††institutetext:  1 Graduate University for Advanced Studies (Sokendai), 1-1
Oho, Tsukuba, Ibaraki 305-0801, Japan
2 Department of Physics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka
819-0395, Japan
3 Department of Physics, Kobe University, Kobe 675-8501, Japan

# Upper bound on the Atiyah-Singer index from tadpole cancellation

Keiya Ishiguro1 Takafumi Kai2 Satsuki Nishimura2 Hajime Otsuka2, and  Maki
Takeuchi3<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We propose an upper bound on the Atiyah-Singer index in the effective action
of string theory. For $E_{8}\times E_{8}^{\prime}$ and $SO(32)$ heterotic
string theories on smooth Calabi-Yau threefolds with line bundles, we find
that the tadpole cancellation and supersymmetry conditions lead to an upper
bound on the generation number of quarks and leptons as well as Higgs
doublets. By taking into account the observed value of four-dimensional gauge
couplings and the supergravity approximation, we explicitly evaluate the bound
on favorable complete intersection Calabi-Yau threefolds. The bound can be
extended to Calabi-Yau threefolds in the Kreuzer-Skarke database. We also put
the upper bound on the Atiyah-Singer index in Type IIB/F-theory
compactifications.

††preprint:

##  1 Introduction

The origin of fermion generations is one of the unsolved problems in the
Standard Model (SM). Higher-dimensional theories have been utilized to
understand the generation structure of chiral fermions. Indeed, background
fluxes and curvatures in the extra-dimensional space will lead to degenerate
chiral fermions in the four-dimensional (4D) low-energy effective action
through the Kaluza-Klein compactifications Candelas:1985en . The degeneracy of
chiral zero modes will be counted by the Atiyah-Singer index theorem, but an
arbitrary value of background fluxes is allowed in general. Thus, there is no
guiding principle to fix the chiral index, and it will be a challenging
problem to answer the generation problem of quarks and leptons.

In string theory, brane charges induced by the background fluxes and
curvatures should be canceled in compact spaces, the so-called tadpole
cancellation conditions. It suggests that the possibility of background fluxes
is finite through the cancellation of brane charges, and the Atiyah-Singer
index determining the number of chiral zero modes is bounded by some
topological quantities. There was an attempt to put the bound on the chiral
index in the context of heterotic $E_{8}\times E_{8}^{\prime}$ string theory
on Calabi-Yau (CY) threefolds with line bundles Buchbinder:2013dna ;
Constantin:2015bea ; GrootNibbelink:2015dvi  under the assumption that the
effective action is described by the 4D ${\cal N}=1$ supergravity and the
four-dimensional coupling constants are finite. This bound was derived by the
requirement of the supersymmetry and anomaly cancellation. It motivates us to
explore the maximal number of quarks and leptons in other corners of string
theories.

The purpose of this paper is to derive a rigorous bound on the Atiyah-Singer
index in the effective action of string theory. We focus on three classes of
string effective action: (i) $E_{8}\times E_{8}^{\prime}$ heterotic line
bundle models incorporating the 4D $SU(5)$ grand unified theory (GUT), (ii)
$SO(32)$ heterotic line bundle models incorporating the 4D Pati-Salam model or
SM-like model, (iii) Type IIB magnetized D7-brane models lift to the F-theory
vacua on compact CY fourfolds. In these models, Abelian fluxes lead to
degenerate chiral zero modes counted by the Atiyah-Singer index theorem, but
the degeneracy will be correlated with the brane charges due to the tadpole
cancellation conditions. Thus, the tadpole charge puts a severe bound on the
chiral index. Our findings about the bound on the index of chiral matters are
summarized as follows:

  * •

For $E_{8}\times E_{8}^{\prime}$ heterotic line bundle models, we find that
the index of quarks and leptons is bounded from above by the tadpole
cancellation condition:

$\displaystyle|N_{\rm gen}|<\alpha|m_{\rm max}|\|c_{2}(T{\cal M})\|,$ (1.1)

where $\alpha$ is the ${\cal O}(1)$ constant, $m_{\rm max}$ is the largest
$U(1)$ flux quanta and $\|c_{2}(T{\cal M})\|$ denotes the Euclidean norm of
the second Chern number of CY threefolds. Note that $m_{\rm max}$ is also
bounded from above by CY topological data through the supersymmetry, K-theory
and tadpole cancellation conditions. When we move to $SO(32)$ heterotic line
bundle models, we arrive at a similar bound on the index of Higgs doublets and
vector-like quarks under the SM gauge group.

  * •

We explicitly check the upper bound on the chiral index on favorable complete
intersection CY threefolds Candelas:1987kf ; Candelas:1987du , taking into
account the finite value of 4D coupling constants. The analysis can be
extended to the CY threefolds in the Kreuzer-Skarke database Kreuzer:2000xy .

  * •

We explore the bound of the chiral index in the context of Type IIB/F-theory
compactifications. Since the index of chiral zero modes arising from the
intersection between two stacks of magnetized D7-branes is determined by
world-volume fluxes on D7-branes, the chiral index is bounded by the brane
charges through the tadpole cancellation condition of D3-branes.

This paper is organized as follows. First, we revisit the $E_{8}\times
E_{8}^{\prime}$ heterotic line bundle models. After summarizing the
consistency conditions in the heterotic model building, we derive the upper
bound on the chiral index in $E_{8}\times E_{8}^{\prime}$ heterotic line
bundle models in Sec. 2. For $SO(32)$ heterotic line bundle models, we
consider two scenarios: (i) 4D Pati-Salam models in Sec. 3.1 and (ii) 4D SM-
like models realized by the so-called hypercharge flux breaking. In addition
to the chiral index of quarks/leptons, the index of vector-like quarks and
Higgs doublets is also bounded from above by geometric quantities of CY
threefolds. In Sec. 4, we analyze the chiral index on magnetized D7-branes in
Type IIB/F-theory compactifications. Sec. 5 is devoted to the conclusions.

##  2 $E_{8}\times E_{8}^{\prime}$ heterotic string theory

We start from $E_{8}\times E_{8}^{\prime}$ heterotic string theory on CY
threefolds with line bundles.111For more details, see, Refs.
Blumenhagen:2005ga ; Blumenhagen:2005pm . The bosonic part of low-energy
effective action up to the order of $\alpha^{\prime}$ is described by

$\displaystyle S_{\rm bos}$
$\displaystyle=\frac{1}{2\kappa_{10}^{2}}\int_{M^{(10)}}e^{-2\phi_{10}}\left[R+4d\phi_{10}\wedge\ast
d\phi_{10}-\frac{1}{2}H\wedge\ast H\right]$
$\displaystyle-\frac{1}{2g_{10}^{2}}\int_{M^{(10)}}e^{-2\phi_{10}}{\rm
tr}(F\wedge\ast
F)-\sum_{s}N_{s}T_{5}\int_{M^{(10)}}B^{(6)}\wedge\delta(\gamma_{s}),$ (2.1)

with

$\displaystyle H=dB^{(2)}-\frac{\alpha^{{}^{\prime}}}{4}(w_{\rm YM}-w_{L}),$
(2.2)

consisting of the Kalb-Ramond two-form $B^{(2)}$ (whose hodge dual is
$B^{(6)}$) and the gauge and gravitational Chern-Simons three-forms, $w_{\rm
YM}$ and $w_{L}$. Here, $\phi_{10}$ denotes the dilaton, and the 10D
gravitational coupling $\kappa_{10}$ and gauge coupling $g_{10}$ are
normalized as $2\kappa_{10}^{2}=(2\pi)^{7}(\alpha^{\prime})^{4}$ and
$g_{10}^{2}=2(2\pi)^{7}(\alpha^{\prime})^{3}$, respectively. The trace of 10D
gauge field strength $F$ is taken in the fundamental representation of
$E_{8}$. Furthermore, we introduce $N_{s}$ stacks of heterotic five-branes
wrapping the holomorphic two-cycles $\gamma_{s}$ whose Poincaré dual four-form
is represented by $\delta(\gamma_{s})$, and the tension of five-branes is
given by $T_{5}=((2\pi)^{5}(\alpha^{\prime})^{3})^{-1}$.

To realize the semi-realistic spectra, we introduce the internal gauge bundle
consisting of multiple line bundles $l_{a}$, each with a structure group
$U(1)\subset E_{8}$, that is,

$\displaystyle W=\bigoplus_{a=1}^{n}l_{a}.$ (2.3)

These line bundles play a role not only in breaking the $E_{8}$ gauge group to
$G\times\Pi_{a}U(1)_{a}$ but also in generating the chiral fermions from the
adjoint representation of $E_{8}$:

$\displaystyle 248\rightarrow\bigoplus_{p}(R_{p},C_{p}),$ (2.4)

under $G$ and $\Pi_{a}U(1)_{a}$, respectively. Indeed, the index of chiral
massless fermions (chiral superfields if the supersymmetry is preserved) in
the representation $R_{p}$ is counted by the Atiyah-Singer index:

$\displaystyle\chi({\cal M},{\cal C}_{p})=\int_{\cal M}\left({\rm
ch}_{3}({\cal C}_{p})+\frac{1}{12}c_{2}(T{\cal M})c_{1}({\cal C}_{p})\right),$
(2.5)

where ${\rm ch}_{3}({\cal C}_{p})$ and $c_{2}(T{\cal M})$ denote the third
Chern character of the internal bundle of each $C_{p}$ and the second Chern
class of the tangent bundle of CY threefolds ${\cal M}$, respectively.
However, the line bundles should satisfy four consistency conditions, as will
be discussed below.

  1. 1.

Tadpole cancellation condition

From the Bianchi identity of the Kalb-Ramond field $B^{(6)}$:

$\displaystyle d(e^{2\phi_{10}}\ast
dB^{(6)})=-\frac{\alpha^{{}^{\prime}}}{4}\left({\rm tr}\bar{F}^{2}-{\rm
tr}{\bar{R}}^{2}-4(2\pi)^{2}\sum_{s}N_{s}\delta(\gamma_{s})\right)$ (2.6)

with $\bar{F}$ and $\bar{R}$ being the internal background field strengths,
the line bundles satisfy the following inequality in cohomology:

$\displaystyle{\rm ch}_{2}(W)+c_{2}(T{\cal
M})=\sum_{s}N_{s}\delta(\gamma_{s})\geq[0].$ (2.7)

Here and in what follows, the trace of $R$ is taken in the fundamental
representation of $SO(1,9)$.

  2. 2.

K-theory condition

The first Chern class of the total bundle $W$ is constrained as

$\displaystyle c_{1}(W)=\sum_{a=1}^{n}n_{a}c_{1}(l_{a})\in H^{2}({\cal
M},2\mathbb{Z})$ (2.8)

to cancel the anomalies in the two-dimensional non-linear sigma model
Witten:1985mj ; Freed:1986zx . This condition will be regarded as the K-theory
condition in Type I string theory Witten:1998cd ; Uranga:2000xp . In what
follows, $n_{a}$ is normalized as one in the analysis of $E_{8}\times
E_{8}^{\prime}$ heterotic string theory, but it depends on the embedding of
line bundles, as shown in Sec. 3 for $SO(32)$ heterotic string theory.
Throughout this paper, we focus on the case with $c_{1}(W)=0$, corresponding
to a structure group $S(U(1)^{n})$.

  3. 3.

Supersymmetry condition

The internal gauge field strength $\bar{F}$ has to be holomorphic, that is,

$\displaystyle\bar{F}_{ij}=\bar{F}_{\bar{i}\bar{j}}=0,\qquad
g^{i\bar{j}}\bar{F}_{i\bar{j}}=0,$ (2.9)

due to the supersymmetry condition. The so-called Hermitian Yang-Mills
equation can be solved when

$\displaystyle\mu(l_{a})=\frac{1}{2l_{s}^{4}}\int_{\cal M}J\wedge J\wedge
c_{1}(l_{a})=0\qquad\forall a,$ (2.10)

at the leading order. Here, $l_{s}$ denotes the string length, and $J$ denotes
the Kähler form of CY threefolds.

  4. 4.

Bound on the 4D gauge coupling

From the dimensional reduction of Einstein-Hilbert term on a six-dimensional
internal manifold, the 4D Planck mass $M_{\rm Pl}$ is extracted as

$\displaystyle M_{\rm Pl}^{2}=e^{-2\langle\phi_{10}\rangle}\frac{{\rm
Vol}({\cal M})}{\kappa_{10}^{2}}=4\pi\frac{{\cal V}}{g_{s}^{2}l_{s}^{2}},$
(2.11)

with $g_{s}=e^{\langle\phi_{10}\rangle}$ being the string coupling. Here, the
CY volume is measured in units of string length, ${\rm Vol}({\cal M})={\cal
V}\,l_{s}^{6}$. Since the 4D gauge coupling is derived from the dimensional
reduction of 10D gauge field strength

$\displaystyle\alpha^{-1}=\frac{4\pi}{g_{4}^{2}}=e^{-2\langle\phi_{10}\rangle}\frac{4\pi{\rm
Vol}({\cal M})}{g_{10}^{2}}=g_{s}^{-2}{\cal V},$ (2.12)

the internal volume is bounded from above by

$\displaystyle{\cal V}=g_{s}^{2}\alpha^{-1}\lesssim 25,$ (2.13)

where we adopt the 4D gauge coupling at the GUT scale, $\alpha^{-1}(M_{\rm
GUT})\simeq 25$ with $M_{\rm GUT}\simeq 2\times 10^{16}$ GeV. We recall that
the CY volume is described by

$\displaystyle{\cal V}=\frac{1}{6}\int_{\cal M}J\wedge J\wedge
J=\frac{1}{6}\sum_{i,j,k}d_{ijk}t^{i}t^{j}t^{k},$ (2.14)

with $J=\sum_{i=1}^{h^{1,1}}t^{i}w_{i}$ in a basis $w_{i}$ of $H^{1,1}({\cal
M},\mathbb{R})$. Here, $t^{i}$ and $d_{ijk}$ denote the Kähler moduli and the
triple intersection numbers of CY manifolds, that is, $d_{ijk}=\int_{\cal
M}w_{i}\wedge w_{j}\wedge w_{k}$, respectively. Note that the Kähler moduli
should reside in $t^{i}>1$ in units of $l_{s}$, otherwise the supergravity
description is not valid. Thus, the bound on the CY volume (2.14) restricts
the sum of intersection number as follows:

$\displaystyle\sum_{i,j,k}d_{ijk}<\sum_{i,j,k}d_{ijk}t^{i}t^{j}t^{k}=6{\cal
V}\leq 150.$ (2.15)

Here, we assume non-negative triple intersection numbers as in the favorable
complete intersection Calabi-Yaus.

In the following, we restrict the index of chiral zero modes by using the
above consistency conditions in $E_{8}\times E_{8}^{\prime}$ heterotic string
theory on CY threefolds with line bundles. In particular, we focus on the
$SU(5)\times S(U(1)^{5})$ GUT as a branch of $E_{8}$.222See Refs.
Anderson:2011ns ; Anderson:2012yf  for a detailed analysis. The net number of
chiral zero modes is then described by

$\displaystyle N_{\rm gen}=-{\rm ind}(W)=\frac{1}{2}\int_{\cal
M}c_{3}(W)=-\sum_{i,j,k}\sum_{a=1}^{n}\frac{d_{ijk}}{6}m_{a}^{i}m_{a}^{j}m_{a}^{k},$
(2.16)

where the first Chern classes are of the form

$\displaystyle c_{1}(l_{a})=\sum_{i=1}^{h^{1,1}}m_{a}^{i}w_{i},$ (2.17)

in the basis $w_{i}$ of $H^{1,1}({\cal M},\mathbb{R})$. Here, $m_{a}^{i}$
denotes the quantized integer, and the number of $U(1)$ is taken as an
arbitrary integer $n$, although $n=5$ in the case of $SU(5)\times S(U(1)^{5})$
GUT. Following Ref. Constantin:2015bea , we present the bound of line bundle
vector $\bm{m_{a}}$. Since the moduli metric:

$\displaystyle G_{ij}$ $\displaystyle=\frac{1}{2(2\pi l_{s})^{6}{\cal
V}}\int_{\rm CY}J_{i}\wedge\ast
J_{j}=-\frac{1}{2}\partial_{i}\partial_{j}\ln{\cal
V}=-3\left(\frac{\kappa_{ij}}{\kappa}-\frac{3\kappa_{i}\kappa_{j}}{2\kappa^{2}}\right),$
(2.18)

with

$\displaystyle\kappa=6{\cal
V}=\sum_{i,j,k}d_{ijk}t^{i}t^{j}t^{k},\qquad\kappa_{i}=\sum_{j,k}d_{ijk}t^{j}t^{k},\qquad\kappa_{ij}=\sum_{k}d_{ijk}t^{k}$
(2.19)

is positive definite in the interior of the Kähler cone, one can derive the
following inequality:

$\displaystyle 0$
$\displaystyle<\sum_{a,i}m_{a}^{i}G_{ij}m_{a}^{j}=-\frac{3}{\kappa}\sum_{a,i}m_{a}^{i}\kappa_{ij}m_{a}^{j}=-\frac{3}{\kappa}\sum_{a,i}d_{ijk}m_{a}^{i}m_{a}^{j}t^{k}=\frac{6}{\kappa}\sum_{i}t^{i}c_{2i}(W)$
$\displaystyle\leq\frac{6}{\kappa}\|{\bm{t}}\|\|c_{2}(T{\cal M})\|,$ (2.20)

where we use the supersymmetry condition (2.10) in the first step and the
tadpole cancellation condition (2.7) in the last step. Here,
$c_{2i}(W)=-\frac{1}{2}\sum_{a,j,k}d_{ijk}m_{a}^{j}m_{a}^{k}$ and $\|\cdot\|$
denotes the Euclidean norm. It suggests the following redefinition of the
moduli metric:

$\displaystyle\widetilde{G}_{ij}=\frac{\kappa}{6\|{\bm{t}}\|}G_{ij},$ (2.21)

from which the inequality is rewritten as

$\displaystyle 0$
$\displaystyle<\sum_{a}m_{a}^{i}\widetilde{G}_{ij}m_{a}^{j}\leq\|c_{2}(T{\cal
M})\|.$ (2.22)

Furthermore, when we define ${\rm Eigen}(\widetilde{G}_{ij})\biggl{|}_{\rm
min}=\lambda_{\rm min}$, the flux vector ${\bm{m}_{a}}$ is bounded from above
by

$\displaystyle\sum_{a}|{\bm{m}}_{a}|^{2}\leq\frac{\|c_{2}(T{\cal
M})\|}{\lambda_{\rm min}},$ (2.23)

as derived in Ref. Constantin:2015bea . For instance, in the case of single
Kähler modulus, i.e., $h^{1,1}=1$, the modulus metric is given by

$\displaystyle G_{11}$
$\displaystyle=-3\left(\frac{\kappa_{11}}{\kappa}-\frac{3\kappa_{1}\kappa_{1}}{2\kappa^{2}}\right)=\frac{3}{2(t^{1})^{2}}$
(2.24)

with

$\displaystyle\kappa=d_{111}(t^{1})^{3},\qquad\kappa_{1}=d_{111}(t^{1})^{2},\qquad\kappa_{11}=d_{111}t^{1}.$
(2.25)

Thus, the flux quanta are bounded from above as

$\displaystyle\sum_{a}(m_{a}^{1})^{2}\leq\frac{4c_{2}(T{\cal M})}{d_{111}}.$
(2.26)

Let us derive the maximal value of the flux quanta $m_{a}^{i}$ in a different
approach of Ref. Constantin:2015bea . By using the so-called K-theory
condition (2.8) which is rewritten in terms of $m_{a}^{i}$333Note that we
focus on $c_{1}(W)=0$ as mentioned below Eq. (2.8).:

$\displaystyle\sum_{a}m_{a}^{i}=0$ (2.27)

for all $i$, it turns out that

$\displaystyle\sum_{a}|{\bm{m}}_{a}|^{2}$
$\displaystyle=\sum_{i=1}^{h^{1,1}}\sum_{b=2}^{n}|m_{b}^{i}|^{2}+|{\bm{m}}_{1}|^{2}$
$\displaystyle\geq\sum_{i=1}^{h^{1,1}}\frac{\left(m_{2}^{i}+\cdots+m_{n}^{i}\right)^{2}}{n-1}+|{\bm{m}}_{1}|^{2}=\sum_{i=1}^{h^{1,1}}\frac{\left(-m_{1}^{i}\right)^{2}}{n-1}+|{\bm{m}}_{1}|^{2}=\frac{n}{n-1}|{\bm{m}}_{1}|^{2},$

where we use the Cauchy-Schwarz inequality. Note that a similar inequality
holds on the other $a$, that is,

$\displaystyle|{\bm{m}}_{a}|^{2}\leq\frac{n-1}{n}\frac{\|c_{2}(T{\cal
M})\|}{\lambda_{\rm min}},$ (2.28)

for all $a$. Thus, when we define $m_{\rm max}$ satisfying
$|m_{a}^{i}|\leq{\rm max}_{a,i}(m_{a}^{i})=:|m_{\rm max}|$ for all $a,i$, we
arrive at

$\displaystyle|m_{\rm max}|^{2}\leq\frac{n-1}{n}\frac{\|c_{2}(T{\cal
M})\|}{\lambda_{\rm min}}-(h^{1,1}-1),$ (2.29)

where we assume $m_{a}^{i}\neq 0$ for all $i$, but it is possible to derive
$m_{a}^{i}=0$ for some $i$. In this case, the term $h^{1,1}-1$ in the above
equation will be modified. When we evaluate the index of chiral modes, we
adopt the above conservative bound of the flux quanta throughout this paper.

We are now ready to derive the upper bound on the Atiyah-Singer index of
chiral zero modes. Since the index of $SU(5)$ matter multiplets is determined
by Eq. (2.16), we find the following inequality:

$\displaystyle|N_{\rm gen}|$
$\displaystyle=\biggl{|}\sum_{a,i,j,k}\frac{d_{ijk}}{6}m_{a}^{i}m_{a}^{j}m_{a}^{k}\biggl{|}\leq\frac{|m_{\rm
max}|}{3}\|c_{2}(W)\|\leq\frac{|m_{\rm max}|}{3}\|c_{2}(T{\cal M})\|,$ (2.30)

where we use the tadpole cancellation in the last step. Thus, the index is
bounded from above by the maximal value of the flux quanta $m_{\rm max}$ and
the second Chern number of the tangent bundle of CY. The authors of Ref.
Constantin:2015bea  derived the index bounded by the CY volume itself, but we
check that our finding bound is stronger than the known bound for all
favorable complete intersection CY threefolds (CICYs).

Here and in what follows, we focus on CICYs defined in the ambient space
$\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{m}}$. The CICYs are
specified by the $m\times R$ configuration matrix Candelas:1987kf ;
Candelas:1987du :

$\displaystyle\begin{matrix}\mathbb{P}^{n_{1}}\\\ \mathbb{P}^{n_{2}}\\\
\vdots\\\ \mathbb{P}^{n_{m}}\\\
\end{matrix}\begin{bmatrix}q_{1}^{1}&q_{2}^{1}&\cdots&q_{R}^{1}\\\
q_{1}^{2}&q_{2}^{2}&\cdots&q_{R}^{2}\\\ \vdots&\vdots&\ddots&\vdots\\\
q_{1}^{m}&q_{2}^{m}&\cdots&q_{R}^{m}\\\ \end{bmatrix}$ (2.31)

where the positive integers $q^{\cal I}_{r}$ $(r=1,\cdots,R)$ denote the
multi-degree of $R$ homogeneous polynomials on the ambient space
$\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{m}}$ with the homogeneous
coordinates of $\mathbb{P}^{n_{\cal I}}$ being $x^{\cal I}_{\alpha}$ for
${\cal I}=1,\cdots,m$ and $\alpha=0,\cdots,n_{\cal I}$. Since the first Chern
class of the tangent bundle can be zero under $\sum_{r=1}^{R}q_{r}^{\cal
I}=n_{\cal I}+1$ for all ${\cal I}$, the CICYs are defined on the common zero
locus of these $R$ polynomials.444The dimension of CICYs is determined by
$\sum_{{\cal I}=1}^{m}n_{\cal I}-R$. In particular, we focus on all
”favorable” CICYs among complete intersection CYs, where the ”favorable” means
that the second cohomology of CY descends from that of ambient space.555 See
for the CICY list, Ref. Lukas .

To calculate the upper bound on the index, we have to calculate $\lambda_{\rm
min}$ which depends on the values of moduli $t^{i}$. In our analysis, we adopt
the universal value of moduli fields, $t:=t^{i}$ for all $i$ whose value is
fixed by a given CY volume ${\cal V}=12$ or ${\cal V}=25$. By using the
explicit values of CY topological data, we plot the index in Fig. 1, where the
CY volume is fixed as ${\cal V}=12$ in the left panel and ${\cal V}=25$ in the
right panel, respectively. We find that there are no viable models on the
large $h^{1,1}$ due to the fact that the volume of favorable CICYs is bounded
from above by the phenomenological requirement (2.14).

Figure 1: The upper bound on the chiral index in $SU(5)$ GUT on upstairs CICYs
${\cal M}$.

So far, we have focused on the so-called ”upstairs” CICYs. However, it was
known that some of the CICYs admit a freely-acting discrete symmetry group
$\Gamma$; one can define the quotient CICYs ${\cal\widetilde{M}}={\cal
M}/\Gamma$ Candelas:1987kf ; Candelas:1987du  on which one can turn on Wilson
lines (see, Ref. Braun:2010vc  for the classification of $\Gamma$).
Remarkably, such discrete quotients decrease the index:

$\displaystyle N_{\rm gen}({\cal\widetilde{M}})=\frac{N_{\rm gen}({\cal
M})}{|\Gamma|},$ (2.32)

although the topological quantities of quotient CICYs are different from
before. Here, we consider a $\Gamma$-equivariant structure for the vector
bundle $V$. For instance, the CY volume also reduces to ${\cal
V}_{\Gamma}:={\cal V}/|\Gamma|$. For more details, see, e.g., Ref.
Anderson:2009mh . In Fig. 2, we plot the chiral index on 195 quotient CICYs
satisfying the volume constraint (2.15). Similarly, the moduli values are
chosen as $t:=t^{i}$ for all $i$ with the CY volume ${\cal V}_{\Gamma}=12$ in
the left panel and ${\cal V}_{\Gamma}=25$ in the right panel of Fig. 2,
respectively. It turns out that the chiral index suppressed by $|\Gamma|$
satisfies $N_{\rm gen}\leq 100$ on quotient CY threefolds.

Figure 2: The upper bound on the chiral index in $SU(5)$ GUT on quotient CICYs
${\cal\widetilde{M}}$.

It is interesting to apply our methods for finding a bound on the index to
other classes of CY models. We study the CY threefolds in the Kreuzer-Skarke
database Kreuzer:2000xy , whose topological data was statistically analyzed in
Ref. Demirtas:2018akl . Note that we focus on favorable CY threefolds in the
Kreuzer-Skarke database as in Ref. Demirtas:2018akl . As seen in figure 1 of
Ref. Demirtas:2018akl  with the large $h^{1,1}$ region, the nonvanishing
triple intersection number behaves as

$\displaystyle\\#(d_{ijk}\neq 0)\gtrsim 6.5h^{1,1}+25,$ (2.33)

for $h^{1,1}\gtrsim 25$666In contrast to Ref. Demirtas:2018akl , we focus on
the CY threefolds rather than their mirror duals.. By introducing a normalized
vector $\mathsf{t}^{i}$, we rewrite the Kähler moduli

$\displaystyle t^{i}=\mathsf{t}^{i}\|{\bm{t}}\|,$ (2.34)

with $\|\mathsf{t}\|=1$, which scales as
$\mathsf{t}^{i}\simeq(h^{1,1})^{-1/2}$ Demirtas:2018akl ; Plauschinn:2021hkp .
Together with Eq. (2.33), the CY volume in the geometrical regime scales as

$\displaystyle{\cal
V}\gtrsim(h^{1,1})^{-1/2}\|{\bm{t}}\|^{3}>(h^{1,1})^{-1/2},$ (2.35)

which will be consistent with the bound on CY volume (2.14) required in the
GUT.

Furthermore, one can derive the non-trivial bound on the CY topological
quantities from the tadpole cancellation. By utilizing the scaling of
$\kappa_{i}\simeq(h^{1,1})^{-1}\|{\bm{t}}\|^{2}$ and the mostly diagonal
$\kappa_{ij}$ Demirtas:2018akl ; Plauschinn:2021hkp , i.e.,
$\kappa_{ii}\simeq(h^{1,1})^{-1/2}\|{\bm{t}}\|$, the sum of the flux vector is
bounded from below by

$\displaystyle\frac{\kappa}{3}\sum_{a,i,j}m_{a}^{i}G_{ij}m_{a}^{j}\geq(h^{1,1})^{1/2}\|{\bm{t}}\|,$
(2.36)

with $h^{1,1}\gg 1$. Combining with Eq. (2.20), we arrive at

$\displaystyle\|c_{2}(T{\cal M})\|\geq\frac{(h^{1,1})^{1/2}}{2}.$ (2.37)

Thus, the second Chern number of CY will be bounded by the hodge number due to
the tadpole cancellation. Note that we have focused on favorable CYs, but it
will be interesting to figure out the behavior of the chiral index on non-
favorable CYs, which is left for future work.

##  3 SO(32) heterotic string

In this section, we discuss the bound on the chiral index in the framework of
$SO(32)$ heterotic string theory on smooth CY threefolds with line
bundles.777For the model building on the toroidal background, see, e.g., Ref.
Abe:2015mua . In particular, we focus on the Pati-Salam group in Sec. 3.1 and
a direct flux breaking, so-called hypercharge flux breaking in Sec. 3.2.

###  3.1 Pati-Salam

Following Ref. Otsuka:2018oyf 888For MSSM-like models in $SO(32)$ heterotic
string theory, see, Ref. GrootNibbelink:2015lme ., the Pati-Salam gauge group
can be embedded into $SO(32)$ as follows:

$\displaystyle SO(32)$ $\displaystyle\rightarrow SO(16)\times SO(16)_{\rm
hid}$ $\displaystyle\rightarrow SU(4)_{C}\times SU(2)_{L}\times
SU(2)_{R}\times\Pi_{c=1}^{2}U(1)_{c}\times U(1)^{\prime}\times SO(16)_{\rm
hid},$ (3.1)

where the Cartan directions of $SO(32)$, $H_{a}$ with $a=1,2,...,16$, are
chosen as $H_{1}-H_{2}$, $H_{1}+H_{2}-2H_{3}$ and $H_{1}+H_{2}+H_{3}-3H_{4}$
for $SU(4)_{C}$, $H_{5}-H_{6}$ for $SU(2)_{L}$, $H_{4}+H_{5}$ for $SU(2)_{R}$,
and $H_{4}-H_{5}$ for $U(1)^{\prime}\simeq S(U(1)_{4}\times U(1)_{5})$,
respectively. On the $H_{a}$ basis, we choose the $SO(32)$ roots
$(\underline{\pm 1,\pm 1,0,\cdots,0})$, where the underline represents all the
possible permutations. $U(1)$s are chosen as

$\displaystyle U(1)_{1}$ $\displaystyle:(0,0,0,0,1,1,0,0;0,\cdots,0),$
$\displaystyle U(1)_{2}$ $\displaystyle:(1,1,1,1,0,0,0,0;0,\cdots,0),$
$\displaystyle U(1)_{3}$ $\displaystyle:(1,1,1,-3,0,0,0,0;0,\cdots,0),$
$\displaystyle U(1)_{4}$ $\displaystyle:(0,0,0,0,0,0,1,0;0,\cdots,0),$
$\displaystyle U(1)_{5}$ $\displaystyle:(0,0,0,0,0,0,0,1;0,\cdots,0),$ (3.2)

respectively. The adjoint representation of $SO(32)$ splits into the spectrum
in the visible and hidden sectors, as shown in Table 1, where the hypercharge
direction is specified as

$\displaystyle
U(1)_{Y}=\frac{1}{6}U(1)_{3}+\frac{1}{2}U(1)_{4}+\frac{1}{2}U(1)_{5}.$ (3.3)

$\left(SU(4)_{C}\times SU(2)_{L}\times SU(2)_{R}\right.$ | $\left(SU(3)_{C}\times SU(2)_{L}\times SO(16)\right.$ |  |
---|---|---|---
$\left.\times SO(16)\right)_{U(1)_{1},U(1)_{2},U(1)_{4},U(1)_{5}}$ | $\left.\right)_{U(1)_{1},U(1)_{2},U(1)_{3},U(1)_{4},U(1)_{5}}$ | Matter | Index
Visible sector
$(4,2,1,1)_{1,1,0,0}$ | $(3,2,1)_{1,1,1,0,0}$ | $Q_{1}$ | $\chi({\cal M},l_{1}\otimes l_{2})$
| $(1,2,1)_{1,1,-3,0,0}$ | $L_{1}$ |
$(4,2,1,1)_{-1,1,0,0}$ | $(3,2,1)_{-1,1,1,0,0}$ | $Q_{2}$ | $\chi({\cal M},l_{1}^{-1}\otimes l_{2})$
| $(1,2,1)_{-1,1,-3,0,0}$ | $L_{2}$ |
$(15,1,1,1)_{0,0,0,0}$ | $({\bar{3}},1,1)_{0,0,-4,0,0}$ | $u_{R_{1}}^{c}$ | $\chi({\cal M},{\cal O}_{\cal M})$
$(6,1,1,1)_{0,2,0,0}$ | $({\bar{3}},1,1)_{0,2,2,0,0}$ | $d_{R_{1}}^{c}$ | $\chi({\cal M},l_{2}^{2})$
| $(3,1,1)_{0,2,-2,0,0}$ | $\bar{d}_{R_{2}}^{c}$ |
$(1,1,1,1)_{2,0,0,0}$ | $(1,1,1)_{2,0,0,0,0}$ | $n_{1}$ | $\chi({\cal M},l_{1}^{2})$
$(\bar{4},1,2,1)_{0,-1,-1,0}$ | $({\bar{3}},1,1)_{0,-1,-1,-1,0}$ | $u_{R_{2}}^{c\,\,4}$ | $\chi({\cal M},l_{2}^{-1}\otimes l_{4}^{-1})$
| $(1,1,1)_{0,-1,3,0,1}$ | $e_{R_{1}}^{c\,\,5}$ |
| $(1,1,1)_{0,-1,3,-1,0}$ | $n_{2}^{c\,4}$ |
| $({\bar{3}},1,1)_{0,-1,-1,0,1}$ | $d_{R_{3}}^{c\,\,5}$ |
$(\overline{4},1,2,1)_{0,-1,1,0}$ | $({\bar{3}},1,1)_{0,-1,-1,1,0}$ | $d_{R_{3}}^{c\,\,4}$ | $\chi({\cal M},l_{2}^{-1}\otimes l_{4})$
| $(1,1,1)_{0,-1,3,1,0}$ | $e_{R_{1}}^{c\,\,4}$ |
| $(1,1,1)_{0,-1,3,0,-1}$ | $n_{2}^{c\,5}$ |
| $({\bar{3}},1,1)_{0,-1,-1,0,-1}$ | $u_{R_{2}}^{c\,\,5}$ |
$(1,2,2,1)_{1,0,-1,0}$ | $(1,2,1)_{1,0,0,-1,0}$ | $L_{3}^{4}$ | $\chi({\cal M},l_{1}\otimes l_{4}^{-1})$
| $(1,2,1)_{1,0,0,0,1}$ | $\bar{L}_{4}^{5}$ |
$(1,2,2,1)_{1,0,1,0}$ | $(1,2,1)_{1,0,0,0,-1}$ | $L_{3}^{5}$ | $\chi({\cal M},l_{1}\otimes l_{4})$
| $(1,2,1)_{1,0,0,1,0}$ | $\bar{L}_{4}^{4}$ |
$(1,1,3,1)_{0,0,0,0}$ | $(1,1,1)_{0,0,0,1,1}$ | $e_{R_{2}}^{c\,\,45}$ | $\chi({\cal M},{\cal O}_{\cal M})$
$(1,1,1,1)_{0,0,2,0}$ | $(1,1,1)_{0,0,0,1,-1}$ | $n_{3}^{c\,45}$ | $\chi({\cal M},l_{4}^{2})$
Hidden sector
$(\bar{4},1,1,16)_{0,-1,0,0}$ | $({\bar{3}},1,16)_{0,-1,-1,0,0}$ | $-$ | $\chi({\cal M},l_{2}^{-1})$
| $(1,1,16)_{0,-1,3,0}$ | $-$ |
$(1,2,1,16)_{1,0,0,0}$ | $(1,2,16)_{1,0,0,0,0}$ | $-$ | $\chi({\cal M},l_{1})$
$(1,1,2,16)_{0,0,1,0}$ | $(1,1,16)_{0,0,0,1,0}$ | $-$ | $\chi({\cal M},l_{4})$
| $(1,1,16)_{0,0,0,0,-1}$ | $-$ |
$(1,1,1,120)_{0,0,0,0}$ | $(1,1,120)_{0,0,0,0,0}$ | $-$ | $\chi({\cal M},{\cal O}_{\cal M})$
Table 1: Massless spectrum and corresponding index under
$\left(SU(4)_{C}\times SU(2)_{L}\times SU(2)_{R}\times
SO(16)\right)_{U(1)_{1},U(1)_{2},U(1)_{4},U(1)_{5}}$ in the first column and
$\left(SU(3)_{C}\times SU(2)_{L}\times
SO(16)\right)_{U(1)_{1},U(1)_{2},U(1)_{3},U(1)_{4},U(1)_{5}}$ in the second
column, respectively. The $U(1)$ charges are specified by the subscript
indices. Here, we represent $U(1)_{4,5}$ charges rather than $U(1)^{\prime}$,
although both are correlated with $l_{5}\simeq l_{4}^{-1}$.

To avoid the existence of chiral exotic modes, we require

$\displaystyle\chi(l_{1})=\chi(l_{2})=\chi(l_{4})=-\chi(l_{5})=0.$ (3.4)

The tadpole cancellation condition is given by

$\displaystyle c_{2}(T{\cal M})$ $\displaystyle\geq-{\rm
ch}_{2}(W)=-2c_{1}^{2}(l_{1})-4c_{1}^{2}(l_{2})-c_{1}^{2}(l_{4})-c_{1}^{2}(l_{5}),$
(3.5)

with

$\displaystyle c_{1}^{2}(l_{a})=\sum_{j,k}d_{ijk}m_{a}^{j}m_{a}^{k}.$ (3.6)

By identifying left-handed quarks $\\{Q_{1},Q_{2}\\}$, right-handed quarks
$\\{u_{R_{2}}^{c\,\,4},u_{R_{2}}^{c\,\,5}\\}$ and vector-like quarks under the
SM gauge group $d_{R_{1}}^{c}$, their generation numbers are counted by

$\displaystyle N_{\rm gen}(Q)$ $\displaystyle:=N_{\rm gen}(Q_{1})+N_{\rm
gen}(Q_{2})=\chi(l_{1}\otimes l_{2})+\chi(l_{1}^{-1}\otimes l_{2}),$
$\displaystyle N_{\rm gen}(u_{R}^{c})$ $\displaystyle:=N_{\rm
gen}(u_{R_{2}}^{c\,\,4})+N_{\rm
gen}(u_{R_{2}}^{c\,\,5})=\chi(l_{2}^{-1}\otimes
l_{4}^{-1})+\chi(l_{2}^{-1}\otimes l_{4}),$ $\displaystyle N_{\rm
gen}^{\rm(vec)}$ $\displaystyle:=N_{\rm gen}(d_{R_{1}}^{c})=\chi(l_{2}^{2}),$
(3.7)

respectively. Note that the index of each field is calculated by using the
Atiyah-Singer index theorem:

$\displaystyle\chi(l_{a}^{p})$
$\displaystyle=\frac{p^{3}}{6}\sum_{i,j,k}d_{ijk}m_{a}^{i}m_{a}^{j}m_{a}^{k}+\frac{p}{12}\sum_{i}m_{a}^{i}c_{2}^{i}(T{\cal
M})=\frac{p^{3}}{6}X_{aaa}+\frac{p}{12}\sum_{i}m_{a}^{i}c_{2}^{i}(T{\cal M}),$
$\displaystyle\chi(l_{a}^{p}\otimes l_{b}^{q})$
$\displaystyle=\chi(l_{a}^{p})+\chi(l_{b}^{q})+\frac{1}{2}(pq^{2}X_{abb}+p^{2}qX_{aab}),$
(3.8)

with

$\displaystyle X_{aaa}=\sum_{i,j,k}d_{ijk}m_{a}^{i}m_{a}^{j}m_{a}^{k},\qquad
X_{abb}=\sum_{i,j,k}d_{ijk}m_{a}^{i}m_{b}^{j}m_{b}^{k},\qquad
X_{aab}=\sum_{i,j,k}d_{ijk}m_{a}^{i}m_{a}^{j}m_{b}^{k}.$ (3.9)

Thus, the generation numbers are rewritten as

$\displaystyle N_{\rm gen}(Q)$
$\displaystyle=2\chi(l_{2})+\frac{1}{2}\left(X_{122}+X_{112}-X_{122}+X_{112}\right)=2\chi(l_{2})+X_{112}=X_{112},$
$\displaystyle N_{\rm gen}(u_{R}^{c})$
$\displaystyle=2\chi(l_{2}^{-1})+\frac{1}{2}\left(-X_{244}-X_{224}-X_{244}+X_{224}\right)=-2\chi(l_{2})-X_{244}=-X_{244},$
$\displaystyle N_{\rm gen}^{({\rm vec})}$
$\displaystyle=\chi(l_{2}^{2})=X_{222},$ (3.10)

where we used Eq. (3.4). The index of left- and right-handed quarks should be
equal, that is, $N_{\rm gen}^{\rm(quark)}=N_{\rm gen}(Q)=N_{\rm
gen}(u_{R}^{c})$. In the following, we will derive the upper bound on these
Atiyah-Singer indices.

From the equality:

$\displaystyle 2N_{\rm gen}(Q)+4N_{\rm gen}^{\rm(vec)}+2N_{\rm
gen}(u_{R}^{c})$ $\displaystyle=2X_{112}+4X_{222}-2X_{244}$
$\displaystyle=m_{2}^{k}\left(2d_{ijk}m_{1}^{i}m_{1}^{j}+4d_{ijk}m_{2}^{i}m_{2}^{j}+2d_{ijk}m_{4}^{i}m_{4}^{j}-4d_{ijk}m_{4}^{i}m_{4}^{j}\right)$
$\displaystyle=-\sum_{k}m_{2}^{k}c_{2}(W)_{k}+4N_{\rm gen}(u_{R}^{c}),$ (3.11)

we arrive at the following formula:

$\displaystyle|N_{\rm
gen}^{\rm(vec)}|=\frac{|\sum_{k}m_{2}^{k}c_{2}(W)_{k}|}{4}\leq\frac{|m_{\rm
max}|\|c_{2}(W)\|}{4}\leq\frac{|m_{\rm max}|\|c_{2}(T{\cal M})\|}{4},$ (3.12)

where we use the tadpole cancellation condition (3.5) and
$|m_{2}^{k}|\leq|m_{\rm max}|$ for all $k$. Thus, the generation number of
vector-like quarks is constrained by

$\displaystyle|N_{\rm gen}^{\rm(vec)}|\leq\frac{|m_{\rm max}|\|c_{2}(T{\cal
M})\|}{4}.$ (3.13)

To identify Higgs fields, let us take $N_{\rm gen}(u_{R_{2}}^{c\,\,4})=0$ for
simplicity.999We obtain the same bound in the case with $N_{\rm
gen}(u_{R_{2}}^{c\,\,5})=0$. Since the following Yukawa couplings of quarks
and leptons are allowed under the gauge symmetries of the low-energy effective
action:

$\displaystyle(Q_{1},\bar{L}_{3}^{5},u_{R_{2}}^{c\,\,5}),\qquad(Q_{2},\bar{L}_{4}^{5},u_{R_{2}}^{c\,\,5}),\qquad$
(3.14)

we identify the up-type Higgs doublets as $\bar{L}_{3,4}^{5}$, which are
vector-like particles under the SM gauge group but are chiral under the extra
$U(1)$ symmetries. The index of Higgs fields is then given by

$\displaystyle N_{\rm gen}(H)$ $\displaystyle:=N_{\rm
gen}(\bar{L}_{4}^{5})+N_{\rm gen}(\bar{L}_{3}^{5})=\chi(l_{1}\otimes
l_{4}^{-1})+\chi(l_{1}^{-1}\otimes l_{4}^{-1})=-X_{114},$ (3.15)

which is correlated with the quark index as

$\displaystyle N_{\rm gen}(H)-2N_{\rm
gen}(u_{R_{2}}^{c\,\,5})+\frac{\sum_{i}m_{4}^{i}(-c_{2}(W)_{i}+c_{2}(T{\cal
M})_{i})}{2}=0.$ (3.16)

Similarly, the tadpole cancellation condition provides the bound on the chiral
index:

$\displaystyle|N_{\rm gen}(H)-2N_{\rm gen}^{\rm(quark)}|\leq\frac{|m_{\rm
max}|}{2}\sum_{s}N_{s}.$ (3.17)

If $N_{\rm gen}^{\rm(quark)}=-3$, the generation number of Higgs fields is
bounded by the brane charge:

$\displaystyle|N_{\rm gen}(H)+6|\leq\frac{|m_{\rm max}|}{2}\sum_{s}N_{s}.$
(3.18)

Thus, an arbitrary number of Higgs fields is not allowed in the context of
$SO(32)$ heterotic line bundle models.

###  3.2 Hypercharge flux

So far, we have focused on the visible sector, equipping the $SU(5)$ or Pati-
Salam gauge symmetries. In this section, we deal with a direct flux breaking,
so-called hypercharge flux breaking in heterotic string theory as discussed in
Refs. Blumenhagen:2005ga ; Blumenhagen:2005pm ; Blumenhagen:2006ux ;
Blumenhagen:2006wj . Since this approach does not require the existence of
Wilson lines, it is applicable to simply-connected manifolds. However, it was
pointed out that the $E_{8}\times E_{8}$ heterotic string with hypercharge
flux does not lead to the SM spectra as a consequence of the large number of
index conditions Anderson:2014hia . This is the reason why we focus on the
direct flux breaking scenario in the context of $SO(32)$ heterotic string
theory.

As systematically analyzed in Ref. Otsuka:2018rki , the SM gauge group can be
directly realized from $SO(16)$ gauge group:

$\displaystyle SO(32)$ $\displaystyle\rightarrow SO(16)\times SO(16)_{\rm
hid}$ $\displaystyle\rightarrow SU(3)_{C}\times
SU(2)_{L}\times\Pi_{a=1}^{5}U(1)_{a}\times SO(16)_{\rm hid},$ (3.19)

where the Cartan directions of $SO(32)$, $H_{a}$ with $a=1,2,...,16$, are
chosen as $H_{1}-H_{2}$ and $H_{2}-H_{3}$ for $SU(3)_{C}$, $H_{4}-H_{5}$ for
$SU(2)_{L}$, respectively. The other $U(1)$s are chosen as

$\displaystyle U(1)_{1}$ $\displaystyle:(1,1,1,0,0,0,0,0;0,\cdots,0),$
$\displaystyle U(1)_{2}$ $\displaystyle:(0,0,0,1,1,0,0,0;0,\cdots,0),$
$\displaystyle U(1)_{3}$ $\displaystyle:(0,0,0,0,0,1,0,0;0,\cdots,0),$
$\displaystyle U(1)_{4}$ $\displaystyle:(0,0,0,0,0,0,1,0;0,\cdots,0),$
$\displaystyle U(1)_{5}$ $\displaystyle:(0,0,0,0,0,0,0,1;0,\cdots,0),$ (3.20)

respectively. The adjoint representation of $SO(32)$ splits into the spectrum
in the visible and hidden sectors, as shown in Table 2, where the hypercharge
direction is specified as

$\displaystyle
U(1)_{Y}=-\frac{1}{6}U(1)_{1}-\frac{1}{2}\biggl{(}U(1)_{3}-U(1)_{4}+U(1)_{5}\biggl{)}.$
(3.21)

To avoid the existence of chiral exotic modes, we require

$\displaystyle\chi(l_{1})=\chi(l_{2})=\chi(l_{3})=\chi(l_{4})=\chi(l_{5})=\chi(l_{1}^{-2})=0.$
(3.22)

The tadpole cancellation condition is given by

$\displaystyle c_{2}(T{\cal M})$ $\displaystyle\geq-{\rm
ch}_{2}(W)=-3c_{1}^{2}(l_{1})-2c_{1}^{2}(l_{2})-c_{1}^{2}(l_{3})-c_{1}^{2}(l_{4})-c_{1}^{2}(l_{5}).$
(3.23)

The generation numbers of left-handed and right-handed quarks/leptons are
counted by

$\displaystyle N_{\rm gen}(Q)$ $\displaystyle:=\chi(l_{1}^{-1}\otimes
l_{2})+\chi(l_{1}^{-1}\otimes l_{2}^{-1}),$ $\displaystyle N_{\rm gen}(L)$
$\displaystyle:=\sum_{s=\pm 1}\chi(l_{2}^{s}\otimes l_{3}^{-1})+\sum_{s=\pm
1}\chi(l_{2}^{s}\otimes l_{4}^{1})+\sum_{s=\pm 1}\chi(l_{2}^{s}\otimes
l_{5}^{1}),$ $\displaystyle N_{\rm gen}(u^{c})$
$\displaystyle:=\chi(l_{1}\otimes l_{3})+\chi(l_{1}\otimes
l_{4}^{-1})+\chi(l_{1}\otimes l_{5}),$ $\displaystyle N_{\rm gen}(d^{c})$
$\displaystyle:=\chi(l_{1}\otimes l_{3}^{-1})+\chi(l_{1}\otimes
l_{4})+\chi(l_{1}\otimes l_{5}^{-1}),$ $\displaystyle N_{\rm gen}(e^{c})$
$\displaystyle:=\chi(l_{3}^{-1}\otimes l_{4})+\chi(l_{3}^{-1}\otimes
l_{5}^{-1})+\chi(l_{4}\otimes l_{5}^{-1}),$ (3.24)

respectively.

Matter | Repr. | Index
---|---|---
Visible sector
$Q_{1}$ | $(3,2)_{-1,1,0,0,0}$ | $\chi({\cal M},l_{1}^{-1}\otimes l_{2})$
$Q_{2}$ | $(3,2)_{-1,-1,0,0,0}$ | $\chi({\cal M},l_{1}^{-1}\otimes l_{2}^{-1})$
$L_{1}$ | $(1,2)_{0,1,1,0,0}$ | $\chi({\cal M},l_{2}\otimes l_{3})$
$L_{2}$ | $(1,2)_{0,-1,1,0,0}$ | $\chi({\cal M},l_{2}^{-1}\otimes l_{3})$
$L_{3}$ | $(1,2)_{0,1,0,-1,0}$ | $\chi({\cal M},l_{2}\otimes l_{4}^{-1})$
$L_{4}$ | $(1,2)_{0,-1,0,-1,0}$ | $\chi({\cal M},l_{2}^{-1}\otimes l_{4}^{-1})$
$L_{5}$ | $(1,2)_{0,1,0,0,1}$ | $\chi({\cal M},l_{2}\otimes l_{5})$
$L_{6}$ | $(1,2)_{0,-1,0,0,1}$ | $\chi({\cal M},l_{2}^{-1}\otimes l_{5})$
$u_{R_{1}}^{c}$ | $({\bar{3}},1)_{1,0,1,0,0}$ | $\chi({\cal M},l_{1}\otimes l_{3})$
$u_{R_{2}}^{c}$ | $({\bar{3}},1)_{1,0,0,-1,0}$ | $\chi({\cal M},l_{1}\otimes l_{4}^{-1})$
$u_{R_{3}}^{c}$ | $({\bar{3}},1)_{1,0,0,0,1}$ | $\chi({\cal M},l_{1}\otimes l_{5})$
$d_{R_{1}}^{c}$ | $({\bar{3}},1)_{1,0,-1,0,0}$ | $\chi({\cal M},l_{1}\otimes l_{3}^{-1})$
$d_{R_{2}}^{c}$ | $({\bar{3}},1)_{1,0,0,1,0}$ | $\chi({\cal M},l_{1}\otimes l_{4})$
$d_{R_{3}}^{c}$ | $({\bar{3}},1)_{1,0,0,0,-1}$ | $\chi({\cal M},l_{1}\otimes l_{5}^{-1})$
$e_{R_{1}}^{c}$ | $(1,1)_{0,0,-1,1,0}$ | $\chi({\cal M},l_{3}^{-1}\otimes l_{4})$
$e_{R_{2}}^{c}$ | $(1,1)_{0,0,-1,0,-1}$ | $\chi({\cal M},l_{3}^{-1}\otimes l_{5}^{-1})$
$e_{R_{3}}^{c}$ | $(1,1)_{0,0,0,1,-1}$ | $\chi({\cal M},l_{4}\otimes l_{5}^{-1})$
$n_{1}^{c}$ | $(1,1)_{0,-2,0,0,0}$ | $\chi({\cal M},l_{2}^{2})$
$n_{2}^{c}$ | $(1,1)_{0,0,-1,-1,0}$ | $\chi({\cal M},l_{3}^{-1}\otimes l_{4}^{-1})$
$n_{3}^{c}$ | $(1,1)_{0,0,-1,0,1}$ | $\chi({\cal M},l_{3}^{-1}\otimes l_{5})$
$n_{4}^{c}$ | $(1,1)_{0,0,0,1,1}$ | $\chi({\cal M},l_{4}\otimes l_{5})$
$\phi$ | $({\bar{3}},1)_{-2,0,0,0,0}$ | $\chi({\cal M},l_{1}^{-2})$
Hidden sector
$-$ | $(3,1,16)_{1,0,0,0}$ | $\chi({\cal M},l_{1})$
$-$ | $(1,2,16)_{0,1,0,0,0}$ | $\chi({\cal M},l_{2})$
$-$ | $(1,1,16)_{0,0,1,0,0}$ | $\chi({\cal M},l_{3})$
$-$ | $(1,1,16)_{0,0,0,1,0}$ | $\chi({\cal M},l_{4})$
$-$ | $(1,1,16)_{0,0,0,0,1}$ | $\chi({\cal M},l_{5})$
$-$ | $(1,1,120)_{0,0,0,0,0}$ | $\chi({\cal M},{\cal O}_{\cal M})$
Table 2: Massless spectrum and corresponding index under
$\left(SU(3)_{C}\times SU(2)_{L}\times
SO(16)\right)_{U(1)_{1},U(1)_{2},U(1)_{3},U(1)_{4},U(1)_{5}}$. Here, the
subscript indices label the $U(1)_{1,2,3,4,5}$ charges.

Note that the hypercharge gauge boson will couple to string axions through the
St$\mathrm{\ddot{u}}$ckelberg couplings. Taking into account the
St$\mathrm{\ddot{u}}$ckelberg couplings of K$\mathrm{\ddot{a}}$hler axions and
dilaton axion, the hypercharge gauge boson $U(1)_{Y}=\sum_{a}f_{a}U(1)_{a}$
becomes massless under101010For more details, see, Ref. Otsuka:2018rki .

$\displaystyle\sum_{a}{\rm tr}(T_{a}^{2})f_{a}m_{a}^{(i)}=0,$ (3.25)
$\displaystyle\sum_{a,b,c,d}{\rm tr}(T_{a}T_{b}T_{c}T_{d})f_{a}X_{bcd}=0,$
(3.26)

where $f_{a}$ is chosen as in Eq. (3.21). Furthermore, flux quanta are
constrained by the K-theory condition (2.8):

$\displaystyle\sum_{a}{\rm tr}(T_{a})m_{a}^{i}=0,$ (3.27)

with $n_{a}={\rm tr}(T_{a})$ in Eq. (2.8). Together with the masslessness
condition of the hypercharge gauge boson (3.25), we obtain

$\displaystyle m_{1}^{i}+m_{2}^{i}+m_{4}^{i}=0,$ (3.28)

which leads to the following equality111111Here, we omit the summation over
$i$.:

$\displaystyle m_{2}^{i}c_{2}(W)_{i}$
$\displaystyle=-m_{2}^{i}(3c_{1}^{2}(l_{1})+2c_{1}^{2}(l_{2})+c_{1}^{2}(l_{3})+c_{1}^{2}(l_{4})+c_{1}^{2}(l_{5}))_{i}$
$\displaystyle=-3X_{112}-2X_{222}+(m_{1}^{i}+m_{4}^{i})(c_{1}^{2}(l_{3})+c_{1}^{2}(l_{4})+c_{1}^{2}(l_{5}))_{i}$
$\displaystyle=-3(N_{\rm gen}(Q_{1})-N_{\rm gen}(Q_{2}))-2X_{222}+N_{\rm
gen}(u^{c})+N_{\rm gen}(d^{c})-m_{4}^{i}c_{2}(W)_{i}$ $\displaystyle\quad-
m_{4}^{i}(3c_{1}^{2}(l_{1})+2c_{1}^{2}(l_{2}))_{i}$ $\displaystyle=-3(N_{\rm
gen}(Q_{1})-N_{\rm gen}(Q_{2}))-2X_{222}+N_{\rm gen}(u^{c})+N_{\rm
gen}(d^{c})-m_{4}^{i}c_{2}(W)_{i}$ $\displaystyle\quad+3(N_{\rm
gen}(u_{R_{2}}^{c})-N_{\rm gen}(d_{R_{2}}^{c}))+2(N_{\rm gen}(L_{3})+N_{\rm
gen}(L_{4})).$ (3.29)

Here, we use Eq. (3.22) and

$\displaystyle N_{\rm gen}(u_{R_{2}}^{c})-N_{\rm gen}(d_{R_{2}}^{c})$
$\displaystyle=-X_{114},$ $\displaystyle N_{\rm gen}(L_{3})+N_{\rm
gen}(L_{4})$ $\displaystyle=-X_{224},$ $\displaystyle N_{\rm
gen}(Q_{1})-N_{\rm gen}(Q_{2})$ $\displaystyle=X_{112},$ $\displaystyle N_{\rm
gen}(u^{c})+N_{\rm gen}(d^{c})$ $\displaystyle=X_{133}+X_{144}+X_{155}.$
(3.30)

Since
$\sum_{i}m_{1}^{i}c_{2}(W)_{i}=-\sum_{i}(m_{2}^{i}+m_{4}^{i})c_{2}(W)_{i}=0$
is satisfied when $N_{\rm gen}(Q)=N_{\rm gen}(u^{c})=N_{\rm gen}(d^{c})$, Eq.
(3.29) is further simplified as

$\displaystyle\sum_{i}m_{2}^{i}c_{2}(T{\cal M})_{i}$ $\displaystyle=3(N_{\rm
gen}(Q_{1})-N_{\rm gen}(Q_{2}))-2N_{\rm gen}(Q)-3(N_{\rm
gen}(u_{R_{2}}^{c})-N_{\rm gen}(d_{R_{2}}^{c}))-2(N_{\rm gen}(L_{3})+N_{\rm
gen}(L_{4})),$ (3.31)

where we use $N_{\rm gen}(Q)=N_{\rm gen}(u^{c})=N_{\rm gen}(d^{c})$ and
$2X_{222}=-\sum_{i}m_{2}^{i}c_{2}(T{\cal M})_{i}$ due to $\chi(l_{2})=0$.
Thus, if $N_{\rm gen}(Q)=N_{\rm gen}(Q_{1})$ with $N_{\rm gen}(Q_{2})=0$, we
obtain a similar bound for the sum of the index of chiral particles, as
discussed in previous sections:

$\displaystyle|N_{\rm gen}(Q)-3(N_{\rm gen}(u_{R_{2}}^{c})-N_{\rm
gen}(d_{R_{2}}^{c}))-2(N_{\rm gen}(L_{3})+N_{\rm gen}(L_{4}))|\leq|m_{\rm
max}|\|c_{2}(T{\cal M})\|.$ (3.32)

Note that the value of indices of $u_{R_{2}}^{c},d_{R_{2}}^{c}$ and $L_{3,4}$
depends on a concrete model, but it is typically of ${\cal O}(N_{\rm
gen}(Q))$.

##  4 Type II string

In this section, we move to the Type II orientifold compactifications. In
particular, we focus on Type IIB orientifold compactifications with O3- and
O7-planes, but the analysis can be extended to the T-dual Type IIA
intersecting D-brane models.

On $N_{a}$ stacks of magnetized D7-branes wrapping holomorphic divisors
$\Gamma_{\rm D7}$ of Calabi-Yau threefolds ${\cal M}$ 121212From the
calibration condition of D7-branes, four-cycles $\Gamma_{\rm D7}$ are
considered the holomorphic divisor Marino:1999af ., world-volume fluxes induce
the degenerate number of chiral zero modes. Similar to the previous sections,
it is possible to embed $U(1)$ gauge fluxes into $U(N_{a})$ on the D7-branes
that do not lie on the O7-planes or $SO(2N_{a})/Sp(2N_{a})$ on the D7-branes
that lie on the O7-planes. Specifically, we focus on $U(N_{a})$ magnetized
D7-branes. Following Ref. Blumenhagen:2008zz , we choose the background gauge
invariant field strength:

$\displaystyle{\cal
F}_{a}=(\bar{F}_{a}^{0}+2\pi\iota^{\ast}B_{2})T_{0}+\sum_{i}\bar{F}_{a}^{i}T_{i},$
(4.1)

where $\iota^{\ast}$ denotes the pull-back from the CY threefolds ${\cal M}$
to $\Gamma_{\rm D7}$. Here, $T_{0}$ and $T_{i}$ denote the generators of
diagonal $U(1)_{a}$ and the other traceless Abelian elements of $U(N_{a})$,
respectively. An orientifold projection $\Omega(-1)^{F_{L}}\sigma$ for $B_{2}$
and ${\cal F}$ is chosen as

$\displaystyle\Omega(-1)^{F_{L}}B_{2}=-B_{2},\qquad\Omega(-1)^{F_{L}}{\cal
F}=-{\cal F},$ (4.2)

and the holomorphic involution $\sigma$ acts on the Kähler form and NS-NS two-
form $B_{2}$ on ${\cal M}$:

$\displaystyle\sigma^{\ast}J=+J,\qquad\sigma^{\ast}B_{2}^{\pm}=\pm
B_{2}^{\pm}.$ (4.3)

Thus, the following internal gauge invariant field strength will be relevant
to the realization of chiral zero modes:

$\displaystyle\overline{{\cal
F}}^{+}=-i(l_{s}^{2}\bar{F}+2\pi\iota^{\ast}B_{2}^{+}\mathbb{I}).$ (4.4)

Note that such background fluxes $\bar{F}$ are assumed to be holomorphic and
preserve the supersymmetry:

$\displaystyle J\wedge\bar{F}=0$ (4.5)

on the divisor $\Gamma_{\rm D7}$, where $J$ denotes the Kähler form of ${\cal
M}$. Furthermore, such magnetized D7-branes as well as O7-planes induce the
D3-brane charge:

$\displaystyle Q_{\rm D7}^{a}$
$\displaystyle=N_{a}\frac{\chi(D_{a})}{24}+\frac{1}{8\pi^{2}}\int_{D_{a}}{\rm
tr}{\cal F}_{a}^{2},$ $\displaystyle Q_{\rm O7}$
$\displaystyle=\frac{\chi(D_{\rm O7})}{6}=\frac{1}{6}\int_{\cal M}c_{2}(D_{\rm
O7})\wedge[D_{\rm O7}],$ (4.6)

where $\chi(D_{a})$ and $\chi(D_{\rm O7})$ denote the Euler characteristic of
divisors $D_{a}$ and $D_{\rm O7}$, respectively. Then, the D3-brane tadpole
cancellation conditions become

$\displaystyle N_{\rm D3}+\frac{N_{\rm flux}}{2}+N_{\rm
gauge}=\frac{\chi(Y_{4})}{24},$ (4.7)

where

$\displaystyle N_{\rm gauge}$
$\displaystyle=-\sum_{a}\frac{1}{8\pi^{2}}\int{\rm tr}{\cal F}_{a}^{2},$
$\displaystyle\frac{\chi(Y_{4})}{24}$
$\displaystyle=\frac{N_{O3}}{4}+\frac{\chi(D_{O7})}{12}+\sum_{a}N_{a}\frac{\chi(D_{a})}{24}.$
(4.8)

Here, we suppose an uplifting of Type IIB orientifolds to the F-theory
compactifications on Calabi-Yau fourfolds $Y_{4}$ with Euler characteristics
$\chi(Y_{4})$.

As discussed in Secs. 2 and 3, we derive the upper bound on the magnetic flux
quanta determining the number of chiral zero modes. Now that the gauge fluxes
are rewritten by

$\displaystyle N_{\rm gauge}$
$\displaystyle=-\frac{1}{2}\sum_{a}N_{a}\int_{D_{a}}c_{1}(l_{a})\wedge
c_{1}(l_{a})=-\frac{1}{2}\sum_{a}N_{a}\int_{\cal M}c_{1}(l_{a})\wedge
c_{1}(l_{a})\wedge[D_{a}]$
$\displaystyle=\frac{1}{2}\sum_{a}N_{a}m_{a}^{2}\kappa_{aaa},$ (4.9)

with $c_{1}(l_{a})=m_{a}[D_{a}]$ and $\kappa_{aaa}:=-\int_{\cal
M}[D_{a}]\wedge[D_{a}]\wedge[D_{a}]$, it is bounded from above by using the
tadpole cancellation condition:

$\displaystyle\frac{{\rm min}_{a}(N_{a}\kappa_{aaa})}{2}\sum_{a}m_{a}^{2}\leq
N_{\rm gauge}=\frac{\chi(Y_{4})}{24}-N_{\rm D3}-\frac{N_{\rm
flux}}{2}\leq\frac{\chi(Y_{4})}{24}.$ (4.10)

Here, we use $N_{\rm flux}\geq 0$ due to the imaginary self-dual condition of
three-form fluxes. The above expression is similar to Eq. (2.28) discussed in
the heterotic line bundle models. When we define $m_{\rm max}$ satisfying
$|m_{a}|\leq{\rm max}_{a}(m_{a}):=|m_{\rm max}|$ for all $a$, we find that

$\displaystyle|m_{\rm max}|^{2}\leq\frac{\chi(Y_{4})}{12\,{\rm
min}_{a}(N_{a}\kappa_{aaa})}-(h^{1,1}(D_{a})-1),$ (4.11)

where we assume $m_{a}^{i}\neq 0$ for all $a$, but it is possible to derive
$m_{a}=0$ for some $a$. Although the term $h^{1,1}(D_{a})-1$ in the above
equation will be modified in this case, we adopt the above conservative bound
of the flux quanta in the following analysis.

Let us discuss the chiral zero modes arising from intersections of two
different stacks of magnetized D7-branes carrying holomorphic line bundles
$l_{a}$ and $l_{b}$, respectively. In particular, we focus on matter fields in
the bifundamental representation $(N_{a},\bar{N}_{b})$, as classified in the
following two classes131313Fore more details, see, e.g., Ref
Blumenhagen:2008zz .:

  1. 1.

Matter divisors

If divisors of two D7-brane coincide with each other, i.e., $D:=D_{a}=D_{b}$,
the index of bifundamental matter fields is counted by the extension group
Katz:2002gh . The chiral index is determined by

$\displaystyle I_{ab}$ $\displaystyle=\int_{\cal
M}[D]\wedge[D]\wedge(c_{1}(l_{a})-c_{1}(l_{b}))$
$\displaystyle=-(m_{a}-m_{b})\kappa_{DDD},$ (4.12)

with $c_{1}(l_{a,b})=m_{a,b}[D]$, $\kappa_{DDD}:=-\int_{\cal
M}[D]\wedge[D]\wedge[D]$. Thus, the chiral index is bounded from above by

$\displaystyle|I_{ab}|\leq 2|m_{\rm max}||\kappa_{DDD}|\leq
2|\kappa_{DDD}|\left(\frac{\chi(Y_{4})}{12\,{\rm
min}_{a}(N_{a}\kappa_{aaa})}-(h^{1,1}_{-}(D_{a}^{+})-1)\right),$ (4.13)

where we use Eq. (4.11) in the last inequality.

  2. 2.

Matter curves

If divisors of two D7-brane are different but intersect over a curve $C$ of
genus $g$, the chiral index is determined by

$\displaystyle I_{ab}$ $\displaystyle=\int_{\cal
M}[D_{a}]\wedge[D_{b}]\wedge(c_{1}(l_{a})-c_{1}(l_{b}))$
$\displaystyle=-m_{a}\kappa_{aab}+m_{b}\kappa_{abb},$ (4.14)

with $c_{1}(l_{a})=m_{a}[D_{a}]$, $c_{1}(l_{b})=m_{b}[D_{b}]$,
$\kappa_{aab}:=-\int_{\cal M}[D_{a}]\wedge[D_{a}]\wedge[D_{b}]$ and
$\kappa_{abb}:=-\int_{\cal M}[D_{a}]\wedge[D_{b}]\wedge[D_{b}]$. By using Eq.
(4.11), we arrive at the upper bound on the chiral index:

$\displaystyle|I_{ab}|$ $\displaystyle\leq 2|m_{\rm max}||{\rm
max}(\kappa_{aab},\kappa_{abb})|$ $\displaystyle\leq 2|{\rm
max}(\kappa_{aab},\kappa_{abb})|\left(\frac{\chi(Y_{4})}{12\,{\rm
min}_{a}(N_{a}\kappa_{aaa})}-(h^{1,1}_{-}(D_{a}^{+})-1)\right).$ (4.15)

For illustrative purposes, let us focus on toroidal orientifolds. The
distribution of flux quanta was explicitly discussed in Ref. Kai:2023ivp  in
the context of Type IIB string theory on
$T^{6}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\prime})$ orientifolds with
$N_{a}$ stacks of magnetized D$(3+2n)$-branes and O-planes. When we turn on
$U(1)_{a}$ magnetic fluxes on $(T^{2})_{i}$:

$\displaystyle\frac{m_{a}^{i}}{2\pi}\int_{(T^{2})_{i}}F_{a}^{i}=n_{a}^{i},$
(4.16)

the wrapping numbers $m_{a}^{i}$ and magnetic fluxes $n_{a}^{i}$ lead to the
net number of chiral zero modes between two stacks $a$ and $b$ of D-branes:

$\displaystyle I_{ab}=\Pi_{i=1}^{3}(n_{a}^{i}m_{b}^{i}-n_{b}^{i}m_{a}^{i}).$
(4.17)

Furthermore, magnetized D-branes carry low-dimensional D-brane charges. In
particular, the D3-brane tadpole charges are satisfied when

$\displaystyle\sum_{a}N_{a}n_{a}^{1}n_{a}^{2}n_{a}^{3}+\frac{1}{2}N_{\rm
flux}=16,$ (4.18)

where we consider 64 O3-planes. As discussed in Ref. Marchesano:2004xz , one
can realize the supersymmetric brane configurations with the SM-like spectra:

$N_{\alpha}$ | Gauge group | ($n_{\alpha}^{1},m_{\alpha}^{1}$) | ($n_{\alpha}^{2},m_{\alpha}^{2}$) | ($n_{\alpha}^{3},m_{\alpha}^{3}$)
---|---|---|---|---
$N_{a}=6$ | $SU(3)_{C}$ | (1,0) | ($g,1$) | ($g,-1$)
$N_{b}=2$ | $USp(2)_{L}$ | (0,1) | ($1,0$) | ($0,-1$)
$N_{c}=2$ | $USp(2)_{R}$ | (0,1) | ($0,-1$) | ($1,0$)
$N_{d}=2$ | $U(1)_{d}$ | (1,0) | ($g,1$) | ($g,-1$)

Here, the magnetic flux $g$ determines the index of quarks and leptons
$I_{ab}=I_{ac}=g$.141414Here, we count the index of both the
$\mathbb{Z}_{2}$-even and -odd modes. From the tadpole cancellation conditions
(4.18), such an index is bounded from above by the orientifold contribution:

$\displaystyle 8g^{2}=16-\frac{N_{\rm flux}}{2}\leq 16,$ (4.19)

i.e., $|I_{ab}|\leq\sqrt{2}$, where the three-form fluxes are supposed to be
the imaginary self-dual fluxes.

So far, we have not considered the existence of a discrete $B$-field, but such
a half-integer flux changes the expression of the tadpole cancellation
conditions. When we turn on $B$-field on the third torus, the D3-brane tadpole
cancellation condition is modified by introducing
$\tilde{n}_{a}^{3}=n_{a}^{3}+\frac{1}{2}m_{a}^{3}$:

$\displaystyle\sum_{a}N_{a}n_{a}^{1}n_{a}^{2}\tilde{n}_{a}^{3}+\frac{1}{2}N_{\rm
flux}=8.$ (4.20)

Following Ref. Kai:2023ivp , one can realize the supersymmetric brane
configurations with the Standard Model-like spectra:

$N_{\alpha}$ | Gauge group | ($n_{\alpha}^{1},m_{\alpha}^{1}$) | ($n_{\alpha}^{2},m_{\alpha}^{2}$) | ($\tilde{n}_{\alpha}^{3},m_{\alpha}^{3}$)
---|---|---|---|---
$N_{a}=8$ | $U(4)_{C}$ | ($0,-1$) | ($1,1$) | ($1/2,1$)
$N_{b}=4$ | $U(2)_{L}$ | ($g,1$) | ($1,0$) | ($1/2,-1$)
$N_{c}=4$ | $U(2)_{R}$ | ($g,-1$) | ($0,1$) | ($1/2,-1$)

Here, the magnetic flux $g$ determines the index of quarks and leptons
$I_{ab}=I_{ca}=g$. From the tadpole cancellation conditions (4.20), such an
index is bounded from above by the orientifold contribution:

$\displaystyle 2g=8-\frac{N_{\rm flux}}{2}\leq 8,$ (4.21)

i.e., $|I_{ab}|\leq 4$, where the three-form fluxes are supposed to be the
imaginary self-dual fluxes.

We restrict ourselves to evaluating the explicit bound in concrete toroidal
orientifolds, but it is interesting to explore the distribution of the chiral
index of matter fields on magnetized D-branes wrapping holomorphic cycles in
Calabi-Yau threefolds. Since the upper bound on the Atiyah-Singer index will
depend on what extent thee-form fluxes $N_{\rm flux}$ distribute inside the
Type IIB/F-theory landscape, we will leave a comprehensive study about the
distribution of the chiral index in the future.

##  5 Conclusions

We studied the index of chiral fermions in the effective action of
$E_{8}\times E_{8}^{\prime}$ and $SO(32)$ heterotic line bundle models, and
magnetized D7-branes in Type IIB/F-theory compactifications. We derived an
analytical expression for the upper bound of the chiral index, which depends
on the topological quantities of CY manifolds. It results in a finite number
of consistent line bundle models subject to phenomenological constraints on
favorable CYs. We focused on a limited class of consistent string models, but
it is interesting to extend our analysis into other corners of the string
landscape in the future.

In $E_{8}\times E_{8}^{\prime}$ and $SO(32)$ heterotic line bundle models, we
evaluated the upper bound on complete intersection CY threefolds. Since the CY
volume is bounded from above by ${\cal V}\leq 25$ to realize the observed
value of 4D gauge couplings, semi-realistic models are realized on some of the
complete intersection CY threefolds. On quotient CY threefolds, the index of
chiral fermions decreases due to the freely-acting discrete symmetry group. It
turned out that the chiral index satisfies $N_{\rm gen}\leq 100$ on quotient
CY threefolds we analyzed. We also commented on the favorable CY threefolds
with the large number of Kähler moduli in the Kreuzer-Skarke database. Note
that for $SO(32)$ heterotic string theory, we found the bound on the chiral
index of Higgs doublets and vector-like quarks in addition to quarks and
leptons. Our finding bound on the chiral index is correlated with the second
Chern number of the tangent bundle of CY threefolds. Such a mutual relation
between the chiral index and the second Chern number was also pointed out in
the context of a machine learning study Otsuka:2020nsk , and it would be
interesting to figure out a physical background of this suggestive behavior.

In Type IIB/F-theory compactifications, we focused on the index of chiral
fermions arising in the intersection between two stacks of magnetized
D7-branes. Since the world-volume fluxes on D7-branes play a role not only in
determining the chiral index but also in inducing the D3-brane charges, the
D3-brane tadpole cancellation condition restricts the maximal value of flux
quanta. It turned out that the chiral index on matter divisors or matter
curves is restricted by the number of D7-brane and topological quantities of
CY manifolds. For concreteness, we analyzed the generation number of quarks
and leptons on $T^{6}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\prime})$
orientifolds. The index is indeed bounded by the orientifold contributions.
Thus, the small tadpole charge leads to a few generations of chiral fermions.

###### Acknowledgements.

The authors would like to thank Andrei Constantin for useful discussions. This
work was supported in part by Kyushu University’s Innovator Fellowship Program
(S.N.), JSPS KAKENHI Grant Numbers JP20K14477 (H.O.), JP 21J20739 (M.T.),
JP22J12877 (K.I.) and JP23H04512 (H.O).

## References

  * (1) P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, _Vacuum configurations for superstrings_ , _Nucl. Phys. B_ 258 (1985) 46.
  * (2) E.I. Buchbinder, A. Constantin and A. Lukas, _The Moduli Space of Heterotic Line Bundle Models: a Case Study for the Tetra-Quadric_ , _JHEP_ 03 (2014) 025 [1311.1941].
  * (3) A. Constantin, A. Lukas and C. Mishra, _The Family Problem: Hints from Heterotic Line Bundle Models_ , _JHEP_ 03 (2016) 173 [1509.02729].
  * (4) S. Groot Nibbelink, O. Loukas, F. Ruehle and P.K.S. Vaudrevange, _Infinite number of MSSMs from heterotic line bundles?_ , _Phys. Rev. D_ 92 (2015) 046002 [1506.00879].
  * (5) P. Candelas, A.M. Dale, C.A. Lutken and R. Schimmrigk, _Complete Intersection Calabi-Yau Manifolds_ , _Nucl. Phys. B_ 298 (1988) 493.
  * (6) P. Candelas, C.A. Lutken and R. Schimmrigk, _COMPLETE INTERSECTION CALABI-YAU MANIFOLDS. 2. THREE GENERATION MANIFOLDS_ , _Nucl. Phys. B_ 306 (1988) 113.
  * (7) M. Kreuzer and H. Skarke, _Complete classification of reflexive polyhedra in four-dimensions_ , _Adv. Theor. Math. Phys._ 4 (2000) 1209 [hep-th/0002240].
  * (8) R. Blumenhagen, G. Honecker and T. Weigand, _Loop-corrected compactifications of the heterotic string with line bundles_ , _JHEP_ 06 (2005) 020 [hep-th/0504232].
  * (9) R. Blumenhagen, G. Honecker and T. Weigand, _Supersymmetric (non-)Abelian bundles in the Type I and SO(32) heterotic string_ , _JHEP_ 08 (2005) 009 [hep-th/0507041].
  * (10) E. Witten, _GLOBAL ANOMALIES IN STRING THEORY_ , in _Symposium on Anomalies, Geometry, Topology_ , 6, 1985.
  * (11) D.S. Freed, _Determinants, Torsion, and Strings_ , _Commun. Math. Phys._ 107 (1986) 483.
  * (12) E. Witten, _D-branes and K-theory_ , _JHEP_ 12 (1998) 019 [hep-th/9810188].
  * (13) A.M. Uranga, _D-brane probes, RR tadpole cancellation and K-theory charge_ , _Nucl. Phys. B_ 598 (2001) 225 [hep-th/0011048].
  * (14) L.B. Anderson, J. Gray, A. Lukas and E. Palti, _Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds_ , _Phys. Rev. D_ 84 (2011) 106005 [1106.4804].
  * (15) L.B. Anderson, J. Gray, A. Lukas and E. Palti, _Heterotic Line Bundle Standard Models_ , _JHEP_ 06 (2012) 113 [1202.1757].
  * (16) A. Lukas et al., http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/cicylist/cicylist.txt.
  * (17) V. Braun, _On Free Quotients of Complete Intersection Calabi-Yau Manifolds_ , _JHEP_ 04 (2011) 005 [1003.3235].
  * (18) L.B. Anderson, J. Gray, Y.-H. He and A. Lukas, _Exploring Positive Monad Bundles And A New Heterotic Standard Model_ , _JHEP_ 02 (2010) 054 [0911.1569].
  * (19) M. Demirtas, C. Long, L. McAllister and M. Stillman, _The Kreuzer-Skarke Axiverse_ , _JHEP_ 04 (2020) 138 [1808.01282].
  * (20) E. Plauschinn, _The tadpole conjecture at large complex-structure_ , _JHEP_ 02 (2022) 206 [2109.00029].
  * (21) H. Abe, T. Kobayashi, H. Otsuka and Y. Takano, _Realistic three-generation models from SO(32) heterotic string theory_ , _JHEP_ 09 (2015) 056 [1503.06770].
  * (22) H. Otsuka, _SO(32) heterotic line bundle models_ , _JHEP_ 05 (2018) 045 [1801.03684].
  * (23) S. Groot Nibbelink, O. Loukas and F. Ruehle, _(MS)SM-like models on smooth Calabi-Yau manifolds from all three heterotic string theories_ , _Fortsch. Phys._ 63 (2015) 609 [1507.07559].
  * (24) R. Blumenhagen, S. Moster and T. Weigand, _Heterotic GUT and standard model vacua from simply connected Calabi-Yau manifolds_ , _Nucl. Phys. B_ 751 (2006) 186 [hep-th/0603015].
  * (25) R. Blumenhagen, S. Moster, R. Reinbacher and T. Weigand, _Massless Spectra of Three Generation U(N) Heterotic String Vacua_ , _JHEP_ 05 (2007) 041 [hep-th/0612039].
  * (26) L.B. Anderson, A. Constantin, S.-J. Lee and A. Lukas, _Hypercharge Flux in Heterotic Compactifications_ , _Phys. Rev. D_ 91 (2015) 046008 [1411.0034].
  * (27) H. Otsuka and K. Takemoto, _$SO(32)$ heterotic standard model vacua in general Calabi-Yau compactifications_, _JHEP_ 11 (2018) 034 [1809.00838].
  * (28) M. Marino, R. Minasian, G.W. Moore and A. Strominger, _Nonlinear instantons from supersymmetric p-branes_ , _JHEP_ 01 (2000) 005 [hep-th/9911206].
  * (29) R. Blumenhagen, V. Braun, T.W. Grimm and T. Weigand, _GUTs in Type IIB Orientifold Compactifications_ , _Nucl. Phys. B_ 815 (2009) 1 [0811.2936].
  * (30) S.H. Katz and E. Sharpe, _D-branes, open string vertex operators, and Ext groups_ , _Adv. Theor. Math. Phys._ 6 (2003) 979 [hep-th/0208104].
  * (31) T. Kai, K. Ishiguro, H. Okada and H. Otsuka, _Flavor, CP and Metaplectic Modular Symmetries in Type IIB Chiral Flux Vacua_ , 2305.19155.
  * (32) F. Marchesano and G. Shiu, _Building MSSM flux vacua_ , _JHEP_ 11 (2004) 041 [hep-th/0409132].
  * (33) H. Otsuka and K. Takemoto, _Deep learning and k-means clustering in heterotic string vacua with line bundles_ , _JHEP_ 05 (2020) 047 [2003.11880].