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\begin{document}
\pagestyle{empty}
\begin{flushright}{\tt hep-th/0109155}\\
TIT/HEP-471\\
%UTHEP-mmm\\
September, 2001
\end{flushright}
\vspace{18pt}

\begin{center}
{\Large  Dualities between K3 fibered Calabi--Yau
 three-folds   } \\ 
\vspace{4pt}
\vspace{16pt}

Mitsuko Abe\fnote{*}{\email}       
 
\vspace{16pt}



{\sl Faculty of Engineering\\
Shibaura  Institute of Technology \\ 
Minato-ku, Tokyo 108-8548, Japan\\}

\center {and}

{\sl Department of Physics\\
Tokyo Institute of Technology \\ 
Oh-okayama, Meguro, Tokyo 152-8551, Japan\\}




\end{center}
\vspace{3cm}


\begin{abstract}
We examine the relations of Calabi--Yau three-folds 
having  the same Hodge numbers, 
$(h^{1,1},h^{2,1})\!=\!(5,185)$ with the various number of K3 fibers.   
We also  argue their implications to string duality. 
\end{abstract}



\vfill
\pagebreak

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\baselineskip=16pt

%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%
\par 
The motivation of our work 
is to seek a way for deriving the physical  non-perturbative 
information from  various kinds of CY3 to look for the 
ultimate unification theory. 
Our work  starts with the conjecture
 about   heterotic-type IIA string duality.
 If a CY3  admits both
K3 and $\text{\bf T}^2$ fibrations  with at least one  
section then type IIA string on the CY3
 is dual to  a Heterotic string on $\text{K}3\!\times\! \text{\bf T}^2$
\cite{vafa1,vafa2,aspinwall,witten0}. 
 
\par
Our concern is to make clear the relation 
between examples of K3 fibered CY3s with $\text{\bf P}^1(1,1)$ base   
and  those  with base  $\text{\bf P}^1(1,s)$. 
Most researches so far have been restricted to
the case of K3-fibration CY3s with base $\text{\bf P}^1(1,1)$
\cite{vafa1,vafa2,aldazabal1,aldazabal2,candelas1,candelas3,bershadsky}.
 
For  CY3   with base $\text{\bf P}^1(1,s)$,  
only few attempts  of  physical application 
   have so far  been   examined \cite{ lerche0,mabe}. 
There are two special points which characterize CY3s treated
in this paper; one is  the existence of extra     
 tensor multiplets in  6-dimensional  intermediate stage and the other  
the existence of multiple  K3 fibrations in CY3s.   

Some  identifications of CY3 phases have been done by using dual polyhedra
\cite{candelas4,candelas0,kreuzer,skarke}.
The method  given by  \cite{hosono1, hosono2} is powerful to 
 see the property of CY3s and their relations.
There are several works about the relation of 
elliptic fibered CY3s with $\text{\bf F}_0$ base and $\text{\bf F}_2$ 
base  \cite{vafa1,witten2,theisen,curio}.
The investigation of the relations in this paper 
is based  on the calculation done by S. Hosono \cite{hosono0} 
and  serves as  an extension  of the earlier works.
Using them, we can discuss the possibility of some extensions  
of the duality  of heterotic string and 
type IIA one for the models with $\text{\bf P}^1(1,s)$ bases 
 and  an aspect  of  heterotic -heterotic string duality.



The organization of this article is  as follows: 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{enumerate}
\item  Introduction 
\item  type IIA/heterotic string duality
\item Toric realizations  of  the four 
$(h^{1,1},h^{2,1})\!=\!(5,185)$    models
\item  The relation among (III), (IV) and (V) models
\item Future problem
\end{enumerate} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%







\section{type IIA -heterotic string duality} 
We consider heterotic string compactified 
on $\text{K}3\!\times\!\text{\bf T}^2$, 
The instanton numbers of the vector bundles on K3 are denoted 
as $k_1$ and $k_2$.    
 They  satisfy
\[ 
  k_1\!+\!k_2\!=\!24,  \qquad (k_1\!=\!12\!+\!n^0,\quad k_2\!=\!12\!-\!n^0) 
\qquad \qquad (1)
\]
($n^0$ is introduced for convenience) \cite{aldazabal1,aldazabal2}.
$G_1\!\times\!G_2$ denotes the perturbative  
 non-Abelian gauge symmetries  in  
heterotic string compactified on $\text{K}3\!\times\!\text{\bf T}^2$.  
$G_1$ and $G_2$  come from each $E_8$.  
\par We list 
 three  series which have  dual,  type IIA  string  on CY3s
\cite{aldazabal1,aldazabal2} \footnote{  
 Furthermore, there are three versions of these three series 
 by changing the kind   of elliptic fiber. 
In this paper, we restrict ourselves to the 
A-chain version \cite{aldazabal2,candelas3},
where the elliptic fiber of the CY3
is $\text{\bf P}(1,2,3)[6]$.   
The extension  to B or  C versions  
with elliptic fiber $\text{\bf P}(1,1,2)[4]$ or $\text{\bf P}(1,1,1)[3]$
should  also be possible.}.


\vspace{0.2cm}

\begin{enumerate}
\item
In one of these series,  the second gauge group  $G_2$ 
  is broken to the 
terminal gauge group which depends on $n^0$.   
The special case of this series with  $G_1\!=\!1$ is called  (I). 
\item The second series consists  of sequential  $G_1$ breaking by VEVs
of hypermultiplets, which we call (II).
\item 
In the third one, the   
$G_2$ part  is  broken to  the terminal gauge group  
  and some additional    tensor multiplets appear    
 keeping $G_1=1$. 
 $(\text{I}^{\dagger})$, (III), (IV) and (V) are included in this series,
which we will describe below. In particular, we will see CY3s for the type IIA duals 
of these four models which have the  same Hodge  
numbers.   
\end{enumerate} 

Use of the index theorem of vector bundles on K3  and the 
anomaly cancellation condition enables us to specify
the number of the massless vector multiplet 
and  the hypermultiplets in 4D \Ntwo SYM theory,
which are expressed by the Hodge numbers of CY3s used in type IIA 
side \cite{aldazabal1,aldazabal2,candelas2}.
Let us denote 
 $n_T\!=\!n_T^0\!+\!\Delta n_T$ with $n_T^0\!=\!1$,  
the number of the tensor multiplets in 6D \None SYM theory, where 
$\Delta n_T$ is the number of shrinking instantons. In the presence of 
shrinking    instantons, eq. (1) with  $n^0=0$ is  modified as  
$k_1\!=\!12$, $k_2\!=\!12\!-\!\Delta n_T$.















\newpage

 
\section{Toric realization of the four 
{\boldmath $(h^{1,1},h^{2,1})\!=\! (5, 185) $
  models } }
 

\vspace{0.4cm}
  
We deal with the four types of CY3 models which have the same Hodge numbers,  
  $(h^{1,1},h^{2,1})\!=\! (5, 185) $   and    
various  numbers  of K3 fibers, each of which  
 can be realized via a hypersurface in a  toric variety\footnote
{For a  review of toric geometry, see  \cite{fulton}.}.    
$S$ in the table 3 denotes  the base surface
with respect to  the elliptic fibration in CY3s. 
They are blow-ups of the $a$th Hirzebruch surface, $\text{\bf F}_a$, 
which we denote by $\text{Bl}(\text{\bf F}_a)$.   
   

\vspace{0.4cm}



\noindent
 $(\text{\bf I}^{\dagger})$
{\tt Model of Candelas, Font, Perevalov and Rajesh} 
\cite{candelas1,candelas3} 
\newline
In the toric realization of this model,  
 the sub dual polyhedron part of one   K3 fiber has been modified from 
those of  case (III) below. 
The features of K3 fibers in CY3s  are the same 
as those of (III).  
\newline
\vspace{0.1cm}
\newline
\noindent
(\text{\bf III}) {\tt Model   of  Candelas, Perevalov and Rajesh}
 \cite{candelas2} 
\newline
The dual polyhedron   
 contains  a  dual  sub polyhedron of K3, 
which    varies  according to  the number  
of tensor multiplets.  It also contains another dual polyhedron of K3.
The latter K3 
is always realized by  $\text{\bf P}^3(1,1,4,6)[12]$. Therefore, 
 CY3s  in  this case admits at most two K3 fibers.  They have     
  $\text{\bf P}^1 (1,1)$ base under first  of the two K3 fibrations. 
We are interested in the $h^{1,1}\!=\!5$ case,
which  has  $\Delta n_T\! =\!2$
and  $\text{\bf P}^3(1,1,4,6)[12]$ as both K3 fibers.
\vspace{0.4cm}
 
\vspace{0.4cm}

\noindent
(\text{\bf IV}) {\tt Model  of  Hosono, Lian  and Yau }  \cite{yau}
\par
CY3s in this  case are 
realized by the weighted projective hypersurfaces 
$
\text{\bf P}^4(1,s,s+1,4s+4,6s+6)[ 12s+12]
$
with  $\{ s\!=\!1,2,3,4,6,8,12 \}$.
These admit a K3 fibration with fiber  
$\text{\bf P}^3(1,1,4,6)[12]$ and 
 base, $\text{\bf P}(1,s)$. In particular,  
$h^{1,1}\!=\!5$ case corresponds to  $s\!=\!\Delta n_T\!=\!2$.
\vspace{0.4cm}

\noindent
(\text{\bf V}) {\tt Model  of Louis et al.} \cite{theisen} 
  

The  CY3s in this case with $h^{1,1}\!=\!5$
admits at most three K3 fibrations. 
All K3 fibers are realized by 
$\text{\bf P}^3(1,1,4,6)[12]$.
The dual polyhedron of (V) 
coincides with that of   $(\I^{\dagger})$ except one vertex:
$(1, 2, 2, 3)^{\III} \rightarrow (-1,-1,2,3)^{\V}$. 



We list the  
four dual polyhedra with $(h^{1,1},h^{2,1}) \!=\!(5, 185)$.  
(We omitted all the points in the codimension one faces, 
which should be removed 
in triangulation.)
\newline
\vspace{0.1cm}

$(\I^{\dagger})$   a   dual polyhedron  in \cite{candelas3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align*}
{} &{} \phantom{\longleftarrow\quad } \text{difference from (III)}  \\
(0)\   (\ph 0,  \ph 0,  \ph 0, \ph 0)\  &{}   \\
(1)\  (\ph 0,  \ph 0,  -1,  \ph 0)\  &{} \\
(2)\  (\ph 1,  \ph 0,  \ph 2,  \ph 3)\  & {} \\
(3)\  (\ph 1,  \ph 2,  \ph 2,  \ph 3)  
\ &\longleftarrow\  (\ph 1, \ph 2, \ph 6, \ph 9 )^{(\III)}\\
(4)\   (\ph 0,  \ph 0,  \ph 0,  -1)\  &{}    \\
(5)\  (-1,  \ph 0,  \ph 2,  \ph 3)\  &{}\\
(6)\ (\ph 0,  -1, \ph 2,  \ph 3)  
\ &\longleftarrow\   (\ph 0, -1, \ph 0, \ph 0 )^{ (\III)} \\
(7)\  (\ph 0,  \ph 1,  \ph 2,  \ph 3)  
\ &\longleftarrow\  (\ph 0, \ph 1, \ph 4, \ph 6 )^{(\III)} \\
(8)\  (\ph 1,  \ph 1,  \ph 2,  \ph 3)  
\ &\longleftarrow\   (\ph 1, \ph 1, \ph 4,\ph  6 )^{(\III)} \\
( 9) \ (\ph 0,  \ph 0,  \ph 2,  \ph 3) \ &{}.       
\end{align*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(III) the  dual polyhedron  in  \cite{candelas2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\allowdisplaybreaks      
\begin{align*}
{}  \text{SL}(4;\text{\bf Z})\ & \text{trans.} {}\\
%- - - - -- - --  --
(0)\quad  (\ph 0,  \ph 0,  \ph 0,  \ph 0) 
\rightarrow&\   (\ph 0, \ph 0, \ph 0, \ph 0),\\
(1)\quad (\ph 0,  \ph 0,  \ph 1,  \ph 2) 
\rightarrow&\  (\ph 0, \ph 0,  -1, \ph 0),\\
(2)\quad  (\ph 1,  \ph 0,  \ph 0,  -1) 
\rightarrow&\  (\ph 1, \ph 0, \ph 2, \ph 3),\\
(3)\quad (\ph 1, \ph 2,  \ph 0,  -1)  
\rightarrow&\   (\ph 1, \ph 2, \ph 6, \ph 9),\\
(4)\quad (\ph 0,  \ph 0,  -1,  -1) 
\rightarrow&\   (\ph 0, \ph 0, \ph 0, -1),\\
(5)\quad  (-1,  \ph 0 , \ph 2,  -1) 
\rightarrow&\   (-1, \ph 0, \ph 2, \ph 3),\\
(6)\quad  (\ph 0,  -1,  \ph 1,  -1) 
\rightarrow&\   (\ph 0, -1, \ph 0, \ph 0),\\
(7)\quad  (\ph 0,  \ph 1 , \ph 1,  -1) 
\rightarrow&\  (\ph 0, \ph 1, \ph 4, \ph 6),\\
(8)\quad  (\ph 1,  \ph 1,  \ph 0,  -1) 
\rightarrow&\  (\ph 1, \ph 1, \ph 4, \ph  6),\\
(9)\quad  (\ph 0,  \ph 0,  \ph 1,  -1) 
\rightarrow&\  (\ph 0, \ph 0, \ph 2,  \ph 3). 
\end{align*}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%\vspace{1cm}
%
(IV) the   dual polyhedron in   \cite{yau}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align*}
{} &\phantom{\longrightarrow\quad}
\text{difference  from (III)} \\
(0)\ (\ph 0,  \ph 0,   \ph 0,   \ph 0)\ &{} \\
(1)\    (\ph 0,  \ph 0,  \ph 0,    -1)\ &{} \\
(2)\    (\ph 0,   \ph 0,  -1,  \ph 0)\ &{} \\
(3)\     (\ph 0,   -1,  \ph 0,   \ph 0)\ &{} \\
(4)\     (-1,\ph 0,   \ph 0,  \ph 0)
\ &\longleftarrow\
(-1,  \ph 0,  \ph 2,  \ph 3 )^{(\III)} \\
(5)\   (\ph 2,  \ph 3, \ 12,  \ 18)
\ &\longleftarrow\
(\ph 1,  \ph 0,  \ph 2,  \ph 3 )^{(\III)}\\
(6)\   (\ph 1,  \ph 2,   \ph 8, \  12)
\ &\longleftarrow\
( \ph 1,  \ph 2,  \ph 6,  \ph 9 )^{(\III)} \\
(7)\  (\ph 0,  \ph 1,  \ph 4,   \ph 6) \ &{} \\  
(8)\  (\ph 1,  \ph 1,   \ph 6,   \ph 9)
\ &\longleftarrow\
(\ph 1,  \ph 1,  \ph 4  ,\ph 6 )^{(\III)}\\
(9)\  (\ph 0,  \ph 0,   \ph 2,   \ph 3) \ &{}        
\end{align*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
(V) the  dual polyhedron  in  \cite{theisen}

It can be obtained by the modifications of (III) or $(\I^{\dagger})$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\allowdisplaybreaks
\begin{align*}
(0)\ (\ph 0,  \ph 0,  \ph 0,  \ph 0)\ &{}\\               
(1)\ (\ph 1,  \ph 1,  \ph 2,  \ph 3) 
\  &\longleftarrow\
 (\ph 1, \ph 1, \ph 4, \ph 6) ^{(\III)} \\ 
(2)\  (\ph 1,  \ph 0,  \ph 2,  \ph 3)\ &{} \\                   
(3)\ (\ph 0,  -1,  \ph 2,  \ph 3)
\   &\longleftarrow\
 (\ph 0, -1, \ph 0, \ph 0)^{(\III)}\\ 
(4)\ (-1, -1, \ph 2, \ph 3)
\    &\longleftarrow\
(\ph 1,  \ph 2,  \ph 6, \ph 9)^{(\III)}\\ 
{} &\phantom{\leftarrow}
          \\
(5)\ (  -1,  \ph 0,  \ph 2, \ph 3)\ &{} \\       
(6)\ (\ph 0,  \ph 1,  \ph 2, \ph 3)
\  & \longleftarrow\
 (\ph 0, \ph 1, \ph 4, \ph 6)^{(\III)}\\ 
(7)\  (\ph 0,  \ph 0,  \ph 2,  \ph 3)
\ &\longleftarrow\  
\text{the point absent in ``17''   {\cite[p.20]{theisen}}}
\\
(8)\ (  \ph 0,  \ph 0,  -1,  \ph 0)\ &{} \\        
(9)\ (  \ph 0,  \ph 0, \ph  0,  -1)\ &{}       
\end{align*}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
There is  a symmetry due to the exchange of three pairs 
of  vertices in the dual polyhedron, which represents  
 the  \text{\bf P}$^1$ basis under the  three K3 fibrations.   
%\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\begin{align*}
\{(5)(-1,0,2,3),(2)(1,0,2,3)\}
&\leftrightarrow
\{(3)(0,-1,2,3),(6)(0,1,2,3)\}\\
&\leftrightarrow 
\{(4)(-1,-1,2,3),(1)(1,1,2,3)\}.
\end{align*}
%\\\\\\\\\\\\\\\\\\\\\\\\\\\
\vspace{1cm}
\par
The puzzle is 
why they have the same Hodge numbers (see table 5).
What relations they obey would be solved.


{}From the view point of  toric geometry,  we see that                            for the case (I$^{\dagger}$)  and the case (III),  the apparent  
correspondence of the  vertices  expressed by  the replacement  
will be related to  the correspondence of the toric  divisors. 
It leads to the equivalence of the  Mori cones of these models.   

                              

For the  case (III) and the  case (IV), the puzzles are rephrased as follows:
 is there any correspondence 
between the vertices of their dual polyhedra,  
 which  relate to the equivalence of the divisors or the Mori vectors?
We would like to derive some  information   to see 
the correspondence of the Mori vectors through the 
relations among vertices in the dual polyhedra for 
future problem such as the correspondence  among  
different CY3s with the same  Hodge numbers higher than $h^{1,1}=5$.





\vspace{0.1cm}
Using these toric data, we can  examine the heterotic-type IIA 
duality.
The heterotic-type IIA string duality for (III)
and (I$^\dagger$)  has been made 
clear in \cite{candelas3,candelas4}, which we review  at first. 
The differences between the Hodge numbers of (I) and (III)
or (I$^\dagger$) with
$n^0\!=\!\Delta n_T$ are given by    
\begin{eqnarray}
\Delta h^{2,1}&=& -(h^{2,1}) \mid_{\text{in (I)}}
+h^{2,1} \mid_{\text{in (III)}}=-29 \Delta h^{1,1},
 \nn \\
\Delta h^{1,1}&=& -(h^{1,1}) \mid_{\text{in (I)}}
+h^{1,1} \mid_{\text{in (III)}}
=\Delta n_T.
\nn
\end{eqnarray}



%-----------------------
 

\noindent
The number of the tensor multiplets 
$n_T$  in (IV) is given by 
%\\\\\\\\\\\\\\\   
\[
n_T=h^{1,1}\left(\text{Bl}(\text{\bf F}_2)\right)-1
=d_1-2d_0-1
=s+1, \ (s\geq 2),
\]
%\\\\\\\\\\\\\\\\\
where $d_i$ denotes the number of 
$i$-dimensional cones of the fan which describes 
the base $\text{Bl}(\text{\bf F}_2)$ as a toric variety. 
(IV) has  $\Delta n_T=2$ for $s=2$.
Also (V)  has  $\Delta n_T=2$.
The Hodge numbers and $n_T$ in (IV) and (V) coincides  
with those in (III).
It seems that there exists heterotic string on 
$\text{K}3\!\times\!\text{\bf T}^2$ 
 dual to  type IIA string on  CY3 of (IV) and (V).








\section{ The relation among (III), (IV) and (V) models}
\par For a  given  dual polyhedron,  
a  CY3  phase is specified by the particular 
triangulation of the polyhedron. The method to derive the 
topological invariants  
has been developed in  \cite{hosono1,hosono2}. 
Here we consider only regular 
triangulations which take into account all the vertices 
except those on faces of codimension one and where all the  
simplices contain the origin. Such a triangulation specifies 
a phase of ambient toric variety of  
 CY3 phases of the underlying conformal field theory. 
There are in general several possible CY3 phases which 
generally lead to topologically different CY3s.
Their Hodge numbers are the same, but the intersection numbers 
and the instanton numbers may be different.  
An important observation  made in \cite{berglund}, however,  is that 
different triangulations ( called phases in this article) 
need not always lead to different CY3.
 In some cases, 
  the difference of  triangulations   
causes  a birational transformation 
of the ambient toric manifolds 
 but it does not affect CY3 hypersurfaces.


The classical  prepotential  ${\cal K}^0$ and $c_2 \cdot \vec J$, 
which are the topological invariants,  characterize the each CY phase.
$\vec J$ is the set of  ample divisors, which generate the K\"ahler cone.   
A sufficient condition of having a  K3 fiber  is that  the 
topological invariant,  $c_2 \cdot  \vec J$ contains 24  \cite{yau}. 
$J_i$ with  $c_2 \cdot   J_i =24$ is a candidate  for 
the dual of heterotic dilaton.
Thus we can see the number of K3 fiber from $c_2 \cdot \vec J$.


The CY3 phase with $\Delta n_T\!=\!0$ in  (III) and 
that with $s\!=\!1$ in  (IV)  have the same Hodge numbers,
 $(h^{1,1},h^{2,1})\!=\!(3,243)$.
The comparison  of them    
has been made  in \cite{vafa1,witten2,theisen}
\footnote{Phases in case (V) with 
    $h^{1,1}\!=\!4$ and $ h^{1,1}\!=\!5$, 
 have been also examined  in \cite{theisen}.}.  

The key point   is that    (III) has the double 
K3 fibrations phase, which is  mapped to  single  K3 fibration phase in (IV)
\footnote{
A review of double K3 fibrations is found in  \cite{gross}}. 
The CY3  with $\Delta n_T\!=\!0$  in (III) has 
$\text{\bf F}_0$   base under the elliptic fibration.
By two ways to construct  K3 fiber for $\text{\bf F}_0$ base
due to the property, 
$\text{\bf F}_0 \!=\! \text{\bf P}^1\!\times\!\text{\bf  P}^1$, 
this CY3 has 
$\text{\bf P}^3(1,1,4,6)[12]$  as double K3 fibers. 
The CY3 with $s\!=\!0$ in (IV) has 
 $\text{\bf F}_2$  base  under the elliptic fibration and 
 has single  K3 fiber,   
$\text{\bf P}^3(1,1,4,6)[12]$. 

Let us see how this fact is reflected in the structure of the 
K\"ahler cones.


Removing two vertices (8)(1,1,4,6) and (3)(1,2,6,9) from  a
dual polyhedron  of  $h^{1,1}\!=\!5$  in section 3, 
one can get one in case (III) with $h^{1,1}\!=\!3$. 
For case (IV), the two points
$(8)(1,1,6,9)$ and 
$(5)(2,3,12,18)$
should be removed from  the  dual polyhedron in section 3  
in order to get one  with $h^{1,1}\!=\!3$.


%One can derive a strucure of the K\"ahler  cone for each phase    
% by triangulations\cite{hosono1,hosono2}.
%This cone can be identified with the K\"ahler cone,  $ \{  J_i \}$ 
%by an  appropriate linear transformation\cite{hosono1, hosono2}.


There are only one phase 
in case (III) specified by the triangulation and  also one phase  
in (IV) for $h^{1,1}=3$.  
\noindent
The Mori vectors  $\{\ell_i \}$ 
in the phase  (III) are     
%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\begin{alignat*}{2}
&( ~a_1,~ a_2,  ~a_4, ~a_5, ~a_6, ~a_7, ~a_9 & )
&    \\
&(\ph 2,    \ph 0,  \ph 3,  \ph 0,  \ph 0,  \ph 0,    \ph 1 &  )
 &=    \ell_1,  \\
&(\ph 0,   \ph 0,  \ph 0, \ph 0, \ph 1,  \ph 1,   -2 & )  &=
\ell_2, \\
&(\ph 0,    \ph 1,  \ph 0,  \ph 1,  \ph 0,  \ph 0,   -2  &)  &=   
\ell_3,  
\end{alignat*}
%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
where $a_i$ is a homogeneous coordinate    
corresponding to $i$th  vertex  in the 
dual polyhedra in sec 3.    
The divisors, $\{J_i\}$ are given by    $J_i = \{ a_j =0\}$ 
for $\exists j$, while 
 $\ell_i$ can be identified with  a Riemann surface. 
The intersection numbers of the toric   divisors and  $\ell_i$ 
is encoded   in the  components   of the  Mori vectors.    


The topological invariants 
 for the double K3 fibered  case in (III) are given by
%\\\\\\\\\\\\\\\\\\\\\\
\begin{align*}
{\cal K}^0{}^{(\III)}&=8\, t_1^3+ 2\, t_1^2  
 t_2+2 \, t_1^2  t_3+  t_1 t_2 t_3,\\
c_2 {\cdot} \vec J^{(\III)}&=(92,24,24).
\end{align*}


In single K3 fibered case of (IV), the  
Mori vectors  and the topological invariants 
 are given by
%\\\\\\\\\\\\\\\\\\\\\\\\\\\
\begin{alignat*}{2}
&( ~a_1,~ a_2,~ a_3,~ a_4,~ a_6,~ a_7,~ a_9   )
 &     \quad 
     \\
%-------------------------------------------------
&(\ph 3, \ph 2,  \ph 0, \ph 0,  \ph 0,  \ph 0,  \ph 1 )  & =  
 {\ell'}_1,& \quad 
  \\
&(\ph 0, \ph 0,  \ph 1,  \ph 0,  \ph 0,  \ph 1,  -2   )&=  
 {\ell'}_2,&  \quad  
   \\ 
&(\ph 0, \ph 0,  \ph 0,  \ph 1,  \ph 1,  -2,  \ph 0   )  & =  
 {\ell'}_3, &  \quad  .
\end{alignat*}
%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\





%\\\\\\\\\\\\\\\\\\\\\\
\begin{align*}
{\cal K}^0{}^{\rm (IV)}&=8 \, {t'}_1{}^3+4\, {t'}_1{}^2  {t'}_2
+2\,  {t'}_1  {t'}_2{}^2
+ 2 \, {t'}_1{}^2  {t'}_3+  {t'}_1
  {t'}_2  {t'}_3,\\
c_2 \cdot \vec J^{(\IV)}&=(92,48,24). 
\end{align*}
%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 


  
The double  K3 fibered CY phase in case (III) can be mapped to single 
K3 fibered one   by the positive integer valued linear transformations 
of $J$.
%\\\\\\\\\\\\\\\\


%\\\\\\\\\\\\\\\\
\[
( J_1, J_2, J_3) \rightarrow  
(\tilde J_1, \tilde  J_2, \tilde J_3)=
( J_1,  J_1+J_2,  J_3). 
\]
%\\\\\\\\\\\\\\\\\

In this single K3-fiber phase in (III), the Mori vectors   
  are given by  
%\\\\\\\\\\\\\\\\\\
\begin{alignat*}{2}
&( ~ a_1, ~ a_2, ~  a_4, ~ a_5, ~ a_6, ~ a_7,
~ a_9 & )
 &     \quad 
     \\
&(\ph 2,  \ph 0, \ph 3, \ph 0, \ph 0,  \ph 0, \ph 1 &  )  
&= \tilde{\ell}_1, \quad   \\
% - - - - - - - - - - - - 
& (\ph 0, \ph 0, \ph 0, \ph 0, \ph 1,  \ph 1,  -2 & ) & =   
\tilde \ell_2, \quad  \\
% - - - - - - - - --  - - - - - - --
&(  \ph 0, \ph 1, \ph 0, \ph 1, -1, -1,  \ph 0  & )  &=  
 \tilde{\ell}_3,\quad. 
\end{alignat*}
%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
  $ c_2 \cdot J$ and  ${\cal K}^0 $ derived from  $\tilde \ell_i$ 
 coincide  with those in the case (IV) derived from  $\ell_i^\prime $,  
 which leads to the matching of the  
Gromov--Witten  invariants of the both models. Recall that  
$\ell_i$, $\tilde \ell_i$ and   $\ell_i^\prime $ are 
the  Mori vectors \cite{mori}
\footnote{ $J=\sum_{i=1}^{5 }t_i J_i$ (${\text {Re}}~ t_i > 0$) is 
the K\"ahler class, $J_i$ being the generator of the K\"ahler cone,  
and $t_i$, $\tilde t_i$ and $t_i^\prime$ the complexified 
K\"ahler moduli. 
$t_i$ can also  be identified with the heterotic modulus under the  
application of heterotic type IIA duality \cite{yau, lerche}.
$\ell_i$ is the dual base of the   K\"ahler cone 
of a CY 3-fold.}.      
The  transformation means   
the restriction of K\"ahler cone of $\text{\bf F}_0$ case with 
double K3 fibers   to that with single K3 fiber. 
These double  K3 fibered  and single K3 fibered phases represent  
one and  the same CY3  with  two  different  local coordinates. 


 The  correspondence  of each Mori vector 
    leads to  the  coincidence of  
the intersection numbers of divisors. 


\vspace{1cm} 

\par
In $\Delta n_T\!=\!2$ case, the  CY phases  have been  
analyzed by Hosono \cite{hosono0} by  the method in  \cite{hosono1,hosono2}.
The situation in this case is   more complicated 
than  that in  $\Delta n_T\!=\!0$ case.  
There are $(\I^{\dagger})$, (III), (IV) and (V) models
\footnote{ The triangulations of model $(\I^{\dagger})$ 
coincide with those of case (III). } \footnote{ 
The phase $\alpha_{10}$ in (V) with three K3 fibers is 
called the   ``17 '' model in \cite{theisen}.}. 
 Four dual polyhedra do not have 
twisted sectors ( = non-toric degree of freedom).
The dual polyhedra do not coincide with each other  by
 SL(4,\text{\bf Z}) transformation.
Nevertheless, in some phases, topological invariants coincide with each other. 


\vspace{1cm}
\par
There are three ways to see the equivalence of these models.
\begin{itemize}
\item
criterion 1 : 

If $c_2 \cdot \vec J $  and   $ {\cal K}^0$ 
match  then it lead to the agreement of 
the Gromov--Witten invariants,  
$ N (\{n_i\})$. 
In this case,  the number of the K3 fibrations in  two phases is the same.


\item
criterion 2 : 

We first compare the  two K\"ahler cones in the same model. 
Then we can  use the transformation matrix of the  K\"ahler cones 
to see the relation of $ N (\{n_i\})$.
If these  invariants   agree with each other then  
these two phases are  the same.
In this case, the true K\"ahler cone 
 contains the union of the two K\"ahler cones. 


\item
criterion 3 : 

To compare the two phases belonging to the different models, 
we must find a  new equivalent  K\"ahler cone pairs. 
They belong to  the different models and   
one of which is  one to compare.
By this procedure,  one can reduce the  problem of a  comparison of 
 two K\"ahler cones   in  two different 
model to that two K\"ahler cones  in the same model.  
 
\end{itemize}






Wall's Theorem says that   
 the agreements of classical invariants,
 $c_2 \cdot  \vec J$ and ${\cal K}^0$,  lead to 
the  agreements  of  topology as well as Gromov--Witten invariants 
\cite{wall} in two CYs.  
Note that neither  the   
Mori vectors nor the  K\"ahler cones  coincide with
each other even when they are equivalent CY3 phases (see appendix 3). 




The identifications of CY3 phases  
 by the criterion 1 is shown in table 5.
The phases in the same line in table 5 have the same 
topological invariants.
%
Furthermore,  we can derive the other relations by 
 the criterion 2 or 3   
among each phases.
There are several  identifications   of the K\"ahler cones   
belonging to the different models.  
 By classifying the  identifications, we can derive some other relation 
between  the phases in the different models. 
For example, the relation between the  phase A in case (IV) 
 with a K3 fiber and  the  phase 5 in case (III) with 
double K3 fibers can be derived,  
(see appendix 3).   The Gromov--Witten invariants in 
phase A and 5 are  also transformed by this matrix. 
For the case of the phases in the same model in case (III), (IV) and (V), 
$N(\{n_i\})$ of  
each phases   can be transformed 
by  integer-valued linear transformations.
These transformations  are derived from the basis  of 
each K\"ahler cone, (see appendix 4)   
\footnote{
In general, the relations of the K\"ahler cones  are not simply  
restrictions even when  they  are in the same phase. 
If M$_{ij}$ in appendix 4 contains only positive integer valued,  
then $J_i$  is the subset of $J_j$ and the relation is the restriction.}.

\vspace{1cm} 
\par 
In conclusion,  all the phases defined  by the 
triangulation in  (III),  (IV) and (V)    
represent one and   the same CY phase.
Each of them corresponds to a local  
coordinate representation    of   the  CY3 phase.  
We   have obtained   the true phase for (IV) model by taking the union  
of the  five K\"ahler cones    
\footnote{ All the five phases in  (IV)  are       
represented by the  simplicial cones. However,  
 the union of five   K\"ahler cones is not simplicial and  has the 
 degenerate basis.  For (III) and (V),  
some phases   are not   simplicial. Therefore,  it is 
difficult to obtain the true phase as the union of   
K\"ahler cones from (III) and (V) models.}. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{1cm}
\par
 In $ \Delta n_T \geq $ 3, their properties  in toric data 
have the difference between case (III) and (IV):  
 twisted sectors 
(  non-toric  freedom  of the deformation
in K\"ahler moduli space ) appear in (IV).  On the other hand, there 
is no twist sector in (III). 
Thus, we can  analyze only the toric degree of freedom for 
the (IV) model :  
{\cal M}$^{\rm (IV)}_{\rm toric }$
\footnote{If one can add some one-cones to the dual polyhedra in (IV) to 
cancel the twist sector then the comparison 
of {\cal M}$^{\rm (IV)}_{\rm total }$ 
and  {\cal M}$^{\rm (III)}_{\rm total }$ is 
possible. The work is in progress.   }.





\newpage

\section{Future problem}
\vspace{0.5cm}


\vspace{1cm}
\par
In this paper, 
we derived the relation of three CY3 models with 
$(h^{1,1} h^{2,1})\!=\!(5, 185)$.  


The comparison of case (III) and case (IV) are interesting, 
since their relation seems to correspond to  the phase transition in  
the  heterotic-heterotic duality.     
The application to type IIB string  side of case (IV) is  
useful to see the non-perturbative property 
rather than case (III). The reason is that in (IV), K3 fiber has no 
elliptic singularity  and  all the enhanced gauge symmetries 
have   non-perturbative origin  in contrast to 
the situation of (III), which has   
the  mixture of the perturbative 
and non-perturbative enhanced gauge symmetries. 
These situations    are easily seen in type IIB side.
 By examinations of these CY3s,
 we can   derive the  physics in strong coupling region,  
which corresponds to the physics  in the  weak coupling region 
of  (III)\footnote{The work about type IIB side is in progress}. 


As an  application to physics,
 a   construction of similar examples of 
double K3 fibrations will   be useful  to 
derive non-perturbative property.

4D $N$\!=\!2 super YM theory  can  also be analyzed as  the 
heterotic strings compactified on 
$K3 \!\times\! \text{\bf T}^2$ in the weak coupling region. 
The threshold correction of case (I) in heterotic string side
 has been given   by the calculation of 
the partition function \cite{moore,kawai, curio2}. 
With the help of the generalized modular forms, 
it is possible to compute it with the perturbative Yukawa coupling 
\cite{moore,kawai, curio2}.
By using the methods of them, we will be able to  derive 
the perturbative  Yukawa couplings for (III)
\footnote{ For the $\Delta n_T=1$ case,  see appendix 8.}.  
It would be very interesting to apply them to models 
with a larger number of extra tensor multiplets. 






\vspace{1cm}

\text{\bf Acknowledgment}
\newline
I   would like to  thank  K. Mohri most  for  
fruitful suggestions and discussions.
I   would like to thank  S. Hosono  for    all  
 calculations   of Mori vectors and topological invariants in this paper 
 including the method of identification 
of phases. I  would like to  thank to N. Sakai  for useful    
discussions. 
 









\newpage

\text{\bf Appendix~ 1~ Yakawa couplings}
 
We follow the result, notations and definitions of \cite{hosono1,hosono2}. 
Let $M$  be a  CY3 and 
$M^\ast$ the mirror of $M$.

The Yukawa coupling of type IIB string side on $M^*$ are given as
the function of the algebraic coordinates $x_i$  by
%\\\\\\\\\\\\\\\\\\
\[
{\cal K}_{x_i,x_j,x_k}
=\int_{M^{*}} \Omega \wedge (  \partial_{x_i}  \partial_{x_j} 
 \partial_{x_k} \Omega ).
\]
%\\\\\\\\\\\\\\\\
The holomorphic three form   $\Omega(x)$  of $M^{*}$
 can be expanded as
as 
%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\[
\Omega(x)
=\sum^{h^{2,1}}_{a=0}
\left(z^a(x)\alpha_a-{\cal G}_a(x) \beta^a\right),
\]
%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
where  $\alpha_a$ and  $\beta^b$
are symplectic basis of $H^3(M^\ast;\text{\bf Z})$;
$z^a$, ${\cal G}_b$ are the period integrals 
with respect to the three-cycles 
dual to $\alpha_a $ and $\beta^b$ respectively.
We thus see that   
the Yukawa couplings can be expressed through these periods.
These periods are obtained as the solutions of 
the Picard--Fuchs (PF) 
differential operators and 
related to the K\"ahler  prepotential. 
The PF differential equation 
are equivalent to the  Gauss--Manin system of $M^{*}$. 

The dual polyhedron associated with $M$, 
which admits a toric realization as is the case of our models,
enables us to derive 
the K\"ahler cones, 
the Mori vectors, 
the PF differential operators  and the mirror maps.

The complexified K\"ahler class $J$ is given by
%\\\\\\\\\\\\\\\\\\\\ 
\[
%U(1) _{ij}&=  ( \ell^T)_i  \times J_j,\\ 
J=\sum^{h^{1,1}}_{i=1}t_i J_i
%=\sum^5_{i=1} \tilde{t}_i h_j
 \in H^2(M;\text{\bf C}),
\] 
%\\\\\\\\\\\\\\\\\\\\
where $\{J_i\}$ are the generators of the K\"ahler cone.

On the other hand, 
the Mori cone is the dual cone of the K\"ahler cone 
generated by the holomorphic curves  $\{\ell_j\}$, which satisfy  
$J_i\cdot\ell_ j=\delta_{ij}$.

Using this intersection pairings,
we can see that
the volume of the curve $\ell=\sum_{j}n_i\ell_j$
measured by  $J$ is    
$\text{vol}_J(\ell)=\sum^{h^{1,1}}_{i=1} n_i t_i$.






The   Yukawa couplings ${\cal K}_{x_i,x_j,x_k}$,
which was originally defined as the functions of the algebraic coordinates 
$\{x_i\}$ above,
can be expressed in terms of 
 the special coordinates $\{t_i\}$    
through the mirror map \cite{hosono1,hosono2}. 
\begin{align}
{\cal K}_{t_i,t_j,t_k}(t)
&={1 \over w_0(x(t))^2 }
\sum_{lmn} 
{\partial x_l \over \partial t_i}
{\partial x_m \over \partial t_j}
{\partial x_n \over \partial t_k}
{\cal K}_{x_l, x_m, x_n}(x(t)),
\nonumber \\
% - - - -- - - - -- - 
&={\cal K}^{0}_{ijk}+ 
\sum_{\{n_l\}}
 N(\{ n_l\} )n_i n_j n_k\,  
\frac{\prod_l q_l^{n_l}}{1\!-\!\prod_l q_l^{n_l}},
\end{align}
%\\\\\\\\\\\\\\\\\
where $q_i\!=\!\text{e}^{2\pi i t_i}$,
${\cal K}^{0}_{ijk}$ is the classical part
of the Yukawa coupling
%\\\\\\\\\\\\\\\\\\\\\\\\\\\
\[
{\cal K}^0_{ijk}:= \frac{1}{3!} d_{ijk} t_i t_j t_k,
\quad 
d_{ijk}=\int_M J_i \wedge J_j \wedge J_k,
\]
%\\\\\\\\\\\\\\\\\\\\\
and the sum in the second term of the second line above
runs over the sigma model instantons $C$:
$n_i\!=\!\int_C J_i$ is its $i$th degree,
and $N(\{n_i\})$ is the instanton number of the rational curves 
with the multidegree $\{n_i\}$.  


%%%%%%%%%%%%%%%%%%%%%%%%%
Let $(J_i,\ell_j)$ and $({J'}_i,{\ell'}_j)$ be the generators of the 
K\"ahler  and the Mori cones of the two CY phases.
If we find the linear transformation between them
%\\\\\\\\\\\\\\\\\\\\\\\\
%\begin{align*}
\[
{J'}_i=\sum_{j=1}^{h^{1,1}}
M_{i,j}J_j,
\quad
{\ell'}_i=\sum_{j=1}^{h^{1,1}}(M^{T})^{-1} {}_{i,j}\ell_j,
%\end{align*}
\]  
%\\\\\\\\\\\\\\\\\\
such that 
\[
N(\{n_i\})=N'(\{ {n'}_i\}),
\quad 
\text{for }\  {n'}_i=\sum_{j=1}^{h^{1,1}}n_j M_{j,i},
\]
then we can regard these two phases to be equivalent.
\footnote{An effective curve $C$ in one phase
may not be effective in the other phase, i.e., 
$[C]$ admits the expansion
$
[C]=\sum_{i=1}^{h^{1,1}}n_i\ell_i
=\sum_{j=1}^{h^{1,1}}{n'}_j{\ell'}_j,
$
where all the $n_i$ are  non-negative, while some of 
${n'}_j$ take  negative values.
In such cases, we compare
$N(\{n_i\})$ with 
$N'(\{ |{n'}_i|\})$.}
%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%\appendix
\text{\bf  Appendix 2~ Linear relations among the verticesin the dual}
\newline
\text{\bf  polyhedra    of three models}

\par
We list some
optional linear relations of one-cones ( called U(1) charge )
 in the dual polyhedra of three models, which are   denoted as 
 $Q_{i} $.   
They are assigned to each vertex of the dual polyhedra in Sec. 3,  whose 
 ordering  follows the one of the vertices  in Sec 3.  
\newline
\vspace{0.4cm}
\newline
transposed U(1) in (III)  : 
{\small
\begin{alignat*}{2}
&( Q_1,~ Q_2, ~Q_3, 
Q_4, Q_5, Q_6,
Q_7, Q_8, Q_9 ) 
& &
%\phantom{(\ph 0,  \ph 0, \ph 0, \ph 0, \ph 0, \ph 0, \ph 0, \ph 0, \ph 0 )}
\\
&(\ph 4, -1, \ph 1, \ph 6, \ph 0, \ph 2, \ph 0, \ph 0, \ph 0 ) 
& \rightarrow 
&(  \ph 8,  \ph 0, \ph 1, \,12, \ph 1, \ph 2, \ph 0, \ph 0, \ph 0 )\\
% - - - - - - - - -- - - - - - - - - - -  - 
&(\ph 4,  \ph 1, \ph 0, \ph 6, \ph 1, \ph 0, \ph 0, \ph 0, \ph 0 )
&       \rightarrow
&(\ph 4, \ph 1, \ph 0, \ph  6, \ph 1, \ph 0, \ph 0, \ph 0, \ph 0 )\\
% - - - - - -- - - - - - - - - - - -- - - - - 
&(\ph 4,  \ph 0, \ph 0, \ph 6, \ph 0, \ph 1, \ph 1, \ph 0, \ph 0 )
&{}        \rightarrow
&(\ph 4, \ph 0, \ph 0,  \ph 6, \ph 0, \ph 1, \ph 1, \ph 0, \ph 0 )\\
% - - - - - - - - - - - - - - - - -- - -
&(\ph 6, \ph 0, \ph 0, \ph 9, \ph 1, \ph 1, \ph 0, \ph 1, \ph 0 )
&{}     \rightarrow
&(\ph 6, \ph 0, \ph 0,  \ph 9, \ph 1, \ph 1, \ph 0, \ph 1, \ph 0 )\\
% - - - - - - - - - -  -- - - - - - -- -- - -
&(\ph 2,  \ph 0, \ph 0, \ph 3, \ph 0, \ph 0, \ph 0, \ph 0, \ph 1 ) 
&{} \rightarrow
&( \ph 2, \ph 0, \ph 0,  \ph 3, \ph 0, \ph 0, \ph 0, \ph 0, \ph 1 ).
\end{alignat*}
}
\vspace{1cm}




 transposed U(1) in (IV): 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small
\begin{align*}
&( Q_1, Q_2, ~Q_3, 
~Q_4, Q_5, Q_6,
Q_7,~ Q_8, Q_9 ) 
 \\
&(18, ~ 12, \po ~ 3,\po ~2, \po ~1, \po ~0, \po~ 0,\po~ 0, \po ~0 )\\
% - - - - - - - - - - - - - - - - -  
&(12,  \po ~8, \po ~2, \po ~1, \po ~0, \po ~1, \po ~0, \po ~0, \po ~0 )\\
% - - - - - - - - - - - - - - - - -  
&(\po 6,  \po ~4, \po ~1, \po ~0, \po ~0, \po ~0, \po ~1, \po ~0, \po ~0 )   \\
% - - - - - - - - - - - - - - - - -  
&(\po 9,  \po ~6, \po ~1, \po ~1, \po ~0, \po ~0, \po ~0, \po ~1, \po ~0 )\\
% - - - - - - - - - - - - - - - - -  
&(\po 3,  \po ~2,\po  ~0, \po ~0, \po ~0, \po ~0, \po ~0, \po ~0, \po ~1 ).
\end{align*}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{1cm}

transposed  U(1) in (V) :
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small
{\allowdisplaybreaks
\begin{align*}
( Q_1, Q_2, Q_3, 
Q_4, Q_5, Q_6,
 Q_7, Q_8, Q_9 ) 
&
\\
(\ph 1,\ph 0, \ph 0, \ph 0, \ph 1, -1, -1, \ph  0,  \ph 0 )
&\longrightarrow 
(\ph 6, \ph 4, \ph 0, \ph 1, \ph 1, \ph 0, \ph 0, \ph 0, \ph 0 )\\
% - - - - - - - - -        
(-1,  \ph 0,   \ph 0,  \ph 0,  -1,   \ph 1,    \ph 2,  \ph 2,  \ph 3 ) 
&\longrightarrow 
(\ph 6, \ph 4, \ph 1, \ph 0, \ph 0, \ph 0, \ph 1, \ph 0, \ph 0 )\\
% - - - - - - - - - -- 
( \ph 2, -2,   \ph 3, -2,   \ph 2,  -1,   -3, -2, -3 )
&\longrightarrow    
(\ph 3, \ph 2, \ph 0, \ph 0, \ph 0, \ph 0, \ph 0, \ph 0, \ph 1 )\\
% - - - - - - - - - - 
(-3,  \ph 2, -2,  \ph 1,  -2,   \ph 2,  \ph 3,  \ph 2,  \ph 3 )
&\longrightarrow  
(\ph 9, \ph 6, \ph 1, \ph 1, \ph 0, \ph 0, \ph 0, \ph 1, \ph 0 )\\
% - - - - - - - - - - - - 
(-2, \ph 1, -2,  \ph 2,  -3,   \ph 2,    \ph 2,  \ph 0,  \ph 0  )
&\longrightarrow  
(\ph 6, \ph 4, \ph 0, \ph 0, \ph 0, \ph 1, \ph 1, \ph 0, \ph 0 ).
\end{align*}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





 

\appendix{\text{\bf Appendix 3~ The relation of the phase A in (IV) and phase e 
in (III) }}


\par
We can not use criterion 1 for the phase A in (IV) since the $C_2 \cdot J$ 
and $\vec J^3$  of  A  do not coincide    
  with those of phases in   (III). 
We use criterion 2 and criterion 3 to show 
that the phase A is equivalent to any phase in (III). 
There are at least two projections    
 to relate the Mori vectors of phases in (III)
and in (IV). One way contains three examples, phase B = phase c,
phase C= phase d, phase D = phase b. The other way contains an example,
phase E = phase a. 
 We can derive the  Mori vectors  
of  a new phase in (III)  which is equivalent to the  
phase A in (IV)  using these projections. The new phase 
 in (III) has the same topological invariants, $c_2 \cdot \vec J$ and 
$\vec J^3$ as those of A phase in (IV).  
The linear relation of the  Mori vectors 
of the new phase in (III) and any phase in (III)  
can be used to see the relation of topological invariants,
$N(\{ n_i \})$.    
For example, we show phase A = phase e by the above method.     
Any phase in (III) is equivalent to each other by method 2 (see appendix 4). 
Therefore, the phase A is equivalent to any  phase in (IV).

We use the  projection  which contains three examples. 
We label each correspondence of the Mori vectors by $f_i$.
\vspace{0.3cm} 



\noindent
%$\{\ell_i\}_{(\text{B})}$ versus $\{\ell_i \}_{({\text c })}$:  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small  
\begin{align*}
\phantom{(\ph 0, \ph 0, \ph 0}\{\ell_i\}_{(\text{B})}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0}
&\phantom{\longleftrightarrow}
\phantom{(\ph 0, \ph 0, \ph 0}\{\ell_i\}_{({\text c })}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0}\\
% - - - - -  - - - - - - - --  -- - - - --
( 3, \ph 2, \ph 0, \ph 0, -1, \ph 1, \ph 0, \ph 1, \ph 0)
&\stackrel{f_1 }{ \longleftrightarrow }  
( 2, -1, \ph 0, \ph 3, \ph 0, \ph 1, \ph 0, \ph 1, \ph 0) 
\\
% - - - - - - - - - - - - - - - - - - - - - -- -         
( 0, \ph 0, \ph 0, \ph 0, \ph 1, -1, \ph 0, -1, \ph 1) 
&\stackrel{f_2}{\longleftrightarrow}
( 0, \ph 1, \ph 0, \ph 0,\ph  0,-1, \ph 0, -1, \ph 1) 
\\ 
% - - - - - - - - - - - - - -  - - - - - - - -  - -        
( 0, \ph 0, \ph 0, \ph 1,\ph  0, \ph 1, -2, \ph 0, \ph 0)
&\stackrel{f_3}{\longleftrightarrow}
( 0, \ph 0, \ph 1, \ph 0, \ph 1, \ph 0, -2, \ph 0, \ph 0)
\\
% - - - - - - - - - - - - - - - - - -  - - - - - - - -         
( 0, \ph 0, \ph 1,\ph  0, \ph 0,\ph  1,\ph  0, -1, -1)
&\stackrel{f_4}{\longleftrightarrow}
( 0, \ph 0, \ph 1, \ph 0, \ph 0, \ph 1, \ph 0, -1, -1)
\\ 
% - - - - - - - - - - -  - - - - - - -        
( 0, \ph 0, \ph 0, \ph 0, \ph 0, -1, \ph 1, \ph 1, -1)
&\stackrel{f_5}{\longleftrightarrow}
( 0, \ph 0, -1, \ph 0, \ph 0, \ph 0, \ph 1, \ph 1, -1).
\end{align*}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\noindent
%$\{\ell_i\}_{(\text{C})}$ versus $\{\ell_i\}_{({\text d})}$:                        
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small
\begin{align*}
\phantom{(\ph 0, \ph 0, \ph 0}\{\ell_i\}_{(\text{C})}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0}
&\phantom{\longleftrightarrow}
\phantom{(\ph 0, \ph 0, \ph 0}\{\ell_i\}_{({\text d })}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0}\\
% - - - -  - -  -- 
 ( 3, \ph 2, \ph 0,  \ph 0,  -1,  \ph 1,    \ph 0,  \ph 1,  \ph 0)
&\stackrel{f_1 }{\longleftrightarrow}
( 2, -1,  \ph 0,  \ph 3,  \ph 0,  \ph 1, \ph  0,  \ph 1,  \ph 0)
\\
% -  - - - - - -  - - - - - - - - - - - - - -  - - - - -         
( 0, \ph 0, \ph 0,  \ph 0,   \ph 1, -2,    \ph 1,  \ph 0,  \ph 0)
&\stackrel{f_6 }{\longleftrightarrow}
( 0,  \ph 1, -1,  \ph 0,  \ph 0, -1,  \ph 1,  \ph 0,  \ph 0)
\\
% - - - - - - -  - - - - - - - - - - - -  -     
( 0, \ph 0, \ph 0,  \ph 1,   \ph 0,   \ph 1,  -2,  \ph 0,  \ph 0)
&\stackrel{f_3 }{\longleftrightarrow}
( 0, \ph 0, \ph 1, \ph 0, \ph 1, \ph 0, -2, \ph 0,  \ph 0)   
\\
% - - - - - - - -  - - - - - - - - -  -         
( 0, \ph 0, \ph 0, -1,  \ph 0,   \ph 0,\ph  1, -1,  \ph 1)  
&\stackrel{f_7 }{\longleftrightarrow}
( 0, \ph 0, \ph 0, \ph 0, -1, \ph 0, \ph 1, -1, \ph 1) 
\\
% - - - - - - - -  - - - - - - - - - - - - - -  - - -         
( 0, \ph 0, \ph 1,  \ph 1,   \ph 0,   \ph 0,    \ph 0,   \ph 1,  -3) 
&\stackrel{f_8 }{\longleftrightarrow} 
( 0, \ph 0, \ph 0, \ph 0, \ph 1, \ph 1, \ph 0, \ph 1, -3) 
\end{align*}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    

\noindent
%$\{\ell_i\}_{(\text{D})}$ versus $\{\ell_i\}_{({\text b})}$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small 
\begin{align*}
\phantom{(\ph 0, \ph 0, \ph 0}\{\ell_i\}_{(\text{D})}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0}
&\phantom{\longleftrightarrow}
\phantom{(\ph 0, \ph 0, \ph 0}\{\ell_i\}_{({\text b})}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0}\\
% - - - - - - - - - - - -- 
 ( 3, \ph 2, \ph 0, \ph 0, -1, \ph 1, \ph 0, \ph 1, \ph 0)
&\stackrel{f_1 }{\longleftrightarrow } 
( 2,  -1, \ph 0, \ph 3, \ph 0, \ph 1, \ph 0, \ph 1, \ph 0)
\\
% - - - - - - - - -  - - - - - -  - - - -         
( 0, \ph 0, -1, \ph 0, \ph 0, -1, \ph 0, \ph 1, \ph 1)
&\stackrel{f_4 }{\longleftrightarrow}  
( 0, \ph 0, -1, \ph 0, \ph 0, -1, \ph 0, \ph 1, \ph 1) 
\\
% - - - - - - - - - - - - - - - - -  -         
( 0, \ph 0, \ph 0, \ph 1, \ph 0, \ph 1, -2, \ph 0, \ph 0)
&\stackrel{f_3 }{\longleftrightarrow } 
( 0, \ph 0,\ph  1, \ph 0, \ph 1, \ph 0, -2, \ph 0, \ph 0)   
\\
% - - - - - - - - - - - - - - - - - - - -          
( 0, \ph 0, \ph 1, \ph 0, \ph 1, \ph 0,\ph  0,  -2, \ph 0)
&\stackrel{f_9 }{\longleftrightarrow} 
( 0, \ph 1, \ph 1, \ph 0, \ph 0, \ph 0, \ph 0, -2, \ph 0) 
\\
% - - - - - - - - - - - - - - -  -         
( 0, \ph 0, \ph 1, \ph 0, \ph 0, \ph 0, \ph 1, \ph 0,  -2)
&\stackrel{f_{10} }{\longleftrightarrow}
( 0, \ph 0, \ph 0, \ph 0, \ph 0, \ph 1, \ph 1, \ph 0, -2)
\end{align*}
} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
%\newpage       
Therefore,  
$\{\ell_i \}$ of the new phase  in (III) side  which is 
equivalent to $\{\ell_i\}_{(\text{A})}$ is:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small 
\begin{align*}
\phantom{(\ph 0, \ph 0, \ph 0}\{\ell_i\}_{(\text{A})}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0}
&\phantom{\longleftrightarrow}
\phantom{(\ph 0, \ph 0, \ph 0}\{\ell_i\}_{({\text{ new}})}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0}\\
( \ph3, \ph 2, \ph 0, \ph 0, \ph 0, \ph 0, \ph 0, \ph 0, \ph 1)
&\stackrel{f_1+f_2 }{\longleftrightarrow} 
( \ph 2,\ph 0,\ph 0,\ph 3,\ph 0,\ph 0, \ph 0,  \ph 0,  \ph 1)
\\
% - - - - - - - - - - - - -  - - - - - - - -   
 ( \ph 0, \ph 0, \ph 0, \ph 0, \ph 1, -2, \ph 1, \ph 0, \ph 0)
&\stackrel{f_6}{\longleftrightarrow}
(\ph 0,\ph 1,-1,\ph 0,\ph 0,-1,\ph 1, \ph 0,\ph 0)
\\
% - - - -- - - - - - - - - - - - - -  - - -
 (\ph 0, \ph 0, \ph 0, \ph 1, \ph 0, \ph 1, -2,\ph  0, \ph 0)
&\stackrel{f_3 }{\longleftrightarrow}
 (\ph 0,\ph 0,\ph 1,\ph 0,\ph 1,\ph 0,-2,\ph 0,\ph 0)   
\\
% - - - - - - -  - - - - - - - -  - - - - - -
 ( \ph 0, \ph 0, \ph 1, \ph 0, \ph 1, \ph 0, \ph 0, -2, \ph 0)
 &\stackrel{f_9 }{\longleftrightarrow}
 ( \ph 0, \ph 1,\ph 1,\ph 0,\ph 0,\ph 0,\ph 0,-2,\ph 0) 
\\
% - - - - - - - - - - - - - - -  - - - - - - -
 (\ph 0, \ph 0, \ph 0, \ph 0, -1, \ph 1, \ph 0, \ph 1, -1)
&\stackrel{f_2 }{\longleftrightarrow }
 (\ph  0,-1,\ph 0,\ph 0,\ph 0, \ph 1,\ph 0,\ph 1,-1)
\end{align*}   
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




    
  
The transformation from the topological invariants of the 
phase e to those of the  phase A is represented
by integer-valued matrix, 
$M_{\text {Ae}}$ : phase e $\rightarrow$ phase A
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[ 
M_{\text{Ae}} := 
\begin{pmatrix}
 1& \ph 0&  \ph 0&  \ph 0&  \ph 1\\    
 0& \ph 0& -1&  \ph 0& -1\\
 0& \ph 0&  \ph 0&  \ph 1&  \ph 1\\
 0& \ph 1&  \ph 0&  \ph 0&  \ph 1\\ 
 0& \ph 0&  \ph 1&  \ph 0&  \ph 0
\end{pmatrix}.
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The correspondence between phase A and e 
is as follows.

$n_{i{\text{ A}}} \leftrightarrow
 \sum_j (M_{\text {Ae}}^T{})^{(-1)}{} _{ij}   
 n^\prime {}_j{}_{\text{(e)}}$.
 $N(\{ n_i\}  )_{(\text{A})} \! =\!
  N^\prime 
(\{(M_{\text{ Ae}}^T ){}^{(-1)}{}_{ij}    {n}^\prime_j \})_{\text {(e)}} 
$, which leads to  
the conclusion : phase A = phase e.  

                                             
For example,  the correspondence  of 
$N( \{  n_i \})$ in   phase A and e are given by   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[
\begin{array}{rrrrrrrr}
N(1, 0, 0, 0, 0)_{(\text{A})}&=&360 & \rightarrow& 
N(1, 0, 0, 0, 1)_{\text{(e)}}&=&360,
\\
N(0, 0, 0, 0, 1)_{(\text{A})}&=&1 &\rightarrow & 
N(0, 0, 1, 0, 0)_{\text{(e)}}&=&1,
\\
N(1, 1, 0, 0, 1)_{(\text{A})}&=&252&\rightarrow&  
N(1, 0, 0, 0, 0)_{\text{(e)}}&=&252,
\\
N(0, 1, 1, 0, 1)_{(\text{A})}&=&1& \rightarrow & 
N(0, 0, 0, 1, 0)_{\text{(e)}}&=&1, 
\\
N(1, 1, 0, 1, 1)_{(\text{A})}&=&252& \rightarrow & 
N(1, 1, 0, 0, 1)_{\text{(e)}}& =& 252.
\\
\nn
\end{array}
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%









        
 The  other 
 correspondence  such as  $\{\ell_i\}_{(\text{E})}= 
\{\ell_i\}_{(1)}$ is 

\noindent
$\{\ell_i\}_{(\text{E})}$: equivalent to  $\{\ell_i \}_{(1)}$ in (III) side :
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small
\begin{align*}
( 3, \ph 2, \ph 0, \ph 0, -1, \ph 1, \ph 0, \ph 1, \ph 0)
&\stackrel{f_1'}{\longleftrightarrow} 
( 2,  \ph 0, -1, \ph 3,  \ph 0, \ph 0,   \ph 1, \ph 1,  \ph 0)
\\
% - - - - - - - - -  - - - - - - - - - - -  -         
( 0, \ph 0, \ph 0, \ph 0, \ph 1, -2, \ph 1, \ph 0, \ph 0) 
&\stackrel{f_2'}{\longleftrightarrow} 
( 0,  \ph 0,  \ph 1, \ph 0,  \ph 1, \ph 0, -2,  \ph 0,  \ph 0)
\\
% - - - - - - - - -  - - - -- - - -  - - - - - -         
( 0, \ph 0, \ph 0, \ph 1, \ph 0, \ph 0, -1, \ph 1, -1) 
&\stackrel{f_3'}{\longleftrightarrow}  
( 0, -1,  \ph 0, \ph 0,  \ph 0, \ph 1,  \ph 0, \ph 1, -1) 
\\
% - - - - - - - - -  - - - - - - -         
( 0, \ph 0, \ph 0, \ph 0, \ph 0, \ph 1, -1, -1, \ph 1) 
&\stackrel{f_4'}{\longleftrightarrow}  
( 0,  \ph 0,  \ph 0, \ph 0, -1, \ph 0,  \ph 1, -1,  \ph 1)
\\
% - - - - - - - - - - - - - - - - - -          
( 0, \ph 0, \ph 1, \ph 0, \ph 0, \ph 0, \ph 1, \ph 0, -2) 
&\stackrel{f_5'}{\longleftrightarrow } 
( 0,  \ph 1,  \ph 0, \ph 0,  \ph 1, \ph 0,  \ph 0,  \ph 0, -2).
\end{align*}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
There is another new phase in (III) which correspond   to phase E 
by using the previous relations.

\noindent
$\{\ell_i\}_{(\text{E})}$:  equivalent  to
$\{\ell_i \}$ of a new phase  in (III) side :
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small
\begin{align*}
 ( 3, \ph 2, \ph 0, \ph 0, -1, \ph 1, \ph 0, \ph 1, \ph 0)
&\stackrel{f_1 }{\longleftrightarrow }
 ( 2,-1, \ph 0, \ph 3, \ph 0, \ph 1, \ph 0, \ph 1, \ph 0)
\\
% - - - - - - - -  - - - -
( 0, \ph 0, \ph 0, \ph 0, \ph 1, -2, \ph 1, \ph 0, \ph 0) 
&\stackrel{f_6 }{\longleftrightarrow }  
( 0, \ph 1, -1, \ph 0, \ph 0, -1, \ph 1, \ph 0, \ph 0)
\\
% - - - - - - - - - - - - - -    
 ( 0, \ph 0, \ph 0, -1, \ph 0, \ph 0, \ph 1, -1, \ph 1) 
&\stackrel{f_7 }{\longleftrightarrow }  
( 0,\ph 0,\ph 0,\ph 0,-1,\ph 0,\ph 1,-1,\ph 1) 
\\
% - - - - - - - - - -- - - - - - 
( 0, \ph 0, \ph 0, \ph 0, \ph 0, -1, \ph 1, \ph 1, -1)
&\stackrel{f_6 }{\longleftrightarrow }
( 0, \ph 0,-1, \ph 0, \ph 0, \ph 0, \ph 1, \ph 1,-1)
\\
% - - - - - - - - - -  - -
( 0, \ph 0, \ph 1, \ph 0, \ph 0, \ph 0, \ph 1, \ph 0, -2)
&\stackrel{f_9 }{\longleftrightarrow } 
( 0, \ph 0, \ph 0, \ph 0, \ph 0, \ph 1, \ph 1, \ph 0,-2).
\end{align*}
 }


 
\appendix{\text{\bf Appendix~4~ 
The transformation   among K\"ahler cones  in (III)}}



\par 
To see the relations among the phases in the same model, we can use 
the transformation matrix of the  Mori vectors in each models. They   
can be derived by the basis of K\"ahler cones.
By using them, we can see  whether each phase has the same 
quantum Yukawa couplings or not, i.e., its topology matches or not.
We listed them in  (III) model. For example,  
$J_{a \cdots  h}^{(\III)}$ denotes the basis of the   K\"ahler cone 
in $a \cdots h$ phase of case (III).  
A linear transformation matrix of generators 
from phase b to phase a is, 
%\\\\\\\\\\\\\\\\\\\\\\\\
\[
M_{\text {ab}}^{(\III)}:
J_{\text b}^{ (\III)}\rightarrow J_{\text a}^{(\III)}.
\]
%\\\\\\\\\\\\\\\\\\\\\\\\\
All phases are transformed to each other by the integer-valued 
linear transformations in (III).   

\par 
 We see that phase d = phase e   by using $M_{\text {ed}}$, for example.
%\\\\\\\\\\
%\begin{align*}
\[
 \{n_i \}_{\text {(d)}}=\{ 
\sum_j (M_{\text {ed}}^T ){}^{(-1)}{}_{ij}  n_j{} _{\text {(e)}}\},
\qquad
N(\{ n_i\})_{\text {(d)}}= N^\prime(\{\sum_j(M_{\text {ed}}^T)^{(-1)}{}_{ij} 
 n_j^\prime{}_{\text {(e)}} \} ),
%\end{align*}
\]
%\\\\\\\\\\\\\\
where the integer-valued 5 $\times 5$ matrix $M_{\text {ed }}$ 
is defined by
%\\\\\\\\\\\\\\
$
(M_{\text {ed}}^T){}^{(-1)}{}_{ij}  \ell_j{} _{\text {(e)}} 
= \ell_i{} _{\text {(d)}}.
$
%\\\\\\\\\\\\\\\\\\\


{\small
\begin{alignat*}{2}
M_{\text {da}}^{\III}&=
\begin{pmatrix}
 1 & \ph 0 &  \ph 0 & \ph 0 & \ph 0\\ 
 1 & \ph 0 & -1 & \ph 0 & \ph 1\\ 
 0 & \ph 1 & -1 & \ph 0 & \ph 1\\
 0 & \ph 0 & -1 & \ph 1 & \ph 1\\ 
 0 & \ph 0 &  \ph 0 & \ph 0 &\ph  1
\end{pmatrix},
& \quad
M_{\text {db}}^{\III}&=
\begin{pmatrix}
 1 &  \ph 0 & \ph 0 & \ph 0 & \ph 0\\
 0 &  \ph 0 & \ph 0 & \ph 1 & \ph 0\\ 
 0 & -1 & \ph 1 & \ph 2 & \ph 0\\ 
 0 & -2 & \ph 0 & \ph 3 & \ph 1\\ 
 0 & -1 & \ph 0 & \ph 1 & \ph 1
\end{pmatrix},
\\
% - - - - - - - - - - - - -  - --  - - - -
M_{\text {dc}}^{\III}
& = \begin{pmatrix}
 1 & \ph 0 & \ph 0 & \ph 0 & \ph 0\\ 
 0 & \ph 1 & \ph 0 & \ph 0 & \ph 0\\ 
 0 & \ph 1 & \ph 1 & \ph 1 & -1\\ 
 0 & \ph 1 & \ph 0 & \ph 2 & -1\\ 
 0 & \ph 0 & \ph 0 & \ph 1 & \ph 0
\end{pmatrix},
& \quad
M_{\text {de}}^{\III} &= 
\begin{pmatrix}
 1 & \ph 0 & \ph  0 & \ph  0 &\ph  0\\ 
 1 & \ph 1 & -1 &  \ph 0 & \ph 0\\ 
 0 & \ph 1 & -1 &  \ph 0 & \ph 1\\
 0 & \ph 2 & -1 & -1 & \ph 1\\ 
 0 & \ph 1 & \ph  0 & \ph  0 &\ph  0
\end{pmatrix},
\\
% - - - - - - - - - - - - - - - - - - - - - 
M_{\text {df}}^{\III} &= 
\begin{pmatrix}
 1 & \ph 0 & \ph 0 & \ph 0 &  \ph 0\\ 
 1 & \ph 1 & \ph 1 & \ph 0 & -1\\ 
 0 & \ph 2 & \ph 1 & \ph 0 & -1\\
 0 & \ph 3 & \ph 1 & \ph 1 & -2\\ 
 0 & \ph 1 & \ph 1 & \ph 1 & -1
\end{pmatrix},
& \quad
M_{\text {dg}}^{\III} &= 
\begin{pmatrix}
 1 & \ph 0 & \ph 0 &  \ph 0 &  \ph 0\\ 
 1 & \ph 0 & \ph 1 & -1 &  \ph 0\\ 
 1 & \ph 1 & \ph 2 & -1 & -1\\
 1 & \ph 0 & \ph 3 & -1 & -1\\ 
 0 & \ph 0 & \ph 1 & \ph  0 & \ph  0
\end{pmatrix},
\\
% - - - - - - --  - - - - - - - - - - - - -
M_{\text {dh}}^{\III} &= 
\begin{pmatrix}
 0 & \ph 0 & \ph 1 & -1 & \ph 1\\ 
 0 & \ph 0 & \ph 0 & \ph 0 & \ph 1\\ 
 0 & \ph 0 & \ph 0 & \ph 1 & \ph 0\\ 
 1  & -1 & \ph 0 & \ph 1 & \ph 0\\ 
 1 & \ph 0 & \ph 0 & \ph 0 & \ph 0
\end{pmatrix}.
& &{} 
\end{alignat*}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




Similarly, one can derive the transformation matrices 
of the  K\"ahler cones in  (IV) and (III). Their transformation 
matrices are also integer valued.
Some examples such as  $ M_{\alpha_i\alpha_j}^{\rm (V)}$ are given by
as follows. 
$\{J_{\alpha_i} ^{(\V)} \}$ denotes the basis of  the  K\"ahler cone in 
$\alpha_i$  phase of case (V).  
A linear transformation  of K\"ahler cone is,
%\\\\\\\\\
\[
M_{\alpha_{i} \alpha_j}^{(\V)}:
J_{\alpha_j}^{(\V)} \rightarrow J_{\alpha_{i}}^{(\V)}. 
\]





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small
\begin{alignat*}{2}
M_{\alpha_{10} \alpha_{5}}^{(\V)}& = 
\begin{pmatrix}
\ph 0 & \ph 0 & \ph 1 & \ph 0 & \ph 0\\ 
\ph 0 & -1 & \ph 1 & \ph 1 & \ph 1\\
\ph 0 & \ph 0 & \ph 0 & \ph 1 & \ph 0\\
\ph 0 & \ph 0 & \ph 0 & \ph 0 & \ph 1\\ 
-1 & \ph 0 & \ph 1 & \ph 1 & \ph 1
\end{pmatrix},
& \quad
M_{\alpha_{10}\alpha_{1}}^{(\V)} &= 
\begin{pmatrix}
\ph 0 & \ph 1 & \ph 0 & \ph 0 & \ph 0\\ 
\ph 0 & \ph 0 & \ph 1 & \ph 0 & \ph 0\\ 
-1 & \ph 1 & \ph 1 & \ph 0 & \ph 0\\
\ph 0 & \ph 0 & \ph 0 & \ph 1 & \ph 0\\ 
\ph 0 & \ph 0 & \ph 0 & \ph 0 & \ph 1
\end{pmatrix},
\\
%  - - - - - - - - -  - - - - - - - 
M_{\alpha_{18} \alpha_1 }^{(\V)} &= 
\begin{pmatrix}
-2 & \ph 3 & \ph 0 & \ph 1 & \ph 1\\
-1 & \ph 2 & -1 & \ph 2 & \ph 1\\ 
-1 & \ph 2 & \ph 0 & \ph 1 &\ph 0\\
\ph  0 & \ph 1 & \ph 0 & \ph 0 & \ph 0\\ 
-1 & \ph 1 & \ph 0 & \ph 1 & \ph 1
\end{pmatrix},
& \quad
M_{\alpha_{3}\alpha_1}^{(\V)} &= 
\begin{pmatrix}
-1 & \ph 2 & \ph 0 & \ph 0 & \ph 1\\ 
\ph 0 & \ph 1 & \ph 1 & -1 & \ph 0\\ 
\ph 0 & \ph 1 & \ph 0 & \ph 0 & \ph 0\\
-1 & \ph 1 & \ph 1 & \ph 0 & \ph 0\\ 
\ph 0 & \ph 0 & \ph 0 & \ph 0 & \ph 1
\end{pmatrix}.
\end{alignat*}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%













%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newpage 
\appendix{\text{\bf Appendix~5~ 
The relation among  CY3 phases in (V)}}


The correspondence of the  Mori vectors in each phase in (V) 
are all explained by some  replacements of  three of 
  vertices pairs, which lead to the correspondence  
of the divisors, see Fig. 1.
\begin{itemize}
\item
case 1
\newline

To get the  Mori vectors in  the  phase $\alpha_8$  
from those in   $\alpha_4$,
$a_1$ and $ a_4$ in the  phase $\alpha_4$    
are replaced with 
$a_5$ and $a_2$  and  vice versa.   

some examples: 
\newline
\noindent



phase $\alpha_4$= phase $\alpha_8$, 
phase $\alpha_7$ = phase $\alpha_{17}$, phase $\alpha_2$= phase $\alpha_{12}$,
\newline
phase $\alpha_1$= phase $\alpha_{15}$ and 
phase $\alpha_{15}$ = phase $\alpha_{16}$.


%$\{\ell_i\}_{(\alpha_4)}^{\V}$ versus  $\{\ell_i \}_{(\alpha_8)}^{\V}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small
\begin{align*}
\phantom{(\ph 0, \ph 0, \ph 0,}\{ \ell_i \}_{(\alpha_4)}^{(\V)}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0)}
&\phantom{\leftrightarrow}
\phantom{(\ph 0, \ph 0, \ph 0,}\{ \ell_i \}_{(\alpha_8)}^{(\V)}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0)}
\\
% - - - - -- - -
(a_1,~ a_2, ~a_3,~ a_4, ~a_5, ~a_6, ~a_7, ~a_8,~ a_9 ) 
&\leftrightarrow 
(a_1,~ a_2,~ a_3,~ a_4, ~a_5, ~a_6,~ a_7, ~a_8,~ a_9)\\
% - - - - - - - - - - - - - 
(-1, \ph 1, -1, \ph 0, \ph 0, \ph 0, \ph 1, \ph 0, \ph 0 ) 
&\leftrightarrow 
(\ph 0, \ph 0, -1, \ph 1, -1, \ph 0, \ph 1, \ph 0, \ph 0)\\
% - - - - - - - - - - - - - 
(\ph 1, -1,\ph 1, \ph 0, \ph 0,\ph 0,\ph 0, \ph 2, \ph 3) 
&\leftrightarrow
(\ph 0, \ph 0, \ph 1, -1, \ph 1, \ph 0, \ph 0, \ph 2, \ph 3)\\
%  - - - - - - - - - - - - - -      
 (\ph 0, \ph 0, \ph 0, \ph 1, -1, \ph 1, -1, \ph 0, \ph 0) 
&\leftrightarrow
(-1, \ph 1, \ph 0, \ph 0, \ph 0, \ph 1, -1, \ph 0, \ph 0) \\
%  - - - - - - - - - -  - - - - - - 
(\ph 0, \ph 0, \ph 1, -1, \ph 1, \ph 0, -1, \ph 0, \ph 0)  
&\leftrightarrow
(\ph 1, -1, \ph 1, \ph 0, \ph 0, \ph 0, -1, \ph 0, \ph 0) \\
% - - - - - - - - - - - - - - - - -     
(\ph 1, \ph 0, \ph 0, \ph 0, \ph 1, -1, -1, \ph 0, \ph 0)  
&\leftrightarrow
(\ph 1, \ph 0, \ph 0, \ph 0, \ph 1, -1, -1, \ph 0, \ph 0).
\end{align*}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item
case 2 




To get the  Mori vectors in  the  phase $\alpha_{16}$  
from those in   $\alpha_{15}$,
$a_1$,  $ a_4$, $a_3$ and  $ a_6$,     
in the  phase $\alpha_{15}$    
are replaced with 
$a_2$, $a_5$, $a_6$ and   $ a_3$ and 
   vice versa.   






some examples:
\newline
phase $\alpha_{11}$= phase $\alpha_{13}$, 
phase $\alpha_2$ = phase $\alpha_{13}$ and 
phase $\alpha_{15}$ = phase $\alpha_{16}$. 


%$\{\ell_i\}_{(\alpha_{15})}^{V}$ versus:
%$\{\ell_i \}_{(\alpha_{16})}^{V}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small 
\begin{align*}
\phantom{(\ph 0, \ph 0, \ph 0,}\{ \ell_i \}_{(\alpha_{15})}^{(\V)}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0)}
&\phantom{\leftrightarrow}
\phantom{(\ph 0, \ph 0, \ph 0,}\{ \ell_i \}_{(\alpha_{16})}^{(\V)}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0)}\\
% - - - - - 
(\ph 1, -1, \ph 0, \ph 0, \ph 0, -1, \ph 1,\ph 0,\ph 0) 
&\leftrightarrow
(\ph 0, \ph 0, \ph 0, -1, \ph 1, -1, \ph 1,\ph 0,\ph 0)\\    
% - - - - - - - -- - - - - - - - 
(-1, \ph 1, \ph 0, \ph 0, \ph 0, \ph 1,\ph 0,\ph 2,\ph 3)   
&\leftrightarrow
(\ph 0, \ph 0, \ph 0, \ph 1, -1, \ph 1, \ph 0,\ph 2,\ph 3)\\
% - - - - - - - -- - - - - - - - 
(\ph 0, \ph 0, \ph 1, -1, \ph 1, \ph 0,-1,\ph 0,\ph 0) 
&\leftrightarrow
(\ph 1, -1, \ph 1, \ph 0, \ph 0, \ph 0, -1,\ph 0,\ph 0) \\
% - - - - - - - -- - - - - - - - 
(\ph 0, \ph 1, -1, \ph 1, \ph 0, \ph 0, -1,\ph 0,\ph 0)  
&\leftrightarrow
(\ph 0, \ph 1, -1, \ph 1, \ph 0, \ph 0,-1,\ph 0,\ph 0) \\
% - - - - - - - -- - - - - - - - 
(\ph 0, \ph 0, \ph 0, \ph 1, -1, \ph 1,-1,\ph 0,\ph 0)  
&\leftrightarrow
(-1, \ph 1, \ph 0, \ph 0, \ph 0, \ph 1,-1,\ph 0,\ph 0).
\end{align*}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item
case 3






To get the  Mori vectors in  the  phase $\alpha_{2}$  
from those in   $\alpha_{11}$,
$a_1$ and  $ a_4$,      
in the  phase $\alpha_{11}$    
are replaced with 
$a_3$ and $a_6$    and  vice versa.   






some examples:

phase $\alpha_2$= phase $\alpha_{ 11}$ and 
phase $\alpha_ 7$= phase $\alpha_ {12}$


%$\{\ell_i\}_{(\alpha_2)}^{(V)}$ versus 
% $\{\ell_i \}_{(\alpha_{11})}^{(V)}$
%%%%%%%%%%%%%%%%%%%%
{\small 
\begin{align*}
\phantom{(\ph 0, \ph 0, \ph 0,}\{ \ell_i \}_{(\alpha_2)}^{(\V)}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0)}
&\phantom{\leftrightarrow}
\phantom{(\ph 0, \ph 0, \ph 0,}\{ \ell_i \}_{(\alpha_{11})}^{(\V)}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0)}\\
% - - - - - - - -- 
 (\ph 0, \ph 0, \ph 0, -1, \ph 1, -1, \ph 1, \ph 0, \ph 0)  
&\leftrightarrow
 (\ph 0, \ph 0, \ph 0, -1, \ph 1, -1, \ph 1, \ph 0, \ph 0)\\   
 (\ph 1, -1, \ph 0, \ph 1, -1, \ph 0, \ph 0, \ph 0, \ph 0)  
&\leftrightarrow
 (\ph 0, -1, \ph 1, \ph 0, -1, \ph 1, \ph 0, \ph 0, \ph 0)\\   
 (-1, \ph 1, \ph 0, \ph 0, \ph 0, \ph 1, \ph 0, \ph 2, \ph 3)    
&\leftrightarrow
(\ph 0,\ph  1, -1, \ph 1, \ph 0, \ph 0, \ph 0, \ph 2, \ph 3) \\  
 (\ph 0, \ph 0, \ph 1, \ph 0, \ph 0, \ph 1, -2, \ph 0, \ph 0)   
&\leftrightarrow
(\ph 1, \ph 0, \ph 0, \ph 1, \ph 0, \ph 0, -2, \ph 0, \ph 0) \\
(\ph 0, \ph 1, -1, \ph 1, \ph 0, \ph 0, -1, \ph 0, \ph 0)     
&\leftrightarrow
 (-1, \ph 1, \ph 0, \ph 0, \ph 0, \ph 1, -1, \ph 0, \ph 0).
\end{align*}
}          
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\item
case 4


To get the  Mori vectors in  the  phase $\alpha_{6}$  
from those in   $\alpha_{5}$,
$a_1$,  $ a_4$, $a_3$ and $a_6$      
in the  phase $\alpha_{5}$    
are replaced with 
$a_4$, $a_1$, $a_2$ and $a_5$    and  vice versa.   



some examples:
\newline
\noindent
phase $\alpha_ 2$ =  phase  $\alpha_7 $, 
phase $\alpha_{12}$ = phase $\alpha_{13}$, 
phase $\alpha_5$ = phase $\alpha_6$, 
phase $\alpha_1$ = phase $\alpha_4$,
phase $\alpha_1$ = phase $\alpha_8$ and 
phase $\alpha_9$ = phase $\alpha_{16}$. 


%$ \{ \ell_i \}_{\alpha_(6)}^{(V)}$ versus
%  $\{\ell_i \}_{\alpha_(6)}^{(V)}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 {\small
\begin{align*}
\phantom{(\ph 0, \ph 0, \ph 0,}\{ \ell_i \}_{(\alpha_3)}^{(\V)}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0)}
&\phantom{\leftrightarrow}
\phantom{(\ph 0, \ph 0, \ph 0,}\{ \ell_i \}_{(\alpha_6)}^{(\V)}
\phantom{\ph 0, \ph 0, \ph 0, \ph 0)}\\
% - - - - -- - - --- - - - - - - -- - -  --
(\ph 0, \ph 0, \ph 0, -1, \ph 1, -1, \ph 1, \ph 0, \ph 0)  
&\leftrightarrow
(-1, \ph 0, \ph 0, \ph 0, -1, \ph 1, \ph 1, \ph 0, \ph 0)\\
% - - - - - - - - - - - - -  - - - - -
(-1, \ph 1, -1, \ph 1, -1, \ph 1, \ph 0, \ph 0, \ph 0)   
&\leftrightarrow
(\ph 1, -1, \ph 1, -1, \ph 1, -1, \ph 0, \ph 0, \ph 0)\\
% - - - - - - - - - -- -
(\ph 1, -1, \ph 1, \ph 0, \ph 0, \ph 0,  \ph 0, \ph 2, \ph 3)   
&\leftrightarrow
 (\ph 0, \ph 1, -1, \ph 1, \ph 0, \ph 0, \ph 0, \ph 2, \ph 3)\\
%  - - - - - - - - - - - -  - -
(\ph 1, \ph 0,  \ph 0,  \ph 1,  \ph 0,  \ph 0,  -2, \ph 0, \ph 0)  
&\leftrightarrow
(\ph 1, \ph 0, \ph 0, \ph 1, \ph 0, \ph 0, -2, \ph 0, \ph 0)\\
% - - - - - - - -  - - -
(\ph 0, \ph 0,  \ph 1,  \ph 0,  \ph 0,  \ph 1,  -2, \ph 0, \ph 0)  
&\leftrightarrow
 (\ph 0, \ph 1, \ph 0, \ph 0, \ph 1, \ph 0, -2, \ph 0, \ph 0).
\end{align*}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{itemize}

\newpage
\appendix\text{\bf Appendix~ 6 The correspondence 
of (III), $(\I^{\dagger})$ and (V)}

The correspondence of the  Mori vectors in (III)  
and those in  $(\I^{\dagger})$
are all explained by the replacements of vertices in 
their dual polyhedra.
 
 
The correspondence of the  Mori vectors in 
(III) and $(V)$ needs 
the modification of the 
correspondence in (III) and $(\I^{\dagger})$ 
due to the  difference of these  dual polyhedra 
though the difference is only one vertex.
This modification is done by using the  symmetry of 
three vertices pairs of (V). 
Furthermore, 
due to this symmetry of three pairs in (V), the correspondences
are not unique  and appear in more complicated ways.  




case 1: phase ${\text a}^{(\III)}$ =phase $\alpha_2^{(\V)}$


%${\ell_i}_{ ({\text a})}^{(\III)}$ 
%${\ell_i}_{ (\alpha_2)}^{(\V)}$


\noindent
%%%%%%%%%%%%%%%%%%%%%%%
{\small 
\begin{align*}
\phantom{(\ph 0,   \ph 0,   \ph 0,  }{\ell_i}_{ ({\text a})}^{(\III)}
\phantom{\ph 0, \ph 0,   \ph 0,   \ph 0)  } &\phantom{\leftrightarrow}
\phantom{(\ph 0,   \ph 0,   \ph 0,  }{\ell_i}_{ (\alpha_2)}^{(\V)}
\phantom{\ph 0, \ph 0,   \ph 0,   \ph 0)  }\\
% - - - - -- - - - - - - -- - - - - - - -
 (\ph 0,   \ph 0,   \ph 0,  \ph 0,  -1,  \ph 0,   \ph 1,  -1,  \ph  1)
&\leftrightarrow 
 (\ph 0,  \ph 0,  \ph 0,  -1,  \ph 1,  -1,  \ph 1,  \ph 0,  \ph 0)\\ 
% - - - - - - - - -
 (\ph 0,   \ph 0,   \ph 1,  \ph 0,   \ph 1,  \ph 0,  -2,   \ph 0,   \ph 0) 
&\leftrightarrow
 (\ph 1,  -1,  \ph 0,  \ph 1,  -1, \ph  0,  \ph 0,  \ph 0,  \ph 0)\\
% - - - - - - - - -
 (\ph 2,   \ph 0,  -1,  \ph 3,   \ph 0, \ph  0,   \ph  1,  \ph 1,   \ph 0)
&\leftrightarrow
 (-1,  \ph 1,  \ph 0,  \ph 0,  \ph 0,  \ph 1,  \ph 0,  \ph 2,  \ph 3) \\
% - - - - - - - - - - - - - -     
 (\ph 0,   \ph 1,   \ph 0,  \ph 0,   \ph 1,  \ph 0,   \ph 0,  \ph  0,  -2)
&\leftrightarrow
 (\ph 0,  \ph 0, \ph  1,  \ph 0, \ph  0,  \ph 1,  -2, \ph  0,  \ph 0)\\
% - - - - - - - - - -
 (\ph 0,  -1,   \ph 0,  \ph 0,   \ph 0,  \ph 1,   \ph 0,   \ph 1,  -1) 
&\leftrightarrow
 (\ph 0,  \ph 1,  -1,  \ph 1,  \ph 0,  \ph 0,  -1,  \ph 0,  \ph 0). 
\end{align*}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


 





 case 2 : phase ${\text e}^{(\III)}$ = phase $\alpha_1^{(\V)}$




%$\{ \ell_i \}_{ ({\text e})}^{(\III)}$
%$\{ \ell_i \}_{ (\alpha_1)}^{(\V)}$



\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small 
\begin{align*}
\phantom{(\ph 0, \ph 0, \ph 0}
\{ \ell_i \}_{\text {(e)}}^{(\III)}
\phantom{\ph 0, \ph 0, \ph 0, \ph}
&\phantom{\leftrightarrow}
\phantom{(\ph 0, \ph 0, \ph 0}
\{ \ell_i \}_{ (\alpha_1)}^{(\V)}
\phantom{\ph 0, \ph 0, \ph 0, \ph}\\
% - - - - - - - - - - - - 
 (\ph 0, \ph 0, \ph 1, \ph 0, \ph 0, \ph 0, -1, -1, \ph 1) 
&\leftrightarrow  
 (\ph 0, -1, \ph 1, -1, \ph 0, \ph 0, \ph 1, \ph 0, \ph 0)
\\
% - - - - - - - - - 
 (\ph 2, \ph 0, -1, \ph 3, \ph 0, \ph 0, \ph 1, \ph 1, \ph 0)  
&\leftrightarrow 
 (\ph 0, \ph 1, -1, \ph 1, \ph 0, \ph 0, \ph 0, \ph 2, \ph 3)
\\
% - - - - - - - - - - - - -
 (\ph 0, \ph 1, \ph 0, \ph 0, \ph 0, \ph 0, \ph 1, -1, -1)   
&\leftrightarrow 
 (\ph 1, \ph 0, \ph 0, \ph 0, \ph 1, -1, -1, \ph 0, \ph 0)
\\
% - - - - - - - - - - - 
 (\ph 0, -1, \ph 0, \ph 0, \ph 0, \ph 1, \ph 0, \ph 1, -1 )   
&\leftrightarrow 
 (-1, \ph 1, \ph 0, \ph 0, \ph 0, \ph 1, -1, \ph 0, \ph 0) 
\\
% - - - - - - - - -
 (\ph 0,  \ph 0, \ph 0, \ph 0, \ph 1, \ph 0, -1, \ph 1,  -1)   
&\leftrightarrow 
 (\ph 0, \ph 0, \ph 0, \ph 1, -1, \ph 1, -1, \ph 0, \ph 0 ). 
\end{align*}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55 










\newpage
\appendix\text{\bf Appendix~7 Topological data of ${\cal K}^0$ and  
$c_2 \cdot \vec J$  }
\newline 
We list thetopological  data of three models
\footnote{all was derived by Hosono.} 

(III) There are 8 phases.
We list the data of all  eight phases.


\begin{itemize}
\item
phase a

$
{\cal K}^0_{({\text a})} = 
     6t_1^3 + 7t_1^2t_2 + 7t_1t_2^2 + 7t_2^3 + 2t_1^2t_3 + 
     2t_1t_2t_3 + 2t_2^2t_3 + 8t_1^2t_4 + 8t_1t_2t_4 + 
     8t_2^2t_4 + 2t_1t_3t_4 + 2t_2t_3t_4 + 8t_1t_4^2 + 
     8t_2t_4^2 + 2t_3t_4^2 + 8t_4^3 + 3t_1^2t_5 + 
     3t_1t_2t_5 + 3t_2^2t_5 + t_1t_3t_5 + t_2t_3t_5 + 
     3t_1t_4t_5 + 3t_2t_4t_5 + t_3t_4t_5 + 3t_4^2t_5 + 
     t_1t_5^2 + t_2t_5^2 + t_4t_5^2;
$

$c_2 \cdot \vec J_{\text {(a)}}$ = \{72, 82, 24, 92, 36\};
% - - - - - - - - - - - - - - - - - -  - - - -
\item
 phase b 

$
{\cal K}^0_{({\text b})} = 
     6t_1^3 + 8t_1^2t_2 + 8t_1t_2^2 + 8t_2^3 + 2t_1^2t_3 + 
     2t_1t_2t_3 + 2t_2^2t_3 + 7t_1^2t_4 + 8t_1t_2t_4 + 
     8t_2^2t_4 + 2t_1t_3t_4 + 2t_2t_3t_4 + 7t_1t_4^2 + 
     8t_2t_4^2 + 2t_3t_4^2 + 7t_4^3 + 4t_1^2t_5 + 
     4t_1t_2t_5 + 4t_2^2t_5 + t_1t_3t_5 + t_2t_3t_5 + 
     4t_1t_4t_5 + 4t_2t_4t_5 + t_3t_4t_5 + 4t_4^2t_5 + 
     2t_1t_5^2 + 2t_2t_5^2 + 2t_4t_5^2;
$

$c_2 \cdot \vec J_{\text{(b)}}$ = \{72, 92, 24, 82, 48\};

\item
 phase c 

$
{\cal K}^0_{({\text c})} = 
     6t_1^3 + 7t_1^2t_2 + 7t_1t_2^2 + 7t_2^3 + 2t_1^2t_3 + 
     2t_1t_2t_3 + 2t_2^2t_3 + 3t_1^2t_4 + 3t_1t_2t_4 + 
     3t_2^2t_4 + t_1t_3t_4 + t_2t_3t_4 + t_1t_4^2 + 
     t_2t_4^2 + 4t_1^2t_5 + 4t_1t_2t_5 + 4t_2^2t_5 + 
     t_1t_3t_5 + t_2t_3t_5 + 2t_1t_4t_5 + 2t_2t_4t_5 + 
     2t_1t_5^2 + 2t_2t_5^2;
$

$c_2 \cdot \vec J_{\text{(c)}}$ = \{72, 82, 24, 36, 48\};

\item
 phase d   

$
{\cal K}^0_{({\text d})} = 
     6t_1^3 + 7t_1^2t_2 + 7t_1t_2^2 + 7t_2^3 + 8t_1^2t_3 + 
     8t_1t_2t_3 + 8t_2^2t_3 + 8t_1t_3^2 + 8t_2t_3^2 + 
     8t_3^3 + 9t_1^2t_4 + 9t_1t_2t_4 + 9t_2^2t_4 + 
     9t_1t_3t_4 + 9t_2t_3t_4 + 9t_3^2t_4 + 9t_1t_4^2 + 
     9t_2t_4^2 + 9t_3t_4^2 + 9t_4^3 + 3t_1^2t_5 + 
     3t_1t_2t_5 + 3t_2^2t_5 + 3t_1t_3t_5 + 3t_2t_3t_5 + 
     3t_3^2t_5 + 3t_1t_4t_5 + 3t_2t_4t_5 + 3t_3t_4t_5 + 
     3t_4^2t_5 + t_1t_5^2 + t_2t_5^2 + t_3t_5^2 + t_4t_5^2;
$

$c_2 \cdot \vec J_{\text{(d)}}$ = \{72, 82, 92, 102, 36\};

\item
 phase e  

$
{\cal K}^0_{({\text e})} = 6t_1^3 + 3t_1^2t_2 + t_1t_2^2 + 2t_1^2t_3 + 
     t_1t_2t_3 + 2t_1^2t_4 + t_1t_2t_4 + t_1t_3t_4 + 
     7t_1^2t_5 + 3t_1t_2t_5 + t_2^2t_5 + 2t_1t_3t_5 + 
     t_2t_3t_5 + 2t_1t_4t_5 + t_2t_4t_5 + t_3t_4t_5 + 
     7t_1t_5^2 + 3t_2t_5^2 + 2t_3t_5^2 + 2t_4t_5^2 + 7t_5^3;
$

$c_2 \cdot \vec J_{\text{(e)}}$ = \{72, 36, 24, 24, 82\};

\item
 phase f  

$
{\cal K}^0_{({\text f})} = 
     6t_1^3 + 7t_1^2t_2 + 7t_1t_2^2 + 7t_2^3 + 2t_1^2t_3 + 
     2t_1t_2t_3 + 2t_2^2t_3 + 2t_1^2t_4 + 2t_1t_2t_4 + 
     2t_2^2t_4 + t_1t_3t_4 + t_2t_3t_4 + 8t_1^2t_5 + 
     8t_1t_2t_5 + 8t_2^2t_5 + 2t_1t_3t_5 + 2t_2t_3t_5 + 
     2t_1t_4t_5 + 2t_2t_4t_5 + t_3t_4t_5 + 8t_1t_5^2 + 
     8t_2t_5^2 + 2t_3t_5^2 + 2t_4t_5^2 + 8t_5^3;
$

$c_2 \cdot \vec J_{\text{(f)}}$ = \{72, 82, 24, 24, 92\};


\item
 phase g  

$
{\cal K}^0_{({\text g})} = 
     6t_1^3 + 2t_1^2t_2 + 3t_1^2t_3 + t_1t_2t_3 + 
     t_1t_3^2 + 2t_1^2t_4 + t_1t_2t_4 + t_1t_3t_4 + 
     4t_1^2t_5 + t_1t_2t_5 + 2t_1t_3t_5 + 2t_1t_4t_5 + 
     2t_1t_5^2;
$

$c_2 \cdot \vec J_{\text{(g)}}$ = \{72, 24, 36, 24, 48\}; 

\item
 phase h  

$
{\cal K}^0_{({\text h})} = t_1^2t_3 + t_1t_2t_3 + 3t_1t_3^2 + 2t_2t_3^2 + 
     7t_3^3 + t_1^2t_4 + t_1t_2t_4 + 3t_1t_3t_4 + 
     2t_2t_3t_4 + 8t_3^2t_4 + 3t_1t_4^2 + 2t_2t_4^2 + 
     8t_3t_4^2 + 8t_4^3 + t_1^2t_5 + t_1t_2t_5 + 
     3t_1t_3t_5 + 2t_2t_3t_5 + 8t_3^2t_5 + 3t_1t_4t_5 + 
     2t_2t_4t_5 + 8t_3t_4t_5 + 8t_4^2t_5 + 3t_1t_5^2 + 
     2t_2t_5^2 + 8t_3t_5^2 + 8t_4t_5^2 + 7t_5^3;
$

$c_2 \cdot \vec J_{\text{(h)}}$ = \{36, 24, 82, 92, 82\};
\end{itemize}


(IV) There are five phases  called  A, B, C, D, E.
We  list only the phase A since the other four phase data coincide with
the data of (III).
\newline
\begin{itemize}
\item
phase A

$
{\cal K}^0_{({\text A})} = 6t_1^3 + 4t_1^2t_2 + 2t_1t_2^2 + 2t_1^2t_3 + 
     t_1t_2t_3 + 3t_1^2t_4 + 2t_1t_2t_4 + t_1t_3t_4 + 
     t_1t_4^2 + 6t_1^2t_5 + 4t_1t_2t_5 + 2t_1t_3t_5 + 
     3t_1t_4t_5 + 6t_1t_5^2;
$

$c_2 \cdot \vec J_{\text{(A)}}$   = \{72, 48, 24, 36, 72\};
\end{itemize}




(V) There are 18 phases in case.
We list only the data of phases 3, 10  and 18, since the  other 15 data 
coincide with those in case (III).
\begin{itemize}
\item
 phase $\alpha_3$  

$
{\cal K}^0_{(\alpha_3)}= 
     7t_1^3 + 8t_1^2t_2 + 8t_1t_2^2 + 7t_2^3 + 7t_1^2t_3 + 
     8t_1t_2t_3 + 7t_2^2t_3 + 7t_1t_3^2 + 7t_2t_3^2 + 
     6t_3^3 + 2t_1^2t_4 + 2t_1t_2t_4 + 2t_2^2t_4 + 
     2t_1t_3t_4 + 2t_2t_3t_4 + 2t_3^2t_4 + 2t_1^2t_5 + 
     2t_1t_2t_5 + 2t_2^2t_5 + 2t_1t_3t_5 + 2t_2t_3t_5 + 
     2t_3^2t_5 + t_1t_4t_5 + t_2t_4t_5 + t_3t_4t_5;
$

$c_2 \cdot \vec J_{(\alpha_3)}$ = \{82, 82, 72, 24, 24\};

\item
  phase $\alpha_{10}$ 

$
{\cal K}^0_{(\alpha_{10})}= 6t_1^3 + 3t_1^2t_2 + t_1t_2^2 + 2t_1^2t_3 + 
     t_1t_2t_3 + 2t_1^2t_4 + t_1t_2t_4 + t_1t_3t_4 + 
     2t_1^2t_5 + t_1t_2t_5 + t_1t_3t_5 + t_1t_4t_5;
$

$c_2 \cdot \vec J_{(\alpha_{10})}$ = \{72, 36, 24, 24, 24\};



\item
  phase $\alpha_{18}$ 

$
{\cal K}^0_{(\alpha_{18})}=
     8t_1^3 + 9t_1^2t_2 + 9t_1t_2^2 + 8t_2^3 + 8t_1^2t_3 + 
     9t_1t_2t_3 + 8t_2^2t_3 + 8t_1t_3^2 + 8t_2t_3^2 + 
     7t_3^3 + 8t_1^2t_4 + 9t_1t_2t_4 + 8t_2^2t_4 + 
     8t_1t_3t_4 + 8t_2t_3t_4 + 7t_3^2t_4 + 8t_1t_4^2 + 
     8t_2t_4^2 + 7t_3t_4^2 + 6t_4^3 + 3t_1^2t_5 + 
     3t_1t_2t_5 + 3t_2^2t_5 + 3t_1t_3t_5 + 3t_2t_3t_5 + 
     3t_3^2t_5 + 3t_1t_4t_5 + 3t_2t_4t_5 + 3t_3t_4t_5 + 
     3t_4^2t_5 + t_1t_5^2 + t_2t_5^2 + t_3t_5^2 + t_4t_5^2;
$

$c_2 \cdot \vec J_{(\alpha_{18})}$ = \{92, 92, 82, 72, 36\};
\end{itemize}



We list three data of quantum corrected ring data 
of  phases d, e and A in tables  6, 7 and 8, which we 
use in appendices 3 and 4
\footnote{We owe all data to S. Hosono and thank him very much.}. 


\vspace{1cm}


\appendix\text{\bf Appendix  8  The perturbative prepotential for the
$S$-$T$-$U$-$V$ model with one extra tensor multiplet} 
\par 
We review the 
perturbative prepotential of the $S$-$T$-$U$-$V$
 model  in  the heterotic string side with  instanton 
numbers,  $(k_1, k_2)=(12, 12)$  \cite{curio2}
\footnote{We have changed the sign 
of ${\cal K}$ in \cite{curio2}.}  
and comment on the relations with the one 
of   $(k_1, k_2)=(12, 11)$ case  and the dual one of      
type IIA  string case. $S$, $T$, $U$ and $V$ represent the vector 
multiplets  except the graviphoton.

The prepotential in Type IIA string  side is 
given   by integrating back of  the Yukawa coupling and  
using  the polylogarithm,  Li$_3$,
\[
{\cal K}^{\text { IIA}}={\cal K}^0+{\cal K}^{\text {inst.}}
={\cal K}^0 -\frac{\chi({\rm CY3})} {2}\zeta(3)+
\frac{1}{(2\pi)^3}\sum_{\{n_i\}}
N(\{n_i\}) {\text{ Li}}_3( \Pi q_i^{n_i} ),
\]
\noindent
where $q_i={\text {e}}^{2\pi i t_i}$. 

The prepotential in heterotic side is  made of  perturbative one,
${\cal K}^{\text P}$ and 
non-perturbative one, ${\cal K}^{\text {NP}}$. 
The perturbative  
one   can be written in terms of the 
supersymmetric index, which is represented by some  appropriate modular 
forms and  Jacobi forms \cite{moore,kawai}. 
Using the expansion coefficients of the supersymmetric index, 
the prepotential is  
\[
{\cal K}^{\text {het}}={\cal K}^{\text {P}}+{\cal K}^{\text {NP}}
={\cal K}^0 
- c(0,0) \zeta(3)+\frac{1}{4\pi^3}\sum_{k,l,b} c(kl,b)
{\text{ Li}}_3(p^k q^l \zeta^b)+{\cal K}^{\text {NP}},
\]
 where $\text{\bf e}[x]:=\exp (2 \pi i x)$. For the 
$S$-$T$-$U$-$V$  model, $p=\text{\bf e}[iT]$, $\zeta=\text{\bf e}[iV]$ 
and $q=\text{\bf e}[iU]$.  



A simple  test of duality  is    
$ 2c(kl, b)=N(kl,b)$ 
 including the case,
 $2c(0,0)=N(kl,b)\mid_{(kl,b)=(0,0)}= - \chi({\text {CY3}})$.
The above  correspondence exists     
only when there is no non-perturbative contribution 
in the heterotic side 
and there are   some identifications of $\{n_i \}$ and $\{k,l,b\}$.
However, for the models with  
 additional tensor multiplets, generation of them  are  due  to  
the non-perturbative effect and they    
can not be seen in  the  
perturbative region of the  heterotic string side. 
Therefore, in general, the relation between 
 $ N (\{n_i \})$  and 
$c(kl, b)$ is non-trivial
\footnote{The contribution of tensor multiplets may also be 
represented by  modular forms.}. 
\par
For example, we treat  the case 
with $(k_1,k_2)=(12,11)$ and $ \Delta n_T=1$, which is   dual to
type IIA string model compactified on the  ``6'' phase 
of CY3 in \cite{theisen}.
This CY3  phase has          $(h^{1,1},h^{2,1})=(4,214)$
and $c_2 \cdot  \vec J=\{82,24,24,92\}$. 
The perturbative side   
 coincides with  the $S$-$T$-$U$-$V$ model of $n=0$ 
with $(k_1,k_2)=(12,12)$ in \cite{curio}. 
This  phase  
is the intermediate one  from the phase  with 
$(h^{1,1}, h^{2,1})=(3,243) $
and $c_2\cdot \vec J =\{92,24,24\}$
to f phase in (III) with $(h^{1,1},h^{2,1})=(5,185)$ 
and $c_2 \cdot \vec J =\{72,82,24,24,92 \}$.
The dual polyhedron  of this model is given by 
(III) in Sec. 3 with (3)(1, 2, 6, 9) removed.


\par
At first,
we review the $S$-$T$-$U$ model 
of the heterotic perturbative side  in \cite{curio}, 
whose type IIA dual 
is compactified on CY3 with  $(h^{1,1},h^{2,1})=(3,243)$. 
\newline
\[
{\text  {Super index}}  
= -2i Z_{2,2} A(\tau)\mid _{h^{1,1}=3}=
-2i Z_{2,2}E_4 E_6/\Delta 
= -2i Z_{2,2}\sum_{n \geq -1}c_{STU}(n)q^n.\]
\newline
The perturbative prepotential from the heterotic string is  
\newline
\[
{\cal K}^{\text {P}}=STU+\frac{1}{3}U^3-\frac{480}{2}\zeta(3) 
+ \frac{1}{4\pi^3}\sum c_{STU}(kl){\text {Li}}_3(\text{\bf e}[kiT+liU])
\] 
 with $2c_{STU}(0) = 480$.
\par
For the $S$-$T$-$U$-$V$ model  in   \cite{curio},

\begin{align*}
{\text { Super~ index}} =-2iZ_{3,2}A(\tau)\mid_{h^{1,1}=4}  
&= -2i Z_{3,2}[
\frac {k_1}{k_1+k_2}E_{4,1} E_6 
+\frac {k_2}{k_1+k_2}E_{4} E_{6,1}]/\Delta 
\\
&=-2i  Z_{3,2} \sum_{M \in\text{\bf Z}, \text{\bf Z}+3/4}c_{4M}(M)q^M 
\end{align*} 
 with $M=4kl-b^2$.
\newline
The perturbative prepotential   
in this case  is 
\newline
\[
{\cal K}^{\text {P}}=STU+ \frac{1}{3}U^3-\frac{1}{3}V^3
+\frac{1}{4\pi^3}  
\sum c _{4kl-b^2}(4kl-b^2){\text {Li}}_3(\text{\bf e}[kiT+liU+biV]).\]
By setting $V=0$, the $S$-$T$-$U$-$V$ model is truncated to the 
$S$-$T$-$U$ model in the heterotic side with     
$c_{STU}(kl)=\sum_b c_n(4kl-b^2)$.

The ${\cal K}^0{}$ in phase 6 in the  type IIA side  dual to this model 
 is in \cite{theisen}.

 
\begin{align}
{\cal  K}^0
& =
7t_1^3 + 2t_1^2t_2 + 2t_1^2t_3 + t_1t_2t_3 + 8t_1^2t_4
\nonumber \\ 
&+ 2t_1t_2t_4 + 2t_1t_3t_4 + t_2t_3t_4 + 
8t_1t_4^2 + 2t_2t_4^2 + 2t_3t_4^2 + 8t_4^3,\nonumber 
\end{align}

\noindent 
where Mori-vectors in the phase  6  are 

%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\begin{alignat*}{2}
&( ~a_1,~ a_2,  ~a_4, ~a_5, ~a_6, ~a_7, ~a_8,~ a_9 & )& \quad \\ 
&(\ph 2,    \ph 1,  \ph 3,  \ph 0,  \ph 0,  \ph 1, -1,     \ph 0 &  )&=\ell 
_1 \quad  \\
&(\ph 0,   \ph 1,  \ph 0, \ph 1, \ph 0,  \ph 0, \ph 0,   -2 & )  &
=\ell_2 \quad 
    \\
&(\ph 0,    \ph 0,  \ph 0,  \ph 0,  \ph 1,  \ph 1, \ph 0,   -2  &)&
=\ell_3   \quad 
 \\
&(\ph 0,    -1,  \ph 0,  \ph 0,  \ph 0,  -1, \ph 1,   \ph 1  &) &
=\ell_4.  \quad
\end{alignat*}
%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\


\noindent
The instanton corrected part of the  prepotential in type IIA side  
is (see table 9  for $N(\{n_i\})$ in phase 6)  


\[
{\cal K}^{\text {inst.}}
= -\frac{420}{2}\zeta(3) + 
\frac{1}{(2 \pi)^3}\sum_{n_1,n_2,n_3,n_4}
N (n_1,n_2,n_3,n_4){\text {Li}}_3(\Pi^4_{i=1}q_i^
{n_i}).
\]
\noindent





To see the correspondence between type IIA string side and  
the heterotic string side, we use the linear transformations 
 from $t_i$ to $ S,T,U,V $ in \cite{theisen}  :
$t_1=V,~ t_2=T-U,~ t_3=S-U,~ t_4=U-V$.
\newline 
By substituting them,
 ${\cal K}^0=STU+ \frac {1}{3} U^3-\frac{1}{3}V^3$.

Using the identification
$kT+lU+bV  = n_1 t_1+n_2t_2+n_4t_4$
and $n_3 = 0$ under the limit $S \rightarrow \infty$, 
 $k=n_2$, $l=n_4-n_2$ and $b=n_1-n_4$ are obtained.
By using them,  

\[
{\cal K}^{\text {inst.}}
= -\frac{420}{2}\zeta(3) + \frac{1}{(2\pi)^3} \sum_{k,l,b}N (k,l,b )
{\text {Li}}_3(e^
{-2\pi (kT+lU+bV)}).
\]
If there is no non-perturbative effect, 
then  the  correspondence,  
$N(k,l,b)=N(4kl-b^2)= 2c_n(4kl-b^2)$ exits.
However, in this case, the above correspondence is modified 
by the non-perturbative effect     except $2c(0,0)=-420=N(0,0)$.  
\par
By substituting 
 $t_4=U,\quad t_2=T-U,\quad t_3=S-U$ and putting   $t_1=V=0$,  
  ${\cal K}^0 $ leads to     
 ${\cal K}^0=STU+\frac{1}{3}U^3 $ with $h^{1,1}=3$ and 
${\cal K }^{\text {inst}}$ also reduces  to   $ h^{1,1}=3$ case   
under the correspondence,  $N(k,l)=\sum_b N(4kl-b^2)
$\cite{theisen,curio2}.
\par
${\cal K}^{ \text { IIA} }$ of  phase f 
 with  $(h^{1,1},h^{2,1})=(5,185)$  
 can be truncated to the one in  6 phase case   by setting $t_1=0$ 
and replacing  
$t_2\rightarrow t_1,~ t_3 \rightarrow t_2,
~t_4 \rightarrow t_3,
~t_5 \rightarrow t_4$ \cite{theisen},
(see table 10 for $N(\{n_i\})$ in phase f).
For the perturbative  heterotic side prepotential, which   
is dual   to type IIA on  f phase, 
the super index will be represented by the modular forms    
of index 2~\cite{Eichler}, which are the product of those 
with index 1.
For example, 
 \begin{align*}
& \frac{1}{\Delta} (E_{4,1}(q, r_1)E_{6,1}(q,r_2)+ E_{6,1}(q, r_1)
E_{4,1}(q,r_2))
\\
&=-360
+\frac{2}{q}
+ 2(\frac{1}{r_2^2}+\frac{1}{r_1^2}+r_1^2+r_2^2)
-32(r_2+r_1+\frac{1}{r_1}+\frac{1}{r_2})
\\
&
+2( \frac{r_1^2}{r_2^2} + \frac{r_2^2}{r_1^2}+r_1^2r_2^2
+\frac{1}{r_1^2r_2^2})q
-9856(\frac{r_1}{r_2} + \frac{r_2}{r_1}+  r_1r_2+\frac{1}{r_1r_2})q
\\
&
-32(\frac {1}{r_1r_2^2}
+\frac{1}{r_1^2r_2}
+\frac{r_1^2}{r_2}
+\frac{r_2^2}{r_1}
+\frac{r_2}{r_1^2}
+\frac{r_1}{r_2^2}
+r_1r_2^2
+r_1^2r_2)q
\\
&
- 33984(\frac{1}{r_2}+\frac{1}{r_1}+r_1+r_2)q
 -360( \frac{1}{r_1^2} + \frac{1}{r_2^2}+r_2^2+r_1^2)q 
-106488q
\\
&
+{\text {higher~order~terms}},
\end{align*}
which satisfies  
$2c(0,0)=360 =-\chi ({\text CY3})=N(0,0)$. 
$E_{4,1}$ and $E_{6,1}$ are the Eisenstein series 
of index 1~\cite{Eichler}\footnote{ 
$r_1$ and $r_2$  may be related  to the additional vector or tensor      
 multiplets to $ T$, $U$ and  $V$ under the limit $S \rightarrow 0$ .}. 



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\newpage

\begin{figure}[h]

\setlength{\unitlength}{0.9mm}

\begin{picture}(150,150)

%-------------------------------------


\put(21,140){\line(1,0){18}}

\put(21,140){\line(1,1){4}}
\put(21,140){\line(1,-1){4}}

\put(40,140){\line(-1,-1){4}}
\put(40, 140){\line(-1,1){4}}


\put(0,140){\{(1)(4)\}}

\put(50,140){\{(5)(2)\}}





\put(25,110){\{ (3)(6)     \}}

\put(50,130){\line(-1,0){4}}
\put(50,130){\line(0,-1){ 4}}
\put(50,130){\line(-1,-1){10}}

\put(40,120){\line(0,1){4}}
\put(40,120){\line(1,0){4}}




\put(20,120){\line(-1,1){10}}

\put(10,130){\line(1,0){4}}
\put(10,130){\line(0,-1){4}}

\put(20,120){\line(0,1){4}}
\put(20,120){\line(-1,0){4}}



\put(70,140){$\Longrightarrow$}


%------------------------------

\put(20, 80){Fig. 1~The correspondence of  three vertex pairs in (V)}

\put(106,140){\line(1,0){18}}

\put(106,140){\line(1,1){4}}
\put(106,140){\line(1,-1){4}}

\put(125,140){\line(-1,-1){4}}
\put(125, 140){\line(-1,1){4}}


\put(80,140){\{$\alpha_{1},\alpha_{4}$\}}

\put(130,140){\{$\alpha_{5},\alpha_{2}$\}}





\put(105,110){\{ $\alpha_{3},\alpha_{6}$     \}}

\put(135,130){\line(-1,0){4}}
\put(135,130){\line(0,-1){ 4}}
\put(135,130){\line(-1,-1){10}}

\put(125,120){\line(0,1){4}}
\put(125,120){\line(1,0){4}}




\put(108,120){\line(-1,1){10}}

\put(98,130){\line(1,0){4}}
\put(98,130){\line(0,-1){4}}

\put(108,120){\line(0,1){4}}
\put(108,120){\line(-1,0){4}}

\end{picture}
\end{figure}









\newpage
%---table 1-------

 
\vspace{0.4cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[h]
\[
\begin{array}
{|l|l|l|l|l|l|}
\hline
\multicolumn{3}{|c|}{\text{CY3s in  (III)}} &
\multicolumn{3}{|c|}{ \text{CY3s in (IV)}}\\
\hline
\Delta n _T  & {S} & \text{K3 fiber} & s 
& {S} & \text{K3 fiber}
\\ \hline
0 &  \text{\bf F}_0  &\text{\bf  P}^3(1,1,4,6)[12]
& 1 &\text{\bf  F}_2&\text{\bf P}^3(1,1,4,6)[12] \\ \hline
2 &  \text{Bl}(\text{\bf F}_2)   & \text{\bf P}^3 (1,1,4,6)[12]
&2&\text{Bl}(\text{\bf F}_2)&\text{\bf P}^3(1,1,4,6)[12] \\ \hline
3 &  \text{Bl}(\text{\bf F}_3)
 & \text{\bf P} ^3(1,2,6,9)[18]&3&\text{Bl}(\text{\bf F}_2)
&\text{\bf P} ^3(1,1,4,6)[12]  \\ \hline
4 & \text{Bl}(\text{\bf F}_4)  & \text{\bf P}^3(1,2,6,9)[18]
&4& \text{Bl}(\text{\bf F}_2)&\text{\bf P}^3(1,1,4,6)[12]  \\ \hline
6 & \text{Bl}(\text{\bf F}_6) 
 & \text{\bf P}^3(1,3,8,12)[24]&6& \text{Bl}(\text{\bf F}_2)
& \text{\bf P}^3(1,1,4,6)[12] \\ \hline
8 & \text{Bl}(\text{\bf F}_8)
  & \text{\bf P}^3(1,4,10,15)[30]&8& \text{Bl}(\text{\bf F}_2)&
\text{\bf P}^3(1,1,4,6)[12]  \\ \hline
12 &  \text{Bl}(\text{\bf F}_{12})
 & \text{\bf P}^3(1,5,12,18)[36]&12&\text{Bl}(\text{\bf F}_2) & 
\text{\bf P}^3(1,1,4,6)[12] \\ \hline
\end{array}
\]
\caption{  The K3 fibrations   of (III) and case (IV)}
\end{table}


%---table 2-----

\begin{table}[h]
\[
\begin{array}{|l|l|l|l|l|l|l|l|l|l|}
\hline 
n^0     &G_2   & h^{1,1}& h^{1,2}& k_1& k_2  
& n_ T^0&\Delta n_T& n_T  
\\ 
\hline
% - - - - - - - -  - - - - - - - - - - -- - - - 
 0    & I             &3      &243     & 12 & 12   &  1&0&1     
\\ \hline
 2    &I             &3       &243     & 12+2 & 12-2&  1&0&1 
\\ \hline
 3    &  A_2   &5       &251     & 12+3 &12-3 &  1&0&1 \\ \hline
 4    &  D_4   &7      &271     & 12+4 & 12-4 &  1&0&1 \\ \hline
 6    &  E_6   &9      &321     & 12+6 & 12-6 &  1&0&1 \\ \hline
 8    &  E_7   &11      &376     & 12+8 & 12-8 &  1&0&1 \\ \hline
 12   &  E_8   &12      &491     & 12+12 & 12-12 &  1&0&1 \\ \hline
\end{array}
\]
\caption{ Hodge and instanton numbers of CY3s in (I)  }
\end{table}


%---table 3 ---- 


%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\begin{table}[h]
\[
\begin{array}{|l|l|l|l|l|l|l|l|}
\hline 
\Delta n_T     & G_2   & h^{1,1}& h^{1,2}& k_1& k_2  
& n_ T^0 & n_T  \\ \hline 
0    &  I             &3      &243     & 12 & 12   &  1&1   
  \\ \hline
2    & I             &5       &185     & 12 & 12-2&  1&3 \\ \hline
3    &  A_2   &8       &164     & 12 &12-3 &  1&4 \\ \hline
4    & D_4   &11      &155     & 12 & 12-4 &  1&5 \\ \hline
6    & E_6   &15      &147     & 12 & 12-6 &  1&7 \\ \hline
8    & E_7   &18      &144     & 12 & 12-8 &  1&9 \\ \hline
12   & E_8   &23      &143     & 12 & 12-12 &  1&13 \\ \hline
\end{array}
\]
\caption{The Hodge and instanton numbers in (III)/(IV) }
\end{table}
%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\




%---table 4-----

\begin{table}[h]
\[
\begin{array}{|c|l|l|}\hline 
{\rm model }     &   \# \{\text{K3 fibers}\} 
&   \# \{\text{phases by triangulation}\}       \\ \hline 
{(\I}^\dagger)      &    \{0,1,2\}   
& \text{\po 8 phases  }  \\ \hline
{(\III)}            &    \{0,1,2\}   
& \text{\po 8 phases  labeled by}\ {\text a}, 
{\dots}, {\text h} \\ \hline
{(\IV)}             &    \{0,1\}     
& \text{\po 5 phases labeled by}\  
\text{A},{\dots}, \text{E}   \\ \hline
{(\V)}              &    \{0,1,2,3\} 
& \text{18 phases labeled by}\ \alpha_1, {\dots}, \alpha_{18}\\ \hline
\end{array}
\]
\caption{ The number of K3 fibers and the phases specified by the  
triangulations in four models}
\end{table}



%---table 5------


%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
{\setlength{\textwidth}{10cm}
\begin{table}[h]
\[
\begin{array}{|c|c|c|c|}
\hline 
{}      & (\IV)   &  (\III)     &  (\V)   
\\ \hline
% - - - - - -
\sharp\{\text{K3 fibers}\}  & s=2  &  \Delta n_{\text {T}}=2
 &  \Delta n_{\text {T}}=2   
\\ \hline 
% - - - - - - - - 
1   & \text{A}    & {}         &            
 \\ \hline
% - - - - - - - 
1   & \text{B}    & {\text c}           &             
\\ \hline
% - - - - - - - 
0   & \text{C}    & {\text d}          & \alpha_{14}          
\\ \hline
% - - - - - - - 
1   & \text{D}   & {\text b}            &      
\\ \hline
% - - - - - - - 
1   & \text{E}    & {\text a}        &  \alpha_2,\alpha_7,
\alpha_{11},\alpha_{12},\alpha_{13},\alpha_{17}       
\\ \hline
% - - - - - - - 
2   &              &  {\text e}          &\alpha_1,\alpha_4,\alpha_8,
\alpha_9,\alpha_{15},\alpha_{16}     
\\ \hline
% - - - - - - - 
2   &              &  {\text f}          &\alpha_5,\alpha_6         
\\ \hline
% - - - - - - - 
2   &             &  {\text g}            &      
\\ \hline
% - - - - - - - 
1   &              &  {\text h}            &      
\\ \hline
% - - - - - - - 
3   &              &               & \alpha_{10}     
\\ \hline
% - - - - - - - 
1   &              &               & \alpha_{18}     
\\ \hline
% - - - - - - - 
0   &              &               &\alpha_{ 3}      
\\ \hline
\end{array}
\]
\caption{ Identification of phases in three models by criterion 1   }
\end{table}
}
%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\


%---table 6------


 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small\allowdisplaybreaks
\begin{table}[h]
\begin{center}
\begin{tabular}{|r|r|r|r|}\hline
% - - - - - - - - - - - - - - - - - - - - -  - - - 
% - -  - - - - - - - - - - - - - - - - -- - 
$\{n_i\}$  & $N(\{ n_i \})$ & $\{n_i\}$ & $N(\{ n_i \})$ 
\\ \hline
% - - - - - - - - - - --
(0, 0, 0, 0, 1 )  & 3 &  (0, 0, 0, 0, 2 ) &  $-6$ 
\\
% - - - - - - - - - --  - - - -
(0, 0, 0, 0, 3 )  & 27 & (0, 0, 0, 0, 4 ) &  $-192$
\\
% - - - - - - - - - - - - - -  - -
(0, 0, 0, 0, 5 ) &  1695 & (0, 0, 0, 1, 0 )&  1 
\\
% - - -  --  - - - -- - - - - - -- - - -  -
(0, 0, 0, 1, 1 ) &  $-2$ & (0, 0, 0, 1, 2 ) &  5 
\\
% - - - - - - - -  - - - - - - - -- - - - - -
(0, 0, 0, 1, 3 ) & $-32$ & (0, 0, 0, 1, 4 ) &  286 
\\
% - - - - - - - - -- - - -- - - -- - - - -
(0, 0, 0, 2, 0 ) &   0 & (0, 0, 0, 2, 2 ) & 0 
\\
% - - - - -- - - -- - - - - -
(0, 0, 0, 2, 3 ) &  7 & (0, 0, 0, 3, 0 ) & 0 
\\
% - - - - - - -  - 
(0, 0, 0, 4, 0 ) &  0  & (0, 0, 0, 5, 0 ) &  0 
\\
% - - - - -- - -- - - - -- - - - - -
(0, 0, 1, 1, 0 ) &  1 & (0, 0, 1, 1, 1 ) & $-2$ 
\\
% - -- - - -- - -  -- - -  -
(0, 0, 1, 1, 2 ) & 5 & (0, 0, 1, 1, 3 ) &  $-32$ 
\\
% - - - - - - -  --
(0, 0, 1, 2, 1 ) &  1 & (0, 0, 1, 2, 2 ) &  $-4$ 
\\
% - - - - -- - - -- - - - -- - - - 
(0, 0, 2, 2, 0 ) & 0 & (0, 1, 1, 1, 0 ) & 1
 \\
% - - -  - -  
(0, 1, 1, 1, 1 )&   $-2$ & (0, 1, 1, 1, 2 ) & 5 
\\
% - - --  - - -  - - - - - -
(0, 1, 1, 2, 1 ) &  1 & (1, 0, 0, 0, 0 ) & 252
\\
% - -- -- - -- - - - - - - -- - - 
(1, 1, 0, 0, 0 ) & 252 &(1, 1, 1, 0, 0 ) & 252 
\\
% - - - - - - - - - 
(1, 1, 1, 1, 0 ) & 360 & (1, 1, 1, 1, 1 )&  $-720$
\\
% - -- - - -- - - -- - - -- -
(2, 0, 0, 0, 0 ) &  $-9252$ & (2, 1, 1, 1, 0 ) &  5130
\\
% - - - - - - - - - - - 
(2, 2, 0, 0, 0 ) & $-9252$ & (3, 0, 0, 0, 0 )&  848628
\\
% - -- -- - - - - - -- -- - 
(4, 0, 0, 0, 0 ) & $-114265008$ & (5, 0, 0, 0, 0 ) & 18958064400
\\ 
\hline
\end{tabular}
\end{center} 
\caption{$N(\{n_i\})$\   of\  phase\ d \ in \ (III)}
\end{table}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





%\newpage




%---table 7-----




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small\allowdisplaybreaks
\begin{table}
\begin{center}
\begin{tabular}{|r|r|r|r|}\hline
$\{ n_i \}$ &  $N(\{n_i\})$  & $\{ n_i \}$ & $N(\{n_i\})$ 
\\ \hline
% - - - - - - - - - - - -
(0, 0, 0, 0, 1 ) &  1   &
 (0, 0, 0, 0, 2 )&  0  \\ 
 (0, 0, 0, 0, 3 )&  0 &     
 (0, 0, 0, 0, 4 )& 0   \\ 
 (0, 0, 0, 0, 5 )&  0 &    
 (0, 0, 0, 1, 0 )&  1 \\
 (0, 0, 0, 1, 1 ) &  1 &    
 (0, 0, 0, 1, 2 ) &  5 \\
 (0, 0, 0, 1, 3 ) &  $-32$ &    
 (0, 0, 0, 1, 4 ) &  286 \\ 
 (0, 0, 0, 2, 0 ) &  0  &   
 (0, 0, 0, 2, 2 ) &  0 \\ 
 (0, 0, 0, 2, 3 )  & 7 &     
 (0, 0, 0, 3, 0 )  &  0   \\
  (0, 0, 0, 4, 0 ) &  0 &   
 (0, 0, 0, 5, 0 ) &  0 \\
 (0, 0, 1, 1, 0 ) &  1 & 
 (0, 0, 1, 1, 1 ) & $ -2$  \\
 (0, 0, 1, 1, 2 ) &  5 &  
 (0, 0, 1, 1, 3 ) &  $-32$  \\
 (0, 0, 1, 2, 1 ) &  1 &  
 (0, 0, 1, 2, 2 ) &  $-4$\\
 (0, 0, 2, 2, 0 ) &  0 & 
 (0, 1, 0, 0, 1 ) & 1 \\
 (0, 1, 0, 1, 1 ) & 1 & 
 (0, 1, 0, 1, 2 ) & $-2$ \\
 (0, 1, 1, 0, 1 ) & 1 & 
 (0, 1, 1, 1, 0 ) & 1 \\
 (0, 1, 1, 1, 1 ) & 1 & 
 (0, 1, 1, 1, 2 ) & $-2$ \\
 (0, 1, 1, 2, 1 ) & 1 &  
 (0, 2, 0, 0, 2 ) &  0\\
 (1, 0, 0, 0, 0 ) &  360 & 
 (1, 0, 0, 0, 1 ) & 252 \\
 (1, 0, 0, 1, 1 ) & 252 & 
 (1, 1, 0, 0, 0 ) & 252 \\
 (1, 1, 0, 0, 1 ) & 252 & 
 (1, 1, 0, 1, 1 ) & 252 \\
 (1, 1, 0, 1, 2 ) & 360 & 
 (1, 1, 1, 0, 0 ) & 252 \\
 (1, 1, 1, 0, 1 ) & 252 &  
 (1, 1, 1, 1, 0 ) & 360  \\
 (1, 1, 1, 1, 1 ) & 252 &  
 (2, 0, 0, 0, 0 ) & 360  \\
 (2, 0, 0, 0, 1 ) & 5130 &  
 (2, 0, 0, 0, 2 ) & $-9252$  \\
 (2, 0, 0, 1, 1 ) & 5130 & 
 (2, 1, 0, 0, 1 ) & 5130 \\
 (2, 1, 0, 1, 1 ) & 5130 &   
 (2, 1, 1, 0, 1 ) & 5130  \\
 (2, 1, 1, 1, 0 ) & 5130 &  
(2, 2, 0, 0, 0 ) & $-9252$ \\
(3, 0, 0, 0, 0 ) & 360 &  
(3, 0, 0, 0, 1 ) & 54760  \\
(3, 0, 0, 0, 2 ) & $-673760$ &  
(3, 0, 0, 1, 1 ) & 54760 \\
(3, 1, 0, 0, 1 ) & 54760 &   
(4, 0, 0, 0, 0 ) & 360 \\
(4, 0, 0, 0, 1 ) & 419895 &   
(5, 0, 0, 0, 0 ) & 360 \\  
\hline
\end{tabular}
\end{center}
\caption{ $N( n_i)$\   of\  phase\  A\  in\ (IV)}
\end{table}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%\newpage


%---table 8----


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
{\small\allowdisplaybreaks
\begin{table}
\begin{center}
\begin{tabular}{|r|r|r|r|}\hline
$\{n_i\}$ &  $N(\{n_i\})$ & 
$\{n_i\}$ & $N(\{ n_i \})$
 \\ \hline
(0,  0,  0,  0,  1 ) & 1 & 
(0,  0,  0,  0,  2 ) & 0 \\
(0,  0,  0,  0,  3 ) &  0 &  
(0,  0,  0,  0,  4 ) &  0 \\
(0,  0,  0,  0,  5 ) &  0 & 
(0,  0,  0,  1,  0 ) &  1 \\
(0,  0,  0,  2,  0 ) &  0 &   
(0,  0,  0,  3,  0 ) &  0 \\
(0,  0,  0,  4,  0 ) &  0  & 
(0,  0,  0,  5,  0 ) &  0 \\
(0,  0,  1,  0,  0 ) &  1  & 
(0,  0,  2,  0,  0 ) &  0 \\
(0,  0,  3,  0,  0 ) &  0  & 
(0,  0,  4,  0,  0 ) &  0 \\
(0,  0,  5,  0,  0 ) &  0  & 
(0,  1,  0,  0,  0 ) &  1 \\
(0,  1,  0,  1,  0 ) &  $-2$ &  
(0,  1,  0,  1,  1 ) &  1 \\
(0,  1,  1,  0,  0 ) &  $-2$ &  
(0,  1,  1,  0,  1 ) &  1 \\
(0,  1,  1,  1,  0 ) &  3  & 
(0,  1,  1,  1,  1 ) &  $-2$ \\
(0,  2,  0,  0,  0 ) &  0 &  
(0,  2,  0,  2,  0 ) &  0 \\
(0,  2,  1,  1,  0 ) &  $-4$ &  
(0,  2,  1,  1,  1 ) &  3 \\
(0,  2,  1,  2,  0 ) &  5 &  
(0,  2,  2,  0,  0 ) &  0 \\
(0,  2,  2,  1,  0 ) &  5 &  
(0,  3,  0,  0,  0 ) &  0 \\
(0,  4,  0,  0,  0 ) &  0 &  
(0,  5,  0,  0,  0 ) &  0 \\
(1,  0,  0,  0,  0 ) &  252 &  
(1,  0,  0,  0,  1 ) &  360 \\
(1,  0,  0,  1,  1 ) &  252 &  
(1,  0,  1,  0,  1 ) &  252 \\
(1,  1,  0,  0,  1 ) &  252 &   
(1,  1,  0,  1,  1 ) &  360 \\
(1,  1,  0,  1,  2 ) &  252 &  
(1,  1,  1,  0,  1 ) &  360 \\
(1,  1,  1,  0,  2 ) &  252 &  
(1,  1,  1,  1,  1 ) &  $-720$ \\
(2,  0,  0,  0,  0 ) &  $-9252$ &  
(2,  0,  0,  0,  1 ) &  5130 \\
(2,  0,  0,  0,  2 ) &  360 &  
(2,  0,  0,  1,  2 ) &  5130 \\
(2,  0,  1,  0,  2 ) &  5130 &  
(2,  1,  0,  0,  2 ) &  5130 \\
(2,  1,  0,  1,  1 ) &  5130 &  
(2,  1,  1,  0,  1 ) &  5130 \\
(3,  0,  0,  0,  0 ) &  848628 &  
(3,  0,  0,  0,  1 ) &  $-673760$ \\
(3,  0,  0,  0,  2 ) &  54760 &  
(4,  0,  0,  0,  0 ) &  $-114265008$ \\
(4,  0,  0,  0,  1 ) &  115243155 &  
(5,  0,  0,  0,  0 ) &  18958064400\\ 
\hline
\end{tabular}
\end{center}
\caption{$N(\{n_i\})$ \  of\  phase\ e\ in\ (III)}
\end{table}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage


%---table 9----

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small\allowdisplaybreaks
\begin{table}[h]
\begin{center}
\begin{tabular}{|r|r|r|r|}\hline
% - - - - - - - - - - - - - - - - - - - - -  - - - 
% - -  - - - - - - - - - - - - - - - - -- - 
$\{n_i\}$  & $N(\{ n_i \})$ & $\{n_i\}$ & $N(\{ n_i \})$ 
\\ \hline
% - - - - - - - - - - --
(0, 0, 0, 1 )  & 1 &  (0, 0, 0, 2 ) &  $0$ 
\\
% - - - - - - - - - --  - - - -
(0, 0, 0, 3 )  & 0 & (0, 0, 1, 0 ) &  $-2$
\\
% - - - - - - - - - - - - - -  - -
(0, 0, 1, 1 ) &  1 & (0, 1, 1,  0 )&  $-4$ 
\\
% - - -  --  - - - -- - - - - - -- - - -  -
(0, 0, 3, 0 ) &  0 & (0, 1, 0, 0 ) &  $-2$ 
\\
% - - - - - - - -  - - - - - - - -- - - - - -
(0, 1, 0, 1 ) & 1 & (0, 1, 2, 0 ) &  $-6$ 
\\
% - - - - - - - - -- - - -- - - -- - - - -
(0, 1, 1, 1 ) &   3 & (0, 2, 0, 0 ) & 0 
\\
% - - - - -- - - -- - - - - -
(0, 3, 0, 0 ) &  0 & (1, 0, 0,  0 ) & 252 
\\
% - - - - - - -  - 
(1, 0, 0, 1 ) &  420  & (1, 0, 1, 1 ) &  420 
\\
% - - - - -- - -- - - - -- - - - - -
(1, 1, 0, 1 ) &  420 & (2, 0, 0, 0 ) & $-9252$ 
\\
% - -- - - -- - -  -- - -  -
(2, 0, 0, 1 ) & 5130 & (3, 0, 0, 0 ) &  $848628$ 
\\
\hline
\end{tabular}
\end{center} 
\caption{$N(\{n_i\})$\   of\  phase\ 6 \  \ }
\end{table}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%---table 10--- 
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small\allowdisplaybreaks
\begin{table}[h]
\begin{center}
\begin{tabular}{|r|r|r|r|}\hline
% - - - - - - - - - - - - - - - - - - - - -  - - - 
% - -  - - - - - - - - - - - - - - - - -- - 
$\{n_i\}$  & $N(\{ n_i \})$ & $\{n_i\}$ & $N(\{ n_i \})$ 
\\ \hline
% - - - - - - - - - - --
(0, 0, 0, 0, 1 )  & 1 &  (0, 0, 0, 0, 2 ) &  $0$ 
\\
% - - - - - - - - - --  - - - -
(0, 0, 0, 0, 3 )  & 0 & (0, 0, 0, 1, 0 ) &  $-2$
\\
% - - - - - - - - - - - - - -  - -
(0, 0, 0, 1,  1 ) &  1 & (0, 1, 0, 0,  1 )&  $1$ 
\\
% - - -  --  - - - -- - - - - - -- - - -  -
(0, 0, 0, 3, 0 ) &  0 & (0, 0, 0, 1, 0 ) & $-2$ 
\\
% - - - - - - - -  - - - - - - - -- - - - - -
(0, 1, 0, 0, 1 ) & 1 & (0, 0, 1, 2, 0 ) &  $-6$ 
\\
% - - - - - - - - -- - - -- - - -- - - - -
(0, 0, 1, 1, 1 ) &   3 & (0, 0, 2, 0, 0 ) & 0 
\\
% - - - - -- - - -- - - - - -
(0, 0, 3, 0, 0 ) &  0 & (1, 0, 0, 0,  0 ) & 252 
\\
% - - - - - - -  - 
(1, 1, 0, 0, 1 ) &  360  & (1, 1, 0, 1, 1 ) &  360 
\\
% - - - - -- - -- - - - -- - - - - -
(1, 1, 0, 1 ,1 ) &  360 & (2, 0, 0, 0, 0 ) & $-9252$ 
\\
% - -- - - -- - -  -- - -  -
(2, 1, 0, 0, 1 ) & 5130 & (3, 0, 0, 0, 0 ) &  $848628$ 
\\
\hline
\end{tabular}
\end{center} 
\caption{$N(\{n_i\})$\   of\  phase\ f \  \ }
\end{table}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{document}
\end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



















































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$B0lIt=$@5CW$7$^$7$?(Btex file$B$rAw$i$;$FD:$-$^$7$?!#(B

page 10  line 7,17, K^0 J-->t
page 11 footnote 8
page 16 acknowledgment
page 17 line 16     t--> J
page 18 line 2      l --> L
        line 11, 12   d= .....
        line 20
        line 22
page 21 line 15

page 23  line 8
page 24  line 5

page 29 30 31  J--> t
page 35 36 37 hep-th $B$NDI2C(B


