Mirror Symmetry and the Web of Landau-Ginzburg String Vacua

Section 5 is largely extended, 23 pages, Latex 2.09, no figure

Abstract:
  We present some mathematical aspects of Landau-Ginzburg string vacua in terms
of toric geometry. The one-to-one correspondence between toric divisors and
some of (-1,1) states in Landau-Ginzburg model is presented for superpotentials
of typical types. The Landau-Ginzburg interpretation of non-toric divisors is
also presented. Using this interpretation, we propose a method to solve the
so-called "twisted sector problem" by orbifold construction. Moreover,this
construction shows that the moduli spaces of the original Landau-Ginzburg
string vacua and their orbifolds are connected. By considering the mirror map
of Landau-Ginzburg models, we obtain the relation between Mori vectors and the
twist operators of our orbifoldization. This consideration enables us to argue
the embedding of the Seiberg-Witten curve in the defining equation of the
Calabi-Yau manifoulds on which the type II string gets compactified. Related
topics concerning the Calabi-Yau fourfolds and the extremal transition are
discussed.

\section{Introduction}

Strings on Calabi-Yau manifold s are promising candidates
for unified theory.
Their physics heavily depends on the topological
properties of Calabi-Yau manifold s.
Recently it is believed that the toric geometry
gives us the most useful technique
to study Calabi-Yau manifold s \cite{vb,hkty1}.

It is also known that Landau-Ginzburg\ models
\cite{iv} can describe the strings
on Calabi-Yau manifold s.
They give us a powerful method for calculating
the (anti-)chiral rings \cite{lvw}
which can be identified with the cohomology rings
on the corresponding Calabi-Yau manifold s,
i.e.$(p,q)$ states of Landau-Ginzburg\ model
can be identified with
$(3-p,q)$ forms on a Calabi-Yau manifold .

In this paper, we study the relation between Landau-Ginzburg\ models
and toric geometry.
First, we find the one-to-one identification
between a toric divisor and a $(-1,1)$ state
which is a ground state in $j^{-l}$ twisted sector,
where $j$ is the $U(1)$ twist of our Landau-Ginzburg\ model.
In the case of Fermat type potential
this identification was already given
by the author
\cite{sa3}.
We try to extend our results to the potentials of
other types, i.e. chain and loop potentials.

In \cite{sa3}
it was not possible to identify
the $(-1,1)$ states written in the form
$\prod_{i}{{X_{i}}^{l_{i}}} \keta {j^{-l}}$
with toric data.
In this paper, however,
 we have succeeded in  the toric interpretation
of these states.
Futhermore we can show
that these states correspond to the
so-called "twisted sector problem" in \cite{hkty1}.
Roughly speaking,
this problem implies that
toric geometry fails to
give us the techniques for describing
the  whole K\"{a}hler moduli space
and the whole complex structure moduli space.
It was pointed out that under certain circumstances
we can solve this problem by deforming a polyhedron
obtained from a defining equation of a Calabi-Yau manifold\ \cite{klemm} .
We find that in the Landau-Ginzburg\ context this construction
corresponds to the orbifold construction of \cite{ls}.
This understanding enables us to propose a useful method
to solve the twisted sector problem.
Furthermore we will show that the Mori vectors,
which are needed to study the moduli space
of Calabi-Yau manifold s,
can be obtained from the relation of
the twist operators of our orbifoldization.
This orbifold construction tells us
that the moduli spaces of the original Landau-Ginzburg\ string vacua
and their orbifolds are connected.
This structure can be shown in terms of
our correspondence
between Landau-Ginzburg\ models and toric geometry.

This paper is organized as follows.
In section 2 we briefly review the construction
of Calabi-Yau manifold s in terms of toric geometry.
The twisted sector problem is explained here.
The Landau-Ginzburg\ models and their mirror symmetry
are given in section 3.
To study mirror symmetry
the useful ingredients are
the phase symmetries of the superpotentials
for the Landau-Ginzburg\ models.
This procedure is explained in this section.
In terms of the phase symmetries the correspondence between
Landau-Ginzburg\ models and toric geometry is presented.

Section 4 is devoted to
the orbifold construction
and its toric aspects.
In the previous section
we have presented the one-to-one identification
between the $(-1,1)$ states and the toric data.
Using this identification,
a useful method to
resolve the twisted sector problem is proposed.
This mehtod is based on the orbifold construction.
For Landau-Ginzburg\ models of our interest,
this orbifold construction
requires the deformation of the superpotential.
This deformation should correspond to
the deformation of toric data.
As a result the moduli spaces of the original
and the orbifoldized theories are connected.
We will show the useful formula
to obtain the appropriate twist operators for this orbifoldization
in terms of the Landau-Ginzburg\ analysis.
The relation between the phase symmetries and the Mori vectors
are also discussed.
Typical examples are studied in detail.

In section 5 we discuss another application of
our toric interpretation of Landau-Ginzburg\ models
in the context of $K3$-fibred Calabi-Yau manifold s and
the Seiberg-Witten geometry.
An interesting example is given
from which
we can obtain a quantum field theory
at the orbifold point in the moduli space.
Finally, in section 6
some speculative remarks are given.

\section{General construction of Calabi-Yau\ hypersurfaces
in toric varietes}

Let us briefly review a method of toric geometry
\cite{vb,hkty1}.
In toric geometry,a pair of
$ (\Delta , \Delta^*) $ gives a Calabi-Yau manifold ,
where $ \Delta $ is a (Newton) polyhedron corresponding to
monomials and $ \Delta^* $ is a dual (or polar) polyhedron
describing the resolution of singularities.
A point on a one- or two-dimensional face
of $\Delta^*$ corresponds to a $(1,1)$ form
coming from resolution.

First, we consider the Fermat type quasi-homogeneous polynomial
in the weighted complex projective space
$WCP^{4}_{(w_{1},w_{2},w_{3},w_{4},w_{5})}[d]$,
where $w_{i}$ are weights and
the degree $ d = \sum_{i = 1}^{5}{w_{i}}$.
The corresponding Calabi-Yau\ hypersurface consists of
monomials $z_i^{d/w_i}\,\,(i=1,\cdots,5)$.
We assume  $w_{1} = 1$,
so that the polyhedron obtained from the hypersurface
is $4$-dimensional.
The associated $4$-dimensional integral convex polyhedron
$\Delta(w)$
is the convex hull of the integral vectors $m$ of the exponents
of all quasi-homogeneous monomials
of degree $d$
shifted by $(-1,\ldots,-1)$, i.e.
$\prod_{i=1}^{5}{z_{i}^{m_{i}+1}}$:
\begin{equation}
\Delta(w):=
\{(m_1,\ldots,m_{5}) \in
\IR^{5}|\sum_{i=1}^{5} w_i m_i=0,m_i\geq-1\} .
\end{equation}
This implies that only the origin is
the point in the interior of $\Delta$.
Its dual polyhedron is defined by
\begin{equation}
\Delta^*=\{\;(x_1,\dots,x_4) \;\vert\;
\sum_{i=1}^4 x_i y_i \geq -1 \;{\rm for \; all \; }
(y_1,\dots,y_4)\in\Delta\;\}.
\end{equation}
In our case it is known that
$ (\Delta , \Delta^*) $
is a reflexive pair.
An $l$-dimensional face
$\Theta \subset \Delta$ can be represented by specifying
its vertices
${\rm v}_{i_1},\cdots,{\rm v}_{i_{k}}$.
Then the dual face $\Theta^*$
is a $(4-l-1)$-dimensional face of $\Delta^*$
and defined by
\begin{equation}
\label{dualface}
\Theta^*=\{ x\in \Delta^*\;
\vert \; (x,{\rm v}_{i_1})=\cdots=(x,{\rm
v}_{i_k})=-1 \},
\end{equation}
where $(*,*)$ is the ordinary inner product.

For models of Fermat type,
we then always obtain as
vertices of $\Delta(w)$
\begin{eqnarray}
\label{fermatpolyhedron}
&\nu_1&=(-1,-1,-1,-1),
\nu_2 \! \! \! =(d/w_2-1, -1, -1, -1),  \
\nu_3=(-1, d/w_3-1, -1, -1), \nonumber \\
&\nu_4&=(-1, -1, d/w_4-1, -1), \
\nu_5 \! \! \! =(-1, -1, -1, d/w_5-1),
\end{eqnarray}
and for the vertices of the dual polyhedron
$\Delta^*(w)$ one finds
\begin{eqnarray}
&\nu_1^*&=(-w_2,-w_3,-w_4,-w_5), \nonumber \\
&\nu_2^*&=(1,0,0,0),  \quad
\nu_3^*=(0,1,0,0), \quad
\nu_4^*=(0,0,1,0), \quad
\nu_5^*=(0,0,0,1).
\end{eqnarray}

Batyrev showed the following useful formula
for non-trivial cohomologies \cite{vb};
\begin{equation}
\label{vb11}
h^{1,1} =
l(\Delta^{*}) - 5
- \sum_{\bf{codimension} \; {\Theta^{*}} = 1}
{l^{'}({\Theta^{*}})}
+\sum_{\bf{codimension} \; \Theta^{*} = 2,
\Theta^{*} \in \Delta^{*}}
{l^{'}(\Theta^{*})
l^{'}(\Theta)}
,
\end{equation}
\begin{equation}
\label{vb21}
h^{2,1} =
l(\Delta) - 5
- \sum_{\bf{codimension} \; {\Theta} = 1}
{l^{'}({\Theta})}
+ \sum_{\bf{codimension} \; \Theta = 2,
\Theta \in \Delta}
{l^{'}(\Theta)
l^{'}(\Theta^{*})}
,
\end{equation}
where
${l({\Delta})}$ ($l(\Delta^{*})$)
denotes the number of integral points
in ${\Delta}$ (${\Delta}^{*}$).
The symbol
${l^{'}({\Theta})}$ ($l^{'}(\Theta^{*})$)
denotes the number of integral points
in the interior of ${\Theta}$ (${\Theta}^{*}$).

The non-zero contributions of
the last terms of (\ref{vb11}) and (\ref{vb21})
are called "twisted sector problem" \cite{hkty1}
If this problem occurs,
toric geometry is not sufficient to
give us the techniques for describing
the  whole K\"{a}hler moduli space
and the whole complex structure moduli space.
In section 4,
we will explain this problem more in depth
and propose a method to circumvent
this undesirable situation.

For the non-Fermat polynomials
which always contain the another type of
monomials ${z_{i}}^{a_{i}}z_{j}$,
it was shown \cite{klemm,ca1} that toric geometry works
in a similar manner.
For example, if the defining equation contains
the monomial ${z_{4}}^{a_{4}}z_{5}$
in stead of $z_{4}^{d/w_{4}}$,
we have to consider the vertex
$(-1,-1,a_{4}-1,0)$ instead of $\nu_{4}$
in (\ref{fermatpolyhedron}).

The defining equation for
a mirror manifold
is obtained as
the zero locus of the Laurent polynomial
\begin{equation}
p^{'} = \sum_{i}{a_{i}{{y_{i}}^{{\nu_{i}}^{*}}}}
\end{equation}
where $y_{i}$ are
coordinates of a canonical torus $({\rm C^{*}})^{4}$
and the coefficients $a_{i}$ are parameters
characterizing the complex structure of the mirror manifold.
If one wants to get the hypersurface as a
quasihomogeneous polynomial,
one should specify an \'{e}tale map \cite{klemm};
\begin{equation}
\label{etale}
y_{i} = \frac{\bar{m_{i}}}
{\bar{X_{1}}\bar{X_{2}}\bar{X_{3}}\bar{X_{4}}\bar{X_{5}}},
\end{equation}
where $\bar{X_{i}}$ are the fields
and $\bar{m_{i}}$ is the appropriate monomial
of the transposed polynomial \cite{bh}
which will be explained in section 3.1
in the context of the Landau-Ginzburg\ superpotentials.

\section{Toric geometrical aspects of Landau-Ginzburg\ string vacua}

\subsection{Landau-Ginzburg\ models and their mirrors}

A Landau-Ginzburg\ model is characterized by a superpotential
$W(X_{i})$ where $X_{i}$ are $N = 2$ chiral superfields.
The Landau-Ginzburg\ orbifolds
\cite{iv} are obtained by
quotienting with an Abelian symmetry group $G$ of
$W{(X_{i})}$,
whose element $g$ acts as an $N \times N$ diagonal matrix,
$g: X_{i} \rightarrow e^{2 \pi i {\tilde{\theta_{i}}^{g}}}X_{i}$.
Here $0 \leq \tilde{\theta_{i}}^{g} < 1$.
Of course the $U(1)$ twist \
$j: X_{i} \rightarrow e^{2 \pi i {q_{i}}}X_{i}$ \
generates the symmetry group of
$W{(X_{i})}$,
where $q_{i} = {w_{i} \over d}$,\ \
 $W(\lambda^{w_{i}} X_{i}) = \lambda^{d}W(X_{i})$ \ and
$\lambda \in \hbox{\,$\inbar\kern-.3em{\rm C}$}^{\ast}$.
In this paper, we further require that
$ w_{1} = 1 $
since the toric description of
the corresponding Calabi-Yau manifold s
and their mirror manifolds
are well-known \cite{vb,hkty1}.
Using the results of Intriligator and Vafa \cite{iv},
we can construct the $(c,c)$ and $(a,c)$ rings,
where $c$ ($a$) denotes chiral (anti-ciral).
Also we could have the left and right $U(1)$ charges of
the ground state \
$\keta h$ \
in the $h$-twisted sector
of the $(a,c)$ ring.
In terms of spectral flow, \ $\keta h$ is mapped to the (c,c)
state $\ketc {h^{\prime}}$ with $h^{\prime} = hj^{-1}$.
Then the charges of the (a,c) ground state of $h$-twisted sector
$ \keta {h} $
are obtained to be
\begin{equation}
\label{uac}
\begin{array}{cc}
\left(\begin{array}{c}
J_{0} \\
\bar{J_{0}}
\end{array} \right) &
\end{array}
\keta {h}
=
\begin{array}{cc}
\left(\begin{array}{c}
{ - \sum_{\tilde{\theta_{i}}^{h^{\prime}}>0}{(1-q_{i}-\tilde{\theta_{i}}^{h^{\prime}})}}
+ \sum_{\tilde{\theta_{i}}^{h^{\prime}}=0} {(2q_{i}-1)}
 \\
{ \sum_{\tilde{\theta_{i}}^{h^{\prime}}>0}{(1-q_{i}-\tilde{\theta_{i}}^{h^{\prime}})}}
\end{array} \right) &
\keta {h}.
\end{array}
\end{equation}

First we consider the Landau-Ginzburg\ models
with the Fermat type superpotentials
\begin{equation}
W_{Fermat} =
X_{1}^{a_{1}} +
X_{2}^{a_{2}} +
\cdots +
X_{n}^{a_{n}},
\end{equation}
and their $(a,c)$ states.
We define the phase symmetries $\rho_{i}$ which act on $X_{i}$ as
\begin{equation}
\rho_{i}X_{i} = e^{-2 \pi i q_{i} }X_{i},
\end{equation}
with trivial action for other fields.
The operator $ \rho_{i} $
can be represented by a diagonal matrix
whose diagonal matrix elements are 1 except for
$ {(\rho_{i})}_{i,i}
= e^{-2 \pi i q_{i} }$.
It is obvious that
\begin{equation}
j^{-1} = \rho_{1} \cdots \rho_{5}.
\end{equation}

In ref.\cite{sa2} the mirror map
for the $(a,c)$ ground states
in the $j^{-l}$-twisted sector
$\keta {j^{-l}}$
are considered.
In the $j^{-l}$-twisted sector, if a field $X_{i}$ is invariant
then
\begin{equation}
\rho_{i}^{-l} = \rho_{i}^{-l_{i}} = {\rm identity},
\end{equation}
where $-l_{i} \equiv -l \ {\rm mod} \ a_{i}$ and one gets
\begin{equation}
\label{key1}
j^{-l} = \prod_{{-l{q_{i}}} \notin \ZZ} {\rho_{i}^{l_{i}}}.
\end{equation}
So, we may represent
${\keta {j^{-l}}}
=\keta {\prod_{{-lq_{i}} \notin \ZZ}{\rho_{i}^{{l_{i}}}}}$.
Furthermore,
we can calculate the $U(1)$ charges of
this ground state
and the result is
\begin{equation}
{( - \sum_{{-lq_{i}} \notin \ZZ}{l_{i}q_{i}},
\sum_{{-lq_{i}} \notin \ZZ}{l_{i}q_{i}})}.
\end{equation}
Using the phase symmmetry $\rho_{i}$ considered above,
one can obtain the following mirror map for Landau-Ginzburg\ model
\cite{sa2};
\begin{equation}
\keta {\prod_{{-lq_{i}} \notin \ZZ}{\rho_{i}^{l_{i}}} } \ \ \
\stackrel{\rm mirror \ partner}{\longleftrightarrow} \ \ \
\prod_{{-lq_{i}} \notin \ZZ}{X_{i}^{l_{i}}}.
\end{equation}

There are potentials of other type,
i.e. so-called chain (tadpole) potential
\begin{equation}
W_{chain} = X_{1}^{a_{1}}X_{2} + \cdots +
X_{n-1}^{a_{n-1} - 1}X_{n} + X_{n}^{a_{n}}
\end{equation}
and loop potential
\begin{equation}
W_{loop} = X_{1}^{a_{1}}X_{2} + \cdots + X_{n}^{a_{n}}X_{1}.
\end{equation}
In general the Landau-Ginzburg\ superpotentials contain
Fermat, chain and loop terms.
To study the mirror theory,
we have to consider the transposed potentials \cite{bh}.
For the chain potentials,
their transposed potentials are
\begin{equation}
\bar{W}_{chain} = \bar{X_{1}}^{a_{1}} + \bar{X_{1}}
\bar{X_{2}}^{a_{2}}
+ \cdots +
{\bar{X}}_{n-1} \bar{X_{n}}^{a_{n}},
\end{equation}
and for the loop potentials
\begin{equation}
\bar{W}_{loop} = \bar{X_{1}} \bar{X_{2}}^{a_{1}} + \cdots
+ {\bar{X}}_{n-1} \bar{X_{n}}^{a_{n-1}}+ \bar{X_{n}}
\bar{X_{1}}^{a_{n}}.
\end{equation}
Their corresponding defining equations
are the hypersurfaces of the mirror manifold.
For the Fermat type potentials
the transposed potentials
are the same form
as the original ones.

The phase symmetreis $\rho_{i}$ of chain and loop potentials
were studied by Kreuzer \cite{kreuzer}.
Let us briefly summarize the results.
The definition of phase transformations is
\begin{equation}
\rho_{i}X_{l} = {\rm exp(2 \pi i}\varphi_{l}^{(i)}{\rm )}X_{l}.
\end{equation}

For the chain potentials, we have
\begin{equation}
\varphi_{l}^{(i)} = \frac{(-1)^{l-i+1}}{a_{i} \cdots a_{l}}
\ \ {\rm for} \ 1 \le l \le i,
\end{equation}
and $\varphi_{i+1}^{(i)} = \cdots = \varphi_{n}^{(i)} = 0$.
Moreover we have the following relations,
\begin{equation}
a_{l}\varphi_{l}^{(i)} + \varphi_{l+1}^{(i)} = - \delta_{l}^{i},
\end{equation}
\begin{equation}
a_{i+1}\varphi_{l}^{(i+1)} + \varphi_{l}^{(i)} = - \delta_{l}^{i+1}.
\end{equation}
From these relations, we can see
\begin{equation}
\label{chain}
(\rho_{i})^{a_{i}} \rho_{i-1} = 1,
\end{equation}
\begin{equation}
q_{i} = - \sum_{j=i}^{n}{\varphi_{i}^{(j)}}.
\end{equation}

For the loop potentials, one obtains
\begin{equation}
\varphi_{i+l}^{(i)} = \frac{(-1)^{n-l}a_{i} \cdots a_{i+l-1}}{\Gamma}
\ \ {\rm for} \ 0 \le l < n,
\end{equation}
where $\Gamma = A - (-1)^{n},\ \ A = a_{1}a_{2} \cdots a_{n}$.
In a similar fashion as chain potentials,
one can see
\begin{equation}
a_{l}\varphi_{l}^{(i)} + \varphi_{l+1}^{(i)} = - \delta_{l+1}^{i},
\end{equation}
\begin{equation}
a_{i}\varphi_{l}^{(i+1)} + \varphi_{l}^{(i)} = - \delta_{l}^{i}
\end{equation}
and
\begin{equation}
\rho_{i} (\rho_{i+1})^{a_{i}} = 1,
\end{equation}
\begin{equation}
q_{i} = - \sum_{l=1}^{n}{\varphi_{i}^{(l)}}.
\end{equation}

For both cases we may represent any twist $h$
uniquely in the form
\begin{equation}
h =
\prod
{\rho_{i}^{\alpha_{i}}}
\end{equation}
where $0 \le \alpha_{i} <  a_{i}$.
A simple calculation shows that
the $U(1)$ charge of the state $\keta h$ is
\begin{equation}
(-\sum_{{\rm all} \ i}{\varphi_{i}},
\sum_{{\rm all} \ i}{\varphi_{i}})
\end{equation}
where
${\varphi_{i}} =
\sum_{l=i}^{n}{{\alpha_{l}}{\varphi_{i}^{(l)}}}$.

Due to the obsevation of the $U(1)$ charge,
one can obtain the following mirror map for Landau-Ginzburg\ model
\cite{kreuzer};
\begin{equation}
\label{lgmirror}
\keta {\prod
{\rho_{i}^{l_{i}}} } \ \ \
\stackrel{\rm mirror \ partner}{\longleftrightarrow} \ \ \
\prod
{\bar{X}_{i}^{l_{i}}},
\end{equation}
where $\bar{X_{i}}$ are the fields of the transposed potential.
The special class of this mirror symmetry, namely
\begin{equation}
\label{specialmirror}
{\rm  (-1,1) \ states} \ \ \keta {j^{-l}} \ \ \
\stackrel{\rm mirror \ partner}{\longleftrightarrow} \ \ \
{\rm  (1,1) \ states} \ \
\prod_{{\varphi_{i}} \notin \ZZ}{\bar{X}_{i}^{l_{i}}} \keta {0},
\end{equation}
is important since the monomials $\prod_{i}{\bar{X}_{i}^{l_{i}}}$
correspond to the monomial deformations
of the complex structure of the mirror manifold.

\subsection{Correspondence between Landau-Ginzburg\ model and toric geometry}

In this section,we study the toric geometrical structure
of  Landau-Ginzburg\  model.
We will show that the Batyrev's formula of
eqs.(\ref{vb11}) and (\ref{vb21})
exactly count the number of
the $(-1,1)$ and $(1,1)$ states, respectively,
as expected.
This analysis enables us to tell
that which states sould correspond to
the twisted sector problem.

First, we consider the Fermat type potentials
and study their toric aspects.
This was already done in \cite{sa3},
so we briefly review the discussion of \cite{sa3}
and make it more precise.
Then the generalization to the chain and loop potentials
will be given.

The eq.(\ref{key1}) is the key equation for our purpose.
This implies
\begin{equation}
\label{key2}
-lq_i = u_i - l_{i}q_{i} \quad \quad {\rm for} \
i=1 \sim 5,
\end{equation}
where $-l_i$ are defined to be
$-a_i+1 \le -l_{i} \le 0$.
Thus $u_i$ are uniquely determined.
Clearly, $u_i$ are integers
and $l_{i}=0$ if $X_{i}$ is invariant
under $j^{-l}$ action.
$u_1$ always vanish
since $w_1 = 1$.

Let $u \equiv (u_2,u_3,u_4,u_5)$
be an integral vector.
It can be shown \cite{sa3}
that $u$ is on a face
of a dual polyhedron $\Delta^*(w)$.
So we should assert
that $u$ is just an integral point inside
$\Delta^*(w)$,
which can be identified with the exceptional divisors.
Through this identification,
we obtain the one-to-one correspondence
between the $(-1,1)$ state $\keta {j^{-l}}$
and the exceptional divisor.
Moreover, once this identification is made,
we can obtain the monomial-divisor mirror map
\cite{agm3}
for Calabi-Yau manifold s.

A simple observation shows that
the above identification is equivalent to
the following identification
\begin{equation}
\rho_{i}^{a_{i}} = 1  \  \  \rightarrow
\ \ (0, \cdots,1, \cdots,0)
\in \Delta^{*},
\end{equation}
where the components of the vectors are
zero except for the $i-$th entry being 1.
More concretely, for the twist $j^{-l}$
eq.(\ref{key2}) implies
\begin{equation}
\label{rhotoric}
\rho_{i}^{l} =
{(\rho_{i}^{a_{i}})}^{-u_{i}}{\rho_{i}^{l_{i}}}.
\end{equation}

If the $(-1,1)$ state
$\keta {j^{-l}}$ with 2 and 3 invariant field under
$j^{-l}$ action,
the toric data $u$ obtained by the above method
lies on the 2 and 1 dimensional face $\Theta^{*}$,
respectively.
If the $(-1,1)$ state $\keta {j^{-l}}$
has only one invariant field,
we can obtain the toric data
$\nu^{*(\alpha)}$
by the same mehtod
and $\nu^{*(\alpha)}$ lies on a codimension 1
(i.e. 3-dimensional) face $\Theta^{*}$.
Moreover, in this case we can further obtain
the new toric data
$\nu^{*(\beta)}$
from the descendent twist $j^{-l+1}$.
If only one field is invariant under $j^{-l+1}$ action,
we can obtain the third toric data
$\nu^{*(\gamma)}$
from the twist $j^{-l+2}$.
This successive process will stop
when the twist $j^{-l+n}$ has 2 or 3 invariant fields.
These results are
in complete agreement with
the subtraction of
$\sum_{\bf{codimension} \; {\Theta^{*}} = 1}
{l^{'}({\Theta^{*}})}$
in the Batyrev's formula
(\ref{vb11}).

It was noticed in \cite{sa3}
that the mirror partners of the $(-1,1)$ states
written in the form
$\prod_{ \tilde{\theta_{i}}^{h^{\prime}}= 0}{{X_{i}}^{l_{i}}}
\keta{h}$
can not be described by the monomials of fields
in the transposed potential.
So it is natural to expect
that  these states correspond to the twisted sector problem.
To prove this, we should note the fact that
the number of $(-1,1)$ states
$\keta {j^{-l}}$ with two invariant fields
(say, $X_{i}$ and $X_{j}$) is $l^{'}({\Theta}^{*})$,
where ${\Theta}^{*}$ is the 2-dimensional dual face of
1-dimensional face ${\Theta}(\nu_{i}, \nu_{j})$.
The toric data
$\nu_{i}$ is the vector whose components are $-1$
except for the $i-$th entry being $a_{i}-1$.
This vector
corresponds to the monomial ${X_{i}}^{a_{i}}$.
If ${\Theta}(\nu_{i}, \nu_{j})$ contains integral points
in its interior,
these points should correspond to the monomials
${X_{i}}^{{\alpha}_{i}}X_{j}^{{\alpha}_{j}}$,
and the number of such monomials is shown to be
$l^{'}(\Theta)$.
Since the $U(1)$ charge of the state
$\keta {j^{-l+1}}$
with two invariant fields is
$(-2+\sum_{lq_{i}\in \ZZ }{q_{i}},\sum_{lq_{i}\in \ZZ }{q_{i}})$
and
${{\alpha}_{i}q_{i}}+{{\alpha}_{j}q_{j}} = 1$,
the states
${X_{i}}^{{\alpha}_{i}-1}{X_{j}}^{{\alpha}_{j}-1}
\keta {j^{-l+1}}$
are $(-1,1)$ states.
One can see that the number of such states is
$l^{'}({\Theta})l^{'}({\Theta}^{*})$
with 2-dimensional dual face
${\Theta}^{*}$.

To generalize the above result
to chain and loop potentials,
we should extend the identification of
(\ref{rhotoric}) to
\begin{equation}
\label{rhotoric1}
\rho_{i}^{a_{i}} = 1  \  \  \rightarrow
\ \ (0, \cdots,1, \cdots,0)
\in \Delta^{*}
\end{equation}
and
\begin{equation}
\label{rhotoric2}
\rho_{i}^{a_{i}}{\rho_{j}} = 1  \  \  \rightarrow
\ \ (0, \cdots,1, \cdots,0)
\in \Delta^{*},
\end{equation}
where the components of the vectors are
zero except for the $i-$th entry being 1
as before.
It can be shown that
these identifications coincide with the equivalence
between the mirror map of (\ref{lgmirror})
and the \'{e}tale map of eq.(\ref{etale}).

Furthermore we can point out
the relation between the Mori vectors
and the phase symmetries.
A Mori cone which consists of Mori vectors
is a dual cone of a K\"ahler cone
and can describe the good coordinates
for a moduli space of complex structure \cite{hkty1}.
A Mori vector represents a relation of the monomials
in a defining equation of a mirror Calabi-Yau manifold .
Since the mirror map (\ref{specialmirror})
connects these monomials
and the $j^{-l}$
twist,
we can expect that a Mori vector translates into
a relation of $j^{-l}$
or ${\rho_{i}}^{l_{i}}$ twists.
Indeed, a following type of  relation,
\begin{equation}
\prod_{i}{\rho_{i}^{l_{i}}} = {\rm identity},
\end{equation}
leads to a Mori vector.
In the later section we will show some examples.

\section{Mirror symmetry and the web of string vacua}

\subsection{Twisted sector problem revisited}

It is a well known fact \cite{vb,hkty1} that in certain situations
not all of the K\"ahler moduli space can be described
by toric divisors of the Calabi-Yau manifold\ ${\cal M}$.
Similarly, the deformation of the defining equation
of the hypersurface of ${\cal M}$ in a toric variety is
in general not sufficient to describe
the complex structure moduli space of ${\cal M}$.
More precisely,
as remarked under (\ref{vb11}) and (\ref{vb21}),
there are non toric divisors in ${\cal M}$ when
\begin{equation}
\label{tw11}
\sum_{\bf{codimension} \; \Theta^{*} = 2,
\Theta^{*} \in \Delta^{*}}
{l^{'}(\Theta^{*})
l^{'}(\Theta)}
\end{equation}
is non-zero.
In a similar fashion, when
\begin{equation}
\label{tw21}
\sum_{\bf{codimension} \; \Theta = 2,
\Theta \in \Delta}
{l^{'}(\Theta)
l^{'}(\Theta^{*})}
\end{equation}
is non-zero
there are non-algebraic deformations
of the complex structure deformations
of the complex structure of ${\cal M}$.

In section 3.2, we have shown that in the Landau-Ginzburg\ context
these numbers (\ref{tw11}) ,(\ref{tw21})
are equivalent to the number of
the states ${\prod_{{\tilde{\theta_{i}}^{h^{\prime}}} \in \ZZ}{X_{i}^{l_{i}}}}\keta {j^{-l}}$
with $U(1)$ charge $(-1,1)$, $(1,1)$
respectively.

\subsection{Orbifold construction and the web of Landau-Ginzburg\ vacua}

In this section, we would like to propose a method
which circumvent
this twisted sector problem
under some circumstances.
As a result
either all of the non-toric divisors in (\ref{tw11})
or the non-algebraic deformations
of the complex structure in (\ref{tw21}),
can be treated by this method.
In the following we will concentrate ourselves on
the K\"{a}hler moduli space
and the corresponding $(-1,1)$ states.

The main idea is orbifold construction,
i.e. we further divide the original theory
by a new discrete symmetry,
whose generator is denoted by $g$,
and project out the unexpected states
${\prod_{{\tilde{\theta_{i}}^{h^{\prime}}} \in \ZZ}{X_{i}^{l_{i}}}}\keta {j^{-l}}$
with $U(1)$ charge $(-1,1)$.
At the same time,
we have new twisted states
$\keta {g^{-m}j^{-l^{'}}}$
with charge $(-1,1)$,
which can be identified with toric data.
According to this orbifoldization,
the superpotential should be deformed,
as discussed in \cite{ls}

On the toric geometry side, after this procedure is done
we have new polyhedra ${\tilde{\Delta}}^{*}$ and ${\tilde{\Delta}}$
such that
\begin{equation}
{\tilde{\Delta}}^{*}
\supset
{{\Delta}^{*}},
\ \
{\tilde{\Delta}}
\subset
{\Delta}.
\end{equation}
This relation implies that
the moduli spaces of the original theory
and the orbifoldized thory are connected
\cite{ca1,ca}.

Can we find such a suitable twist?
In the following, we  will show that
under some circumstances
we can find the appropriate $g$ for our construction.

Consider the state
$\prod_{ \tilde{\theta_{i}}^{h^{\prime}}= 0}{{X_{i}}^{\alpha_{i}}}
\keta{j^{-l+1}}$
where $0 \le \alpha_{i} \le {{a_{i}}-2}$.
The $U(1)$  charge of this state
is obtained to be
\begin{equation}
\label{uac1}
\left(\begin{array}{c}
{ - \sum_{\tilde{\theta_{i}}^{h^{\prime}}>0}{(1-q_{i}-\tilde{\theta_{i}}^{h^{\prime}})}}
+ \sum_{\tilde{\theta_{i}}^{h^{\prime}}=0} {(2q_{i}-1)}
+\sum_{\tilde{\theta_{i}}^{h^{\prime}}=0} {{\alpha_{i}}{q_{i}}}
 \\
{ \sum_{\tilde{\theta_{i}}^{h^{\prime}}>0}{(1-q_{i}-\tilde{\theta_{i}}^{h^{\prime}})}
+\sum_{\tilde{\theta_{i}}^{h^{\prime}}=0} {{\alpha_{i}}{q_{i}}}
}
\end{array} \right) ,
\end{equation}
where $h = j^{-l}$ and $h^{'} = h{j^{-1}}$.

In this case the appropriate $g$
is found to be
\begin{equation}
\label{defg}
g^{-1} =
\prod_{\tilde{\theta_{i}}^{h^{\prime}} = 0}{{\rho_{i}}^{{{\alpha}_{i}}+1}}.
\end{equation}
Using this $g$,
the new twisted state
$\keta{{g^{-1}}{j^{-l+1}}}$
will appear and its $U(1)$ charge is obtained to be
$(-Q,Q)$,
where
$Q =
{  \sum_{\tilde{\theta_{i}}^{h^{\prime}}>0}{(1-q_{i}-\tilde{\theta_{i}}^{h^{\prime}})}}
+ \sum_{\tilde{\theta_{i}}^{h^{\prime}}=0} {{\alpha_{i}}{q_{i}}}.
$

We see that if
\begin{equation}
\sum_{\tilde{\theta_{i}}^{h^{\prime}} = 0}{(2q_{i} - 1)} +
\sum_{\tilde{\theta_{i}}^{h^{\prime}} = 0}{{\alpha_{i}}{q_{i}}}
=
- \sum_{\tilde{\theta_{i}}^{h^{\prime}} = 0}{{\alpha_{i}}{q_{i}}},
\end{equation}
then the two states
$\prod_{ \tilde{\theta_{i}}^{h^{\prime}}= 0}{{X_{i}}^{\alpha_{i}}}
\keta{j^{-l+1}}$
and
$\keta{{g^{-1}}{j^{-l+1}}}$
have same $U(1)$ charge.
To obtain the toric data of the orbifoldized theory,
we should use the identifications of (\ref{rhotoric1})
and (\ref{rhotoric2}) in $j^{-l+1}$.

\subsection{Examples}

As a first example, we consider the Landau-Ginzburg\ model with the
superpotential
\begin{equation}
\label{w21}
W_{1} =
X_{1}^{12}+X_{2}^{12}+X_{3}^{12}+X_{4}^{4}+X_{5}^{2},
\end{equation}
with $U(1)$ charges of $X_{i}$ being
\begin{equation}
( \
\frac{1}{12}, \
\frac{1}{12}, \
\frac{1}{12}, \
\frac{1}{4}, \
\frac{1}{2} \
),
\end{equation}
which corresponds to the hypersurface
in $WCP_{(1,1,1,3,6)}^{4}[12]$ and were studied in ref.\cite{klemm}.

We find that there are three $(-1,1)$ states
$\keta {j^{-2}}$,
$\keta {j^{-4}}$ and
$X_{4} {\keta {j^{-3}}}$.
From the first two of these states
we can easily obtain the toric data and
the results are displayed in Table \ref{w1t},
where Dim. means the dimension of the dual face
on which the point $\nu^{*}$ lies.
The reader should remind
that for Calabi-Yau\ threefolds,
a point lying on a face of Dim. = 1 or 2
corresponds to a toric divisors
and
$\nu_0^*=(0,0,0,0)$
corresponds to the canonical divisor.
Note that since only one field is invariant under
the twist $j^{-2}$,
we should consider the descendent twist $j^{-1}$
to obtain the toric data completely,
as explained.

\begin{table}[htbp]
\[ \begin{tabular}{||c|c|c|c||} \hline
(a,c) state &
twist & $\nu_j^*$ vector & Dim. \\ \hline \hline
$\keta {j^{-2}}$ &
${j^{-2}}$ &
$\nu_6^*=(0,0,0,-1)$ &
$ 3 $
\\ \hline
 &
${j^{-1}}$ &
$\nu_0^*=(0,0,0,0)$ &
$ 4 $
\\ \hline
$\keta {j^{-4}}$ &
${j^{-4}}$ &
$\nu_7^*=(0,0,-1,-2)$ &
$ 2 $
\\ \hline
\end{tabular} \]
\caption{The toric data of
the $(-1,1)$ states
in the
Landau-Ginzburg\ orbifold of $W_{1}$}
\label{w1t}
\end{table}

The remaining $(-1,1)$ state represented by
$X_{4} {\keta {j^{-3}}}$
should correspond to so-called
twisted sector problem in toric geometry.
divisors.
To solve this twisted sector problem,
we will use the mehtod proposed in the section 4.2.
Let us remind that
the essential ingredient is an orbifold construction.
If one finds an appropriate orbifoldization,
the $(-1,1)$ states such as
$X_{4} {\keta {j^{-3}}}$
are projected out and new $(-1,1)$ states will appear
in the new twisted sectors.
For our example, we can find such an appropriate orbifoldization.

Consider the following superpotential
\begin{equation}
\label{w2}
W_{2} =
Y_{1}^{12}+Y_{2}^{12}+Y_{3}^{12}+Y_{4}^{2}Y_{5}+Y_{5}^{2}.
\end{equation}
Note that the set of fields $Y_{i}$ have
the same $U(1)$ charges as $X_{i}$'s.
So $W_{2}$-theory has the same topological numbers as
$W_{1}$-theory.
Hence one may identify the fields $X_{i}$ and $Y_{i}$;
\begin{equation}
X_{i} = Y_{i} \ \ {\rm for} \ \ i = 1,2,3, \ \ \
X_{4}^{4} = Y_{5}^{2}, \ \ X_{5}^{2} = Y_{4}^{2}Y_{5}.
\end{equation}
For one-to-one identification, we should further identify
\begin{equation}
\label{z2}
X_{4} \sim -X_{4},\ \ \
X_{5} \sim -X_{5}.
\end{equation}
Since this twist is not included
in the $U(1)$ twist $j^{-l}$,
 $W_{2}$-theoy can be considered as
$\ZZ_{2}$ orbifold of $W_{1}$-theory
\cite{ls}.
We denote by $g$ this $\ZZ_{2}$ twist.
From the results of section 4.2,
we find the appropriate $g$ of (\ref{defg})
is,
\begin{equation}
g^{-1} = {{\rho_{4}}^2}{\rho_{5}},
\end{equation}
this agrees with $\ZZ_{2}$ identification (\ref{z2}).
The new $(-1,1)$ state is
$\keta {g^{-1}j^{-3}}$.
Below one finds our results.
The twisted sector problem has resolved
as expected.

\begin{table}[htbp]
\[ \begin{tabular}{||c|c|c|c||} \hline
(a,c) state &
twists &
$\nu_j^*$ vector & Dim. \\ \hline \hline
$\keta {j^{-2}}$ &
${j^{-2}}$ &
$\nu_6^*=(0,0,0,-1)$ &
$ 3 $
\\ \hline
 &
${j^{-1}}$ &
$\nu_0^*=(0,0,0,0)$ &
$ 4 $
\\ \hline
$\keta {j^{-4}}$ &
${j^{-4}}$ &
$\nu_7^*=(0,0,-1,-2)$ &
$ 2 $
\\ \hline
$\keta {g^{-1}j^{-3}}$ &
${g^{-1}j^{-3}}$ &
$\nu_{11}^*=(0,0,-1,-1)$ &
$ 3 $
\\ \hline
  &
${g^{-1}j^{-2}}$ &
$\nu_{10}^*=(0,0,-1,0)$ &
$ 3 $
\\ \hline
  &
${g^{-1}j^{-1}}$ &
$\nu_9^*=(0,0,-1,1)$ &
$ 3 $
\\ \hline
  &
${g^{-1}}$ &
$\nu_8^*=(0,0,0,1)$ &
$ 1 $
\\ \hline
\end{tabular} \]
\caption{The toric data of
the $(-1,1)$ states in
the Landau-Ginzburg\ orbifold of $W_{2}$}
\label{w2t}
\end{table}

Our results shows that the toric data $(\Delta_{i}, \Delta_{i}^{*})$
which
correspond to $W_{i}$-theory satisfy
\begin{equation}
\Delta_{1} \supset \Delta_{2}, \ \
{\rm and} \ \ \Delta_{1}^{*} \subset \Delta_{2}^{*}.
\end{equation}
These properties of toric data imply that
the mouli spaces of these two theories are connected
\cite{ca1,ca}.
From our construction, the singular $\ZZ_{2}$ orbifold point
should be considered that
the superpotential is $W_{2}$ and $(-1,1)$ states are
$\keta {j^{-2}}$ and
$\keta {j^{-4}}$.
Thus on the singular orbifold point,
the corresponding toric data are
$(\Delta_{2}, \Delta_{1}^{*})$.
This is not a dual pair
so that one could not obtain a smooth Calabi-Yau manifold .
However, this is a boundary point which connects
the moduli spaces of the two theories described by
the superpotentials $W_{1}$ and $W_{2}$.
Our orbifold construction corresponds to the toric construction
in refs.\cite{klemm,ca},
in which the structures of moduli spaces of Calabi-Yau manifold s
are studied.

Then, we consider the relation
between Mori vectors and the twist operators.
In the original theory whose potential is $W_{1}$,
there is
a $(-1,1)$ state $\keta {j^{-4}}$
whose toric data is $\nu_{7}^{*}$.
In terms of the mirror map for Landau-Ginzburg\ model
(\ref{specialmirror}), the relation
$({j^{-4}})^{3} = {j^{-12}} = {\rm identity}$
translates into
$({X_{1}}^{4}{X_{2}}^{4}{X_{3}}^{4})^{3}
 = ({X_{1}}^{12})({X_{2}}^{12})({X_{3}}^{12})$.
This implies the Mori vector
$l_{1}^{(1)} =
(0;1,1,1,0,0,-3,0).$
In a similar manner,
${j^{-4}} = (j^{-1})^{4}$
translates into
$({X_{1}^{4}}{X_{2}^{4}}{X_{3}^{4}})(X_{4}^{4})
(X_{5}^{2})^{2}
= {({X_{1}}{X_{2}}{X_{3}}{X_{4}}
{X_{5}})}^{4}$
and we get the Mori vector
$l_{1}^{(2)} =
(-4;0,0,0,1,2,1,0).$

In the orbifoldized theory
whose potential is $W_{2}$,
we have a new operator set
$\{ {\tilde{\rho}}_{i} \}$.
From the eq.(\ref{chain}), we have
\begin{equation}
\label{rhoalg2}
{\tilde{\rho_{1}}}^{12} = {\tilde{\rho_{2}}}^{12}
= {\tilde{\rho_{3}}}^{12} = {\tilde{\rho_{4}}}^{2}
= {\tilde{\rho_{4}}}{\tilde{\rho_{5}}}^{2}
= {\rm identity}.
\end{equation}
Note that from these basic relations of
the operators ${\tilde{\rho_{i}}}$,
we have the new relation
$ {({{\tilde{\rho}}_{4}}{{\tilde{\rho}}_{5}}^{2})}^{2}
{({\tilde{\rho_{4}}}^{2})}^{-1} =
{{\tilde{\rho}}_{5}}^{4} =
{\rm identity}$.
This relation
$ {({{\tilde{\rho}}_{4}}{{\tilde{\rho}}_{5}}^{2})}^{2}
{({\tilde{\rho_{4}}}^{2})}^{-1} =
{{\tilde{\rho}}_{5}}^{4} =
{\rm identity}$
gives us the new Mori vector.
In terms of the mirror map for Landau-Ginzburg\ model
(\ref{specialmirror}),
we can obtain
$({\bar{Y_{4}}}{\bar{Y_{5}}}^{2})^{2}
= ({\bar{Y_{4}}}^{2})
({\bar{Y_{5}}}^{4}).
$
The corresponding Mori vector is
$l_{2}^{(1)} =
(0;0,0,0,1,-2,0,1).$
In this orbifold model,
${j^{-4}} = (j^{-1})^{4}$
translates into the Mori vector
$l_{2}^{(2)} =
(-4;0,0,0,0,4,1,-1).$
As a result, the Mori vectors satisfy the relation
$l_{1}^{(2)} = l_{2}^{(1)} + l_{2}^{(2)}$
\cite{klemm}.
We can understand this relation
as the change of the set of twist operators
$\{ {\rho}_{i} \}$
of the phase symmetry,
or equivalently, as the orbifoldization.

As a second example, we consider
the hypersurface in $WCP_{(1,2,3,6,6)}[18]$
which was considered in \cite{agm,sa3}.
In this case the orbifoldized potential
contains a loop part.
The defining superpotential is
\begin{equation}
\label{w3}
W_{3} =
X_{1}^{18}+X_{2}^{9}+X_{3}^{6}+X_{4}^{3}+X_{5}^{3}.
\end{equation}
There are seven $(-1,1)$ states
and two of them correspond to
the twisted sector problem, i.e.
$X_{4} \keta{j^{-2}}$
and
$X_{5} \keta{j^{-2}}$.
The appropriate twist is
\begin{equation}
g^{-1} = {\rho_{4}}^{2}{\rho_{5}},
\end{equation}
and this generates the $\ZZ_{3}$ being
\begin{equation}
X_{4} \sim \alpha^{2} X_{4},\ \ \
X_{5} \sim \alpha X_{5}
\end{equation}
where $\alpha$ is a cubic loot of unity.
After the orbifoldization, the deformed superpotential includes
a loop potential and obtained to be
\begin{equation}
\label{w4}
W_{4} =
Y_{1}^{18}+Y_{2}^{9}+Y_{3}^{6}+Y_{4}^{2}Y_{5}+Y_{4}Y_{5}^{2}.
\end{equation}

In the orbifoldized theory,
the new $(-1,1)$ states are
$ \keta {g^{-1}j^{-2}}$ and
$ \keta {g^{-2}j^{-2}}$.
The corresponding toric data are
displayed in Table \ref{w4t}
with the descendent twists.
Our results show that
the twisted sector problem
has circumvented as expected.

\begin{table}[htbp]
\[ \begin{tabular}{||c|c|c|c||} \hline
(a,c) state &
twists &
$\nu_j^*$ vector & Dim. \\ \hline \hline
$ \keta {g^{-1}j^{-2}}$ &
${g^{-1}j^{-2}}$ &
$\nu_6^*=(0,0,-1,0)$ &
$ 3 $
\\ \hline
  &
${g^{-1}j^{-1}}$ &
$\nu_8^*=(0,0,1,-1)$ &
$ 3 $
\\ \hline
  &
${g^{-1}}$ &
$\nu_{10}^*=(0,0,1,0)$ &
$ 1 $
\\ \hline
$\keta {g^{-2}j^{-2}}$ &
${g^{-2}j^{-2}}$ &
$\nu_7^*=(0,0,0,-1)$ &
$ 3 $
\\ \hline
  &
${g^{-2}j^{-1}}$ &
$\nu_9^*=(0,0,-1,1)$ &
$ 3 $
\\ \hline
  &
${g^{-1}}$ &
$\nu_{11}^*=(0,0,0,1)$ &
$ 1 $
\\ \hline
\end{tabular} \]
\caption{The toric data of
Landau-Ginzburg\ orbifold of $W_{4}$}
\label{w4t}
\end{table}

Our third example \cite{ca1} is
the hypersurface in $WCP_{(1,2,2,2,7)}[14]$.
The corresponding superpotential is
\begin{equation}
\label{w5}
W_{5} =
X_{1}^{14}+X_{2}^{7}+X_{3}^{7}+X_{4}^{7}+X_{5}^{2},
\end{equation}
and there are two $(-1,1)$ states
$\keta {j^{-2}}$ and $\keta {j^{-8}}$.
Once we identify
\begin{equation}
\label{identifyw5}
X_{1} \sim -X_{1}, \ \
X_{5} \sim -X_{5},
\end{equation}
then we obtain the orbifoldized potential
\begin{equation}
\label{w6}
W_{6} =
Y_{1}^{7}+Y_{2}^{7}+Y_{3}^{7}+Y_{4}^{7}+Y_{1}Y_{5}^{2}.
\end{equation}
The corresponding hypersurface is
in $WCP_{(1,1,1,1,3)}[7]$.
This theory includes
two $(-1,1)$ states
$\keta {j^{-1}}$ and $\keta {j^{-2}}$.
Though their weights are diferent,
these two theories should be equivalent
since the identification (\ref{identifyw5})
is a part of projective equivalnce of $WCP_{(1,2,2,2,7)}[14]$,
\begin{equation}
(X_{1},X_{2},X_{3},X_{4},X_{5})
 \sim ({\lambda}X_{1},{\lambda}^{2}X_{2},
{\lambda}^{2}X_{3},{\lambda}^{2}X_{4},{\lambda}^{7}X_{5}),
\end{equation}
with ${\lambda} = -1$.

The vertices of the dual polyhedron $\Delta_{5}^{*}$
corresponding to the $W_{5}$-theory are
\begin{eqnarray*}
&(-2,-2,-2,-7)&,  \ \
(1,0,0,0), \ \
(0,1,0,0), \\
&(0,0,1,0)&, \ \
(0,0,0,1), \ \
(0,0,0,-1), \\
&(-1,-1,-1,-3)&, \ \
(-1,-1,-1,-4).
\end{eqnarray*}
The toric data $(0,0,0,-1)$ and
$(-1,-1,-1,-4)$ correspond to
the two $(-1,1)$ states
$\keta {j^{-2}}$ and $\keta {j^{-8}}$, respectively.
The toric data $(-1,-1,-1,-3)$ can be identified
with the descendent twist $j^{-7}$.

Turning our attention to the $W_{6}$-theory,
the vertices of $\Delta_{6}^{*}$ are
\begin{eqnarray*}
&(-1,-1,-1,-3)&,  \ \
(1,0,0,0), \ \
(0,1,0,0), \\
&(0,0,1,0)&, \ \
(0,0,0,1).
\end{eqnarray*}

At first sight, the sets of toric data for
these two theories are different.
However, we can show that
the toric data
which are obtained from $W_{6}$-theory
are the same as
the ones from $W_{5}$-thory.
The results are displayed in Table \ref{w6t}

\begin{table}[htbp]
\[ \begin{tabular}{||c|c|c|c||} \hline
(a,c) state &
twists &
$\nu_j^*$ vector & Dim. \\ \hline \hline
$ \keta {j^{-2}}$ &
${j^{-2}}$ &
$\nu_6^*=(0,0,0,-1)$ &
$ 3 $
\\ \hline
  &
${j^{-1}}$ &
$\nu_7^*=(-1,-1,-1,-4)$ &
$ 3 $
\\ \hline
  &
${j^{0}}$ &
$\nu_8^*=(-2,-2,-2,-7)$ &
$ 1 $
\\ \hline
\end{tabular} \]
\caption{The toric data of
the $(-1,1)$ states in
the Landau-Ginzburg\ orbifold of $W_{6}$}
\label{w6t}
\end{table}

As a final example,
we consider the hypersurface
in $WCP_{(1,1,1,2,5)}[10]_{-288}^{1}$,
where the superscript denotes the number of $(1,1)$ forms
and the subscript implies the Euler number.
The corresponding superpotential is
\begin{equation}
\label{w7}
W_{7} =
X_{1}^{10}+X_{2}^{10}+X_{3}^{10}+X_{4}^{5}+X_{5}^{2}.
\end{equation}
This theory contains only one $(-1,1)$ state
$\keta {j^{-1}}$
whose toric data is the origin
$(0,0,0,0)$
in the dual polyhedron ${\Delta}^{*}$.
The identification
\begin{equation}
\label{identifyw7}
X_{3} \sim -X_{3}, \ \
X_{5} \sim -X_{5},
\end{equation}
leads to the orbifoldized theory whose potential is
\begin{equation}
\label{w8}
W_{8} =
Y_{1}^{10}+Y_{2}^{10}+Y_{3}^{5}+Y_{4}^{5}+Y_{3}Y_{5}^{2},
\end{equation}
which corresponds to the hypersurface
in $WCP_{(1,1,2,2,4)}[10]_{-192}^{3}$.
This theory contains the additional
two $(-1,1)$ states, i.e.
$\keta {j^{-2}}$
and
$\keta {j^{-5}}$
whose toric data are
$(0,0,0,-1)$ and
$(0,-1,-1,-2)$, respectively.
Our results implies that
\begin{equation}
\Delta_{{7}} \supset \Delta_{{8}}, \ \
{\rm and} \ \ \Delta_{{7}}^{*} \subset \Delta_{{8}}^{*}.
\end{equation}
These relations show that
the moduli spaces of these two theories are connected,
even though their numbers of states,
namely their mass spectra,
are different.
This should be considered as
an example of the extremal transition
between the topologically different
Calabi-Yau manifold s.

\section{Application to type II strings}

In the context of string duality,
the $K3$-fibred Calabi-Yau manifold s are actvily studied.
The conjectured duality \cite{kv} is
that the heterotic string on $K3 \times {T}^{2}$
is dual to the type II string on the Calabi-Yau manifold\
which admits $K3$-fibration.
Such type II strings are the promising candidates
to obtain the Seiberg-Witten gauge theory \cite{seiwit}
by taking the suitable low energy limit \cite{kkv,w,kmv}.

In this section, we consider the simplest $K3$-fibred Calabi-Yau manifold s
constructed in \cite{klm},
i.e. the Fermat type Calabi-Yau manifold s in
$WCP_{(1,1,2w_{3},2w_{4},2w_{5})}[2d]$
whose $K3$-fibre are
$WCP_{(1,w_{3},w_{4},w_{5})}[d]$,
where $d = 1 + w_{3} + w_{4} + w_{5}$.
For these models,
our Landau-Ginzburg\ analysis shows that
the $(-1,1)$ state
$\keta {j^{-d}}$
always exists
and that
this state corresponds to
the toric data $\nu_{6}^* = (0,-w_{3},-w_{4},-w_{5})$.
Since the dual polyhedron ${\Delta}^{*}$ contains the two vertices
$\nu_{1}^{*} = (-1,-2w_{3},-2w_{4},-2w_{5})$
and
$\nu_{2}^{*} = (1,0,0,0)$,
we see $\nu_{6}^{*} = {\frac{1}{2}} (\nu_{1}^{*} + \nu_{2}^{*})$.
This corresponds to the relation
$(j^{-d})^{2} = j^{-2d}
$.
In other words,
$\nu_{1}^{*}$,
$\nu_{2}^{*}$ and
$\nu_{6}^{*}$
correspond to the monomials
$X_{1}^{2d}$,
$X_{2}^{2d}$ and
$X_{1}^{d}X_{2}^{d}$, respectively,
of the defining equation
of the mirror manifold
on which a type IIB string
gets compactified.
The existence of these monomials is
necessary to obtain
the Seiberg-Witten geometry \cite{kkv,w,kmv}.
More precisely,
the defining equation of the
$K3$-fibred
Calabi-Yau manifold
can be written as
\begin{equation}
p = {\frac{1}{2d}}{X_{1}^{2d}}
+ {\frac{1}{2d}}{X_{2}^{2d}}
+ {\frac{1}{d \sqrt{c}}}{X_{1}^{d}X_{2}^{d}}
+ \tilde{W}({\frac{X_{1}X_{2}}{c^{1/d}}},X_{3},X_{4},X_{5}),
\end{equation}
where $c$ is a one of the parameters
which characterize the complex structures.
After suitable reparametrization,
one obtains the desirable form of \cite{w}
which can represent the embedding of
the Seiberg-Witten curve
in terms of the ALE space.
If the $K3$ fibre admits the elliptic fibration,
we have more $(-1,1)$ state written in the form
$\keta {j^{- \frac{2d}{n}}}$
with the integer $n > 2$.

Let us remind that our orbifold example
$W_{2}$ of (\ref{w2})
gives us a new set of relations of
twist orerators,
or equivalently a new set of Mori vectors.
This fact suggests the interesting possibility
that after the appropriate orbifoldization,
we will get certain Calabi-Yau manifold s
in which the method of
geometric engineering proposed in \cite{kmv}
will adequately work.

As a simple and interesting example,
consider the superpotential
\begin{equation}
\label{w}
W =
X_{1}^{12}+X_{2}^{12}+X_{3}^{6}+X_{4}^{6}+X_{5}^{2}.
\end{equation}
This model corresponds to the Calabi-Yau\ manifold
whose hypersurface is embedded in
$WCP_{(1,1,2,2,6)}[12]^{2}_{-252}$.
This hypersurface admits $K3$-fibration.
This model contains two $(-1,1)$ states
$\keta {j^{-2}}$ and $\keta {j^{-6}}$.
Their toric data are $(0,0,0,0)$
and $(0,-1,-1,-3)$, respectively.
The dual polyhedron $\Delta^{*}_{W}$
consists of the following vertices
\begin{eqnarray*}
&\nu^*_1=(-1,-2,-2,-6)&,  \ \
\nu^*_2=(1,0,0,0), \ \
\nu^*_3=(0,1,0,0), \\
&\nu^*_4=(0,0,1,0)&, \ \
\nu^*_5=(0,0,0,1), \ \
\nu^*_6=(0,-1,-1,-3),
\end{eqnarray*}
and we have two Mori vectors
$l^{(1)}_{1}=(0;1,1,0,0,0,-2,0)$ and
$l^{(2)}_{1}=(-6;0,0,1,1,3,1,0)$
(the last entry 0 is unnecessary at present,
but we have added it for later use).
These Mori vectors $l^{(1)}_{1}$ and $l^{(2)}_{1}$
correspond to the two relations of the $U(1)$ twist
operators
$(j^{-6})^{2} = j^{-12}$ and
$j^{-6} = (j^{-1})^{6}$ respectively,
as noted in the previous section.
Due to the these Mori vectors,
the method of geometric engineering in \cite{kmv}
does not work as it stands.
However after the appropriate orbifoldization,
we can apply that method
to obtain a quantum field theory.

Consider the $\ZZ_{2}$ orbifoldization
\begin{equation}
X_{2} \sim -X_{2}, \ \
X_{5} \sim -X_{5}.
\end{equation}
We then obtain the orbifoldized superpotential
\begin{equation}
\label{wprime}
W^{\prime} =
Y_{1}^{12}+Y_{2}^{6}+Y_{3}^{6}+Y_{4}^{6}+Y_{2}Y_{5}^{2}.
\end{equation}
The corresponding hypersurface is embedded
in $WCP_{(1,2,2,2,5)}[12]_{-180}^{4}$.
Note that the corresponding hypersurface does not admit
$K3$-fibration.
But the hypersurface which corresponds to
the following transposed potential
\begin{equation}
\label{tildewprime}
\tilde{W^{\prime}} =
\tilde{Y_{1}}^{12}+\tilde{Y_{2}}^{6}\tilde{Y_{5}}
+\tilde{Y_{3}}^{6}+\tilde{Y_{4}}^{6}+\tilde{Y_{5}}^{2}
\end{equation}
does admit  $K3$-fibration,
since that hypersurface is embedded in
$WCP_{(1,1,2,2,6)}[12]$.
The original Calabi-Yau\ manifold is embedded
in the same weighted projective space.
This is the general property for our orbifoldization
that the transposed hypersurface of the orbifoldized one
is embedded in the same weighted projective space
as the original hypersurface is embedded in.
This fact is important.
To obtain a quantum field theory,
we have only to consider
the local mirror symmetry of \cite{kmv}.
So at least one of the hypersurfaces corrresponding to
${W^{\prime}}$ and $\tilde{W^{\prime}}$
should have the $K3$-fibration structure.

At the orbifold point,
we still have the two $(-1,1)$ states
$\keta {j^{-2}}$ and $\keta {j^{-6}}$
so that the dual polyhedron $\Delta^{*}_{W}$
is unchanged.
However once we consider the orbifoldized potential
(\ref{wprime}),
we obtain a new set of operators
$\{ {\tilde{\rho}}_{i} \}$ satisfying
\begin{equation}
{\tilde{\rho_{1}}}^{12}
= {\tilde{\rho_{2}}}^{6}{\tilde{\rho_{5}}}
= {\tilde{\rho_{3}}}^{6} = {\tilde{\rho_{4}}}^{6}
= {\tilde{\rho_{5}}}^{2}
= {\rm identity}.
\end{equation}
Since the new relation of operators
${({{\tilde{\rho}}_{2}}^{6}{{\tilde{\rho}}_{5}})}^{2}
= {{{\tilde{\rho}}_{2}}^{12}}{{{\tilde{\rho}}_{5}}^{2}}
= {\rm identity}$ holds,
we have a new toric data
$\nu^{*}_{7} = (2,0,0,-1)$
which corresponds to
${{{\tilde{\rho}}_{2}}^{12}}
= {\rm identity}$.
The insertion of $\nu^{*}_7$ into the dual polyhedron
$\Delta^{*}_{W}$ implies the toric blow-up.
As a result, we obtain a new set of Mori vectors
\begin{equation}
l^{(1)}_{2}=(0;1,1,0,0,0,-2,0) = l^{(1)}_{1},
\end {equation}
\begin{equation}
l^{(2)}_{2}=(0;0,-2,0,0,1,0,1),
\end {equation}
\begin{equation}
l^{(3)}_{2}=(-6;0,2,1,1,2,1,-1).
\end {equation}
The orbifold relation is
$l^{(2)}_{1} = l^{(2)}_{2} + l^{(3)}_{2}$.
Using $l^{(1)}_{2}$ and $l^{(2)}_{2}$,
we can obtain a quantum field theory
by the geometric engineering of \cite{kmv}.

\section{Discussions}

In this paper, we have given
the toric interpretation of
Landau-Ginzburg\ models.
This interpretation gives us the insight
and the useful techniques
to resolve the twisted sector problem.
The orbifold construction is a powerful tool
for solving the twisted sector problem.
Using mirror symmetry,
we can study somewhat large class of
the K\"{a}hler moduli space.
Our construction shows that
through the orbifold point
the different Landau-Ginzburg\ string vacua are connected.
It is discussed
that a relation of twist operators for orbifoldization
have a natural interpretation as a Mori vector.

It is natural to expect
that Landau-Ginzburg\ models with $c = 12$ correspond to
Calabi-Yau\ fourfolds.
Recently, Calabi-Yau\ fourfolds are studied in the context of F-theory
\cite{mv,bs},
i.e. a F-theory on a Calabi-Yau\ fourfold is a  dual candidate
for a heterotic string on a Calabi-Yau\ threefold.
Our results of correspondence between the Landau-Ginzburg\ model and toric
geometry can be easily extended to Calabi-Yau\ fourfolds
and give us some insight for those theories.

In \cite{mayr} it is pointed out that for Calabi-Yau\ fourfolds
on which F-theory can compactify,
the same difficulties as twisted sector problem
for threefolds may occur
when one considers the non-perturbative superpotential
\cite{witten}.
We hope that our approach for solving the twisted sector problem
will be useful to improve the unexpected situations
of Calabi-Yau\ fourfolds.

Recently, some attention has been paid to
the extremal transition \cite{extremal}.
It is interesting to consider the relation
between our orbifold construction and such
extremal transition.
In the context of F-theory,
the 4-cycle transition is studied.
For example \cite{alg},
the Fermat hypersurface in $WCP_{(1,2,2,3,4)}[12]_{74}^{2}$
has the 4-cycle transition
by which this theory transforms to
the Fermat hypersurface in $WCP_{(1,1,1,1,2)}[6]_{103}^{1}$.
It is easy to see that
this transition can not be considered as our orbifold construction.
So, it is interesting to consider
whether our orbifold construction gives us
a new class of examples as extremal transition.

\vspace{1cm}

{\it Acknowledgements} :
The author would like to thank H. Itoyama
for valuable discussions
and for careful reading of this manuscript.

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