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#
Heterotic Complex Structure Moduli Stabilization for Elliptically Fibered
Calabi-Yau Manifolds

Wei Cui1, Mohsen Karkheiran2

###### Abstract

Complex structure moduli of a Calabi-Yau threefold in $N=1$ supersymmetric
heterotic compactifications can be stabilized by holomorphic vector bundles.
The stabilized moduli are determined by a computation of Atiyah class. In this
paper, we study how this mechanism work in the context of elliptically fibered
Calabi-Yau manifolds where complex structure moduli space contains two kinds
of moduli, ones from base and ones from fibration. With spectral cover
bundles, we find three types of situations when holomorphicity of bundles is
determined by algebraic cycles supported on special choice of complex
structure, which allows us to stabilize both of these two moduli. We present
concrete examples for each type and develop practical tools to analyze the
stabilized moduli. Finally, by checking the holomorphicity of the four-flux
and/or local Higgs bundle data in F-theory, we briefly study the dual complex
structure moduli stabilization scenarios.

1Physics Department, Robeson Hall, Virginia Tech, Blacksburg, VA 24061, USA
2Center for Theoretical Physics of the Universe, Institute for Basic Science,
Daejeon 34051, South Korea
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 The Atiyah Class
    1. 2.1 A Rank Two Extension Bundle
  3. 3 Complex Structure Moduli Stabilization Mechanisms
    1. 3.1 Type I: “Jumping” Extension Group
    2. 3.2 Type II: “Jumping” Picard Group
    3. 3.3 Type III: Emergent Algebraic Cycle
  4. 4 Discussion and Comments on F-Theory Dual
  5. 5 Conclusion
  6. A Complex Structure Moduli Space of Elliptically Fibered Calabi-Yau Threefold
  7. B Type II and Type II Fourier-Mukai Calculations
  8. C Computation Details of “Jumping” Ext
  9. D Approach for Identifying Reducible Deformations

##  1 Introduction

Compactification of $E_{8}\times E_{8}$ heterotic string [1, 2] on a Calabi-
Yau threefold with a vector bundle is a promising approach to derive the
Standard Model of the particle physics in four-dimensional spacetime [6, 5, 4,
3, 7, 8, 9, 10, 11]. To preserve supersymmetry, the bundle should be
holomorphic, slope zero and poly-stable. The fluctuations of the Calabi-Yau’s
complex structure and Kahler structure and the bundle’s fluctuations appear as
moduli particles in the effective theory. The holomorphic property of the
bundle has been used to stabilize complex structure moduli in these models
[12, 13, 14, 15, 16]. The idea is that only some of the deformations of
complex structure are consistent with the supersymmetry. The others will force
the bundle to become non-holomorphic. This will generate an non-trivial
F-terms for such deformations in the effective theory and the complex
structure moduli corresponding to them are stabilized. As shown in [14], this
mechanism together with a similar one due to bundle poly-stability [17, 18,
19, 21, 20] is an effective way to stabilize geometric moduli in heterotic
Calabi-Yau compactification.

Deformations of complex structure preserving bundle holomorphy are described
by Atiyah class [22]. The computation of it is in general difficult. However,
for holomorphic bundles in which the necessary ingredients in the construction
have a complex structure dependence, these deformations can be calculated
efficiently by a “bottom up” approach. For example, extension bundles [5] are
defined by specifying an element in Ext group. As shown in [12, 13], Ext group
can jump in dimension for special choice of complex structure, which stabilize
some of complex structure moduli locally on special locus in moduli space.
Instead of Atiyah class, the stabilized moduli can be identified quickly by a
computation of “jumping” Ext. As a result, the ”bottom up” approaches are
equivalent to calculation of Atiyah class and are efficient tools for complex
structure moduli stabilization.

In this work, we will focus on Calabi-Yau threefold with an elliptic
fibration111The Calabi-Yau manifolds considered here are all in the
Weierstrass form. This paper’s ideas will be valid for any elliptic fibration,
as long as there is at least one holomorphic section. However, when all of the
sections are rational, one will need more careful analysis [33, 34]. [23, 24,
25, 26, 27, 28, 29, 30] and study how its complex structure moduli can be
stabilized by bundle holomorphy. Bundles on elliptically fibered Calabi-Yau
threefold can be constructed by spectral cover [31, 32, 33, 34], which can be
mapped to the extension or monad bundles by a Fourier-Mukai transformation.
The reason to work with spectral cover bundle is that poly-stability, a
property that is difficult to check for other constructions, is
straightforward to check in terms of spectral data at least as long as the
spectral cover is smooth. Furthermore, for technical reasons, the models
considered so far are restricted to $SU(2)$ bundles, whereas working with
spectral data does not restrict the rank of the bundle.222It is interesting to
see how many of the moduli fields can be stabilized in the phenomenologically
interesting models using the methods presented in this paper. However, we only
explain the general ideas here and leave such questions for a future project.
Besides, since spectral data specify the Heterotic models, the Heterotic
model’s moduli stabilization process can be translated to an associated
process in the dual F-theory model. Hence the relation with moduli
stabilization scenarios in F-theory/type II models will be manifest.

Complex structure moduli of an elliptically fibered Calabi-Yau threefold $X$
can be either the complex structure moduli of the base $B$ or complex
structure moduli of the fibration $F$. As we will see later, both of them can
be stabilized using holomorphic bundles. Specifically, we find three types of
approaches to do that but, from the perspective of spectral cover bundles,
they all correspond to emergent algebraic cycles in $X$. In more technical
terms, we look for the Noether-Lefschetz loci in the complex structure moduli
and specify the spectral data so that the associated vector bundle stays
holomorphic on these loci. For type I case, we consider an $SU(2)$ extension
bundle with Ext group jumping on special locus in complex structure moduli
space of $B$. By Fourier-Mukai transformation, elements in jumping Ext group
corresponds to new algebraic curves in the spectral cover. One can observe
that, variations of complex structure away from the special locus in moduli
space will make algebraic curve disappear and spectral cover is not algebraic
anymore. Hence the corresponding vector bundle becomes non-holomorphic and the
associated complex structure moduli of $B$ are stabilized.

In type II approach, we find that Picard group of $B$ jumps when the complex
structure of the base takes some “special” form. Suppose the holomorphic
bundle is defined with the new algebraic cycles in “jumping” Picard group.
Then, variations of complex structure that does not preserve this “special”
form of complex structure will make the bundle non-holomorphic and the
corresponding moduli are stabilized. Besides complex structure moduli of the
base, holomorphic bundles can also stabilize moduli of the fibration. The idea
of type III approach is similar with type II one, in which taking a “special”
complex structure gives new algebraic cycles in the base. The difference is
that this time, the algebraic curves appear in the total space of the
fibration. Suppose the spectral data are defined in term of these emergent
algebraic cycles. Then, variations of complex structure away from the
“special” form will make algebraic cycles non-algebraic and hence the bundle
becomes non-holomorphic and the associated moduli are stabilized. Notice that
the last two approaches does not require the bundle is constructed as an
extension and can be applicable to more general bundles. We give explicit
examples for all three approaches, and similar to “jumping” Ext, we develop
practical tools to identify variations of complex structure that does not
preserve the “special” form used above and thus obtain the stabilized moduli
in type II and type III approaches.

The image of these moduli stabilization scenarios are also interesting. We
will show the type I example (which is the usual stabilization example by rank
$2$ bundles) the F-theory complex structure moduli is stabilized by a four-
flux which in some way is related to the isolated vertical fibers of the
spectral cover. In type II example the flux that is derived from the spectral
line bundle is responsible for the moduli stabilization. Finally the four-flux
in type II example is actually the horizontal algebraic four-flux constructed
in [35]. A recent work on complex structure moduli stabilization also used the
idea of algebraic cycles is discussed in [36].

The organization of this paper is as follows: In Section 2, we review how the
holomorphic property of a vector bundle can be used for complex structure
moduli stabilization in Calabi-Yau compactification. We study this mechanism
in terms of the the Atiyah class and illustrate it with a $SU(2)$ extension
bundle. In Section 3, we introduce three approaches to stabilize the complex
structure moduli of an elliptically fibered Calabi-Yau threefold with
holomorphic bundles in spectral cover construction. In Section 4, we discuss
how these approaches in heterotic compactification manifest themselves in
F-theory. In Section 5, we conclude our work. Some of the technical
discussions are presented in the appendices.

##  2 The Atiyah Class

Let $X$ be a Calabi-Yau threefold with a vector bundle $V$. To preserve $N=1$
supersymmetry, the bundle should have a connection satisfying the following
(zero slope) Hermitian Yang-Mills equations,

$F_{ab}=F_{{\bar{a}}{\bar{b}}}=0,\;g^{a\bar{b}}F_{a\bar{b}}=0,$ (2.1)

where $F$ is the gauge field strength associated with that connection and $a$
and $\bar{b}$ are holomorphic and anti-holomorphic indices on the Calabi-Yau
manifold. The first equation in (2.1) implies that the bundle is holomorphic
[37]. It is the property that will be used for moduli stabilization. The
second equation in (2.1) means that the bundle is poly-stable [38, 39]. We
will always keep $V$ poly-stable when studying variations of complex structure
of $X$. As shown in [12, 13], one can do that as long as the bundle is
holomorphic.

We start with a supersymmetric vacuum where the initial Kähler, complex
structure and bundle moduli are chosen to obey the Hermitian Yang-Mills
equation in (2.1). Notice that the initial objects will be denoted with a
superscript “$(0)$” and their deformation will be denoted with a $\delta$ in
front. From the deformation theory of compact complex manifolds [22, 40, 41,
42, 43], the space of simultaneous holomorphic deformations of $X$ and $V$ is
measured by $H^{1}({\cal{Q}})$ with $Q$ satisfying the Atiyah sequence

$0\to V\otimes V^{\vee}\to{\cal Q}\stackrel{{\scriptstyle\pi}}{{\to}}TX\to
0\;.$ (2.2)

The extension class is

$\alpha=[F^{(0)1,1}]\in H^{1}(V\otimes V^{\vee}\otimes TX^{\vee})$ (2.3)

the $(1,1)$ component of the initial field strength $F^{(0)}$ of the
background, referred to as the “Atiyah class” [22]. From the long exact
sequence of (2.2), we have

$0\to H^{1}(V\otimes V^{\vee})\to H^{1}({\cal Q})\stackrel{{\scriptstyle
d\pi}}{{\to}}H^{1}(TX)\stackrel{{\scriptstyle\alpha}}{{\to}}H^{2}(V\otimes
V^{\vee})\to\ldots\;.$ (2.4)

The simultaneous holomorphic deformations space $H^{1}({\cal Q})$ is given by

$H^{1}({\cal Q})=H^{1}(V\otimes V^{\vee})\oplus{\rm Ker}(\alpha)\ .$ (2.5)

Notice that we have used the fact that $H^{0}(TX)=0$ for the stable bundle
$TX$ in (2.4).

As we can see in (2.5), all bundle moduli in $H^{1}(V\otimes V^{\vee})$ belong
to $H^{1}({\cal Q})$ while for complex structure moduli, only those that are
in the kernel of the Atiyah class are elements of $H^{1}({\cal Q})$ and should
be regarded as the “true” moduli in the effective theory. The other ones have
non-zero images in $H^{2}(V\otimes V^{\vee})$, which lead to a non-vanishing
field strengths $F^{0,2}$ in the background and make $V$ non-holomorphic.
Thus, variations of complex structure along those directions are stabilized by
a nontrivial F-terms in superpotential. The number of stabilized moduli is
counted by the dimension of ${\rm Im}(\alpha)$. From the sequence in (2.4), it
is bounded by $h^{2}(V\otimes V^{\vee})$, so there are at most $h^{2}(V\otimes
V^{\vee})$ complex structure moduli that can be stabilized by the holomorphic
bundle.

The above mechanism can be made more precise if we write the elements of
$H^{1}(TX)$ explicitly as $\nu=\delta{\mathfrak{z}}^{I}v_{I{\bar{a}}}^{c}$
where $v_{I}$ are tangent bundle valued harmonic one-forms and
$\delta\mathfrak{z}^{I}$ are variations of the complex structure moduli
$\mathfrak{z}^{I}$. By the sequence (2.4), elements in the kernel of Atiyah
class should satisfy

$\delta{\mathfrak{z}}^{I}v^{c}_{I[{\bar{a}}}F^{(0)}_{|c|{\bar{b}}]}=D^{(0)}_{[{\bar{a}}}\Lambda_{{\bar{b}}]}\,$
(2.6)

where $\Lambda$ is a bundle-valued one-form and $D^{(0)}$ is the covariant
derivative with respect to the initial connection $A^{(0)}$. Notice that the
right hand side of the equation above is an exact 2-form in $H^{2}(X,V\otimes
V^{\vee})$. In fact, the bundle-valued one-form $\Lambda$ can be understood as
deformation of the connection $\Lambda_{\bar{b}}=\delta A_{\bar{b}}$. Thus,
for deformations of complex structure $\delta{\mathfrak{z}}^{I}$ that are in
${\rm Ker}(\alpha)$, there exist a deformation of the connection $\delta A$
such that the the equation (2.6) is satisfied and $V$ can remain holomorphic.
However, other deformations will make $V$ non-holomorphic and the associated
moduli are stabilized.

In conclusion, with Atiyah class defined in (2.3), we can determine which
deformations of the complex structure in $H^{1}(TX)$ can keep $V$ holomorphic
and which are not. The number of these independent deformations are given by
the dimension of ${\rm Ker}(\alpha)$ and ${\rm Im}(\alpha)$ respectively. For
most general bundles, they can be computed directly from the long exact
sequence in (2.4) known as the “top down” approach. However, for some classes
of holomorphic bundles constructed with ingredients that are only well-defined
for special complex structure, there is a quick way to determine the ${\rm
Ker}(\alpha)$ and ${\rm Im}(\alpha)$. This “bottom up” approach is equivalent
333 For the rank two vector bundles constructed as extension of a line bundle
and its dual, it was proved in [13] that the Atiyah class computation is
rigorously equivalent to an analysis of “Jumping” of Ext group. This is the
“bottom up” approach for extension bundles. to the direct computation of
$H^{1}({\cal Q})$ and is much easier computationally [13]. We will illustrate
this method with a simple extension bundle in the following subsection.

###  2.1 A Rank Two Extension Bundle

We assume that the Calabi-Yau threefold $X$ is defined as a hypersurface in a
product of projective spaces ${\cal
A}=\mathbb{P}^{n_{1}}\times\ldots\times\mathbb{P}^{n_{m}}$. We will further
ask that it is “favorable”, i.e. the Picard group of $X$ is spanned by the
restriction to $X$ of ambient divisors, $D_{i}$, associated with the
hyperplane class in $\mathbb{P}^{n_{i}}$. The defining polynomial of $X$ is
given by $p_{0}\in H^{0}({\cal A},N)$. Here $N$ is the normal bundle of the
hypersurface. Notice that the defining polynomial $p_{0}$ is a redundant
description of the complex structure. The independent ones can be obtained
from the computation of $H^{1}(X,TX)$, which will be discussed in section 3.

Consider a rank two bundle $V$ defined as an extension of a line bundle $L$
and its dual

$0\to L\to V\to L^{\vee}\to 0,$ (2.7)

with extension class $\phi\in Ext^{1}(L^{\vee},L)=H^{1}(L^{2})$. If $\phi\neq
0$, the extension is non-trivial and the bundle defined is a an $SU(2)$
indecomposable bundle, otherwise, the extension is trivial and the bundle
splits into $V=L\oplus L^{\vee}$. However, non-trivial extension does not
always exist on $X$ because the Ext group $H^{1}(L^{2})$ that the extension
class takes value in depends on the choice of complex structure. We will be
interested in a situation where $H^{1}(L^{2})$ is non-trivial only for some
special choices of complex structure of $X$ and $H^{1}(L^{2})=0$ for others.
Thus, the indecomposable $SU(2)$ bundle is well-defined on special locus in
the moduli space and it splits when the complex structure is deformed away
from this locus. This setting will play a key role in the moduli stabilization
discussed later.

Let’s assume that the initial complex structure is chosen such that the
$H^{1}(L^{2})$ jumps in dimension and the bundle defined in (2.7) is a
indecomposable $SU(2)$ bundle. At the same time, one needs to choose the
initial Kähler form $\omega$ of $X$ so that $V$ is poly-stable. Although poly-
stability is in general difficult to check, for this simple extension bundle,
it can be determined relatively easier 444It have been shown in [19], it is a
sufficient condition for $V$ being poly-stable. by the following criterion
[19]. $V$ is poly-stable if the Kähler form $\omega$ satisfies $\mu(L)<0$
where $\mu(L)$ is the slope of $L$ defined by

$\mu(L)=\frac{1}{rank(V)}\int_{X}c_{1}(L)\wedge\omega\wedge\omega.$ (2.8)

With this condition, we will fix the initial Kähler moduli of $X$ so that $V$
is poly-stable in this example.

Now, let’s consider the complex structure deformations with respect to this
background. From the equation (2.3), deformations that are consistent with the
holomorphic bundle is measured by the kernel of the Atiyah class $\alpha\in
H^{1}(V\otimes V^{\vee}\otimes TX^{\vee})$. For this example, it can be shown
[19] that $\alpha$ is an element of $H^{1}(X,L^{2}\otimes TX^{\vee})$, a
subset of $H^{1}(V\otimes V^{\vee}\otimes TX^{\vee})$. Thus, the part of
sequence in (2.4) containing $\alpha$ becomes

$\alpha:H^{1}(X,TX)\to H^{2}(X,L^{2}).$ (2.9)

By Bott-Borel-Weil Theorem [37], one can express the source $H^{1}(X,TX)$ and
target $H^{2}(X,L^{2})$ in terms of homogeneous polynomials. The kernel of the
map $\alpha$ follows by a direct computation of these polynomials. However,
non-trivial extension does not always exist on $X$ because the Ext group
$H^{1}(L^{2})$ that the extension class takes value in depends on the choice
of complex structure. We will be interested in a situation where
$H^{1}(L^{2})$ is non-trivial only for some special choices of complex
structure of $X$ and $H^{1}(L^{2})=0$ for others. Thus, the indecomposable
$SU(2)$ bundle is well-defined on special locus in the moduli space and it
splits when the complex structure is deformed away from this locus. This
setting will play a key role in the moduli stabilization discussed later.

The idea of “bottom up” approach [19] is that instead of the direct
calculation of ${\rm Ker}(\alpha)$ outlined above, the same information can be
obtained by studying the “jumping” of Ext group $H^{1}(X,L^{2})$.
Specifically, recall that the indecomposable bundle $V$ considered here is
defined on the special locus in the moduli space where Ext group
$H^{1}(X,L^{2})$ jumps, while for generic complex structure, the Ext group
$H^{1}(X,L^{2})$ is trivial and $V$ splits. Consider the deformation of
complex structure $p_{0}\to p_{0}+\delta p$. If $\delta P$ is within the
special locus of moduli space, the $V$ is still indecomposable and
holomorphic. However, if $\delta P$ is generic, the Ext group becomes
$H^{1}(X,L^{2})=0$. It seems that in this case the only choice for extension
class is $\phi=0$ and the bundle becomes a direct sum $L\oplus L^{\vee}$. In
fact, the extension class $\phi\in{H^{1}}(X,L^{2})$, does not change when we
vary the complex structure moduli. Instead, $\phi$ is simply no longer a
closed $(0,1)$-form with respect to the new complex structure. It means that
the extension bundle $V$ after generic deformation $\delta p$ is not
holomorphic anymore. In this example, we find that deformations of complex
structure that has “jumping” Ext group are the same deformations that make $V$
holomorphic. While deformations of complex structure that has trivial Ext
group corresponds to ones that make $V$ non-holomorphic. Thus, we see clearly
that the calculation of “jumping” Ext is equivalent to the Atiyah computation
in this example.

Next, let’s briefly describe how to identify the special locus where the Ext
group $H^{1}(X,L^{2})$ jumps in moduli space. By the Koszul sequence [37], the
line bundle $L^{2}$ on $X$ can be expressed in terms of bundles on the ambient
space ${\cal A}$. For the manifold considered here, it is

$0\to N^{\vee}\otimes L_{{\cal A}}^{2}\stackrel{{\scriptstyle
p_{0}}}{{\rightarrow}}L_{{\cal A}}^{2}\to L^{2}\to 0\ ,$ (2.10)

where $N^{\vee}$ is the dual of the normal bundle of $X$. The cohomology
$H^{1}(X,L^{2})$ can be computed via the associated long exact sequence

$0\to H^{1}(X,L^{2})\to H^{2}({\cal A},N^{\vee}\otimes L_{{\cal
A}}^{2})\stackrel{{\scriptstyle p_{0}}}{{\rightarrow}}H^{2}({\cal A},L_{{\cal
A}}^{2})\to H^{2}(X,L^{2})\to 0\ .$ (2.11)

Here $p_{0}$ is the defining polynomial of $X$. Thus, we have
$H^{1}(X,L^{2})=\textnormal{Ker}(p_{0})$ and
$H^{1}(X,{L^{\vee}}^{2})=\textnormal{Coker}(p_{0})$. For generic defining
polynomial, both $\textnormal{Ker}(p_{0})$ and $\textnormal{Coker}(p_{0})$ are
trivial. While for the special choice of $p_{0}$, they can “jump” together so
that the index $\textnormal{Ind}(L^{2})=-h^{1}(X,L^{2})+h^{2}(X,L^{2})$ is
preserved [13]. For the variation of complex structure $p\to p_{0}+\delta p$,
the number of independent variations that make $h^{1}(X,L^{2})$ jumps is the
same as the dimension of ${\rm Ker}(\alpha)$ computed from the equation (2.9).
While the other variations of complex structure with $h^{1}(X,L^{2})=0$
corresponds to ${\rm Im}(\alpha)$ in the Atiyah computation.

This moduli stabilization mechanism can also be understood in the low-energy
effective theory. Let $\mathfrak{z}_{0}$ be the initial complex structure
moduli chosen within the special locus of moduli space where Ext group of the
bundle defined in (2.7) jumps. The non-trivial extension defines a $SU(2)$
bundle. The low-energy gauge group is $E_{7}$. However, if the extension class
is trivial, $V$ splits and the low energy the low-energy gauge group is
enhanced by an anomalous $U(1)$ factor to $E_{7}\times U(1)$ [17, 19, 18]. The
bundle moduli (and matter fields) are charged under the enhanced $U(1)$
symmetry denoted by

$C^{i}_{+}\in H^{1}(L^{2}),\qquad C^{j}_{-}\in H^{1}({L^{\vee}}^{2}),$ (2.12)

with the subscript $\pm$ indicating the $U(1)$ charge. We will take the
extension class $\left<C^{i}_{+}\right>\neq 0$ not far from the zero in
$H^{1}(X,L^{2})$ and keep $\left<C^{j}_{-}\right>=0$ in the following.

The nontrivial F-term comes from the Gukov-Vafa-Witten superpotential [48,
49]. For bundle considered in (2.7), one can show that superpotential from the
contribution of bundle moduli is [13]

$W=\lambda_{ij}(\mathfrak{z}_{0})C_{+}^{i}C_{-}^{j}+\dots$ (2.13)

where the coefficients $\lambda_{ij}$ depends on the complex structure moduli
$\mathfrak{z}_{0}^{I}\in H^{1}(X,TX)$ explicitly. Note that for generic
complex structure, the holomorphic function
$\lambda_{ij}(\mathfrak{z}_{0})\neq 0$. As a result, the fields $C^{i}_{+}$
and $C^{j}_{-}$ are massive. However, on the special locus in complex
structure moduli space, $\lambda_{ij}(\mathfrak{z}_{0})=0$, which indicates
that $C^{i}_{+}$ and $C^{j}_{-}$ are massless. The non-trivial vevs of them
define the non-Abelian $SU(2)$ bundle [19]. This corresponds the “jump” of the
Ext group described above.

The F-terms of the bundle moduli follows directly from the superpotential
above. In particular, the F-terms of $C_{+}^{i}$ vanish while the other one

$F_{C_{-}}=\frac{\partial W}{\partial
C_{-}^{j}}=\lambda_{ij}(\mathfrak{z}_{0})\left\langle
C_{+}^{i}\right\rangle\;.$ (2.14)

As discussed above, since the initial complex structure is chosen in the
special locus of the moduli space, hence, $\lambda_{ij}(\mathfrak{z}_{0})=0$.
Consider the fluctuation
$\mathfrak{z}_{0}^{I}\to\mathfrak{z}_{0}^{I}+\delta\mathfrak{z}$. For the
fluctuations within the sub-locus where $H^{1}(X,L^{2})\neq 0$ (denoted by
$\mathfrak{z}^{a}_{\parallel}$), the new complex structure is still in the
sub-locus and hence $\lambda_{ij}(\mathfrak{z}_{0})=0$ and $F_{C_{-}}$
vanishes. However, for the fluctuations away from this sub-locus (denoted by
$\mathfrak{z}^{a}_{\perp}$), the new complex structure is a generic element in
the complex structure moduli space and $\lambda_{ij}(\mathfrak{z}_{0})\neq 0$.
Up to the quadratic order in the field fluctuations, we have the non-zero
F-term of $F_{C_{-}}$ and the potential is

$V=|\frac{\partial\lambda_{ij}(\mathfrak{z}_{0})}{\partial{\mathfrak{z}^{I}_{\perp}}}\langle
C^{i}_{+}\rangle|^{2}|\delta\mathfrak{z}_{\perp}^{I}|^{2}+\dots\ .$ (2.15)

This gives the mass of the moduli $\mathfrak{z}^{a}_{\perp}$ and these degree
of freedoms are stabilized. The number of these moduli is the same as the
$\textnormal{Im}(\alpha)$ computed in the Atiyah sequence.

In summary, we review the complex structure moduli stabilization in Calabi-Yau
compactification due to the presence of holomorphic bundles. The “true” moduli
are those keeping the holomorphy of the bundle, measured by the kernel of
Atiyah class $\alpha$, while the other moduli are stabilized. With a simple
rank two extension bundle (2.7), we demonstrate that
$\textnormal{Ker}(\alpha)$ can be calculated by an analysis of “jumping” Ext
group, which is equivalent to the direct computation from the sequence (2.4)
and much easier computationally. Next, we will focus on the moduli
stabilization on an elliptically fibered threefold and study how this
mechanism works in that context.

##  3 Complex Structure Moduli Stabilization Mechanisms

An elliptically fibered Calabi–Yau threefold $X$ consists of a base $B$, which
is a complex two-surface, and an analytic map

$\pi:X\to B$ (3.1)

with the property that for a generic point $b\in B$, the fiber
$E_{b}=\pi^{-1}(b)$ is an elliptic curve. In addition, we will require that
there exist a global section $\sigma:B\to X$ that assigns to every point $b\in
B$ the zero element $\sigma(b)=p\in E_{b}$ discussed below. Let $K_{B}$ be the
canonical bundle of $B$. The elliptic fibration is described by the
Weierstrass form

$\displaystyle y^{2}-x^{3}-fxz^{4}-gz^{6}=0,$ (3.2)

where $f$ and $g$ are sections of $K_{B}^{-4}$ and $K_{B}^{-6}$, and $(x,y,z)$
are global sections of $\mathcal{O}_{X}(2\sigma)\otimes K_{B}^{-2}$,
$\mathcal{O}_{X}(3\sigma)\otimes K_{B}^{-3}$ , $\mathcal{O}_{X}(\sigma)$
respectively.

Consider a $SU(n)$ vector bundle $V$ on $X$. The Chern character of $V$ can be
written as the following general form,

$\displaystyle ch(V)=n+\sigma\pi^{*}\eta+\omega[f]+\frac{1}{2}c_{3}(V),$ (3.3)

where $n$ is the rank of the bundle, $\sigma$ is the section of the fibration,
$\eta\in H^{2}(B,{\mathbb{Z}})$ is a divisor in the base, $\omega$ in an
element in $H^{2,2}(B,{\mathbb{Z}})$, and $[f]$ is the fiber class of the
elliptic fibration. $V$ can be constructed by spectral cover method [31, 32].
It consists of a $n$ sheeted cover of $B$ called spectral cover ${\cal
S}_{n}$. It is given by finite morphism $\pi_{S}:{\cal S}_{n}\to B$. The class
of ${\cal S}_{n}$ can then be written as

$\displaystyle[{\cal S}_{n}]=n\sigma+\pi^{*}(\eta)\in
H^{2}({Z_{3}},{\mathbb{Z}}).$ (3.4)

In affine coordinates, where $z=1$, we write

$\mathcal{S}_{n}=a_{0}+a_{2}x+a_{3}y+a_{4}x^{2}+a_{5}x^{2}y+\dots+a_{n}x^{n/2}=0$
(3.5)

If $n$ is odd the last term is $a_{n}x^{(n-3)/2}y$. In addition, we need to
specify a line bundle $N\to{\cal S}_{n}$ over the spectral cover called
spectral line bundle (or a rank one coherent sheaf generally), and it must
satisfy some certain topological conditions. The set of data $(\cal S,N)$ is
called spectral data. It is shown [31, 32] that there is a one-to-one
correspondence between holomorphic vector bundles and the spectral data,

$\displaystyle V\leftrightarrow(\mathcal{S}_{n},N).$ (3.6)

For an elliptically fibered Calabi-Yau threefold, the complex structure moduli
space contains the moduli from the base $B$ and the moduli from the fibration.
In fact, one can show that the moduli space of the base $H^{1}(B,TB)$ is a
subset of $H^{1}(X,TX)$. The detail of the derivation can be found in Appendix
A. The complex structure moduli space of the base $H^{1}(B,TB)$ can be
computed from the Euler sequence and adjunction formula. Suppose $B$ is
defined as a hypersurface of some ambient space $A$, where $A$ can be the
direct product of projective spaces
$\mathbb{P}^{m_{1}}\times\ldots\times\mathbb{P}^{m_{n}}$ or in general toric
varieties. The defining polynomial of $B$ is $p_{0}\in H^{0}({A},N)$ for some
ample line bundle $N$. Then the tangent bundle is described by

$\displaystyle 0\to O_{B}^{\oplus n}\stackrel{{\scriptstyle
l_{1}}}{{\rightarrow}}$
$\displaystyle\bigoplus_{i=1,\dots,n}O_{B}(D_{i})^{\oplus(n_{i}+1)}\to
T{A}|_{B}\to 0$ (3.7) $\displaystyle 0\to TB\to
T{A}|_{B}\stackrel{{\scriptstyle l_{2}}}{{\rightarrow}}N\to 0$

where $D_{i}$ are the restriction of the hyperplane divisors of each
projective factor of the ambient space $A$ and the polynomial maps
$l_{1},l_{2}$ satisfy $l_{2}\circ l_{1}=p_{0}$. The complex structure moduli
space $H^{1}(B,TB)$ follows from the associated long exact sequence. From the
long exact sequence of cohomology groups, we have

$0\to H^{0}(B,TB)\to H^{0}(B,TA)\stackrel{{\scriptstyle
l_{2}}}{{\rightarrow}}H^{0}(B,N)\to H^{1}(B,TB)\to
H^{1}(B,TA)\stackrel{{\scriptstyle l_{2}}}{{\rightarrow}}H^{1}(B,N)\to\ldots$

In general, it contains the polynomial deformations and the non-polynomial
ones. We will only focus on the polynomial part
$H^{1}_{\textnormal{poly}}(B,TB)\subset H^{1}(B,TB)$ which is given by

$H^{1}_{\textnormal{poly}}(B,TB)=\frac{H^{0}(B,N)}{\textnormal{Im}(l_{2})}\ .$
(3.8)

The fibration $F$ used in this work has the Weierstrass form. Its complex
structure moduli can be computed in the similar manner.

We wish to use these spectral data to find new examples for the complex
structure moduli stabilization. There are three types of examples that we are
going to consider. In the first type some complex structure deformations are
obstructed simply because the spectral cover cannot be defined globally
everywhere in moduli space. In the second type, however, the Picard group of
the base space jumps over some loci in the complex structure moduli, and if
the spectral data depends explicitly on these new divisors, the deformations
normal to this loci is obstructed. In the third type, the Picard group of the
Calabi-Yau threefold jumps. More clearly, some specific algebraic curves
inside the ambient four-fold will lie inside the Calabi-Yau threefold. For
some specific bundles, these algebraic curves lie inside the spectral
cover.555Notation: From now on, we will denote the divisors in the base
without writing the pullback $\pi^{*}$ on front, e.g. we will write $\eta$
instead of $\pi^{*}\eta$. We also denote the spectral cover simply by $[\cal
S]$ or $\cal S$ without emphasizing on its degree.

###  3.1 Type I: “Jumping” Extension Group

Consider the rank two extension bundle $V$ defined in (2.7). As discussed in
Section 2, on special loci in complex structure moduli space, cohomology
$H^{2}(X,V\otimes V^{\vee})$ jumps in dimension and the rank of Atiyah class
$\alpha$ in (2.4) jumps as well. Hence, complex structure deformations
perpendicular to this special loci will make $V$ non-holomorphic and the
associated moduli are stabilized [14]. In this subsection we consider a
similar set-up over an elliptically fibered Calabi-Yau threefold
$\pi:X\rightarrow B$, and by analyzing such bundles using spectral
cover/Fourier-Mukai construction, one gets a better insight about how exactly
such bundles become non-holomorphic.

As in [33] we take $\mathcal{L}=\mathcal{O}_{X}(-\sigma+D_{b})$. The extension
group of $V$ is given by

$\displaystyle Ext_{X}^{1}(\mathcal{L}^{\vee},\mathcal{L})$
$\displaystyle=H^{1}(X,\mathcal{O}_{X}(-2\sigma+2D_{b}))$ (3.9)
$\displaystyle\simeq H^{0}(B,\mathcal{O}_{B}(2D_{b}+K_{B}))\oplus
H^{0}(B,\mathcal{O}_{B}(2D_{b}-K_{b})),$

where the last isomorphism is derived by the Leray spectral sequence relative
to the morphism $\pi$. It is also not hard to show [14] that the (holomorphic)
deformations of $V$ i.e., $H^{1}(X,V^{\vee}\otimes V)\simeq
H^{2}(X,V^{\vee}\otimes V)$ depends on the extension group above.

On the other hand, over the elliptically fibered Calabi-Yau $X$ the vector
bundle $V$ can be parameterized by the spectral data $(\mathcal{S},N)$ [31,
32], where $\cal S$ is a double cover of the base $B$ and $N$ is a rank one
coherent sheaf over $\cal S$. When $\cal S$ is smooth, $N$ is simply a line
bundle. The divisor class of $\cal S$ is determined by the topology of $V$,

$\displaystyle[\cal S]$ $\displaystyle=2\sigma+\eta,$ (3.10)
$\displaystyle\eta$ $\displaystyle=2D_{b}+c_{1}(B).$ (3.11)

Also the general form of the algebraic equation of the spectral cover (when
$X$ is given by the Weierstrass fibration) is given by,

$\displaystyle\mathcal{S}=a_{2}Z^{2}+a_{0}X,$ (3.12) $\displaystyle a_{0}\in
H^{0}(B,\mathcal{O}_{B}(c_{1}(B)+2D_{b})),\quad a_{2}\in
H^{0}(B,\mathcal{O}_{B}(-c_{1}(B)+2D_{b})).$ (3.13)

By computing the Fourier-Mukai transformation of $V$, it is shown [33] that
the polynomials $a_{2}$ and $a_{0}$ can be identified with the elements of the
extension group (3.9). Therefore in the spectral cover language the loci in
the complex structure moduli where the group $H^{2}(X,V^{\vee}\otimes V)$
jumps correspond to the loci where it is possible to add new terms in the
polynomials $a_{2}$ and $a_{0}$. Once one add such terms in the spectral cover
the complex structure moduli is stabilized. Since the the complex structure
deformations that turns off such terms make the spectral cover non-algebraic
and hence the associated vector bundle, non-holomorphic.

#### Example

To be more concrete consider a Weierstrass elliptically fibered Calabi-Yau
threefold $X$ over the following surface,

$\displaystyle B=\left[\begin{array}[]{c|ccc}\mathbb{P}^{1}_{x}&2\\\
\mathbb{P}^{2}_{y}&2\end{array}\right],\quad c_{1}(B)=H_{2}.$ (3.16)

Let us choose $D_{b}=-H_{1}+2H_{2}$, where $H_{1}$ and $H_{2}$ are hyperplanes
in $\mathbb{P}^{1}_{x}$ and $\mathbb{P}^{2}_{y}$ respectively. Therefore

$\displaystyle a_{0}$ $\displaystyle\in H^{0}(B,\mathcal{O}_{B}(-2,5))\;,$
(3.17) $\displaystyle a_{2}$ $\displaystyle\in
H^{0}(B,\mathcal{O}_{B}(-2,3))\;.$ (3.18)

Generic defining equation $P$ for $B$ can be written as

$\displaystyle P=x_{1}^{2}P_{1}(y)+x_{2}^{2}P_{2}(y)+x_{1}x_{2}P_{3}(y),$
(3.19)

where $P_{1}$, $P_{2}$, $P_{3}$ are degree two homogeneous polynomials in y.
However the cohomology of line bundles over $B$ can jump over codimension one
loci in complex structure moduli of the base $B$. In particular the cohomology
of $\mathcal{O}_{B}(-2,3)$ and $\mathcal{O}_{B}(-2,5)$ can jump as,666The
surface $B$ is still smooth over this special complex structure locus.

$\displaystyle h^{0}(B,O(-2,3))=\begin{cases}0,&P\quad\text{generic}\\\
3,&P=x_{1}^{2}P_{1}(y)+x_{2}^{2}P_{2}(y)\end{cases},$ (3.20) $\displaystyle
h^{0}(B,O(-2,5))=\begin{cases}9,&P\quad\text{generic}\\\
13,&P=x_{1}^{2}P_{1}(y)+x_{2}^{2}P_{2}(y)\end{cases}.$

The computation details are in the Appendix C. This means the spectral cover
in general can be written as,

$\displaystyle\cal S=\begin{cases}a_{0}Z^{2},&P\quad\text{generic}\\\
a_{0}Z^{2}+a_{2}X,&P=x_{1}^{2}P_{1}(y)+x_{2}^{2}P_{2}(y),\end{cases}$ (3.21)

where in the second case $a_{0}$ is a rational polynomial [50] which is
holomorphic over the hypersurface $P=0$,

$\displaystyle
a_{2}=\frac{P_{1}(y)}{x_{2}^{2}}y_{1},\frac{P_{1}(y)}{x_{2}^{2}}y_{2},\frac{P_{1}(y)}{x_{2}^{2}}y_{3}.$
(3.22)

Similarly, the extra terms for the special defining equation $P=0$ is
generally of form $\frac{P_{1}(y)}{x_{2}^{2}}f_{3}(y)$. Where $f_{3}$ is
degree three polynomial in $y_{1},y_{2},y_{3}$, but we don’t need the detailed
form of it. Therefore in the complex structure locus where
$P=x_{1}^{2}P_{1}(y)+x_{2}^{2}P_{2}(y)$ the extension group (3.9) jumps, and
it is possible turn on these new terms. This corresponds to add the new terms
in $a_{0}$ and $a_{2}$ in the spectral cover. We consider the new terms in
$a_{0}$ and $a_{2}$ in turn.

First consider the $a_{2}$ term. When we turn on terms (3.22), the spectral
cover is smooth. As it is shown in [31, 32, 33] generally the spectral line
bundle $N$ is

$\displaystyle N=\mathcal{O}_{S}(\sigma+D_{b}).$ (3.23)

Therefore in general $N$ depends on the divisor $\sigma|_{S}$. The image of
this curve on $B$ is the curve $(a_{2}=0)$. When one turns on the term
$x_{1}x_{2}P_{3}(y)$ in (3.29) This curve “disappears”. In other words, away
form the locus (in the complex structure moduli space) where $P_{3}(y)$ in
(3.29) is nonzero the divisor $-2H_{1}+3H_{2}$ is not effective anymore. This
means after this complex structure deformation the curve $(a_{2}=0)$ mixes
with $(2,0)$ or $(0,2)$ cycles. Therefore the corresponding curve
$\sigma|_{S}$ on the spectral cover is not algebraic anymore and mixes with
$(2,0)$ or $(0,2)$ cycles. Hence the corresponding vector bundle becomes non-
holomorphic.

Next, suppose $a_{2}=0$. Instead we turn on the extra terms in $a_{0}$. In
this case the spectral cover777In this case the spectral cover is a union of
two copies of the zero section $Z=0$ (which are “infinitesimally close” to
each other) and the vertical surface $a_{2}=0$. $S=a_{0}Z^{2}$. Similar to the
previous case if $a_{0}$ depends on the terms
$\frac{P_{1}(y)}{x_{2}^{2}}f_{3}(y)$, the corresponding curve $a_{0}$ becomes
non-algebraic after the complex structure deformation. Therefore the
corresponding vector bundle $V$ becomes non-holomorphic after the deformation.

So the usual stabilization scenario of complex structure moduli by rank two
vector bundles [14, 16] translates to the question of whether the coefficients
of the spectral cover equation exists globally. Therefore one may generalize
this type of stabilization to vector bundles of arbitrary ranks simply by
asking if a holomorphic vector bundle with the Chern character
$ch(V)=n+\sigma\eta+\omega[f]$ is given, then does the divisor
$[S]=n\sigma+\eta$ have global section everywhere in the complex structure
moduli?

Now, we will see how many complex structure moduli are stabilized in this
example. if the initial defining polynomial is fixed by equation (C.7), the
complex structure moduli space $H^{1}(B,TB)$ can be computed by equation (3.7)
and (3.8). It follows that the space of polynomial deformations
$H^{1}_{\textnormal{poly}}(B,TB)$ has dimension $6$ with basis
$\\{d_{i}\\},i=1,2,\ldots,6$. As discussed above, if any monomials in a
polynomial deformation of $P$ contains $x_{0}x_{1}$ term and spectral cover
depends on the extra terms in (3.20), then $\cal S$ will be non-algebraic
after the deformation and $V$ is not holomorphic any more. It turns out that
there are $4$ such deformations in $\\{d_{1},d_{2},\ldots,d_{6}\\}$.888Note
that by an argument from linear algebra, this result does not change by choice
of basis in $H^{1}_{\textnormal{poly}}(B,TB)$. Therefore, we find $4$ out of
$6$ complex structure moduli of $B$ are stabilized.

###  3.2 Type II: “Jumping” Picard Group

For elliptically fibered Calabi-Yau threefold, the Picard group of the base
$B$ can jump in dimension, which can be used for moduli stabilization. The
idea is that look for the loci in the complex structure moduli that Picard
group jumps (Noether-Lefschetz loci) and choose the spectral data to depend on
the “new” elements of the Picard group. Therefore deforming the complex
structure back is impossible because the corresponding vector bundle becomes
non-holomorphic. We will explain this approach by means of an example.

#### Example

More concretely consider the following surface

$\displaystyle
B^{0}=\begin{array}[]{cccccc|c}x_{1}&x_{2}&z_{1}&z_{2}&u_{1}&u_{2}&P\\\
\hline\cr 1&1&0&0&0&0&1\\\ 0&1&1&1&0&0&3\\\ 0&1&0&1&1&1&3\end{array}$ (3.28)

Where P is generically given by

$\displaystyle
P=x_{1}(z_{1}^{3}f_{3}(u)+z_{1}^{2}z_{2}f_{2}(u)+z_{1}z_{2}^{2}f_{1}+z_{2}^{3}f_{0})+x_{2}(z_{1}^{2}g_{2}(u)+z_{1}z_{2}g_{1}(u)+z_{2}^{2}g_{0})=0.$
(3.29)

Here, $f_{d}(u)$ and $g_{d}(u)$ represent degree $d$ homogeneous polynomials
in $\\{u_{1},u_{2}\\}$. This is actually the blow-up of $\mathbb{F}_{1}$ at
the following six points

$\displaystyle
z_{1}^{3}f_{3}(u)+z_{1}^{2}z_{2}f_{2}(u)+z_{1}z_{2}^{2}f_{1}+z_{2}^{3}f_{0}=z_{1}^{2}g_{2}(u)+z_{1}z_{2}g_{1}(u)+z_{2}^{2}g_{0}=0.$
(3.30)

However the Picard number of $B$ is three ($\rho=3$), but $h^{1,1}=8$. In
other words this surface is not favorable. This is because the six points are
blown up “at the same time”. Note that the exceptional divisor can be
identified by $-J_{1}+2J_{2}+2J_{3}$ is effective and has self intersection
$(-6)$. The global section of the corresponding line bundle is given as

$\displaystyle
e=\frac{z_{1}^{2}g_{2}(u)+z_{1}z_{2}g_{1}(u)+z_{2}^{2}g_{0}}{x_{1}}.$ (3.31)

The zero locus $e=0$ corresponds to a union of six $\mathbb{P}^{1}$’s each one
is a $(-1)$ curve.

Now let’s tune the complex structure of the base to999The hypersurface is
still smooth after this tuning.

$\displaystyle
P=x_{1}(z_{1}^{3}f_{3}(u)+z_{1}^{2}z_{2}f_{2}(u)+z_{1}z_{2}^{2}f_{1}+z_{2}^{3}f_{0})+x_{2}(z_{1}h_{1}(u)+z_{2}h_{0})(z_{1}l_{1}(u)+z_{2}l_{0})=0.$
(3.32)

Now there are two independent set of three curves over

$\displaystyle
z_{1}^{3}f_{3}(u)+z_{1}^{2}z_{2}f_{2}(u)+z_{1}z_{2}^{2}f_{1}+z_{2}^{3}f_{0}=z_{1}h_{1}(u)+z_{2}h_{0}=0,$

and over

$\displaystyle
z_{1}^{3}f_{3}(u)+z_{1}^{2}z_{2}f_{2}(u)+z_{1}z_{2}^{2}f_{1}+z_{2}^{3}f_{0}=z_{1}l_{1}(u)+z_{2}l_{0}=0.$
(3.33)

So the Picard number jumps to 4. To make this more clear one can rewrite the
same geometry in the following way

$\displaystyle B^{1}$ $\displaystyle=$
$\displaystyle\begin{array}[]{cccccccc|cc}y_{1}&y_{2}&x_{1}&x_{2}&z_{1}&z_{2}&u_{1}&u_{2}&P_{1}&P_{2}\\\
\hline\cr 1&1&0&0&0&0&0&0&1&1\\\ 0&0&1&1&0&0&0&0&1&0\\\ 0&2&0&1&1&1&0&0&2&3\\\
0&2&0&1&0&1&1&1&2&3\end{array}.$ (3.39) $\displaystyle P_{1}$ $\displaystyle=$
$\displaystyle y_{1}x_{2}(z_{1}h_{1}(u)+z_{2}h_{0})+y_{2}x_{1}=0,$
$\displaystyle P_{2}$ $\displaystyle=$ $\displaystyle-
y_{1}(z_{1}^{3}f_{3}(u)+z_{1}^{2}z_{2}f_{2}(u)+z_{1}z_{2}^{2}f_{1}+z_{2}^{3}f_{0})+y_{2}(z_{1}l_{1}(u)+z_{2}l_{0})=0.$
(3.40)

The geometry above is equivalent to (3.32) by eliminating $y_{1}$ and $y_{2}$.
The two $(-1)$-curves and their global sections can be identified as

$\displaystyle e_{1}=-J_{1}+J_{3}+J_{4},\qquad$ $\displaystyle
a_{1}=\frac{z_{1}l_{1}(u)+z_{2}l_{0}}{y_{1}},$ (3.41) $\displaystyle
e_{2}=J_{1}-J_{2}+J_{3}+J_{4},\quad$ $\displaystyle
a_{2}=\frac{y_{1}(z_{1}h_{1}(u)+z_{2}h_{0})}{x_{1}}.$ (3.42)

Now, we want to use this jump in $Pic(B)$ to stabilize the complex structure
deformation that makes the divisor $e_{1}+e_{2}$ irreducible. To do this,
consider for example a holomorphic vector bundle with Chern character

$\displaystyle ch(V)=n-(\sigma(\eta_{0}+e_{1})+\omega[f])+\dots,$ (3.43)

where $\eta_{0}$ depends on $e_{1}+e_{2}$. Due to the special form of
$\eta=\eta_{0}+e_{1}$, the spectral cover of $V$ is reducible. Therefore the
spectral sheaf $\cal N$ is therefore defined as the extension,

$\displaystyle 0\rightarrow{\cal L}_{1}\rightarrow{\cal N}\rightarrow{\cal
L}_{2}\rightarrow 0,$ (3.44)

where ${\cal L}_{1}$ is a line bundle supported on $\pi^{*}e_{1}$, and ${\cal
L}_{2}$ is a line bundle supported over smooth surface in the divisor class
$n\sigma+\eta_{0}$. After a calculation similar to [33] one can show,101010The
term $\lambda_{1}e_{1}$ in $c_{1}(\mathcal{L}_{2})$ have to be compensated
with a term $-\lambda_{1}{\cal S}_{0}$ in $c_{1}(\mathcal{L}_{1})$. They
contribute to $c_{3}(V)$ and $\omega$, but since these are not important for
the main purpose of this paper, we ignore further details.

$\displaystyle c_{1}({\cal L}_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}(c_{1}(B)-[{\cal
S}_{0}])+\lambda\left(n\sigma-\eta_{0}+nc_{1}(B)\right)+\lambda_{1}e_{1}.$
(3.45)

The details to derive it is in Appendix B. What is important here is the
dependence of the spectral sheaf on $e_{1}$. This means deforming the complex
structure back to the point where $e_{1}+e_{2}$ is irreducible is impossible
for two reason. First the spectral cover $[{\cal S}]=[{\cal S}_{0}]+e_{1}$
becomes non-effective, i.e. one cannot define the spectral cover globally.
Second if $\lambda_{1}\neq 0$ the line bundle ${\cal L}_{2}$ over ${\cal
S}_{0}$ becomes non-holomorphic as we deform the complex structure.

We will study how the complex structure moduli of $B^{0}$ are stabilized in
this example. Set the initial defining polynomial $P^{(0)}$ to be the special
polynomial defined in (3.32). To simply the notation, let’s write it as
$P^{(0)}=x_{1}F_{1}^{(0)}+x_{2}F_{2}^{(0)}$ where
$F_{1}^{(0)}=z_{1}^{3}f_{3}(u)+z_{1}^{2}z_{2}f_{2}(u)+z_{1}z_{2}^{2}f_{1}+z_{2}^{3}f_{0}$
and $F_{2}^{(0)}$ is a reducible polynomial given by

$\displaystyle
F_{2}^{(0)}=\left(A_{1}u_{1}z_{1}+A_{2}u_{2}z_{1}+A_{3}z_{2}\right)\left(B_{1}u_{1}z_{1}+B_{2}u_{2}z_{1}+B_{3}z_{2}\right),$
(3.46)

with $A_{i}$’s and $B_{i}$’s the coefficients. As discussed above, the base
$B^{0}$ admits two algebraic divisors $e_{1}$ and $e_{2}$ given by equation
(3.41). With them, one can define a holomorphic bundle $V$.

Now, consider the polynomial deformations of $P^{(0)}$ given by $\delta
P=x_{1}\delta F_{1}+x_{2}\delta F_{2}$. At first, we keep our discussion in
general and allow $\delta P$ to be any elements in $H^{0}(B^{0},N)$.
Eventually, we will restrict them to be the deformation in
$H^{1}_{\textnormal{poly}}(B^{0},TB^{0})$. To keep $V$ holomorphic, the term
$F_{2}^{(0)}+\delta F_{2}$ in deformed defining polynomial $P^{(0)}+\delta P$
should remain reducible. Deformations satisfying this condition are called
reducible deformations. Before studying complex structure moduli
stabilization, one needs to find all the reducible deformations.

Obviously, $\delta P$ with $\delta F_{2}=0$ are reducible deformations because
they have no effect to $F_{2}^{(0)}$. Let $M^{1}_{\;\;r}$ be the space of all
such deformations. For deformations with $\delta F_{2}\neq 0$, it is not clear
if $F_{2}^{(0)}+\delta F_{2}$ remain reducible. However, we know that the
deformed $F_{2}$ term, if reducible, should be able to be written in the form
of (3.46). The first-order deformation $\delta F_{2}$ just shifts the initial
coefficients in (3.46) to a set of new coefficients $A_{i}+\delta A_{i}$ and
$B_{i}+\delta B_{i}$. We will see this in details in Appendix D. Notice that
if $\delta F_{2}$ is a deformations that keeps of $F_{2}^{(0)}+\delta F_{2}$
reducible, then $\delta P=x_{2}\delta F_{2}$ is a reducible deformation of
$P^{(0)}$. Without losing clarity, these $\delta F_{2}$ will also be called
reducible deformations.

To obtain the reducible deformations of $F_{2}^{(0)}$, it is sufficient to
calculate what is the $\delta F_{2}$ obtained by a shift of each coefficients.
For example, the reducible deformation corresponding to a shift of the first
coefficient $A_{1}+\delta A_{1}$ is obtained by

$\displaystyle F_{2}$ $\displaystyle=$ $\displaystyle\left((A_{1}+\delta
A_{1})u_{1}z_{1}+A_{2}u_{2}z_{1}+A_{3}z_{2}\right)\left(B_{1}u_{1}z_{1}+B_{2}u_{2}z_{1}+B_{3}z_{2}\right)$
$\displaystyle=$ $\displaystyle F_{2}^{(0)}+\delta
A_{1}u_{1}z_{1}\left(B_{1}u_{1}z_{1}+B_{2}u_{2}z_{1}+B_{3}z_{2}\right)$
$\displaystyle=$ $\displaystyle F_{2}^{(0)}+\delta A_{1}e^{A}_{1}\;.$

In the similar manner, one can obtain the reducible deformations from shifting
the other coefficients

$\displaystyle
e^{A}_{1}=u_{1}z_{1}\left(B_{1}u_{1}z_{1}+B_{2}u_{2}z_{1}+B_{3}z_{2}\right),$
$\displaystyle
e^{B}_{1}=u_{1}z_{1}\left(A_{1}u_{1}z_{1}+A_{2}u_{2}z_{1}+A_{3}z_{2}\right)$
$\displaystyle
e^{A}_{2}=u_{2}z_{1}\left(B_{1}u_{1}z_{1}+B_{2}u_{2}z_{1}+B_{3}z_{2}\right),$
$\displaystyle
e^{B}_{2}=u_{2}z_{1}\left(A_{1}u_{1}z_{1}+A_{2}u_{2}z_{1}+A_{3}z_{2}\right)$
$\displaystyle
e^{A}_{3}=z_{2}\left(B_{1}u_{1}z_{1}+B_{2}u_{2}z_{1}+B_{3}z_{2}\right),\quad$
$\displaystyle
e^{B}_{3}=z_{2}\left(A_{1}u_{1}z_{1}+A_{2}u_{2}z_{1}+A_{3}z_{2}\right)$

The linear combination of $\\{e^{A}_{i}\\}$ and $\\{e^{B}_{j}\\}$ give a
reducible deformations $\delta F_{2}$. However, there is a relation among them

$\displaystyle F_{2}^{(0)}+A_{1}e^{A}_{1}+A_{2}e^{A}_{2}+A_{3}e^{A}_{3}$
$\displaystyle=$ $\displaystyle
F_{2}^{(0)}+B_{1}e^{B}_{1}+B_{2}e^{B}_{2}+B_{3}e^{B}_{3}$ (3.48)

as one can easily check with the equation (3.46) and (3.2). Thus, only $5$ of
$\\{e^{A}_{i}\\}$ and $\\{e^{B}_{j}\\}$ are independent and, when multiplying
them by $x_{2}$, they span a vector space denoted by $M^{2}_{\;\;r}$
containing all the reducible deformations with $\delta F_{2}\neq 0$.
Therefore, the space of all the reducible deformations for $P^{(0)}$ is
$M_{r}=M^{1}_{\;\;r}\cup M^{2}_{\;\;r}$.

The complex structure moduli space $H^{1}(B^{0},TB^{0})$ is computed with the
equation (3.7) and (3.8). It follows that the space of polynomial deformations
$H^{1}_{\textnormal{poly}}(B^{0},TB^{0})$ has dimension $5$ 111111There are
also non-polynomials deformations, but we will not consider them here.,
spanned by homogeneous polynomials $d_{i},i=1,2,\ldots,5$. The intersection
$H^{1}_{\textnormal{poly}}(B^{0},TB^{0})\cap M_{r}$ contains all the
polynomial deformations from $B^{0}$ that keep $V$ holomorphic and contribute
to the nontrivial kernel of Atiyah class in (2.4). The dimension of this
intersection determines the number of the ”true” moduli in the effective
theory while the moduli in the complement of the intersection are stabilized.
In our example, $M_{r}\cap H^{1}_{\textnormal{poly}}(B^{0},TB^{0})$ is a $4$
dimensional space. Thus, $1$ complex structure moduli is stabilized.121212One
could guess this result by comparing the complex structure of $B^{0}$ and
$B^{1}$. Since the complex structure of $B^{1}$ is “frozen” in the sense that
it is impossible to “glue” the divisors $e_{1}$ and $e_{2}$ into a single
divisor $e$, but any other complex structure deformation of $B^{0}$ is in a
one-one correspondence with the complex structure deformations of $B^{1}$. So
the space $H^{1}(TB^{1})$ actually corresponds to the flat (unstabilized)
directions. Finally one can check that
$h^{1}_{poly}(B^{1},TB^{1})-h^{1}_{poly}(B^{0},TB^{0})=-1$ as expected.

###  3.3 Type III: Emergent Algebraic Cycle

In the final example we let an algebraic curve to exist in total elliptic
fibration after tuning the complex structure of the Calabi-Yau threefold. As
in the previous example, the complex structure can be stabilized by choosing a
vector bundle such that it’s topology depends on this new algebraic curve.

Consider an elliptic fibered Calabi-Yau threefold $\pi:X\rightarrow B$ with
Weierstrass fibration given by

$\displaystyle F^{(0)}_{W}=y^{2}-x^{3}-fxz^{4}-gz^{6}=0\;,\quad g=\alpha
g^{\prime}\;,$ (3.49)

where $f\in H^{0}(B,K_{B}^{-4})$ and $g\in H^{0}(B,K_{B}^{-6})$ chosen to be a
product of two polynomials $\alpha$ and $g^{\prime}$ with correct degree. This
reducible polynomial $g$ allow us to define an algebraic 2-cycle on $X$

$\displaystyle[C]=\\{x=y=\alpha=0\\}.$ (3.50)

Note that this algebraic cycle is independent of the intersection of divisors
$\sigma\cdot\pi^{*}D_{b}$ and the fiber class $[f]$. Now consider the spectral
cover of a general rank $n$ vector bundle $V_{n}$

$\displaystyle\mathcal{S}(V_{n})=a_{0}z^{n}+a_{2}z^{n-2}x+\dots+a_{n}x^{n/2}.$
(3.51)

The 2-cycle $[C]$ intersect with spectral cover at finite number of points.
However after the choosing $a_{0}$ to be reducible

$\displaystyle a_{0}=\alpha a_{0},$ (3.52)

the curve $[C]$ can lie inside the spectral cover. So if the spectral sheaf
$N$ depends on this new divisor inside $\mathcal{S}(V_{n})$ we can stabilize
the complex structure moduli, simply because any deformation that removes
$[C]$ from $\mathcal{S}(V_{n})$ makes $N$ and hence the bundle $V_{n}$ non-
holomorphic. However a Fourier-Mukai analysis as in [33] shows that if the
Chern classes of the vector bundle does not depend on the 2-cycle $[C]$, then
the spectral line bundle also does not depend on the new divisor $[C]$ in
$\mathcal{S}(V_{n})$, and therefore the complex structure moduli cannot be
stabilized. So suppose the Chern character of $V_{n}$ is of the form (See
Appendix B for details)

$\displaystyle
Ch(V_{n})=n-\left(\sigma\cdot\eta+\omega[f]+[C]\right)+\frac{1}{2}c_{3}(V_{n}).$
(3.53)

By doing a similar calculations as in [33] we can derive the topological
constraints on the divisor class of $S(V_{n})$, Chern classes of $V_{n}$ and
$\mathcal{L}$. We will not repeat them here, however we just mention the part
of the results that we will need,131313Note that we should choose $\alpha$
such that $\frac{1}{n}\pi_{*}[C]$ be integral.

$\displaystyle[\mathcal{S}(V_{n})]$ $\displaystyle=$ $\displaystyle
n\sigma+\eta+\pi_{*}[C],$ (3.54) $\displaystyle c_{1}(\mathcal{N})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left([S(V_{n})]-c_{1}(B)\right)+\lambda\left(n\sigma-\eta+nc_{1}(B)\right)+\left(\frac{1}{n}\pi_{*}[C]-[C]\right).$
(3.55)

Therefore $\mathcal{N}$ explicitly depends on the divisor $[C]$ in the
spectral cover. This dependence (partially) stabilize both complex structure
and vector bundle moduli.

Consider a deformation of Weierstrass fibration $F^{(0)}_{W}\to
F^{(0)}_{W}+\delta gz^{6}$, which induces a deformation to function $g$ by

$\displaystyle\alpha g^{\prime}$ $\displaystyle\longrightarrow$ $\displaystyle
g.$ (3.56)

If the deformed $g$ can be expressed as a product of two polynomials (having
the same degree with $\alpha$ and $g^{\prime}$), then the algebraic curve
$[C]$ defined in (3.50) still exist and the bundle $V$ remain holomorphic.
However, if the function $g$ after deformation is irreducible, the curve $[C]$
is not algebraic anymore, and the vector bundle defined by the spectral line
bundle (3.55) which explicitly depends on $[C]$ becomes non-holomorphic.
Therefore the associated complex structure moduli of $F_{W}$ can be
stabilized. As we will explain in Section 4, these polynomial deformations
maps to complex structure moduli in the F-theory dual picture, and they are
stabilized by a four-flux.

#### Example

Let $\pi:X\rightarrow\mathbb{P}^{2}$ be an elliptically fibered Calabi-Yau
threefold with base $\mathbb{P}^{2}$. The fibration is Weierstrass introduced
in equation (3.2) with $f\in H^{0}(\mathbb{P}^{2},O(12))$ and $g\in
H^{0}(\mathbb{P}^{2},O(18))$. Let $\\{u_{1},u_{2},u_{3}\\}$ be the homogeneous
coordinates of $\mathbb{P}^{2}$. Choose the initial fibration to be The
initial fibration is chosen to be

$f^{(0)}=F_{12}(u),\quad g^{(0)}=\alpha F_{16}(u)$ (3.57)

where $F_{12}$ and $F_{16}$ are generic polynomials with degree $12$ and $16$
and $\alpha$ is a degree $2$ homogeneous polynomial. As shown in (3.50), the
fibration with reducible $g$ admits an algebraic curve $[C]$ in
$\mathbb{P}^{2}$, which will be used to define the topology of vector bundles.

Consider a $SU(2)$ extension bundle $V$ over $X$. The spectral cover
construction of $V$ introduced in equation (3.12) is $S=a_{2}z^{2}+a_{0}x$
where $x,z$ are coordinates of the fibration. If $D_{b}=O(2)$, then $a_{2}$
and $a_{0}$ should be global sections of $O_{B}(7)$ and $O_{B}(1)$. Let’s take
them to be

$\displaystyle a_{0}$ $\displaystyle=F_{1}(u_{1},u_{2},u_{3}),\quad
a_{2}=\alpha F_{5}(u_{1},u_{2},u_{3})$ (3.58)

where $F_{d}$ is a generic homogeneous polynomial with degree $d$. As
discussed in (3.53) and (3.55), if the Chern character of $V$ depends on the
algebraic 2-cycle $[C]$, then the Chern class of spectral sheaf depends on
$[C]$ explicitly. Thus, the holomorphy of $V$ depends on the existence of the
algebraic cycle $[C]$ that will be used to stabilize complex structure moduli
in a second.

We will study how many complex structure moduli can be stabilized in this
example. The clliptically fibered Calabi-Yau manifold $X$ can be regarded as

$\displaystyle X=\begin{array}[]{cccccc|c}u_{1}&u_{2}&u_{3}&x&y&z&P\\\
\hline\cr 1&1&1&6&9&0&18\\\ 0&0&0&2&3&1&6\end{array}$ (3.62)

With it, one can compute the complex structure moduli space $H^{1}(X,TX)$ from
the equation (3.7) and (3.8). Again, we will only focus on the polynomial
deformations $H^{1}_{\textnormal{poly}}(X,TX)$ which has dimension $272$.
Since the base $\mathbb{P}^{2}$ is rigid, all the complex structure moduli are
coming from the fibration, i.e. deformations of $f$ and $g$ 141414Indeed, the
degree of freedom of $f$ and $g$ is
$h^{0}(\mathbb{P}^{2},O(12))+h^{0}(\mathbb{P}^{2},O(18))=91+190=281$.
Considering the coordinates redefinition of $\mathbb{P}^{2}$, one find the
independent degree of freedom of $f$ and $g$ is $272$, which matches the
number of complex structure moduli counted in $h^{1}(X,TX)=272$..

As discussed above, deformations that keep $V$ holomorphic are those preserve
the reducibility of $g=\alpha F_{16}(u)$. Again, we will call them reducible
deformations. Obviously, any polynomial without $z^{6}$ terms is reducible
deformation. While, the reducible deformations with $z^{6}$ terms can be
calculated from the method in Appendix D. We find that these deformations can
be generated from $159$ polynomial deformations $e^{A}_{i},i=1,2,\dots,6$ and
$e^{B}_{j},i=1,2,\dots,153$, which are obtained by deforming the coefficients
in $\alpha$ and $F_{16}(u)$ respectively. In fact, the space of these
deformations denoted by $M^{2}_{\;\;r}$ has dimension $158$ because there is a
nontrivial relation in (D.8) between $\\{e^{A}_{i}\\}$’s and
$\\{e^{A}_{j}\\}$’s.

Finally, we will see how many polynomial deformations
$H^{1}_{\textnormal{poly}}(X,TX)$ are stabilized. One needs to split $272$
deformations found earlier into two groups. The former ones does not terms
proportional to $z^{6}$ which have no effect on $g$ and should be considered
as the moduli of the effective theory. There are $82$ of them. The latter ones
contain terms proportional to $z^{6}$ and could have an effect on the
reducibility of $g$. It turns out that there are $190$ such polynomials. Their
coefficients of $z^{6}$ is denoted by $d_{i},i=1,2,\ldots,190$. Then, one
needs to study how many of them or their linear combinations can give rise to
reducibility preserving deformations in $M^{2}_{\;\;r}$. Direct computations
shows that all $158$ polynomials in $M^{2}_{\;\;r}$ can be expressed in the
complex structure moduli $\\{d_{i}\\}$. Thus, there are $241$ polynomial
deformations that preserves the reducible of $g$ while there are $32$ complex
structure moduli are stabilized.

##  4 Discussion and Comments on F-Theory Dual

In this section we briefly describe the F-theory translation of the Heterotic
examples mentioned before. When spectral cover is smooth the duality procedure
is well known [51]. The geometry of the F-theory half-Calabi-Yau fourfold
(more precisely $dP_{8}$ fibration) is given by (for a $SU(n)$ bundle with
$n\leq 5$),

$\displaystyle Y^{2}=X^{3}+fXv^{4}+gv^{6}+uv^{5-n}S_{n}(X,Y,v),$ (4.1)

where the term in the parenthesis is the spectral cover of a $SU(n)$ vector
bundle (with $z$ is replaced by $v$), and $(X,Y,u,v)$ are the homogeneous
coordinates of $\mathbb{WP}^{3211}$. In addition using the cylinder map one
constructs the four-flux. The details can be found in [51]. It is only
sufficient mention that the part of the $c_{1}(\mathcal{L})$ that lies in the
kernel of $\pi_{S}:S\rightarrow B$, lifts to the four-flux in F-theory
language,

$\displaystyle\begin{array}[]{c}c_{1}(\mathcal{L})=\frac{1}{2}\left(c_{1}(B)-[S]\right)+\gamma\\\
\pi_{S*}\gamma=0\end{array}\Bigg{\\}}\Rightarrow\gamma\longrightarrow
G_{4}\sim P_{Cyl}^{*}\gamma,$ (4.4)

where $P_{Cyl}$ is the cylinder map which is a $\mathbb{P}^{1}$ fibration over
the spectral cover $S_{n}$,

$\displaystyle P_{Cyl}:\\{{S_{n}(X,Y,v)=0}\subset
X_{F}\\}\xrightarrow{\mathbb{P}^{1}}S_{n}(x,y,z)\subset X_{Het}.$ (4.5)

In terms of the local 7-brane data, there is a doublet $(E,\Phi)$, called
Higgs bundles, which consists of a holomorphic bundle $E$ on $B$ (where
7-brane wraps) and a “Higgs” field $\Phi\in H^{0}(E\otimes E^{*}\otimes
K(\eta))$ which satisfy the Hitchin system equations [52]. One can study the
Higgs bundles by spectral data,

$\displaystyle(E,\Phi)\leftrightarrow(S_{H},\mathcal{L}_{H}),$ (4.6)

where $S_{H}$ is a finite cover of $B$ defined as subvariety of $Tot(K)$,

$\displaystyle S_{H}=Det\left(\lambda 1-\Phi\right)=0,$ (4.7)

where $\lambda$ is the tautological section of $Tot(K)$, and $\mathcal{L}_{H}$
is a line bundle over $S_{H}$. The Higgs bundle $(E,\Phi)$ can be
reconstructed by spectral data as,

$\displaystyle E=\pi_{*}\mathcal{L}_{H},\quad\Phi=\pi_{*}\lambda,$ (4.8)
$\displaystyle\pi_{S}:S_{H}\rightarrow B.$ (4.9)

The Higgs bundle spectral data is clearly very similar to the Heterotic
spectral data derived from the Fourier transform of $V$. They are indeed
related to each other by cylinder construction [53].

Finally note that what is responsible for the complex structure is the F-term
derived from the Gukov-Vafa-Witten like superpotentials, which takes the
following forms in Heterotic, F-theory and 7-brane

Heterotic | $\leftrightarrow$ | F-theory | $\leftrightarrow$ | 7-brane
---|---|---|---|---
$W=\int_{X_{Het}}\omega_{3}\wedge\Omega_{3}$ | $\leftrightarrow$ | $W=\int_{X_{F}}G_{4}\wedge\Omega_{4}$ | $\leftrightarrow$ | $W=\int_{B}Tr(F_{2}\wedge\Phi)$

Where $\Omega$ corresponds to the complex structure top-form and $F_{2}$ is
the curvature two-form of $E$. With this introduction, we move forward to
explain briefly the “image” of the moduli stabilization scenarios of Heterotic
string in the dual F-theory. Locally over the 7-brane, the story is similar to
the Heterotic version. Basically, the deforming away from special loci in
complex structure moduli of the base make the Higgs bundle non-holomorphic,
and the term $F_{2}\wedge\Phi$ in the superpotential makes makes some
components of $\Phi$ massive. Generally, since the Higgs field corresponds to
the fluctuations of the 7-brane in the normal directions, this means that away
from some special loci in the complex structure moduli of $B$, the 7-brane
cannot “wiggle” in specific ways. This can be seen more clearly, by noting
that the coefficients of the spectral cover can be identified with the
Casimirs of the Higgs field.

The above arguments were local, but in the following we will mainly focus one
the effect of these deformations on the $G_{4}$ flux. In particular the type
III example can only be explained in terms of the global $G_{4}$ flux
stabilization rather than local two-flux.

#### Type I

We already know in the extension class of $V$ in (2.7) appeared as the new
terms in $a_{0}$ and $a_{2}$ in (3.9). By the well-known arguments reviewed
briefly above, these jumps in $a_{0}$ and $a_{2}$ corresponds to new umps in
Casimirs of $\Phi$, in other words, new zero modes in fluctuations of the
7-brane.

So new zero modes in the moduli space of the vector bundle in Heterotic
string, corresponds to new zero modes in deformations of the 7-brane in
F-theory.

To construct the $G_{4}$ flux, assume the spectral cover is smooth,

$\displaystyle S=a_{2}x+a_{0}z^{2}.$

Then by the introduction above, $G_{4}$ flux is related to the divisor
$\gamma$ in $c_{1}(\mathcal{N})$. But $\gamma=0$ for $SU(2)$
bundles.151515$\gamma$ induces non-zero chirality in the effective theory, but
for $SU(2)$ bundles, the chirality must be zero. The vanishing of $\gamma$ can
also be seen from direct spectral cover calculations. So one may naively
expect that the $G_{4}$ flux must vanish in this case. But, note that in this
case, there are finite number of “vertical components” (i.e. isolated elliptic
curves) inside the spectral cover located above the points $a_{0}=a_{2}=0$ in
the base $B$. In the dual F-theory geometry, such vertical components become
4-cycles $C_{4}$ by cylinder map which are (at least topologically) elliptic
curves times a $\mathbb{P}^{1}$.

One can use these cycles to construct $G_{4}$ flux,

$\displaystyle G_{4}\longrightarrow C_{4}+\text{Corrections},$ (4.10)

where the corrections are terms we need to add such that

$\displaystyle G_{4}\cdot\sigma\cdot\pi^{*}D_{base}=0.$ (4.11)

Note that this $G_{4}$ flux only exists when the terms $a_{2}$ and $a_{0}$ in
the spectral cover are not zero. Therefore, when we move to a generic point in
the complex structure moduli where there are no jumps in $a_{2}$ and $a_{0}$,
the $G_{4}$ flux constructed above “disappears” i.e. $G_{4}$ becomes non-
holomorphic, which by the Gukov-Vafa-Witten superpotential makes the
corresponding complex structure deformation massive.161616The Calabi-Yau four-
fold of F-theory has $(4,6,12)$ singularity over the points of vertical
components of $S$. Naively one may intend to blow these singularities up, but
this is only possible when the Heterotic “bundle” (coherent sheaf) is a direct
sum of a vector bundle and small instantons localized on the elliptic curves.

#### Type II

In this case the spectral cover is smooth, and as explained in subsection 3.2,
after the complex structure deformation the spectral cover is not algebraic
anymore. So an argument similar to the one mentioned above is (when $a_{2}\neq
0$) also applies in this case. In other words, the dual Higgs bundle spectral
data on the F-theory side is also required to be holomorphic, and these
deformations are obstructed by the F-term equations in the 7-brane worldvolume
theory.

In addition, $\gamma$ term in (4.4) can be non-zero in general (for $n>2$).
The general form of $\gamma$ in [31, 32] is

$\displaystyle\gamma=\lambda(n\sigma-\eta+nc_{1}(B))|_{S}.$ (4.12)

The curves (divisors) $\sigma|_{S}$ and $\eta_{S}$, and hence $\gamma$ and
$G_{4}$ in (4.4), are holomorphic as long as the spectral cover is
holomorphic. Therefore, if the coefficients of the spectral cover depend on
the new terms that appear in special loci in the complex structure moduli,
then it is impossible to move away from those points since $\gamma$ and hence
the corresponding $G_{4}$ flux becomes non-holomorphic.

In the particular example we considered in this paper $\gamma$ depends on the
new divisor $e_{1}$,

$\displaystyle\gamma=\gamma_{0}+\lambda_{1}e_{1}.$ (4.13)

Of course $\pi_{s*}\gamma\neq 0$ now, so one should modify the corresponding
four-flux, but the point is that this four-flux is only holomorphic on the
jumping locus.

#### Type III

In this type the the $\gamma$ is modified as171717Note that
$\pi_{S*}\left(\frac{1}{n}\pi_{*}[C]-[C]\right)=0$.

$\displaystyle\gamma=\lambda\left(n\sigma-\eta+nc_{1}(B)\right)+\left(\frac{1}{n}\pi_{*}[C]-[C]\right).$
(4.14)

As mentioned before on the Heterotic side the dependence of (3.55) on $[C]$
stabilize both the complex structure moduli and the vector bundle moduli. But
in F-theory dual both (3.49) and (3.52) corresponds to complex structure
deformation on the F-theory side.

The corresponding (horizontal) four-flux that is responsible for stabilizing
the F-theory complex structure is already studied [35]. A quick look at the
defining relation for the F-theory geometry (4.1) makes it clear that with
this particular constraints over the geometry of the Heterotic Calabi-Yau and
spectral cover, there is an algebraic four-cycle,

$\displaystyle C_{4}:=\\{X=Y=\alpha=0\\}\subset X_{F},$ (4.15)

which can be used to construct an “algebraic four-flux”. It is also clear that
$C_{4}$ is actually a $\mathbb{P}^{1}$ fibration (cylinder) over the algebraic
curve $[C]$ in the Heterotic geometry. Therefore we can identify $C_{4}$ with
the $P_{Cyl}^{*}[C]$ part of the four-flux induced by the spectral cover.

Finally note that contrary to type I and II, the two-flux localised inside the
7-brane is not responsible for stabilizing the complex structure moduli. This
is because the vector bundle in the Heterotic side is identified with the
spectral data living in the compact elliptic fibration over $B$, while the
Higgs bundle on the 7-brane is identified with spectral data that live in the
non-compact fibration $Tot(K_{B})$. Therefore in local F-theory/Heterotic
duality only the part of Heterotic spectral data in the patch that contains
the zero section is identified with the Higgs bundle spectral data, in other
words,

$\displaystyle(\mathcal{S}_{Het},\mathcal{N}_{Het})|_{\sigma\text{-patch}}\longleftrightarrow(\mathcal{S}_{Higgs},\mathcal{N}_{Higgs}).$
(4.16)

However, remember that

$\displaystyle\sigma\cdot[C]=0.$ (4.17)

##  5 Conclusion

In this paper, we have studied how the complex structure moduli of
elliptically fibered Calabi-Yau threefold can be stabilized by holomorphic
bundles in the context of a heterotic compactification. Complex structure
moduli of elliptically fibered Calabi-Yau include moduli from both the base
and the fibration. We find three approaches In terms of spectral cover
bundles, we find both of these moduli can be stabilized by the use of
algebraic cycles. The idea is that on special locus in the moduli space of
base (fibration), there exist emergent algebraic cycles. If one defines We
develop efficient tools to identify the stabilized moduli and present concrete
examples for all three approaches. As a result, we show that complex structure
moduli can be effectively stabilized by bundle holomorphy.

As mentioned in the introduction, one of the advantages of working with
spectral covers is the manifest connection with the F-theory and hence type II
models. In other words, the complex structure moduli stabilization scenarios
are already well known, at least conceptually, in both type II and Heterotic
string, but apparently they look very different with each other. But both of
these theories can be seen as different limits of F-theory. So by describing
the moduli stabilization in terms of spectral data, we could see what should
be the corresponding stabilization scenario in the dual F/type II theory. In
particular we have shown that in the type I/II examples the dual stabilization
is made possible by the flux inside the 7-brane worldvolume, while in the type
III example the dual F-theory geometry is stabilized by a horizontal algebraic
$G_{4}$ flux.

## Acknowledgments

W. Cui would like to thank James Gray for useful discussion.

##  Appendix A Complex Structure Moduli Space of Elliptically Fibered Calabi-
Yau Threefold

fibered In this appendix we briefly discuss how the complex structure of the
base $H^{1}(B,TB)$ is embedded into the complex structure of the Calabi-Yau
threefold. We assume $B$ is a Fano surface ($K_{B}^{-1}$ is ample) and
$h^{1}(\mathcal{O}_{B})=0$.

One can use the adjunction sequence for the section, $\sigma$, of the
fibration to relate the tangent bundle of the base $TB$ with the tangent
bundle of the Calabi-Yau,

$\displaystyle 0\rightarrow TB\rightarrow
TX|_{B}\rightarrow\mathcal{N}_{B/X}=K_{B}\rightarrow 0.$ (A.1)

From the sequence above one can see, as long as the base $B$ is simply
connected, $H^{1}(TB)\simeq H^{1}(B,TX|_{B})$. So we should find the relation
between $TX|_{B}$ and $TX$,

$\displaystyle 0\rightarrow TX\otimes\mathcal{O}_{X}(-\sigma)\rightarrow
TX\rightarrow TX|_{B}\rightarrow 0.$ (A.2)

The corresponding long exact sequence is,

$\displaystyle 0$ $\displaystyle\rightarrow
H^{0}(TX\otimes\mathcal{O}_{X}(-\sigma))\rightarrow H^{0}(TX)\rightarrow
H^{0}(TX|_{B})\rightarrow$ (A.3) $\displaystyle\rightarrow
H^{1}(TX\otimes\mathcal{O}_{X}(-\sigma))\rightarrow H^{1}(TX)\rightarrow
H^{1}(TX|_{B})\rightarrow$ $\displaystyle\rightarrow
H^{2}(TX\otimes\mathcal{O}_{X}(-\sigma))\rightarrow H^{2}(TX)\rightarrow
H^{2}(TX|_{B})\rightarrow$ $\displaystyle\rightarrow
H^{3}(TX\otimes\mathcal{O}_{X}(-\sigma))\rightarrow H^{3}(TX)\rightarrow 0.$

We will show bellow that as long as $K_{B}^{-1}$ is ample, the homeomorphism
$H^{1}(X,TX|_{B})\simeq H^{1}(B,TB)\rightarrow
H^{2}(TX\otimes\mathcal{O}_{X}(-\sigma))$ have to be zero. To see this we will
use the following sequences,

$\displaystyle 0\rightarrow TX\otimes\mathcal{O}_{X}(-\sigma)\rightarrow
T\mathcal{A}|_{X}\otimes\mathcal{O}_{X}(-\sigma)\rightarrow\mathcal{N}|_{X}\otimes\mathcal{O}_{X}(-\sigma)\rightarrow
0,$ $\displaystyle 0\rightarrow T_{\bar{\pi}}|_{X}\rightarrow
T\mathcal{A}|_{X}\rightarrow\pi^{*}TB\rightarrow 0,$ $\displaystyle
0\rightarrow\mathcal{O}_{X}\rightarrow\mathcal{O}_{X}(3\sigma-3K_{B})\oplus\mathcal{O}_{X}(2\sigma-2K_{B})\oplus\mathcal{O}_{X}(\sigma)\rightarrow
T_{\bar{\pi}}|_{X}\rightarrow 0,$

where $\bar{\pi}:\mathcal{A}\rightarrow B$ is the ambient space that the
Calabi-Yau is embedded inside it.181818Note that $\mathcal{A}$ is a
$\mathbb{P}^{231}$ fibration. Now, by the last sequence above, one can show

$\displaystyle\pi_{*}(T_{\bar{\pi}}|_{X}\otimes\mathcal{O}_{X}(-\sigma))=K_{B}^{-3}\otimes(K_{B}^{2}\oplus\mathcal{O}_{B})\oplus
K_{B}^{-2}\oplus\mathcal{O}_{B}.$

On the other hand, one can use the second sequence to show,

$\displaystyle
R\pi_{*}\left(T\mathcal{A}|_{X}\otimes\mathcal{O}_{X}(-\sigma)\right)=\pi_{*}(T_{\bar{\pi}}|_{X}\otimes\mathcal{O}_{X}(-\sigma))\oplus
TB\otimes K_{B}[-1].$

Finally the first short exact sequence, and the result above leads
to,191919Note that $\pi_{*}\left(TX\otimes\mathcal{O}_{X}(-\sigma)\right)=0$
because $TX$ is a stable bundle, and one can show its restriction on generic
fiber of $X$ is given by non-trivial extension of trivial bundles. Therefore
$TX\otimes\mathcal{O}_{X}(-\sigma)$ doesn’t have any global section on generic
fibers. Naively, we may say
$\pi_{*}\left(TX\otimes\mathcal{O}_{X}(-\sigma)\right)$ should be a torsion
sheaf on $B$ (probably non-zero only on the discriminant curve of $X$), but
since $\pi$ is flat and $TX\otimes\mathcal{O}_{X}(-\sigma)$ locally free, the
pushforward $\pi_{*}\left(TX\otimes\mathcal{O}_{X}(-\sigma)\right)$ must be
locally free, and hence should vanish everywhere.

$\displaystyle
0\rightarrow\pi_{*}(T_{\bar{\pi}}|_{X}\otimes\mathcal{O}_{X}(-\sigma))\rightarrow(\pi_{*}\mathcal{N}|_{X}\otimes\mathcal{O}_{X}(-\sigma))\rightarrow
R^{1}\pi_{*}(TX\otimes\mathcal{O}_{X}(-\sigma))\rightarrow TB\otimes
K_{B}\rightarrow 0,$

or, by considering the cokernel of the second map, we can rewrite this
sequence as

$\displaystyle 0\rightarrow K_{B}^{-10}\otimes\mathcal{I}_{24K^{2}}\rightarrow
R^{1}\pi_{*}(TX\otimes\mathcal{O}_{X}(-\sigma))\rightarrow TB\otimes
K_{B}\rightarrow 0,$ (A.4)

where, $\mathcal{I}_{24K^{2}}$ is the ideal sheaf of a finite number of points
given by the intersection of divisors $(-6K_{B})\cdot(-4K_{B})$. Now we can
talk about the cohomologies of $TX\otimes\mathcal{O}_{X}(-\sigma)$ in (A.3).
By Leray spectral sequence,

$\displaystyle
E_{2}^{p,q}:=H^{p}(B,R^{q}\pi_{*}(TX\otimes\mathcal{O}_{X}(-\sigma)))\Rightarrow
H^{p+q}(X,TX\otimes\mathcal{O}_{X}(-\sigma))).$

Since $R^{0}\pi_{*}(TX\otimes\mathcal{O}_{X}(-\sigma))=0$ then

$\displaystyle
H^{p}(X,TX\otimes\mathcal{O}_{X}(-\sigma)))=H^{p-1}(B,R^{1}\pi_{*}(TX\otimes\mathcal{O}_{X}(-\sigma))).$

The cohomology long exact sequence of the equation (A.4) leads to,

$\displaystyle 0$ $\displaystyle\rightarrow H^{0}(K_{B}^{-10})\rightarrow
H^{0}(R^{1}\pi_{*}(TX\otimes\mathcal{O}_{X}(-\sigma)))\rightarrow
H^{0}(TB\otimes K_{B})\rightarrow$ $\displaystyle\rightarrow
H^{1}(K_{B}^{-10})\rightarrow
H^{1}(R^{1}\pi_{*}(TX\otimes\mathcal{O}_{X}(-\sigma)))\rightarrow
H^{1}(TB\otimes K_{B})\rightarrow$ $\displaystyle\rightarrow
H^{2}(K_{B}^{-10})\rightarrow
H^{2}(R^{1}\pi_{*}(TX\otimes\mathcal{O}_{X}(-\sigma)))\rightarrow
H^{2}(TB\otimes K_{B})\rightarrow 0$

By the ampleness assumption of $K_{B}^{-1}$ we expect $H^{i}(B,K_{B}^{-10})=0$
for $i\neq 0$. So

$\displaystyle H^{2}(X,TX\otimes\mathcal{O}_{X}(-\sigma)))$ $\displaystyle=$
$\displaystyle
H^{1}(R^{1}\pi_{*}(TX\otimes\mathcal{O}_{X}(-\sigma)))=H^{1}(TB\otimes
K_{B})=H^{1,1}(B),$ $\displaystyle
H^{3}(X,TX\otimes\mathcal{O}_{X}(-\sigma)))$ $\displaystyle=$ $\displaystyle
H^{2}(R^{1}\pi_{*}(TX\otimes\mathcal{O}_{X}(-\sigma)))=H^{2}(TB\otimes
K_{B})=H^{0}(B,TB^{*}).$

At the end note that in general $h^{2}(X,TX)=h^{1,1}(X)=1+h^{1,1}(B)$, and
also from (A.1) we can show $H^{2}(TX|_{B})=H^{2}(K_{B})\oplus
H^{2}(TB)=\mathbb{C}\oplus H^{2}(TB)$. Then to get the dimension of
$H^{2}(TX)$ in (A.3) correct, all of the elements of
$H^{2}(X,TX\otimes\mathcal{O}_{X}(-\sigma)))=H^{1,1}(B)$ should inject into
$H^{2}(TX)$. Therefore we get a surjection (at least as long as
$H^{1}(\mathcal{O}_{B})=0$),

$\displaystyle H^{1}(TX)\rightarrow H^{1}(TX|_{B})\simeq H^{1}(TB)\rightarrow
0.$

We conclude that the elements of $H^{1}(TB)$ corresponds to a subspace of the
complex structure deformation of the Calabi-Yau $H^{1}(TX)$.

##  Appendix B Type II and Type II Fourier-Mukai Calculations

For completeness, in this section, we present some of the Fourier-Mukai
calculations’ details as much as is necessary here. Following [33] we can
compute the Chern characters of the Fourier-Mukai $\Phi(V)$ of V. In type II
example it would be,

$\displaystyle
Ch(\Phi(V))=-[\mathcal{S}]+\left([\mathcal{S}]\cdot\left(\frac{C_{1}(B)}{2}\right)+\frac{1}{2}c_{3}(V)[f]\right),$
(B.1)

where $[\mathcal{S}]=\eta+n\sigma$. In type II example we chose
$\eta=eta_{0}+e_{1}$, and with this choice the algebraic equation of
$[\mathcal{S}]$ will be reducible. Therefore the spectral line bundle
$\mathcal{N}$ will be given as an extension,

$\displaystyle 0\longrightarrow
i_{e_{1}*}\mathcal{L}_{1}\longrightarrow\mathcal{N}\longrightarrow
i_{S_{0}*}\mathcal{L}_{2}\longrightarrow 0,$ (B.2)

where $i_{e_{1}}$ and $i_{S_{0}}$ are respectively the inclusion morphisms of
the surfaces $\pi^{*}e_{1}$ and $S_{0}$ with divisor class $n\sigma+\eta_{0}$.
Now one can use,

$\displaystyle\Phi(V)=i_{S*}\mathcal{N}[-1]\Longrightarrow
Ch(\Phi(V))=-Ch(i_{S*}\mathcal{N})=-Ch(i_{e_{1}*}\mathcal{L}_{1})-Ch(i_{S_{0}*}\mathcal{L}_{2}).$
(B.3)

Next we can use the Grothendieck-Riemann-Rock to compute,

$\displaystyle Ch(i_{e_{1}*}\mathcal{L}_{1})$ $\displaystyle=$
$\displaystyle[e_{1}]\cdot\left(\frac{e^{c_{1}(\mathcal{L}_{1})}}{Td(\mathcal{O}(e_{1}))}\right),$
(B.4) $\displaystyle Ch(i_{S_{0}*}\mathcal{L}_{2})$ $\displaystyle=$
$\displaystyle[S_{0}]\cdot\left(\frac{s^{c_{1}(\mathcal{L}_{2})}}{Td(\mathcal{O}(S_{0}))}\right).$
(B.5)

An order by order comparison of the formulas written above leads to the
following result,

$\displaystyle c_{1}({\cal L}_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}(c_{1}(B)-[{\cal
S}_{0}])+\lambda\left(n\sigma-\eta_{0}+nc_{1}(B)\right)+\lambda_{1}e_{1}.$
(B.6)

A similar calculation for the type III example would lead to,

$\displaystyle
Ch(\Phi(V))=-[n\sigma+\eta+\pi^{*}\pi_{*}[C]]+[n\sigma+\eta+\pi^{*}\pi_{*}[C]]\cdot\left(\frac{c_{1}(B)}{2}-\frac{1}{2}\pi^{*}\pi_{*}[C]\right)+[C]+\dots.$
(B.7)

Again order by order comparison of this with the Chern character of the
spectral sheaf over a smooth surface $i_{S*}\mathcal{N}$ shows that first of
all the divisor class of the spectral cover is

$\displaystyle[S]=n\sigma+\eta+\pi^{*}\pi_{*}[C],$ (B.8)

second Chern class of the line bundle $\mathcal{N}$ must be,

$\displaystyle c_{1}(\mathcal{N})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left([S(V_{n})]-c_{1}(B)\right)+\lambda\left(n\sigma-\eta+nc_{1}(B)\right)+\left(\frac{1}{n}\pi_{*}[C]-[C]\right).$
(B.9)

##  Appendix C Computation Details of “Jumping” Ext

In this Appendix, we briefly outline the details of the calculation about the
“jumping” of the line bundle cohomology and the number of moduli stabilized in
Subsection 3.1. The base $B$ of the elliptic Calabi-Yau threefold $X$ is
defined in (3.16). The defining polynomial $p_{0}$ is an degree $(2,2)$
polynomial in $\\{x_{0},x_{1}\\}$ and $\\{y_{0},y_{1},y_{2}\\}$. The
coefficients of $p_{0}$ is a redundant description of the complex structure of
$B$. We will extract the independent ones from them later.

The rank $2$ extension bundle $V$ is introduced in (2.7) with extension group
$Ext(L,L^{\vee})=H^{1}(X,L^{2})$. If the threefold $X$ admits an elliptic
fibration, this group can be expressed as

$H^{1}(X,L^{2})=H^{0}(B,L_{1}\oplus L_{2}),$ (C.1)

where $L_{1}=O(D_{b}-C_{1}(B))$ and $L_{2}=O(D_{b}+C_{1}(B))$ are line bundles
and $D_{b}$ is a divisor on the base. Thus, the Ext group of $V$ is a direct
sum of $H^{0}(B,L_{1})$ and $H^{0}(B,L_{2})$. We will study the “jumping” of
both of them in the following.

The line bundle cohomology on the base $B$ follows from a similar procedure
discussed in Subsection 2.1. As before, we can write the $L_{1}$ and $L_{2}$
in terms of the line bundles of the ambient space by Koszul sequence [37]. The
associated long exact sequence is

$0\to H^{0}(B,L)\to H^{1}({\cal A},N^{\vee}\otimes L_{{\cal
A}})\stackrel{{\scriptstyle p_{0}}}{{\rightarrow}}H^{1}({\cal A},L_{{\cal
A}})\to H^{1}(B,L)\to 0\ .$ (C.2)

Notice that, instead of $H^{1}(X,L^{2})$, we are interested in the
$H^{0}(B,L)$ here. For our example considered in Section 3, $L_{1}=O(-2,3)$
and $L_{2}=O(-2,5)$. From the equation (C.2), we find

$\displaystyle H^{0}(X,L_{1})$
$\displaystyle=\textnormal{Ker}\left[H^{1}(A,O(-4,1))\stackrel{{\scriptstyle
p_{0}}}{{\rightarrow}}H^{1}(A,O(-2,3))\right]$ (C.3) $\displaystyle
H^{0}(X,L_{2})$
$\displaystyle=\textnormal{Ker}\left[H^{1}(A,O(-4,3))\stackrel{{\scriptstyle
p_{0}}}{{\rightarrow}}H^{1}(A,O(-2,5))\right]$ (C.4)

To determine the kernel, we express the cohomology of the source and the
target in terms of their Bott-Borel-Weil polynomial representations. For the
first one in Equation (C.3), they are

$\displaystyle H^{1}(\mathbb{P}^{1}\times\mathbb{P}^{2},O(-4,1))$
$\displaystyle=\\{\frac{y_{0}}{x_{0}^{2}},\frac{y_{1}}{x_{0}^{2}},\ldots\frac{y_{0}}{x_{0}x_{1}},\ldots\frac{y_{2}}{x_{1}^{2}}\\}$
(C.5) $\displaystyle H^{1}(\mathbb{P}^{1}\times\mathbb{P}^{2},O(-2,3))$
$\displaystyle=\\{y_{0}^{3},y_{0}^{2}y_{1},\ldots y_{2}^{3}\\}$ (C.6)

and $p_{0}$ is the degree $(2,2)$ defining polynomial. Note that the
multiplication of $\frac{y_{0}}{x_{0}x_{1}}$ and $x_{0}^{2}y_{0}^{2}$ maps to
zero in the target because the resulting monomials have the wrong degree and
should not be considered as an appropriate element of the target. The same is
for a map like $x_{1}^{2}y_{0}^{2}$. Thus, the most general form of the
$p_{0}$ that have non-trivial kernel is given by

$p_{0}=x_{1}^{2}P_{1}(y)+x_{2}^{2}P_{2}(y),$ (C.7)

where $P_{1}(y)$ and $P_{2}(y)$ are arbitrary degree $2$ polynomials in
$\\{y_{0},y_{1},y_{2}\\}$. There are three independent monomials with the
denominator $x_{0}x_{1}$. Therefore, the dimension of the kernel “jumps” to
$3$ when the defining polynomial has the form in (C.7) and it vanishes when
$p_{0}$ is a generic polynomial. By the same approach, we also compute the
“jumping” of $H^{0}(X,L_{2})$. The results are summarized in Equation (3.20)
that is used for complex structure moduli stabilization in Section 3.

##  Appendix D Approach for Identifying Reducible Deformations

In Subsection 3.2, we find that the space of reducible deformations
$M^{2}_{\;\;r}$ is generated by $5$ out of $6$ polynomials given in (3.2).
There, we use the fact that polynomial deformation $\delta F_{2}$, if
reducible, is equivalent to shifts of the coefficients in $F^{(0)}_{2}$. We
will see why it is true and verify that $M^{2}_{\;\;r}$ contains all the
possible reducible deformations. In addition, we will generalize the method
introduced in Subsection 3.2 to find reducible deformations of other initial
polynomials.

Let’s expand the initial $F^{(0)}_{2}$ defined in (3.46) as

$\displaystyle
F^{(0)}_{2}=(au_{1}^{2}+bu_{1}u_{2}+cu^{2}_{2})z_{1}^{2}+(du_{1}+eu_{2})z_{1}z_{2}+fz^{2}_{2}\;.$
(D.1)

where coefficients $a,b,c,d,e,f$ are given by

$\displaystyle a=A_{1}B_{1}\;,\qquad$ $\displaystyle
b=A_{1}B_{2}+A_{2}B_{1}\;,\quad c=A_{2}B_{2}\;,$ $\displaystyle
d=A_{1}B_{3}+A_{3}B_{1}\;,\quad$ $\displaystyle
e=A_{2}B_{3}+A_{3}B_{2}\;,\quad f=A_{3}B_{3}\;$

with $A_{1},A_{2},A_{3},B_{1},B_{2},B_{3}$ the coefficient of $F^{(0)}_{2}$ in
the factored form. It means that given a polynomial with coefficients
$a,b,c,d,e,f$, if one can find a set of $\\{A_{i}\\}$ and $\\{B_{i}\\}$ solves
the equation (D), the polynomial is reducible. This is the reducible condition
that will be used later. Obviously, $F^{(0)}_{2}$ is expressed by the
following $6$ monomials

$(u_{1}^{2}z_{1}^{2},u_{1}u_{2}z_{1}^{2},u^{2}_{2}z_{1}^{2},u_{1}z_{1}z_{2},u_{2}z_{1}z_{2},z^{2}_{2})\;.$
(D.3)

Thus, there are at most $6$ polynomial deformations for $F^{(0)}_{2}$. Each of
those is given by the deformation of a monomial. We will study which of these
monomial deformation or their combinations $\delta F_{2}$ can keep
$F^{(0)}_{2}+\delta F_{2}$ reducible.

Suppose we deform the initial $F^{(0)}_{2}$ by the first monomial $\delta
au_{1}^{2}z_{1}^{2}$. The coefficients of the deformed polynomial
$F^{(0)}_{2}+\delta au_{1}^{2}z_{1}^{2}$ becomes $\\{a+\delta a,b,c,d,e,f\\}$,
which does not satisfy the reducible condition in (D). Thus, the deformed
polynomial is not reducible. To restore the reducibility, one needs to adjust
the coefficients $\\{A_{i}\\}$ and $\\{B_{i}\\}$ so that they give rise to the
new parameters $\\{a+\delta a,b,c,d,e,f\\}$ via the condition (D). Since
$a=A_{1}B_{1}$, it can be done by simply redefining $A_{1}$ to be
$A^{\prime}_{1}=A_{1}+\frac{\delta a}{B_{1}}$. Now, $a+\delta
a=A^{\prime}_{1}B_{1}$. To satisfy the reducible condition (D), one needs to
replace the $A_{1}$ with $A^{\prime}_{1}$ in all other equations involving
$A_{1}$, which are $b=A^{\prime}_{1}B_{2}+A_{2}B_{1}$ and
$d=A^{\prime}_{1}B_{3}+A_{3}B_{1}$. The redefinition is equivalent to shift
$b\to b+(\frac{B_{2}}{B_{1}})\delta a$ and $d\to d+(\frac{B_{3}}{B_{1}})\delta
a$. That means, to preserve the reducibility, whenever we deform the $F$ by
$\delta au_{1}^{2}z_{1}^{2}$, we have to add the extra terms
$(\frac{B_{2}}{B_{1}})u_{1}u_{2}z_{1}^{2}\delta a$ and
$(\frac{B_{3}}{B_{1}})u_{1}z_{1}z_{2}\delta a$ as well. The reducible
deformation after a redefinition of $A_{1}$ is

$e_{1}=u_{1}^{2}z_{1}^{2}+(\frac{B_{2}}{B_{1}})u_{1}u_{2}z_{1}^{2}+(\frac{B_{3}}{B_{1}})u_{1}z_{1}z_{2},$
(D.4)

which, by a factor of $B_{1}$, is the same deformations $e^{A}_{1}=B_{1}e_{1}$
as the one obtained by a shift of $A_{1}$ in (3.2).

Besides the redefinition of $A_{1}$, to absorb the deformation $\delta
au_{1}^{2}z_{1}^{2}$, one can also consider the redefinition of the $B_{1}$ as
$B^{\prime}_{1}=B_{1}+\frac{\delta a}{A_{1}}$. From the relation (D), one has
to redefine the $b$ and $d$ by shifting $b\to b+(\frac{A_{2}}{A_{1}})\delta a$
and $d\to d+(\frac{A_{3}}{A_{1}})\delta a$, which is given by

$e_{2}=u_{1}^{2}z_{1}^{2}+(\frac{A_{2}}{A_{1}})u_{1}u_{2}z_{1}^{2}+(\frac{A_{3}}{A_{1}})u_{1}z_{1}z_{2}.$
(D.5)

Again, up to a factor of $A_{1}$, it is the same deformation obtained by a
shift of $B_{1}$. Following the same analysis, we consider all $6$ monomial
deformations in (D.3). We find that all these deformations can be absorbed by
the shifts of the coefficients $\\{A_{i}\\}$ and $\\{B_{i}\\}$, which gives
the same reducible deformations as ones in $M^{2}_{\;\;r}$.

The reducible deformations considered above are obtained by the redefinition
of one coefficients. For example, $e_{1}$ and $e_{2}$ are generated by a shift
of $A_{1}$ and $B_{1}$. One may suspect that can we get more reducible
deformations by redefining $A_{1}$ and $B_{1}$ at the same time? For example,
given the monomial deformation $\delta au_{1}^{2}z_{1}^{2}$, one can absorb it
by shifting the $A_{1}$ and $B_{1}$ at the same time by $\delta
A_{1}=\frac{\delta a}{2B_{1}}$ and $\delta B_{1}=\frac{\delta a}{2A_{1}}$. It
is easy to check that, to first-order, the reducible condition (D) is
satisfied. In particular, $a+\delta a=(A_{1}+\delta A_{1})(B_{1}+\delta
B_{1})$. The corresponding reducible deformations $E$ follow by shifting the
$A_{1}$ and $B_{1}$ in all other equations in (D). One can show that
$E=\frac{1}{2}e_{1}+\frac{1}{2}e_{2}$, which means that $E$ is included in
$M^{2}_{\;\;r}$ and redefining multiple coefficients at the same time does not
give new reducible deformations. Therefore, we verify that $M^{2}_{\;\;r}$
contains all the possible reducible deformations and they are generated by the
shifts of each coefficient in (3.46) independently.

We generalize the above method to study the reducible deformations of
polynomial other than (3.32). Suppose the initial $F_{2}^{(0)}$ is a product
of a polynomial with $m$ terms and a polynomial with $n$ terms

$F_{2}^{(0)}=(A_{1}m^{A}_{1}+A_{2}m^{A}_{2}+\ldots+A_{r}m^{A}_{r})(B_{1}m^{B}_{1}+B_{2}m^{B}_{2}+\ldots+B_{s}m^{B}_{s})$
(D.6)

where $\\{m^{A}_{i}\\},i=1,2,\ldots,r$ and $\\{m^{B}_{j}\\},j=1,2,\ldots,s$
are monomials and $\\{A_{i}\\}$ and $\\{B_{j}\\}$ are the corresponding
coefficients. As before, the reducible deformations can be obtained by shifts
of each coefficient in $\\{A_{i}\\}$ and $\\{B_{j}\\}$,

$\displaystyle
e^{A}_{1}=(B_{1}m^{B}_{1}+B_{2}m^{B}_{2}+\ldots+B_{s}m^{B}_{s})m^{A}_{1},\quad$
$\displaystyle
e^{B}_{1}=(A_{1}m^{A}_{1}+A_{2}m^{A}_{2}+\ldots+A_{r}m^{A}_{r})m^{B}_{1}$
$\displaystyle
e^{A}_{2}=(B_{1}m^{B}_{1}+B_{2}m^{B}_{2}+\ldots+B_{s}m^{B}_{s})m^{A}_{2},\quad$
$\displaystyle
e^{B}_{2}=(A_{1}m^{A}_{1}+A_{2}m^{A}_{2}+\ldots+A_{r}m^{A}_{r})m^{B}_{2},$
$\displaystyle\qquad\vdots$ $\displaystyle\hskip 113.81102pt\vdots$
$\displaystyle
e^{A}_{r}=(B_{1}m^{B}_{1}+B_{2}m^{B}_{2}+\ldots+B_{s}m^{B}_{s})m^{A}_{r},\quad$
$\displaystyle
e^{B}_{s}=(A_{1}m^{A}_{1}+A_{2}m^{A}_{2}+\ldots+A_{r}m^{A}_{r})m^{B}_{s}\;.$

There are totally $(r+s)$ such polynomials. As the equation (3.48), one can
verify that these deformations satisfy the relation

$F_{2}^{(0)}+\left(A_{1}e^{A}_{1}+A_{2}e^{A}_{2}+\ldots+A_{r}e^{A}_{r}\right)=F_{2}^{(0)}+\left(B_{1}e^{B}_{1}+B_{2}e^{B}_{2}+\ldots+B_{s}e^{B}_{s}\right)\;.$
(D.8)

Thus, there are $(r+s-1)$ independent polynomials and they form a basis of
$M^{2}_{\;\;r}$.

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