KUNS-2882

EPHOU-21-007

4D effective action from non-Abelian DBI action
with magnetic flux background

Yoshihiko<EMAIL_ADDRESS>Tetsutaro
<EMAIL_ADDRESS>Tatsuo
<EMAIL_ADDRESS>
Shintaro<EMAIL_ADDRESS>and Rei Takahashi2

1Department of Physics, Kyoto University, Kyoto 606-8502, Japan
2Department of Physics, Keio University, Yokohama 223-8533, Japan
3Department of Physics, Hokkaido University, Sapporo 060-0810, Japan

###### Contents

  1. 1 Introduction
  2. 2 Non-Abelian DBI action on magnetized extra dimensions
    1. 2.1 Magnetized D9-branes on the six-dimensional torus
    2. 2.2 Flux and matter zero modes
      1. 2.2.1 Symmetry breaking of $U(3)\to U(1)_{a}\times U(1)_{b}\times U(1)_{c}$ and degeneracy
      2. 2.2.2 Supersymmetry conditions on the background fluxes
      3. 2.2.3 Matter zero modes in SUSY theories
  3. 3 SUSY effective action of $U(1)_{a}\times U(1)_{b}\times U(1)_{c}$ theory
    1. 3.1 Gauge couplings
    2. 3.2 Kähler metric of chiral matters
    3. 3.3 Scalar quartic term in the F-term scalar potential
  4. 4 Summary and discussions
  5. A Details of the calculations
    1. A.1 Gauge couplings
    2. A.2 Kinetic terms
      1. A.2.1 Kähler metric of charged matters
      2. A.2.2 Kähler metric of open string moduli
    3. A.3 Quartic terms
      1. A.3.1 F-term potential
  6. B Comments on the Yukawa type superpotential

##  1 Introduction

Superstring theory is an attractive candidate for a unified theory consistent
with quantum gravity. The theory can provide us with a theoretical framework
to describe all the interactions and chiral matters such as quarks and leptons
as well as the Higgs field. The string theory can predict the existence of
extra dimensions and D-branes. Dynamics of low energy excitations on D-branes
is described by gauge theories. Compactification of string theory on tori is
one of simple ways to obtain four-dimensional (4D) effective field theories
but these are nonchiral, while the Standard Model is chiral. The chiral nature
of matter fields is realized by introducing magnetic fluxes on the world
volume of D-branes in the compact extra dimensions [1, 2, 3, 4]. Even in
toroidal compactfications, magnetic fluxes realize 4D chiral theory. Orbifold
compactification with magnetic fluxes is also studied in Refs. [5, 6, 7]. The
number of chiral generations is determined by the size of the magnetic flux on
compact extra dimensions.111The number of chiral generations also depends on
twisted boundary conditions, discrete Wilson lines, and Scherk-Schwarz phase
in orbifold models.  Three-generation models have been classified in Refs. [8,
9, 10]. Moreover, as the zero-mode functions of the Dirac (Laplace) operator
are quasilocalized in compact space and Yukawa couplings as well as higher
order couplings are written by overlap integration among their zero mode
functions, hierarchical couplings can be realized [11, 12]. The realization of
quark and lepton masses and their mixing angles was studied in Refs. [13, 14,
15, 16]. Furthermore, their flavor structure is controlled by modular symmetry
[17, 18, 19, 20, 21, 22, 23]. Thus, compactification with magnetic background
fluxes is one of practical methods to derive realistic particle physics from
string theory.

4D low energy effective theories have often been constructed through
compactification of higher-dimensional super Yang-Mills (SYM) theory with the
canonical kinetic term [11, 24]. On the other hand, the Dirac-Born-Infeld
(DBI) action [25, 26] with the Chern-Simons (CS) terms [27, 28, 29, 30, 31,
32, 33] describes the dynamics of massless open string modes on the D-branes.
At the lowest order of the gauge field strength $F$, the DBI action reduces to
Yang-Mills theory. However, the DBI action can describe more stringy D-brane
natures, e.g. T duality. For non-Abelian DBI action, higher order terms of the
gauge field strength are less-known owing to its noncommutativity [34, 35, 36,
37, 38, 39, 40] and it is also less known to compute explicitly 4D effective
theories via compactification on a magnetized torus. This naturally motivates
us to study dimensional reduction of the non-Abelian DBI action for including
higher order corrections. Our purpose in this paper is to compute 4D
$\mathcal{N}=1$ supersymmetric effective action from ten-dimensional (10D)
non-Abelian DBI action compactified on the magnetized six-dimensional torus
with focus on terms up to $\mathcal{O}({F}^{4})$:

$\displaystyle\mathcal{L}_{\text{4D}}=\int_{\mathbb{T}^{6}}d^{6}y\,\mathcal{L}_{\text{non-
Abelian
DBI}}\sim\int_{\mathbb{T}^{6}}d^{6}y\,(\mathop{\mathrm{tr}}\nolimits{F}^{2}+\mathop{\mathrm{tr}}\nolimits{F}^{4})\quad\text{where
}\hat{{F}}_{y^{i}y^{j}}\neq 0.$ (1.1)

Here $y^{i}~{}(i=1,2,\ldots,6)$ denote the coordinates in the extra six
dimensions and $\hat{F}$ is the background flux. Hereafter, we drop the
Neveu–Schwarz-Neveu–Schwarz (NSNS) two-form potential for simplicity
throughout this paper. We ignore also CS terms in the D-brane action since
they mainly contribute to topological terms and supersymmetry (SUSY) breaking
terms which vanish for supersymmetric vacua with canceled tadpoles. We focus
on bosonic part of non-Abelian DBI action in this paper, since fermions can be
naturally introduced with SUSY.

In 4D action, we show the matter Kähler metric, gauge kinetic function, and
superpotential in supergravity (SUGRA) through a systematic study of
dimensional reduction. The DBI correction of $\mathcal{O}({F}^{4})$
contributes only to the matter Kähler metric and gauge kinetic function. It
turns out that there exists a new flux contribution to the matter Kähler
potential, while gauge kinetic functions and holomorphic Yukawa couplings in
the superpotential are consistent with previous works. Such a new flux
correction to the Kähler metric has been often neglected, although a flux
contribution to gauge coupling is frequently discussed for the coupling
unification. We take flux corrections into account consistently in this sense
and show a concrete dependence on fluxes in the Kähler potential of chiral
matters. Also, that of open string moduli, which was discussed in Refs. [41,
42, 43], is shown in Appendix A. Such consistent treatment may become
important to study swampland conjectures [44] with effective field theories
(see [45] for a review). The new matter Kähler metric is independent of flavor
but depends on the fluxes, 4D dilaton, Kähler moduli and complex structure
moduli, and will be always positive definite if an induced Ramond-Ramond (RR)
charge of the D-branes on which matters are living are positive. The
contribution of the matter Kähler metric to the scalar potential is shown to
be consistent with the SUGRA formulation, and the superpotential is read from
scalar quartic interaction.

The paper is organized as follows. In Sec. 2, we give a brief review of the
non-Abelian DBI action and a magnetized torus. In Sec. 3, we derive 4D
supersymmetric low energy effective action from the DBI action compactified on
a magnetized torus. The results turn out to be consistent with 4D SUGRA
formulation. Sec. 4 is devoted to the summary and discussion. In Appendixes A
and B, we give the details of the calculations.

##  2 Non-Abelian DBI action on magnetized extra dimensions

In this section, we introduce the DBI action and summarize our setup of flux
compactification of the DBI action on a six-dimensional torus.

The dynamics of massless open string modes on the D$p$-brane is described by
the DBI action with the CS terms. The DBI action for Abelian gauge theory is
expressed as

$\displaystyle S_{\text{DBI}}[g_{MN},\varphi,A_{M}]=-T_{p}\int
d^{p+1}\xi\,e^{-\varphi}\sqrt{-\det_{p+1}\bigl{(}g_{MN}+2\pi\alpha^{\prime}F_{MN}\bigr{)}},$
(2.1)

where $M,N=0,1,\ldots,p$ stand for the indices of the $(p+1)$-dimensional
world volume of D$p$-brane, and $g_{MN}$ is the pull back of the bulk metric
on the D-brane. $\alpha^{\prime}$ denotes the Regge slope, and $F_{MN}$ is the
gauge field strength on the D$p$-brane,
$F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}$. $\varphi$ denotes the 10D
dilaton field and $T_{p}$ is the brane tension given by
$T_{p}=2\pi/\ell_{s}^{p+1}=2\pi/(2\pi\alpha^{\prime 1/2})^{p+1}$, where
$\ell_{s}=2\pi\alpha^{\prime 1/2}$ is the string length. The superpartner
fermions are dropped here for simplicity. The DBI action (2.1) is known to be
robust for an Abelian gauge theory living on a single D-brane.

A non-Abelian gauge theory is realized on a stack of D-branes. The author in
Ref. [46] proposed the non-Abelian version of the DBI action with a
prescription of the symmetrized trace, while terms higher than ${\cal
O}({F}^{6})$ in the non-Abelian DBI (NDBI) action are still ambiguous owing to
its noncommutativity [34, 35, 36, 37, 38, 39, 40]. As the extension of Eq.
(2.1), NDBI action is given by [46]

$\displaystyle S_{\text{NDBI}}=-T_{p}\int
d^{p+1}\xi\,e^{-\varphi}\mathop{\mathrm{str}}\nolimits\sqrt{-\det_{p+1}(g_{MN}+2\pi\alpha^{\prime}F_{MN})}.$
(2.2)

Here $F_{MN}$ is the field strength of non-Abelian gauge field,
$F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}+i[A_{M},A_{N}]$, and
“$\mathop{\mathrm{str}}\nolimits$” denotes the symmetrized trace,

$\displaystyle\mathop{\mathrm{str}}\nolimits(T_{1}\cdots
T_{n})=\frac{1}{n!}\mathop{\mathrm{tr}}\nolimits\bigl{[}T_{1}\cdots
T_{n}+(\text{permutations})\bigr{]}.$ (2.3)

Hereafter, we consider space-filling D9-branes ($p=9$) for concreteness
because the Lagrangian in the bosonic part consists only of the gauge field.
We focus on terms up to ${\cal O}({F}^{4})$.

###  2.1 Magnetized D9-branes on the six-dimensional torus

We introduce background fluxes on a stack of D9-branes compactified on a six-
dimensional torus. Let us consider a six-dimensional torus consisting of three
two-dimensional tori as the extra dimension
$\mathbb{T}^{6}=\prod_{i=1}^{3}\mathbb{T}^{2}_{i}$. The 10D metric of
$M_{4}\times\prod_{i=1}^{3}\mathbb{T}^{2}_{i}$ is given by

$\displaystyle
ds_{10}^{2}=e^{2\Phi}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\ell_{s}^{2}\sum_{i=1}^{3}g_{mn}^{(i)}dy_{i}^{m}dy_{i}^{n},\quad
g^{(i)}_{mn}=e^{2\sigma_{i}}\begin{pmatrix}1&\tau^{(i)}_{R}\\\
\tau^{(i)}_{R}&|\tau^{(i)}|^{2}\end{pmatrix},$ (2.4)

where $\mu,\nu=0,1,2,3$, $\eta_{\mu\nu}=\mathop{\mathrm{diag}}(-1,1,1,1)$ is
the Minkowski metric and
$\tau^{(i)}=\tau^{(i)}_{R}+i\,\tau^{(i)}_{I}~{}(i=1,2,3)$ is the complex
structure modulus on the $i$th torus $\mathbb{T}^{2}_{i}$.
$y^{m}_{i}~{}(m=1,2)$ denotes the coordinate on $\mathbb{T}^{2}_{i}$ and
$0\leq y_{i}^{m}\leq 1$, where $y$’s are normalized by the string length. The
volume of the $i$th torus in the string length unit reads

$\displaystyle\mathop{\mathrm{Vol}}(\mathbb{T}^{2}_{i})=\sqrt{g^{(i)}}=\mathcal{A}^{(i)}=e^{2\sigma_{i}}\tau^{(i)}_{I}.$
(2.5)

Hence, $e^{2\sigma_{i}}$ is regarded as a volume modulus of
$\mathbb{T}^{2}_{i}$. For the 4D Einstein frame, we have introduced the 4D
dilaton $\Phi$,

$\displaystyle\Phi=\varphi-\frac{1}{2}\log\prod_{i}\mathcal{A}^{(i)}=\varphi-\frac{1}{2}\log\mathop{\mathrm{Vol}}(\mathbb{T}^{6}),$
(2.6)

where
$\mathop{\mathrm{Vol}}(\mathbb{T}^{6})=\mathcal{A}^{(1)}\mathcal{A}^{(2)}\mathcal{A}^{(3)}$
is the volume of $\mathbb{T}^{6}$. With the complex coordinate on the $i$th
torus

$\displaystyle dz_{i}=dy_{i}^{1}+\tau^{(i)}dy_{i}^{2},\quad i=1,2,3,$ (2.7)

the 10D metric is rewritten as

$\displaystyle
ds_{10}^{2}=e^{2\Phi}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\ell_{s}^{2}\sum_{i=1}^{3}e^{2\sigma_{i}}dz_{i}d\overline{z_{i}}.$
(2.8)

Thus, the metric on the $\mathbb{T}^{2}_{i}$ in the the complex basis is given
by

$\displaystyle
g_{i\overline{j}}=\ell_{s}^{2}\frac{e^{2\sigma_{i}}}{2}\delta_{i\overline{j}}.$
(2.9)

We shall focus on a stack of the space-filling D9-branes on the factorized
torus $\prod_{i=1}^{3}\mathbb{T}^{2}_{i}$ with nontrivial background fluxes on
the D-branes. The NDBI action (2.2) expanded up to $\mathcal{O}({F}^{4})$ is
given by [46]

$\displaystyle S_{\text{NDBI}}\approx$ $\displaystyle~{}-T_{9}\int
d^{10}X\sqrt{-\det
g_{MN}}\,e^{-\varphi}\frac{(2\pi\alpha^{\prime})^{2}}{4}\mathop{\mathrm{tr}}\nolimits\biggl{[}{F}_{MN}F_{MN}-\frac{(2\pi\alpha^{\prime})^{2}}{3}\biggl{(}{F}_{KL}{F}_{LM}{F}_{NK}{F}_{MN}$
$\displaystyle+\frac{1}{2}{F}_{KL}{F}_{LM}{F}_{MN}{F}_{NK}-\frac{1}{4}{F}_{KL}{F}_{KL}{F}_{MN}{F}_{MN}-\frac{1}{8}{F}_{KL}{F}_{MN}{F}_{KL}{F}_{MN}\biggr{)}+\mathcal{O}({F}^{6})\biggr{]},$
(2.10)

where the metric is omitted in contracting indices of the gauge field
strength, e.g. ${F}_{MN}{F}_{MN}\coloneqq g^{MK}g^{NL}{F}_{MN}{F}_{KL}$. $X$
denotes the bulk coordinate in 10D. The normalization of gauge group generator
is assumed to be given by
$\mathop{\mathrm{tr}}\nolimits(T^{a}T^{b})=\delta^{ab}$. The quadratic term
$\mathop{\mathrm{tr}}\nolimits{F}_{MN}^{2}$ can reduce to the well-known Yang-
Mills action with the canonical kinetic term.

With respect to the background fluxes on the D9-branes, it is assumed that
only the fluxes on the extra six dimension have nonzero values,

$\displaystyle{F}_{MN}\ni\hat{F}_{y^{m}_{i}y^{n}_{j}},\quad\text{where}~{}\hat{F}_{y^{1}_{i}y^{2}_{i}}\neq
0.$ (2.11)

Here, the background flux $\hat{F}$ is taken to be diagonal with respect to
the torus index, i.e., $\hat{F}_{y^{1}_{i}y^{2}_{i}}\neq 0$ for $i=1,2,3$ and
$\hat{F}_{y^{m}_{i}y^{n}_{j}}=0$ for $i\neq j$. In the complex basis,
nonvanishing components of the fluxes are given by

$\displaystyle\hat{F}_{z_{i}\overline{z}_{i}}=\frac{\partial
y_{i}^{m}}{\partial z_{i}}\frac{\partial
y_{i}^{n}}{\partial\overline{z}_{i}}\hat{F}_{y^{m}_{i}y^{n}_{i}},~{}~{}~{}i=1,2,3.$
(2.12)

See Appendix A for details. This is consistent with the SUSY condition as
discussed later.

###  2.2 Flux and matter zero modes

Although fermions are neglected so far, they exist in the presence of the
SUSY. We briefly review a zero (massless) mode solution of the Dirac equation
on the $\mathbb{T}^{2}$ with $U(1)$ magnetic flux [11]. A generalization of
the solution to the $\mathbb{T}^{6}$ case is discussed later.

The background magnetic flux on $\mathbb{T}^{2}$ in the string length unit is
given by

$\displaystyle\int_{\mathbb{T}^{2}}\frac{\hat{F}}{2\pi}=M~{}\to~{}\hat{F}=\frac{\pi
iM}{\tau_{I}}dz\wedge d\overline{z},\quad M\in{\mathbb{Z}}.$ (2.13)

Then, the gauge potential can be written as

$\displaystyle\hat{A}(z)=\frac{\pi
M}{\tau_{I}}\mathop{\mathrm{Im}}(\overline{z}\,dz).$ (2.14)

A large gauge transformation associated with translations on the torus is
given by

$\displaystyle\hat{A}(z+1)=\hat{A}(z)+d\bigg{(}\frac{\pi M}{\tau_{I}}{\rm
Im}(z)\bigg{)},\quad\hat{A}(z+\tau)=\hat{A}(z)+d\bigg{(}\frac{\pi
M}{\tau_{I}}{\rm Im}(\overline{\tau}z)\bigg{)}.$ (2.15)

Let us consider the spinor $\psi$ on the $\mathbb{T}^{2}$ with unit charge
$q=1$, where

$\displaystyle\psi=\begin{pmatrix}\psi_{+}(z)\\\ \psi_{-}(z)\end{pmatrix}.$
(2.16)

Here, $\pm$ denotes the eigenvalue of $SO(2)$ spinor algebra associated with
the torus (chirality). The gauge transformation acts on the spinor as

$\displaystyle\psi(z+1)=\exp\bigg{[}i\frac{\pi M}{\tau_{I}}{\rm
Im}(z)\bigg{]}\psi(z),\quad\psi(z+\tau)=\exp\bigg{[}i\frac{\pi
M}{\tau_{I}}{\rm Im}(\overline{\tau}z)\bigg{]}\psi(z).$ (2.17)

With these two boundary conditions, we solve the Dirac equation
$i\not{D}\psi=0$ on the $\mathbb{T}^{2}$. It is noted that the spinor becomes
a single-valued function up to the gauge transformation when
$M\in{\mathbb{Z}}$. For $M>0$, only $\psi_{+}$ is a normalizable zero modes,
which is $|M|$-fold degenerate; similarly, only $\psi_{-}$ is a normalizable
$|M|$-fold degenerate zero mode for $M<0$. Hence, the effective theory becomes
chiral in the low energy limit. Explicitly, for $M>0$ the $\psi_{+}$ is
written as

$\displaystyle\psi_{{\mathbb{T}}^{2}}^{A,M}=\Theta^{A,M}(z)\coloneqq\mathcal{N}_{M}\exp\bigg{[}\pi
iMz\frac{{\rm Im}(z)}{\tau_{I}}\bigg{]}\vartheta\begin{bmatrix}\frac{A}{M}\\\
0\end{bmatrix}(Mz,M\tau),\quad A=0,1,\cdots,M-1.$ (2.18)

Here, $\mathcal{N}_{M}$ is the normalization constant, $A$ labels the number
of degeneracy, i.e. flavor, and $\vartheta$ is the Jacobi theta function

$\displaystyle\vartheta\begin{bmatrix}a\\\
b\end{bmatrix}(\nu,\tau)\coloneqq\sum_{l\in{\mathbb{Z}}}e^{\pi
i(a+l)^{2}\tau}e^{2\pi i(a+l)(\nu+b)}.$ (2.19)

The normalization of $\psi_{+}$ reads

$\displaystyle\int_{\mathbb{T}^{2}}d^{2}y\sqrt{g_{2}}\,\overline{\Theta^{A,M}(z)}\Theta^{B,M}(z)=\delta^{AB}\frac{(\mathcal{N}_{M})^{2}\mathcal{A}}{\sqrt{2\tau_{I}|M|}},$
(2.20)

and we choose the following condition222The normalization factor for $M=0$ is
$\mathcal{N}=1/\sqrt{\mathcal{A}}$.

$\displaystyle(\mathcal{N}_{M})^{2}=\frac{\sqrt{2\tau_{I}|M|}}{\mathcal{A}}$
(2.21)

such that
$\int_{\mathbb{T}^{2}}d^{2}y\sqrt{g_{2}}\,\overline{\Theta^{A,M}(z)}\Theta^{B,M}(z)=\delta^{AB}$.
Here, we used dimensionless coordinate $z$. For $M<0$, the normalizable
solution of $\psi_{-}$ is written as

$\displaystyle\psi_{{\mathbb{T}}^{2}}^{A,M}=\overline{\Theta^{A,|M|}(z)},\quad
A=0,1,\ldots,|M|-1,$ (2.22)

where the normalization constant is the same as that for $M>0$. Thus, a
signature of $M$ is associated with the chirality of fermion.

####  2.2.1 Symmetry breaking of $U(3)\to U(1)_{a}\times U(1)_{b}\times
U(1)_{c}$ and degeneracy

It is easy to extend the above solution to a 10D theory compactified on
${\mathbb{T}}^{6}=\prod_{i=1}^{3}\mathbb{T}^{2}_{i}$ with non-Abelian gauge
symmetries of our interest. In the 10D SYM theory, there exist gauge fields
$A_{M}$ and their superpartner gluinos $\lambda^{(10)}$.

It is necessary to take into account of background fluxes to identify which
zero modes survive in 4D theory. We give the following background fluxes in a
non-Abelian gauge theory:

$\displaystyle\hat{F}_{z_{i}\overline{z}_{i}}\eqqcolon\hat{F}_{i\overline{i}}=\frac{i\pi
M^{(i)}}{\tau_{I}^{(i)}},\quad i=1,2,3.$ (2.23)

Here, $M^{(i)}$ is a matrix-valued constant, and gives the gauge symmetry
which can survive in the 4D theory through $[M^{(i)},A_{\mu}]=0$. Otherwise,
gauge fields become massive. For simplicity, we hereafter focus on the case in
which the $U(3)$ gauge group in 10D is broken to $U(1)_{a}\times
U(1)_{b}\times U(1)_{c}$ in 4D by the diagonal background fluxes,

$\displaystyle\frac{1}{2\pi}\int_{\mathbb{T}^{2}_{i}}dz_{i}\wedge
d\overline{z}_{i}\,\hat{F}_{i\overline{i}}=M^{(i)}=\begin{pmatrix}M^{(i)}_{a}&&\\\
&M^{(i)}_{b}&\\\ &&M^{(i)}_{c}\end{pmatrix},\quad
M^{(i)}_{a,b,c}\in\mathbb{Z},$ (2.24)

where the fluxes are similarly quantized for a charged zero mode to have a
single-valued function on the each $\mathbb{T}^{2}$ up to gauge
transformation. Replacing unity with identity matrix in Eq. (2.24) can realize
4D non-Abelian gauge symmetries. It is noted that gauge fields and gluinos in
10D are both adjoint representations, in which they are coupled to the fluxes
with a commutator through their covariant derivatives. Hence, the degeneracy
of fermion zero modes $I_{\alpha\beta}$ depends on the difference of fluxes
between two gauge groups on each torus [11],

$\displaystyle
I_{\alpha\beta}\coloneqq\prod_{i=1}^{3}I^{(i)}_{\alpha\beta},\qquad
I^{(i)}_{\alpha\beta}\coloneqq
M_{\alpha}^{(i)}-M_{\beta}^{(i)}\quad(\alpha,~{}\beta=a,~{}b,~{}c),$ (2.25)

for matter with a charge of $(1,-1)$ against a $U(1)_{\alpha}\times
U(1)_{\beta}$ gauge group. It is noted that the definition of
$I_{\alpha\beta}$ gives

$\displaystyle I_{ab}^{(i)}+I_{bc}^{(i)}+I_{ca}^{(i)}=0.$ (2.26)

This equation can determine a relative signature among $I_{\alpha\beta}$’s.
Next, we show the SUSY condition for avoiding tachyons, and visit concrete
zero mode functions.

####  2.2.2 Supersymmetry conditions on the background fluxes

We consider the condition for background fluxes to preserve 4D $\mathcal{N}=1$
SUSY for realizing chiral theories. The SUSY transformation of 10D fermions
should vanish to preserve the 4D SUSY. Then, the condition of the background
fluxes at $\mathcal{O}(\hat{F}^{2})$ on the complex manifold reads [47, 48,
41, 11, 49, 50]:

$\displaystyle g^{i\overline{j}}\hat{F}_{i\overline{j}}=0,$ (2.27)
$\displaystyle\hat{F}_{ij}=\hat{F}_{\overline{i}\overline{j}}=0.$ (2.28)

It is noted that an additional term of
$\hat{F}_{z_{1}\overline{z}_{1}}\hat{F}_{z_{2}\overline{z}_{2}}\hat{F}_{z_{3}\overline{z}_{3}}$
to the rhs of Eq. (2.27) is required for the calibration condition of
magnetized D-branes with DBI action. However, the above condition is
sufficient to us since we focus on the terms of
$\mathcal{O}(F^{4})\ni\hat{F}^{2}\times(\mathrm{fluctuations})$ in the
Lagrangian. Higher order corrections in
$\mathcal{O}(F^{6})\ni\hat{F}^{4}\times(\mathrm{fluctuations})$ neglected in
this paper can modify the condition to the terms involved in $F^{4}$. In our
case, the former condition (2.27) is satisfied when

$\displaystyle\sum_{i=1}^{3}\frac{M^{(i)}_{\alpha}}{\mathcal{A}^{(i)}}=0.\qquad(\alpha=a,b,c).$
(2.29)

The latter condition on the vanishing holomorphic flux condition (2.28) is
satisfied when we consider the diagonal fluxes in the torus index. Then
tachyons are absent in the effective theories since their mass squared is
proportional to [51, 11, 5]

$\displaystyle\sum_{i=1}^{3}\frac{M^{(i)}_{\alpha}-M^{(i)}_{\beta}}{\mathcal{A}^{(i)}}=\sum_{i=1}^{3}\frac{I^{(i)}_{\alpha\beta}}{\mathcal{A}^{(i)}}=0.$
(2.30)

This equation can also determine a relative signature among
$I_{\alpha\beta}^{(i)}$’s with fixed $\alpha$ and $\beta$ on top of Eq.
(2.26). For later convenience, we introduce the notation of the flux divided
by the torus area as

$\displaystyle
m^{(i)}_{\alpha}\coloneqq\frac{M^{(i)}_{\alpha}}{\mathcal{A}^{(i)}}\qquad(\alpha=a,b,c).$
(2.31)

####  2.2.3 Matter zero modes in SUSY theories

We shall consider zero mode functions on ${\mathbb{T}}^{6}$ in the presence of
4D SUSY. Let us take 10D chirality of the gluino $\lambda^{(10)}$ as [24]

$\displaystyle\Gamma\lambda^{(10)}=+\lambda^{(10)}.$ (2.32)

Then, the gluino is decomposed into the irreducible spinor representation with
$SO(2)^{3}$ that is the Cartan subalgebra of $SO(6)$,

$\displaystyle\lambda_{0}\coloneqq\lambda_{+++},~{}~{}~{}\lambda_{1}\coloneqq\lambda_{+--},~{}~{}~{}\lambda_{2}\coloneqq\lambda_{-+-},~{}~{}~{}\lambda_{3}\coloneqq\lambda_{--+},$
(2.33)

where $\pm$ denotes the eigenvalues of $SO(2)^{3}$ spinor algebra
(chiralities). 10D gauge fields $A_{M}$ can be decomposed similarly into

$\displaystyle
A_{\mu},~{}~{}~{}A_{z_{1}},~{}~{}~{}A_{z_{2}},~{}~{}~{}A_{z_{3}},$ (2.34)

where
$A_{z_{i}}=\frac{i}{2\tau_{I}^{(i)}}(\overline{\tau^{(i)}}A_{y_{i}^{1}}-A_{y_{i}^{2}})$.
In 4D ${\cal N}=1$ SUSY theories, a vector multiplet $V$ consists of $A_{\mu}$
and $\lambda_{0}$, whereas chiral multiplets $\Phi_{i}$ can consist of
fluctuations of $A_{z_{i}}$ and $\lambda_{i}~{}(i=1,2,3)$. When the background
fluxes preserve the 4D SUSY in flat spacetime, bosonic partners have the same
zero mode function as fermions’ [11, 24]. Then, the zero mode function of the
massless gauge multiplet $V$ is independent of coordinates $y$ since there
exists no coupling to the fluxes in the zero mode equation, i.e.,
$[M^{(i)},A_{\mu}]=0$. For the chiral multiplets $\Phi_{i}(x)$, the zero mode
functions $\phi_{i}(y)$ are given by products of those on each torus:

$\displaystyle\Phi_{i}^{\text{10D}}(x,y)$
$\displaystyle=\sum_{{\mathbb{A}}}\Phi_{i}^{{\mathbb{A}},I_{\alpha\beta}}(x)\otimes\phi_{i}^{{\mathbb{A}},I_{\alpha\beta}}(y)~{}+~{}({\rm
massive~{}modes}),$ (2.35)
$\displaystyle\phi_{i}^{{\mathbb{A}},I_{\alpha\beta}}(y)$
$\displaystyle=\bigg{(}\prod_{r=1}^{3}\phi_{i,{\mathbb{T}}_{r}^{2}}^{A^{(r)},I_{\alpha\beta}^{(r)}}(y_{r})\bigg{)}.$
(2.36)

Here, for $I_{\alpha\beta}\neq 0$,

$\displaystyle\phi_{i,{\mathbb{T}}_{r}^{2}}^{A^{(r)},I_{\alpha\beta}^{(r)}}=\begin{cases}\Theta^{A^{(r)},I_{\alpha\beta}^{(r)}}(z_{r})~{}{\rm
with}~{}\tau^{(r)}&(r=i~{}\&~{}I_{\alpha\beta}^{(r)}>0),\\\
\overline{\Theta^{A^{(r)},|I_{\alpha\beta}^{(r)}|}(z_{r})}~{}{\rm
with}~{}\overline{\tau^{(r)}}&(r\neq i~{}\&~{}I_{\alpha\beta}^{(r)}<0),\\\
0&({\rm other~{}cases}).\end{cases}$ (2.37)

This is consistent with chiralities in Eq. (2.33). $A^{(r)}$ is the index of
flavor on each torus: $A^{(r)}=0,1,\ldots|I_{\alpha\beta}^{(r)}|-1$ ,and
hence, the total flavor index is
${\mathbb{A}}=0,1,\ldots,|I_{\alpha\beta}|-1$. It is noted that matter
$\Phi_{i}^{{\mathbb{A}},I_{\alpha\beta}}$ has a charge of $(1,-1)$ against the
$U(1)_{\alpha}\times U(1)_{\beta}$ gauge group.

Without loss of generality, we assume that

$\displaystyle I_{ab}^{(1)}>0,\qquad I_{ab}^{(2),(3)}<0,$ $\displaystyle
I_{bc}^{(2)}>0,\qquad I_{bc}^{(1),(3)}<0,$ (2.38) $\displaystyle
I_{ca}^{(3)}>0,\qquad I_{ca}^{(1),(2)}<0,$

to satisfy Eqs. (2.26) and (2.30). This is also consistent with decomposition
of Eq. (2.33) as below. As noted, we have the gauge symmetry breaking of
$U(3)\to U(1)_{a}\times U(1)_{b}\times U(1)_{c}$. Then, the fluctuations of
10D gauge fields are decomposed into 4D zero (massless) modes, which are,
namely, gauge fields and complex scalars charged under the 4D gauge
symmetries:

$\displaystyle a_{\mu}=\begin{pmatrix}a^{a}_{\mu}&&\\\ &a^{b}_{\mu}&\\\
&&a^{c}_{\mu}\end{pmatrix},$ (2.39) $\displaystyle
a_{z_{i}}=\begin{pmatrix}&a^{ab}_{i}&\\\ &&a^{bc}_{i}\\\
a^{ca}_{i}&&\end{pmatrix}\eqqcolon\begin{pmatrix}&A_{i}\phi_{i}^{ab}\delta_{i1}&\\\
&&B_{i}\phi_{i}^{bc}\delta_{i2}\\\
C_{i}\phi_{i}^{ca}\delta_{i3}&&\end{pmatrix},$ (2.40)

where $a_{M}$ denotes fluctuations of the 10D gauge fields; $a^{a,b,c}_{\mu}$
are the 4D gauge fields associated with $U(1)_{a,b,c}$ symmetries, $A_{i}$,
$B_{i}$ and $C_{i}$ denote 4D complex scalars.
$(\phi_{i}^{ab},\phi_{i}^{bc},\phi_{i}^{ca}):=(\phi_{i}^{I_{ab}},\phi_{i}^{I_{bc}},\phi_{i}^{I_{ca}})$
show zero mode functions relevant to each complex scalar and we suppressed the
flavor index. These scalars have bifundamental charges against $U(1)_{a}\times
U(1)_{b}\times U(1)_{c}$ symmetries, $Q(A_{i})=(1,-1,0)$, $Q(B_{i})=(0,1,-1)$
and $Q(C_{i})=(-1,0,1)$, respectively, where $Q({\rm scalar})$ denotes the
$U(1)$ charges of the scalar. According to Eqs. (2.37) and (2.38), the
surviving zero modes in 4D are only

$\displaystyle
A_{1}^{\mathbb{A}},~{}B_{2}^{\mathbb{B}}~{}\text{and}~{}C_{3}^{\mathbb{C}}.$
(2.41)

Here, $\mathbb{A},~{}\mathbb{B},$ and $\mathbb{C}$ are the flavor indices, and
their zero mode functions surviving in 4D are written as

$\displaystyle\phi_{1}^{{\mathbb{A}},ab}$
$\displaystyle=\Theta^{A^{(1)},I_{ab}^{(1)}}(z_{1})\otimes\overline{\Theta^{A^{(2)},|I_{ab}^{(2)}|}(z_{2})}\otimes\overline{\Theta^{A^{(3)},|I_{ab}^{(3)}|}(z_{3})},$
$\displaystyle\phi_{2}^{{\mathbb{B}},bc}$
$\displaystyle=\overline{\Theta^{B^{(1)},|I_{bc}^{(1)}|}(z_{1})}\otimes\Theta^{B^{(2)},I_{bc}^{(2)}}(z_{2})\otimes\overline{\Theta^{B^{(3)},|I_{bc}^{(3)}|}(z_{3})},$
(2.42) $\displaystyle\phi_{3}^{{\mathbb{C}},ca}$
$\displaystyle=\overline{\Theta^{C^{(1)},|I_{ca}^{(1)}|}(z_{1})}\otimes\overline{\Theta^{C^{(2)},|I_{ca}^{(2)}|}(z_{2})}\otimes\Theta^{C^{(3)},I_{ca}^{(3)}}(z_{3}),$

where $A^{(r)}=0,1,\cdots|I_{ab}^{(r)}|-1$,
$B^{(r)}=0,1,\cdots,|I_{bc}^{(r)}|-1$, and
$C^{(r)}=0,1,\cdots,|I_{ca}^{(r)}|-1$ ($r=1,2,3$):
${\mathbb{A}}=0,1,\cdots|I_{ab}|-1$, ${\mathbb{B}}=0,1,\cdots,|I_{bc}|-1$, and
${\mathbb{C}}=0,1,\cdots,|I_{ca}|-1$. The normalization factor of
$\Theta^{A^{(1)},I^{(1)}_{ab}}(z_{1})$ is denoted as
$\mathcal{N}^{1}_{I^{(1)}_{ab}}$ for instance. From Eq. (2.21), these zero
mode functions are normalized as

$\displaystyle\int_{\mathbb{T}^{6}}d^{6}y\sqrt{g_{6}}\,\phi^{\mathbb{A},ab}_{1}\overline{\phi^{\mathbb{A}^{\prime},ab}_{1}}=\delta_{\mathbb{A},\mathbb{A}^{\prime}}.$
(2.43)

$\phi$’s are zero mode solutions for 10D SYM with the canonical kinetic term.
In the case with the NDBI action, there are corrections of fluxes to this zero
mode solution. Since the flux is constant to the coordinates of a six-
dimensional torus, the corrections are expected to change the normalization of
the matter Kähler metric. In this paper, for simplicity, we neglect higher
order interactions with derivatives in 4D theories such as $|A|^{2}|\partial
A|^{2}$ or $|\partial A|^{4}$, where $A$ is a 4D complex scalar in a chiral
matter multiplet.

##  3 SUSY effective action of $U(1)_{a}\times U(1)_{b}\times U(1)_{c}$ theory

In this section, we will exhibit 4D SUSY effective action derived from the 10D
NDBI action, focusing on the bosonic sector. As noted already, we assume to
start with 10D $U(3)$ gauge symmetry which is broken to $U(1)_{a}\times
U(1)_{b}\times U(1)_{c}$ by the background flux of Eq. (2.24).

We can read the 4D gauge couplings, Kähler metrics of the chiral matters and
scalar quartic couplings, after substituting the fields of Eqs. (2.39), (2.40)
and the metric (2.8) into the NDBI action (2.10). For later convenience, we
define closed string moduli [11]:

$\displaystyle s$ $\displaystyle\coloneqq e^{-\varphi}{\cal A}^{(1)}{\cal
A}^{(2)}{\cal A}^{(3)}=e^{-\varphi}{\rm Vol}(\mathbb{T}^{6}),$ (3.1)
$\displaystyle t_{i}$ $\displaystyle\coloneqq e^{-\varphi}{\cal
A}^{(i)}=e^{-\varphi}{\rm Vol}(\mathbb{T}_{i}^{2}),$ (3.2) $\displaystyle
U_{i}$ $\displaystyle\coloneqq i\overline{\tau^{(i)}},\quad
u_{i}\coloneqq\mathop{\mathrm{Re}}(U_{i})=\tau^{(i)}_{I},$ (3.3)

where $s$ is the 4D dilaton, and $t_{i}$ are the Kähler moduli. $U_{i}$ stand
for the complex structure moduli of ${\mathbb{T}}^{2}_{i}$ in the SUGRA basis.
In combination with axions descended from RR tensors, the above moduli
constitute the complexified dilaton $S$ and the Kähler moduli $T_{i}$. The
Kähler potential of these closed string moduli $K^{(0)}$ is given by

$\displaystyle
K^{(0)}=-\log(S+\overline{S})-\sum_{i=1}^{3}\log(T_{i}+\overline{T_{i}})-\sum_{i=1}^{3}\log(U_{i}+\overline{U_{i}}).$
(3.4)

4D effective action of chiral matters is written with these closed string
moduli as seen below. See Appendix A for details of the computation.

###  3.1 Gauge couplings

The gauge couplings of $U(1)_{a}\times U(1)_{b}\times U(1)_{c}$ are read from
the coefficient of the gauge kinetic term. The canonical kinetic term
$\mathcal{L}_{\text{4D}}\ni\int
d^{6}y\sqrt{g_{6}}\,e^{-\varphi}\mathop{\mathrm{tr}}\nolimits(f_{\mu\nu})^{2}$
gives the leading contribution without fluxes, whereas the flux-corrected
contributions come from333A contribution of $\int
d^{6}y\sqrt{g_{6}}\,e^{-\varphi}{\rm
tr}[\hat{F}_{i\overline{j}}f_{\mu\nu}{\hat{F}}^{\overline{j}i}f^{\mu\nu}]$ is
included because $[\hat{F}_{i\overline{j}},f_{\mu\nu}]=0.$
$\mathcal{L}_{\text{4D}}\ni\int d^{6}y\sqrt{g_{6}}\,e^{-\varphi}{\rm
tr}[({\hat{F}}_{i\overline{j}}{\hat{F}}^{\overline{j}i})(f_{\mu\nu})^{2}]$.
Here, $f_{\mu\nu}$ is the fluctuation of the 10D gauge field strength of the
$U(3)$ gauge symmetry with the 4D subscripts. The former kinetic term depends
on $e^{-\varphi}{\rm Vol}(\mathbb{T}^{6})=s$ and the latter includes $s\times
m^{2}$, where $m$ is moduli-dependent flux defined in Eq. (2.31). Thus, we
find

$\displaystyle S_{\text{NDBI}}\ni-\frac{1}{2\pi}\int
d^{4}x\sqrt{-g_{4}}\,\frac{1}{4g_{a}^{2}}(f^{a}_{\mu\nu})^{2},$ (3.5)

where $f_{\mu\nu}^{a}=\partial_{\mu}a_{\nu}^{a}-\partial_{\nu}a_{\mu}^{a}$ is
the field strength for the $U(1)_{a}$, and the gauge coupling for the
$U(1)_{a}$ group is

$\displaystyle\frac{1}{g_{a}^{2}}=$
$\displaystyle~{}s\biggl{[}1+\frac{1}{2}\sum_{i=1}^{3}(m^{(i)}_{a})^{2}\biggr{]}$
(3.6) $\displaystyle=$
$\displaystyle~{}s-t_{1}M^{(2)}_{a}M^{(3)}_{a}-t_{2}M^{(1)}_{a}M^{(3)}_{a}-t_{3}M^{(1)}_{a}M^{(2)}_{a}.$
(3.7)

In the second line, the SUSY condition (2.29) is used. The results for
$U(1)_{b}$ and $U(1)_{c}$ symmetries are similar to that of the $U(1)_{a}$.
This is a well-known result of the D-brane models [41, 42, 43, 50] and is
regarded as the real part of a corresponding holomorphic gauge coupling
$f_{a}$,

$\displaystyle\mathop{\mathrm{Re}}(f_{a})$
$\displaystyle=\frac{1}{g_{a}^{2}},$ (3.8) $\displaystyle f_{a}$
$\displaystyle=S-T_{1}M^{(2)}_{a}M^{(3)}_{a}-T_{2}M^{(1)}_{a}M^{(3)}_{a}-T_{3}M^{(1)}_{a}M^{(2)}_{a}.$
(3.9)

The expansion in fluxes is valid when
$s>t_{i}|M^{(j)}_{a}M^{(k)}_{a}|~{}(i\neq j\neq k\neq i)$. Then a gauge
coupling will become weak for large vacuum expectation values of moduli. It is
noted that terms dependent on $T_{i}$ can be positive contributions to the
gauge coupling when an induced D5-brane charge $-M^{(j)}_{a}M^{(k)}_{a}$,
which is carried by a magnetized D9-brane, is positive.444The induced charge
and its contribution to a holomorphic gauge coupling are seen from CS term on
a D9-brane, $\int_{D9}(C_{6}+\frac{1}{2}C_{2}\wedge f\wedge
f)\wedge\hat{F}\wedge\hat{F}$, where $C_{2}$ and $C_{6}$ are RR two-from and
six-form potential.

###  3.2 Kähler metric of chiral matters

The coefficient of a scalar kinetic term gives the Kähler metric for chiral
matter in SUSY theories. The kinetic terms with the leading contribution
without fluxes are read from $\mathcal{L}_{\text{4D}}\ni\int
d^{6}y\,\sqrt{g_{6}}e^{2\Phi-\varphi}\mathop{\mathrm{tr}}\nolimits(f_{\mu
i}f^{\mu i})$, whereas the next leading contributions with fluxes are roughly
given by a combination of $\mathcal{L}_{\text{4D}}\ni\int
d^{6}y\sqrt{g_{6}}\,e^{2\Phi-\varphi}\mathop{\mathrm{tr}}\nolimits(\hat{F}_{j\overline{k}}\hat{F}^{\overline{k}j}f_{\mu
i}f^{\mu i}+\hat{F}_{j\overline{k}}f_{\mu i}\hat{F}^{\overline{k}j}f^{\mu i})$
and similar terms. Here, $f_{\mu i}\coloneqq f_{\mu z^{i}}$ is the fluctuation
of 10D field strength and includes the 4D kinetic term of a scalar
fluctuation, e.g. $\partial_{\mu}A^{\mathbb{A}}_{i}$, where $A_{i}$ is given
in Eq. (2.40) with the intersection number (2.38).555We have generalized
$A_{1}$ to $A_{i}$ with any $i$.  A factor $e^{2\Phi}$ originates from the 4D
Einstein frame metric $g_{\mu\nu}=e^{-2\Phi}\tilde{g}_{\mu\nu}$ in the kinetic
term $\sqrt{-\tilde{g}_{4}}\times\tilde{g}^{\mu\nu}g^{i\overline{i}}f_{\mu
i}f_{\nu\overline{i}}$, where $\tilde{g}_{\mu\nu}$ is the Jordan frame metric,
$ds_{10}^{2}\ni
e^{2\Phi}g_{\mu\nu}dx^{\mu}dx^{\nu}\eqqcolon\tilde{g}_{\mu\nu}dx^{\mu}dx^{\nu}$
as in Eq. (2.4). For instance, we roughly estimate

$\displaystyle\int
d^{6}y\,\sqrt{g_{6}}e^{2\Phi-\varphi}\mathop{\mathrm{tr}}\nolimits(f_{\mu
i}f^{\mu i})$ $\displaystyle\sim
e^{2\Phi-\varphi}g^{i\overline{i}}|\partial_{\mu}A^{\mathbb{A}}_{i}|^{2}\int
d^{6}y\sqrt{g_{6}}\,|\phi_{i}^{\mathbb{A},ab}|^{2}$
$\displaystyle\sim\frac{2u_{i}}{t_{i}\mathop{\mathrm{Vol}}(\mathbb{T}^{6})}|\partial_{\mu}A^{\mathbb{A}}_{i}|^{2},$
(3.10)

for terms without fluxes and

$\displaystyle\int d^{6}y\,\sqrt{g_{6}}e^{2\Phi-\varphi}{\rm
tr}(\hat{F}_{j\overline{k}}\hat{F}^{\overline{k}j}f_{\mu i}f^{\mu i})$
$\displaystyle\sim
e^{2\Phi-\varphi}\hat{F}_{j\overline{k}}\hat{F}^{\overline{k}j}g^{i\overline{i}}|\partial_{\mu}A^{\mathbb{A}}_{i}|^{2}\int
d^{6}y\sqrt{g_{6}}\,|\phi_{i}^{\mathbb{A},ab}|^{2}$ $\displaystyle\sim
m^{2}\times\frac{2u_{i}}{t_{i}\mathop{\mathrm{Vol}}(\mathbb{T}^{6})}|\partial_{\mu}A^{\mathbb{A}}_{i}|^{2},$
(3.11)

for the flux-corrected terms with the moduli-dependent fluxes $m$ in Eq.
(2.31). Here, $\phi_{i}^{\mathbb{A},ab}$ is the zero mode function for $A_{i}$
in the magnetized extra dimension, and we used
$g^{i\overline{i}}=2e^{-\varphi}\frac{u_{i}}{t_{i}}$ and the normalization of
$\phi_{i}^{\mathbb{A},ab}$ in Eq. (2.43). In addition, let us rescale the
matter field as $A_{i}\to\alpha_{ab}^{(i)}A_{i}$ so that matter superpotential
becomes a holomorphic function of the moduli and the matter Kähler metric
results in a real function of the moduli [24], where

$\displaystyle\alpha^{(i)}_{\alpha\beta}=\frac{1}{\sqrt{2^{2}u_{i}}}\frac{\sqrt{\mathop{\mathrm{Vol}}(\mathbb{T}^{6})}}{(2^{3}u_{1}u_{2}u_{3})^{1/4}}\biggl{(}\frac{|I^{(i)}_{\alpha\beta}|}{\prod_{r\neq
i}|I^{(r)}_{\alpha\beta}|}\biggr{)}^{1/4},~{}~{}~{}\alpha,\beta=a,b,c,$ (3.12)

for $I_{ab}I_{bc}I_{ca}\neq 0$. Then, the metric for $A_{i}$,
$\mathcal{Z}^{i}_{ab}$, is obtained as

$\displaystyle S_{\text{NDBI}}\ni-\frac{1}{2\pi}\int
d^{4}x\sqrt{-g_{4}}\,\mathcal{Z}^{i}_{ab}|D_{\mu}A^{\mathbb{A}}_{i}|^{2},$
(3.13)

where
$D_{\mu}A^{\mathbb{A}}_{i}=(\partial_{\mu}+ia_{\mu}^{a}-ia_{\mu}^{b})A^{\mathbb{A}}_{i}$,
and

$\displaystyle\mathcal{Z}^{i}_{ab}=$ $\displaystyle
Z^{i}_{ab}\times\biggl{[}1-\frac{1}{6}\bigl{(}2m^{(j)}_{a}m^{(k)}_{a}+2m^{(j)}_{b}m^{(k)}_{b}+m^{(j)}_{a}m^{(k)}_{b}+m^{(j)}_{b}m^{(k)}_{a}\bigr{)}\biggr{]}\quad(i\neq
j\neq k\neq i)$ $\displaystyle=$ $\displaystyle
Z^{i}_{ab}\times\biggl{[}1-\frac{t_{i}}{6s}\bigl{(}2M^{(j)}_{a}M^{(k)}_{a}+2M^{(j)}_{b}M^{(k)}_{b}+M^{(j)}_{a}M^{(k)}_{b}+M^{(j)}_{b}M^{(k)}_{a}\bigr{)}\biggr{]}\quad(i\neq
j\neq k\neq i),$ (3.14) $\displaystyle Z^{i}_{ab}\coloneqq$
$\displaystyle\frac{2u_{i}}{t_{i}\mathop{\mathrm{Vol}}(\mathbb{T}^{6})}\bigl{(}\alpha^{(i)}_{ab}\bigr{)}^{2}=\frac{1}{2t_{i}}\biggl{(}\prod_{k=1}^{3}\frac{1}{\sqrt{2u_{k}}}\biggr{)}\sqrt{\frac{|I^{((i))}_{ab}|}{\prod_{j\neq
i}|I^{((j))}_{ab}|}}.$ (3.15)

Here, we used the SUSY condition in the computation. It is noted that
$Z^{(i)}_{ab}$ is the metric obtained 10D SYM with the canonical kinetic term
on the magnetized extra dimension [11, 24] and that the above
$\mathcal{Z}^{(i)}_{ab}$ in Eq. (3.14) is symmetric under exchange of $a$ and
$b$ and independent of labels of flavor. This is also rewritten with
complexified moduli and intersection numbers as

$\displaystyle\mathcal{Z}^{i}_{ab}=$ $\displaystyle
Z^{i}_{ab}\times\biggl{[}1+\frac{(T_{i}+\overline{T_{i}})}{6(S+\overline{S})}\bigl{(}I^{(j)}_{ab}I^{(k)}_{ab}-3M^{(j)}_{a}M^{(k)}_{a}-3M^{(j)}_{b}M^{(k)}_{b}\bigr{)}\biggr{]}\quad(i\neq
j\neq k\neq i),$ (3.16) $\displaystyle Z^{i}_{ab}$
$\displaystyle=\frac{1}{T_{i}+\overline{T_{i}}}\biggl{(}\prod_{k=1}^{3}\frac{1}{\sqrt{(U_{k}+\overline{U_{k}})}}\biggr{)}\sqrt{\frac{|I^{(i)}_{ab}|}{\prod_{j\neq
i}|I^{(j)}_{ab}|}}.$ (3.17)

The expansion in fluxes is valid when $s>t_{i}|M^{(j)}M^{(k)}|~{}(i\neq j\neq
k\neq i)$, and this is similar to the case of a gauge coupling. Then the
metric $\mathcal{Z}^{i}_{ab}$ can be positive definite in SUSY theories when
induced D5-brane charges, $-M^{(j)}_{a}M^{(k)}_{a}$ and
$-M^{(j)}_{b}M^{(k)}_{b}$, are positive, even if flux corrections become
large. This is because a sign of the product of intersection numbers,
$I^{(j)}_{ab}I^{(k)}_{ab}$, is always positive owing to a chirality of $A_{i}$
multiplet. A similar Kähler potential which depends on $S$ is obtained in type
II theories with string scattering amplitudes [41, 42, 43] and is found also
in Heterotic M-theory [52] with an effective field theory approach.

The Kähler metrics for the other fields are systematically given by the cyclic
replacement of the label of the tori and gauge groups.

###  3.3 Scalar quartic term in the F-term scalar potential

Let us check if the Kähler metric in the previous subsection is correct by
showing the scalar potential. We derive scalar quartic couplings in the F-term
potential from NDBI action and compare it with the SUGRA description. For
concreteness, we focus on $A_{1}B_{2}\overline{A_{1}}\overline{B_{2}}$ term
included in the potential. This is related to the Yukawa coupling in the
superpotential and hence is restricted by holomorphy. On the other hand, there
is another type of quartic terms of $|A_{1}|^{4}$ that is associated with D
term. The D-term scalar potential is less constrained than that of F term and
hence we do not discuss the details in this paper for simplicity.

The leading term in flux expansion of the F-term scalar potential which
consists of multiplication of the holomorphic function and its complex
conjugate one is estimated from

$\displaystyle 2\pi\mathcal{L}_{\text{4D}}\ni-V_{F}\ni 2\int
d^{6}y\sqrt{g_{6}}\,e^{4\Phi-\varphi}g^{i\overline{i}}g^{j\overline{j}}\mathop{\mathrm{tr}}\nolimits[a_{i},a_{j}][a_{\overline{i}},a_{\overline{j}}].$

Here, $V_{F}$ denotes the F-term scalar potential, and we drop the covariant
derivative on zero modes since we focus on a scalar quartic term.666It is
noted that
$D_{z_{i}}a_{z_{j}}=\partial_{z_{i}}a_{z_{j}}+i[\hat{A}_{z_{i}},a_{z_{j}}]=0$
for $i\neq j$ and
$D_{\overline{z}_{i}}a_{z_{i}}=\partial_{\overline{z}_{i}}a_{z_{i}}+i[\hat{A}_{\overline{z}_{i}},a_{z_{i}}]=0$
for zero modes [11, 24]. Terms proportional to $D_{z_{i}}a_{z_{i}}$ and
$D_{\overline{z}_{i}}a_{z_{j}}~{}(i\neq j)$ for zero modes will contribute to
4D action as a moduli-dependent Fayet-Illiopoulos D-term, which will be
vanishing if the SUSY condition is preserved.  A factor of $e^{4\Phi}$
originates from $\sqrt{\tilde{g}_{4}}$ in the 4D effective action with the
Einstein frame metric $g_{\mu\nu}=e^{-2\Phi}\tilde{g}_{\mu\nu}$. The term of
$A^{\mathbb{A}}_{1}B^{\mathbb{B}}_{2}\overline{A^{\mathbb{A}^{\prime}}_{1}}\overline{B^{\mathbb{B}^{\prime}}_{2}}$
including flux corrections arises from those proportional to
$[a_{1},a_{2}][a_{\overline{1}},a_{\overline{2}}]$,

$\displaystyle V_{F}\ni$
$\displaystyle~{}2\frac{e^{3\varphi}}{(\mathop{\mathrm{Vol}}(\mathbb{T}^{6}))^{2}}\,g^{1\overline{1}}g^{2\overline{2}}\biggl{[}1+\frac{1}{6}\Bigl{(}2m^{(1)}_{a}m^{(2)}_{a}+2m^{(1)}_{c}m^{(2)}_{c}+m^{(1)}_{a}m^{(2)}_{c}+m^{(1)}_{c}m^{(2)}_{a}\Bigr{)}\biggr{]}$
$\displaystyle~{}\times(\alpha^{(1)}_{ab})^{2}\times(\alpha^{(2)}_{bc})^{2}\times
A^{\mathbb{A}}_{1}B^{\mathbb{B}}_{2}\overline{A^{\mathbb{A}^{\prime}}_{1}}\overline{B^{\mathbb{B}^{\prime}}_{2}}\times\biggl{(}\int
d^{6}y\sqrt{g_{6}}\,\phi^{\mathbb{A},ab}_{1}\phi^{\mathbb{B},bc}_{2}\overline{\phi^{\mathbb{A}^{\prime},ab}_{1}}\overline{\phi^{\mathbb{B}^{\prime},bc}_{2}}\biggr{)}$
(3.18) $\displaystyle=$ $\displaystyle
A^{\mathbb{A}}_{1}B^{\mathbb{B}}_{2}\overline{A^{\mathbb{A}^{\prime}}_{1}}\overline{B^{\mathbb{B}^{\prime}}_{2}}\times\frac{2Z^{3}_{ca}}{\mathcal{Z}^{3}_{ca}}\frac{e^{3\varphi}}{(\mathop{\mathrm{Vol}}(\mathbb{T}^{6}))^{2}}g^{1\overline{1}}g^{2\overline{2}}(\alpha^{(1)}_{ab})^{2}(\alpha^{(2)}_{bc})^{2}\biggl{(}\int
d^{6}\sqrt{g_{6}}\,\phi^{\mathbb{A},ab}_{1}\phi^{\mathbb{B},bc}_{2}\overline{\phi^{\mathbb{A}^{\prime},ab}_{1}}\overline{\phi^{\mathbb{B}^{\prime},bc}_{2}}\biggr{)},$
(3.19)

where Eq. (3.14) is used, and
$(\alpha^{(1)}_{ab})^{2}\times(\alpha^{(2)}_{bc})^{2}$ comes from the
rescaling of $A_{1}^{\mathbb{A}}\to\alpha^{(1)}_{ab}A^{\mathbb{A}}_{1}$ and
$B^{\mathbb{B}}_{2}\to\alpha^{(2)}_{bc}B^{\mathbb{B}}_{2}$ for the SUGRA
basis. Since
$Z^{3}_{ca}=e^{2\Phi-\varphi}g^{3\overline{3}}(\alpha^{(3)}_{ca})^{2}$ and
$\mathop{\mathrm{Vol}}(\mathbb{T}^{6})=e^{-2\Phi+2\varphi}$, this potential is
also written as

$\displaystyle V_{F}\ni
A^{\mathbb{A}}_{1}B^{\mathbb{B}}_{2}\overline{A^{\mathbb{A}^{\prime}}_{1}}\overline{B^{\mathbb{B}^{\prime}}_{2}}\times\frac{e^{K^{(0)}}}{\mathcal{Z}^{3}_{ca}}\biggl{(}\sqrt{2}e^{-K^{(0)}/2}e^{3\Phi-\varphi}\frac{\alpha^{(1)}_{ab}\alpha^{(2)}_{bc}\alpha^{(3)}_{ca}}{\sqrt{g_{1\overline{1}}g_{2\overline{2}}g_{3\overline{3}}}}\biggr{)}^{2}\int
d^{6}y\sqrt{g_{6}}\phi^{\mathbb{A},ab}_{1}\phi^{\mathbb{B},bc}_{2}\overline{\phi^{\mathbb{A}^{\prime},ab}_{1}}\overline{\phi^{\mathbb{B}^{\prime},bc}_{2}}.$
(3.20)

Here, $e^{K^{(0)}}=1/(2^{7}st_{1}t_{2}t_{3}u_{1}u_{2}u_{3})$ and
$g^{i\overline{i}}=1/g_{i\overline{i}}$.

Before carrying out the integration of four zero mode functions, we introduce
a holomorphic Yukawa coupling $W_{\mathbb{A}\mathbb{B}\mathbb{C}}$ with an
integration of three zero mode functions, since the former integration is
written as the square of the absolute value of the latter one. As discussed in
Ref. [11], a holomorphic Yukawa coupling is expressed as

$\displaystyle W_{\mathbb{A}\mathbb{B}\mathbb{C}}\coloneqq$
$\displaystyle~{}\sqrt{2}e^{-K_{0}/2}\alpha^{(1)}_{ab}\alpha^{(2)}_{bc}\alpha^{(3)}_{ca}\frac{e^{3\Phi-\varphi}}{\sqrt{g_{1\overline{1}}g_{2\overline{2}}g_{3\overline{3}}}}\int
d^{6}y\sqrt{g_{6}}\,\phi^{\mathbb{A},ab}_{1}\phi^{\mathbb{B},bc}_{2}\phi^{\mathbb{C},ca}_{3}$
(3.21) $\displaystyle=$ $\displaystyle
2\prod_{r=1}^{3}W_{A^{(r)}B^{(r)}C^{(r)}},$ (3.22)

where holomorphic function of $W^{(r)}_{A^{(r)}B^{(r)}C^{(r)}}~{}(r=1,2,3)$ is
given by

$\displaystyle W_{A^{(1)}B^{(1)}C^{(1)}}\coloneqq$
$\displaystyle~{}\overline{\vartheta\begin{bmatrix}\frac{B^{(1)}|I^{(1)}_{ca}|-C^{(1)}|I^{(1)}_{bc}|+m^{(1)}I^{(1)}_{bc}I^{(1)}_{ca}}{|I^{(1)}_{ab}I^{(1)}_{bc}I^{(1)}_{ca}|}\\\
0\end{bmatrix}(0,i\overline{U_{1}}|I^{(1)}_{ab}I^{(1)}_{bc}I^{(1)}_{ca}|)},$
(3.23) $\displaystyle W_{A^{(2)}B^{(2)}C^{(2)}}\coloneqq$
$\displaystyle~{}\overline{\vartheta\begin{bmatrix}\frac{C^{(2)}|I^{(2)}_{ab}|-A^{(2)}|I^{(2)}_{ca}|+m^{(2)}|I^{(2)}_{ab}I^{(2)}_{ca}|}{|I^{(2)}_{ab}I^{(2)}_{bc}I^{(2)}_{ca}|}\\\
0\end{bmatrix}(0,i\overline{U_{2}}|I^{(2)}_{ab}I^{(2)}_{bc}I^{(2)}_{ca}|)},$
(3.24) $\displaystyle W_{A^{(3)}B^{(3)}C^{(3)}}\coloneqq$
$\displaystyle~{}\overline{\vartheta\begin{bmatrix}\frac{A^{(3)}|I^{(3)}_{bc}|-B^{(3)}|I^{(3)}_{ab}|+m^{(3)}|I^{(3)}_{ab}I^{(3)}_{bc}|}{|I^{(3)}_{ab}I^{(3)}_{bc}I^{(3)}_{ca}|}\\\
0\end{bmatrix}(0,i\overline{U_{3}}|I^{(3)}_{ab}I^{(3)}_{bc}I^{(3)}_{ca}|)},$
(3.25)

and

$\displaystyle A^{(1)}=B^{(1)}+C^{(1)}+m^{(1)}|I^{(1)}_{bc}|,\quad
m^{(1)}=0,1,\ldots,I^{(1)}_{ab}-1,$ $\displaystyle
B^{(2)}=A^{(2)}+C^{(2)}+m^{(2)}|I^{(2)}_{ca}|,\quad
m^{(2)}=0,1,\ldots,I^{(2)}_{bc}-1,$ (3.26) $\displaystyle
C^{(3)}=A^{(3)}+B^{(3)}+m^{(3)}|I^{(3)}_{ab}|,\quad
m^{(3)}=0,1,.\ldots,I^{(3)}_{ca}-1.$

It is noted that this coupling depends on the complex structure moduli $U_{i}$
via the argument of the theta function. The coefficient in Eq. (3.21) is
chosen such that the Yukawa coupling becomes a holomorphic function consistent
with the SUGRA formulation as noted already (see also Appendix B).

To evaluate the zero mode integral in the rhs of Eq. (3.20), we first rewrite
the integral as

$\displaystyle\int_{\mathbb{T}^{6}}d^{6}y\sqrt{g_{6}}\,\phi^{\mathbb{A},ab}_{1}\phi^{\mathbb{B},bc}_{2}\overline{\phi^{\mathbb{B}^{\prime},bc}}\overline{\phi^{\mathbb{A}^{\prime},ab}}$
$\displaystyle=\int_{\mathbb{T}^{6}}d^{6}y\sqrt{g_{6}}\,\phi^{\mathbb{A},ab}_{1}(y)\phi^{\mathbb{B},bc}_{2}(y)\int_{\mathbb{T}^{6}}d^{6}y^{\prime}\sqrt{g_{6}}\,\overline{\phi^{\mathbb{B}^{\prime},bc}}(y^{\prime})\overline{\phi^{\mathbb{A}^{\prime},ab}}(y^{\prime})\times\frac{1}{\sqrt{g_{6}}}\delta(y-y^{\prime})$
(3.27)

and use the following completeness relation [12]777The integration on the
third torus is straightforward even without the completeness relation. We
obtain the result of $|W_{A^{(3)}B^{(3)}C^{(3)}}|^{2}$ explicitly consistent
with the SUGRA formulation after the integration because both
$\phi^{\mathbb{A},ab}_{1}$ and $\phi^{\mathbb{B},bc}_{2}$ have the (almost)
antiholomorphic solution on the third torus. :

$\displaystyle\sum_{n\geq
0,{\mathbb{C}}}\Xi_{n}^{{\mathbb{C}},ca}(y)\overline{\Xi_{n}^{{\mathbb{C}},ca}(y^{\prime})}=\frac{1}{\sqrt{g_{6}}}\delta(y-y^{\prime}).$
(3.28)

Here, $\Xi_{n}^{\mathbb{C},ca}$ are the eigenfunctions of the Dirac equation
with the magnetic flux of $I^{(i)}_{ca}=M^{(i)}_{c}-M^{(i)}_{a}$ on each
torus, and $n$ denotes the label of the Landau level including the zero mode.
The degeneracy is given by $|I_{ca}|$. These functions are assumed to be
normalized as

$\displaystyle\int_{\mathbb{T}^{6}}d^{6}y\sqrt{g_{6}}\,\overline{\Xi_{m}^{{\mathbb{C}}^{\prime},ca}}\Xi_{n}^{{\mathbb{C}},ca}=\delta_{m,n}\delta_{{\mathbb{C}},{\mathbb{C}}^{\prime}}.$
(3.29)

Massive modes in Landau level are orthogonal to zero modes, so Eq. (3.27)
becomes

$\displaystyle\sum_{{\mathbb{C}}}\int_{\mathbb{T}^{6}}d^{6}y\sqrt{g_{6}}\,\phi^{\mathbb{A},ab}_{1}(y)\phi^{\mathbb{B},bc}_{2}(y)\phi^{\mathbb{C},ca}_{3}(y)\times\int_{\mathbb{T}^{6}}d^{6}y^{\prime}\sqrt{g_{6}}\,\overline{\phi^{\mathbb{A}^{\prime},ab}}(y^{\prime})\overline{\phi^{\mathbb{B}^{\prime},bc}}(y^{\prime})\overline{\phi^{\mathbb{C},ca}_{3}}(y^{\prime}).$

Thus, this is evaluated as

$\displaystyle\int_{\mathbb{T}^{6}}d^{6}y\sqrt{g_{6}}\,\phi^{\mathbb{A},ab}_{1}\phi^{\mathbb{B},bc}_{2}\overline{\phi^{\mathbb{B}^{\prime},bc}}\overline{\phi^{\mathbb{A}^{\prime},ab}}$
$\displaystyle=\frac{1}{2^{2}}\frac{\sqrt{2^{3}u_{1}u_{2}u_{3}}}{\mathop{\mathrm{Vol}}(\mathbb{T}^{6})}\frac{\sqrt{I_{ab}I_{bc}I_{ca}}}{I^{(1)}_{ab}I^{(2)}_{bc}I^{(3)}_{ca}}\sum_{\mathbb{C}}W_{\mathbb{A}\mathbb{B}\mathbb{C}}\overline{W_{\mathbb{A}^{\prime}\mathbb{B}^{\prime}\mathbb{C}}}$
$\displaystyle=\biggl{(}\sqrt{2}e^{-K^{(0)}/2}e^{3\Phi-\varphi}\frac{\alpha^{(1)}_{ab}\alpha^{(2)}_{bc}\alpha^{(3)}_{ca}}{\sqrt{g_{1\overline{1}}g_{2\overline{2}}g_{3\overline{3}}}}\biggr{)}^{-2}\sum_{\mathbb{C}}W_{\mathbb{A}\mathbb{B}\mathbb{C}}\overline{W_{\mathbb{A}^{\prime}\mathbb{B}^{\prime}\mathbb{C}}},$
(3.30)

where flavor labels including $\mathbb{A}^{\prime}$ and $\mathbb{B}^{\prime}$
satisfy Eq. (3.26) and a factor $1/2^{2}$ comes from the normalization 2 of
$W_{\mathbb{A}\mathbb{B}\mathbb{C}}$ in Eq. (3.22). Using this result, Eq.
(3.20) becomes

$\displaystyle V_{F}\ni\frac{e^{K^{(0)}}}{\mathcal{Z}^{3}_{ca}}\times
A^{\mathbb{A}}_{1}B^{\mathbb{B}}_{2}\overline{A^{\mathbb{A}^{\prime}}_{1}}\overline{B^{\mathbb{B}^{\prime}}_{2}}\times\sum_{\mathbb{C}}W_{\mathbb{A}\mathbb{B}\mathbb{C}}\overline{W_{\mathbb{A}^{\prime}\mathbb{B}^{\prime}\mathbb{C}}}.$
(3.31)

Suppose that the superpotential is given by

$\displaystyle
W=\sum_{\mathbb{A},\mathbb{B},\mathbb{C}}W_{\mathbb{ABC}}A^{\mathbb{A}}_{1}B^{\mathbb{B}}_{2}C^{\mathbb{C}}_{3},$
(3.32)

where $W_{\mathbb{A}\mathbb{B}\mathbb{C}}$ is the holomorphic Yukawa coupling
defined in Eq. (3.21). This superpotential is discussed also in Refs. [11,
24]. With this superpotential, the above scalar potential turns out to be
written based on the SUGRA formulation:

$\displaystyle V_{F}$
$\displaystyle\ni\frac{e^{K^{(0)}}}{\mathcal{Z}^{3}_{ca}}\sum_{\mathbb{C}}\bigl{(}\partial_{C_{3}^{\mathbb{C}}}W)\overline{\bigl{(}\partial_{C_{3}^{\mathbb{C}}}W\bigr{)}}$
(3.33) $\displaystyle\ni\frac{e^{K^{(0)}}}{\mathcal{Z}^{3}_{ca}}\times
A^{\mathbb{A}}_{1}B^{\mathbb{B}}_{2}\overline{A^{\mathbb{A}^{\prime}}_{1}}\overline{B^{\mathbb{B}^{\prime}}_{2}}\times\sum_{\mathbb{C}}W_{\mathbb{ABC}}\overline{W_{\mathbb{A}^{\prime}\mathbb{B}^{\prime}\mathbb{C}}}.$
(3.34)

Thus, the Kähler metric derived from the NDBI action is consistent with the
scalar potential based on the SUGRA formulation.

##  4 Summary and discussions

4D $\mathcal{N}=1$ supersymmetric effective action is systematically derived
from the 10D NDBI action on a six-dimensional magnetized torus. The 10D action
is expanded in the series of fluxes up to ${\cal O}(F^{4})$ with a symmetrized
trace prescription. The eigenfunctions of the Dirac equations on the torus are
explicitly written with using the Jacobi theta function and contribute to the
4D effective action as an integrand in the extra dimension. We calculated the
flux corrections systematically to the matter Kähler metrics, the gauge
couplings and the holomorphic superpotential via scalar quartic couplings in
the F-term potential. Our finding is a new flux correction appearing in the
Kähler metrics of Eqs. (3.16) and (3.17) in a flavor-independent way. The new
matter Kähler metric depends on the fluxes, 4D dilaton, Kähler moduli, and
complex structure moduli and will be always positive definite if an induced RR
charge of the D-branes on which matters are living are positive. A
contribution of the new matter Kähler metric to the F-term scalar potential
turns out to be consistent with the SUGRA formula. The gauge coupling in Eq.
(3.9) and the holomorphic superpotential in Eq. (3.32) are consistent with the
previous works.

Phenomenologically, matter Kähler metrics contribute to physical Yukawa
couplings in a flavor-independent way. If fluxes on a stack of D-branes on
which quarks in the Standard Model are living are different from those on
which leptons are living, differences in their Kähler metrics will be induced
and could explain the mass difference between quarks and leptons. If fluxes
are common both in the quark sector and lepton one as in the Pati-Salam like
D-brane models, such an explanation will be difficult in toroidal
compactifications. As for SUSY breaking effects to chiral matters, even if
vacuum expectation values of F components of $T_{i}$ and $U_{i}$ are much
smaller than that of $S$, the flux corrections depending on $S$ in the Kähler
metrics can generate sizable soft terms in comparison with cases without the
corrections [43].

In this work, we consider the SUSY condition of (2.29). However, if the
configuration of D9-branes is supersymmetric, this condition will be modified
as

$\displaystyle\sum_{i}\frac{M^{(i)}_{\alpha}}{\mathcal{A}^{(i)}}=\prod_{j=1}^{3}\frac{M^{(j)}_{\alpha}}{\mathcal{A}^{(j)}},\quad\alpha=a,b,c.$
(4.1)

It will be worthwhile studying the D-term potential including the Fayet-
Illiopoulos term. Further, imposing this SUSY condition on the D9-brane action
requires higher order corrections to the Lagrangian. For instance, ${\cal
O}(F^{6})$ terms are required for the SUSY condition when we focus on ${\cal
O}(F^{4})$ terms as in this paper. We could identify a part of ${\cal
O}(F^{6})$ then. To include higher order interactions with derivatives can be
important to study swampland conjectures with effective field theories.

## Acknowledgments

This work is supported in part by JSPS Grant-in-Aid for Scientific Research
KAKENHI Grant No. JP20J11901 (Y.A.) and MEXT KAKENHI Grant No. JP19H04605
(T.K.).

##  Appendix A Details of the calculations

In this section, we show the details of the calculations of the NDBI action
and use the action in Eq. (2.10) and the metric ansatz in Eq. (2.8). Using
these, we read the gauge couplings, matter kinetic terms, and quartic terms of
the scalar potential for the fluctuations in Eqs. (2.39) and (2.40) around the
background fluxes in Eq. (2.24). Then, the 10D field strength in Eq. (2.10) is
expressed as

$\displaystyle F_{MN}=\hat{F}_{MN}+f_{MN},$ (A.1)

where $\hat{F}_{MN}$ denotes the background flux with the background gauge
field $\hat{A}_{z_{i}}$ and the fluctuation $f_{MN}$ is given by

$\displaystyle f_{\mu\nu}=\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu},\quad
f_{\mu
i}=\partial_{\mu}a_{z_{i}}+i[a_{\mu},a_{z_{i}}]+i[a_{\mu},\hat{A}_{z_{i}}],$
(A.2) $\displaystyle
f_{ij}=\partial_{z_{i}}a_{z_{j}}+i[\hat{A}_{z_{i}},a_{z_{j}}]-\partial_{z_{j}}a_{z_{i}}-i[\hat{A}_{z_{j}},a_{z_{i}}]+i[a_{z_{i}},a_{z_{j}}],$
(A.3) $\displaystyle
f_{i\overline{j}}=\partial_{z_{i}}a_{\overline{z}_{j}}+i[\hat{A}_{z_{i}},a_{\overline{z}_{j}}]-\partial_{\overline{z}_{j}}a_{z_{i}}-i[\hat{A}_{\overline{z}_{j}},a_{z_{i}}]+i[a_{z_{i}},a_{\overline{z}_{j}}].$
(A.4)

Here, $\hat{A}_{z_{i}}$ denotes the background gauge field, and $a_{M}$
denotes the fluctuation. In addition, let us introduce the following quantity
for simplicity:

$\displaystyle\hat{G}_{j}=g^{j\overline{j}}\hat{F}_{j\overline{j}}=i\frac{2u_{j}}{\ell_{s}^{2}\mathcal{A}^{(j)}}\frac{\pi}{u_{j}}M^{(i)}=\frac{i}{2\pi\alpha^{\prime}}m^{(j)};\quad
m^{(j)}\coloneqq\frac{M^{(j)}}{\mathcal{A}^{(j)}},$ (A.5)

where $j=1,2,3$, the summation with respect to $j$ is not taken, and $M^{(j)}$
is given by Eq. (2.24). The SUSY condition (2.27) is rewritten as the
condition of $\hat{G}_{j}$ as

$\displaystyle\hat{G}_{1}+\hat{G}_{2}+\hat{G}_{3}=0.$ (A.6)

In the following parts, we focus just on $f_{\mu\nu}$,
$\partial_{\mu}a_{z_{i}}\in f_{\mu i}$, $i[a_{z_{i}},a_{z_{j}}]\in f_{ij}$ and
$i[a_{z_{i}},a_{\overline{z}_{j}}]\in f_{i\overline{j}}$ to calculate the
effective action. Derivative terms of
$D_{z_{i}}a_{z_{j}}=\partial_{z_{i}}a_{z_{j}}+i[\hat{A}_{z_{i}},a_{z_{j}}]~{}(i\neq
j)$ and
$D_{\overline{z}_{i}}a_{z_{i}}=\partial_{\overline{z}_{i}}a_{z_{i}}+i[\hat{A}_{\overline{z}_{i}},a_{z_{i}}]$
are vanishing for zero modes [11, 24]. Terms proportional to
$D_{z_{i}}a_{z_{i}}$ and $D_{\overline{z}_{i}}a_{z_{j}}~{}(i\neq j)$ for zero
modes will contribute to 4D action as a moduli-dependent Fayet-Illiopoulos
D-term, which will be vanishing if the SUSY condition is preserved.

###  A.1 Gauge couplings

The gauge coupling is read from the coefficient of the gauge kinetic term. Due
to the index structure of Eq. (2.10), only its third and forth terms in
$\mathcal{O}(F^{4})$ contribute to the gauge kinetic terms, and then the
expansion of the NDBI action is calculated as

$\displaystyle 2\pi\mathcal{L}_{\text{NDBI}}\ni$ $\displaystyle-\int
d^{6}y\sqrt{g_{6}}\,\frac{e^{4\Phi-\varphi}}{4}e^{-4\Phi}\biggl{[}\mathop{\mathrm{tr}}\nolimits
f_{\mu\nu}f^{\mu\nu}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2\times\frac{(2\pi\alpha^{\prime})^{2}}{8}\frac{1}{3}\mathop{\mathrm{tr}}\nolimits\Bigl{(}2\hat{F}_{j\overline{k}}\hat{F}^{j\overline{k}}f_{\mu\nu}f^{\mu\nu}+\hat{F}_{j\overline{k}}f_{\mu\nu}\hat{F}^{j\overline{k}}f^{\mu\nu}\Bigr{)}\biggr{]}$
(A.7) $\displaystyle=$ $\displaystyle-\int
d^{6}y\sqrt{g_{6}}\,\frac{e^{-\varphi}}{4}\biggl{[}\mathop{\mathrm{tr}}\nolimits
f_{\mu\nu}f^{\mu\nu}-\frac{(2\pi\alpha^{\prime})^{2}}{6}\sum_{k}\mathop{\mathrm{tr}}\nolimits\Bigl{(}2\hat{G}_{k}\hat{G}_{k}f_{\mu\nu}f^{\mu\nu}+\hat{G}_{k}f_{\mu\nu}\hat{G}_{k}f^{\mu\nu}\Bigr{)}\biggr{]},$
(A.8)

where we used the fact that the background flux is diagonal,
$\hat{F}_{j\overline{k}}=\hat{F}_{k\overline{k}}\delta_{kj}$. Since these
fluxes are assumed to be Abelian, the Lagrangian reduces to

$\displaystyle 2\pi\mathcal{L}_{\text{NDBI}}$ $\displaystyle\ni-\int
d^{6}y\sqrt{g_{6}}\frac{e^{-\varphi}}{4}\mathop{\mathrm{tr}}\nolimits\biggl{[}\Bigl{(}1-\frac{(2\pi\alpha^{\prime})^{2}}{2}\sum_{k}\hat{G}_{k}\hat{G}_{k}\Bigr{)}f_{\mu\nu}f^{\mu\nu}\biggr{]}$
(A.9)
$\displaystyle=-\frac{s}{4}\mathop{\mathrm{tr}}\nolimits\biggl{[}\Bigl{(}1+\frac{1}{2}\sum_{k}(m^{(k)})^{2}\Bigr{)}f_{\mu\nu}f^{\mu\nu}\biggr{]},$
(A.10)

where we used $\int d^{6}y\sqrt{g_{6}}e^{-\varphi}=s$.

###  A.2 Kinetic terms

The scalar kinetic terms come from those proportional to $f_{\mu
i}f_{\nu\overline{i}}g^{\mu\nu}g^{i\overline{i}}$. Such terms including flux
corrections are given by

$\displaystyle 2\pi\mathcal{L}_{\text{NDBI}}\ni$ $\displaystyle-\int
d^{6}y\sqrt{g_{6}}\,\frac{e^{2\Phi-\varphi}}{4}\biggl{[}4\sum_{i}\mathop{\mathrm{tr}}\nolimits
f_{\mu i}f_{\nu\overline{i}}g^{\mu\nu}g^{i\overline{i}}$
$\displaystyle~{}-\frac{(2\pi\alpha^{\prime})^{2}}{3}Z_{K1}-\frac{(2\pi\alpha^{\prime})^{2}}{6}Z_{K_{2}}+\frac{(2\pi\alpha^{\prime})^{2}}{12}Z_{K3}+\frac{(2\pi\alpha^{\prime})^{2}}{24}Z_{K_{4}}\biggr{]},$
(A.11)

where

$\displaystyle Z_{K1}=$
$\displaystyle-\sum_{i}g^{i\overline{i}}g^{\mu\nu}\mathop{\mathrm{tr}}\nolimits\bigl{[}2\hat{G}_{i}\hat{G}_{i}(f_{\mu
i}f_{\nu\overline{i}}+f_{\mu\overline{i}}f_{\nu i})+4\hat{G}_{i}f_{\mu
i}\hat{G}_{i}f_{\nu\overline{i}}\bigr{]},$ (A.12) $\displaystyle Z_{K2}=$
$\displaystyle-4\sum_{i}g^{i\overline{i}}g^{\mu\nu}\mathop{\mathrm{tr}}\nolimits\bigl{[}\hat{G}_{i}\hat{G}_{i}(f_{\mu
i}f_{\nu\overline{i}}+f_{\mu\overline{i}}f_{\nu i})\bigr{]},$ (A.13)
$\displaystyle Z_{K3}=$
$\displaystyle-8\sum_{i}\sum_{k}g^{i\overline{i}}g^{\mu\nu}\mathop{\mathrm{tr}}\nolimits\bigl{[}\hat{G}_{k}\hat{G}_{k}(f_{\mu
i}f_{\nu\overline{i}}+f_{\mu\overline{i}}f_{\nu i})\bigr{]},$ (A.14)
$\displaystyle Z_{K4}=$
$\displaystyle-16\sum_{i}\sum_{k}g^{i\overline{i}}g^{\mu\nu}\mathop{\mathrm{tr}}\nolimits\bigl{[}\hat{G}_{k}f_{\mu
i}\hat{G}_{k}f_{\nu\overline{i}}\bigr{]}.$ (A.15)

####  A.2.1 Kähler metric of charged matters

With the background and the fluctuations substituted into the above equations,
it turns out that the Kähler metric of chiral matter $A_{i}$ is given by

$\displaystyle 2\pi\mathcal{L}_{\text{NDBI}}\ni$
$\displaystyle-\frac{2u_{i}}{t_{i}\mathop{\mathrm{Vol}}(\mathbb{T}^{6})}\biggl{[}1-\frac{t_{i}}{6s}\bigl{(}2M^{(j)}_{a}M^{(k)}_{a}+2M^{(j)}_{b}M^{(k)}_{b}+M^{(j)}_{a}M^{(k)}_{b}+M^{(j)}_{b}M^{(k)}_{a}\bigr{)}\biggr{]}|\partial_{\mu}A_{i}|^{2}$
(A.16)

with $i\neq j\neq k\neq i$ and $i=1$ for a fixed choice of intersection number
in this paper. The Kähler metrics for the other fields are systematically
given by the cyclic replacement of the label of the tori and gauge groups.

####  A.2.2 Kähler metric of open string moduli

A diagonal part of gauge fluctuation $a_{i}$ is open string modulus
$a_{i}^{b}\coloneqq a_{i}^{bb}$. Its Kähler metric can be read from Eq.
(A.11),

$\displaystyle 2\pi\mathcal{L}_{\text{NDBI}}$
$\displaystyle\ni-\frac{2}{(2t_{i})(2u_{i})}\biggl{[}1-\frac{t_{i}}{s}M^{(j)}_{b}M^{(k)}_{b}\biggr{]}|\partial_{\mu}a^{b}_{i}|^{2},$
(A.17) $\displaystyle{\cal Z}_{bb}^{i}$
$\displaystyle=\frac{2}{(T_{i}+\overline{T}_{i})(U_{i}+\overline{U}_{i})}\biggl{[}1-\frac{(T_{i}+\overline{T}_{i})}{(S+\overline{S})}M^{(j)}_{b}M^{(k)}_{b}\biggr{]}.$
(A.18)

It is noted that a flux correction in this result is obtained also by
replacing $M_{a}$ with $M_{b}$. This matches the result discussed in Refs.
[53, 41, 42, 43]. The positivity condition on the kinetic term of the open
string modulus is same as that of the gauge coupling.

###  A.3 Quartic terms

Scalar quartic terms originate from those including
$g^{i\overline{i}}g^{j\overline{j}}\bigl{(}f_{ij}f_{\overline{i}\overline{j}}+f_{i\overline{j}}f_{\overline{i}j}\bigr{)}$,
where $f_{ij}\coloneqq i[a_{i},a_{j}]$ and $f_{i\overline{j}}\coloneqq
i[a_{i},a_{\overline{j}}]$. We can read such terms from NDBI action,

$\displaystyle 2\pi\mathcal{L}_{\text{NDBI}}\ni$ $\displaystyle-\int
d^{6}y\sqrt{g_{6}}\,\frac{e^{4\Phi-\varphi}}{4}\biggl{[}\sum_{i,j}2g^{i\overline{i}}g^{j\overline{j}}\bigl{(}\mathop{\mathrm{tr}}\nolimits
f_{ij}f_{\overline{i}\overline{j}}+\mathop{\mathrm{tr}}\nolimits
f_{i\overline{j}}f_{\overline{i}j}\bigr{)}$
$\displaystyle~{}-\frac{(2\pi\alpha^{\prime})^{2}}{3}K_{1}-\frac{(2\pi\alpha^{\prime})^{2}}{6}K_{2}+\frac{(2\pi\alpha^{\prime})^{2}}{12}K_{3}+\frac{(2\pi\alpha^{\prime})^{2}}{24}K_{4}\biggr{]},$
(A.19)

and $K_{p=1,2,3,4}$ are decomposed to two parts: one is $K_{p,F}$ containing
$f_{ij}f_{\overline{i}\overline{j}}$ and the other is $K_{p,D}$ containing
$f_{i\overline{j}}f_{\overline{i}j}$. Explicitly they are given by

$\displaystyle K_{1,F}=$
$\displaystyle\sum_{i,j}g^{i\overline{i}}g^{j\overline{j}}\mathop{\mathrm{tr}}\nolimits\Bigl{\\{}2\hat{G}_{i}\hat{G}_{i}(f_{ij}f_{\overline{ji}}+f_{\overline{ij}}f_{ji})+4\hat{G}_{i}f_{ij}\hat{G}_{i}f_{\overline{ji}}$
$\displaystyle~{}-(\hat{G}_{i}\hat{G}_{j}+\hat{G}_{j}\hat{G}_{i})(f_{ij}f_{\overline{ji}}+f_{\overline{ij}}f_{ji})\Bigr{\\}},$
(A.20) $\displaystyle K_{1,D}=$
$\displaystyle\sum_{i,j}g^{i\overline{i}}g^{j\overline{j}}\mathop{\mathrm{tr}}\nolimits\Bigl{\\{}2\hat{G}_{i}\hat{G}_{i}(f_{j\overline{i}}f_{i\overline{j}}+f_{\overline{j}i}f_{\overline{i}j})+4\hat{G}_{i}f_{i\overline{j}}\hat{G}_{i}f_{j\overline{i}}$
$\displaystyle~{}+(\hat{G}_{i}\hat{G}_{j}+\hat{G}_{j}\hat{G}_{i})(f_{i\overline{j}}f_{j\overline{i}}+f_{\overline{i}j}f_{\overline{j}i})\Bigr{\\}},$
(A.21) $\displaystyle K_{2,F}=$
$\displaystyle\sum_{i,j}g^{i\overline{i}}g^{j\overline{j}}\mathop{\mathrm{tr}}\nolimits\Bigl{\\{}4\hat{G}_{i}\hat{G}_{i}(f_{ij}f_{\overline{ji}}+f_{\overline{ij}}f_{ji})-4\hat{G}_{i}f_{ij}\hat{G}_{j}f_{\overline{ji}}\Bigr{\\}},$
(A.22) $\displaystyle K_{2,D}=$
$\displaystyle\sum_{i,j}g^{i\overline{i}}g^{j\overline{j}}\mathop{\mathrm{tr}}\nolimits\Bigl{\\{}4\hat{G}_{i}\hat{G}_{i}(f_{i\overline{j}}f_{j\overline{i}}+f_{\overline{i}j}f_{\overline{j}i})+2(\hat{G}_{i}f_{i\overline{j}}\hat{G}_{j}f_{j\overline{i}}+\hat{G}_{i}f_{\overline{i}j}\hat{G}_{j}f_{\overline{j}i})\Bigr{\\}},$
(A.23) $\displaystyle K_{3,F}=$
$\displaystyle\sum_{i,j}g^{i\overline{i}}g^{j\overline{j}}\mathop{\mathrm{tr}}\nolimits\Bigl{\\{}4\bigl{(}\sum_{k}\hat{G}_{k}\hat{G}_{k}\bigr{)}(f_{ij}f_{\overline{ji}}+f_{\overline{ij}}f_{ji})\Bigr{\\}},$
(A.24) $\displaystyle K_{3,D}=$
$\displaystyle\sum_{i,j}g^{i\overline{i}}g^{j\overline{j}}\mathop{\mathrm{tr}}\nolimits\Bigl{\\{}4\bigl{(}\sum_{k}\hat{G}_{k}\hat{G}_{k}\bigr{)}(f_{i\overline{j}}f_{j\overline{i}}+f_{\overline{i}j}f_{\overline{j}i})$
$\displaystyle~{}+4\bigl{(}\hat{G}_{i}f_{i\overline{i}}\hat{G}_{j}f_{j\overline{j}}+\hat{G}_{i}\hat{G}_{j}f_{j\overline{j}}f_{i\overline{i}}+f_{i\overline{i}}\hat{G}_{i}f_{j\overline{j}}\hat{G}_{j}+\hat{G}_{i}f_{i\overline{i}}f_{j\overline{j}}\hat{G}_{j}\bigr{)}\Bigr{\\}}$
(A.25) $\displaystyle K_{4,F}=$
$\displaystyle\sum_{i,j}8g^{i\overline{i}}g^{j\overline{j}}\mathop{\mathrm{tr}}\nolimits\Bigl{\\{}\sum_{k}\hat{G}_{k}f_{ij}\hat{G}_{k}f_{\overline{ji}}\Bigr{\\}},$
(A.26) $\displaystyle K_{4,D}=$
$\displaystyle\sum_{i,j}g^{i\overline{i}}g^{j\overline{j}}\mathop{\mathrm{tr}}\nolimits\Bigr{\\{}8\sum_{k}\hat{G}_{k}f_{i\overline{j}}\hat{G}_{k}f_{j\overline{i}}+4\bigl{(}\hat{G}_{i}\hat{G}_{j}f_{i\overline{i}}f_{j\overline{j}}$
$\displaystyle~{}+\hat{G}_{i}f_{j\overline{j}}f_{i\overline{i}}\hat{G}_{j}+f_{i\overline{i}}\hat{G}_{j}\hat{G}_{i}f_{j\overline{j}}+f_{i\overline{i}}f_{j\overline{j}}\hat{G}_{i}\hat{G}_{j}\bigr{)}\Bigr{\\}}.$
(A.27)

If we want to get the specific quartic coupling such as
$A^{\mathbb{A}}_{1}B^{\mathbb{B}}_{2}\overline{B^{\mathbb{B}^{\prime}}_{2}}\overline{A^{\mathbb{A}^{\prime}}_{1}}$,
one needs to choose a term with a fixed index like
$f_{12}f_{\overline{1}\overline{2}}$.

####  A.3.1 F-term potential

The terms containing $f_{ij}~{}(i\neq j)$ contribute to the scalar F-term
potential. When the background fluxes satisfy the SUSY condition of Eq. (2.27)
or (A.6), we can show that $K_{p,D}~{}(p=1,2,3,4)$ does not include
$f_{i\overline{j}}~{}(i\neq j)$ and hence does not contribute to the F-term
scalar potential. In the leading contribution in the flux expansion, however,
$\mathop{\mathrm{tr}}\nolimits[a_{i},a_{\overline{j}}][a_{j},a_{\overline{i}}]$
is shown to have
$\mathop{\mathrm{tr}}\nolimits[a_{i},a_{j}][a_{\overline{i}},a_{\overline{j}}]$
contributing to the F-term potential through the Jacobi identity as [54]

$\displaystyle\mathop{\mathrm{tr}}\nolimits[a_{i},a_{\overline{j}}][a_{j},a_{\overline{i}}]+\mathop{\mathrm{tr}}\nolimits[a_{i},a_{\overline{i}}][a_{j},a_{\overline{j}}]+\mathop{\mathrm{tr}}\nolimits[a_{i},a_{j}][a_{\overline{i}},a_{\overline{j}}]=0.$
(A.28)

Then, the F-term potential from NDBI action is given by

$\displaystyle 2\pi\mathcal{L}_{\text{NDBI}}\ni$
$\displaystyle~{}-\frac{1}{4}\int
d^{6}y\sqrt{g_{6}}\,e^{4\Phi-\varphi}\sum_{i<j}g^{i\overline{i}}g^{j\overline{j}}\mathop{\mathrm{tr}}\nolimits\biggl{\\{}8f_{ij}f_{\overline{i}\overline{j}}-\frac{4(2\alpha^{\prime})^{2}}{3}\Bigl{[}\hat{G}_{i}f_{ij}\hat{G}_{j}f_{\overline{i}\overline{j}}+\hat{G}_{j}f_{ij}\hat{G}_{i}f_{\overline{i}\overline{j}}$
$\displaystyle~{}+(\hat{G}_{i}\hat{G}_{j}+\hat{G}_{j}\hat{G}_{i})(f_{ij}f_{\overline{i}\overline{j}}+f_{\overline{i}\overline{j}}f_{ij})\Bigr{]}\biggr{\\}}.$
(A.29)

Thus, we can get Eq. (3.19) by substituting the flux background and
fluctuations into this Lagrangian.

##  Appendix B Comments on the Yukawa type superpotential

Here, we show that a factor 2 in Eq. (3.22) is consistent with the SUGRA
formulation. Let us consider the following Yukawa type superpotential with
introduction of a coefficient $w$:

$\displaystyle
W_{w}=w{\bm{\vartheta}_{\mathbb{A}\mathbb{B}\mathbb{C}}}A_{1}^{\mathbb{A}}B_{2}^{\mathbb{B}}C_{3}^{\mathbb{C}}=\frac{w}{\sqrt{2}}\frac{e^{-K_{0}/2}\alpha^{(1)}_{ab}\alpha^{(2)}_{bc}\alpha^{(3)}_{ca}e^{3\Phi-\varphi}}{\sqrt{g_{1\overline{1}}g_{2\overline{2}}g_{3\overline{3}}}}\int
d^{6}\sqrt{g_{6}}\phi^{\mathbb{A},ab}_{1}\phi^{\mathbb{B},bc}_{2}\phi^{\mathbb{C},ca}_{3}A^{\mathbb{A}}B^{\mathbb{B}}C^{\mathbb{C}},$
(B.1)

where

$\displaystyle{\bm{\vartheta}_{\mathbb{A}\mathbb{B}\mathbb{C}}}\coloneqq\prod_{r=1}^{3}W_{A^{(r)}B^{(r)}C^{(r)}}.$
(B.2)

With this superpotential, the scalar potential of
$|A^{\mathbb{A}}_{1}B^{\mathbb{B}}_{2}|^{2}$ derived from the NDBI action is
expressed as

$\displaystyle
A^{\mathbb{A}}_{1}B^{\mathbb{B}}_{2}\overline{A^{\mathbb{A}^{\prime}}_{1}}\overline{B^{\mathbb{B}^{\prime}}_{2}}\times\frac{2}{\mathcal{Z}^{3}_{ca}}e^{K_{0}}\biggl{(}\frac{e^{-K_{0}/2}\alpha^{(1)}_{ab}\alpha^{(2)}_{bc}\alpha^{(3)}_{ca}e^{3\Phi-\varphi}}{\sqrt{g_{1\overline{1}}g_{2\overline{2}}g_{3\overline{3}}}}\biggr{)}^{2}\int
d^{6}y\sqrt{g_{6}}\phi^{\mathbb{A},ab}_{1}\phi^{\mathbb{B},bc}_{2}\overline{\phi^{\mathbb{B}^{\prime},bc}_{2}}\overline{\phi^{\mathbb{A}^{\prime},ab}_{1}}$
(B.3) $\displaystyle\in$
$\displaystyle\biggl{(}\frac{2}{w}\biggr{)}^{2}\frac{1}{\mathcal{Z}^{3}_{ca}}e^{K_{0}}|\partial_{C_{3}}W_{w}|^{2},$
(B.4)

which implies that $w=2$ makes this SUGRA potential be equal to Eq. (3.20)
derived from NDBI action.

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