# Hidden Sectors from Multiple Line Bundles
for the $B-L$ MSSM

Anthony Ashmore,a,b Sebastian Dumitruc and Burt A. Ovrutc
aEnrico Fermi Institute & Kadanoff Center for Theoretical Physics
University of Chicago, Chicago, IL 60637, USA
bSorbonne Université, LPTHE,
F-75005 Paris, France
cDepartment of Physics, University of Pennsylvania
Philadelphia, PA 19104, USA

We give a formalism for constructing hidden sector bundles as extensions of
sums of line bundles in heterotic $M$-theory. Although this construction is
generic, we present it within the context of the specific Schoen threefold
that leads to the physically realistic $B-L$ MSSM model. We discuss the
embedding of the line bundles, the existence of the extension bundle, and a
number of necessary conditions for the resulting bundle to be slope-stable and
thus $N=1$ supersymmetric. An explicit example is presented, where two line
bundles are embedded into the $SU(3)$ factor of the $E_{6}\times SU(3)$
maximal subgroup of the hidden sector $E_{8}$ gauge group, and then enhanced
to a non-Abelian $SU(3)$ bundle by extension. For this example, there are in
fact six inequivalent extension branches, significantly generalizing that
space of solutions compared with hidden sectors constructed from a single line
bundle.

††footnotetext<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 $B-L$ MSSM and Universal Constraints
    1. 2.1 $B-L$ MSSM Vacuum
    2. 2.2 Universal Physical Constraints
  3. 3 Two Line Bundle Embedding
    1. 3.1 Line Bundle Embeddings
    2. 3.2 Embedding Constraints for Two Line Bundles
    3. 3.3 Line Bundles Scan
  4. 4 Different Extension Branches
    1. 4.1 Choosing a Different Branch
    2. 4.2 Full Branch Structure Scan
  5. 5 Conclusion
  6. A Conventions
  7. B Anomaly Cancellation
  8. C Linearization Constraints
  9. D Genus-One Corrected FI Terms
  10. E Gauge Threshold Corrections
  11. F Subbundles of Isomorphic Extension Bundles

##  1 Introduction

Hořava and Witten were the first to put eleven-dimensional $M$-theory on a
$S^{1}/\mathbb{Z}_{2}$ orbifold [1, 2]. Consistency of the theory requires a
ten-dimensional orbifold plane at each end of the $S^{1}/\mathbb{Z}_{2}$
interval, with each plane carrying an $E_{8}$ gauge connection. The degrees of
freedom arising from each orbifold plane are usually referred to as the
“observable” and “hidden” sectors respectively. One can compactify
Hořava–Witten theory on a Calabi–Yau threefold to give five-dimensional
“heterotic $M$-theory”. Integrating out the modes on the
$S^{1}/\mathbb{Z}_{2}$ interval, one then finds a four-dimensional effective
theory with an $E_{8}\times E_{8}$ gauge symmetry [3, 4, 5, 6, 7]. The action
of this theory matches that of the standard heterotic string, however the mass
scales are governed by the Calabi–Yau radius and the orbifold length, which
allow one to more easily obtain the unification gauge couplings and mass scale
required in phenomenologically realistic GUT models. Other aspects of such
theories, such as such as the spontaneous breaking of their supersymmetry [8,
9, 10, 11, 12, 13], the role of five-branes in the orbifold interval [14, 15,
16, 17, 18, 19, 20] and the methods for stabilizing moduli [21, 22, 23, 24],
have also been discussed in the literature.

Within this context there have been a large number of physically promising
low-energy models constructed as top-down string models [25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35]. In particular, in [26, 36, 25, 37] it was shown that
one can obtain the exact MSSM spectrum in Minkowski space by compactifying on
a certain Schoen Calabi–Yau threefold equipped with a particular slope-stable
$SU(4)$ gauge bundle. The resulting low-energy field content consists of three
families of quarks and leptons with three right-handed neutrino chiral
supermultiplets, one per family, and a Higgs-Higgs conjugate pair of chiral
superfields [26, 36, 25]. There are no vector-like pairs and no exotic fields.
However, in addition to the gauge group $SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}$ of the MSSM, there is an extra gauged $U(1)_{B-L}$. Hence, this is
referred to as the $B-L$ MSSM model [38, 39, 40, 41, 42]. A number of papers
[43, 44, 45, 46, 47, 48, 49, 50] discuss the physics of breaking this
additional $U(1)_{B-L}$ symmetry above the electroweak scale.

For completeness, it is essential that there also exists a slope-stable gauge
bundle on the hidden sector plane of the $B-L$ MSSM with the correct
properties to ensure, for example, anomaly cancellation. The majority of
previous work has focussed on constructing a realistic observable sector with
the view that compatible hidden sectors would be found later. This view was
supported by the results of [37], which showed that hidden sector bundles are
compatible with the Bogomolov inequality and so stable hidden sector bundles
likely exist. The goal of this and several recent papers is to fill this gap
by constructing these hidden sectors for the $B-L$ MSSM.

There have been many previous works on constructing and classifying heterotic
line bundle backgrounds in both supersymmetric and non-supersymmetric theories
– see [32, 51, 52, 53, 31, 33, 34, 35, 54, 55, 56, 57, 58, 59, 60] and
references therein for a selection of these. In the context of the $B-L$ MSSM,
viable hidden sector bundles were proposed in [61], built from both single
line bundles and direct sums of line bundles. Unfortunately, this was within
the context of the weakly coupled heterotic string and so the values obtained
for the unification scale and associated gauge couplings were not compatible
with their expected values. In a previous work [62], we rectified this problem
by moving to the strongly coupled heterotic string. We presented a hidden
sector bundle built from a single line bundle $L$ via an induced rank-two
bundle $L\oplus L^{-1}$ which, promisingly, satisfied all of the required
“vacuum” constraints. We found a substantial region of Kähler moduli space in
which the $S^{1}/{\mathbb{Z}}_{2}$ orbifold scale was sufficiently large
compared to the average Calabi–Yau radius, and where the effective strong
coupling parameter was large enough, to obtain the correct values for the
observable sector $SO(10)$ unification scale and gauge couplings. This hidden
sector model, however, came with a caveat: although the results were computed
only to order $\kappa_{11}^{4/3}$, the region of Kähler moduli space in which
one is required to work to guarantee slope-stability generically leads to a
large effective coupling parameter. Even though, by definition, we are working
within the context of “strongly coupled” heterotic theory and, hence, higher-
order corrections are not expected to be negligible, the large coupling
parameter required in our case enhances this concern. That is, the linear
approximation which is standard in heterotic M-theory theory and which was
used in the $L\oplus L^{-1}$ bundle analysis became less trustworthy.

This issue was explored in [62] by computing the theory to the next order,
that is, ${\cal{O}}(\kappa_{11}^{6/3})$. The results were reassuring since the
${\cal{O}}(\kappa_{11}^{6/3})$ corrections both strengthened and even improved
upon the ${\cal{O}}(\kappa_{11}^{4/3})$ results. Be that as it may, this issue
requires further investigation. With this in mind, an alternative construction
of the hidden sector bundle from a single line bundle $L$ was preliminarily
investigated in [62]. Instead of the bundle $L\oplus L^{-1}$ discussed
earlier, one now moves in Kähler moduli space so that the slope of $L^{-1}$
becomes negative. This significantly reduces the size of the effective
coupling parameter and, hence, makes the ${\cal{O}}(\kappa_{11}^{4/3})$
expansion more trustworthy. To preserve supersymmetry, one must move in bundle
moduli space to a rank-two hidden sector bundle $V$, defined by the extension
$0\rightarrow L^{-1}\rightarrow V\rightarrow L\rightarrow 0$. One must then
show that the bundle $V$ is Bogomolov stable (a necessary but not sufficient
condition for stability) and, second, that the extension is non-trivial; that
is, that $H^{1}(L^{-2})\neq 0$. A preliminary examination of this was carried
out in [62]. It was found that while the stability condition is relatively
easy to satisfy, proving that the extension is non-trivial required a
mathematical analysis beyond the scope of that paper. So, this approach also
requires further investigation.

The purpose of the current work is to improve on these results. Our aim is to
find new hidden sector bundles for which the linear approximation can be
trusted while still satisfying all of the phenomenological and vacuum
constraints laid out in [62]. We focus on hidden sectors built from two line
bundles, where the $U(1)\times U(1)$ structure group embeds as $U(1)\times
U(1)\subset SU(3)\subset SU(3)\times E_{6}\subset E_{8}$. We will show that
there are a large number of examples which solve all of the phenomenological
and vacuum constraints, as well as passing a number of non-trivial stability
checks – including Bogomolov stability. Deciding whether or not these
candidate hidden sectors are viable is then reduced to checking whether
certain $\operatorname{Ext}^{1}$ groups are non-vanishing and then carrying
out a more rigorous check of slope stability. Both of these analyses are
beyond the scope of the present work – we will to return to this in future
publications. Our results indicate that it is very promising to build hidden
sectors from two line bundles, and that using three or more line bundles would
lead to even more examples.

In Section 2, we briefly review how the $B-L$ MSSM arises in heterotic
$M$-theory, with a focus on the physical constraints we impose that are
independent of the explicit construction of the hidden sector. In Section 3,
we outline how one can construct a hidden sector bundle from multiple line
bundles using the formalism of “line bundle vectors” [63]. For specificity, we
focus on embedding two line bundles into the $SU(3)$ factor of $SU(3)\times
E_{6}\subset E_{8}$, and compare the line bundle vector formalism with an
explicit embedding via $U(1)$ subgroups of $SU(3)$. The anomaly condition and
the formalism for computing the low-energy spectrum are then presented in this
context. We discuss how to deform the corresponding Whitney sum bundle to an
irreducible $SU(3)$ bundle, and give a number of necessary conditions for this
non-Abelian bundle to be slope-stable. We then scan over possible choices of
pairs of line bundles and reduce the question of whether or not the
corresponding hidden sector bundles exist to the calculation of certain
bundle-valued cohomologies. In Section 4, we extend this analysis to other
branches in bundle moduli space and perform a scan over line bundles once
more. We find a large number of candidate line bundles that could lead to
viable hidden sectors. The appendices contain a discussion of anomaly
cancellation, the linearized approximation for heterotic M-theory, the genus-
one corrected Fayet–Iliopoulos terms for $U(1)$s, expressions for the gauge
couplings, and a discussion of the subbundles of the extension bundle in each
extension class.

##  2 $B-L$ MSSM and Universal Constraints

The Hořava–Witten vacuum is an $S^{1}/\mathbb{Z}_{2}$ orbifold of M-theory
with two ten-dimensional planes, one at each fixed point, separated by a one-
dimensional interval. Each fixed plane, the observable and hidden sector
planes respectively, has an $E_{8}$ gauge group. Heterotic M-theory is
obtained by compactifying six of the ten remaining dimensions on a Calabi–Yau
threefold. Heterotic M-theory compactification has been discussed widely in
the literature – see, for example, [64, 26, 25, 36]. We now present a brief
outline of the aspects of the $B-L$ MSSM vacuum of heterotic M-theory that are
relevant for the present paper.

###  2.1 $B-L$ MSSM Vacuum

The first step in constructing the $B-L$ vacuum of heterotic M-theory is to
compactify six of the ten dimensions on a particular Calabi–Yau manifold –
specifically, a Schoen threefold quotiented by a freely acting
$\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ [65, 64]. Secondly, on the Schoen
threefold of the observable plane, one places a holomorphic vector bundle with
structure group $SU(4)\subset E_{8}$. The connection on the $SU(4)$ bundle
breaks the $E_{8}$ group down to

$E_{8}\to Spin(10)$ (2.1)

in four dimensions, leading to a $Spin(10)$ “grand unified” group in the
observable sector. This GUT group is then broken further at a scale $\langle
M_{U}\rangle$ of order $10^{16}\text{ GeV}$ to the gauge group of the $B-L$
MSSM by turning on two Wilson lines, each associated with a different
$\mathbb{Z}_{3}$ factor of the $\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ holonomy.
This preserves the $N=1$ supersymmetry of the four-dimensional effective
theory, but breaks the observable sector group to

$Spin(10)\to SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}\times U(1)_{B-L}\ ,$
(2.2)

which is the gauge group of the MSSM with an additional gauged $U(1)_{B-L}$
symmetry. The spectrum of the $B-L$ MSSM is determined by the structure of the
Schoen threefold [65, 36] and the precise choice of $SU(4)$ bundle [37]. More
details can be found in [25, 26]. The spectrum contains exactly the three
quark and lepton families of the MSSM, including three additional right-handed
neutrino chiral multiplets, one per family. It also contains the conventional
$H_{u}$ and $H_{d}$ Higgs doublet supermultiplets. There are no exotic fields
or vector-like pairs.

The observable and hidden sectors are separated by a one-dimensional interval
$S^{1}/\mathbb{Z}_{2}$. Within this interval, parallel to the orbifold planes,
the theory allows for the existence of multiple five-branes. In our previous
paper [62], for simplicity, we assumed that all such branes coalesced into a
single five-brane located near the hidden sector plane. We will make the same
assumptions in the present work. To preserve $N=1$ supersymmetry, this five-
brane must wrap a holomorphic cycle in the Schoen threefold. This is
equivalent to the topological class $W$ of the five-brane being “effective”;
that is,

$W_{i}\geq 0\ ,\quad i=1,2,3\ .$ (2.3)

The numbers $W_{i}$ which characterize the five-brane class are defined in
Appendix B.

The remaining ingredient of the $B-L$ MSSM vacuum is the hidden sector. In
previous work in [62], we constructed a hidden sector using a single line
bundle. However, the main goal of this present work is to construct slope-
stable hidden sector bundles built from two line bundles. As we will show,
this greatly increases the number of viable $B-L$ MSSM hidden sectors. This
work will be presented in Sections 3 and 4.

An important aspect of the $B-L$ MSSM vacuum is the set of geometric moduli
associated with it. The relevant moduli are the following. The Schoen
threefold on which the Hořava–Witten theory is compactified is parametrized by
the three real, positive Kähler moduli, $(a^{1},a^{2},a^{3})$. These
determine, for example, the volume of the Calabi–Yau threefold and, hence, are
associated with the compactification scale. In addition, the
$S^{1}/\mathbb{Z}_{2}$ interval between the observable and hidden sectors is
parametrized by a single real modulus, $\hat{R}$. This determines the physical
length, and hence the mass scale, of the interval. In [62], we derived a set
of constraints on the values that $(a^{1},a^{2},a^{3})$ and $\hat{R}$ must
satisfy, thus restricting us to a specific region inside the positive Kähler
cone. These constraints are a mixture of geometrical and phenomenological
constraints, such as slope-stability of the observable sector $SU(4)$ bundle
or ensuring the gauge couplings are compatible with experimental bounds. Some
of these constraints can depend on the choice of hidden sector bundle, but
only weakly through the effective five-brane class $W_{i}$. We call this set
of constraints “universal”, and present them in the next subsection. On the
other hand, the constraints required to construct a consistent hidden sector
bundle are independent of these universal physical constraints. These “hidden
sector bundle constraints” will be discussed within the context of line bundle
hidden sectors in Sections 3 and 4. Our goal is to find hidden sector gauge
bundles which are slope-stable inside the region of Kähler moduli space carved
out by the universal set of constraints. More specifically, we will search for
such bundles in the sub-region of this moduli space where the linear
approximation to the vacuum is strictly valid – that is, in the region of
relatively small effective coupling.

###  2.2 Universal Physical Constraints

The universal physical constraints, and the geometrical and phenomenological
requirements leading to them, were discussed in detail in [62]. In this
subsection, we briefly review them since they impose important constraints on
the physically allowed region of Kähler and $\hat{R}$ moduli space.

  1. 1.

In order to preserve $N=1$ supersymmetry in the four-dimensional effective
theory, the $SU(4)$ bundle must be both slope-stable and have vanishing slope
[36, 37]. As proven in detail in [37], the $SU(4)$ bundle in the observable
sector of the $B-L$ MSSM is slope-stable, and, hence, admits a connection that
satisfies the Hermitian Yang–Mills (HYM) equations, in the regions of the
positive Kähler cone defined by

$\begin{gathered}\mathopen{}\mathclose{{}\left(a^{1}<a^{2}\leq\sqrt{\tfrac{5}{2}}a^{1}\quad\text{and}\quad
a^{3}<\frac{-(a^{1})^{2}-3a^{1}a^{2}+(a^{2})^{2}}{6a^{1}-6a^{2}}}\right)\quad\text{or}\\\
\mathopen{}\mathclose{{}\left(\sqrt{\tfrac{5}{2}}a^{1}<a^{2}<2a^{1}\quad\text{and}\quad\frac{2(a^{2})^{2}-5(a^{1})^{2}}{30a^{1}-12a^{2}}<a^{3}<\frac{-(a^{1})^{2}-3a^{1}a^{2}+(a^{2})^{2}}{6a^{1}-6a^{2}}}\right)\
.\end{gathered}$ (2.4)

  2. 2.

The squares of the “unified” gauge couplings in both the observable and hidden
sectors must be positive definite. As shown in Appendix E, these conditions
can be written as

$\displaystyle
d_{ijk}a^{i}a^{j}a^{k}+3\frac{\epsilon_{S}^{\prime}\hat{R}}{V^{1/3}}\mathopen{}\mathclose{{}\left(\tfrac{2}{3}a^{1}-\tfrac{1}{3}a^{2}+4a^{3}+(\tfrac{1}{2}-\lambda)^{2}W_{i}a^{i}}\right)$
$\displaystyle>0\ ,$ (2.5) $\displaystyle
d_{ijk}a^{i}a^{j}a^{k}-3\frac{\epsilon_{S}^{\prime}\hat{R}}{V^{1/3}}\mathopen{}\mathclose{{}\left(\tfrac{2}{3}a^{1}-\tfrac{1}{3}a^{2}+4a^{3}+(1-(\tfrac{1}{2}+\lambda)^{2})W_{i}a^{i}}\right)$
$\displaystyle>0\ ,$ (2.6)

where $z=\tfrac{1}{2}+\lambda$ gives the position of the five-brane in the
interval. Following [62, 66], in this paper we set the five-brane close to the
hidden wall at $\lambda=0.49$. Note that for this choice,
$(1-\tfrac{\lambda}{2})^{2}\sim{10}^{-4}$ and
$(1-(\tfrac{1}{2}+\lambda))^{2}\sim{10}^{-2}$. Therefore, to a good
approximation, the terms proportional to the five-brane charges $W_{i}$ drop
out from the expressions above.

  3. 3.

The four-dimensional effective theory is derived by first compactifying on a
Calabi–Yau threefold to give a five-dimensional theory, and then reducing
further on the $S^{1}/{\mathbb{Z}}_{2}$ interval. For this to be consistent,
we require that the length of the interval is sufficiently large compared to
the average Calabi–Yau radius. This condition takes the form

$\frac{\pi\rho{{\widehat{R}}}V^{-1/3}}{(vV)^{1/6}}>1\ .$ (2.7)

  4. 4.

We want our top-down model to give reasonable four-dimensional physics, which
leads to a number of “phenomenological” constraints. One such constraint is
that the $Spin(10)$ grand unification scale, $\langle M_{U}\rangle$, and the
associated unified gauge coupling in the observable sector,
$\langle\alpha_{u}\rangle=\langle g^{(1)}\rangle^{2}/4\pi$, be consistent with
phenomenologically acceptable values for these quantities. As discussed in
[62], a reasonable choice for these quantities, which we will use in this
paper, is

$\langle M_{U}\rangle=3.15\times 10^{16}~{}\text{GeV}\
,\qquad\langle\alpha_{u}\rangle=\frac{1}{26.46}\ .$ (2.8)

  5. 5.

The reduction on the orbifold interval uses a linearized approximation to the
five-dimensional BPS solution of heterotic M-theory [5]. In Appendix B we show
that this requirement takes the form

$2\epsilon_{S}^{\prime}\frac{{\widehat{R}}}{V^{1/3}}\mathopen{}\mathclose{{}\left|\beta_{i}^{(0)}\big{(}z-\tfrac{1}{2}\big{)}-\tfrac{1}{2}W_{i}(\tfrac{1}{2}-\lambda)^{2}}\right|\ll|d_{ijk}a^{j}a^{k}|\
,\quad z\in[0,\lambda+\tfrac{1}{2}]\ ,$ (2.9)
$2\epsilon_{S}^{\prime}\frac{{\widehat{R}}}{V^{1/3}}\mathopen{}\mathclose{{}\left|(\beta_{i}^{(0)}+W_{i})\big{(}z-\tfrac{1}{2}\big{)}-\tfrac{1}{2}W_{i}(\tfrac{1}{2}+\lambda)^{2}}\right|\ll|d_{ijk}a^{j}a^{k}|\
,\quad z\in[\lambda+\tfrac{1}{2},1]\ .$ (2.10)

The $d_{ijk}$ are the intersection numbers for the specific Schoen threefold –
see (B.5). With the five-brane placed near to the hidden wall, $\lambda\approx
0.5$, these two conditions simplify to

$d_{ijk}a^{j}a^{k}\gg\epsilon_{S}^{\prime}\frac{{\widehat{R}}}{V^{1/3}}\beta_{i}^{(0)}\
.$ (2.11)

Specifically, (2.11) is valid for the choice choice of $\lambda=0.49$ used
in[62, 66] and to simplify (2.5) and (2.6) above.

Figure 1: Working in unity gauge, we show the entire region of Kähler moduli
space where the universal conditions (2.4)–(2.8) are satisfied. To a good
approximation, this subspace is independent of the hidden sector bundle. In
the “red” region, the theory is very strongly coupled. Within this region, the
effective expansion parameter $\epsilon_{S}^{\text{eff}}$ is greater than 1.
The “blue” region represents the region of Kähler moduli space in which, in
addition to all the universal conditions being satisfied, the linearization
constraint (2.11) is also valid – that is, where
$\epsilon_{S}^{\text{eff}}\lesssim 0.2$. The intermediate region, where
$0.2<\epsilon_{S}^{\text{eff}}<1$, is displayed in “yellow”. The main focus of
this paper is to find a hidden sector bundle whose region of stability in
moduli space intersects the “blue” region.

As discussed in [62, 67], the actual strong coupling expansion parameter used
in the linearized approximation to the heterotic M-theory vacuum is given by

$\epsilon_{S}^{\rm eff}=\frac{\epsilon_{S}^{\prime}\hat{R}}{V}\ .$ (2.12)

We point out that this is, up to a constant factor of order one, precisely the
strong coupling parameter presented in equation (1.3) of [68]. Since
$V=\frac{1}{6}d_{ijk}a^{i}a^{j}a^{k}$, it follows that
$\epsilon_{S}^{\text{eff}}$ is both Kähler and $\hat{R}$ moduli dependent. We
find that the linearization conditions (2.10) and (2.11), as well as the the
linearized approximation to the square of the gauge couplings, given in (2.5)
and (2.6), are satisfied only in the “weak coupling” region of moduli space
where $\epsilon_{S}^{\text{eff}}\lesssim 0.2$. On the other hand, the
linearization conditions and, hence, the linearized approximation clearly
breaks down in the “strong coupling” regime of moduli space where
$\epsilon_{S}^{\text{eff}}\gtrsim 1$.

Note that these all of the above constraints are invariant under the rescaling

$a^{i}\rightarrow\mu a^{i}\
,\qquad\epsilon^{\prime}_{S}{\widehat{R}}\rightarrow\mu^{3}\epsilon^{\prime}_{S}{\widehat{R}}\
,$ (2.13)

for $\mu\in\mathbb{R}^{+}$ and so one can absorb the coupling
$\epsilon^{\prime}_{S}{{\widehat{R}}}/V^{1/3}$ into the definition of the
moduli. This is equivalent to setting

$\frac{\epsilon^{\prime}_{S}{{\widehat{R}}}}{V^{1/3}}=1\ .$ (2.14)

We refer to this choice as “unity” gauge [61, 62]. Making use of this
simplification, we can find the subregion of the Kähler cone in which the
universal constraints (2.4)–(2.8) are all satisfied. We display this as the
entire colored region in Figure 1. In [62, 66] we were forced to work in a
very strongly coupled regime since we could not preserve supersymmetry in the
hidden sector unless the genus-one corrected slope of the line bundles
vanished. Such a cancellation is possible only if the effective expansion
parameter $\epsilon_{S}^{\text{eff}}$ was larger than 1. It follows that the
linearization constraint (2.11) is not satisfied and, hence, the linearized
approximation to various quantities, such as the gauge couplings in (2.5) and
(2.6), are uncertain. The region of Kähler moduli space in which
$\epsilon_{S}^{\text{eff}}>1$ is shown in “red” in Figure 1. We want to move
away from this very strongly coupled regime and find hidden sector bundles for
which $\epsilon_{S}^{\text{eff}}\lesssim 0.2$ and, hence, the linearization
constraint (2.11) and linearized approximations are valid. The region of
Kähler moduli space in which this holds is shown in “blue” in Figure 1.
Finally, the “intermediate” region, where $0.2<\epsilon_{S}^{\text{eff}}<1$,
is displayed in “yellow”. The focus of this paper is to find hidden sector
bundles which can be stable when the Kähler moduli $(a^{1},a^{2},a^{3})$ are
in the blue region – and so satisfy all of the universal constraints as well
as the linearization condition.

##  3 Two Line Bundle Embedding

We showed in [62, 66] that vacuum configurations with hidden sectors built
from a single line bundle require that the genus-one corrected slope of the
line bundle vanishes. It seems that such a configuration always pushes us into
a strongly coupled regime in which the accuracy of the linear approximation
used to derive the effective four-dimensional theory is uncertain. Here, we
begin with a general overview of constructing Abelian line bundle backgrounds,
before specializing our discussion to hidden sectors built from two line
bundles embedded into the hidden $E_{8}$ gauge group. As an example of how
this works, we will then focus on embeddings which lead to the breaking
pattern $E_{8}\to E_{6}\times U(1)\times U(1)$.

###  3.1 Line Bundle Embeddings

A particularly simple set of hidden sector bundles are those constructed from
line bundles. These are defined by a set of line bundles and the embedding of
their corresponding ${U}(1)$ groups into ${E}_{8}$. Given a set of line
bundles, there are multiple inequivalent ways to embed their Abelian gauge
connections into the ten-dimensional hidden ${E}_{8}$ connection. A
particularly useful formalism for describing these embeddings is using “line
bundle vectors”. Following [63], the Abelian gauge connections can be embedded
in the hidden sector ${E}_{8}$ by expanding the curvature $F_{E_{8}}$ as

$\frac{F_{E_{8}}}{2\pi}=\frac{1}{v^{1/3}}\omega_{i}H_{i}\ ,$ (3.1)

where the coefficients $H_{i}$ are matrices valued in the Lie algebra of
${E}_{8}$. As in [62, 66], the $\omega_{i}$ are the three harmonic
$(1,1)$-forms that span the $H^{1,1}(X,\mathbb{C})$ cohomology on the Schoen
threefold $X$, with their intersection numbers $d_{ijk}$ given by (B.5). Since
the background is Abelian, one can expand the coefficients as

$H_{i}=V_{i}^{I}H_{I}\ ,$ (3.2)

where $I=1,\ldots,8$ runs over the Cartan subalgebra of the hidden ${E}_{8}$.
Here the $H_{I}$ denote the Cartan generators of the ${SO}(16)\subset{E}_{8}$
maximal subgroup, normalised so that

$\operatorname{tr}H_{I}H_{J}=2\,\delta_{IJ}\ ,$ (3.3)

where the trace is $1/30$ of the trace over the $\bm{\underline{248}}$ of
${E}_{8}$ or, equivalently, taken in the fundamental $\bm{\underline{16}}$
representation of ${SO}(16)$.111This agrees with the normalisation in [63]
after noting that their trace is taken in the fundamental of an SU group
rather than SO. See [55, Appendix A] for more details. The eight-component
vectors $\bm{V}_{i}=V_{i}^{I}$ are known as _line bundle vectors_. Given a
choice of Cartan generators, an Abelian hidden sector bundle is completely
specified by a choice of three line bundles vectors $\bm{V}_{i}$, $i=1,2,3$.
For example, $\bm{V}_{1}$ then encodes how much $\omega_{1}$ contributes to
the curvature $F_{E_{8}}$. This is somewhat abstract at the moment, but we
will see how this works with an explicit example later.

As noted in [34], the flux $F_{E_{8}}$ has to be quantized when evaluated on a
string state and integrated over any curve dual to a divisor defined by a sum
of the $\omega_{i}$. Since the string states are characterized by weight
vectors that lie on the ${E}_{8}$ root lattice $\Lambda$, the line bundle
vectors must also lie on the root lattice, that is $\bm{V}_{i}\in\Lambda.$
Following the conventions of [69, 70], the ${E}_{8}$ root lattice is given by
the set of points $\Lambda\in\mathbb{R}^{8}$ such that all eight coordinates
are integers or half-integers (but not a mix of the two), and the coordinates
sum to an even integer. This constrains the form of the line bundle vectors
and ensures that the curvature of the resulting hidden sector bundle obeys
flux quantisation.

The second Chern character of the hidden sector bundle $\mathcal{V}^{(2)}$
constructed from the line bundles is given by

$\text{ch}_{2}(\mathcal{V}^{(2)})=\frac{1}{16\pi^{2}}\operatorname{tr}F_{E_{8}}\wedge
F_{E_{8}}=\tfrac{1}{2}(\bm{V}_{i}\cdot\bm{V}_{j})\,\frac{1}{v^{2/3}}\omega_{i}\wedge\omega_{j}\
,$ (3.4)

where $\bm{V}_{i}\cdot\bm{V}_{j}=V_{i}^{I}V_{j}^{I}$ is the Euclidean scalar
product between the $i^{\text{th}}$ and $j^{\text{th}}$ line bundle vectors.
Since $c_{1}(\mathcal{V})=0$ – following from the fact that the generators
$H_{I}$ are traceless – the second Chern class is given by
$c_{2}(\mathcal{V}^{(2)})=-\text{ch}_{2}(\mathcal{V}^{(2)})$. Wedging with
$\omega_{i}$ and integrating over $X$, one finds that the second Chern numbers
of $\mathcal{V}^{(2)}$ are

$c_{2,i}(\mathcal{V}^{(2)})\equiv\frac{1}{v^{1/3}}\int_{X}c_{2}(\mathcal{V}^{(2)})\wedge\omega_{i}=-\tfrac{1}{2}d_{ijk}(\bm{V}_{j}\cdot\bm{V}_{k})\
.$ (3.5)

The unbroken gauge group $G$ in four dimensions is given by the commutant of
the structure group of $\mathcal{V}^{(2)}$ with the ten-dimensional gauge
group. The _non-Abelian_ part of $G$ can be computed by finding all
roots222The 240 roots are given by vectors $\bm{r}$ that lie on the root
lattice with length squared equal to 2;
$\\{\bm{r}\in\Lambda\;|\;\bm{r}\cdot\bm{r}=2\\}$. $\bm{r}$ of
$\mathfrak{e}_{8}$ that are orthogonal to all of the line bundle vectors:

$H_{i}(\bm{r})\equiv\bm{V}_{i}\cdot\bm{r}=0\ ,\quad\text{for all
}i=1,\ldots,h^{1,1}\ .$ (3.6)

This ensures that the components of the ${E}_{8}$ connection that form the
four-dimensional connection are uncharged with respect to $G$. Since we are
considering bundles with an Abelian structure group, there may also be
${U}(1)$ factors in $G$ (since they commute with themselves).

One can also calculate the chiral part of the matter spectrum in the resulting
four-dimensional theory by computing the Euler characteristic $\chi$ for the
bundles in which the various matter fields transform. This can be done using
the multiplicity operator $\mathcal{N}$ [71]. More precisely, the Euler
characteristic $\chi(V)$ for a bundle $V$ whose sections transform in the
representation $\bm{\underline{R}}$ is calculated by $\mathcal{N}(\bm{r})$,
where $\bm{r}$ is the $\mathfrak{e}_{8}$ root corresponding to the highest
weight of $\bm{\underline{R}}$. That is, given a decomposition of the
$\bm{\underline{248}}$ into representations $\bm{\underline{R}}$, each
$\bm{\underline{R}}$ is characterized by some highest weight, which
corresponds to some root of $\mathfrak{e}_{8}$ (since the roots are the
weights of the $\bm{\underline{248}}$). To compute $\chi(V)$, one simply finds
the $\mathfrak{e}_{8}$ root $\bm{r}$ that the highest weight of
$\bm{\underline{R}}$ corresponds to and then evaluates $\mathcal{N}(\bm{r})$
as

$\chi(V)=\mathcal{N}(\bm{r})\equiv\tfrac{1}{12}c_{2,i}(X)H_{i}(\bm{r})+\tfrac{1}{6}d_{ijk}H_{i}(\bm{r})H_{j}(\bm{r})H_{k}(\bm{r})\
,$ (3.7)

where the second Chern numbers of the Schoen threefold are
$c_{2,i}(X)=(4,4,0)_{i}$. Note that we will often abuse notation and write
$\chi(\bm{\underline{R}})$ for the Euler characteristic of the bundle $V$
transforming in the $\bm{\underline{R}}$ representation. With these
conventions, a left chiral fermion zero-mode in four dimensions has
$\mathcal{N}(\bm{r})<0$.

###  3.2 Embedding Constraints for Two Line Bundles

As mentioned above, since our Schoen threefold has $h^{1,1}=3$, we need to
specify three line bundle vectors $\bm{V}_{i}$. To be concrete, we now
consider the explicit example of a line bundle background that breaks
${E}_{8}$ to ${E}_{6}\times{U}(1)\times{U}(1)$. We will describe this bundle
using both the line bundle vector description above and a more standard
description.

It is useful to decompose the $\bm{V}_{i}$ into a set of line bundle data
$(m^{i},n^{i})$ and a set of linearly independent, eight-component basis
vectors $(\bm{t}_{1},\bm{t}_{2})$. The line bundle vectors can then be written
as

$\bm{V}_{i}=m^{i}\bm{t}_{1}+n^{i}\bm{t}_{2}\ .$ (3.8)

To match with our previous conventions in the case of the single line bundle,
we take the generators of the ${U}(1)\times{U}(1)$ structure group to be
$(-\bm{t}_{1},-\bm{t}_{2})$. One then finds that matter fields transform
according to

$\bm{\underline{1}}_{-1,0}\sim\mathcal{O}_{X}(m^{1},m^{2},m^{3})=L_{1}\
,\qquad\bm{\underline{1}}_{0,-1}\sim\mathcal{O}_{X}(n^{1},n^{2},n^{3})=L_{2}\
,$ (3.9)

From this we see that $(m^{i},n^{i})$ specify the line bundles, while
$(\bm{t}_{1},\bm{t}_{2})$ give the embedding of ${U}(1)\times{U}(1)$ into the
hidden sector ${E}_{8}$. If we want to consider the embedding of a single
${U}(1)$ into ${E}_{8}$, one takes $\bm{t}_{2}=\bm{0}$, while more ${U}(1)$s
could be embedded by including more basis vectors.333This should be compared
with the discussion in [63, Section 7.2], where they give an example of a
${U}(1)\times{U}(1)$ bundle with three line bundle vectors and a single
relation between them, implying that they can be written in a basis with two
linearly independent generators. More details on the generators and our
conventions can be found in Appendix A. The eight-component basis vectors are
taken to be

$\bm{t}_{1}=(0,0,0,0,0,-1,1,0)\ ,\qquad\bm{t}_{2}=(0,0,0,0,0,-1,-1,-2)\ ,$
(3.10)

where the two ${U}(1)$ groups are generated by $(-\bm{t}_{1},-\bm{t}_{2})$.
Note that this choice obeys flux quantisation when the entries of $m^{i}$ and
$n^{i}$ are integers. Note also that one must have $m^{1}+m^{2}\mod 3=0$ for
equivariance of the line bundle $L_{1}$, with the same condition for the
$n^{i}$ as well.

It is easy to see which simple roots of ${E}_{8}$ are broken by this choice –
one simply takes the inner product of each $\bm{V}_{i}$ with the simple roots
in (A.1). In particular, $\bm{t}_{1}$ breaks a combination of $\alpha_{6}$ and
$\alpha_{7}$, while $\bm{t}_{2}$ breaks $\alpha_{6}$ alone. Together, they
break both $\alpha_{6}$ and $\alpha_{7}$,444One can also see this by
transforming the basis vectors to the $\omega$-basis, where they are given by
$(0,0,0,0,0,-1,2,0)$ and $(0,0,0,0,0,-1,0,0)$. suggesting that the unbroken
gauge group will be ${E}_{6}$. One can check this explicitly using (3.6) from
which one sees that 72 roots of ${E}_{8}$ are annihilated by the line bundle
vectors, which then form the 72 roots of the unbroken ${E}_{6}$ which commutes
with ${U}(1)\times{U}(1)$ inside ${E}_{8}$. Using the _Mathematica_ package
LieART [69, 70], one can then find the decomposition of the adjoint
representation of ${E}_{8}$:

$\begin{split}\bm{\underline{248}}&=\bm{\underline{78}}_{0,0}+2\times\bm{\underline{1}}_{0,0}+\bm{\underline{1}}_{2,0}+\bm{\underline{1}}_{-2,0}+\bm{\underline{1}}_{1,3}+\bm{\underline{1}}_{-1,3}+\bm{\underline{1}}_{1,-3}+\bm{\underline{1}}_{-1,-3}\\\
&\mathrel{\phantom{{=}}{}}+\bm{\underline{27}}_{1,-1}+\bm{\underline{27}}_{-1,-1}+\overline{\bm{\underline{27}}}_{1,1}+\overline{\bm{\underline{27}}}_{-1,1}+\bm{\underline{27}}_{0,2}+\overline{\bm{\underline{27}}}_{0,-2}\
.\\\ \end{split}$ (3.11)

This particular breaking pattern can be obtained in a more conventional manner
by first breaking $E_{8}\rightarrow E_{6}\times SU(3)$, under which the
adjoint representation of $E_{8}$ decomposes as

$\bf{248}=(\bm{\underline{78}},\bm{\underline{1}})+(\bm{\underline{1}},\bm{\underline{8}})+(\bm{\underline{27}},\overline{\bm{\underline{3}}})+(\overline{\bm{\underline{27}}},\bm{\underline{3}})\
.$ (3.12)

Breaking $SU(3)$ further to $SU(2)\times U(1)$, the $SU(3)$ representations
that appear above decompose as

$\begin{split}\bm{\underline{3}}&=\bm{\underline{2}}_{1}+\bm{\underline{1}}_{-2}\
,\\\
\overline{\bm{\underline{3}}}&=\bm{\underline{2}}_{-1}+\bm{\underline{1}}_{2}\
,\\\
\bm{\underline{8}}&=\bm{\underline{3}}_{0}+\bm{\underline{2}}_{3}+\bm{\underline{2}}_{-3}+\bm{\underline{1}}_{0}\
.\\\ \end{split}$ (3.13)

Finally, we break the $SU(2)$ further to $U(1)$, with the $SU(2)$
representations above decomposing as

$\displaystyle\bm{\underline{2}}$
$\displaystyle=\bm{\underline{1}}_{1}+\bm{\underline{1}}_{-1}\ ,$ (3.14)
$\displaystyle\bm{\underline{3}}$
$\displaystyle=\bm{\underline{1}}_{2}+\bm{\underline{1}}_{0}+\bm{\underline{1}}_{-2}\
.$ (3.15)

Putting this together, one sees that the adjoint representation of $E_{8}$
decomposes under this $E_{6}\times U(1)\times U(1)$ exactly as in (3.11)
above. We thus have two equivalent descriptions of the embedding of this
$U(1)\times U(1)$ in $E_{8}$, either via the line bundle vectors defined by
(3.10) or via a chain of subgroups starting from the maximal subgroup
$E_{6}\times SU(3)$.

We can find the $E_{8}$ connection that corresponds to this breaking pattern
by first building the $SU(3)$ connection induced by the breaking
$SU(3)\rightarrow S(U(1)\times U(1)\times U(1))\simeq U(1)\times U(1)$ and two
$U(1)$ connections, $A^{(1)}_{U(1)}$ and $A^{(2)}_{U(1)}$. The two $U(1)$s
embed into $SU(3)$ as

$(\text{e}^{\text{i}\phi_{1}},\text{e}^{\text{i}\phi_{2}})\hookrightarrow\begin{pmatrix}\text{e}^{-2\text{i}\phi_{2}}&0\\\
0&\text{e}^{\text{i}\phi_{2}}\begin{pmatrix}\text{e}^{\text{i}\phi_{1}}&0\\\
0&\text{e}^{-\text{i}\phi_{1}}\end{pmatrix}\end{pmatrix}\ .$ (3.16)

Using this, we can build a connection associated with a rank three bundle
$V_{\bm{\underline{3}}}$ as

$\begin{split}A_{SU(3)}=\begin{pmatrix}-2A^{(2)}_{U(1)}&0&0\\\
0&A^{(1)}_{U(1)}+A^{(2)}_{U(1)}&0\\\
0&0&-A^{(1)}_{U(1)}+A^{(2)}_{U(1)}\end{pmatrix}\ ,\\\ \end{split}$ (3.17)

where $A^{(1)}_{U(1)}$ and $A^{(2)}_{U(1)}$ are the $L_{1}$ and $L_{2}$ line
bundle connections respectively. The form of connection (3.17) implies that
the rank three bundle is the Whitney sum

$\begin{split}V_{\bm{\underline{3}}}&=L_{2}^{-2}\oplus\mathopen{}\mathclose{{}\left((L_{1}\oplus
L_{1}^{-1})\otimes L_{2}}\right)\\\ &=L_{2}^{-2}\oplus L_{1}L_{2}\oplus
L_{1}^{-1}L_{2}\equiv\mathcal{F}\oplus\mathcal{K}\oplus\mathcal{E}\
,\end{split}$ (3.18)

where we have defined

$\mathcal{F}=L_{2}^{-2}\ ,\qquad\mathcal{K}=L_{1}L_{2}\
,\qquad\mathcal{E}=L_{1}^{-1}L_{2}\ .$ (3.19)

From the form of the $SU(3)$ connection in (3.17), one can read off that the
$U(1)$ connections associated with the line bundles $\mathcal{F}$,
$\mathcal{K}$ and $\mathcal{E}$ are

$A^{\mathcal{F}}_{U(1)}=-2A^{(2)}_{U(1)}\ ,\qquad
A^{\mathcal{K}}_{U(1)}=A^{(1)}_{U(1)}+A^{(2)}_{U(1)}\ ,\qquad
A^{\mathcal{E}}_{U(1)}=-A^{(1)}_{U(1)}+A^{(2)}_{U(1)}$ (3.20)

respectively. Note that
$A^{\mathcal{E}}_{U(1)}+A^{\mathcal{F}}_{U(1)}+A^{\mathcal{K}}_{U(1)}=0$ and,
therefore, that

$c_{1}(\mathcal{F})+c_{1}(\mathcal{K})+c_{1}(\mathcal{E})=0\ ,$ (3.21)

which is simply the condition that $V_{\bm{\underline{3}}}$ is an $SU(3)$
bundle.

The DUY theorem [72, 73] implies that there exists a connection $A_{SU(3)}$
that solves the HYM equation if
$V_{\bm{\underline{3}}}=\mathcal{F}\oplus\mathcal{K}\oplus\mathcal{E}$ is
poly-stable. Since the slope of any $SU(N)$ bundle such as
$V_{\bm{\underline{3}}}$ vanishes, it follows that $V_{\bm{\underline{3}}}$
can be poly-stable only if the slopes of each of its subbundles vanish as
well. Hence, the hidden sector bundle $V_{\bm{\underline{3}}}$ given in (3.18)
will be slope poly-stable if

$\mu(\mathcal{F})=\mu(\mathcal{K})=0\ .$ (3.22)

Note that the slope of $\mathcal{E}$ vanishes automatically if the slopes of
$\mathcal{F}$ and $\mathcal{K}$ do, so the condition above is sufficient.

The $SU(3)$ connection in (3.17) embeds further into an $E_{8}$ connection
such that it commutes with $E_{6}$ as in (3.12). The embedding of the line
bundle connections into the hidden $E_{8}$ is then given by

$(A^{(1)}_{U(1)},A^{(2)}_{U(1)})\hookrightarrow
A_{E_{8}}=A^{(1)}_{U(1)}Q_{1}+A^{(2)}_{U(1)}Q_{2}\ ,$ (3.23)

where $Q_{1}$ and $Q_{2}$ are elements of the $E_{8}$ algebra whose traces
obey

$\tfrac{1}{4}\operatorname{tr}Q_{1}^{2}=1\
,\qquad\tfrac{1}{4}\operatorname{tr}Q_{2}^{2}=3\
,\qquad\operatorname{tr}Q_{1}Q_{2}=0\ ,$ (3.24)

which can also be read off from the decomposition in (3.11). We see that
$Q_{1}$ and $Q_{2}$ contain the charges associated with each of the two $U(1)$
subgroups.

We can also see this from the line bundle vector description as follows. Since
the curvature can be expanded in the Cartan generators as in (3.1), we can
write

$F_{E_{8}}=\frac{2\pi}{v^{1/3}}\omega_{i}H_{i}=\frac{2\pi}{v^{1/3}}\omega_{i}(m^{i}t^{I}_{1}+n^{i}t^{I}_{2})H_{I}=(F_{U(1)}^{(1)}t^{I}_{1}+F_{U(1)}^{(2)}t^{I}_{2})H_{I}\
,$ (3.25)

where we have identified $F_{U(1)}^{(1)}=2\pi v^{-1/3}m^{i}\omega_{i}$ as the
curvature of the line bundle $L_{1}=\mathcal{O}_{X}(m^{1},m^{2},m^{3})$
defined in (3.9), and similarly for $L_{2}$. Comparing with (3.23), we read
off that $Q_{1}=t_{1}^{I}H_{I}$ and $Q_{2}=t_{2}^{I}H_{I}$. It is then simple
to repeat the calculation of the traces to give

$\tfrac{1}{4}\operatorname{tr}Q_{1}^{2}=\tfrac{1}{2}\bm{t}_{1}\cdot\bm{t}_{1}=1\
,\qquad\tfrac{1}{4}\operatorname{tr}Q_{2}^{2}=\tfrac{1}{2}\bm{t}_{2}\cdot\bm{t}_{2}=3\
,\qquad\tfrac{1}{4}\operatorname{tr}Q_{1}Q_{2}=\tfrac{1}{2}\bm{t}_{1}\cdot\bm{t}_{2}=0\
,$ (3.26)

where we have used the normalisation of the generators in (3.3). We see that
this agrees with (3.24) upon inserting the definitions from (3.8) and (3.10).

### Anomaly Condition

The second Chern character of the hidden sector bundle is

$\operatorname{ch}_{2}(\mathcal{V}^{(2)})=\frac{1}{16\pi^{2}}\operatorname{tr}F_{E_{8}}\wedge
F_{E_{8}},$ (3.27)

where $F_{E_{8}}$ is the curvature of the $E_{8}$ connection induced from the
two line bundle connections in (3.23). Expanding out, we find

$\operatorname{ch}_{2}(\mathcal{V}^{(2)})=\frac{1}{16\pi^{2}}\mathopen{}\mathclose{{}\left(\operatorname{tr}Q_{1}^{2}F_{U(1)}^{(1)}\wedge
F_{U(1)}^{(1)}+\operatorname{tr}Q_{2}^{2}F_{U(1)}^{(2)}\wedge
F_{U(1)}^{(2)}+2\operatorname{tr}Q_{1}Q_{2}F_{U(1)}^{(1)}\wedge
F_{U(1)}^{(2)}}\right)\ ,$ (3.28)

where the $F_{U(1)}^{(i)}$ are given in terms of the first Chern class of each
line bundle as $c_{1}^{i}=F^{(i)}_{U(1)}/2\pi$. Again denoting the $L_{1}$ and
$L_{2}$ line bundles by

$L_{1}=\mathcal{O}_{X}(m^{1},m^{2},m^{3})\ ,\qquad
L_{2}=\mathcal{O}_{X}(n^{1},n^{2},n^{3})\ ,$ (3.29)

the second Chern character of $\mathcal{V}^{(2)}$ is then simply

$\begin{split}\operatorname{ch}_{2}(\mathcal{V}^{(2)})&=\,c_{1}(L_{1})\wedge
c_{1}(L_{1})+3c_{1}(L_{2})\wedge c_{1}(L_{2})\\\
&=\frac{1}{v^{2/3}}(m^{1}\omega_{1}+m^{2}\omega_{2}+m^{3}\omega_{3})^{2}+\frac{3}{v^{2/3}}(n^{1}\omega_{1}+n^{2}\omega_{2}+n^{3}\omega_{3})^{2}\
,\end{split}$ (3.30)

where we used the trace relations from (3.24). It then follows from the
discussion in Appendix B that the anomaly condition is given by

$W_{i}=\mathopen{}\mathclose{{}\left(\tfrac{4}{3},\tfrac{7}{3},-4}\right)_{i}+d_{ijk}m^{j}m^{k}+3d_{ijk}n^{j}n^{k}\
.$ (3.31)

Furthermore, to be consistent with $N=1$ supersymmetry, $W_{i}$ must satisfy
the constraint given in (2.3); that is, $W_{i}\geq 0$ for $i=1,2,3$.

Again we can compare this with the line bundle vector formalism. We have
already given the second Chern character in (3.4). Expanding this out and
using the trace relations in (3.26), one can check that it reproduces (3.30)
above. Furthermore, the anomaly cancellation condition can then be written as

$W_{i}=(\tfrac{4}{3},\tfrac{7}{3},-4)_{i}+\tfrac{1}{2}d_{ijk}(\bm{V}_{j}\cdot\bm{V}_{k})\
,$ (3.32)

which agrees with (3.31) upon using the definitions from (3.8) and (3.10).

### Low-Energy Fields

As usual, low-energy matter superfields fields arising from the decomposition
in eq. (3.11) are associated with bundle-valued cohomologies on the Calabi–Yau
threefold. Using the identification (3.19), we find that the $E_{6}$ singlets
with non-zero $U(1)$ charges are associated with

$\displaystyle\bm{\underline{1}}_{0,2}$ $\displaystyle\sim
H^{\bullet}(X,\mathcal{F})=H^{\bullet}(X,L_{2}^{-2})\ ,$ (3.33)
$\displaystyle\bm{\underline{1}}_{-1,-1}$ $\displaystyle\sim
H^{\bullet}(X,\mathcal{K})=H^{\bullet}(X,L_{1}L_{2})\ ,$ (3.34)
$\displaystyle\bm{\underline{1}}_{1,-1}$ $\displaystyle\sim
H^{\bullet}(X,\mathcal{E})=H^{\bullet}(X,L_{1}^{-1}L_{2})\ .$ (3.35)

To find the cohomologies for the other representations in the decomposition,
we can either use the fact that all $SU(3)$ representations can be obtained
from the fundamental $\bm{\underline{3}}$ and its conjugate
$\overline{\bm{\underline{3}}}$, and then use the decomposition of the
$\bm{\underline{3}}$ in terms of $U(1)\times U(1)$, or we can just count the
charges in (3.11) and use the above identifications to find the corresponding
cohomology. The representations we obtain for the $E_{6}\times U(1)\times
U(1)$ low-energy group, as well as their corresponding cohomologies, are shown
in Table 1. Note that each representation $\bm{\underline{R}}$ in Table 1 has
an associated cohomology of the form

$H^{\bullet}(X,L_{\bm{\underline{R}}})=H^{\bullet}(X,L_{1}^{-q_{\bm{\underline{R}}}}\otimes
L_{2}^{-p_{\bm{\underline{R}}}})\ ,$ (3.36)

where $q_{\bm{\underline{R}}}$ and $p_{\bm{\underline{R}}}$ are the charges of
$\bm{\underline{R}}$ for each of the two $U(1)$ groups.

Let us now consider the low-energy matter spectrum. For fields in the
$\bm{\underline{R}}$ representation, with associated line bundle
$L_{\bm{\underline{R}}}$, the Euler characteristic
$\chi(L_{\bm{\underline{R}}})$ counts the chiral asymmetry. For a line bundle
of the form
$L_{\bm{\underline{R}}}=\mathcal{O}_{X}(l^{1}_{\bm{\underline{R}}},l^{2}_{\bm{\underline{R}}},l^{3}_{\bm{\underline{R}}})$,
the Euler characteristic is given by

$\chi(L_{\bm{\underline{R}}})=\sum_{i=0}^{3}(-1)^{i}h^{i}(X,L_{\bm{\underline{R}}})=\int_{X}\operatorname{ch}(L_{\bm{\underline{R}}})\wedge\operatorname{Td}(X)\
,$ (3.37)

where $\operatorname{ch}(L_{\bm{\underline{R}}})$ is the Chern character of
$L_{\bm{\underline{R}}}$ and $\operatorname{Td}(X)$ is the Todd class of the
tangent bundle of $X$. On the Schoen manifold we are considering, this
simplifies to

$\chi(L_{\bm{\underline{R}}})=\tfrac{1}{3}(l^{1}_{\bm{\underline{R}}}+l^{2}_{\bm{\underline{R}}})+\tfrac{1}{6}d_{ijk}l^{i}_{\bm{\underline{R}}}l^{j}_{\bm{\underline{R}}}l^{k}_{\bm{\underline{R}}}\
.$ (3.38)

Using the intersection numbers $d_{ijk}$ given in (B.5), this expression
becomes

$\chi(L_{\bm{\underline{R}}})=\tfrac{1}{6}\mathopen{}\mathclose{{}\left(l^{1}_{\bm{\underline{R}}}l^{2}_{\bm{\underline{R}}}(l^{1}_{\bm{\underline{R}}}+l^{2}_{\bm{\underline{R}}}+6l^{3}_{\bm{\underline{R}}})+2l^{1}_{\bm{\underline{R}}}+2l^{2}_{\bm{\underline{R}}}}\right)\
.$ (3.39)

The numbers $l_{\bm{\underline{R}}}^{i}$ characterizing the line bundle
$L_{\bm{\underline{R}}}$ depend on the low-energy representation
$\bm{\underline{R}}$. For our line bundle embedding and a representation
$\bm{\underline{R}}$ with $U(1)$ charges $q_{\bm{\underline{R}}}$ and
$p_{\bm{\underline{R}}}$, the Euler characteristic is given by
$\chi(L_{\bm{\underline{R}}})=\chi({L_{1}^{-q_{\bm{\underline{R}}}}L_{2}^{-p_{\bm{\underline{R}}}}})$.
Defining $L_{1}$ and $L_{2}$ as in (3.29), one finds that

$l_{\bm{\underline{R}}}^{i}=-q_{\bm{\underline{R}}}m^{i}-p_{\bm{\underline{R}}}n^{i}\
,\quad i=1,2,3\ .$ (3.40)

It is then simple to evaluate the Euler characteristic for each representation
by substituting the values of $l_{\bm{\underline{R}}}^{i}$ into (3.39). For
example, for the line bundle $L_{1}^{-1}L_{2}^{-3}$ associated with the
representation $\bm{\underline{1}}_{1,3}$ in Table 1, the $l^{i}$ are

$l^{i}=-m^{i}-3n^{i}\ .$ (3.41)

The Euler characteristic is then given by

$\chi(\bm{\underline{1}}_{1,3})\equiv\chi(L_{1}^{-1}L_{2}^{-3})=\tfrac{1}{6}\mathopen{}\mathclose{{}\left(l^{1}l^{2}(l^{1}+l^{2}+6l^{3})+2l^{1}+2l^{2}}\right)\
.$ (3.42)

It is straightforward to check that this agrees with the Euler characteristic
as computed using the multiplicity operator in (3.7). For example, the highest
weight of $\bm{\underline{1}}_{1,3}$ can be obtained by projecting the root
$\bm{r}=(0,0,0,0,0,1,0,1)$ of $\mathfrak{e}_{8}$. Using this in (3.7), one
finds that $\mathcal{N}(\bm{r})$ evaluates to (3.42).

The matter fields associated with any given representation and cohomology in
Table 1 are either 1) chiral superfields if the corresponding Euler
characteristic $\chi<0$ or 2) anti-chiral superfields if $\chi>0$. For any
given entry in Table 1, the sign of the Euler characteristic depends on the
choice of line bundles $L_{1}$ and $L_{2}$. Hence, the same entry in the Table
can correspond to either a chiral superfield or an anti-chiral superfield
depending on the circumstances of the solution. Note, however, that if a
representation, such as ${\underline{\bf 1}}_{1,3}$, corresponds to chiral
superfields then the conjugate representation, such as ${\underline{\bf
1}}_{-1,-3}$, corresponds to anti-chiral superfields. With this in mind, in
the last column of Table 1 we have assigned a specific symbol to the matter
supermultiplet of each representation. Having done that, we note that each
such superfield contains a complex scalar field component. In the following,
it is convenient to abuse notation and denote each superfield and its complex
scalar field component using the same symbol. Whether we are referring to the
full supermultiplet or its scalar component will be clear from the context.

$E_{6}\times U(1)\times U(1)$ | Cohomology | Field Name
---|---|---
$\bm{\underline{1}}_{1,3}$ | $H^{\bullet}(X,\mathcal{F}\otimes\mathcal{K}^{*}=L_{1}^{-1}L_{2}^{-3})$ | $C_{1}$
$\bm{\underline{1}}_{-1,-3}$ | $H^{\bullet}(X,\mathcal{F}^{*}\otimes\mathcal{K}=L_{1}L_{2}^{3})$ | $\tilde{C}_{1}$
$\bm{\underline{1}}_{-1,3}$ | $H^{\bullet}(X,\mathcal{F}\otimes\mathcal{E}^{*}=L_{1}L_{2}^{-3})$ | $C_{2}$
$\bm{\underline{1}}_{1,-3}$ | $H^{\bullet}(X,\mathcal{F}^{*}\otimes\mathcal{E}=L_{1}^{-1}L_{2}^{3})$ | $\tilde{C}_{2}$
$\bm{\underline{1}}_{2,0}$ | $H^{\bullet}(X,\mathcal{K}^{*}\otimes\mathcal{E}=L_{1}^{-2})$ | $C_{3}$
$\bm{\underline{1}}_{-2,0}$ | $H^{\bullet}(X,\mathcal{K}\otimes\mathcal{E}^{*}=L_{1}^{2})$ | $\tilde{C}_{3}$
$\bm{\underline{27}}_{0,-2}$ | $H^{\bullet}(X,\mathcal{F}^{*}=L_{2}^{2})$ | $f_{1}$
$\bm{\underline{27}}_{1,1}$ | $H^{\bullet}(X,\mathcal{K}^{*}=L_{1}^{-1}L_{2}^{-1})$ | $f_{2}$
$\bm{\underline{27}}_{-1,1}$ | $H^{\bullet}(X,\mathcal{E}^{*}=L_{1}L_{2}^{-1})$ | $f_{3}$
$\bm{\underline{27}}_{0,2}$ | $H^{\bullet}(X,\mathcal{F}=L_{2}^{-2})$ | $\tilde{f}_{1}$
$\bm{\underline{27}}_{-1,-1}$ | $H^{\bullet}(X,\mathcal{K}=L_{1}L_{2})$ | $\tilde{f}_{2}$
$\bm{\underline{27}}_{1,-1}$ | $H^{\bullet}(X,\mathcal{E}=L_{1}^{-1}L_{2})$ | $\tilde{f}_{3}$
Table 1: Low-energy representations of $E_{6}\times U(1)\times U(1)$ and their
associated cohomologies. $L_{1}$ is a line bundle of the form
$L_{1}=\mathcal{O}_{X}(m^{1},m^{2},m^{3})$, while we write
$L_{2}=\mathcal{O}_{X}(n^{1},n^{2},n^{3})$ for $L_{2}$ . The entries in the
third column correspond to either a chiral or an anti-chiral supermultiplet if
the Euler characteristic of the associated line bundle is negative or positive
respectively. Hence, the supermultiplets corresponding to a line bundle and
its inverse bundle are conjugates of each other. Note that the fields $C_{i}$
and $\tilde{C}_{i}$ are singlets under $E_{6}$. Deforming the bundle
$V_{\bm{\underline{3}}}$ away from the decomposable locus is equivalent to
turning on different combinations of VEVs for the scalar components of these
supermultiplets in the effective theory.

### Extension Bundle

In this section, we study how to deform the Whitney sum bundle defined in eq.
(3.18) away from the decomposable locus to construct an irreducible ${SU}(3)$
bundle. We will also discuss how this construction appears in the effective
field theory.

Consider the pair of exact sequences

$\begin{split}0&\rightarrow\mathcal{F}\rightarrow
W\rightarrow\mathcal{E}\rightarrow 0\ ,\\\ 0&\rightarrow\mathcal{K}\rightarrow
V^{\prime}_{\bm{\underline{3}}}\rightarrow W\rightarrow 0\ .\end{split}$
(3.43)

These define an $SU(3)$ bundle $V^{\prime}_{\bm{\underline{3}}}$, since
$c_{1}(V^{\prime}_{\bm{\underline{3}}})=0$. The first extension can be non-
trivial – that is, $W$ is not simply $\mathcal{F}\oplus\mathcal{E}$ – if and
only if the Ext group
$\operatorname{Ext}^{1}(\mathcal{E},\mathcal{F})=H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})$
is non-trivial. Similarly, the second extension can be non-trivial if and only
if $\operatorname{Ext}^{1}(W,\mathcal{K})=H^{1}(X,\mathcal{K}\otimes W^{*})$
is non-trivial. It is relatively easy to see that a non-trivial extension
class for the first extension sequence corresponds to turning on a VEV for the
field $C_{2}$ in the effective theory. Indeed, from Table 1 we see that
$H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})$ counts the number of $C_{2}$
fields. It is straightforward, if not obvious, to see that the extension class
for the second sequence is equivalent to turning on a VEV for the field
$\tilde{C}_{3}$. To prove this, note that one can show that
$\operatorname{Ext}^{1}(W,\mathcal{K})=H^{1}(X,\mathcal{K}\otimes
W^{*})=H^{1}(X,\mathcal{K}\otimes\mathcal{E}^{*})$ [74].555Specifically, in
Section 3.3 of [74] it is shown that $H^{1}(X,\mathcal{K}\otimes
W^{*})=H^{1}(X,\mathcal{K}\otimes\mathcal{E}^{*})+\operatorname{ker}\delta$,
where $\operatorname{ker}\delta\in
H^{1}(X,\mathcal{K}\otimes\mathcal{F}^{*})$. Note that this branch is confined
to the zero element of $H^{1}(X,\mathcal{K}\otimes\mathcal{F}^{*})$. It
follows from Table 1 that $H^{1}(X,\mathcal{K}\otimes\mathcal{E}^{*})$ counts
the number of $\tilde{C}_{3}$ fields. We learn that the pair of non-trivial
extensions in (3.43) corresponds to turning on VEVs for the fields $C_{2}$ and
$\tilde{C}_{3}$ in the effective theory,

$\langle C_{2}\rangle\neq 0\ ,\qquad\langle\tilde{C}_{3}\rangle\neq 0\ ,$
(3.44)

while the VEVs of the other charged matter fields are set to zero. Having
allowed for these VEVs, one should check that the F-term and D-term
constraints are satisfied so that the vacuum of the four-dimensional theory
preserves supersymmetry. As shown in [74], one can turn on this combination of
VEVs while still having the superpotential and its derivative vanish – that
is, the effective theory satisfies the F-flatness requirement. Furthermore,
using the generic expression for a D-term given in [74], we see that fields
$C_{2}$ and $\tilde{C}_{3}$ have the correct charges to cancel the FI terms
associated with the two $U(1)$ bundles $\mathcal{F}$ and $\mathcal{K}$, as
defined in Appendix D. Therefore, the theory is also D-flat.

Note that there is another pair of extension sequences which correspond to the
same non-zero elements of $H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})$ and
$H^{1}(X,\mathcal{K}\otimes\mathcal{E}^{*})$, namely

$\displaystyle\begin{split}0&\rightarrow\mathcal{K}\rightarrow
W^{\prime}\rightarrow\mathcal{E}\rightarrow 0\ ,\\\
0&\rightarrow\mathcal{F}\rightarrow\mathcal{V}^{\prime}_{\bm{\underline{3}}}\rightarrow
W^{\prime}\rightarrow 0\
,\end{split}\begin{split}\operatorname{Ext}^{1}(\mathcal{E},\mathcal{K})&=H^{1}(X,\mathcal{K}\otimes\mathcal{E}^{*})\neq
0\ ,\\\
\operatorname{Ext}^{1}(W^{\prime},\mathcal{F})&=H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})\neq
0\ .\end{split}$ (3.45)

This seemingly gives us a different $SU(3)$ bundle
$\mathcal{V}^{\prime}_{\bm{\underline{3}}}$ for the same set of VEVs. However,
as shown in [74], this new bundle is actually isomorphic to the above, that is

$\mathcal{V}^{\prime}_{\bm{\underline{3}}}\simeq
V^{\prime}_{\bm{\underline{3}}}\ .$ (3.46)

From an effective theory perspective, both pairs of sequences correspond to
turning on VEVs for the low-energy fields $\tilde{C}_{2}$ and $C_{3}$. Roughly
speaking, the choice between (3.43) or (3.45) corresponds to turning on VEVs
for $C_{2}$ first and then $\tilde{C}_{3}$ or vice versa, respectively.

As we will discuss in Section 4.1 there are actually six inequivalent
_branches_ along which one can, in principle, deform the $SU(3)$ bundle away
from the decomposable locus. From the low-energy perspective, deforming the
bundle along one branch or another comes from turning on different pairs of
VEVs of the $E_{6}$ singlet fields $C_{i}$ and $\tilde{C}_{i}$ which are both
F-flat and D-flat. The extensions in (3.43) or (3.45) give one possible
branch. We will study the full branch structure for a hidden sector built from
two line bundles in detail in Sections 4.1 and 4.2, but for the moment let us
continue focusing on the $\langle C_{2}\rangle,\langle\tilde{C}_{3}\rangle\neq
0$ branch. We have seen that a non-trivial extension is possible if and only
if the cohomology groups $H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})$ and
$H^{1}(X,\mathcal{K}\otimes\mathcal{E}^{*})$ are non-empty, that is

$h^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})>0\ ,\qquad
h^{1}(X,\mathcal{K}\otimes\mathcal{E}^{*})>0\ .$ (3.47)

It is useful to observe that the case where either of the line bundles
$\mathcal{F}\otimes\mathcal{E}^{*}$ or $\mathcal{K}\otimes\mathcal{E}^{*}$ is
ample is eliminated from the start, since ample line bundles have vanishing
first cohomology groups on a Calabi–Yau manifold. Hence, one must choose the
line bundles $L_{1}$, $L_{2}$ so that $\mathcal{F}\otimes\mathcal{E}^{*}$ and
$\mathcal{K}\otimes\mathcal{E}^{*}$ are not ample. That being said, computing
the cohomology groups $H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})$ and
$H^{1}(X,\mathcal{K}\otimes\mathcal{E}^{*})$ directly on our Schoen manifold
$X$ is a difficult task, which far exceeds the scope of the present paper.
This will be discussed in future publications.

### Bundle Stability

The conditions above ensure that an extension is possible, which corresponds
to the existence of non-zero VEVs of certain charged matter fields in the
four-dimensional effective theory. The existence of the deformed $SU(3)$
bundle is not sufficient to ensure supersymmetry however. One must also
require that the new bundle admits a connection that solves the HYM equation;
that is, the bundle must be slope-stable. Therefore, the next question one
must ask, assuming the extension exists, is whether the resulting bundle is
slope-stable. As we have emphasized, checking stability of a bundle is
generally a difficult calculation which, at the moment, cannot be done
algorithmically on our particular Schoen threefold. Instead, we will focus on
some necessary conditions; specifically that some obvious subbundles have
negative slope and that the Bogomolov inequality is satisfied. This last
requirement is significant since often the Bogomolov inequality is the only
obstruction to finding a slope-stable bundle [37].

From the first sequence in (3.43), we learn that there is an embedding

$\mathcal{K}\hookrightarrow V^{\prime}_{\bm{\underline{3}}}\ .$ (3.48)

That is, the line bundle $\mathcal{K}$ injects into
$V_{\bm{\underline{3}}}^{\prime}$. Since $V_{\bm{\underline{3}}}^{\prime}$ has
vanishing slope, the first necessary condition for stability is that
$\mathcal{K}$ has negative tree-level slope:

$\mu(\mathcal{K})=\mu(L_{1}L_{2})<0\quad\Rightarrow\quad
d_{ijk}(m^{i}+n^{j})a^{j}a^{k}<0\ ,$ (3.49)

Furthermore, it would appear from (3.45) and (3.46), and is proven in Appendix
F, that the bundle $\mathcal{F}$ also injects into
$V^{\prime}_{\bm{\underline{3}}}$,

$\mathcal{F}\hookrightarrow V^{\prime}_{\bm{\underline{3}}}\ ,$ (3.50)

and so the slope of $\mu(\mathcal{F})$ must also be negative:

$\mu(\mathcal{F})=\mu(L_{2}^{-2})<0\quad\Rightarrow\quad-2d_{ijk}n^{j}a^{j}a^{k}<0\
.$ (3.51)

Given a choice of $L_{1}$ and $L_{2}$ such that (3.49) and (3.51) are
satisfied, the final necessary condition that we impose is that
$V_{\bm{\underline{3}}}^{\prime}$ satisfies the Bogomolov inequality

$\int c_{2}(V_{\bm{\underline{3}}}^{\prime})\wedge\omega\geq 0\ .$ (3.52)

A bundle is stable only if it satisfies this highly non-trivial constraint.
Thankfully, since $c_{2}$ is topological, this can be computed from the data
of the Whitney sum bundle
$V_{\bm{\underline{3}}}=\mathcal{F}\oplus\mathcal{K}\oplus\mathcal{E}$. The
total Chern class is

$\begin{split}c(V_{\bm{\underline{3}}}^{\prime})\equiv
c(V_{\bm{\underline{3}}})&=c(L_{2}^{-2}\oplus L_{1}L_{2}\oplus
L_{1}^{-1}L_{2})\\\ &=1-3c_{1}^{2}(L_{2})-c_{1}^{2}(L_{1})+\dots\end{split}$
(3.53)

from which we see that
$c_{2}(V_{\bm{\underline{3}}}^{\prime})=3c_{1}^{2}(L_{2})-c_{1}^{2}(L_{2})$.
It follows that condition (3.52) becomes

$d_{ijk}m^{i}m^{j}a^{k}+3d_{ijk}n^{i}n^{j}a^{k}\leq 0\ .$ (3.54)

We emphasize that since the Kähler form $\omega$ appears in the calculation of
the two slopes and the Bogomolov inequality, all three of these conditions
depend on where one is in the Kähler cone, which is also restricted by the
physical constraints discussed in Section 2.2.

###  3.3 Line Bundles Scan

We are now in a position to look for an appropriate hidden sector bundle,
which has a non-trivial extension $V_{\bm{\underline{3}}}^{\prime}$ that might
be stable away from the decomposable locus. In addition to the physical
constraints (2.4)–(2.11), we impose the constraints derived so far in this
section. Specifically, we want to find two line bundles
$L_{1}=\mathcal{O}_{X}(m^{1},m^{2},m^{3})$ and
$L_{2}=\mathcal{O}_{X}(n^{1},n^{2},n^{3})$ such that

  1. 1.

Both $L_{1}$ and $L_{2}$ are equivariant:

$(m_{1}+m_{2})\operatorname{mod}3=0\ ,\quad(n_{1}+n_{2})\operatorname{mod}3=0\
.$ (3.55)

  2. 2.

The five-brane class is effective:

$W_{i}=\mathopen{}\mathclose{{}\left(\tfrac{4}{3},\tfrac{7}{3},-4}\right)_{i}+d_{ijk}m^{j}m^{k}+3d_{ijk}n^{j}n^{k}\geq
0\ .$ (3.56)

  3. 3.

The cohomology groups
$H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})=H^{1}(X,L_{1}L_{2}^{-3})$ and
$H^{1}(X,\mathcal{K}\otimes\mathcal{E}^{*})=H^{1}(X,L_{1}^{2})$ are non-zero.
In this scan we will simply impose the necessary condition that
$\mathcal{F}\otimes\mathcal{E}^{*}=L_{1}L_{2}^{-3}$ and
$\mathcal{K}\otimes\mathcal{E}^{*}=L_{1}^{2}$ are not ample.

  4. 4.

The extended bundle $V_{\bm{\underline{3}}}^{\prime}$ is stable. The necessary
conditions that we impose are:

(i) The slopes of the line bundles $\mathcal{F}$ and $\mathcal{K}$ are
negative:

$d_{ijk}(m^{i}+n^{i})a^{j}a^{k}<0\ ,\qquad-2d_{ijk}n^{i}a^{j}a^{k}<0\ .$
(3.57)

(ii) The extension bundle $V_{\bm{\underline{3}}}^{\prime}$ satisfies the
Bogomolov inequality

$d_{ijk}m^{i}m^{j}a^{k}+3d_{ijk}n^{i}n^{j}a^{k}\leq 0\ .$ (3.58)

Figure 2:  The colored region is the subspace of Kähler moduli space which, in
addition to satisfying conditions (2.4)-(2.8) outlined in Section 2, also
restricts the strong coupling parameter to the smaller values
$\epsilon_{S}^{\text{eff}}\lesssim 0.2$. Hence, any point in this region also
satisfies the linearization constraint (2.11). This subspace does not depend
on the hidden sector bundle. The “orange” regime is a subspace of this colored
region, in which the non-trivial $SU(3)$ bundle
$V_{\bm{\underline{3}}}^{\prime}$, defined as an extension of the two line
bundles $L_{1}=\mathcal{O}_{X}(-5,-1,1)$ and $L_{2}=\mathcal{O}_{X}(2,1,-1)$,
can be slope-stable. Note that one can show that the locus where the
decomposable Whitney sum bundle $V_{\bm{\underline{3}}}$ is poly-stable lies
outside the colored region shown in Figure 1 and, hence, does not satisfy all
of the universal constraints.

We performed a systematic scan over all possible pairs of line bundles, with
$|m^{i}|\leq 15$, $|n^{i}|\leq 15$ for $i=1,2,3$. The values of
$(a^{1},a^{2},a^{3})$ we sample sit inside the “blue” subspace of the Kähler
cone shown in Figure 1. We find a number of pairs of line bundles which lead
to a solution satisfying all of these constraints. These include the line
bundles

$L_{1}=\mathcal{O}_{X}(-5,-1,1)\ ,\qquad L_{2}=\mathcal{O}_{X}(2,1,-1).$
(3.59)

Before giving the other examples, let us analyze this case in more detail. For
this pair of line bundles, we find that the class of the five-brane is

$W_{i}=(2,0,18)_{i}\ ,$ (3.60)

which is indeed effective ($W_{i}\geq 0$). Furthermore, the bundles
$L_{1}L_{2}^{-3}=\mathcal{O}_{X}(-11,-4,4)$ and
$L_{1}^{2}=\mathcal{O}_{X}(-10,-2,2)$, associated with the
$\bm{\underline{1}}_{-1,3}$ and the $\bm{\underline{1}}_{-2,0}$
representations respectively, are clearly non-ample. However, we do not know
if the extensions in (3.43) exist without computing the cohomologies

$\displaystyle
H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})=H^{1}(X,L_{1}L_{2}^{-3})=H^{1}(X,\mathcal{O}_{X}(-11,-4,4)).$
(3.61) $\displaystyle H^{1}(X,\mathcal{K}\otimes\mathcal{E}^{*})\equiv
H^{1}(X,L_{1}^{2})=H^{1}(X,\mathcal{O}_{X}(-10,-2,2))\ .$ (3.62)

Assuming these cohomologies are non-empty, it is simple to check that the
necessary conditions for stability are also satisfied.

In order to display the region of stability of the extension
$V_{\bm{\underline{3}}}^{\prime}$ of the line bundles (3.59), we sample values
$(a^{1},a^{2},a^{3})$ inside the “blue” region in Figure 1 and check the
stability criteria we gave in equations (3.57) and (3.58). Importantly, we
choose to restrict our discussion to the “blue” region in Figure 1 since in
this subspace, by definition, the strong coupling parameter satisfies
$\epsilon_{S}^{\text{eff}}\lesssim 0.2$ and, hence, is relatively small. As
discussed in Section 2, this implies that the linearization constraint (2.11)
is satisfied. The subset of the blue region in Figure 1 where the extension
bundle can be slope-stable is displayed as the “orange” subspace in Figure 2.
Having found stable bundles $V_{\bm{\underline{3}}}^{\prime}$ inside this
“orange”

(a)

(b)

Figure 3: In this Figure, for the line bundles
$L_{1}=\mathcal{O}_{X}(-5,-1,1)$ and $L_{2}=\mathcal{O}_{X}(2,1,-1)$, we plot
both the ratio of the genus-one correction to the tree-level value of the FI-
term associated with the bundles $\mathcal{F}$ in (a) and $\mathcal{K}$ in
(b), against the ratio of the fifth-dimensional length
$\pi\rho\hat{R}V^{-1/3}$ to the average Calabi–Yau radius $(vV)^{-1/6}$
(displayed on the horizontal axis). We also show the values of the effective
strong coupling parameter
$\epsilon_{S}^{\text{eff}}=\epsilon_{S}^{\prime}\hat{R}/V$ using the colored
shading. Points in the plot are sampled from inside the “orange” region of
Figure 2. Across the solution space, the ratios from (3.64) and (3.65) are
less that $0.2$. Furthermore, in both cases, the effective strong coupling
parameter $\epsilon_{S}^{\text{eff}}$ is no larger than $0.2$, suggesting that
the linear approximation is accurate.

region allows us to move away from the decomposable locus and, importantly,
away from the region of Kähler moduli space in which the theory is very
strongly coupled.

In addition to the strong coupling parameter satisfying
$\epsilon_{S}^{\text{eff}}\lesssim 0.2$, there are two other criteria that
specify whether or not the strong coupling corrections are relatively small.
First, let us analyze the ratio of the genus one corrections to the tree level
value of the Fayet-Iliopoulos term associated with each of the two line
generic bundles $\mathcal{F}$ and $\mathcal{K}$. As shown in Appendix D, the
FI-terms associated with $\mathcal{F}$ and $\mathcal{K}$ are given by

$\begin{split}FI_{\mathcal{F}}&=\frac{1}{2}\frac{\epsilon_{S}\epsilon_{R}^{2}}{\kappa_{4}^{2}}\frac{1}{\hat{R}V^{2/3}}\mathopen{}\mathclose{{}\left[-2d_{ijk}n^{i}a^{j}a^{k}+\frac{2\epsilon_{S}^{\prime}\hat{R}}{V^{1/3}}n^{i}\Big{(}\big{(}\tfrac{2}{3},-\tfrac{1}{3},4\big{)}\big{|}_{i}+\mathopen{}\mathclose{{}\left(1-(1+\tfrac{\lambda}{2})^{2}}\right)W_{i}\Big{)}}\right]\
,\\\
FI_{\mathcal{K}}&=\frac{1}{2}\frac{\epsilon_{S}\epsilon_{R}^{2}}{\kappa_{4}^{2}}\frac{1}{\hat{R}V^{2/3}}\bigg{[}d_{ijk}(m^{i}+n^{i})a^{j}a^{k}\\\
&\mathrel{\phantom{{=}}{}}\qquad\qquad-\frac{\epsilon_{S}^{\prime}\hat{R}}{V^{1/3}}(m^{i}+n^{i})\Big{(}\big{(}\tfrac{2}{3},-\tfrac{1}{3},4\big{)}\big{|}_{i}+\mathopen{}\mathclose{{}\left(1-(1+\tfrac{\lambda}{2})^{2}}\right)W_{i}\Big{)}\bigg{]}\
.\\\ \end{split}$ (3.63)

Making use of “unity” gauge where one sets
$\epsilon_{S}^{\prime}\hat{R}/V^{1/3}=1$, we find that this ratio is given by

$\displaystyle\mathopen{}\mathclose{{}\left|\frac{\text{Genus-
One}}{\text{Tree-
Level}}}\right|_{\mathcal{F}}=\frac{\mathopen{}\mathclose{{}\left|-2n^{i}\Big{(}\big{(}\tfrac{2}{3},-\tfrac{1}{3},4\big{)}\big{|}_{i}+\mathopen{}\mathclose{{}\left(1-(1+\tfrac{\lambda}{2})^{2}}\right)W_{i}\Big{)}}\right|}{|2d_{ijk}n^{i}a^{j}a^{k}|}\
,$ (3.64)

$\displaystyle\mathopen{}\mathclose{{}\left|\frac{\text{Genus-
One}}{\text{Tree-
Level}}}\right|_{\mathcal{K}}=\frac{\mathopen{}\mathclose{{}\left|(m^{i}+n^{i})\Big{(}\big{(}\tfrac{2}{3},-\tfrac{1}{3},4\big{)}\big{|}_{i}+\mathopen{}\mathclose{{}\left(1-(1+\tfrac{\lambda}{2})^{2}}\right)W_{i}\Big{)}}\right|}{|d_{ijk}(m^{i}+n^{i})a^{j}a^{k}|}$
(3.65)

for $\mathcal{F}$ and $\mathcal{K}$ respectively. A second criterion that the
strong coupling corrections are relatively small is that ratio between the
orbifold length and the average Calabi–Yau radius, given by

$\frac{\pi\rho\hat{R}V^{-1/3}}{(vV)^{-1/6}}$ (3.66)

where $V=\frac{1}{6}d_{ijk}a^{i}a^{j}a^{k}$, should not exceed a value of
$\sim 2.$ In [62] it was shown that increasing the orbifold length, so that
the hidden and observable sectors are more separated, also increased the
effective expansion parameter, causing the linear approximation to become less
accurate. For an effective expansion parameter of order 1, the orbifold length
is roughly 12 times larger than the Calabi–Yau scale.

Returning to our explicit solution with line bundles
$L_{1}=\mathcal{O}_{X}(-5,-1,1)$ and $L_{2}=\mathcal{O}_{X}(2,1,-1)$ given in
(3.59), we now plot (3.64), (3.65) and (3.66), as well as
$\epsilon_{S}^{\text{eff}}=\epsilon_{S}^{\prime}\hat{R}/V$, over all points in
the “orange” region of Figure 2. We find that across the “orange” solution
space, in the region where the ratio between the orbifold length and the
average Calabi–Yau radius is less than 2, the genus-one corrections to the FI
terms of $\mathcal{F}$ and $\mathcal{K}$ in eq. (3.63) are less than $0.2$
times the tree-level values. Finally, the effective strong coupling parameter
$\epsilon_{S}^{\text{eff}}$ is no larger than $0.2$, as expected since we are
within the “blue” region of Figure 1. We conclude that for Kähler moduli in
the “orange” regime of Figure 2, all three criteria for relatively small
$\kappa_{11}^{4/3}$ corrections are satisfied. It is of interest to compare
these results to the single line bundle case studied in [62], where cancelling
the tree-level FI term against the genus-one correction was necessary to
obtain a poly-stable hidden sector bundle. In that context, ${|\text{Genus-
One}|}/{|\text{Tree-Level}|}|\approx 1$, while the effective expansion
parameter $\epsilon_{S}^{\text{eff}}=\epsilon_{S}^{\prime}\hat{R}/V$ was also
of order 1. Hence, in that case, the linear approximation was less
trustworthy. That is, the results shown in Figure 3 are well within the range
of applicability of the linear approximation, which is a significant
improvement over [62].

For a particular branch of the hidden sector bundle construction, we have
found at least one example of two line bundles $(L_{1},L_{2})$ which can
potentially solve all of the geometrical, phenomenological and dimensional
reduction constraints we have outlined in this paper. The problem of finding
viable hidden sectors has thus been reduced to the calculation of the
$H^{1}(X,L_{1}L_{2}^{-3})$ and $H^{1}(X,L_{1}^{2})$ cohomologies (modulo
whether the necessary conditions for stability that we have imposed do indeed
lead to stable bundles). The two bundles $L_{1}=\mathcal{O}_{X}(-5,-1,1)$ and
$L_{2}=\mathcal{O}_{X}(2,1,-1)$ that we have discussed in detail in this
section are not unique – there are actually a large number of pairs of line
bundles that satisfy the same constraints. We show a subset of them in Table
2, restricted to the range $|m^{i}|\leq 15$, $|n^{i}|\leq 15$, with $i=1,2,3$.

Working with two line bundles has introduced more degrees of freedom into our
system and, implicitly, more flexibility in finding solutions than was present
in the single line bundle case given in [62]. With this extra freedom, we have
potentially succeeded in building a non-trivial $SU(3)$ bundle
$V_{\bm{\underline{3}}}^{\prime}$, which is stable within the physically
viable region of the Kähler cone.

$L_{1}$ | $L_{2}$
---|---
$\mathcal{O}_{X}(-11,-10,4)$ | $\mathcal{O}_{X}(-2,14,-2)$
$\mathcal{O}_{X}(-8,-7,3)$ | $\mathcal{O}_{X}(-1,13,-2)$
$\mathcal{O}_{X}(-5,-4,2)$ | $\mathcal{O}_{X}(2,-2,0)$
$\mathcal{O}_{X}(-5,-1,1)$ | $\mathcal{O}_{X}(2,1,-1)$
Table 2: A subset of pairs of line bundles that lead to solutions of the full
set of the constraints on the branch $\langle
C_{2}\rangle,\langle\tilde{C}_{3}\rangle\neq 0$. The solution space which
corresponds to $L_{1}=\mathcal{O}_{X}(-5,-1,1)$ and
$L_{2}=\mathcal{O}_{X}(2,1,-1)$ was shown in Figure 2. We have restricted our
scan to line bundles $L_{1}=\mathcal{O}_{X}(m^{1},m^{2},m^{3})$ and
$L_{2}=\mathcal{O}_{X}(n^{1},n^{2},n^{3})$ for which $|m^{i}|\leq 15$,
$|n^{i}|\leq 15$.

##  4 Different Extension Branches

Branch | Field VEVs | Sequences | Ext
---|---|---|---
1 | $\langle C_{2}\rangle,\langle\tilde{C}_{3}\rangle\neq 0$ | $\begin{matrix}0\rightarrow\mathcal{F}\rightarrow W\rightarrow\mathcal{E}\rightarrow 0\\\ 0\rightarrow\mathcal{K}\rightarrow V_{\bm{\underline{3}}}^{\prime}\rightarrow W\rightarrow 0\end{matrix}$ | $\begin{matrix}H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})\\\ H^{1}(X,\mathcal{K}\otimes\mathcal{E}^{*})\end{matrix}$
2 | $\langle C_{3}\rangle,\langle\tilde{C}_{2}\rangle\neq 0$ | $\begin{matrix}0\rightarrow\mathcal{E}\rightarrow W^{(2)}\rightarrow\mathcal{K}\rightarrow 0\\\ 0\rightarrow W^{(2)}\rightarrow{V_{\bm{\underline{3}}}^{\prime}}^{(2)}\rightarrow\mathcal{F}\rightarrow 0\end{matrix}$ | $\begin{matrix}H^{1}(X,\mathcal{E}\otimes\mathcal{K}^{*})\\\ H^{1}(X,\mathcal{E}\otimes\mathcal{F}^{*})\end{matrix}$
3 | $\langle C_{2}\rangle,\langle C_{1}\rangle\neq 0$ | $\begin{matrix}0\rightarrow\mathcal{F}\rightarrow W^{(3)}\rightarrow\mathcal{E}\rightarrow 0\\\ 0\rightarrow W^{(3)}\rightarrow{V_{\bm{\underline{3}}}^{\prime}}^{(3)}\rightarrow\mathcal{K}\rightarrow 0\end{matrix}$ | $\begin{matrix}H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})\\\ H^{1}(X,\mathcal{F}\otimes\mathcal{K}^{*})\end{matrix}$
4 | $\langle\tilde{C}_{1}\rangle,\langle\tilde{C}_{2}\rangle\neq 0$ | $\begin{matrix}0\rightarrow\mathcal{K}\rightarrow W^{(4)}\rightarrow\mathcal{F}\rightarrow 0\\\ 0\rightarrow\mathcal{E}\rightarrow{V_{\bm{\underline{3}}}^{\prime}}^{(4)}\rightarrow W^{(4)}\rightarrow 0\end{matrix}$ | $\begin{matrix}H^{1}(X,\mathcal{K}\otimes\mathcal{F}^{*})\\\ H^{1}(X,\mathcal{E}\otimes\mathcal{F}^{*})\end{matrix}$
5 | $\langle C_{3}\rangle,\langle C_{1}\rangle\neq 0$ | $\begin{matrix}0\rightarrow\mathcal{E}\rightarrow W^{(5)}\rightarrow\mathcal{K}\rightarrow 0\\\ 0\rightarrow\mathcal{F}\rightarrow{V_{\bm{\underline{3}}}^{\prime}}^{(5)}\rightarrow W^{(5)}\rightarrow 0\end{matrix}$ | $\begin{matrix}H^{1}(X,\mathcal{E}\otimes\mathcal{K}^{*})\\\ H^{1}(X,\mathcal{F}\otimes\mathcal{K}^{*})\end{matrix}$
6 | $\langle\tilde{C}_{1}\rangle,\langle\tilde{C}_{3}\rangle\neq 0$ | $\begin{matrix}0\rightarrow\mathcal{K}\rightarrow W^{(6)}\rightarrow\mathcal{F}\rightarrow 0\\\ 0\rightarrow W^{(6)}\rightarrow{V_{\bm{\underline{3}}}^{\prime}}^{(6)}\rightarrow\mathcal{E}\rightarrow 0\end{matrix}$ | $\begin{matrix}H^{1}(X,\mathcal{K}\otimes\mathcal{F}^{*})\\\ H^{1}(X,\mathcal{K}\otimes\mathcal{E}^{*})\end{matrix}$
Table 3: The six extension branches of the split Whitney sum bundle
$V_{\bm{\underline{3}}}=\mathcal{F}\oplus\mathcal{K}\oplus\mathcal{E}$. For
each branch, there is also a second pair of sequences which corresponds to
switching the order of the extensions. The resulting bundles can be shown to
be isomorphic [74] and, hence, we do not display them.

In the example in the previous section, we showed it might be possible to
extend the decomposable bundle $V_{\bm{\underline{3}}}$, defined in (3.18), to
a non-trivial stable $SU(3)$ bundle $V_{\bm{\underline{3}}}^{\prime}$ via the
extension sequences defined in (3.43). As we discussed, this extension is
equivalent to turning on VEVs for the fields $C_{2}$ and $\tilde{C}_{3}$ in
the four-dimensional effective theory. We now ask if we can still solve the
system of vacuum constraints if we chose a different extension sequence or,
equivalently, if we chose to turn on VEVs for different combinations of $C$
and $\tilde{C}$ fields.

First of all, not all combinations of pairs of VEVs are allowed. The
F-flatness conditions, coming from the vanishing of the superpotential and its
first derivative, reduce the fifteen combinations of pairs of VEVs to six
[74]. In Table 3 we give the allowed VEVs for each branch and the
corresponding extension sequences. The branch we studied in the previous
section corresponds to branch one in this table. Let us now analyse the
conditions for a non-trivial extension and stability for the remaining
branches.

###  4.1 Choosing a Different Branch

Consider first turning on VEVs for the fields $C_{1}$ and $C_{2}$

$\langle C_{1}\rangle\neq 0\ ,\qquad\langle C_{2}\rangle\neq 0\ ,$ (4.1)

while all other VEVs are set to zero. This combination of VEVs corresponds to
the third branch in Table 3. The $U(1)\times U(1)$ bundle
$V_{\bm{\underline{3}}}$ is deformed to an irreducible $SU(3)$ bundle
${V_{\bm{\underline{3}}}^{\prime}}^{(3)}$–where the superscript (3) indicates
that one is working in the third branch of Table 3–via two extension
sequences,

$\begin{split}\langle C_{2}\rangle\neq
0\colon\qquad&0\rightarrow\mathcal{F}\rightarrow
W^{(3)}\rightarrow\mathcal{E}\rightarrow 0\ ,\\\ \langle C_{1}\rangle\neq
0\colon\qquad&0\rightarrow
W^{(3)}\rightarrow{V^{\prime}}^{(3)}_{\bm{\underline{3}}}\rightarrow\mathcal{K}\rightarrow
0\ .\end{split}$ (4.2)

In this case, a non-trivial extension
${V^{\prime}}^{(3)}_{\bm{\underline{3}}}$ exists if the Ext groups
$\operatorname{Ext}^{1}(\mathcal{E},\mathcal{F})=H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})$
and
$\operatorname{Ext}^{1}(\mathcal{K},\mathcal{F})=H^{1}(X,\mathcal{F}\otimes\mathcal{K}^{*})$
are non-trivial. As for branch one, there is an alternative set of sequences
which define an isomorphic bundle,

$\begin{split}\langle C_{1}\rangle\neq
0\colon\qquad&0\rightarrow\mathcal{F}\rightarrow{W^{\prime}}^{(3)}\rightarrow\mathcal{K}\rightarrow
0\ ,\\\ \langle C_{2}\rangle\neq 0\colon\qquad&0\rightarrow
W^{(3)\prime}\rightarrow{\mathcal{V}^{\prime}}^{(3)}_{\bm{\underline{3}}}\rightarrow\mathcal{E}\rightarrow
0\ .\end{split}$ (4.3)

Again, the cohomology groups $H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})$ and
$H^{1}(X,\mathcal{F}\otimes\mathcal{K}^{*})$ must be non-vanishing. The final
$SU(3)$ bundles are isomorphic,
${V^{\prime}}^{(3)}_{\bm{\underline{3}}}\sim{\mathcal{V}^{\prime}}^{(3)}_{\bm{\underline{3}}}$.

It follows from Table 1 that the fields $C_{1}$ and $C_{2}$ correspond to

$\displaystyle\mathcal{F}\otimes\mathcal{K}^{*}=L_{1}^{-1}L_{2}^{-3}$
$\displaystyle=\mathcal{O}_{X}(-m_{1}-3n_{1},-m_{2}-3n_{2},-m_{3}-3n_{3})\ ,$
(4.4) $\displaystyle\mathcal{F}\otimes\mathcal{E}^{*}=L_{1}L_{2}^{-3}$
$\displaystyle=\mathcal{O}_{X}(m_{1}-3n_{1},m_{2}-3n_{2},m_{3}-3n_{3})$ (4.5)

respectively. Hence, as discussed above, in order for the cohomology groups
$H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})$ and
$H^{1}(X,\mathcal{F}\otimes\mathcal{K}^{*})$ to be non-vanishing, one must
impose the constraint that the line bundles $L_{1}L_{2}^{-3}$ and
$L_{1}^{-1}L_{2}^{-3}$ be non-ample.

Assuming a non-trivial extension exists, we also want
${V_{\bm{\underline{3}}}^{\prime}}^{(3)}$ to be slope-stable. Looking at the
sequences in eq. (4.2), we see that the line bundle $\mathcal{F}$ injects into
$W^{(3)}$, which itself injects into
${V_{\bm{\underline{3}}}^{\prime}}^{(3)}$. Hence $\mathcal{F}$ is a subbundle
of ${V_{\bm{\underline{3}}}^{\prime}}^{(3)}$. Since
${V_{\bm{\underline{3}}}^{\prime}}^{(3)}$ has vanishing slope, it can be
slope-stable only if $\mathcal{F}$ has negative slope at tree-level:

$\displaystyle\mu(\mathcal{F})=\mu(L_{2}^{-1})<0\quad\Rightarrow\quad-2d_{ijk}n^{i}a^{j}a^{k}<0\
.$ (4.6)

We also obtain a condition on the slope of $\mathcal{K}$ from the same
sequences in eq. (4.2). From the second line we learn that

$c_{1}(W^{(3)})+c_{1}(\mathcal{K})=c_{1}({V_{\bm{\underline{3}}}^{\prime}}^{(3)})=0\
.$ (4.7)

Recall that the slope of a bundle $\mathcal{L}$ is given by

$\mu(\mathcal{L})=\frac{1}{\operatorname{rk}\mathcal{L}}\int_{X}\omega\wedge\omega\wedge
c_{1}(\mathcal{L})\ ,$ (4.8)

where $\operatorname{rk}\mathcal{L}$ is the rank of the bundle $\mathcal{L}$.
Noting that $\operatorname{rk}\mathcal{K}=1$, $\operatorname{rk}{W}^{(3)}=2$
and $\operatorname{rk}V_{\bm{\underline{3}}}^{\prime(3)}=3$, it follows from
(4.7) that

$2\mu(W^{(3)})+\mu(\mathcal{K})=3\mu({V_{\bm{\underline{3}}}^{\prime}}^{(3)})=0\
.$ (4.9)

Since $W^{(3)}$ is a subbundle of ${V_{\bm{\underline{3}}}^{\prime}}^{(3)}$,
it must have a negative slope in order for $V_{\bm{\underline{3}}}^{\prime}$
to be slope-stable. Therefore, one must take the slope of $\mathcal{K}$ to be
positive

$\displaystyle\mu(\mathcal{K})=\mu(L_{1}L_{2})>0\quad\Rightarrow\quad
2d_{ijk}(m^{i}+n^{i})a^{j}a^{k}>0\ .$ (4.10)

From the alternative sequences in (4.3), we learn that

$2\mu(W^{(3)\prime})+\mu(\mathcal{E})=3\mu({\mathcal{V}_{\bm{\underline{3}}}^{\prime}}^{(3)})=0\
.$ (4.11)

Again, since $W^{(3)\prime}$ is a subbundle of
${\mathcal{V}_{\bm{\underline{3}}}^{\prime}}^{(3)}$, its slope must be
negative and, therefore, $\mathcal{E}$ must have positive slope. That is

$\displaystyle\mu(\mathcal{E})=\mu(L_{1}^{-1}L_{2})>0\quad\Rightarrow\quad
d_{ijk}(-m^{i}+n^{i})a^{j}a^{k}>0\ .$ (4.12)

Putting this all together, three necessary conditions for the slope-stability
of ${V_{\bm{\underline{3}}}^{\prime}}^{(3)}$ are

$\mu(\mathcal{F})<0\ ,\quad\mu(\mathcal{K})>0\ ,\quad\mu(\mathcal{E})>0\ .$
(4.13)

Of course, in addition to these three conditions, it is necessary to require
that $V_{\bm{\underline{3}}}^{\prime}$ satisfy the Bogomolov inequality. That
is,

$\int c_{2}({V_{\bm{\underline{3}}}^{\prime}}^{(3)})\wedge\omega\geq
0\quad\Rightarrow\quad d_{ijk}m^{i}m^{j}a^{k}+3d_{ijk}n^{i}n^{j}a^{k}\leq 0\
.$ (4.14)

Note that the form of the Bogomolov inequality is the same as in (3.54). Of
course, all of the universal physical constraints remain unchanged.

Figure 4:  The colored region is the subspace of Kähler moduli space which, in
addition to satisfying conditions (2.4)-(2.8) outlined in Section 2, also
restricts the strong coupling parameter to the smaller values
$\epsilon_{S}^{\text{eff}}\lesssim 0.2$. Hence, any point in this region also
satisfies the linearization constraint (2.11). This subspace does not depend
on the hidden sector bundle. The “orange” regime is a subspace of this colored
region, in which the non-trivial $SU(3)$ bundle
$V_{\bm{\underline{3}}}^{\prime}$, defined as an extension of the two line
bundles $L_{1}=\mathcal{O}_{X}(-1,-2,1)$ and $L_{2}=\mathcal{O}_{X}(6,0,-1)$,
can be slope-stable. Note that one can show that the locus where the
decomposable Whitney sum bundle $V_{\bm{\underline{3}}}$ is poly-stable lies
outside the colored region shown in Figure 1 and, hence, does not satisfy all
of the universal constraints.

Once again, we perform a systematic scan over all pairs of line bundles with
$|m^{i}|,|n^{i}|\leq 15$ for $i=1,2,3$. The values of $(a^{1},a^{2},a^{3})$
that we sample sit inside the “blue” region of the Kähler cone given in Figure
1. The survey produces multiple solutions, among which are

$L_{1}=\mathcal{O}_{X}(-1,-2,1)\ ,\qquad L_{2}=\mathcal{O}_{X}(6,0,-1)\ .$
(4.15)

For this pair of line bundles, we find that the class of the five-brane is

$W_{i}=(0,2,0)_{i}\ ,$ (4.16)

which is effective since $W_{i}\geq 0$ for $i=1,2,3$. Furthermore, the line
bundles $L_{1}^{-1}L_{2}^{-3}=\mathcal{O}_{X}(-19,-2,4)$ and
$L_{1}L_{2}^{-3}=\mathcal{O}_{X}(-17,2,2)$, associated with the cohomologies
of the $\bm{\underline{1}}_{1,3}$ and $\bm{\underline{1}}_{-1,3}$
representations respectively, are clearly non-ample. This ensures that the
relevant sequence extension cohomologies $H^{1}(X,L_{1}^{-1}L_{2}^{-3})$ and
$H^{1}(X,L_{1}L_{2}^{-3})$ are potentially non-vanishing. As discussed above,
these cohomologies need to be computed individually to ensure that they are,
in fact, non-zero. However, as mentioned earlier, such computations are beyond
the scope of the present paper. Assuming that the existence of the extension
can be proven, we can check the necessary conditions for stability. We sample
values $(a^{1},a^{2},a^{3})$ inside the “blue” region in Figure 1 and check
that the stability criteria derived for this third extension branch, that is,
equations (4.6), (4.10) and (4.14), are satisfied. The subspace of the “blue”
region where this extended bundle is potentially slope-stable is displayed in
“orange” in Figure 4.

There are other pairs of bundles, in addition to (4.15), that satisfy all
constraints when we deform the decomposable bundle along the third branch. We
display the full results of our survey in Table 5, along with the analogous
solutions for the remaining four branches as well. Again, whether or not each
of these solutions exists depends on the pair of $H^{1}$ cohomologies
associated with the extension sequences being non-vanishing.

###  4.2 Full Branch Structure Scan

In Sections 3.3 and 4.1 we derived the sets of constraints for the first and
the third extension branches, respectively. The constraints for the remaining
four branches can be derived in similar fashion and so we will not give any
details. Instead, in Table 4 we give the necessary conditions on the slopes of
the bundles $\mathcal{F}$, $\mathcal{K}$ and $\mathcal{E}$ for each branch.
The columns in this table indicate the required signs of the slopes. For
example, when we move away from the decomposable locus along the first branch,
both $\mathcal{F}$ and $\mathcal{K}$ must have negative slope for the
resulting $SU(3)$ bundle to be stable.

Branch | $\mu(\mathcal{F})$ | $\mu(\mathcal{K})$ | $\mu(\mathcal{E})$
---|---|---|---
1 | $-$ | $-$ | $+$
2 | $+$ | $+$ | $-$
3 | $-$ | $+$ | $+$
4 | $+$ | $-$ | $-$
5 | $-$ | $+$ | $-$
6 | $+$ | $-$ | $+$
Table 4: Necessary stability conditions for the slopes of $\mathcal{F}$,
$\mathcal{K}$ and the $\mathcal{E}$.

We have already given the set of solutions for the first branch in Table 2. We
now perform similar scans for each of the remaining extension branches and
display the results of the scan in Table 5. The first two columns of Table 5
show the pairs of line bundles $L_{1}$ and $L_{2}$ which satisfy the necessary
stability conditions inside the “blue” region shown in Figure 1. We show only
those bundles with $|m^{i}|,|n^{i}|\leq 10$ to save space666Note that in
Sections 3 and 4 we scanned over the larger intervals $|m^{i}|,|n^{i}|\leq
15$. Hence, for example, the last two entries in Table 2 do not appear in
Table 5.. In the third and fourth columns of Table 5 we give the line bundles
associated with the fields $C$ and $\tilde{C}$ which get VEVs in the effective
theory. Therefore, the cohomology groups $H^{1}$ which are associated with
these bundles should be non-zero to allow for a non-trivial extension bundle.

Branch 1 | $L_{1}$ | $L_{2}$ | $\mathcal{F}\otimes\mathcal{E}^{*}$ | $\mathcal{K}\otimes\mathcal{E}^{*}$
---|---|---|---|---
| $(-5,-4,2)$ | $(-2,-2,0)$ | $(-11,2,2)$ | $(-10,-8,4)$
| $(-5,-1,1)$ | $(2,1,-1)$ | $(-11,-4,4)$ | $(-10,-2,2)$
Branch 2 | $L_{1}$ | $L_{2}$ | $\mathcal{E}\otimes\mathcal{F}^{*}$ | $\mathcal{E}\otimes\mathcal{K}^{*}$
| $(1,5,-1)$ | $(-2,-1,1)$ | $(-7,-8,4)$ | $(-2,-10,2)$
| $(5,1,-1)$ | $(-2,-1,1)$ | $(-11,-4,4)$ | $(-10,-2,2)$
| $(5,4,-2)$ | $(-2,2,0)$ | $(-11,2,2)$ | $(-10,-8,4)$
Branch 3 | $L_{1}$ | $L_{2}$ | $\mathcal{F}\otimes\mathcal{E}^{*}$ | $\mathcal{F}\otimes\mathcal{K}^{*}$
| $(-2,-1,1)$ | $(6,0,-1)$ | $(-20,-1,4)$ | $(-16,1,2)$
| $(-1,-8,4)$ | $(0,6,0)$ | $(-1,-26,4)$ | $(1,-10,-4)$
| $(-1,-2,1)$ | $(6,0,-1)$ | $(-19,-2,4)$ | $(-17,2,2)$
| $(1,2,-1)$ | $(6,0,-1)$ | $(-17,2,2)$ | $(-19,-2,4)$
| $(1,8,-4)$ | $(0,6,0)$ | $(1,-10,-4)$ | $(-1,-26,4)$
| $(2,1,-1)$ | $(6,0,-1)$ | $(-16,1,2)$ | $(-20,-1,4)$
Branch 4 | $L_{1}$ | $L_{2}$ | $\mathcal{E}\otimes\mathcal{F}^{*}$ | $\mathcal{K}\otimes\mathcal{F}^{*}$
| $(-2,-1,1)$ | $(-6,0,1)$ | $(-16,1,2)$ | $(-20,-1,4)$
| $(-1,-8,4)$ | $(0,-6,0)$ | $(1,-10,-4)$ | $(-1,-26,4)$
| $(-1,-2,1)$ | $(-6,0,1)$ | $(-17,2,2)$ | $(-19,-2,4)$
| $(1,2,-1)$ | $(-6,0,1)$ | $(-19,-2,4)$ | $(-17,2,2)$
| $(1,8,-4)$ | $(0,-6,0)$ | $(-1,-26,4)$ | $(1,-10,-4)$
| $(2,1,-1)$ | $(-6,0,1)$ | $(-20,-1,4)$ | $(-16,1,2)$
Branch 5 | $L_{1}$ | $L_{2}$ | $\mathcal{E}\otimes\mathcal{K}^{*}$ | $\mathcal{F}\otimes\mathcal{K}^{*}$
| $(1,-5,-1)$ | $(2,1,-1)$ | $(-2,-10,2)$ | $(-7,-8,4)$
| $(5,1,-1)$ | $(2,1,-1)$ | $(-10,-2,2)$ | $(-11,-4,4)$
| $(5,4,-2)$ | $(2,-2,0)$ | $(-10,-8,4)$ | $(-11,2,2)$
Branch 6 | $L_{1}$ | $L_{2}$ | $\mathcal{K}\otimes\mathcal{E}^{*}$ | $\mathcal{K}\otimes\mathcal{F}^{*}$
| $(-5,-4,2)$ | $(-2,2,0)$ | $(-10,-8,4)$ | $(-11,2,2)$
| $(-5,-1,1)$ | $(-2,-1,1)$ | $(-10,-2,2)$ | $(-11,-4,4)$
| $(-2,-10,1)$ | $(1,2,-3)$ | $(-4,-20,2)$ | $(1,-4,-8)$
| $(-1,-5,1)$ | $(-2,-1,1)$ | $(-2,-10,2)$ | $(-7,-8,4)$
Table 5: In the first two columns, we list pairs of line bundles $L_{1}$ and
$L_{2}$ which satisfy the necessary stability conditions inside the “blue”
region shown in Figure 1. In the third and forth columns, we present the line
bundles associated with the charged matter fields which get VEVs. Note that we
have used an abbreviated notation to represent the line bundles,
$\mathcal{O}_{X}(a,b,c)\equiv(a,b,c)$.

Showing that any of these extensions is non-zero would encourage a full check
of stability for the relevant extension bundle, as they would give a plausible
hidden sector for the $B-L$ MSSM. Compared with the construction of hidden
sectors from a single line bundle presented in [62], we have many more
possible solutions when using two line bundles.

##  5 Conclusion

In this paper, we attempted to build a hidden sector bundle using two line
bundles together with an explicit embedding into $E_{8}$, via the two line
bundle vectors given in (3.10) or, equivalently, by embedding them into the
$SU(3)$ factor of the maximal subgroup $E_{6}\times SU(3)$. It should be
clear, however, that the analysis and methods developed in this paper can be
applied to models with hidden sectors built from different embeddings into
$E_{8}$. For example, consider two line bundles embedded into $E_{8}$, not
with the generators given in (3.10), but, rather, with two different linearly
independent generators

$\displaystyle\bm{t}_{1}$ $\displaystyle=(0,0,0,0,0,0,0,-2)\ ,$ (5.1)
$\displaystyle\bm{t}_{2}$ $\displaystyle=(0,0,0,0,0,0,-2,0)\ ,$ (5.2)

with the two different line bundle vectors given by

$\bm{V}_{i}=m^{i}\bm{t}_{1}+n^{i}\bm{t}_{2}\ \quad i=1,2,3\ .$ (5.3)

The non-Abelian commutant of $U(1)\times U(1)$ inside $E_{8}$ is then
$SO(12)$. Using the _Mathematica_ package LieART [69, 70], one can then find
the decomposition of the adjoint representation of ${E}_{8}$ with respect to
$SO(12)\times U(1)\times U(1)$. It is given by

$\begin{split}\bm{\underline{248}}&=\bm{\underline{66}}_{0,0}+2\times\bm{\underline{1}}_{0,0}+\bm{\underline{1}}_{2,0}+\bm{\underline{1}}_{-2,0}+\bm{\underline{1}}_{0,2}+\bm{\underline{1}}_{0,-2}\\\
&\mathrel{\phantom{{=}}{}}+\bm{\underline{32}}_{1,0}+\bm{\underline{32}}_{-1,0}+\bm{\underline{32}}_{1,1}+\bm{\underline{32}}_{1,-1}\\\
&\mathrel{\phantom{{=}}{}}+\bm{\underline{12}}_{1,1}+\bm{\underline{12}}_{-1,1}+\bm{\underline{12}}_{1,-1}+\bm{\underline{12}}_{-1,1}\
.\end{split}$ (5.4)

More prosaically, this breaking pattern can be obtained by first breaking
$E_{8}$ to $E_{7}\times SU(2)$, under which the adjoint representation of
$E_{8}$ decomposes as

$\bm{\underline{248}}=(\bm{\underline{133}},\bm{\underline{1}})+(\bm{\underline{56}},\bm{\underline{2}})+(\bm{\underline{1}},\bm{\underline{3}})\
.$ (5.5)

Breaking $E_{7}$ further to $SO(12)\times SU(2)$, the $E_{7}$ representations
that appear above decompose as

$\begin{split}\bm{\underline{133}}&=(\bm{\underline{66}},\bm{\underline{1}})+(\bm{\underline{32}},\bm{\underline{2}})+(\bm{\underline{1}},\bm{\underline{3}})\
,\\\
\bm{\underline{56}}&=(\bm{\underline{12}},\bm{\underline{2}})+(\bm{\underline{32}},\bm{\underline{1}})\
.\\\ \end{split}$ (5.6)

Finally, we break both $SU(2)$ groups down to $U(1)$s, with the fundamental
representation of $SU(2)$ decomposing as
$\bm{\underline{2}}=\bm{\underline{1}}_{1}+\bm{\underline{1}}_{-1}$. Putting
this together, one sees that the adjoint representation of $E_{8}$ decomposes
under $SO(12)\times U(1)\times U(1)$ exactly as in (5.4) above.

Each of the two line bundles $L_{1}$ and $L_{2}$ associated with the two
$U(1)$ factors are embedded into a different $SU(2)$ bundle. At the
decomposable locus, we have a reducible rank-four bundle

$\begin{split}V_{\bm{\underline{2}},\bm{\underline{2}}}=(L_{1}^{-1}\oplus
L_{1})\oplus(L_{2}^{-1}\oplus L_{2})\ ,\end{split}$ (5.7)

with $U(1)\times U(1)$ structure group. In this case, the only singlets under
the non-Abelian $SO(12)$ factor that are charged under the two $U(1)$s are
$\bm{\underline{1}}_{2,0}$, $\bm{\underline{1}}_{0,2}$ and their conjugates.
These give charged singlet matter fields whose VEVs may be used to set the
D-terms associated with the low-energy $U(1)$s to zero. Turning on a
combination of two VEVs and, hence, extending the associated exact sequences,
would correspond to deforming the hidden sector bundle with $U(1)\times U(1)$
structure group into one with ${SU}(2)\times{SU}(2)$ instead. If each $SU(2)$
bundle can be made slope-stable, the extended version of
$V_{\bm{\underline{2}},\bm{\underline{2}}}$ will then be slope poly-stable.

We conclude that there are clearly a large number of different ways of
consistently embedding two line bundles into $E_{8}$ and, hence, many
different hidden sector bundles for the $B-L$ MSSM that can be constructed in
this manner. It is also clear that there are a large number of hidden sectors
that one can construct using three or more line bundles with various
embeddings into $E_{8}$. We hope to come back to this in future work.

#### Acknowledgements

AA is supported by the European Union’s Horizon 2020 research and innovation
program under the Marie Skłodowska-Curie grant agreement No. 838776. SD is
supported in part by research grant DOE No. DESC0007901. BAO is supported in
part by the research grant DOE No. DESC0007901 and SAS Account
020-0188-2-010202-6603-0338.

##  Appendix A Conventions

We follow the Dynkin convention for labelling the simple roots of
$\mathfrak{e}_{8}$, in agreement with the _Mathematica_ package LieART [69,
70] which we used in many of the line bundle vector calculations. In
particular, we number the nodes of the Dynkin diagram as in Figure 5. We
mostly work in the “orthogonal basis” $\\{e_{a}\\}$ with $a=1,\ldots,8$ (see
[69] for more details), where the components of the root vectors are given
with respect to an orthogonal basis. In particular, the line bundle vectors
$\bm{V}_{i}$ and roots $\bm{r}$ in the main text are expressed in the
orthogonal basis. The eight simple roots $\alpha_{I}=\alpha_{I}^{a}e_{a}$ of
$\mathfrak{e}_{8}$ are given in this basis by

$\displaystyle\alpha_{1}^{a}$
$\displaystyle=\tfrac{1}{2}(1,-1,-1,-1,-1,-1,-1,1)\ ,$
$\displaystyle\alpha_{5}^{a}$ $\displaystyle=(0,0,0,-1,1,0,0,0)\ ,$ (A.1)
$\displaystyle\alpha_{2}^{a}$ $\displaystyle=(-1,1,0,0,0,0,0,0)\ ,$
$\displaystyle\alpha_{6}^{a}$ $\displaystyle=(0,0,0,0,-1,1,0,0)\ ,$
$\displaystyle\alpha_{3}^{a}$ $\displaystyle=(0,-1,1,0,0,0,0,0)\ ,$
$\displaystyle\alpha_{7}^{a}$ $\displaystyle=(0,0,0,0,0,-1,1,0)\ ,$
$\displaystyle\alpha_{4}^{a}$ $\displaystyle=(0,0,-1,1,0,0,0,0)\ ,$
$\displaystyle\alpha_{8}^{a}$ $\displaystyle=(1,1,0,0,0,0,0,0)\ .$

In addition to the orthogonal basis we also have the $\alpha$-basis and the
$\omega$-basis. The $\alpha$-basis is the basis of simple roots. This has the
advantage that it shows precisely how a given root is made from a sum of
simple roots. In this basis, the components of the simple roots are given by

$\tilde{\alpha}_{1}^{a}=(1,0,0,0,0,0,0,0)\
,\qquad\tilde{\alpha}_{2}^{a}=(0,1,0,0,0,0,0,0)\ ,$ (A.2)

and so on, so that
$\alpha_{I}=\tilde{\alpha}_{I}^{a}\alpha_{a}=\delta_{I}^{a}\alpha_{a}$.
Finally, the $\omega$-basis is the basis of fundamental weights, also known as
the Dynkin basis. This basis is such that the simple roots correspond to the
rows of the Cartan matrix $A_{ab}$ (for $\mathfrak{e}_{8}$ in this case), so
that, for example, the first two simple roots can be written as

$\hat{\alpha}_{1}^{a}=(2,-1,0,0,0,0,0,0)\
,\qquad\hat{\alpha}_{2}^{a}=(-1,2,-1,0,0,0,0,0)\ ,$ (A.3)

where we have written the simple roots in the $\omega$-basis as
$\alpha_{I}=\hat{\alpha}_{I}^{a}\omega_{a}$. Note that for algebras (such as
$\mathfrak{e}_{8}$) whose roots are all of length 2, the $\alpha$-basis and
$\omega$-basis are dual

$(\alpha_{a},\omega_{b})=\delta_{ab}\ .$ (A.4)

In particular, this implies

$(\alpha_{I},\alpha_{J})=\tilde{\alpha}_{I}^{a}\hat{\alpha}_{J}^{b}(\alpha_{a},\omega_{b})=\delta_{I}^{a}\hat{\alpha}_{J}^{b}\delta_{ab}=A_{IJ}\
.$ (A.5)

The transformations between these bases are given by

$\alpha_{a}=\sum_{b}A_{ab}\omega_{b}\
,\qquad\omega_{a}=\sum_{b}\hat{\Omega}_{ab}e_{b}\ ,$ (A.6)

where $\hat{\Omega}_{ab}$ is the matrix whose rows are the fundamental weights
in the orthogonal basis and $A_{ab}$ is the Cartan matrix of
$\mathfrak{e}_{8}$, given by

$A_{ab}=\mathopen{}\mathclose{{}\left(\begin{array}[]{cccccccc}2&-1&0&0&0&0&0&0\\\
-1&2&-1&0&0&0&0&0\\\ 0&-1&2&-1&0&0&0&-1\\\ 0&0&-1&2&-1&0&0&0\\\
0&0&0&-1&2&-1&0&0\\\ 0&0&0&0&-1&2&-1&0\\\ 0&0&0&0&0&-1&2&0\\\
0&0&-1&0&0&0&0&2\end{array}}\right)\ .$ (A.7)

Figure 5: Dynkin diagrams of the classical Lie groups. We follow the Dynkin
convention for labelling simple roots, as used in [69, 70].

Given the transformations between the various bases, it is simple to write
down line bundle vectors that break particular combinations of the simple
roots. Consider the inner product of the line bundle vector $V$ with the
$I^{\text{th}}$ simple root $\alpha_{I}$:

$V\cdot\alpha_{I}=\hat{V}^{a}\tilde{\alpha}_{I}^{b}(\omega_{a},\alpha_{b})=\hat{V}^{a}\delta_{I}^{b}\delta_{ab}=\hat{V}_{I}\
.$ (A.8)

This means that the inner product of a line bundle vector $V$ with a simple
root $\alpha_{I}$ is given by the $I^{\text{th}}$ component of the line bundle
vector written in the $\omega$-basis. If, for example, we want to pick a line
bundle vector that breaks the first simple root of $\mathfrak{e}_{8}$ and is
orthogonal to the others, we can take $\hat{V}^{a}=(1,0,0,0,0,0,0,0)$.
Transforming back to the orthogonal basis (which is the basis used in the main
text), we would then have

$\bm{V}=(0,0,0,0,0,0,0,2)\ .$ (A.9)

This would lead to an unbroken $SO(14)$ group.

##  Appendix B Anomaly Cancellation

As discussed in [5, 6], anomaly cancellation in heterotic M-theory requires
that

$c_{2}(TX)-c_{2}(\mathcal{V}^{(1)})-\operatorname{ch}_{2}(\mathcal{V}^{(2)})-W=0\
,$ (B.1)

where $\mathcal{V}^{(1)}$ is the observable $SU(4)$ bundle,
$\mathcal{V}^{(2)}$ is the hidden sector bundle – whose composition is the
main interest in this paper – $TX$ is the tangent bundle of the
compactification threefold, while $W$ is the effective class of the single
five-brane between the hidden and observable sector. Using results for the
second Chern class of the observable sector tangent bundle and gauge bundle of
the $B-L$ MSSM given in [67], we find that

$\frac{1}{v^{1/3}}\int_{X}\mathopen{}\mathclose{{}\left(c_{2}(TX)-c_{2}(\mathcal{V}^{(1)})}\right)\wedge\omega_{i}=\mathopen{}\mathclose{{}\left(\tfrac{4}{3},\tfrac{7}{3},-4}\right)_{i}\
.$ (B.2)

Similarly, it was shown in [67] that for a generic hidden sector vector bundle
of the form

$\mathcal{V}^{(2)}=\mathcal{V}_{N}\oplus\mathcal{L},\quad\mathcal{L}=\bigoplus_{r=1}^{R}L_{r}$
(B.3)

where $\mathcal{V}_{N}$ is a slope-stable, non-Abelian bundle and each
$L_{r}=\mathcal{O}_{X}(l_{r}^{1},l_{r}^{2},l_{r}^{3})$ is a holomorphic line
bundle with structure group U(1), that

$\frac{1}{v^{1/3}}\int_{X}\mathopen{}\mathclose{{}\left(-\rm{ch}_{2}(\mathcal{V}^{(2)})}\right)\wedge\omega_{i}=-d_{ijk}c_{N}^{jk}+\sum_{r=1}^{R}a_{r}d_{ijk}l_{r}^{j}l_{r}^{k}\
.$ (B.4)

The coefficients $d_{ijk}$ are the intersection numbers associated with our
specific Schoen threefold. They were given in [66, 67], for example. However,
since they are used extensively in this paper, we present them here again for
completeness. They are given by

$(d_{ijk})=\mathopen{}\mathclose{{}\left(\begin{array}[]{ccc}\mathopen{}\mathclose{{}\left(0,\tfrac{1}{3},0}\right)&\mathopen{}\mathclose{{}\left(\tfrac{1}{3},\tfrac{1}{3},1}\right)&\mathopen{}\mathclose{{}\left(0,1,0}\right)\\\
\mathopen{}\mathclose{{}\left(\tfrac{1}{3},\tfrac{1}{3},1}\right)&\mathopen{}\mathclose{{}\left(\tfrac{1}{3},0,0}\right)&\mathopen{}\mathclose{{}\left(1,0,0}\right)\\\
\mathopen{}\mathclose{{}\left(0,1,0}\right)&\mathopen{}\mathclose{{}\left(1,0,0}\right)&\mathopen{}\mathclose{{}\left(0,0,0}\right)\end{array}}\right).$
(B.5)

The $(i,j)$-th entry in the matrix corresponds to the triplet
$(d_{ijk})_{k=1,2,3}$. The coefficient

$a_{r}=\frac{1}{4}\operatorname{tr}Q_{r}^{2}\ ,$ (B.6)

with $Q_{r}$ the generator of the $r$-th $U(1)$ factor embedding into the
$\underline{\bf{248}}$ representation of the hidden sector $E_{8}$. If we
define

$W_{i}=\frac{1}{v^{1/3}}\int_{X}{W\wedge\omega_{i}}\ ,$ (B.7)

it follows that the anomaly condition (B.1) can be re-expressed as

$W_{i}=\big{(}\tfrac{4}{3},\tfrac{7}{3},-4\big{)}\big{|}_{i}-d_{ijk}c_{N}^{jk}+\sum_{r=1}^{R}a_{r}d_{ijk}l_{r}^{j}l_{r}^{k}\
.$ (B.8)

As discussed in [67], in order for each line bundle $L_{r}$ to arise from a
$\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ equivariant line bundle on the covering
space of X, the integers $l_{r}^{i}$, $i=1,2,3$ of
$L_{r}=\mathcal{O}_{X}(l_{r}^{1},l_{r}^{2},l_{r}^{3})$ must satisfy that
constraint that

$(l_{r}^{1}+l_{r}^{2})~{}{\rm{mod}}~{}3=0\ .$ (B.9)

Furthermore, in order to preserve $N=1$ supersymmetry $W$ must be an effective
class; that is, each component $W_{i}$, $i=1,2,3$ must be non-negative.

In Section 3, we study a hidden sector composed of two line bundles
$L_{1}=\mathcal{O}_{X}(m^{1},m^{2},m^{3})$ and
$L_{2}=\mathcal{O}_{X}(n^{1},n^{2},n^{3})$ embedded into an $SU(3)\subset
E_{8}$ connection with no non-Abelian bundle factor. In this case, the second
Chern character of the hidden sector bundle is found to be to be

$\begin{split}\operatorname{ch}_{2}(\mathcal{V}^{(2)})=&\,c_{1}(L_{1})\wedge
c_{1}(L_{1})+3c_{1}(L_{2})\wedge c_{1}(L_{2})\\\
&=\frac{1}{v^{2/3}}(m^{1}\omega_{1}+m^{2}\omega_{2}+m^{3}\omega_{3})^{2}+\frac{3}{v^{2/3}}(n^{1}\omega_{1}+n^{2}\omega_{2}+n^{3}\omega_{3})^{2}\
.\end{split}$ (B.10)

The anomaly condition then takes the form

$W_{i}=\big{(}\tfrac{4}{3},\tfrac{7}{3},-4\big{)}\big{|}_{i}+d_{ijk}m^{j}m^{k}+3d_{ijk}n^{j}n^{k}\
.$ (B.11)

Finally, we note that in our model with a single five brane the
$\beta^{(n)}_{i}$ charges have the form

$\displaystyle\beta^{(0)}_{i}$
$\displaystyle=\big{(}\tfrac{2}{3},-\tfrac{1}{3},4\big{)}\big{|}_{i}\ ,$
(B.12) $\displaystyle\beta^{(1)}$ $\displaystyle=W_{i}\ ,$ (B.13)
$\displaystyle\beta^{N+1}_{i}$ $\displaystyle=-\beta_{i}^{(0)}-W_{i}\ .$
(B.14)

##  Appendix C Linearization Constraints

The five-dimensional effective theory of heterotic M-theory, obtained by
reducing Hořava–Witten theory on the Calabi–Yau threefold, admits a BPS double
domain wall solution with five-branes in the bulk space. This double domain
wall was analyzed in in detail in a series of papers [75, 5, 14, 4, 3, 18]. We
will summarize some of the relevant results here, and we will make use of the
notation outlined in [62].

The detailed structure of the linearized double domain wall depends on the
solution of three non-linear equations discussed in [5, 76]. These can be
approximately solved by expanding to linear order777This set of non-linear
equations was also solved to second order in [62]. in the expansion parameter

$\epsilon^{\text{eff}}_{S}=\frac{\epsilon_{S}^{\prime}\hat{R}}{V}\ .$ (C.1)

It was shown in [5] that the conditions for the validity of the linear
approximation then break into two parts. Written in terms of the averaged
moduli, these are

$2\epsilon_{S}^{\prime}\frac{{\widehat{R}}}{V^{1/3}}\mathopen{}\mathclose{{}\left|\beta_{i}^{(0)}\big{(}z-\tfrac{1}{2}\big{)}-\frac{1}{2}W_{i}(\tfrac{1}{2}-\lambda)^{2}}\right|\ll\mathopen{}\mathclose{{}\left|d_{ijk}a^{j}a^{k}}\right|,\quad
z\in[0,z_{1}]$ (C.2)

and

$2\epsilon_{S}^{\prime}\frac{{\widehat{R}}}{V^{1/3}}\mathopen{}\mathclose{{}\left|(\beta_{i}^{(0)}+W_{i})\big{(}z-\tfrac{1}{2}\big{)}-\frac{1}{2}W_{i}(\tfrac{1}{2}+\lambda)^{2}}\right|\ll\mathopen{}\mathclose{{}\left|d_{ijk}a^{j}a^{k}}\right|,\quad
z\in[z_{1},1].$ (C.3)

Here we define $z=\frac{x^{11}}{\pi\rho}$, where $x^{11}$ is the coordinate
along the dimension which separates the observable and the hidden sector in
the BPS state, while $\pi\rho$ is the separation parameter between these
sectors. In our setup, $z\in[0,1]$. The observable sector and the hidden
sector are located at $z=0$ and $z=1$, respectively, while the five brane is
set at $z=\lambda+\tfrac{1}{2}$. In the absence of a mechanism to fix the
five-brane position, $\lambda$ is another modulus in our system. For
consistency with our previous work[62], we fix the position of the five-brane
close to the hidden sector, at $\lambda=0.49$.

##  Appendix D Genus-One Corrected FI Terms

As is commonly known, a ${U}(1)$ symmetry that appears in the both the
internal and four-dimensional gauge generate a D-term potential proportional
to an FI term associated with the $U(1)$ bundle [77, 78, 79, 80] The
expression for the genus one corrected FI term associated with a $U(1)$ bundle
$L$ on the hidden sector was computed in [81, 80] within the context of the
weakly coupled heterotic string. It was then shown in [47] that in the
strongly coupled limit this expression becomes

$FI_{L}=\frac{a_{L}}{2}\frac{\epsilon_{S}\epsilon_{R}^{2}}{\kappa_{4}^{2}}\frac{1}{\hat{R}V^{2/3}}\mathopen{}\mathclose{{}\left[\mu(L)+\frac{\epsilon_{S}^{\prime}\hat{R}}{V^{1/3}}\int_{X}c_{1}(L)\wedge\mathopen{}\mathclose{{}\left(J^{(N+1)}+\sum^{N}_{n=1}z_{n}^{2}J^{(n)}}\right)}\right]\
,$ (D.1)

where the complex two-forms $J$ are defined in [47] and $n$ runs over all
five-branes in the bulk interval. In our set up, we only have one five brane
at position $z=\lambda+\frac{1}{2}$ with the source term given by $J^{1}=W$.
The coefficient $a_{L}$ depends on the exact embedding of the line bundle $L$
associated with the FI term into the hidden sector $E_{8}$. In the case of a
hidden sector with a single line bundle, there is one FI term associated with
it. In this case, the coefficient $a_{L}$ is simply equal to the coefficient
$a$ derived for the second Chern character. For the embedding $U(1)\rightarrow
SU(2)\rightarrow E_{8}$, we found in [62] that $a_{L}=1$.

In the case of a hidden sector with two line bundles, we have two $FI$ terms.
Each is associated with one of the two line bundles $\mathcal{F}$ and
$\mathcal{K}$ defined in the decomposition

$V_{\bm{\underline{3}}}=\mathcal{F}\oplus\mathcal{K}\oplus\mathcal{E}$ (D.2)

of the $SU(3)$ bundle $V_{\bm{\underline{3}}}$ at the decomposable locus. Note
that the line bundle $\mathcal{E}$ depends on $\mathcal{K}$ and $\mathcal{F}$
such that $c_{1}(\mathcal{E})=-c_{1}(\mathcal{F})-c_{1}(\mathcal{K})$. Hence,
$V_{\bm{\underline{3}}}$ has the structure group $S(U(1)\times U(1)\times
U(1))\sim U(1)\times U(1)$ at the stability wall. The genus-one corrected FI
terms in this case are

$\begin{split}&FI_{\mathcal{F}}=\frac{a_{\mathcal{F}}}{2}\frac{\epsilon_{S}\epsilon_{R}^{2}}{\kappa_{4}^{2}}\frac{1}{\hat{R}V^{2/3}}\mathopen{}\mathclose{{}\left[\mu(\mathcal{F})+\frac{\epsilon_{S}^{\prime}\hat{R}}{V^{1/3}}\int_{X}c_{1}(\mathcal{F})\wedge\mathopen{}\mathclose{{}\left(J^{(N+1)}+\sum^{N}_{n=1}z_{n}^{2}J^{(n)}}\right)}\right]\
,\\\
&FI_{\mathcal{K}}=\frac{a_{\mathcal{K}}}{2}\frac{\epsilon_{S}\epsilon_{R}^{2}}{\kappa_{4}^{2}}\frac{1}{\hat{R}V^{2/3}}\mathopen{}\mathclose{{}\left[\mu(\mathcal{K})+\frac{\epsilon_{S}^{\prime}\hat{R}}{V^{1/3}}\int_{X}c_{1}(\mathcal{K})\wedge\mathopen{}\mathclose{{}\left(J^{(N+1)}+\sum^{N}_{n=1}z_{n}^{2}J^{(n)}}\right)}\right]\
.\end{split}$ (D.3)

Note that the $FI$ terms in the effective theory are associated with the line
bundles $\mathcal{F}=L_{2}^{-2}$ and $\mathcal{K}=L_{1}L_{2}$ and _not_ with
the bundles $L_{1}=\mathcal{O}_{X}(m^{1},m^{2},m^{3})$ and
$L_{2}=\mathcal{O}_{X}(n^{1},n^{2},n^{3})$. Hence, the coefficients
$a_{\mathcal{F}}$ and $a_{\mathcal{K}}$ in front of the expressions in
(LABEL:eq:B3_App) depend on the how the bundles $\mathcal{F}$ and
$\mathcal{K}$ embed into the $E_{8}$ connection. This calculation was trivial
in the single bundles case, because we parametrized directly all the equations
in terms of the bundle $L$ associated with the $FI$ term. For the two line
bundle case of interest in this paper, one can read off the generators
$Q_{\mathcal{F}}$ and $Q_{\mathcal{K}}$ associated with their embedding into
$E_{8}$ from Table 1. These are given by

$\displaystyle
Q_{\mathcal{F}}=(1,-1,2,-2,-1,1,-\operatorname{id}_{27},0\times\operatorname{id}_{27},\operatorname{id}_{27},\operatorname{id}_{27},0\times\operatorname{id}_{27},-\operatorname{id}_{27})\
,$ (D.4) $\displaystyle
Q_{\mathcal{K}}=(-1,1,1,-1,-2,2,0\times\operatorname{id}_{27},-\operatorname{id}_{27},\operatorname{id}_{27},0\times\operatorname{id}_{27},\operatorname{id}_{27},-\operatorname{id}_{27})\
.$ (D.5)

Hence, we get

$\displaystyle
a_{\mathcal{F}}=\frac{1}{4}\operatorname{tr}Q_{\mathcal{F}}^{2}=1,$ (D.6)
$\displaystyle
a_{\mathcal{K}}=\frac{1}{4}\operatorname{tr}Q_{\mathcal{K}}^{2}=1\ .$ (D.7)

As a side note, one might realize that we could have parametrized our
equations in terms of the bundles $\mathcal{F}$ and $\mathcal{K}$ directly and
write

$\mathcal{F}=\mathcal{O}_{X}(p^{1},p^{2},p^{3})\
,\quad\mathcal{K}=\mathcal{O}_{X}(r^{1},r^{2},r^{3})\ ,$ (D.8)

such that

$\mathcal{E}=\mathcal{O}_{X}(-(p^{1}+r^{1}),-(p^{2}+r^{2}),-(p^{3}+r^{3}))\ .$
(D.9)

This description is related to the one we have used by the transformations

$p^{i}=-2m^{i}\ ,\quad r^{i}=m^{i}+n^{i}\ ,\quad i=1,2,3\ .$ (D.10)

When using this description, however, the constraint equations derived in the
main text become considerably more complicated. In particular, note that
$\operatorname{tr}Q_{\mathcal{F}}Q_{\mathcal{K}}\neq 0$. This non-zero mixing
term leads to more convoluted expressions for the second Chern character
$\operatorname{ch}(\mathcal{V}^{(2)})$ and any equations that contain it, such
as the anomaly condition. Of course, the two descriptions are equivalent and
lead to the same results in the end. However, we will not use this second
parametrization in this paper.

For the two line bundles system with
$L_{1}=\mathcal{O}_{X}(m^{1},m^{2},m^{3})$ and
$L_{2}=\mathcal{O}_{X}(n^{1},n^{2},n^{3})$, we obtain

$\begin{split}FI_{\mathcal{F}}=\frac{1}{2}\frac{\epsilon_{S}\epsilon_{R}^{2}}{\kappa_{4}^{2}}\frac{1}{\hat{R}V^{2/3}}&\mathopen{}\mathclose{{}\left[-2d_{ijk}n^{i}a^{j}a^{k}+\frac{2\epsilon_{S}^{\prime}\hat{R}}{V^{1/3}}n^{i}\Big{(}\big{(}\tfrac{2}{3},-\tfrac{1}{3},4\big{)}\big{|}_{i}+\mathopen{}\mathclose{{}\left(1-(1+\tfrac{\lambda}{2})^{2}}\right)W_{i}\Big{)}}\right]\
,\\\
FI_{\mathcal{K}}=\frac{1}{2}\frac{\epsilon_{S}\epsilon_{R}^{2}}{\kappa_{4}^{2}}\frac{1}{\hat{R}V^{2/3}}&\bigg{[}d_{ijk}(m^{j}+n_{j})^{i}a^{j}a^{k}\\\
&-\frac{\epsilon_{S}^{\prime}\hat{R}}{V^{1/3}}(m^{i}+n^{i})\Big{(}\big{(}\tfrac{2}{3},-\tfrac{1}{3},4\big{)}\big{|}_{i}+\mathopen{}\mathclose{{}\left(1-(1+\tfrac{\lambda}{2})^{2}}\right)W_{i}\Big{)}\bigg{]}\
.\\\ \end{split}$ (D.11)

$W_{i}$ is given in eq. (B.11) for a system with two line bundles embedded
into $SU(3)$.

##  Appendix E Gauge Threshold Corrections

The gauge couplings of the non-anomalous components of the $d=4$ gauge group,
in both the observable and hidden sectors, have been computed to order
$\kappa_{11}^{4/3}$ in [14]. Written in terms of the Kähler moduli $a^{i}$,
these are given by

$\frac{4\pi}{(g^{(1)})^{2}}\propto
V(1+\epsilon_{S}^{\prime}\frac{{\widehat{R}}}{2V^{4/3}}\sum_{n=0}^{N}(1-z_{n})^{2}a^{i}\beta^{(n)}_{i})$
(E.1)

and

$\frac{4\pi}{(g^{(2)})^{2}}\propto
V(1+\epsilon_{S}^{\prime}\frac{{\widehat{R}}}{2V^{4/3}}\sum_{n=1}^{N+1}z_{n}^{2}a^{i}\beta^{(n)}_{i})$
(E.2)

respectively. The positive definite constant of proportionality is identical
for both gauge couplings and is not relevant to the present discussion. It is
important to note that the effective parameter of the $\kappa_{11}^{2/3}$
expansion is, as discussed above, $\epsilon_{S}^{\rm
eff}=\epsilon_{S}^{\prime}\frac{{\widehat{R}}}{V}$.

Consistency of the $d=4$ effective theory requires both $(g^{(1)})^{2}$ and
$(g^{(2)})^{2}$ to be positive. Expressing
$V=\tfrac{1}{6}d_{ijk}a^{i}a^{j}a^{k}$ and the $\beta^{n}$ charges as in eq
(B.14), we can write these two constraints in terms of the Kähler moduli:

$\displaystyle(g^{(1)})^{2}>0\>\Rightarrow\>$ $\displaystyle
d_{ijk}a^{i}a^{j}a^{k}+3\frac{\epsilon_{S}^{\prime}\hat{R}}{V^{1/3}}\big{(}\tfrac{2}{3}a^{1}-\tfrac{1}{3}a^{2}+4a^{3}+(\tfrac{1}{2}-\lambda)^{2}W_{i}a^{i}\big{)}>0\
,$ (E.3) $\displaystyle(g^{(2)})^{2}>0\>\Rightarrow\>$ $\displaystyle
d_{ijk}a^{i}a^{j}a^{k}-3\frac{\epsilon_{S}^{\prime}\hat{R}}{V^{1/3}}\big{(}\tfrac{2}{3}a^{1}-\tfrac{1}{3}a^{2}+4a^{3}+(1-(\tfrac{1}{2}+\lambda))^{2}W_{i}a^{i}\big{)}>0\
.$ (E.4)

Changing the model we use for the hidden sector bundle is reflected solely in
the expression for $W_{i}$. $W_{i}$ is given in eq (B.8) for a generic hidden
sector vector bundle of the form (B.3).

##  Appendix F Subbundles of Isomorphic Extension Bundles

In Table 3, we presented the six different extension branches for deforming
the Whitney sum
$V_{\bm{\underline{3}}}=\mathcal{F}\oplus\mathcal{K}\oplus\mathcal{E}$ away
from the decomposable locus. For each such branch, there are two different
pairs of sequences which lead, however, to isomorphic $SU(3)$ bundles. Because
of this isomorphism, in Table 3 we presented only one pair of these sequences.
Here, however, we need to discuss both of them. For simplicity, let us
restrict the discussion to the first extension branch only. However, the
conclusions will apply to the remaining five branches as well. Let us briefly
review the two sets of extension sequences in the first extension branch.
These are

$\begin{split}\operatorname{Ext}^{1}(\mathcal{E},\mathcal{F})=H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})\neq
0\quad\Rightarrow\quad&0\rightarrow\mathcal{F}\rightarrow
W\rightarrow\mathcal{E}\rightarrow 0\ ,\\\
\operatorname{Ext}^{1}(W,\mathcal{K})=H^{1}(X,\mathcal{K}\otimes\mathcal{E}^{*})\neq
0\quad\Rightarrow\quad&0\rightarrow\mathcal{K}\rightarrow
V^{\prime}_{\bm{\underline{3}}}\rightarrow W\rightarrow 0\ ,\end{split}$ (F.1)

or

$\begin{split}\operatorname{Ext}^{1}(\mathcal{E},\mathcal{K})=H^{1}(X,\mathcal{K}\otimes\mathcal{E}^{*})\neq
0\quad\Rightarrow\quad&0\rightarrow\mathcal{K}\rightarrow
W^{\prime}\rightarrow\mathcal{E}\rightarrow 0\ ,\\\
\operatorname{Ext}^{1}(W^{\prime},\mathcal{F})=H^{1}(X,\mathcal{F}\otimes\mathcal{E}^{*})\neq
0\quad\Rightarrow\quad&0\rightarrow\mathcal{F}\rightarrow\mathcal{V}^{\prime}_{\bm{\underline{3}}}\rightarrow
W^{\prime}\rightarrow 0\ .\end{split}$ (F.2)

However, following the calculation in [74], it can be shown that the resulting
$SU(3)$ bundles are actually isomorphic
$V^{\prime}_{\bm{\underline{3}}}\simeq\mathcal{V}_{\bm{\underline{3}}}^{\prime}$
and so it does not matter which extension one uses.

From the first set definition of the extension in (F.1), we learn there is an
embedding

$\mathcal{K}\hookrightarrow V^{\prime}_{\bm{\underline{3}}}\ ,$ (F.3)

and, hence, $\mathcal{K}$ is a rank-one subbundle of
$V_{\bm{\underline{3}}}^{\prime}$. This means
$V_{\bm{\underline{3}}}^{\prime}$ is stable only if the slope of $\mathcal{K}$
is less than the slope of $V_{\bm{\underline{3}}}^{\prime}$ (which vanishes):

$\mu(\mathcal{K})<0\ .$ (F.4)

From the second definition of the extension in F.2, we learn that
$\mathcal{F}$ is a subbundle of $\mathcal{V}_{3}^{\prime}$, which itself is
isomorphic to $V_{\bm{\underline{3}}}^{\prime}$. An obvious question is
whether $\mathcal{F}$ is thus a subbundle of $V_{\bm{\underline{3}}}^{\prime}$
as well, which would then constrain the slope of $\mathcal{F}$ to be negative.

Recall that a sheaf $\mathcal{F}$ is a sub-sheaf of $V$ if it has smaller rank
and and there exists an embedding $\mathcal{F}\hookrightarrow V$ [82]. The
space of homomorphisms from $\mathcal{F}$ to $V$, denoted
$\operatorname{Hom}_{X}(\mathcal{F},V)$, is then isomorphic to the space of
global sections $H^{0}(X,\mathcal{F}^{*}\otimes V)$. If $V$ is an $SU(N)$
bundle, it is stable if all its sub-sheaves $\mathcal{F}$ have negative slope.
Hence, we have that

$\begin{gathered}V\text{ is stable}\\\ \Updownarrow\\\
\mu(\mathcal{F})<0\quad\forall\,\mathcal{F}\text{ with
}0<\operatorname{rk}\mathcal{F}<\operatorname{rk}V\text{ and
}H^{0}(X,\mathcal{F}^{*}\otimes V)\neq 0\ .\end{gathered}$ (F.5)

Applying this statement to our case, the bundle $\mathcal{F}$ is a subbundle
of $V_{\bm{\underline{3}}}^{\prime}$ if we can find a homomorphism
$\mathcal{F}\hookrightarrow V^{\prime}_{\bm{\underline{3}}}$ or, equivalently,
if

$\operatorname{Hom}_{X}(\mathcal{F},V^{\prime}_{\bm{\underline{3}}})=H^{0}(X,\mathcal{F}^{*}\otimes
V^{\prime}_{\bm{\underline{3}}})\neq 0\ .$ (F.6)

In the following, we will show that such a homomorphism does indeed exist. Let
us start by tensoring with $\mathcal{F}^{*}$ the sequences

$0\rightarrow\mathcal{K}\rightarrow V^{\prime}_{\bm{\underline{3}}}\rightarrow
W\rightarrow 0\ $ (F.7)

and

$0\rightarrow\mathcal{F}\rightarrow W\rightarrow\mathcal{E}\rightarrow 0\ $
(F.8)

to obtain

$\begin{split}&0\rightarrow\mathcal{F}^{*}\otimes\mathcal{K}\rightarrow\mathcal{F}^{*}\otimes
V^{\prime}_{\bm{\underline{3}}}\rightarrow\mathcal{F}^{*}\otimes W\rightarrow
0\ ,\\\
&0\rightarrow\mathcal{F}^{*}\otimes\mathcal{F}\rightarrow\mathcal{F}^{*}\otimes
W\rightarrow\mathcal{F}^{*}\otimes\mathcal{E}\rightarrow 0\ .\end{split}$
(F.9)

Taking long exact sequences in cohomology of these, gives

$\begin{split}0&\rightarrow
H^{0}(X,\mathcal{F}^{*}\otimes\mathcal{K})\rightarrow
H^{0}(X,\mathcal{F}^{*}\otimes V^{\prime}_{\bm{\underline{3}}})\rightarrow
H^{0}(X,\mathcal{F}^{*}\otimes W)\\\
&\xrightarrow{\delta_{1}}H^{1}(X,\mathcal{F}^{*}\otimes\mathcal{K})\rightarrow\dots\
,\\\ \end{split}$ (F.10)

and

$\begin{split}0&\rightarrow
H^{0}(X,\mathcal{F}^{*}\otimes\mathcal{F})\rightarrow
H^{0}(X,\mathcal{F}^{*}\otimes W)\rightarrow
H^{0}(X,\mathcal{F}^{*}\otimes\mathcal{E})\\\
&\xrightarrow{\delta_{2}}H^{1}(X,\mathcal{F}^{*}\otimes\mathcal{F})\rightarrow\dots\
.\\\ \end{split}$ (F.11)

For a line bundle and its dual we have

$\begin{split}H^{0}(X,\mathcal{F}^{*}\otimes\mathcal{F})=H^{0}(X,\mathcal{O}_{X})=\mathbb{C}\
,\\\ H^{1}(X,\mathcal{F}^{*}\otimes\mathcal{F})=H^{1}(X,\mathcal{O}_{X})=0\
.\end{split}$ (F.12)

Furthermore, if the line bundles $\mathcal{F}^{*}\otimes\mathcal{K}$ and
$\mathcal{F}^{*}\otimes\mathcal{E}$ have negative slopes somewhere in the
Kähler cone, the zeroth cohomology classes

$H^{0}(X,\mathcal{F}^{*}\otimes\mathcal{K})=0\ ,\qquad
H^{0}(X,\mathcal{F}^{*}\otimes\mathcal{E})=0\ ,$ (F.13)

vanish, as explained in Footnote 4 of [32]. It can be shown that for our
particular Schoen manifold, this condition is always satisfied if the line
bundles $\mathcal{F}^{*}\otimes\mathcal{K}=L_{1}L_{2}^{3}$ and
$\mathcal{F}^{*}\otimes\mathcal{E}=L_{1}^{-1}L_{2}^{3}$ have both positive and
negative entries $m^{i}+3n^{i}$ and $-m^{i}+3n^{i}$, when written as
$\mathcal{F}^{*}\otimes\mathcal{K}=\mathcal{O}_{X}(m^{1}+3n^{1},m^{2}+3n^{2},m^{3}+3n^{3})$
and
$\mathcal{F}^{*}\otimes\mathcal{E}=\mathcal{O}_{X}(-m^{1}+3n^{1},-m^{2}+3n^{2},-m^{3}+3n^{3})$.
This is generally the case for the line bundles we sample. In particular, it
can be checked that the negative slopes condition is valid for the all the
line bundle configurations from Table 2.

Hence equations (F.10) and (F.11) become

$0\rightarrow H^{0}(X,\mathcal{F}^{*}\otimes
V^{\prime}_{\bm{\underline{3}}})\rightarrow H^{0}(X,\mathcal{F}^{*}\otimes
W)\xrightarrow{\delta_{1}}H^{1}(X,\mathcal{F}^{*}\otimes\mathcal{K})\rightarrow\dots\
,\\\ $ (F.14)

and

$0\rightarrow\mathbb{C}\rightarrow H^{0}(X,\mathcal{F}^{*}\otimes
W)\rightarrow 0$ (F.15)

respectively.

From the sequence (F.14) we learn that $H^{0}(X,\mathcal{F}^{*}\otimes
V^{\prime}_{\bm{\underline{3}}})=\operatorname{ker}\delta_{1}$. Therefore, to
evaluate $H^{0}(X,\mathcal{F}^{*}\otimes V^{\prime}_{\bm{\underline{3}}})$, we
must first analyze the coboundary map $\delta_{1}$. First note that from eq.
(F.15) we learn that $H^{0}(X,\mathcal{F}^{*}\otimes W)=\mathbb{C}$.
Furthermore, we have that the fields $\tilde{C}_{1}$ are counted by
$H^{1}(X,\mathcal{F}^{*}\otimes\mathcal{K})$. In the chosen vacuum branch all
VEVs for the $\tilde{C}_{1}$ fields vanish. Since $\delta_{1}$ is determined
by the vacuum state configuration, it follows that it maps only to the zero
element of $H^{1}(X,\mathcal{F}^{*}\otimes\mathcal{K})$. Hence,
$\operatorname{ker}\delta_{1}=H^{0}(X,\mathcal{F}^{*}\otimes W)=\mathbb{C}$.

Putting this together, we conclude that

$H^{0}(X,\mathcal{F}^{*}\otimes V^{\prime}_{\bm{\underline{3}}})=\mathbb{C}\
,$ (F.16)

which is indeed non-zero. Therefore, according to (F.6), there exists a
homomorphism $\mathcal{F}\hookrightarrow V^{\prime}_{\bm{\underline{3}}}$ such
that $\mathcal{F}$ is a subbundle of $V^{\prime}_{\bm{\underline{3}}}$.
Sequences (F.1) and (F.2) then tell us that both $\mathcal{F}$ and
$\mathcal{K}$ are subbundles of $V_{\bm{\underline{3}}}^{\prime}$. Therefore,
according to (F.5), $V_{\bm{\underline{3}}}^{\prime}$ is stable only if

$\mu({\mathcal{F}})<0\quad\text{and}\quad\mu({\mathcal{K}})<0\ .$ (F.17)

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