text:
\section{Introduction}

Considerable progress has been made in the last two years
in understanding flavor structure in F-theory compactifications.
Supersymmetric compactifications of F-theory to 3+1
dimensional space-time is described primarily by a set of data
$(X_4, G^{(4)})$, where $X_4$ is an elliptic fibered Calabi--Yau
4-fold and $G^{(4)}$ a 4-form flux on it. Non-Abelian gauge theories
like SU(5)$_{\rm GUT}$ unified theories can arise from singular geometry
of $X_4$.
It is difficult to extract physics directly from singular geometry, but
low-energy effective theory associated with the singular local
geometry of $X_4$ can be described by using non-Abelian gauge theory
in 7+1 dimensions \cite{KV} to some level of approximation.
It has been the key in the progress in the understanding of
flavor structure in F-theory to use this gauge-theory effective
description \cite{DW-1, BHV-1}.
See also \cite{Hayashi-2, DW-3, Tsuchiya, Harvard-nonC, Conlon, Hayashi-3App}.

In supersymmetric compactification for realistic models,
dimension-4 proton decay operators
\begin{equation}
 \Delta W \ni \bar{\bf 5}_M \cdot {\bf 10} \cdot \bar{\bf 5}_M
\label{eq:dim-4}
\end{equation}
have to be brought under control. An obvious solution is
\begin{itemize}
\item [(i)] to consider a compactification with $(X_4, G^{(4)})$ that has
  a $\mathbb{Z}_2$ symmetry \cite{Tsuchiya}.
\end{itemize}
The $\mathbb{Z}_2$ symmetry will become an $R$-parity (or equivalently
a matter parity) in the low-energy effective theory below the
Kaluza--Klein scale. Other solutions to the problem include
\begin{itemize}
 \item [(ii)] rank-5 GUT scenario, that is, to consider $\mathop{\rm SU}(6)$ or
 $\mathop{\rm SO}(10)$ GUT models \cite{TW-1},
 \item [(iii)] factorized spectral surface scenario
  \cite{Tsuchiya, Caltech-0906}
     (see also \cite{BHV-2, DW-2, Harvard-nu, Harvard-E8}), and
 \item [(iv)] spontaneous $R$-parity violation scenario \cite{TW-1, Tsuchiya}
(see also \cite{Blhg-0908}).
\end{itemize}
The scenario (ii) is a special case of (iii), and is also a special case
of (iv).

The factorized spectral surface scenario (iii), however, is not
without a theoretical concern \cite{Tsuchiya}. The dimension-4 proton
decay problem is so severe that the coupling (\ref{eq:dim-4}) should
be extremely small even if they are present.
The question is whether we have a theoretical framework in which
the scenario can be formulated rigorously and even an extremely small
contribution to (\ref{eq:dim-4}) can be studied.
At least so far, the answer is no.
It is, thus, important to study to what extent this scenario works,
and that is what we do in this article.

Heart of the idea of the factorized spectral surface scenario is to
consider a factorization limit of the spectral surface so that
\begin{itemize}
 \item matter fields $\bar{\bf 5}_M$ and Higgs fields $\bar{\bf 5}_H$
       are associated with irreducible components of the spectral
       surface for fields in the $\bar{\bf 5}$ representation
       of $\mathop{\rm SU}(5)_{\rm GUT}$, and
 \item an unbroken U(1) symmetry remains in the low-energy effective
       theory below the Kaluza--Klein scale, and the
       operator (\ref{eq:dim-4}) is ruled out because of the symmetry.
\end{itemize}
This scenario has been formulated by using the gauge theory description
in 7+1 dimensions.
Because of the approximate nature of the gauge theory description,
factorization is not rigorously well-defined; there are also some
corrections that have not been captured in the gauge theory
description on 7+1 dimensions \cite{Tsuchiya}. We will explain
these limitations of the gauge theory description on 7+1 dimensions
more in detail in section \ref{sec:effect_beyond_gauge_theory},
although these materials have almost been spelled out in \cite{Tsuchiya,
Caltech-0906} (see also \cite{DW-3}).

In section \ref{sec:mod}, we study monodromy of 2-cycles in $X_4$
instead of a gauge theory on 7+1 dimensions, in order to find out
whether an unbroken U(1) symmetry remains in the low-energy
effective theory. We conclude in section \ref{sec:physics}
that the proton decay operators (\ref{eq:dim-4}) are expected to be
generated in the factorized spectral surface scenario, but we also
list some loopholes in the argument.
The appendix \ref{sec:calculation} explains calculation of the
monodromy of 2-cycles using language of string junction in detail;
all the necessary techniques in the appendix \ref{sec:calculation},
however, are already available in the literature (e.g., \cite{jcn-KcMd}).
The monodromy matrices obtained in this way in F-theory language
can be understood much more transparently in dual Heterotic language;
this is the subject of the appendix \ref{sec:Het}.

\section{Effects Beyond Gauge Theory in 7+1 Dimension}
\label{sec:effect_beyond_gauge_theory}

The gauge theory description on 7+1 dimensions in F-theory captures
geometry of a local neighbourhood of a discriminant locus of an elliptic
fibration \cite{KV}.
When a geometry of an elliptically fibered Calabi--Yau 4-fold
$\pi_X: X_4 \rightarrow B_3$ is locally approximately an ALE space of
A--D--E type in the direction transverse to the discriminant locus,
gauge theory of the same A--D--E type describes the physics associated with
this local geometry.
Thus, by construction, the gauge theory description has limited power
in capturing the physics of geometry in the transverse direction;
only physics associated with {\it non-compact} (local) geometry that is
{\it approximately} ALE fibration can be captured {\it approximately}.
When it comes to the dimension-4 proton decay problem, this approximate
nature of the gauge theory description of F-theory can be a problem.

To see this more explicitly, let us consider a local geometry given by the
generalized Weierstrass form
\begin{eqnarray}
y^2 & = & x^3 - A_1 y x + A_2 x^2 - A_3 y + A_4 x + A_6,
\label{eq:local-defeq}
\end{eqnarray}
where $(x,y)$ are coordinates of the elliptic fiber of $X_4$, and
$A_{1,\cdots,4,6}$ are regarded as functions locally on $B_3$.
For an ${\rm SU}(5)_{\rm GUT}$ GUT model \cite{6authors}, we need to
consider a case where $A_{1,\cdots, 4,6}$ are in the form of
\begin{equation}
 A_k = (-)^k \left[
  z^{k-1} a_{6-k} + z^k a'_{6-k} + z^{k+1} a''_{6-k} + \cdots \right],
\label{eq:Ak-expnsn}
\end{equation}
where $z$ is a local coordinate of a base manifold $B_3$;
the $z=0$ locus becomes an irreducible component $S_{GUT}$ of the
discriminant locus $\Delta$. $a_{0,2,3,4,5}$ and $a'_{0,2,3,4,5}$ are
coefficients that depend on two local coordinates on $S_{GUT}$.
Consider, for example, a case\footnote{This can be interpreted as
considering a region of $S_{GUT}$ where the condition
(\ref{eq:eK-scale}) is satisfied.}\raisebox{4pt}{,}\footnote{
Additional scaling may hold in some regions of $S_{GUT}$.
For example, along the matter curve $a_5 = 0$,
\begin{equation}
a_{5,*}=\lambda \tilde{a}_{5,*}, \qquad \tilde{a}_{5,*} \sim {\cal O}(1)
\end{equation}
is satisfied for $\lambda \ll 1$. One of the four 2-cycles corresponding
to the simple roots of the structure group SU(5)$_{\rm str}$ becomes
much smaller than all the others in this region, and this corresonds to
a hierarchical symmetry breaking $E_8 \rightarrow D_5 \rightarrow A_4$.
The rank-1 extended gauge theories in \cite{KV} and rank-2 extended
gauge theories in \cite{BHV-1, Hayashi-2} describe physics of local
geometries that have such scalings.
We also introduce a hierarchical symmetry breaking in the choice of
a base point of monodromy analysis in (\ref{eq:reference_point})
in this article.
}
when all of $a_{2,3,4,5}$ are small in a way indicated by
\begin{equation}
 a_{5}=\epsilon_{K}^{5}a_{5,*},\;\;a_{4}=\epsilon_{K}^{4}a_{4,*},\;\;
     a_{3}=\epsilon_{K}^{3}a_{3,*},\;\;a_{2}=\epsilon_{K}^{2}a_{2,*}
\label{eq:eK-scale}
\end{equation}
for $a_{5,*}$, $a_{4,*}$, $a_{3,*}$, $a_{2,*}$ of all $\sim {\cal O}(1)$.
$\epsilon_K$ is a small parameter that is different from zero.
We know that this geometry nearly has an $E_8$ singularity around $z=0$;
to be more precise, there are four vanishing 2-cycles right at $z=0$,
and there are four small 2-cycles near $z=0$. One can focus on
the geometry of these small 2-cycles by choosing a new set of
coordinates $(x', y', z')$:
\begin{equation}
x=\epsilon_{K}^{10}x^{\prime},\;\;
y=\epsilon_{K}^{15}y^{\prime},\;\;
z=\epsilon_{K}^{6}z^{\prime}.
\label{eq:coordinate-scale}
\end{equation}
With these new coordinates and coefficients $a_{r,*} \sim {\cal O}(1)$
($r=0,2,3,4,5$), the local defining equation (\ref{eq:local-defeq})
becomes exactly the $E_8$ singularity with relevant deformation
parameters \cite{KM}, with correction terms whose coefficients are
suppressed by positive power of the small number $\epsilon_K$.
The gauge theory description focuses on the geometry of these small
2-cycles (i.e., forgets about the rest of the geometry), and further
make an approximation of ignoring the correction terms
that are small but present \cite{Hayashi-2} (see also \cite{DW-3,
Caltech-0906}). Even when the $\epsilon_K$ scaling of the coefficients
is assumed as above to maximize the number of
small 2-cycles captured in a gauge theory description, the gauge
theory description is only an approximate description of the
compact geometry $X_4$ by construction \cite{Tsuchiya}.
Let us call this {\bf Problem A}.

We may have another difficulty in justifying the factorized spectral
surface scenario in the gauge theory description.
To see this, let us remind ourselves of the following.
Since $\bar{\bf 5}_M$ and $\bar{\bf 5}_H$ multiplets are zero modes
appearing in the low-energy effective theory below the Kaluza--Klein
scale, the factorization of the spectral surface should be defined
{\it globally} on the GUT divisor $S_{GUT}$.
Field theory local model in an open patch $U_a \subset S_{GUT}$, on the other
hand, captures very small 2-cycles in the ALE fiber in addition to
the four vanishing 2-cycles in $A_4$ singularity, and hence the rank of
gauge group in $U_a$ may be different from the rank in another
open patch $U_b \subset S_{GUT}$ \cite{KV, BHV-1, DW-2, Hayashi-2}; these
local descriptions with gauge groups of varying rank are glued
together approximately in overlapping regions of open patches
$\{ U_a \}_{a\in A}$ to cover the entire ALE-fibered geometry over
$S_{GUT}$.
The global factorization of the spectral surface, however, cannot be
well-defined, when the rank of the spectral surface varies from
one patch to another. This problem can be avoided if we consider
a Higgs bundle that has a fixed rank over the entire $S_{GUT}$.

Fixed rank Higgs bundle description over the entire $S_{GUT}$, however,
often breaks down somewhere in $S_{GUT}$. For example, a rank-$k$ spectral
surface $C$ is defined globally in $K_S$ by
\begin{equation}
 a_{5-k} \xi^k + \cdots + a_4 \xi + a_5 = 0,
\label{eq:spec-surf-1}
\end{equation}
where $\xi$ is the fiber coordinate of the canonical bundle
$\pi_{K_S}: K_S \rightarrow S$,
if $a_r$'s ($5-k \leq r \leq 5$) are global holomorphic sections of line
bundles ${\cal O}(r K_S + \eta)$ for some divisor $\eta$ in $S_{GUT}$.
$\pi_{K_S}|_C: C \rightarrow S$ is
an $k$-fold cover at generic points of $S_{GUT}$, but when the coefficient
of the highest degree term $a_{5-k}$ vanishes, one of the $k$ solutions
of (\ref{eq:spec-surf-1}) shoots off to infinity in the fiber direction of
$\pi_{K_S} : K_S \rightarrow S_{GUT}$. This behavior of the
spectral surface indicates that one of (very) small 2-cycles becomes
relatively large there.\footnote{\label{fn:eff-KS}
The vev of the Higgs field $\varphi$ is obtained by
a holomorphic 4-form on a Calabi--Yau 4-fold on 2-cycles. The $\varphi$
vev being large means that the period integral over the corresponding
2-cycle is large \cite{DW-1, Tsuchiya}. } It is not a
sensible approximation to keep the physics associated with all the
2-cycles in (\ref{eq:spec-surf-1}) while ignoring all others
in a local neighbourhood around the zero locus of $a_{5-k}$ .
As long as the divisor $(5-k)K_S + \eta$ is effective,\footnote{
The divisor $5K_S+\eta$ needs to be effective so that the matter curve
for the SU(5)$_{\rm GUT}$-{\bf 10} representation field is effective.
If $-K_S$ is effective, as in del Pezzo surfaces and Hirzebruch surfaces,
then $(5-k)K_S + \eta$ is also effective. Thus, this problem is
unavoidable for surfaces $S_{GUT}$ with effective $-K_S$.
See section \ref{sec:physics} for comments on $S_{GUT}$ with
effective $K_S$ (rather than effective $-K_S$).}
this does happen somewhere in $S_{GUT}$.
The problem in this neighbourhood can be fixed (c.f. \cite{BHV-1}) by adopting a Higgs
bundle that is one-rank higher, a rank-$(k+1)$ Higgs bundle, as long as
$a_{4-k}$ is not identically zero.\footnote{We will come back to a loophole
here in section \ref{subsec:loophole}.} We still encounter the same problem
around the zero locus of $a_{4-k}$. One could still crank the rank
of gauge group up, once again. But $E_8$ gauge group is maximal in
the E series of the A--D--E classification, and we come to a dead end.

We could consider a Calabi--Yau 4-fold $X_4$ for F-theory compactification
which is approximately an ALE fibration of $E_8$ type over $S_{GUT}$ in the
neighbourhood along $S_{GUT}$.
This is done by choosing the coefficients indicated as in
(\ref{eq:eK-scale}) \cite{DW-3}.
When we take GUT gauge group as SU(5)$_{\rm GUT}$, we have a 5-fold
spectral cover,
\begin{equation}
a_{0}\xi^{5}+a_{2}\xi^{3}+a_{3}\xi^{2}+a_{4}\xi+a_{5}=0.
\label{eq:5-fold}
\end{equation}
Factorization conditions can be imposed on (\ref{eq:5-fold}) in an
$E_8$ gauge theory defined globally on $S_{GUT}$ \cite{Caltech-0906}.
That is the best we hope to do within gauge theories on 7+1 dimensions.
In a region of $S_{GUT}$ around a point where $a_0$ vanishes, however,
two roots of (\ref{eq:5-fold}) become large, as
\begin{equation}
\xi \sim \pm i \sqrt{(a_2/a_0)}.
\label{eq:shoot-spec-surf}
\end{equation}
The $E_8$ gauge theory is not a sensible approximation in the local
neighbourhood of the $a_0 = 0$ locus, because the corresponding two
2-cycles are no longer relatively small.
In a region slightly away from the $a_0 = 0$ locus, one can see
that the two roots (and hence the Higgs field vevs $\varphi$ in
two diagonal entries) are exchanged when the phase of $a_0$ changes
by $2\pi$. This phenomenon indicates that the two 2-cycles not just become
large, but also are twisted by a monodromy around the $a_0=0$ locus.
2-cycles that are not captured by the $E_8$ gauge theory may also
be involved in this monodromy, because there is no clear separation
between the 2-cycles within $E_8$ and those that are not around
the $a_0=0$ locus.
In order to study geometry and physical consequences associated with
this behavior, one has to go beyond $E_8$ gauge theory on 7+1 dimensions.
We call this {\bf Problem B}.

Clearly we need a theoretical idea how to study whether the operator
(\ref{eq:dim-4}) is generated or not; a new idea should keep the
successful aspects of the factorized spectral surface scenario, while
it should not rely on the gauge theory descriptions in 7+1 dimensions.
While spectral surfaces can be defined only within the gauge theory
descriptions, the essence of the scenario is to keep an unbroken
U(1) symmetry in the low-energy effective theory below the
Kaluza--Klein scale.
We can thus focus on a question whether an unbroken U(1) symmetry
is maintained (or how to find one) in F-theory compactifications.
Noting that topological 2-forms of certain type in $X_4$ yield U(1)
vector fields in 3+1 dimensions \cite{MV2, DRS},\footnote{
These vector fields can be massive, yet global U(1) symmetries may remain
in the effective theory. That is the situation we are interested in.}
and that the vector fields are always accompanied by
U(1) symmetries\footnote{The similar idea that an invariant 2-cycle
under the monodoromy group gives rise to an unbroken U(1) symmetry, has been
applied to the system of D5-branes wrapped on 2-cycles in an ALE fibered
geometry \cite{CKV}.},
we see that a solution\footnote{One might alternatively think of factorizing
the discriminant
as a generalization of factorizing the spectral surface. However, this
idea does not work from the very beginning.
Global factorization of the {\it spectral surface} makes sense as a solution
to the dimension-4 proton decay problem,
because the spectral surface for the $\mathop{\rm SU}(5)_{\rm GUT}$-$\bar{\bf 5}$
representation is factorized into 2 irreducible branches around
points of down-type Yukawa coupling ($D_6$ singularity enhancement
points).
As studied in section 4.3 of \cite{Hayashi-2}, however,
the {\it discriminant locus} does not have this property.
See Figure~10.(b) of \cite{Hayashi-2}.
} is to focus on geometry of $X_4$, instead of gauge theory associated
with the canonical bundle $K_S$ on $S_{GUT}$.

Since we are interested in U(1) symmetries under which
$\mathop{\rm SU}(5)_{\rm GUT}$-charged matter fields on $S_{GUT}$ are charged,
we are interested in 2-forms of $X_4$ which have components dual to
2-cycles in the local ALE fiber approximation along $S_{GUT}$ in $X_4$.
It is a topological problem whether or not an extra U(1) symmetry
remains unbroken. We keep tracks of topological 2-forms / 2-cycles
to address this problem; it is not necessary to restrict our
attention only to 2-cycles contained in one of the $A$--$D$--$E$ types,
or to 2-cycles that are relatively small. All the 2-cycles in $X_4$
can be treated in an equal footing, and hence, we are free from all
the problems in the gauge theory description on 7+1 dimensions.
The factorization condition of the spectral surface for an unbroken
U(1) symmetry is replaced by a condition that there is at least one\footnote{
To be more precise, the condition is the existence of an extra 6-cycle
[equivalently its Poincare dual] of $X_4$ contributing to $H^{1,1}(X)$
other than $B_3$ or those in $\pi_X^* (H^{1,1}(B_3))$ \cite{MV2, DRS}.
We will come back to this issue in section \ref{subsec:non-K3}.}
monodromy-invariant 2-cycle over $S_{GUT}$.

\section{Monodromy of 2-Cycles}
\label{sec:mod}

In this section, we study monodromy of 2-cycles in a Calabi--Yau
4-fold $X_4$ and discuss whether an unbroken U(1) symmetry remains
or not when the spectral surface satisfies a factorization condition.
In section \ref{subsec:mod_gauge}, we show that certain subgroup
of the monodromy corresponds to what we expect from the gauge theory
descriptions on 7+1 dimensions. This subgroup reduces to a smaller
one when the spectral surface is in the factorization limit, and there
is a monodromy invariant 2-cycle [and hence an unbroken U(1)] at this
level of analysis. This guarantees that we can keep the heart of the
idea of the factorized spectral surface scenario without relying on
the gauge theory description.
Section \ref{subsec:str_sing} explains the structure
of the full monodromy group; the subgroup in section \ref{subsec:mod_gauge}
is a proper subgroup of the full monodromy group. We study monodromy
of 2-cycles for some other generators in
section \ref{subsec:monod_beyond_gauge}, and find that there is no
monodromy-invariant 2-cycles when the full monodromy group is taken
into consideration. This means that there is no unbroken U(1) symmetry
in the effective theory in the factorized limit of the spectral surface.

\subsection{The model and the notation}

In the study of monodromy of 2-cycles in
sections \ref{subsec:mod_gauge}--\ref{subsec:monod_beyond_gauge}, we only
consider a special case where a Calabi--Yau 4-fold $X_4$ is a K3
fibration over $S_{GUT}$: $\pi'_X: X_4 \rightarrow S_{GUT}$.
One of the advantages of this $X_4$ as a global fibration on $S_{GUT}$ is
that the U(1) vector fields in 3+1 dimensions, and hence the
non-$H^2(B_3)$ non-$H^0(B_3; R^2 \pi_{X*} \mathbb{Z})$ components of $H^2(X_4; \mathbb{Z})$ can
be studied by $H^0(S_{GUT}; R^2 \pi'_{X*} \mathbb{Z})$ \cite{CD, DW-1, Hayashi-1}.
Global sections of the {\it local} system $R^2 \pi'_{X*} \mathbb{Z}$ correspond
precisely to the monodromy invariant 2-cycles in the fiber of
$\pi'_{X}: X_4 \rightarrow S_{GUT}$, and hence the problem can be
formulated as a local theory on 7+1 dimensions (if not as a local
{\it gauge} theory on 7+1 dimensions). Although this advantage may appear
to be available only for a special case, global fibration of
$X_4$ over $S_{GUT}$, we consider otherwise. Topological 2-cycles
in a neighbourhood of $S_{GUT}$ can be captured by a local system
like $R^2 \pi'_{X*} \mathbb{Z}$ at least locally, if not globally.
Since the monodromy among 2-cycles is about a non-trivial local
behavior of such a local system at special points (which we call monodromy
locus later in this article), one only has to examine local behavior of such
local systems to find out whether a U(1) symmetry is projected out
or not. Thus, although the following presentation in this section
may appear to rely exclusively on a $X_4$ that is a global fibration
on $S_{GUT}$, we do not loose generality as a study of F-theory
compactifications. Discussion on non-K3 fibered cases is found in
section \ref{subsec:non-K3}.

The other advantage is that we know a lot about 2-cycles of
K3 manifold, so that we can make our presentation very concrete.
Since the monodromy around $a_0 = 0$ locus might involve not just
2-cycles within $E_8$ but also other 2-cycles in the direction
transverse to the GUT divisor $S_{GUT}$, we certainly need a geometry
for analysis where such an ``other 2-cycle'' is identified.
With a compact K3 manifold in the transverse direction, we have
all the necessary techniques. On the other hand, we do not loose
generality by this specific choice, because we are interested
only in the local behavior of monodromy among 2-cycles in $E_8$ and
those that are ``adjacent'' to them.

In principle, we could study monodromy of 2-cycles in a configuration
of real interest, where the $A_4$ singularity is along $S_{GUT}$, and
a factorization condition is imposed on the 5-fold spectral
cover (\ref{eq:5-fold}).
Instead of this realistic $E_8 \rightarrow \mathop{\rm SU}(5)_{\rm GUT}$ symmetry
breaking model, however, we study monodromy of 2-cycles in an
$E_8 \rightarrow E_6$ symmetry breaking model. 2+1 factorization
condition can be imposed on the 3-fold spectral cover
\begin{equation}
a_0 \xi^3 + a_2 \xi + a_3 = 0.
\label{eq:rank3_spect}
\end{equation}
Moreover, since the 2-cycles associated with the hidden sector is
not essential in our problem, we consider a configuration with an
unbroken hidden $E_8$ symmetry.
Namely, we study monodromy of 2-cycles in $X_4$ whose K3 fibration
structure is given by
\begin{equation}
y^2 = x^3 + (a_2 z^3 + f_0 z^4) x
+ \left( \frac{1}{4} a_3^2 z^4 + a_0 z^5 + g_0 z^6
+ a''_0 z^7 \right) \, .
\label{eq:model}
\end{equation}
Here, $(x,y)$ are coordinates of the elliptic fiber of $X_4$,
$z$ an inhomogeneous coordinate of the base $\mathbb{P}^1$ of K3
and $a_{0,2,3}, a''_0, f_0, g_0$ are sections of some line bundles
on $S_{GUT}$.

When F-theory is compactified on a K3 manifold that is an elliptic
fibration $\pi_{K3}: K3 \rightarrow \mathbb{P}^1$ with a section, then there
are 20 2-cycles dual to $H^1(\mathbb{P}^1; R^1 \pi_{K3 *} \mathbb{Z})$ and corresponding
U(1) symmetries in 7+1 dimensions.
Such topological 2-cycles can be described by string junctions on the
base $\mathbb{P}^1$. All the necessary details in how to find independent
20 configuration of such junctions \cite{jcn-KcMd} are reviewed
in the appendix \ref{sec:calculation}.

In order to calculate monodromy of 2-cycles, or equivalently to see
how the local system $R^2 \pi'_{X*} \mathbb{Z}$ is twisted, one first has to
take a base point $b \in S_{GUT}$, and set a basis in the (co)homology
group of $H_2(\pi^{'-1}_X(b); \mathbb{Z}) \simeq H_2({\rm K3}; \mathbb{Z})$.
For a loop $\gamma$ on $S_{GUT}$ that starts and ends on the base
point $b$, we can trace the basis elements $H_2(\pi^{'-1}_{X}(p); \mathbb{Z})$ for points
$p$ on the loop $\gamma$, starting from the base point $b$ until
we come back to the base point again. The calculation of monodromy
is practically carried out by tracing string junction configuration
that correspond to the basis elements of
$H_2(\pi^{'-1}_X(b); \mathbb{Z})$. We only have to trace along the loop $\gamma$
the motion of discriminant locus points---$[p,q]$ 7-branes---in the
complex $z$ plane. All the necessary techniques in this monodromy analysis
is quite standard in Type IIB string theory. We thus present detailed procedure
of practical calculation of monodromy for only three loops in the
appendix \ref{sec:calculation}; for all other loops, only the results
are presented in this article.

It is convenient to introduce a notation for a set of independent
2-cycles of the K3 fiber and also to assign names to individual
$[p,q]$ 7-branes at a base point $b$. To the 24 discriminant
points, we assign the following names: $A8$--$A1$, $B$, $C1$, $C2$,
$D$, $A8'$--$A1'$, $B'$, $C1'$, $C2'$ and $D'$.
The appendix \ref{sec:calculation} explains the choice of the base point
$b$, location of the the 24 7-branes in the $z$ plane, as well as
their $[p,q]$ charges for the base point. We assign names (notation)
such as $C_{A76}$, $C_{A65}$ etc. to string junction configurations and
corresponding 2-cycles of ${\rm K3} = \pi^{'-1}(b)$. All the details
are written in the appendix \ref{sec:calculation}, but the most relevant
part of the information is summarized in Figure \ref{fig:2-cycles}.
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.3]{2-cycles_K3_light.eps}
\caption{The independent 2-cycles of a K3 surface. Dotted lines
 represent the branch cuts of 7-branes. Black blobs represent
 $A$-branes, open circles are B-branes, open boxes correspond to
C-branes and open triangles are ``D''-branes whose $[p,q]$ charge
is [3, 1]. $A8^{\prime} \sim D^{\prime}$ 7-branes have the same $[p,q]$
charges as those without a $'$, and are ordered from left to right as in
 the same way as $A8 \sim D$ 7-branes.
7-branes from $A7$ to $C2$ and those from $A7'$ to $C2'$ constitute
separate sets of $[A^7BC^2]$ 7-brane configuration of $E_8$.
}
\label{fig:2-cycles}
\end{center}
\end{figure}

There is no unique choice of a basis of $H_2({\rm K3}; \mathbb{Z})$,
but one of the most convenient choices is to use the 2-cycles
listed in Table~\ref{tab:2-cycles}.
\begin{table}[tb]
 \begin{center}
  \begin{tabular}{l|l} \hline
 2-cycles inside $E_8$ & $C_{A76}, C_{A65}, C_{A54}, C_{A43}, C_{A32},
  C_{A21}, C_{ABC}, C_{C12}$ \\ \hline
 2-cycles outside $E_8$ & $C_{\alpha}^1$, $C_\alpha^2$, $C_{\beta}^1$,
  $C_\beta^2$ \\ \hline
 those for hidden $E_8$ & $C'_{A76}, C'_{A65}, C'_{A54}, C'_{A43},
C'_{A32}, C'_{A21}, C'_{ABC}, C'_{C12}$ \\ \hline
  \end{tabular}
\caption{A convenient choice of a basis of $H_2(\pi^{'-1}(b); \mathbb{Z})$. }
\label{tab:2-cycles}
 \end{center}
\end{table}
Among them, $C_\alpha^{1,2}$, $C_\beta^{2}$ need to be defined
by linear combinations of string junction configurations like those in
Figure~\ref{fig:2-cycles}; see (\ref{eq:basis1}, \ref{eq:basis2}).
The intersection form becomes (\ref{eq:K3-int-form}).
The intersection form within the eight 2-cycles in the first row of
Table \ref{tab:2-cycles} is described by the Dynkin diagram of $E_8$
shown in Figure~\ref{fig:dynkin}.
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.3]{e8_dynkin_l.eps}
\caption{2-cycles inside $E_8$ and their intersection.
Note that $C_{-\theta}$ is not the same as $C_{A87}$;
see footnote \ref{fn:K3-RES} for more.}
\label{fig:dynkin}
\end{center}
\end{figure}

As we study monodromy of 2-cycles in (\ref{eq:model}) that has
unbroken $E_6$ symmetry (split form $E_6$ singularity) \cite{6authors},
monodromy matrices must be trivial on a subspace generated purely by
$C_{A54}, C_{A43}, C_{A32}, C_{A21}, C_{ABC}, C_{C12}$, the simple
roots of $E_6$. Thus, we can choose $C_{-\theta}$ as a basis element
instead of $C_{A65}$ as in Table~\ref{tab:2-cycles}, where
$C_{-\theta}$ is a linear combination purely of the 2-cycles in the
first row of Table~\ref{tab:2-cycles},
\begin{equation}
C_{-\theta} = -(2 C_{A76} + 3 C_{A65} + 4 C_{A54} + 5 C_{A43} + 6 C_{A32} + 4
C_{ABC} + 3 C_{A21} + 2 C_{C12}) \, .
\label{eq:C-theta-def}
\end{equation}
The six 2-cycles of $E_6$ and the two 2-cycles $C_{A76}$ and
$C_{-\theta}$ for the ``structure group'' $\mathop{\rm SU}(3)_{\rm str}$
are orthogonal in the intersection form.
Since we restrict ourselves in a region of moduli space where
the hidden $E_8$ 2-cycles do not participate in the monodromy,
the monodromy can be represented by $6 \times 6$ matrices
acting on a vector space generated by $C_{A76}, C_{-\theta}, C_\alpha^1,
C_\alpha^2, C_\beta^1$ and $C_\beta^2$.

The calculation of monodromy boils down to following the trace
of the $[p,q]$ 7-branes along paths in $S_{GUT}$. Thus, an explicit
expressions for the discriminant of \eqref{eq:model} is very useful:
\begin{equation}
\Delta = z^8 \Delta'
\end{equation}
where $\Delta'$ is given by
\begin{equation}
\begin{split}
\Delta' &= \frac{27}{16} a_3^4
 + \left( \frac{27}{2} a_0 a_3^2 + 4 a_2^3 \right) z
 + \left( \frac{27}{2} a_3^2 g_0 + 27 a_0^2 + 12 a_2^2 f_0 \right) z^2 \\
 & \quad + \left( 54 a_0 g_0 + 12 a_2 f_0^2 + \frac{27}{2} a_3^2 a''_0 \right) z^3
  + (4 f_0^3 + 27 g_0^2 + 54 a_0 a''_0) z^4 \\
 & \quad + 54 a''_0 g_0 z^5 + 27 a^{''2}_0 z^6 \, .
\end{split}
\label{eq:discriminant}
\end{equation}
In this model, eight 7-branes $A5$--$A1$ and $B$, $C1$, $C2$ are at
an $E_6$ singularity at $z=0$ \cite{Johansen, GZ}, and ten 7-branes
$A7'$--$A1'$, $B'$, $C1'$and $C2'$ form an $E_8$ singularity at $z = \infty$.
The six other 7-branes, $A6$, $A7$, $A8$, $D$, $A8'$ and $D'$, are
the zero points of $\Delta'$ above.
Along a path $\gamma$ in $S_{GUT}$, values of all the coefficients
$a_{0,2,3}$, $f_0$, $g_0$ and $a''_0$ change, and hence the six 7-branes
change their locations in the $z$-plane. We can thus determine
the monodromy matrix by following the 7-branes and string junctions.

In sections \ref{subsec:mod_gauge}--\ref{subsec:monod_beyond_gauge},
we present monodromy matrices obtained as a result of such a
calculation. Although the monodromy for a loop $\gamma$ in $S_{GUT}$
is ultimately what we are interested in, it is a better idea to
study monodromy of 2-cycles in a little more abstract level;
we will study monodromy of 2-cycles for
loops in a moduli space of an elliptic K3 manifold.
Since any loops in $S_{GUT}$ are mapped by the choice of sections
$a_{0,2,3}$, $f_0$ etc. to loops in the moduli space of the elliptic K3 manifold,
the monodromy group for the loops on $S_{GUT}$ is obtained by a pull-back
of the monodromy group for the loops on the elliptic K3 moduli space
by the sections. Such questions as whether an unbroken U(1) symmetry
exists can be answered at the level of monodromy on the elliptic
K3 moduli space, independent of specific details of $S_{GUT}$ or
sections $a_{0,2,3}$ etc. on it.

\subsection{The monodromy in 8D gauge theory region}
\label{subsec:mod_gauge}

In the factorization limit of the spectral surface, the monodromy
group of 2-cycles captured in the gauge theory description on
7+1 dimensions is reduced; there is no question about this statement
in the deformation of $A_{N+M}$ singularity to $A_M$. This statement
is believed to be true also in deformation of $E_{6,7,8}$ type singularity
to $A_4$ \cite{Hayashi-2, Tsuchiya} (see also \cite{DW-1}), for example,
but it has not been confirmed directly so far by looking at 2-cycles
in $X_4$. In this section \ref{subsec:mod_gauge}, we study explicitly
the geometry of 2-cycles in $X_4$ when the spectral surface is at the
factorization limit, and confirm that the statement above is correct.
This justifies our strategy to replace the factorization limit of
spectral surfaces by existence of monodromy invariant 2-cycles.

At the same time, we will see that the monodromy of 2-cycles that appear
in the gauge theory description on 7+1 dimensions is only a proper
subgroup of the entire monodromy group of the whole theory. Thus, it
will be clear what one overlooks in the gauge theory description on
7+1 dimensions.

The family of elliptic K3 manifold (\ref{eq:model}) is parametrized
by $(a_{0}, a_2, a_3, f_0, g_0, a''_0)$, among which two are redundant
because of the freedom to rescale the coordinates $(x,y,z)$.
Instead of using a set of ``gauge invariant'' parameters of this
moduli space such as $(a_0a''_0/g_0^2)$ and
$(a_2/a_0)(4f_0^3 + 27 g_0^2)^{1/6}$, we fix a gauge by fixing the
values of $f_0$ (or $g_0$) and $a''_0$.

The full scope of the problem is to consider all possible loops
in the moduli space of (\ref{eq:model}) and determine the monodromy
of 2-cycles of the K3 for the loops. We restrict our attention in this
section, however, to a subset of the moduli space
\begin{equation}
 a_0 = a_{0,*} \epsilon_\eta, \quad
 a_2 = a_{2,*} \epsilon_K^2 \epsilon_\eta, \quad
 a_3 = a_{3,*} \epsilon_K^3 \epsilon_\eta,
\label{eq:scaling-rg}
\end{equation}
where parameters $a_{r, *} \in \mathbb{C}$ and $g_0 \in \mathbb{C}$ are at most
${\cal O}(1)$; $0 \neq |\epsilon_K| \ll 1$ and
$0 \neq |\epsilon_\eta| \ll 1$ are fixed numbers, and
we fix the ``gauge'' by setting $f_0 = -1$ and $a''_0 = \epsilon_\eta$.
Because $|\epsilon_\eta| \ll 1$, the elliptic K3 manifold is always
close to the stable degeneration limit \cite{MV2, FMW} within
the subset specified above.
We call this subset scaling region.
It helps in ensuring the validity of the gauge theory description
in 7+1 dimensions, but not quite, to set $\epsilon_K$ as a small number.
We will see this in detail in the rest of this section in F-theory
language. The appendix \ref{sec:Het} will provide a very clear explanation
of this in the dual Heterotic language.
The base point of the moduli space is chosen within this subset, and
only loops that stay within this subset are considered in this section.
Although this is only to find part of the monodromy group, we will see
that this partial result is interesting enough from mathematical and
physical points of view, and is also sufficient in drawing conclusions
for practical purposes.

We will call a subset of the scaling region characterized by
\begin{equation}
 |\epsilon_K|^{2} \ll |a_{0,*}| \sim {\cal O}(1)
\label{eq:gauge-th-rg}
\end{equation}
as an {\it 8D gauge theory region}. We will see in this
section \ref{subsec:mod_gauge} that all the loops (except the one
mentioned at the end of section \ref{subsec:monod_beyond_gauge})
that stay within this `` 8D gauge theory region'' yield monodromies of
2-cycles that are expected from the gauge theory descriptions on
7+1 dimensions.

For any points in the 8D gauge theory region of the moduli space,
the six 7-branes specified by $\Delta' = 0$ are distributed in
the complex $z$ plane as follows; there are two 7-branes in the
region $z \sim {\cal O}(\epsilon_K^6 \epsilon_\eta)$, two
in the region $z \sim {\cal O}(\epsilon_\eta)$, and two
in $z \sim {\cal O}(\epsilon_\eta^{-1})$. These three groups
of 7-branes remain distinct along any loops that stay within
the 8D gauge theory region of the moduli space. It is, thus,
appropriate to identify that the two 7-branes in the
${\cal O}(\epsilon_\eta^{-1})$ region as the 7-branes $A8'$ and $D'$,
those in the ${\cal O}(\epsilon_\eta)$ region as $A8$ and $D$, and
those in the ${\cal O}(\epsilon_K^6 \epsilon_\eta)$ region as
$A6$ and $A7$, respectively.
Over the entire 8D gauge theory region, the positions
of the 7-branes A6, A7 are effectively given by the zero points of
\begin{equation}
\Delta'_{\mathrm{lower-quadr}} = \frac{27}{16} a_3^4
 + \left( \frac{27}{2} a_0 a_3^2 + 4 a_2^3 \right) z
 + 27 a_0^2 z^2 \, ,
\label{eq:discriminant_lqp}
\end{equation}
which is approximately the first three terms of the right hand side
of \eqref{eq:discriminant}.

The string junction configuration (and the corresponding 2-cycles)
$C_{A76}$ and $C_{-\theta}$ correspond to the two roots of
$\mathop{\rm SU}(3)_{\rm str}$, the commutant of $E_6$ within $E_8$.
As a point moves along a loop in (the 8D gauge theory region of) the
moduli space, the coefficients $a_{0,2,3}$ in (\ref{eq:discriminant_lqp})
change, and the positions of the $A7$ and $A6$ 7-branes move in the
complex $z$-plane. Consequently the branch cuts and the string junction
configurations also have to change continuously. Monodromy of the
two 2-cycles $C_{A76}$ and $C_{-\theta}$ should reproduce what we expect
from the 8D gauge theory descriptions.

Such loops in the moduli space should avoid a locus where more than
one 7-branes come on top of one another, because the elliptic K3 ``manifold''
becomes singular and it is subtle to talk of ``topological 2-cycles'' there.
The degeneration of the A6 and A7 7-brane positions are characterized
by the discriminant locus of the quadratic equation
$\Delta'_{\mathrm{lower-quadr}} = 0$:
\begin{equation}
\tilde \Delta_{\mathrm{lower-quadr}} =
 4 a_2^3 (27 a_0 a_3^2 + 4 a_2^3) = 0.
\label{eq:tilde_discriminant_lqp}
\end{equation}
As one moves on a loop around such a locus of degeneration of the 7-brane
positions, $\tilde{\Delta} = 0$, more than one 7-branes may exchange
their positions at the end of the loop. Moreover, branch cuts may have to be
rearranged in case of degeneration of mutually non-local $[p,q]$ 7-branes.
Thus, the locus of 7-brane degeneration is where monodromy of 2-cycles
could be generated, and hence we call $\tilde{\Delta} = 0$ locus as
the monodromy locus.

The monodromy locus in the 8D gauge theory region is given
by (\ref{eq:tilde_discriminant_lqp})
[see also section \ref{subsec:str_sing}].
Interestingly the second factor of \eqref{eq:tilde_discriminant_lqp} is
the same as the ramification locus of the spectral
surface \eqref{eq:rank3_spect}.
Since we expect a monodromy among 2-cycles along the ramification
locus of the spectral surface in the gauge theory description on
7+1 dimensions, it is more than natural that the same factor
appears in the monodromy locus that we introduced in the language
of $X_4$. The first factor of the monodromy locus, $a_2 = 0$ with
a multiplicity 3, on the other hand, does not appear in the
gauge theory description. We thus move on to calculate the monodromy
matrices explicitly to clarify the relation between the two
descriptions.

\subsubsection{Monodromy without factorization}
\label{sssec:wo-factorization}

We begin with a general choice of complex structure without a
factorization of (\ref{eq:rank3_spect}) in
section \ref{sssec:wo-factorization}. From the gauge theory
description on 7+1 dimensions, we expect that the monodromy of
the 2-cycles $C_{A76}$ and $C_{-\theta}$ is the full Weyl group of
$\mathop{\rm SU}(3)_{\rm str}$, $S_3$, and there is no mixing between
the 2-cycles in the visible $E_8$ and others in $H_2({\rm K3} ; \mathbb{Z})$.
Moreover, we expect that the monodromy is generated along
the ramification locus $(27 a_0 a_3^2 + 4 a_2^3) = 0$, but
nowhere else. These expectations are indeed confirmed by
explicit calculations of the monodromy, as we explain in
the following.

Let us pick a base point in the moduli space, and fix it once
and for all. All the loops start and end at the base point.
We choose it in the 8D gauge theory region:
\begin{equation}
(a_0,a_2,a_3) = (a_{0,0}, a_{2,0}, a_{3,0}) \equiv
 (1, \epsilon_K^2, i  \delta \epsilon_K^3) \epsilon_\eta,
\label{eq:reference_point}
\end{equation}
with real positive small numbers $\epsilon_K$ and $\epsilon_\eta$.
$\delta$ is also a small real positive number, which does not play
an important role in this article. We set $g_0 = g_{0,0} \equiv 1$
at the base point.
Arrangement of the branch cuts and the $[p,q]$ charges of the
7-branes are described explicitly in the appendix \ref{sec:calculation}
for this choice of the base point.

Although we could study monodromy of all the loops in the 8D gauge theory
region of the moduli space parametrized by
$(a_0, a_2, a_3, g_{0})$, it just introduces a mess that
is not essential to the problem.
Instead, let us take a slice of constant $a_0 = a_{0,0}$, $a_3 = a_{3,0}$ and
$g_{0} = g_{0,0}$ of the moduli space, and consider loops in the
complex $a_2$-plane.

On the $a_2$-plane, the monodromy locus
$\tilde \Delta_{\mathrm{lower-quadr}} = 0$
consists of one point $a_{2-A}$ with a multiplicity 3, and three points
$a_{2-1}$, $a_{2-2}$ and $a_{2-3}$ with multiplicity 1.
\begin{figure}[tb]
 \begin{center}
  \includegraphics[scale=0.3]{a2_gauge_l.eps}
  \caption{
The monodromy locus $\tilde \Delta_{\mathrm{lower-quadr}} = 0$
in (\ref{eq:tilde_discriminant_lqp}) on the $a_2$-plane and independent
loops around these points. $a_2 = a_{2-A}$ is a triple point and
correspond to the first factor $a_2 = 0$ of
$\tilde{\Delta}_{\mathrm{lower-quadr}}$. $a_2 = a_{2-1,2,3}$ are
single points, correspond to the second factor of
$\tilde{\Delta}_{\mathrm{lower-quadr}}$.
The base point is denoted by a cross mark.}
  \label{fig:a2-plane_e_K}
 \end{center}
\end{figure}
Figure \ref{fig:a2-plane_e_K} shows four independent non-trivial
loops in the $a_2$-plane. We calculate the monodromy for these
four loops. The calculation itself is straightforward, and we
just quote the results in this article; calculations for
$\gamma_{2-A}$ and $\gamma_{2-2}$ here and $\gamma_{0-4}$ that
appears later, however, are explained explicitly in the appendix
\ref{sec:calculation} in detail as samples.

It turns out that the monodromy associated with $\gamma_{2-A}$
is trivial.
This is consistent with the expectation that the monodromy is
generated only at the second factor of the monodromy
locus (\ref{eq:tilde_discriminant_lqp}), not at the $a_2 = 0$ factor.

Next we consider the loops $\gamma_{2-1}, \gamma_{2-2}$ and $\gamma_{2-3}$.
The monodromy associated with $\gamma_{2-2}$ is explicitly computed in
the appendix \ref{sec:calculation} to give the result
\begin{equation}
\begin{split}
\tilde C_{A65} &= C_{-\theta} + C_{A65} + C_{A76}\, , \\
\tilde C_{A76} &= - C_{-\theta} \, , \\
\tilde C_{-\theta} &= - C_{A76},
\end{split}
\label{eq:monod_2-2}
\end{equation}
while $C_{\alpha}^{1,2}$ and $C_{\beta}^{1,2}$ are left invariant.
Here $\tilde {C}_i$ denotes the transformed 2-cycle of $C_i$ after going
along $\gamma_{2-2}$.
Similarly, the monodromies associated with $\gamma_{2-1}$ and
$\gamma_{2-3}$ can be computed. The result for $\gamma_{2-1}$ is
\begin{equation}
\begin{split}
\tilde C_{A65} &= C_{A76} + C_{A65} \, , \\
\tilde C_{A76} &= - C_{A76} \, , \\
\tilde C_{-\theta} &= C_{A76} + C_{-\theta}
\end{split}
\label{eq:monod_2-1}
\end{equation}
and, for $\gamma_{2-3}$, the same monodromy as \eqref{eq:monod_2-2}
is obtained. Thus, the monodromy for these three loops act only on the
2-dimensional subspace of $H_2({\rm K3} \;; \mathbb{Z})$ generated by
$C_{A76}$ and $C_{-\theta}$, and are trivial
on all the six generators of the visible $E_6$, all the eight generators
of the hidden $E_8$ and all the four 2-cycles $C_{\alpha,\beta}^{1,2}$
in the middle. These monodromies were generated essentially around
the $(27 a_0 a_3^2 + 4 a_2^3) = 0$ locus.
This is also consistent with the expectation.

It is not difficult to see that these monodromy transformation
on the visible $E_8$ 2-cycles are regarded as Weyl reflections of
$\mathop{\rm SU}(3)_{\rm str} \subset E_8$. The monodromy $\rho(\gamma_{2-1})$
is a Weyl reflection generated by a root $C_{A76}$, $W_{C_{A76}}$,
and $\rho(\gamma_{2-2})$ and $\rho(\gamma_{2-3})$ are
$W_{C_{A76} + C_{-\theta}}$.
When they are represented on a three-element basis,
$(C_{A65}, C_{A65}+C_{A76}, C_{A65}+C_{A76}+C_{-\theta})$, they become
\begin{equation}
\rho_{\mathbf{3}} (\gamma_{2-1}) =
\begin{pmatrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{pmatrix}
\label{eq:monod_2-1_matrix}
\end{equation}
\begin{equation}
\rho_{\mathbf{3}} (\gamma_{2-2}) = \rho_{\mathbf{3}} (\gamma_{2-3}) =
\begin{pmatrix}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\end{pmatrix}.
\label{eq:monod_2-2_matrix}
\end{equation}
Thus, they generate the permutation group $S_3$, the full Weyl group
of $\mathop{\rm SU}(3)_{\rm str}$, just as expected in the gauge theory description
without a factorization of the spectral surface.

\subsubsection{Monodromy with factorization}
\label{sssec:w-factorization}

It is known that in the gauge theory description on 7+1 dimensions,
if we impose a factorization condition on a spectral surface,
then the structure group is reduced and an unbroken U(1) symmetry appears.
In the remainder of this subsection,
we will see that this fact can be described in terms of the
monodromies of 2-cycles as well.

On the spectral surface \eqref{eq:rank3_spect}, let us impose the
2+1 factorization condition \cite{TW-1, Caltech-0906}
\begin{equation}
 0 = a_0 \xi^3 + a_2 \xi + a_3 = (c_0 \xi^2 + c_1 \xi + c_2)(d_0 \xi + d_1)
\label{eq:2+1}
\end{equation}
for some sections $c_{0,1,2}$ and $d_{0,1}$ on $S_{GUT}$.
These sections have to satisfy a condition \cite{Caltech-0906}
\begin{equation}
 c_0 d_1 + c_1 d_0 = 0,
\label{eq:vanish-a1}
\end{equation}
and we adopt a solution to this condition, $d_0 = c_0$ and
$d_1 = - c_1$. Thus,
\begin{align}
a_0 &= c_0^2 \, , \label{eq:factorization_cond1} \\
a_2 &= c_0 c_2 - c_1^2 \, ,\label{eq:factorization_cond2} \\
a_3 &= - c_1 c_2 \, .
\label{eq:factorization_cond3}
\end{align}
We now consider a family of elliptic K3 manifold parametrized
by $(c_0, c_1, c_2, g_0)$, and study monodromy of 2-cycles
for loops in this moduli space.

The 8D gauge theory region in the $(c_0, c_1, c_2, g_0)$ space is
characterized by
\begin{equation}
 c_0 = c_{0,*} \epsilon^{1/2}_\eta, \quad
 c_1 = c_{1,*} \epsilon_K \epsilon^{1/2}_\eta, \quad
 c_2 = c_{2,*} \epsilon_K^2 \epsilon^{1/2}_\eta,
\label{eq:scaling-rg-c}
\end{equation}
where parameters $c_{r,*} \in \mathbb{C}$ ($r=0,1,2$) and $g_0 \in \mathbb{C}$ are
at most ${\cal O}(1)$. $\epsilon_K$ and $\epsilon_\eta$ are small
but non-zero parameters as before. We further require that
\begin{equation}
 \epsilon_K \ll |c_{0,*}| \sim {\cal O}(1).
\end{equation}
We will take a base point as
\begin{equation}
 (c_0, c_1, c_2) = (c_{0,0}, c_{1,0}, c_{2,0}) \equiv
  (1, -i \epsilon_K, \delta' \epsilon_K^2) \epsilon_{\eta}^{1/2},
\label{eq:reference_point_c}
\end{equation}
and $g_{0} = g_{0,0} = 1$; real positive values are used for small
numbers $\epsilon_K$, $\epsilon_\eta$ and $\delta'$, so that this
base point is mapped to the base point in the $(a_0, a_2, a_3, g_0)$
parameter space
through (\ref{eq:factorization_cond1}--\ref{eq:factorization_cond3}).

The monodromy locus in the 8D gauge theory region in the $(c_0,c_1,c_2, g_0)$
parameter space is given by simply rewriting (\ref{eq:tilde_discriminant_lqp})
by (\ref{eq:factorization_cond1}--\ref{eq:factorization_cond3}).
It factorizes as
\begin{equation}
\tilde \Delta_{\mathrm{lower-quad}} = 4
 (c_0 c_2 - c_1^2)^3 (4 c_0 c_2 - c_1^2) (c_0 c_2 + 2 c_1^2)^2 \, .
\label{eq:tilde_discriminant_lqp_factorize}
\end{equation}

If we focus on a constant $c_0$, $c_1$, $g_0$ slice (complex
$c_2$-plane) of the moduli space that contains the base point,
then the monodromy locus $\tilde{\Delta}_{\mathrm{lower-quar}} = 0$
consists of three points; see Figure~\ref{fig:c2-plane_e_K}.
\begin{figure}[tb]
 \begin{center}
  \includegraphics[scale=0.3]{c2_gauge_l.eps}
  \caption{The monodromy locus $\tilde \Delta_{\mathrm{lower-quad}} = 0$
on the $c_2$-plane and independent loops around these points.
The cross mark is the base point $c_2 = c_{2,0}$.}
  \label{fig:c2-plane_e_K}
 \end{center}
\end{figure}
One of them $c_{2-TR}$ is a triple point,
another $c_{2-SG}$ a single point, and the other
$c_{2-DB}$ a double point; they correspond to the zero locus of
the first, second and the last factor
of (\ref{eq:tilde_discriminant_lqp_factorize}), respectively.
Three independent non-trivial loops are contained in this slice
(Figure~\ref{fig:c2-plane_e_K}), and we study the monodromy of 2-cycles
for these loops.

Since the loops $\gamma_{2-TR}, \gamma_{2-SG}$ and $\gamma_{2-DB}$ are
mapped, respectively, to $\gamma_{2-A}, \gamma_{2-3}$ and
$(\gamma_{2-1})^2$ by
(\ref{eq:factorization_cond1}--\ref{eq:factorization_cond3}) topologically,
the monodromies are given by $\rho(\gamma_{2-TR}) = \rho(\gamma_{2-A}) =
{\rm id}.$,
$\rho(\gamma_{2-SG}) = \rho(\gamma_{2-3})$ and
$\rho(\gamma_{2-DB}) = (\rho(\gamma_{2-1}))^2 = {\rm id}.$.
Therefore, the full monodromy group of the 2-cycles for these loops
is generated only by
\begin{equation}
 \rho(\gamma_{2-SG}) = W_{C_{A76}+C_{-\theta}} \simeq \mathbb{Z}_2.
\label{eq:mod_gauge_fctr}
\end{equation}
The monodromy group is reduced from the full Weyl group $S_3$ of
$\mathop{\rm SU}(3)_{\rm str}$ to its $\mathbb{Z}_2$ subgroup, and the
2-cycle $C_{A76}$ is monodromy invariant.

The reduction of the monodromy group is a direct consequence of the
factorization limit (\ref{eq:2+1}); the $(27 a_0 a_3^2 + 4 a_2^3) = 0$
component of the monodromy locus factorized as
in (\ref{eq:tilde_discriminant_lqp_factorize}), and one of the
irreducible pieces become multiplicity two. That is essential in
making the monodromy $\rho(\gamma_{c_{2-DB}})$ trivial.
The remaining monodromy is generated essentially around
$(c_1^2 - 4 c_0 c_2) = 0$, which is precisely the ramification
locus of the 2-fold spectral cover in the 2+1 factorization limit.
All these results obtained in terms of monodromy of 2-cycles in
elliptic K3 manifold agree with the expectation from the gauge theory
description on 7+1 dimensions.

\subsection{The ``Full'' Monodromy Group}
\label{subsec:str_sing}

In the previous subsection, we focused on the $a_2$-plane in the
8D gauge theory region and reproduced the expected monodromy group
by looking directly at the monodromy of 2-cycles.
Thus, we can use the monodromy of the 2-cycles to study physics/geometry
that cannot be described precisely in the gauge theory description
on 7+1 dimensions.

As discussed in section \ref{sec:effect_beyond_gauge_theory},
the gauge theory description on 7+1 dimensions have two independent
difficulties. The problem A was that one has no choice but to drop
terms that either are higher order in the $z'$ coordinate expansion,
or have coefficients suppressed by $\epsilon_K$, in order
to fit the geometry into an $E_8$ gauge theory. This approximation
corresponds to using (\ref{eq:discriminant_lqp}) instead
of (\ref{eq:discriminant}) by dropping higher order terms.
Thus, this problem of the gauge theory description can be overcome
by simply repeating the same monodromy analysis
for (\ref{eq:discriminant}) by keeping higher order terms.

The other difficulty of the gauge theory description, the problem B,
was how to formulate a region near the $a_0 = 0$ locus in $S_{GUT}$.
In order to study the geometry of an elliptic K3 manifold near
the $a_0 = 0$ region of the moduli space, we only have to follow
loops that step into the $a_0 \simeq 0$ region and calculate the monodromy,
just like we did for loops that stay within the 8D gauge theory region.

In order to study the full monodromy in the scaling
region (\ref{eq:scaling-rg}) without staying strictly in the
8D gauge theory region (\ref{eq:gauge-th-rg}), it is no longer
possible to maintain only the lower quadratic terms from
(\ref{eq:discriminant}).
As $|a_{0,*}|$ becomes much smaller than 1 and comes closer
to $|\epsilon_K^2|$, one of the two 7-branes in the
$z \sim {\cal O}(\epsilon_K^6 \epsilon_\eta)$ region behaves
as
\begin{equation}
 z_i \simeq - \frac{4}{27}\frac{a_2^3}{a_0^2} \sim
  {\cal O}\left(\epsilon_\eta \epsilon_K^6 \right) \times
  \frac{1}{a_{0,*}^2}.
\label{eq:A7-behavior}
\end{equation}
This behavior in terms of the discriminant locus corresponds
to (\ref{eq:shoot-spec-surf}) in terms of the spectral surface.
At the same time, the two 7-branes in the
$z \sim {\cal O}(\epsilon_\eta)$ region move as
$z_i \sim a_0 = \epsilon_\eta a_{0,*}$.
Thus, the three 7-branes come close to one another when
$a_{0,*}$ is as small as $\epsilon_K^2$. Those 7-branes are
no longer clearly separated into the groups of [A6 + A7] and
[A8 + D]. We should use at least quartic polynomial part of
(\ref{eq:discriminant}).

Monodromy of 2-cycles are (potentially) generated only around a locus
in the moduli space where more than one 7-branes come on top of
the other(s). Thus, the monodromy locus on the moduli space is
characterized by the discriminant locus $\tilde{\Delta} = 0$ of
a polynomial $\Delta'(z) = 0$ in (\ref{eq:discriminant}) that
determines the positions of the 7-branes.
$\tilde{\Delta} = 0$ defines a divisor in the moduli space.
Monodromy of 2-cycles should be calculated for loops on the moduli
space that stay away from the monodromy locus. Therefore, the
full monodromy group is a representation of the fundamental group
of the moduli space without the monodromy divisor.
There is no essential difficulty in approaching the
$a_0 \simeq 0$ region, or incorporating higher order terms of
the $\Delta'(z)$ polynomial.

As long as we stay within the scaling region (\ref{eq:scaling-rg})
with an $a''_0 = \epsilon_\eta \neq 0$ gauge, however, the problem
becomes a little easier. The two 7-branes [A8$'$ + D$'$] stay within
the $z \sim {\cal O}(\epsilon_\eta^{-1})$ region, and are frozen out
there;  we thus only need to follow the behavior of the four other 7-branes.
We can now use
\begin{equation}
\begin{split}
\Delta'_{\eta} &= \frac{27}{16} a_3^4
 + \left( \frac{27}{2} a_0 a_3^2 + 4 a_2^3 \right) z
 + \left( \frac{27}{2} a_3^2 g_0 + 27 a_0^2 + 12 a_2^2 f_0 \right) z^2 \\
 & \quad + (54 a_0 g_0 + 12 a_2 f_0^2) z^3
  + (4 f_0^3 + 27 g_0^2) z^4 \,
\end{split}
\label{eq:discriminant_four}
\end{equation}
instead of (\ref{eq:discriminant}) or (\ref{eq:discriminant_lqp}).

The monodromy divisor in the moduli space is the discriminant
locus of $\Delta'_\eta(z) = 0$:
\begin{equation}
\tilde \Delta_{\eta} = - 19683 A^3 B = 0,
\label{eq:tilde_discriminant}
\end{equation}
where $A$ and $B$ are given by
\begin{align}
A &= 4 a_0 a_2 f_0 - a_3^2 f_0^2 - 4 a_2^2 g_0 \: , \label{eq:A} \\
B &= 4 a_0^3 a_2^3 + 27 a_0^4 a_3^2 + 4 a_0 a_2^5 f_0
 + 30 a_0^2 a_2^2 a_3^2 f_0 - a_2^4 a_3^2 f_0^2 - 24 a_0 a_2 a_3^4 f_0^2 \notag
 \\
  & \quad  + 4 a_3^6 f_0^3 - 4 a_2^6 g_0 - 36 a_0 a_2^3 a_3^2 g_0
   - 54 a_0^2 a_3^4 g_0 + 18 a_2^2 a_3^4 f_0 g_0 + 27 a_3^6 g_0^2
   \: . \label{eq:B}
\end{align}
$\tilde{\Delta}_{\mathrm{lower-quadr}}$ in (\ref{eq:tilde_discriminant_lqp})
is reproduced\footnote{
The triple point $a_{2-A}$ and the three points $a_{2-1,2,3}$
on the $a_2$-plane are obtained as the roots of $A = 0$ and $B=0$;
all of $a_0$, $a_3$ and $g_0$ are treated as fixed numbers here.
Although $B$ is an order-six polynomial of $a_2$, the remaining three
roots are not in the scaling region $a_{2,*} \lesssim {\cal O}(1)$.}
by keeping only the leading order terms in $\epsilon_K$
in \eqref{eq:A} and \eqref{eq:B}:
\begin{align}
A_{\mathrm{leading}} &= 4 a_0 a_2 f_0 \, , \\
B_{\mathrm{leading}} &= a_0^3 (4 a_2^3 + 27 a_0 a_3^2) \label{eq:B_leading} \,.
\end{align}

In order to see study the monodromy associated with
the $a_0 \simeq 0$ region, it is useful to take a slice
of the moduli space at constant $a_2$, $a_3$ and $g_0$ with
a free $a_0$, and look at the distribution of the monodromy
divisor in the complex $a_0$ plane.
We take a slice that contains the base point (\ref{eq:reference_point}).
The $B = 0$ component of the monodromy locus appears in the
$a_0$ plane as in Figure~ \ref{fig:a0-plane_e_K}.
\begin{figure}[tb]
 \begin{center}
\begin{tabular}{cc}
  \includegraphics[scale=0.25]{a0_gauge_l.eps} &
  \includegraphics[scale=0.25]{a0_loops_l.eps} \\
 (a) & (b)
\end{tabular}
  \caption{The $B=0$ component of the monodromy locus consists of four points
on the $a_0$-plane containing the base point (cross mark). Among them,
only $a_{0-out}$ lies in the 8D gauge theory region
$|\epsilon_K^2 \epsilon_\eta| \ll |a_0|$,
and all others $a_{0-4,5,6} \sim {\cal O}(\epsilon_K^2 \epsilon_\eta)$ are
located outside of the 8D gauge theory region (in a shaded region in (a)).
}
\label{fig:a0-plane_e_K}
 \end{center}
\end{figure}
There is only one monodromy locus $a_{0-out}$ in the 8D gauge theory region
$|\epsilon_K^2 \epsilon_\eta| \ll |a_0|$, but three new types of
monodromy loci appear in the $a_0 \sim {\cal O}(\epsilon_K^2 \epsilon_\eta)$
region.
We can think of various loops drawn in Figure~\ref{fig:a0-plane_e_K};
it is clearly a question of interest what the monodromy of 2-cycles
will be for those loops.

Note that we do not have to consider the $A=0$ component of the
monodromy locus any more. In section \ref{subsec:mod_gauge}, we see
that the monodromy of the loop $\gamma_{2-A}$ is trivial.
For any loops $\gamma_A$ in the moduli space (except $A=0$ and $B=0$)
that are homotopic to a loop of the form
\begin{equation}
 	\gamma_{A}=\gamma_{B}^{-1}\circ\gamma_{2-A}\circ\gamma_{B}
\end{equation}
for some loop $\gamma_B$, $\rho(\gamma_A) = {\rm id}.$, because
$\rho(\gamma_{2-A}) = {\rm id}.$.
Thus, we focus only on the monodromy of the loops which go around the
$B=0$ component (\ref{eq:B}) in the following.

Among the loops in the $a_0$ plane in Figure~\ref{fig:a0-plane_e_K},
$\gamma_{0-out}$ and $\gamma_{0-\theta}$ stay within the 8D gauge theory
region of the moduli space. Direct computation of the monodromy
$\rho(\gamma_{0-out})$ shows that
\begin{equation}
 \rho(\gamma_{0-out}) = W_{C_{A76}} = \rho(\gamma_{2-1}).
\label{eq:0-out}
\end{equation}
This is actually expected, because the loop $\gamma_{0-out}$ is
homotopic to $\gamma_{2-1}$ in the $(a_0, a_2, a_3, g_0)$ moduli space
without the $B = 0$ monodromy locus. One can also see that
$\gamma_{2-2} \sim \gamma_{0-\theta} \circ \gamma_{0-out} \circ
\gamma_{0-\theta}^{-1}$ and
$\gamma_{2-3} \sim \gamma_{0-\theta}^{-1} \circ \gamma_{0-out} \circ
\gamma_{0-\theta}$, and hence the following relations
\begin{eqnarray}
  \rho(\gamma_{0-\theta})\rho(\gamma_{0-out})\rho(\gamma_{0-\theta})^{-1}
 &=& \rho(\gamma_{2-2}) = W_{C_{A76}+C_{-\theta}}, \label{eq:0-theta1}\\
  \rho(\gamma_{0-\theta})^{-1}\rho(\gamma_{0-out})\rho(\gamma_{0-\theta})
 &=& \rho(\gamma_{2-3}) = W_{C_{A76}+C_{-\theta}}  \label{eq:0-theta2}
\end{eqnarray}
should hold true. We confirmed these relations by explicit computation
of the monodromy\footnote{\label{fn:0-theta}
The monodromy matrix $\rho(\gamma_{0-\theta})$
splits into a 2 by 2 block on {\rm Span}$_{\mathbb{Z}} \{C_{A76}, C_{-\theta}\}$
and a 4 by 4 block ${\rm Span}_{\mathbb{Z}} \{ C_{\alpha, \beta}^{1,2} \}$.
It is a Weyl reflection $W_{C_{-\theta}}$ in the former. In the latter,
$\tilde{C}_{\alpha, \beta}^{1} = C_{\alpha, \beta}^1$,
$\tilde{C}_\alpha^2 = C_\alpha^2 - C_\beta^1$ and
$\tilde{C}_\beta^2 = C_\beta^2 + C_\alpha^1$. The monodromy still splits
between the visible $E_8$ sector and others for this loop
$\gamma_{0-\theta}$.}
$\rho(\gamma_{0-\theta})$.
Other loops in Figure~\ref{fig:a0-plane_e_K} in the $a_0$ plane,
which go away from the 8D gauge theory region of the moduli space,
are not homotopic at least apparently to the loops whose monodromy
we have already calculated.

The full monodromy group of the 2-cycles is a representation of the
fundamental group of the $(a_0, a_2, a_3, g_0)$ moduli space from which
the $B=0$ monodromy locus is deleted, and this is what we are interested
in. The monodromy group observed in the gauge theory description on
7+1 dimensions, however, correspond to the representation of a subgroup
generated only by loops that stay within the 8D gauge theory region
of the moduli space. The monodromy group on the 2-cycles split into direct
product of the one on ${\rm Span}_{\mathbb{Z}} \{C_{A76}, C_{-\theta}\}$ and
the one on ${\rm Span}_{\mathbb{Z}} \{ C_{\alpha, \beta}^{1,2} \}$ at this
level of analysis.
The loops such as $\gamma_{0-4,5,6}$ in Figure~\ref{fig:a0-plane_e_K},
however, may not be contained in this subgroup, and in general, the
full monodromy group is larger than the expectation from the gauge
theory description on 7+1 dimensions.
Exactly the same thing can be said about the monodromy group
of 2-cycles of a family of elliptic K3 manifold parametrized by
$(c_0, c_1, c_2, g_0)$.

\subsection{The monodromy beyond the 8D gauge theory region}
\label{subsec:monod_beyond_gauge}

We first show that the full monodromy group of the family
(\ref{eq:model}) with $(a_0, a_2, a_3, g_0)$ moduli space is
indeed larger than the monodromy group expected from the gauge
theory description on 7+1 dimensions. This is done by
calculating the monodromy of the loops $\gamma_{0-4,5,6}$ in
Figure~\ref{fig:a0-plane_e_K}; this is to probe physics and geometry
of a region of small $a_0$, where the $E_8$ gauge theory description on
7+1 dimensions breaks down. We then move on to study the monodromy group
of the family for the factorization limit parametrized
by $(c_0, c_1, c_2, g_0)$.

The monodromy for the loops $\gamma_{0,4,5,6}$ can be computed by
the same method as in the preceding sections; the calculation for
the loop $\gamma_{0-4}$ is demonstrated explicitly in the
appendix \ref{sec:calculation}. It turns out that the
monodromy of $\gamma_{0-4}$ is
\begin{eqnarray}
\tilde{C}_{A65}&=&C_{A65},\;\;\;\;\;\; \tilde{C}_{A76}=C_{A76}+C_{-\theta}-C_{\alpha}^{1}, \nonumber \\
\tilde{C}_{\alpha}^{1}&=&C_{\alpha}^{1},\;\;\;\;\;\; \tilde{C}_{\alpha}^{2}=C_{\alpha}^{2}+C_{\alpha}^{1}-C_{-\theta}, \nonumber \\
\tilde{C}_{\beta}^{1}&=&C_{\beta}^{1},\;\;\;\;\;\; \tilde{C}_{\beta}^{2}=C_{\beta}^{2},
	\label{eq:0-4}
\end{eqnarray}
or equivalently,
\begin{equation}
 \rho(\gamma_{0-4}) =
  \left(
\begin{array}{cc|cc|c}
 1 &    &   &    & \\
 1 & -1 &   & -1 & \\
\hline
-1 &  2 & 1 &  1 & \\
   &    &   &  1 & \\
\hline
   &    &   &    & {\bf 1}_{2 \times 2}
\end{array}
  \right) \label{eq:0-4-mtrx}
\end{equation}
when we choose the basis as $(C_{A76}, C_{-\theta}, C_\alpha^1,
C_\alpha^2, C_\beta^1, C_\beta^2)$. The monodromy matrix
$\rho(\gamma_{0-4})$ is trivial on the 2-cycles in the visible $E_6$ and
hidden $E_8$. Similarly, the monodromy for $\gamma_{0-5}$ is given by
	\begin{eqnarray}
	\tilde{C}_{A65}&=&C_{A65},\;\;\;\;\;\;\tilde{C}_{A76}=C_{A76}+C_{-\theta}-C_{\alpha}^{1}+C_{\beta}^{1},\nonumber \\
	\tilde{C}_{\alpha}^{1}&=&C_{\alpha}^{1},\;\;\;\;\;\; \tilde{C}_{\alpha}^{2}=C_{\alpha}^{2}+C_{\alpha}^{1}-C_{\beta}^{1}-C_{-\theta},\nonumber \\
	\tilde{C}_{\beta}^{1}&=&C_{\beta}^{1},\;\;\;\;\;\; \tilde{C}_{\beta}^{2}=C_{\beta}^{2}-C_{\alpha}^{1}+C_{\beta}^{1}+C_{-\theta},
	\label{eq:0-5}
	\end{eqnarray}
and, finally, the one for $\gamma_{0-6}$ is
	\begin{eqnarray}
	\tilde{C}_{A65}&=&C_{A65},\;\;\;\;\;\;\tilde{C}_{A76}=C_{A76}+C_{-\theta}-C_{\beta}^{1},\nonumber\\
	\tilde{C}_{\alpha}^{1}&=&C_{\alpha}^{1},\;\;\;\;\;\; \tilde{C}_{\alpha}^{2}=C_{\alpha}^{2}, \nonumber \\
	\tilde{C}_{\beta}^{1}&=&C_{\beta}^{1},\;\;\;\;\;\; \tilde{C}_{\beta}^{2}=C_{\beta}^{2}+C_{\beta}^{1}-C_{-\theta}.
	\label{eq:0-6}
	\end{eqnarray}
The 2-cycles inside the $E_{8}$ root lattice and
$C_{\alpha,\beta}^{1,2}$
are mixed up under the monodromy $\rho(\gamma_{0-4,5,6})$.
This clearly shows that the loops $\gamma_{0-4,5,6}$ are not
homotopic to the loops that stay within the 8D gauge theory region
of the moduli space. More importantly, $\rho(\gamma_{0-4,5,6})$
should be added to the list of generators of the full monodromy
group; the full monodromy group is no longer the product of the
$\mathop{\rm SU}(3)_{\rm str}$ Weyl group $S_3$ on the visible $E_8$ and something
acting on the four 2-cycles $C_{\alpha,\beta}^{1,2}$, but the full
monodromy group is larger\footnote{
The only non-vanishing entries in the 4th and 6th rows of the
monodromy matrices (\ref{eq:0-4}--\ref{eq:0-6}) are in the
4th and 6th column, respectively, and are all ``$1$''.
This is an artifact of restricting the moduli space to
the scaling region with $|\epsilon_\eta| \ll 1$.
The period integrals over $C_{\alpha, \beta}^2$ are large and
those for others small for $|\epsilon_\eta| \ll 1$. This is why
$C_{\alpha, \beta}^{2}$ cannot mix into other 2-cycles.
See also the appendix \ref{sec:Het}.
When one considers the full monodromy group for a family without
the restriction of $|\epsilon_\eta| \ll 1$, new generators will
appear, and we expect that this special feature will disappear. }
than that.

The subject of our real interest is the monodromy group of 2-cycles
at the factorization limit (\ref{eq:2+1}--\ref{eq:factorization_cond3}).
Although we have seen that the monodromy subgroup generated by loops
in the constant $(c_0, c_1, g_0)$ slice in the 8D gauge theory region
is the $\mathbb{Z}_2$ subgroup of the Weyl group $S_3$, there may be other
generators in the full monodromy group, and the full monodromy group
may be larger than $\mathbb{Z}_2$. To find out candidates for such a generator,
let us take a constant $(c_1, c_2, g_0)$ slice of the moduli space and
look at the complex $c_0$ plane.

The monodromy locus $B = 0$ in (\ref{eq:B}) should now be rewritten in
terms of $c_0, c_1, c_2$. At the leading order in $\epsilon_K$, it
becomes
\begin{equation}
 B_{{\rm leading}} = (c_0)^6 (4 c_0 c_2 - c_1^2)(c_0 c_2 + c_1^2)^2,
\end{equation}
and hence it appears as in Figure~\ref{fig:c0-plane_e_K}~(a).
\begin{figure}[tb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=0.3]{c0_gauge_l.eps} &
\includegraphics[scale=0.3]{c0_l.eps} \\
(a) & (b)
\end{tabular}
\caption{The monodromy locus appearing in the constant $(c_1, c_2, g_0)$
 slice of the moduli space, and non-trivial loops that go around them.
We have chosen a slice that contains the base
point (\ref{eq:reference_point_c}).
The panel (a) is a coarse picture based on $B_{\rm leading} = 0$, where
higher order corrections in $\epsilon_K$ are ignored. The three loops,
$\gamma_{0-DB}$, $\gamma_{0-SG}$ and $\gamma_{0-0}$ stay within the
8D gauge theory region, away from the shaded region
$|c_{0,*}| \lesssim \epsilon_K$. The panel (b) is a fine picture where
higher order terms in $\epsilon_K$ are not dropped from $B = 0$.
The $c_0 = 0$ monodromy point with the multiplicity 6
in $B_{\rm leading} = 0$ (gray blob in the panel (a)) now splits
into 6 distinct points; we can thus find six loops
$\gamma_{0-4,5,6}^{1,2}$ in moduli space. The $c_{0-DB}$ monodromy point
with multiplicity 2 in the panel (a) also splits into two points
$c_{0-DB}^{1,2}$, and two loops $\gamma_{0-DB}^{1,2}$ are found.
}
\label{fig:c0-plane_e_K}
\end{center}
\end{figure}
However, it was our motivation to study the geometry directly instead of
gauge theory descriptions, to take account of higher order
corrections in $\epsilon_K$, and also to study the region with small
$c_0$. With all the higher order terms of $B = 0$ maintained,
Figure~\ref{fig:c0-plane_e_K}~(b) is the precise picture of the
monodromy locus appearing in the $c_0$ plane.
It is important to note that the double point $c_{0-DB}$ in
Figure~\ref{fig:c0-plane_e_K}~(a) splits into two points,
$c_{0-DB}^{1,2}$ in Figure~\ref{fig:c0-plane_e_K}~(b), due
to the higher order terms in $\epsilon_K$ in $B = 0$.
The split in the $c_0$ plane is approximately
\begin{equation}
	\Delta c_{0-DB}^{1,2}=\pm \frac{\sqrt{3f}}{2} c_{2} \sim
	 \mathcal{O}(\epsilon_{K}^{2}\epsilon_{\eta}^{\frac{1}{2}}),
\end{equation}
which shows clearly that this is a higher order effect in $\epsilon_K$.
The split is certainly small, but still non-zero.
Thus, the loops $\gamma^{1,2}_{0-DB}$ that go around either one of
them can be separately non-trivial topologically in the moduli space;
they may even become extra generators of the monodromy group.
Each one of the $a_{0}=a_{0-4,5,6}$ monodromy locus points in the
$a_0$ plane (Figure~\ref{fig:a0-plane_e_K}) splits into a pair
$c_{0-4}^{1,2}, c_{0-5}^{1,2}, c_{0-6}^{1,2} \sim
\mathcal{O}(\epsilon_{K}\epsilon_{\eta}^{\frac{1}{2}})$
in the $c_0$ plane, because of the factorization condition
(\ref{eq:factorization_cond1}).
There are six loops $\gamma_{0-4,5,6}^{1,2}$ that go around them
in the $c_0$ plane, and they might also generate extra monodromy
of the 2-cycles.

We found by numerical study that the motion of 7-branes in the $z$ plane
along the loops $\gamma_{0-4,5,6}^{1,2}$ in the $c_{0}$-plane is
topologically the same\footnote{This is easily guessed by the
map (\ref{eq:factorization_cond1}--\ref{eq:factorization_cond3}).}
as those along certain combinations of the loops $\gamma_{0-4,5,6}$
in the $a_{0}$-plane. Thus, the monodromy of the loops
$\gamma^{1,2}_{0-4,5,6}$ in the $c_0$ plane is given by
	\begin{eqnarray}
	\rho(\gamma_{0-4}^{1})&=&\rho(\gamma_{0-4}), \;\;\;\;\;\; \rho(\gamma_{0-4}^{2})=\rho(\gamma_{0-4}), \nonumber\\
	\rho(\gamma_{0-5}^{1})&=&\rho(\gamma_{0-5}), \;\;\;\;\;\; \rho(\gamma_{0-5}^{2})=\rho(\gamma_{0-4})\rho(\gamma_{0-5})\rho(\gamma_{0-4})^{-1}, \nonumber\\
	\rho(\gamma_{0-6}^{1})&=&\rho(\gamma_{0-6}), \;\;\;\;\;\; \rho(\gamma_{0-6}^{2})=\rho(\gamma_{0-4})^{-1}\rho(\gamma_{0-6})\rho(\gamma_{0-4}).
	\label{eq:mod_full}
	\end{eqnarray}
This clearly shows that the loops digging into the $c_0 \simeq 0$
region give rise to monodromy of 2-cycles that have not been observed
in the gauge theory descriptions on 7+1 dimensions;
2-cycles within the visible $E_8$ and those that are not are mixed up
under the monodromy (\ref{eq:mod_full}).

An U(1) symmetry that survives all these monodromies correspond to
a six-component row vector that remain invariant after multiplying
any one of these monodromy matrices (\ref{eq:mod_gauge_fctr},
\ref{eq:mod_full}) from the right. There is none.
We therefore conclude that the U(1) symmetry remaining in the
S[U(2)$\times$U(1)] Higgs bundle compactification is broken in the
full geometry of F-theory compactification that has a region with
$c_0 = 0$ on $S_{GUT}$. The problem B is indeed a problem.

To determine the monodromy associated with the loops
$\gamma_{0-DB}^{1,2}$, one only needs to note that the motion of
7-branes in the $z$ plane along these loops
are topologically the same as those along the loop $\gamma_{0-out}$
in the $a_{0}$-plane. Thus,
\begin{equation}
	\rho(\gamma_{0-DB}^{1,2})=\rho(\gamma_{0-out})=W_{C_{A76}}.
	\label{eq:0-DB_1,2}
\end{equation}
These loops generate monodromy of 2-cycles in the visible $E_8$;
combining this new generator and the one (\ref{eq:mod_gauge_fctr})
that we already know in the gauge theory description on 7+1 dimensions,
the whole $S_3$ Weyl group is generated.
We have thought that the monodromy of 2-cycles is reduced from
$S_3$ to $\mathbb{Z}_2$ in the factorization limit of the spectral surface,
but because of the $\epsilon_K$-suppressed higher order terms that
were simply ignored in the 8D gauge theory description, actually
the monodromy is not reduced from $S_3$.
We conclude that there is no unbroken U(1) symmetry with non-trivial
components in the visible $E_8$ that survives the monodromy generated
by both (\ref{eq:mod_gauge_fctr}) and (\ref{eq:0-DB_1,2}).
The problem A is a problem indeed.

To summarize,\footnote{
It is useful for sanity check to exploit the relations following
from the homotopy equivalence of various loops in the moduli space,
as we did in the $(a_0, a_2)$ moduli space in section \ref{subsec:str_sing}.
In the $(c_0, c_2)$ moduli space, we have relations
\begin{eqnarray}
& &
\gamma_{2-DB} \sim \gamma_{0-DB} \sim \gamma_{0-DB}^1 \circ
\gamma_{0-DB}^2, \label{eq:hom-1} \\
& &
\gamma_{2-SG} \sim \gamma_{0-SG}, \label{eq:hom-2} \\
& &
\gamma_{0-6} \circ \gamma_{0-4} \circ \gamma_{0-5} \sim \gamma_{0-\theta},
 \label{eq:hom-3}\\
& &
\gamma_{0-6}^1 \circ \gamma_{0-5}^2 \circ \gamma_{0-4}^1 \circ
\gamma_{0-4}^2 \circ \gamma_{0-6}^2 \circ \gamma_{0-5}^1 \sim
\gamma_{0-0}. \label{eq:hom-4}
\end{eqnarray}
Our results in section \ref{subsec:mod_gauge} and (\ref{eq:0-DB_1,2})
both consistently yield $[W_{C_{A76}}]^2 = [\rho(\gamma_{2-1})]^2$
in (\ref{eq:hom-1}).
Direct computation of $\rho(\gamma_{0-SG})$ indeed turned out to be
the same as $\rho_{2-SG}$ in (\ref{eq:mod_gauge_fctr}).
The homotopy relation (\ref{eq:hom-3}) in the monodromy matrices
is confirmed directly by using the results (\ref{eq:0-4}--\ref{eq:0-6})
and that in footnote \ref{fn:0-theta}.
Finally, the product of (\ref{eq:mod_full}) in the order specified in
the left-hand side of (\ref{eq:hom-4}) becomes
$[\rho(\gamma_{0-6})\rho(\gamma_{0-4})\rho(\gamma_{0-5})]^2$, which
is equal to $[\rho(\gamma_{0-\theta})]^2$, because of (\ref{eq:hom-3}).
This should be the same as $\rho(\gamma_{0-0})$, because
the loop $\gamma_{0-0}$ in the $c_0$ plane is mapped to
$\gamma_{0-\theta} \circ \gamma_{0-\theta}$ in the $a_0$ plane
through (\ref{eq:factorization_cond1}).
All these consistency checks as a whole provides confidence in the results
of our calculation. }
the full monodromy group contains at least
(\ref{eq:mod_gauge_fctr}),
(\ref{eq:mod_full}) and (\ref{eq:0-DB_1,2}) as generators, maybe more,
when the spectral surface is in the factorization limit
(\ref{eq:factorization_cond1}--\ref{eq:factorization_cond3}).
There is no unbroken U(1) symmetry with non-trivial components
in the visible $E_8$ under the full monodromy group.
Thus, we cannot expect an unbroken U(1) symmetry in the low-energy
effective theory in ruling out the dimension-4 proton decay operators.

\subsection{Non-K3-fibred 4-fold $X_4$}
\label{subsec:non-K3}

We have so far
used a Calabi--Yau 4-fold $X_4$ that is a K3-fibration on $S_{GUT}$.
However, this is just for a concreteness. The monodromy analysis
certainly needs to be phrased for individual cases for $X_4$'s that
are not K3-fibration over $S_{GUT}$. But the essence of the problem A
and B, and the essence of analysis remain the same.
Let us take simplest non-K3 fibered models as examples:
$B_3 = \mathbb{P}^3$, and $S_{GUT}$ is a quadratic ($d = 2$) or cubic ($d=3$)
surface of $B_3$, the models discussed in \cite{BHV-2}.
First, $A_k$ in (\ref{eq:local-defeq}) are set to be \cite{Caltech-0904}
\begin{equation}
A_k = s^{k-1} \tilde{A}_k, \qquad
\tilde{A}_{k} \in
\Gamma(B_3; {\cal O}_{B_3}( - k K_{B_3}-(k-1)S_{GUT})),
\label{eq:global-choice-Ak}
\end{equation}
where $s$ is a global holomorphic section of
${\cal O}_{B_3}(S_{GUT}) \simeq {\cal O}_{\mathbb{P}^3}(d H)$ whose
zero locus is the GUT divisor $S_{GUT}$. $a_{6-k}$ on $S_{GUT}$ in
(\ref{eq:Ak-expnsn}) correspond to $\tilde{A}_k|_{S_{GUT}}$.
Suppose that a point $p_0 \equiv [0:0:0:1] \in \mathbb{P}^3$ is not contained in
$S_{GUT}$. The base manifold except this point, $\mathbb{P}^3 \backslash p_0$
is covered by three patches $\simeq \mathbb{C}^3$, which constitute
${\cal O}_{\mathbb{P}^2}(1)$. Restriction of the original elliptic fibration
over $B_3 = \mathbb{P}^3$ to that over ${\cal O}_{\mathbb{P}^2}(1)$, combined with
a projection $\pi_0: {\cal O}_{\mathbb{P}^2}(1) \rightarrow \mathbb{P}^2$ defines
a complex surface fibration
\begin{equation}
\pi_0 \circ \pi_X: \pi^{-1}_X \left( {\cal O}_{\mathbb{P}^2}(1) \right)
  \rightarrow \mathbb{P}^2.
\label{eq:nonK3-fib}
\end{equation}
The fiber of this map is an elliptic fibration over $\mathbb{C}$, with 48
discriminant points on the $\mathbb{C}$ plane. Although the number of
$[p, q]$ 7-branes is not the same as in the case of an elliptic K3
manifold, independent topological 2-cycles and their intersection form
of the fiber complex surface can be worked out by using the techniques
in \cite{jcn-intersection, jcn-KcMd}.
Monodromy can be studied for loops\footnote{Since
$\pi_0|_{S_{GUT}}: S_{GUT} \rightarrow \mathbb{P}^2$ is a $d$-fold covering,
one can pull-back the complex-surface fibration (\ref{eq:nonK3-fib}) from
$\mathbb{P}^2$ to $S_{GUT}$. The monodromy analysis can then be phrased for
loops in $S_{GUT}$.} in $\mathbb{P}^2$.
When $d \neq 1$, other points $p_i$'s in $B_3 = \mathbb{P}^3$ should also
be chosen so that the analysis for
$\pi_i: \mathbb{P}^3 \backslash p_i \simeq {\cal O}_{\mathbb{P}^2} (1) \rightarrow \mathbb{P}^2$
is carried out and all the ramification locus of the projection
$\pi_i|_{S_{GUT}}: S_{GUT} \rightarrow \mathbb{P}^2$ is covered by some of the
analysis of $\mathbb{P}^3 \backslash p_i$.
Generalization from these examples to, e.g., toric $B_3$ (c.f. \cite{Yau}),
is straightforward.

The problem A arises from the difference between
$a_{6-k}=\tilde{A}_k|_{S_{GUT}}$ and $\tilde{A}_k$, and the problem B arises
essentially because of the local behavior of 2-cycles wherever $a_0$
vanishes. Thus, we expect similar phenomena also in the case of
$B_3 = \mathbb{P}^3$, though we have not done the analysis. Since more than
one points of $S_{GUT}$ are projected onto the same point in $\mathbb{P}^2$ for
$d \neq 1$, 2-cycles in $E_8$ on a point of $S_{GUT}$ may mix with 2-cycles
in $E_8$ on another point of $S_{GUT}$ in the context of problem B.

\section{Consequences in Physics}
\label{sec:physics}

\subsection{U(1) Violating State-Mixing and Trilinear Couplings}
\label{subsec:M2brane}

We have studied a model of $E_8 \rightarrow E_6$ symmetry breaking with
a spectral surface in the 2+1 factorization limit in the previous
section, as a toy model of $E_8 \rightarrow {\rm SU}(5)_{\rm GUT}$
symmetry breaking in a certain factorization limit. On the contrary
to the expectation in gauge theory in 7+1 dimensions, the study shows
that there is no monodromy-invariant 2-cycle, and hence there is no
unbroken U(1) symmetry in the low-energy effective theory below
the Kaluza--Klein scale.
Without an unbroken U(1) symmetry, we generally expect that dimension-4
proton decay operators will be generated. In this section, we discuss
whether the dimension-4 proton decay operators are really generated from
known interactions in string theory.

In F-theory, charged matter fields come from the degree of freedom of
M2-branes wrapping on the 2-cycles corresponding to the representation
of charged matters.
Trilinear interactions are (likely to be) generated, if the sum of 2-cycles for
the three fields is topologically trivial.
Thus, by using this criterion, we can check whether dimension-4 proton
decay operators are generated or not.

In the $E_8 \rightarrow E_6$ symmetry breaking model with the 2+1
factorization, the matter curve for $E_6$-{\bf 27} representation
splits into two irreducible pieces; one is characterized by
$c_2 = 0$, and the other by $d_1 = - c_1 = 0$. Based on the gauge theory
description on $S_{GUT}$, one would expect that the charged matter
fields in the $E_6 \times {\rm U}(1)$-${\bf 27}_{+1}$ representation
are localized
along the $c_2 = 0$ curve in $S_{GUT}$, and those in the ${\bf 27}_{-2}$
along the $c_1 = 0$ curve. Ignoring the monodromy that we studied in
section \ref{subsec:monod_beyond_gauge}, the 2-cycle $(C_{A76}+C_{A65})$
vanishes along the $c_1 = 0$ curve, while either $C_{A65}$ or
$(C_{-\theta}+C_{A76}+C_{A65})$ does along $c_2 = 0$.
$4 c_0 c_2 - c_1^2 = 0$ is the ramification locus of the 2-fold
spectral cover, where monodromy $W_{C_{A76}+C_{-\theta}}$ exchanges $C_{A65}$ and
$(C_{-\theta}+C_{A76}+C_{A65})$. See Figure~\ref{fig:S-GUT-E6}.
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.3]{S_GUT_l.eps}
  \caption{\label{fig:S-GUT-E6}A schematic picture of matter curves
and monodromy loci on $S_{GUT}$. The ramification locus
$c_1^2 - 4 c_2 c_0 \simeq 0$ and a locus $(2c_1^2 + c_2 c_0) \simeq 0$
with multiplicity 2 both approach the points of enhanced $E_8$
singularity. The multiplicity-2 branch, however, has a small split, and
consists of two branches with multiplicity 1.
Furthermore, all the three branches form a single irreducible component
monodromy locus $B=0$ around $(c_1, c_2) \simeq (0,0)$. The monodromy
locus $B = 0$ also intersect the matter curves at $(c_1,c_0) \simeq (0,0)$
and $(c_2,c_0) \simeq (0,0)$. }
\end{center}
\end{figure}
The trilinear interaction
$\Delta W = {\bf 27}_{+1} \cdot {\bf 27}_{+1} \cdot {\bf 27}_{-2}$
is generated at $E_8$ singularity enhancement points $c_1 = c_2 = 0$,
because all the three 2-cycles vanish simultaneously there, and
they satisfy
\begin{equation}
 (C_{-\theta}+C_{A76}+C_{A65}) + C_{A65} + (C_{A76}+C_{A65}) \equiv 0
  \quad ({\rm mod~}E_6 ).
\label{eq:cycle-sum}
\end{equation}
This Yukawa coupling is invariant under the U(1) symmetry.

We know, however, that there are more monodromy among the 2-cycles.
Under the monodromy (\ref{eq:0-DB_1,2}), the 2-cycle $C_{A65}$ for
${\bf 27}_{+1}$ turns into $(C_{A76}+C_{A65})$ for ${\bf 27}_{-2}$ and
vice versa. The matter fields in the $E_6$-${\bf 27}$ representation
cannot remain pure eigenstates of the U(1) symmetry. This mixing
among states with different U(1) charges (and hence the monodromy)
takes place around the monodromy locus
\begin{equation}
 (c_{0,*})^4 (c_{0,*} c_{2,*} + 2 c_{1,*}^2)^2 \simeq
 12 f_0 (\epsilon_K)^4 (c_{1,*}^2)^4,
\end{equation}
which is a part of $B=0$ locus, just like the ramification locus
$4c_0c_2-c_1^2 \simeq 0$.
Since this branch also approach the $c_1 = c_2 = 0$ point just like
the ramification locus (see Figure~\ref{fig:S-GUT-E6}), we do not
have a reason to believe that the mixing (monodromy) due to this
branch cancels, while that of the ramification locus remain.
The mixing may be suppressed by some function of $\epsilon_K$,
but we cannot take $\epsilon_K$ to be literally zero
in (\ref{eq:eK-scale}, \ref{eq:coordinate-scale}, \ref{eq:scaling-rg},
\ref{eq:scaling-rg-c}) in $\mathop{\rm SU}(5)_{\rm GUT}$ models.
Therefore, all sorts of U(1) violating trilinear interactions
$\Delta W = {\bf 27} \cdot {\bf 27} \cdot {\bf 27}$ will be
generated, unless there is a cancellation.

There are other groups of regions in $S_{GUT}$ where the matter curves
$c_1 = 0$ or $c_2 = 0$ encounters the $B = 0$ monodromy locus.
One is $c_0 \simeq 0$ on $c_1 \simeq 0$ ($c_{2,*} \neq 0$), and the other
is $c_0 \simeq 0$ along $c_2 \simeq 0$ ($c_{1,*} \neq 0$). The monodromy
we studied in section \ref{subsec:monod_beyond_gauge} is relevant to
the latter.
The 2-cycle $(C_{-\theta}+C_{A76}+C_{A65})$ for ${\bf 27}_{+1}$ and
$(C_{A76}+C_{A65})$ for ${\bf 27}_{-2}$ are exchanged under the
monodromy (\ref{eq:mod_full}) modulo 2-cycles $C_{\alpha, \beta}^{1,2}$.
Thus, the ``${\bf 27}_{+1}$'' fields will have non-vanishing
$[C_{A76}+C_{A65}]$ (mod $C_{\alpha, \beta}^{1,2}$) component
and vice versa. The mixing, however, will presumably be suppressed
exponentially, because it is 2-cycles of finite size, rather than
vanishing 2-cycles, that are exchanged under the monodromy at
$c_0 \simeq 0$.

The U(1) symmetry violating Yukawa couplings ${\bf 27} \cdot {\bf 27}
\cdot {\bf 27}$ are generated only when the sum of three topological
2-cycles vanish in $H_2({\rm K3}; \mathbb{Z})$, just like
in (\ref{eq:cycle-sum}). Thus, just a single monodromy out of
(\ref{eq:mod_full}) acting on $(C_{-\theta}+C_{A76}+C_{A65})$ does not
generate such a U(1) violating coupling ${\bf 27}_{}$;
\begin{equation}
 (C_{-\theta}+C_{A76}+C_{A65}) + C_{A65}
  +(\tilde{C}_{-\theta}+\tilde{C}_{A76}+\tilde{C}_{A65}) \neq 0
\end{equation}
in $H_2({\rm K3}; \mathbb{Z})$ (mod $E_6$). After exploiting all the
monodromies available in a 4-fold $X_4$ that is a K3 fibration on
$S_{GUT}$,\footnote{Note, for example, that both $f_0$ and $g_0$ vanish
at some points on $S_{GUT}$, where one cannot say $\epsilon_\eta$ is
small. Although we did not study monodromy associated with non
$|\epsilon_\eta| \ll 1$ region, generically we should expect such a
monodromy in a compact model.} however, all $C_{\alpha, \beta}^{1,2}$
will be mixed up and twisted over $S_{GUT}$, and eventually such a
U(1) violating coupling will be generated, although the coupling
constant may be highly suppressed. In the case of non-K3 fibred $X_4$,
$H_2({\rm K3}; \mathbb{Z})$ just has to be replaced by $H_2$ of the
complex-surface fibration of (\ref{eq:nonK3-fib}).

The renormalizable proton decay operators are not the only consequence
of the U(1) symmetry-breaking state mixing. Factorized spectral surface
limit has been used for dimension-5 proton decay problem \cite{Caltech-0906},
or for the purpose of exploiting \cite{Hayashi-3, Cordova}
the idea of flavor structure in \cite{HV-08Nov, Harvard-nonC,
Conlon, Marchesano}.
The discussion so far implies that it is impossible to confine
$\bar{\bf 5}_H$ and ${\bf 5}_H$ to separate irreducible pieces of
$\mathop{\rm SU}(5)_{\rm GUT}$ matter curves completely. This may not be a
big problem, as opposed to the dimension-4 proton decay problem,
however, because the dimension-5 proton decay problem only requires
a small amount of suppression; complete separation is not necessary.
In the context of flavor structure of Yukawa couplings, the three copies
of the visible sector matter fields $\bar{\bf 5}_M$ or
${\bf 10}_M$ may not be able to have wavefunctions strictly in
a single irreducible component of the factorized matter curves, even
at the factorization limit of the spectral surface. This means
that the up-type Yukawa matrix of the low-energy effective theory below
the Kaluza--Klein scale receives contributions from more than
one point of enhanced $E_6$ singularity \cite{Hayashi-3, Cordova}. If the
state mixing is small, then this mixing is not terribly bad, and may
even contribute in generating smaller eigenvalues; the flavor structure
generated in this way is not necessarily similar to the one we observe
in the Standard Model, however.

It should be remembered, however, that our analysis employed a 2+1
factorization in the $E_8 \rightarrow E_6$ symmetry breaking.
Thus, it will not be a terribly bad guess to expect similar results
for the 4+1 and 3+2 factorization in the $E_8 \rightarrow \mathop{\rm SU}(5)_{\rm GUT}$
symmetry breaking \cite{TW-1, Tsuchiya, Caltech-0906}.
When another type of factorization (or monodromy subgroup of $S_5$) is
employed, as in \cite{Harvard-nu, Harvard-E8}, separate study is necessary,
especially because it is not clear how the problem A will look like
in such a factorization limit.

\subsection{Loopholes in This Argument}
\label{subsec:loophole}

The discussion so far hints that the dimension-4 proton decay operators
are generated in the factorized spectral surface scenario. There are
some loopholes in the argument, however. At the end of this article,
here, we list up some loopholes that come to our minds. The list can
also be taken as possible ways to save the factorized spectral surface
solution.

First of all, the argument so far only showed that
\begin{itemize}
 \item there is no unbroken
U(1) symmetry (apart from exceptional cases that we discuss later) in
the low-energy effective theory that we hope would exclude the
dimension-4 proton decay operators, and
 \item the picture of interactions as recombination of M2-branes without
       a change in the total topology does not exclude the U(1)
       violating operators.
\end{itemize}
It is not that we have calculated the coefficients of such operators,
and in fact, we do not even have a theoretical framework to calculate
the coefficients within F-theory. A gauge theory on 7+1 dimensions
cannot handle this. A possible direction is to exploit the Type
IIB--M-theory duality or F-theory--Heterotic duality; dual descriptions
may be used to see whether cancellation mechanism is likely to exist,
or to make an estimate of the coefficients. That will tell us how small
$\epsilon_K$ should be.
Such a study is beyond the scope of this article, however.

If one literally sets $\epsilon_K = 0$, instead of fine-tuning it to be
sufficiently small, then $S_{GUT}$ becomes a locus of $E_8$ singularity.
If a vector bundle on $S_{GUT}$ has a structure group that is smaller
than $\mathop{\rm SU}(5)_{\rm str} \times \mathop{\rm U}(1)_Y$ so that an extra U(1) factor is
contained, then an unbroken U(1) symmetry remains in the low-energy
effective theory and prevent proton decay. The chirality of various
charged matter fields in this case, however, are determined simply
by intersection numbers of first Chern classes of those bundles and
$K_{S_{GUT}}$ \cite{TW-1, DW-1, BHV-1, Hayashi-1}, and existence of
exotics is predicted easily \cite{TW-1, BHV-2, Caltech-0906}.
It is also obvious that theories of flavor structure like those
in \cite{Hayashi-2, HV-08Nov, Font, Harvard-nonC, Hayashi-3, Cordova}
are not applied to such a case, because only $A_4$ singularity
is assumed along $S_{GUT}$.

Secondly, one will notice that our analysis in section \ref{sec:mod}
is based on a
choice (\ref{eq:factorization_cond1}--\ref{eq:factorization_cond3}) of
the solution to (\ref{eq:vanish-a1}). The condition (\ref{eq:vanish-a1}),
however, can be solved in the form of
\begin{equation}
 c_0 = p s, \quad d_1 = - q r, \quad c_1 = p r, \quad d_0 = q s
\end{equation}
for some global holomorphic sections
\begin{eqnarray}
& & s \in \Gamma(S_{GUT}; {\cal O}_S(\eta_0)), \quad
 p \in \Gamma(S_{GUT}; {\cal O}_S(\eta'_1)), \quad
 q \in \Gamma(S_{GUT}; {\cal O}_S(\eta'_2)), \\
& & r \in \Gamma(S_{GUT}; {\cal O}_S(\eta_0 + K_{S_{GUT}}));
\end{eqnarray}
here, $\eta'_{1,2}$ and $\eta_0$ are some divisors on $S_{GUT}$, and
$c_2 \in \Gamma(S_{GUT}; {\cal O}_S(\eta'_1 + \eta_0 + 2 K_{S_{GUT}}))$.
What we studied in section \ref{sec:mod} is the monodromy associated
with $s=0$ locus.

What if the global section $s$ does not have a zero locus?
This is possible if the line bundle ${\cal O}_S(\eta_0)$ is trivial;
$s$ can now take a constant non-zero value over the entire $S_{GUT}$.
Suppose that this topological condition is satisfied. If we further
assume that the divisor $K_{S_{GUT}}$ is not effective, as in del Pezzo
surfaces and Hirzebruch surfaces, then there is no non-trivial
global holomorphic section $r$. We have to set $r=0$.
This means that
\begin{equation}
c_1 = d_1 = 0
\end{equation}
over the entire $S_{GUT}$, which is nothing but the case we already
listed as the rank-5 GUT scenario (ii) in Introduction. The dimension-4
proton decay problem can be solved completely in this scenario, because
of an unbroken (or spontaneously broken) U(1) symmetry in the low-energy
effective theory; an extra 2-cycle is along $S_{GUT}$, and a semi-local
geometry of this form along $S_{GUT}$ is sufficient in ensuring the proton
stability \cite{TW-1}. The singularity along $S_{GUT}$ is either $D_5$ or $A_5$
in this case, and theories of flavor structure \cite{Hayashi-2, HV-08Nov,
Font, Harvard-nonC, Hayashi-3} are not applied here. Exotic-matter
free conditions have also been studied for rank-5 GUT
scenarios \cite{BHV-2, Chung}.

If the divisor $K_{S_{GUT}}$ is effective, on the other hand,
we can introduce a different set of topological conditions: all of
line bundles ${\cal O}_S(\eta_0)$ and ${\cal O}_S(\eta'_{1,2})$ are
trivial, so that there is no zero locus in $s$, $p$ and $q$.
This is possible for an effective $K_{S_{GUT}}$, because the matter
curves belong to topological classes of effective divisors.
Now there is no $a_0 = 0$ locus, and at least the problem B is gone,
under this set of topological conditions.

An obvious loophole in the argument that follows (\ref{eq:spec-surf-1})
is that the coefficient of the highest degree term $a_{5-k}$ may not
have a zero locus. We have already exploited a case that $a_0$
is constant and non-zero, and a remaining alternative is a case\footnote{
\label{fn:ex}
As an example, one can consider an elliptic fibration
$\pi_X : X_3 \rightarrow B_2 = \mathbb{P}^2$, and take $S_{GUT}$ to be the zero
locus of a homogeneous function $s$ of degree 4 on $B_2= \mathbb{P}^2$.
The line bundle for $\tilde{A}_4$ is trivial on $B_2$, and hence
$a_2 = \tilde{A}_4|_{S_{GUT}}$ has to be constant on $S_{GUT}$.
One will also see that $\tilde{A}_6 = 0$ everywhere on $B_2$.
The divisor $K_{S_{GUT}}$ on $S_{GUT}$ is effective. }
where $a_2$ is constant and non-zero everywhere on $S_{GUT}$.
This is possible only when $K_{S_{GUT}}$ is effective on $S_{GUT}$,
and $a_0 \in \Gamma(S_{GUT}; {\cal O}_S(- 2 K_{S_{GUT}}))$ vanishes
as a consequence.  An $E_7$ gauge theory\footnote{$E_7$ is a minimal choice
in obtaining all the matter fields and their trilinear interactions of
the supersymmetric standard Models \cite{TW-1, Bourjaily-1, Harvard-E8}.
Thus, it is not an option for realistic GUT models to assume
that $a_3$ is constant and non-zero over $S_{GUT}$.} is well-defined over the
entire $S_{GUT} \times \mathbb{R}^{3,1}$. There is no mixing between 2-cycles
in $E_7$ and those outside, and the problem~B is gone in this case.
What is more intriguing in this model is that there is no need to impose
a factorization condition like (\ref{eq:2+1}) by hand. The commutant
of $\mathop{\rm SU}(5)_{\rm GUT}$ in $E_7$ is $\mathop{\rm U}(3)$. The decomposition
\begin{eqnarray}
 {\rm Res}^{E_7}_{\mathop{\rm U}(3) \times \mathop{\rm SU}(5)_{\rm GUT}} {\bf adj.} & = &
 ({\bf adj.}, {\bf 1}) + ({\bf 1}, {\bf adj.}) \nonumber \\
 & + &
 \left[ ({\bf 3}, \overline{\bf 10}) + (\wedge^2 {\bf 3}, {\bf 5})
        + (\wedge^3 {\bf 3}, \bar{\bf 5}) \right] + {\rm h.c.}
\end{eqnarray}
shows that there are two different kinds of charged matter fields
in the $\mathop{\rm SU}(5)_{\rm GUT}$-$\bar{\bf 5}$ representation. For a fully
generic 3-fold spectral cover globally defined on $S_{GUT}$,
$({\bf 3},\overline{\bf 10})+(\bar{\bf 3},{\bf 10})$ matter fields
are localized along the $a_5=0$ locus, while
$(\wedge^2 {\bf 3}, {\bf 5})+{\rm h.c.}$ and
$(\wedge^3 {\bf 3}, \bar{\bf 5})+{\rm h.c.}$ matter fields are localized
along the $(-a_2 a_5 + a_4 a_3)=0$ and $a_3=0$ loci, respectively.
This factorization / decomposition automatically takes place under this
topological condition.\footnote{We thank Cumrun Vafa for discussion.}
Because of the commutation relation of the $E_7$ Lie algebra,
$(\wedge^2 {\bf 3}, {\bf 5})$ is identified with the origin of the
up-type Higgs multiplet. If $(\wedge^3 {\bf 3}, \bar{\bf 5})$ and
$(\wedge^2 \bar{\bf 3}, \bar{\bf 5})$ are identified with
$\bar{\bf 5}_H$ and $\bar{\bf 5}_M$, respectively,\footnote{The exotic
free condition and successful doublet--triplet splitting cannot be
realized simultaneously in this identification, however \cite{Caltech-0906}.
If the two components are identified in the opposite way with
$\bar{\bf 5}_H$ and $\bar{\bf 5}_M$, the interaction
(\ref{eq:Yukawa-in-E7}) gives rise to the trilinear interaction
$\Delta W = S \bar{{\bf 5}}_H {\bf 5}_H$ of the next-to-minimal supersymmetric
Standard Models, and a candidate for the right-handed neutrinos is lost.}
then the moduli
$({\bf adj.}, {\bf 1})$ for the $\mathop{\rm U}(3)$ spectral surface may become
right-handed neutrinos \cite{Harvard-nu, Tsuchiya}.
The up-type Yukawa couplings and down-type/charged lepton Yukawa couplings
originate from
\begin{equation}
 \Delta W = \left(\wedge^4 {\bf 8}\right)^{MNPQ} \cdot
  {\bf adj.}^{I}_{\; H} \cdot \left(\wedge^4 {\bf 8}\right)^{HJKL} \;
   \epsilon_{IJKLMNPQ}
\end{equation}
in the language of $\mathfrak{su}(8) \subset \mathfrak{e}_7$, and
the neutrino Yukawa couplings from \cite{TW-1}
\begin{equation}
 \Delta W = \mathop{\rm tr} {}_{\mathfrak{su}(8)}
    \left( {\bf adj.}, [ {\bf adj.}, {\bf adj.} ] \right).
\label{eq:Yukawa-in-E7}
\end{equation}
It is worthwhile to study the monodromy for this case, e.g, using the example
in footnote \ref{fn:ex}, to see how the problem A will look like, because
the ``problem A'' may be quite different in nature, or may even be absent,
in such a case without a need for imposing factorization condition by hand.

Finally, there are also loopholes that resort to continuous fine-tuning
of complex structure parameters. Flux compactifications should ultimately
explain such a tuning. At the level of gauge theory description on 7+1
dimensions, we introduced a parametrization of the complex structure
moduli $a_0$, $a_2$ and $a_3$ by $c_{0,1,2}$ as
in (\ref{eq:factorization_cond1}--\ref{eq:factorization_cond3}),
so that the spectral surface factorizes in the new parametrization.
But the ultimate goal is to reduce the monodromy of 2-cycles so that
an unbroken U(1) symmetry is maintained in the low-energy effective
theory.\footnote{Strictly speaking, we do not need a full U(1) symmetry,
or equivalently a monodromy-invariant 2-cycle. If a torsion
component survives the monodromy, then some selection rule will
remain, as we discussed in section \ref{subsec:M2brane}.
} Thus, the generalized version of the solution is to consider a
parametrization of the complex structure moduli $(a_{0,2,3}, f_0, g_0, a''_0)$
of $X_4$ so that the polynomial $B$ in (\ref{eq:B}) is factorized
appropriately in the new parameters. The geometry for compactification
should be given by holomorphic sections from $S_{GUT}$ to the new
parameter space.
It will be straightforward to work out the expression for ``$B$''
in the case of $E_8 \rightarrow {\rm SU}(5)_{\rm GUT}$ symmetry breaking.

In the case of $E_8 \rightarrow E_6$ symmetry breaking, the polynomial $B$
should be factorized so that the split between
$c^1_{2-DB}$--$c^1_{0-DB}$ and $c^2_{2-DB}$--$c^2_{0-DB}$ disappears and they
form a monodromy locus of multiplicity 2.
The monodromy (\ref{eq:0-DB_1,2}) is gone in this limit;
this is a fine-tuning solution to the problem A.

A hint for a fine-tuning solution to the problem B comes from
Heterotic dual description. In Heterotic string language, one can
tune complex structure moduli so that the elliptic fibred Calabi--Yau 3-fold
admits a non-trivial section; moduli of spectral surfaces of $V_1$ and $V_2$
may be tuned so that the line bundle ${\rm det} \; V_1$ corresponds to
the non-trivial section.\footnote{We thank Ron Donagi for discussion.}
In other words, this is to consider the factorization limit
of (\ref{eq:Het-spec-surf}) instead of the factorization limit
of (\ref{eq:5-fold}, \ref{eq:rank3_spect}). There still remains a problem
of determining the $\epsilon_\eta$ suppressed corrections, because the
spectral surface picture relies on supergravity + $E_8 \times E_8$
super Yang--Mills approximation in the Heterotic description (see
the appendix \ref{sec:Het} for more). In F-theory language, this
tuning, we expect, will correspond to factorization of $B$ so that
all the six branches $c^{1,2}_{0-4,5,6}$ somehow come on top of one another
in the new parametrization space to form a monodromy locus of
multiplicity 6. The monodromy, then, is $\rho(\gamma_{0-0})$, which we
know is trivial on the $E_8$ part of the 2-cycle. We do not have a
concrete picture of how to modify the parametrization (\ref{eq:factorization_cond1}--\ref{eq:factorization_cond3}) systematically to obtain the new
parametrization, however.

  \section*{Acknowledgements}

TW thanks Ron Donagi, Sergei Galkin, Joseph Marsano, Cumrun Vafa and
Martijn Wijnholt for stimulating discussion, useful comments and
communications. This work started when two of the authors (TK and TW)
were staying at YITP, Kyoto University, during a program
``Branes, Strings and Black Holes,'' September--November, 2009.
TW thanks Caltech theory group and KITP, UC Santa Barbara, for hospitality,
where he stayed during the final stage of this project.
This work was supported in part
by JSPS Research Fellowships for Young Scientists (HH),
by a Grant-in-Aid \#19540268 from the MEXT of Japan (TK),
by Global COE Program ``the Physical Sciences Frontier'', MEXT, Japan (YT),
by WPI Initiative, MEXT, Japan and by the NSF under Grant
No. PHY05-51164 (TW).


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