{t_{1}^{\prime}}$ $16$ $\theta_{15}$ $t_{1}-t_{5}$ $0$ $-M_{5_{3}}$
$-M_{5_{3}}\nu^{c}$ $27_{t_{1}^{\prime}}$ $10$ $5_{1}$ $-t_{1}-t_{3}$
$-\tilde{N}$ $-M_{5_{3}}$ $-M_{5_{3}}D+(-M_{5_{3}}-\tilde{N})H_{u}$
$27_{t_{1}^{\prime}}$ $10$ $\overline{5}_{2}$ $t_{1}+t_{4}$ $\tilde{N}$
$-M_{5_{3}}$ $-M_{5_{3}}\overline{D}+(-M_{5_{3}}+\tilde{N})H_{d}$
$27_{t_{1}^{\prime}}$ $1$ $\theta_{14}$ $t_{1}-t_{4}$ $0$ $-M_{5_{3}}$
$-M_{5_{3}}S$ $27_{t_{3}^{\prime}}$ $16$ $\overline{5}_{5}$ $t_{3}+t_{5}$
$-\tilde{N}$ $M_{5_{H_{u}}}$ $M_{5_{H_{u}}}d^{c}+(M_{5_{H_{u}}}-\tilde{N})L$
$27_{t_{3}^{\prime}}$ $16$ $10_{2}$ $t_{3}$ $\tilde{N}$ $M_{5_{H_{u}}}$
$M_{5_{H_{u}}}Q+(M_{5_{H_{u}}}-\tilde{N})u^{c}+(M_{5_{H_{u}}}+\tilde{N})e^{c}$
$27_{t_{3}^{\prime}}$ $16$ $\theta_{35}$ $t_{3}-t_{5}$ $0$ $M_{5_{H_{u}}}$
$M_{5_{H_{u}}}\nu^{c}$ $27_{t_{3}^{\prime}}$ $10$ $5_{H_{u}}$ $-2t_{1}$
$\tilde{N}$ $M_{5_{H_{u}}}$ $M_{5_{H_{u}}}D+(M_{5_{H_{u}}}+\tilde{N})H_{u}$
$27_{t_{3}^{\prime}}$ $10$ $\overline{5}_{4}$ $t_{3}+t_{4}$ $-\tilde{N}$
$M_{5_{H_{u}}}$ $M_{5_{H_{u}}}\overline{D}+(M_{5_{H_{u}}}-\tilde{N})H_{d}$
$27_{t_{3}^{\prime}}$ $1$ $\theta_{34}$ $t_{3}-t_{4}$ $0$ $M_{5_{H_{u}}}$
$M_{5_{H_{u}}}S$

Table 2: Complete $27$s of $E_{6}$ and their $SO(10)$ and $SU(5)$
decompositions. The indices of the $SU(5)$ non-trivial states $10,5$ refer to
the labelling of the corresponding matter curve (we use the notation of [30]).
We impose the extra conditions on the integers $N_{Y}$ and $M_{U(1)}$ from the
requirement of having complete 27s of $E_{6}$ and no 78 matter. The $SU(5)$
matter states decompose into SM states as $\overline{5}\rightarrow d^{c},L$
and $10\rightarrow Q,u^{c},e^{c}$ with right-handed neutrinos
$1\rightarrow\nu^{c}$, while the $SU(5)$ Higgs states decompose as
$5\rightarrow D,H_{u}$ and $\overline{5}\rightarrow\overline{D},H_{d}$, where
$D,\overline{D}$ are exotic colour triplets and antitriplets. We identify RH
neutrinos as $\nu^{c}=\theta_{15,35}$ and extra singlets from the 27 as
$S=\theta_{14,34}$.

In the present work, an explicit model is constructed by choosing the fluxes
given in table 3.

$M_{10_{M}}$ | $M_{5_{3}}$ | $M_{5_{1}}$ | $M_{5_{2}}$ | $M_{10_{2}}$ | $M_{5_{5}}$ | $M_{5_{4}}$ | $M_{H_{u}}$ | $M_{\theta_{15}}$ | $\tilde{N}$
---|---|---|---|---|---|---|---|---|---
4 | $-4$ | 3 | $-3$ | $-1$ | 1 | 0 | 0 | 2 | 1
Table 3: The choice of Fluxes used in this model.

This choice leads to the spectrum given in table 4 where both the down quarks
and leptons originate from $27_{t_{1}^{\prime}}$.

$E_{6}$ $SO(10)$ $SU(5)$ Weight vector $N_{Y}$ $M_{U(1)}$ SM particle content
Low energy spectrum $27_{t_{1}^{\prime}}$ $16$ $\overline{5}_{3}$
$t_{1}+t_{5}$ $1$ $4$ $4d^{c}+5L$ $3d^{c}+3L$ $27_{t_{1}^{\prime}}$ $16$
$10_{M}$ $t_{1}$ $-1$ $4$ $4Q+5u^{c}+3e^{c}$ $3Q+3u^{c}+3e^{c}$
$27_{t_{1}^{\prime}}$ $16$ $\theta_{15}$ $t_{1}-t_{5}$ $0$ $3$ $3\nu^{c}$ -
$27_{t_{1}^{\prime}}$ $10$ $5_{1}$ $-t_{1}-t_{3}$ $-1$ $3$ $3D+2H_{u}$ -
$27_{t_{1}^{\prime}}$ $10$ $\overline{5}_{2}$ $t_{1}+t_{4}$ $1$ $3$
$3\overline{D}+4H_{d}$ $H_{d}$ $27_{t_{3}^{\prime}}$ $16$ $\overline{5}_{5}$
$t_{3}+t_{5}$ $-1$ $-1$ $\overline{d^{c}}+2\overline{L}$ -
$27_{t_{3}^{\prime}}$ $16$ $10_{2}$ $t_{3}$ $1$ $-1$
$\overline{Q}+2\bar{u^{c}}$ - $27_{t_{3}^{\prime}}$ $16$ $\theta_{35}$
$t_{3}-t_{5}$ $0$ $0$ $-$ - $27_{t_{3}^{\prime}}$ $10$ $5_{H_{u}}$ $-2t_{1}$
$1$ $0$ $H_{u}$ $H_{u}$ $27_{t_{3}^{\prime}}$ $10$ $\overline{5}_{4}$
$t_{3}+t_{4}$ $-1$ $0$ $\overline{H_{d}}$ - $27_{t_{3}^{\prime}}$ $1$
$\theta_{34}$ $t_{3}-t_{4}$ $0$ $1$ $\theta_{34}$ - - $1$ $\theta_{31}$
$t_{3}-t_{1}$ $0$ $4$ $\theta_{31}$ - - $1$ $\theta_{53}$ $t_{5}-t_{3}$ $0$
$1$ $\theta_{53}$ - - $1$ $\theta_{14}$ $t_{1}-t_{4}$ $0$ $3$ $\theta_{14}$ -
- $1$ $\theta_{45}$ $t_{4}-t_{5}$ $0$ $2$ $\theta_{45}$ -

Table 4: Complete $27$s of $E_{6}$ and their $SO(10)$ and $SU(5)$
decompositions. We use the notation of ref [30] for the indices of the $SU(5)$
states and impose the extra conditions on the integers $N_{Y}$ and $M_{U(1)}$
from the requirement of having complete 27s of $E_{6}$ and no 78 matter. The
$SU(5)$ matter states decompose into SM states as $\overline{5}\rightarrow
d^{c},L$ and $10\rightarrow Q,u^{c},e^{c}$ with right-handed neutrinos
$1\rightarrow\nu^{c}$, while the $SU(5)$ Higgs states decompose as
$5\rightarrow D,H_{u}$ and $\overline{5}\rightarrow\overline{D},H_{d}$, where
$D,\overline{D}$ are exotic colour triplets and antitriplets. We identify RH
neutrinos as $\nu^{c}=\theta_{15}$. Extra singlets are needed to given mass to
neutrinos and exotics and to ensure F- and D- flatness.

As we have argued in the previous section, the states supported on a matter
curve will inherit modular symmetry properties related to the complex
structure moduli parametrising that curve. Therefore, states supported on a
given curve are expected to have the same modular weights and to furnish full
representations of the discrete modular group that survives the
compactification. Imposing these modular symmetry properties in the above
representations, a version of the model presented above with non-trivial
discrete modular group $S_{4}$ can be written as

$\displaystyle\mathcal{W}$
$\displaystyle=\alpha\;(u^{c}_{1,2}Q_{1,2}Y_{1}^{(4)})_{1}H_{u}+\beta\;(u^{c}_{1,2}Q_{1,2}Y_{2}^{(4)})_{1}H_{u}+\gamma\;(u^{c}_{3}Q_{3}Y_{1}^{(4)})_{1}H_{u}+\delta\;(u^{c}_{1,2}Q_{3}Y_{2}^{(4)})_{1}H_{u}$
$\displaystyle+\bigg{(}\alpha^{\prime}\;(d^{c}_{1,2}Q_{1,2}Y_{1}^{(6)})_{1}H_{d}+\beta^{\prime}\;(d^{c}_{1,2}Q_{1,2}Y_{2}^{(6)})_{1}H_{d}+\gamma_{1}^{\prime}\;(d^{c}_{3}Q_{1,2}Y_{2,1}^{(8)})_{1}H_{d}$
$\displaystyle\qquad+\gamma^{\prime}\;(d^{c}_{3}Q_{1,2}Y_{2,2}^{(8)})_{1}H_{d}++\delta^{\prime}\;(d^{c}_{1,2}Q_{3}Y_{2}^{(6)})_{1}H_{d}+\epsilon^{\prime}\;(d^{c}_{3}Q_{3}Y_{1}^{(8)})_{1}H_{d}\bigg{)}\dfrac{\theta_{31}}{M},$
(106)

where $M$ is the F-Theory characteristic compacfication scale and we will set,
for simplicity, ${\theta_{31}}/{M}\simeq 1$ as we expect the VEVs of the
singlets to be close to the scale $M$ and this quantity can be reabsorbed into
the definition of the primed coefficients.

MSSM fields |  Matter Curves |  Charge |  $S_{4}$ |  k
---|---|---|---|---
$Q_{1,2},u^{c}_{1,2},e^{c}_{1,2}$ |  $10_{M}$ |  $t_{1}$ |  $2$ |  $2$
$Q_{3},u^{c}_{3},e^{c}_{3}$ |  $10_{M}$ |  $t_{1}$ |  $1$ |  $2$
$d^{c}_{1,2},L_{1,2}$ |  $\bar{5}_{3}$ |  $t_{1}+t_{5}$ |  $2$ |  $4$
$d^{c}_{3},L_{3}$ |  $\bar{5}_{3}$ |  $t_{1}+t_{5}$ |  $1$ |  $6$
$H_{u}$ |  $5_{H_{u}}$ |  $-2t_{1}$ |  $1$ |  $0$
$H_{d}$ |  $5_{2}$ |  $t_{1}+t_{4}$ |  $1$ |  $0$
$\nu^{c}$ |  $\theta_{15}$ |  $t_{1}-t_{5}$ |  $3$ |  $0$
Table 5: Perpendicular charges, modular weights, and $S_{4}$ discrete modular
group representations associated with the matter curves hosting the model from
table 4.

According to the superpotential (106), the up-type quarks Yukawa matrix is
given by

$\displaystyle\lambda_{u}=\begin{pmatrix}\alpha\left(Y_{1}^{2}+Y_{2}^{2}\right)-\beta\left(Y_{2}^{2}-Y_{1}^{2}\right)&2\beta
Y_{1}Y_{2}&\delta\left(Y_{2}^{2}-Y_{1}^{2}\right)\\\ 2\beta
Y_{1}Y_{2}&\alpha\left(Y_{1}^{2}+Y_{2}^{2}\right)+\beta\left(Y_{2}^{2}-Y_{1}^{2}\right)&2\delta
Y_{1}Y_{2}\\\ 0&0&\gamma\left(Y_{1}^{2}+Y_{2}^{2}\right)\end{pmatrix}~{},$
(107)

and for the down-type quarks, the relevant Yukawa matrix is written as

$\displaystyle\lambda_{d}=\left(\begin{smallmatrix}\alpha^{\prime}Y_{1}\left(3Y_{2}^{2}-Y_{1}^{2}\right)-\beta^{\prime}Y_{1}\left(Y_{1}^{2}+Y_{2}^{2}\right)&\beta^{\prime}Y_{2}\left(Y_{1}^{2}+Y_{2}^{2}\right)&\delta^{\prime}Y_{1}\left(Y_{1}^{2}+Y_{2}^{2}\right)\\\
\beta^{\prime}Y_{2}\left(Y_{1}^{2}+Y_{2}^{2}\right)&\alpha^{\prime}Y_{1}\left(3Y_{2}^{2}-Y_{1}^{2}\right)+\beta^{\prime}Y_{1}\left(Y_{1}^{2}+Y_{2}^{2}\right)&\delta^{\prime}Y_{2}\left(Y_{1}^{2}+Y_{2}^{2}\right)\\\
\gamma^{\prime}\left(Y_{1}^{2}-3Y_{2}^{2}\right)Y_{1}^{2}+\gamma_{1}^{\prime}\left(Y_{2}^{2}-Y_{1}^{2}\right)\left(Y_{1}^{2}+Y_{2}^{2}\right)&\gamma^{\prime}Y_{1}Y_{2}\left(Y_{1}^{2}-3Y_{2}^{2}\right)+2\gamma_{1}^{\prime}Y_{1}Y_{2}\left(Y_{1}^{2}+Y_{2}^{2}\right)&\epsilon^{\prime}\left(Y_{1}^{2}+Y_{2}^{2}\right){}^{2}\end{smallmatrix}\right)\
.$ (108)

The charged leptons have the same Yukawa matrix structure as the down-type
quarks. However, inspecting the spectrum of F-Theory zero modes in table 4, we
see that the three families of $L$, $Q$, $e^{c}$, and $d^{c}$ descend from
different linear combinations of UV states from F-Theory zero modes.
Therefore, the superpotential coefficients for the down-type quarks and the
charged leptons are not the same, leading to a realisation of a Georgi-
Jarlskog mechanism [31]. We then write down the charged leptons Yukawa matrix
as

$\displaystyle\lambda_{L}=\left(\begin{smallmatrix}\alpha^{\prime\prime}Y_{1}\left(3Y_{2}^{2}-Y_{1}^{2}\right)-\beta^{\prime\prime}Y_{1}\left(Y_{1}^{2}+Y_{2}^{2}\right)&\beta^{\prime\prime}Y_{2}\left(Y_{1}^{2}+Y_{2}^{2}\right)&\delta^{\prime\prime}Y_{1}\left(Y_{1}^{2}+Y_{2}^{2}\right)\\\
\beta^{\prime\prime}Y_{2}\left(Y_{1}^{2}+Y_{2}^{2}\right)&\alpha^{\prime\prime}Y_{1}\left(3Y_{2}^{2}-Y_{1}^{2}\right)+\beta^{\prime\prime}Y_{1}\left(Y_{1}^{2}+Y_{2}^{2}\right)&\delta^{\prime\prime}Y_{2}\left(Y_{1}^{2}+Y_{2}^{2}\right)\\\
\gamma^{\prime\prime}\left(Y_{1}^{2}-3Y_{2}^{2}\right)Y_{1}^{2}+\gamma_{1}^{\prime\prime}\left(Y_{2}^{2}-Y_{1}^{2}\right)\left(Y_{1}^{2}+Y_{2}^{2}\right)&\gamma^{\prime\prime}Y_{1}Y_{2}\left(Y_{1}^{2}-3Y_{2}^{2}\right)+2\gamma_{1}^{\prime\prime}Y_{1}Y_{2}\left(Y_{1}^{2}+Y_{2}^{2}\right)&\epsilon^{\prime\prime}\left(Y_{1}^{2}+Y_{2}^{2}\right){}^{2}\end{smallmatrix}\right)\
,$ (109)

where the modular form components have the same dependence on $\tau_{d}$ as
those appearing in the down-type quark Yukawa matrix.

In the following discussion, we are going to sketch a scenario in which
conjugate right-handed neutrinos are identified with the singlets
$\theta_{15}$, which are included in the particle spectrum of the F-Theory
model. Since these fields are considered as degrees of freedom that lie in the
transverse space of the matter curves [32], this fact leads us to consider the
case that they do not carry any modular weight. However, a simple model is
presented here in which the singlets transform as a triplet under the $S_{4}$
modular symmetry. In addition to the singlets mentioned before, more degrees
of freedom are needed to give a Majorana mass to $\theta_{15}$, leading to the
implementation of a (type-I) seesaw scenario for the light neutrino masses. An
important condition is that the additional singlets of the model have to
cancel the perpendicular charges of coupling.

The superpotential, following the transformation properties of Table (5), is
written as:

$\displaystyle\mathcal{W}_{\nu}=\zeta\;(\nu^{c}L_{1,2}Y^{(4)}_{3})_{1}H_{u}+\eta\;(\nu^{c}L_{3}Y^{(6)}_{3})_{1}H_{u}+\lambda\;(\nu^{c}\nu^{c})_{1}\;\dfrac{\theta_{53}^{2}\theta_{31}^{2}}{M^{3}},$
(110)

where in the last coupling stands for the Majorana mass term of the conjugate
right-handed neutrinos. Given the first two couplings the Yukawa matrix
responsible for the neutrino Dirac mass can be written as:

$\displaystyle\lambda_{\nu}=\begin{pmatrix}-2\zeta Y_{2}Y_{3}&0&\eta
Y_{1}(Y_{4}^{2}-Y_{5}^{2})\\\
-\dfrac{1}{2}\zeta(\sqrt{3}Y_{1}Y_{4}+Y_{2}Y_{5})&\dfrac{\sqrt{3}}{2}\zeta(\sqrt{3}Y_{1}Y_{5}+Y_{2}Y_{4})&-\eta
Y_{3}(Y_{1}Y_{4}+\sqrt{3}Y_{2}Y_{5})\\\
-\dfrac{1}{2}\zeta(\sqrt{3}Y_{1}Y_{5}+Y_{2}Y_{4})&\dfrac{\sqrt{3}}{2}\zeta(\sqrt{3}Y_{1}Y_{4}+Y_{2}Y_{5})&\eta
Y_{3}(Y_{1}Y_{5}+\sqrt{3}Y_{2}Y_{4})\end{pmatrix}~{},$ (111)

where the modular form components depend on the same modulus of the up-type
quark, $\tau_{u}$, and the conjugate right-handed neutrino Majorana mass
matrix can be easily read out as

$\displaystyle M_{R}=\begin{pmatrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix}\lambda\dfrac{\theta_{53}^{2}\theta_{31}^{2}}{M^{3}},$ (112)

where we will take
$\lambda\theta_{53}^{2}\theta_{31}^{2}/M^{3}=\tilde{\lambda}M_{GUT}$.

Given the above matrices, we could implement a type-I seesaw mechanism in our
model to explain the neutrino masses. The light neutrino mass matrix is given
by:

$\displaystyle M_{\nu}=-M_{D}^{T}M_{R}^{-1}M_{D}~{},$ (113)

where $M_{D}=v\lambda_{\nu}$, with $v=173$ GeV being the Standard Model Higgs
vacuum expectation value.

###  4.1 Numerical study

We now perform a brief numerical study to find whether the model presented
above and explicitly stated by the superpotential in eq. 106 can provide a
good fit to quark masses and mixing. To do so, we will compare the model
predictions against the values of the quark masses and mixing data at the GUT
scale, which, for $\tan\beta=5$, can be found in table 6. The neutrino data
are taken from the latest NuFit 5.3, [33] and is shown in table 7 alongside
the charged lepton Yukawa eigenvalues.

Quark and CKM Data
---
$y_{d}$ | $(4.81\pm 1.06)\times 10^{-6}$ | $\theta_{12}$ | $13.027^{\circ}\pm 0.0814^{\circ}$
$y_{s}$ | $(9.52\pm 1.03)\times 10^{-5}$ | $\theta_{23}$ | $2.054^{\circ}\pm 0.384^{\circ}$
$y_{b}$ | $(6.95\pm 0.175)\times 10^{-3}$ | $\theta_{13}$ | $0.1802^{\circ}\pm 0.0281^{\circ}$
$y_{u}$ | $(2.92\pm 1.81)\times 10^{-6}$ | $\delta_{CP}$ | $69.21^{\circ}\pm 6.19^{\circ}$
$y_{c}$ | $(1.43\pm 0.100)\times 10^{-3}$ |  |
$y_{t}$ | $0.534\pm 0.0341$ |  |
Table 6: Quark and CKM data [34, 35, 36]. Lepton and PMNS Data
---
$y_{e}$ | $(1.97\pm 0.024)\times 10^{-6}$ | $\sin^{2}\theta^{L}_{12}$ | $0.307\pm 0.012$
$y_{\mu}$ | $(4.16\pm 0.05)\times 10^{-4}$ | $\sin^{2}\theta^{L}_{23}$ | $0.572\pm 0.023$
$y_{\tau}$ | $(7.07\pm 0.073)\times 10^{-3}$ | $\sin^{2}\theta^{L}_{13}$ | $(2.203\pm 0.58)\times 10^{-2}$
$\Delta m^{2}_{12}$ | $(7.41\pm 0.21)\times 10^{-5}\text{ eV}^{2}$ | $\delta^{L}_{CP}$ | $197^{\circ}\pm 41^{\circ}$
$\Delta m^{2}_{13}$ | $(2.511\pm 0.027)\times 10^{-3}\text{ eV}^{2}$ |  |
Table 7: Lepton and PMNS data. Neutrino masses are given in normal ordering
[34, 35, 36, 33]. When the uncertainty interval is asymmetric, the larger
values was taken in the analysis for the Gaussian likelihood profile.

We use the effective Yukawa coupling matrices for the quarks, eqs. 107 and
108, as well as for the neutrinos, eqs. 111 and 112, to compute the
predictions and compare them to the data in tables 6 and 7. Although the
coefficients of the superpotential are in principle calculable in F-Theory
(see, for example, [37, 25, 28] for Yukawa couplings and [38] for R-Parity
violating terms), in this work we will consider these coefficients as free
parameters and leave the study of their computation for future work.
Additionally, we also have the dependency on the complex structure moduli
fields parametrising the geometry of the matter curves, from which the matter
fields inherit their discrete modular symmetry properties. Since up- and down-
type quark Yukawas emerge at different intersection points in the internal
geometry between different curves, the geometry describing each Yukawa
coupling is in general different from each other and parametrised by its own
modulus, i.e. the components of the modular forms appearing in the up- and
down-type Yukawas can depend on different moduli fields, $\tau_{u}$ and
$\tau_{d}$, respectively. However, the charged leptons (neutrino) Yukawa
matrix arises from the same intersection as the down-type (up-type) quark
Yukawas and should therefore depend on the same modulus. Therefore, our
(effective) parametric freedom encompasses:

  * •

Four complex coefficients ($\alpha$, $\beta$, $\delta$, $\gamma$) and a
complex modulus ($\tau_{u}$) for the up-type Yukawa matrix,

  * •

three complex coefficients ($\zeta$, $\eta$, $\tilde{\lambda}$) for the
neutrino sector (as well as a dependency on $\tau_{u}$),

  * •

six coefficients ($\alpha^{\prime}$, $\beta^{\prime}$, $\gamma^{\prime}$,
$\gamma^{\prime}_{1}$, $\epsilon^{\prime}$) and one complex modulus
($\tau_{d}$) for the down-type Yukawa matrix,

  * •

six coefficients ($\alpha^{\prime\prime}$, $\beta^{\prime\prime}$,
$\gamma^{\prime\prime}$, $\gamma^{\prime\prime}_{1}$,
$\epsilon^{\prime\prime}$) for the charged lepton Yukawa matrix (as well as a
dependency on $\tau_{d}$).

This sums up to a total of 19 complex parameters, or 38 real parameters.
Although this seems to over-parameterise our problem, as we only have 19
observables in tables 6 and 7, we must reiterate that the complex coefficients
are in principle calculable in F-Theory and that the analysis present here
simplifies this step.

To find whether we can jointly fit all observables, we employ an artificial
intelligence search algorithm called Covariant Matrix Approximation
Evolutionary Strategy (CMAES) [39], which was first proposed in [40] to
simplify the task of finding valid points in highly constrained
multidimensional BSM parameter spaces.666See also [41] for a recent
application to the $Z_{3}$ 3HDM, where CMAES was shown to have up to nine
orders of magnitude improvement in sampling efficiency over random sampling.
CMAES can be seen as a population-based optimisation algorithm that can find
minima of any arbitrary function, irrespective of its continuity and
differentiability. Therefore, we will use CMAES to minimise the minus log-
likelihood of the data, $D$, given a point of the parameter space,
$\theta$,777Or, equivalently, to minimise the sum of the $\chi^{2}$.

$-llh(D|\theta)=\sum_{i}\frac{(\bar{\mu}_{i}-\mu_{i}(\theta))^{2}}{2\sigma_{i}^{2}}\
,$ (114)

where $i$ runs over the observables, $\mu_{i}(\theta)$ is the prediction for
the observable $i$ given a parameter space point $\theta$, the data, $D$, are
comprised of the set of tuples $\\{(\bar{\mu}_{i},\sigma_{i})\\}$, where
$\bar{\mu}_{i}$, $\sigma_{i}$ are, respectively, the central and 1-$\sigma$
uncertainty values of the observables and are listed in tables 6 and 7, and we
have assumed a Gaussian profile likelihood for the data. We implemented CMAES
using the python package cmaes [42], and we performed 1000 independent runs,
each running until converged to a minimum of eq. 114, and kept all points
whose observable predictions were within 3-$\sigma$.888This methodology is
justified by the fact that our goal is not to draw a complete portrait of the
parameter space, but rather to find examples of viable points.. The parameters
of our model were bounded, so that the superpotential coefficients remain
perturbative and the moduli take values in their fundamental domain with an
upper bound on the imaginary part

$\left\\{\tau_{i}\in\mathbb{C},\ s.t.\ |\Re{(\tau_{i})}|\leq
0.5\wedge\sqrt{1-\Re{(\tau_{i})}^{2}}\leq\Im{(\tau_{i})}\leq 10\right\\}\
,i=u,\ d\ \ .$ (115)

Multiple successful runs converged, generating $18\times 10^{6}$ points that
fit all observables within 3-$\sigma$. The best point across all runs, that
minimises the eq. 114 at a value $1.15\times 10^{-15}$ (i.e., effectively with
vanishing $\chi^{2}$ or likelihood of $1$), is given by the set of parameters
(to up to one decimal digit)

$\displaystyle\left(\alpha,\beta,\delta,\gamma\right)=$
$\displaystyle(-1.8\times 10^{-3}+1.8\times 10^{-5}i,4.5\times
10^{-5}-1.4\times 10^{-4}i,$ $\displaystyle 3.2\times 10^{-4}+1.8\times
10^{-3}i,1.8\times 10^{-1}+4.0\times 10^{-2}i)$
$\displaystyle\left(\lambda^{\prime},\beta^{\prime},\gamma^{\prime},\gamma_{1}^{\prime},\epsilon^{\prime}\right)=$
$\displaystyle(2.1\times 10^{-5}-8.8\times 10^{-8}i,-3.3\times
10^{-5}+3.2\times 10^{-8}i,$ $\displaystyle-4.4\times 10^{-5}+7.2\times
10^{-5}i,-2.3\times 10^{-4}-2.3\times 10^{-4}i,$ $\displaystyle-7.7\times
10^{-5}+1.4\times 10^{-4}i,1.3\times 10^{-4}-4.6\times 10^{-3}i)$
$\displaystyle(\zeta,\eta,\tilde{\lambda})=$ $\displaystyle(-6.1\times
10^{-2}+9.1\times 10^{-1}i,-1.5\times 10^{-1}+7.2\times 10^{-3}i,$
$\displaystyle 1.8\times 10^{-1}+5.3\times 10^{-2}i)$
$\displaystyle\left(\lambda^{\prime\prime},\beta^{\prime\prime},\gamma^{\prime\prime},\gamma_{1}^{\prime\prime},\epsilon^{\prime\prime}\right)=$
$\displaystyle(7.5\times 10^{-5}+1.4\times 10^{-7}i,2.7\times
10^{-4}-7.3\times 10^{-7}i,$ $\displaystyle-1.0\times 10^{-3}+1.9\times
10^{-3}i,1.2\times 10^{-4}+2.6\times 10^{-8}i,$ $\displaystyle-2.9\times
10^{-3}+4.1\times 10^{-5}i,4.2\times 10^{-4}-1.7\times 10^{-3}i)$
$\displaystyle\tau_{u}=$ $\displaystyle-4.1\times 10^{-1}+9.1\times 10^{-1}i$
$\displaystyle\tau_{d}=$ $\displaystyle-5.0\times 10^{-1}+1.2i\ ,$

where we organised the parameters by mass sector. We notice that the point
above requires some hierarchy between superpotential coefficients which should
be around the same order, e.g. $|\gamma|\sim\mathcal{O}(1)$ whereas
$|\alpha|\sim\mathcal{O}(10^{-3})$. This hierarchy between coefficients of
operators arising from the intersection of the same matter curves at the same
intersection point is at odds with our F-Theory expectations, which requires
further study involving their explicit computation.

In fig. 1 we show the values of the moduli field that were obtained by CMAES,
where we see that lower values of the imaginary part of the moduli are
preferred, and most points have $\Im(\tau_{i})\lesssim 2$. We omit scatter
plots for the remaining parameters as these are, in principle, computable in
F-Theory, and the details of their numerical realisation are left to future
study. We also note that one should not attempt to make statistical
interpretations of the results of CMAES, as it is not an algorithm designed to
populate a posterior (as Monte Carlo Markov Chains do in Bayesian inference)
as it produces points through the path of quickest descent of the loss
function (and therefore the points should also not be used for frequentist
interpretations as one usually does with random sampling). However, all points
are within 3-$\sigma$ of all observables and therefore have a very high
likelihood, or, conversely, a very small $\chi^{2}$.

Figure 1: $\tau_{u}$ and $\tau_{d}$ values for the CMAES scan. All the points
hold predictions within 3-$\sigma$. The red star point represents the best fit
point, LABEL:eq:best_point. Dashed line represents the boundary of the
fundamental domain.

We first look at the results pertaining to the quark data. In fig. 2 we can
observe the resulting values for the up-type quark Yukawa eigenvalues of
points obtained, and in fig. 3 we present the equivalent plots for the down-
type quarks. We see that many points can be arbitrarly close to the central
value, but also span the region within the 3-$\sigma$ limits, showing that the
model produces a good fit to the data. The same can be observed in fig. 4 for
the CKM mixing angle and CP violating phase.

Figure 2: Up-type quark Yukawa eigenvalues obtained for the CMAES scan. All
the points hold predictions within 3-$\sigma$. The red star point represents
the best fit point, LABEL:eq:best_point. The dashed (full) lines represent the
central value (3-$\sigma$ bounds) from table 6. Figure 3: Down-type quark
Yukawa eigenvalues obtained for the CMAES scan. All the points hold
predictions within 3-$\sigma$. The red star point represents the best fit
point, LABEL:eq:best_point. The dashed (full) lines represent the central
value (3-$\sigma$ bounds) from table 6. Figure 4: CKM angles and CP phase
obtained for the CMAES scan. All the points hold predictions within
3-$\sigma$. The red star point represents the best fit point,
LABEL:eq:best_point. The dashed (full) lines represent the central value
(3-$\sigma$ bounds) from table 6.

In figs. 5, 6 and 7 we can arrive at similar conclusions regarding the charged
lepton Yukawa eigenvalues, neutrinos squared mass differences, PMNS mixing
angles, and CP violating phase.

Figure 5: Charged leptons Yukawa eigenvalues obtained for the CMAES scan. All
the points hold predictions within 3-$\sigma$. The red star point represents
the best fit point, LABEL:eq:best_point. The dashed (full) lines represent the
central value (3-$\sigma$ bounds) from table 7. Figure 6: Neutrino squared
mass differences obtained for the CMAES scan. All the points hold predictions
within 3-$\sigma$. The red star point represents the best fit point,
LABEL:eq:best_point. The dashed (full) lines represent the central value
(3-$\sigma$ bounds) from table 7. Figure 7: PMNS angles and CP phase obtained
for the CMAES scan. All the points hold predictions within 3-$\sigma$. The red
star point represents the best fit point, LABEL:eq:best_point. The dashed
(full) lines represent the central value (3-$\sigma$ bounds) from table 7.

The above results show that our model can fit the data very well, with the
best point having a likelihood close to unity or, conversely, a vanishingly
small $\chi^{2}$. However, the problem is over-parametrised by the number of
superpotential coefficients, which, although in principle calculable in
F-Theory, are considered free parameters in this analysis. To assess whether
we can reduce the parametric freedom, we considered alternative scenarios with
reduced parametric freedom with respect to the moduli. In our first
alternative scenario, we fixed the moduli to take the same values (i.e.
$\tau_{u}=\tau_{d}$ but otherwise allowed the moduli to take values in the
fundamental domain eq. 115) even though our F-Theory construction naturally
provides distinct moduli for each Yukawa type. The scans converged
successfully as before, from which we can conclude that our model does not
require two independent moduli to fit the data.

For the second case, we fixed the moduli to special values $\tau_{u},\
\tau_{d}\in\\{i,\ i\infty,\omega=\exp{(2\pi i/3)}\\}$ (but not necessarily
equal). In this scenario, CMAES failed to find points that fit the data. To
further study this scenario, we restricted the problem to only fit the quark
data, and even then the best-case scenario was for the configuration
$\tau_{u}=i\infty$, $\tau_{d}=i$, for which we were able to fit all the
observables within 3-$\sigma$ except for the $\theta_{12}$ angle of the CKM
matrix. The fact that the best-case scenario relies on $\tau_{u}=i\infty$
suggests that it is indeed not possible to find good points that fit all the
data with the moduli stabilised at special values, as we have seen in fig. 1
that the scans showed a preference for small values of $\Im(\tau_{i})$.

Therefore, we conclude that, despite being over-parameterised, the model works
with fewer parameters although we lack F-Theoretical motivations to restrict
their number. The realisation of a model with fewer parameters within the
framework presented in this paper is left to future work, as well as the
computation of the superpotential coefficients.

##  5 Summary and Conclusions

In this work we have shown that finite modular family symmetry can emerge from
local F-Theory constructions. Such theories provide a rich framework for the
application of modular family symmetry using bottom-up constructions. As a
first example of the new approach, we have analysed in detail a concrete
F-Theory Fluxed $SU(5)$ GUT with modular $S_{4}$ family symmetry. The model
fits the fermion mass and mixing data very well and serves as a demonstration
of the bottom-up approach to modular family symmetry in F-Theory fluxed GUTs.
Indeed this example represents the first F-Theory fluxed GUT with discrete
modular family symmetry.

To accomplish this, we first discussed in detail the origins of discrete
modular symmetries in F-Theory in section 3, and analysed an explicit model in
section 4. In section 3 we argued that in F-Theory the Yukawa couplings should
inherit modular symmetry properties from the axio-dilaton and the complex
structure moduli of the theory. In section 3.1 we revisited an explicit Type
IIB construction where the vacuum transforms non-trivially under the normal
congruence subgroup of the modular symmetry, breaking it to a discrete modular
group. This construction has the important feature that the modular symmetry
associated with the axio-dilaton, initially independent of the modular
symmetries of the background orbifold, is aligned with the modular symmetry of
one of the directions of the orbifold in supersymmetric preserving vacuum.

The result from section 3.1 is crucial as the Yukawa couplings in F-Theory are
known expected on the axio-dilaton. In section 3.2 this is explicitly
discussed, and we argue that if S-duality is to be maintained by the
perturbative superpotential, then the fields must transform under the action
of the axio-dilaton modular symmetry.

The conclusions of sections 3.1 and 3.2 are further expanded in section 3.3
where we discussed the explicit constructions of Yukawa couplings in F-Theory
as integrals of overlapping wavefunctions around the intersection of matter
curves, each a Riemannian surface. In this construction, the wavefunctions are
solutions to EoM of the compactified YM theory on the 7-branes, which the
solutions have an undetermined holomorphic dependency on the coordinates of
the matter curve. This observation leads us to conclude that the structure of
the Yukawa couplings will depend on the complex structure moduli of the matter
curves in a way that, if these curves exhibit modular symmetries, then the
Yukawa couplings can be modular forms associated with these symmetries.

Equipped with the insights gathered in section 3, we finally set out to
explicitly construct a fluxed GUT with discrete modular symmetry in section 4.
The model discussed is an extension of a model already presented in the
literature, where the fields are now furnishing a representation of a discrete
modular family group, in our case $S_{4}$, and carry modular weights.
Additionally, the Yukawa matrices, arising from the overlapping wavefunction
integrals at matter curve intersections, are promoted to modular forms of
appropriate weight. In section 4.1 we show that the model fits the data very
well, with the best point having vanishing $\chi^{2}$, although it is over-
parameterised with respect to superpotential coefficients.

In conclusion, we have shown that finite modular family symmetry can plausibly
emerge from local F-Theory constructions. As a first example of this approach,
we have analysed a concrete F-Theory Fluxed $SU(5)$ GUT example with modular
$S_{4}$ family symmetry, opening up new avenues of model building in this rich
framework. Although this approach looks promising, we have identified aspects
that need further detailed exploration, such as the explicit construction of
global F-Theory models with matter curves exhibiting modular symmetries and
the explicit computation of the Yukawa couplings in the presence of such
geometries which are in principle calculable in F-Theory. We leave these
issues for future work.

## Acknowledgements

MCR is supported by the STFC under Grant No. ST/T001011/1. SFK acknowledges
the STFC Consolidated Grant ST/L000296/1 and the European Union’s Horizon 2020
Research and Innovation programme under Marie Skłodowska-Curie grant agreement
HIDDeN European ITN project (H2020-MSCA-ITN-2019//860881-HIDDeN). GKL would
like to thank the staff of the Mainz Institute for Theoretical Physics (MITP)
for their kind hospitality where part of this work has been carried out.

##  Appendix A Appendix

###  A.1 Definitions of modular forms

For the modular group $\Gamma_{4}\cong S_{4}$, there are five linearly
independent modular forms of the lowest non-trivial weight $k=2$, denoted as
$Y_{i}(\tau)$ for $i=1,2,..,5$, which form a doublet 2 and a triplet
$3^{\prime}$ under the modular $S_{4}$ symmetry transformations, namely,

$\displaystyle Y_{2}^{(2)}=\begin{pmatrix}Y_{1}(\tau)\\\
Y_{2}(\tau)\end{pmatrix},\quad
Y_{3^{\prime}}^{(2)}=\begin{pmatrix}Y_{3}(\tau)\\\ Y_{4}(\tau)\\\
Y_{5}(\tau)\end{pmatrix}~{},$ (117)

$\displaystyle Y_{1}^{(4)}=Y_{1}^{2}+Y^{2}_{2},\quad
Y_{2}^{(4)}=\begin{pmatrix}Y_{2}^{2}-Y_{1}^{2}\\\
2Y_{1}Y_{2}\end{pmatrix},\;Y_{3}^{(4)}=\begin{pmatrix}-2Y_{2}Y_{3}\\\
\sqrt{3}Y_{1}Y_{5}+Y_{2}Y_{4}\\\
\sqrt{3}Y_{1}Y_{4}+Y_{2}Y_{5}\end{pmatrix},\;Y_{3^{\prime}}^{(4)}=\begin{pmatrix}2Y_{1}Y_{3}\\\
\sqrt{3}Y_{2}Y_{5}-Y_{1}Y_{4}\\\
\sqrt{3}Y_{2}Y_{4}-Y_{1}Y_{5}\end{pmatrix}~{},$ (118)

$\displaystyle Y_{1}^{(6)}=Y_{1}(3Y_{2}^{2}-Y_{1}^{2}),\quad
Y_{1^{\prime}}^{(6)}=Y_{2}(3Y_{1}^{2}-Y_{2}^{2})~{},$ (119)

$\displaystyle Y_{2}^{(6)}=(Y_{1}^{2}+Y_{2}^{2})\begin{pmatrix}Y_{1}\\\
Y_{2}\end{pmatrix},\;\;Y_{3}^{(6)}=\begin{pmatrix}Y_{1}(Y_{4}^{2}-Y_{5}^{2})\\\
Y_{3}(Y_{1}Y_{5}+\sqrt{3}Y_{2}Y_{4})\\\
-Y_{3}(Y_{1}Y_{4}+\sqrt{3}Y_{2}Y_{5})\end{pmatrix}~{},$ (120)

$\displaystyle
Y_{3^{\prime},1}^{(6)}=(Y_{1}^{2}+Y_{2}^{2})\begin{pmatrix}Y_{3}\\\ Y_{4}\\\
Y_{5}\end{pmatrix},\quad
Y_{3^{\prime},2}^{(6)}=\begin{pmatrix}Y_{2}(Y_{5}^{2}-Y_{4}^{2})\\\
-Y_{3}(Y_{2}Y_{5}-\sqrt{3}Y_{1}Y_{4})\\\
Y_{3}(Y_{2}Y_{4}-\sqrt{3}Y_{1}Y_{5})\end{pmatrix}~{},$ (121)

and

$\displaystyle Y_{1}^{(8)}=(Y_{1}^{2}+Y_{2}^{2})^{2},\quad
Y_{2,1}^{(8)}=(Y_{1}^{2}+Y_{2}^{2})\begin{pmatrix}Y_{2}^{2}-Y_{1}^{2}\\\
2Y_{1}Y_{2}\end{pmatrix},\quad
Y_{2,1}^{(8)}=(Y_{1}^{2}-3Y_{2}^{2})\begin{pmatrix}Y_{1}^{2}\\\
Y_{1}Y_{2}\end{pmatrix}~{}.$ (122)

The expressions of modular forms can be derived with the help of the Dedekind
$\eta$ function

$\displaystyle\eta(\eta)=q^{1/24}\Pi_{n=1}^{\infty}(1-q^{n}),\;\;q=e^{2\pi
i\tau},$ (123)

and its derivative

$\displaystyle Y(\alpha_{1},..,\alpha_{6}|\tau)\equiv\frac{d}{d\tau}\big{[}$
$\displaystyle\alpha_{1}\log\eta(\tau+\frac{1}{2})+\alpha_{2}\log\eta(4\tau)+\alpha_{3}\log\eta(\frac{\tau}{4})+\alpha_{4}\log\eta(\frac{\tau+1}{4})+$
$\displaystyle+\alpha_{5}\log\eta(\frac{\tau+2}{4})+\alpha_{6}\log\eta(\frac{\tau+3}{4})\big{]},$
(124)

with the coefficients $\alpha_{i}$ (for
$i=1,2,\textperiodcentered\textperiodcentered\textperiodcentered,6)$
fulfilling
$\alpha_{1}+\textperiodcentered\textperiodcentered\textperiodcentered+\alpha_{6}=0$.
More explicitly, we have

$\displaystyle Y_{1}(\tau)\equiv iY(1,1,-1/2,-1/2,-1/2,-1/2|\tau),$
$\displaystyle Y_{2}(\tau)\equiv
iY(0,0,\sqrt{3}/2,-\sqrt{3}/2,\sqrt{3}/2,-\sqrt{3}/2|\tau),$ $\displaystyle
Y_{3}(\tau)\equiv iY(1,-1,0,0,0,0|\tau),$ $\displaystyle Y_{4}(\tau)\equiv
iY(0,0,-1/\sqrt{2},i/\sqrt{2},1/\sqrt{2},-i/\sqrt{2}|\tau),$ $\displaystyle
Y_{5}(\tau)\equiv
iY(0,0,-1/\sqrt{2},-i/\sqrt{2},1/\sqrt{2},i/\sqrt{2}|\tau)~{}.$ (125)

In the above computations, we have used the following Fourier expansions of
the modular forms:

$\displaystyle Y_{1}$
$\displaystyle=-3\pi\left(39q^{9}+3q^{8}+24q^{7}+12q^{6}+18q^{5}+3q^{4}+12q^{3}+3q^{2}+3q+\frac{1}{8}\right),$
$\displaystyle Y_{2}$
$\displaystyle=3\sqrt{3}\pi\sqrt{q}\left(18q^{8}+24q^{7}+14q^{6}+12q^{5}+13q^{4}+8q^{3}+6q^{2}+4q+1\right),$
$\displaystyle Y_{3}$
$\displaystyle=\pi\left(26q^{9}+6q^{8}-16q^{7}+24q^{6}-12q^{5}+6q^{4}-8q^{3}+6q^{2}-2q+\frac{1}{4}\right),$
$\displaystyle Y_{4}$
$\displaystyle=-\sqrt{2}\pi\sqrt[4]{q}\left(48q^{8}+30q^{7}+31q^{6}+32q^{5}+18q^{4}+14q^{3}+13q^{2}+6q+1\right),$
$\displaystyle Y_{5}$ $\displaystyle=-4\sqrt{2}\pi
q^{3/4}\left(12q^{8}+8q^{7}+10q^{6}+6q^{5}+5q^{4}+6q^{3}+3q^{2}+2q+1\right)~{}.$
(126)

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