A.C. Avram, P. Candelas, D. Jancic, M. Mandelberg

On the Connectedness of the Moduli Space of Calabi--Yau Manifolds

22 pages plain TeX, Tables and references added

Abstract:
  We show that the moduli space of all Calabi-Yau manifolds that can be
realized as hypersurfaces described by a transverse polynomial in a four
dimensional weighted projective space, is connected. This is achieved by
exploiting techniques of toric geometry and the construction of Batyrev that
relate Calabi-Yau manifolds to reflexive polyhedra. Taken together with the
previously known fact that the moduli space of all CICY's is connected, and is
moreover connected to the moduli space of the present class of Calabi-Yau
manifolds (since the quintic threefold P_4[5] is both CICY and a hypersurface
in a weighted P_4, this strongly suggests that the moduli space of all simply
connected Calabi-Yau manifolds is connected. It is of interest that singular
Calabi-Yau manifolds corresponding to the points in which the moduli spaces
meet are often, for the present class, more singular than the conifolds that
connect the moduli spaces of CICY's.

\section{intro}{Introduction}
 %
It has been known for some time \cite{{\rReid,\rCDLS,\rRolling,\rAGM}} that the
moduli spaces of
some Calabi--Yau manifolds\ meet along boundary components that correspond to certain singular
manifolds.  Differently put: certain singular Calabi--Yau manifolds\ can be approached as limits
of deformation classes of topologically distinct manifolds.  In
\cite{\rReid} M. Reid
made a bold conjecture that the parameter space of threefolds with vanishing
first Chern class is connected.  In ~\cite{{\rRolling,\rBeast}} it was shown
that the parameter space
of all CICY's is connected.  CICY's are complete intersection Calabi-Yau
manifolds~\cite{{\rCDLS,\rBeast,\rTris,\rPGTH}}: Calabi--Yau manifolds\ that can be realized as
complete intersections of polynomials
defined on products of projective spaces.  The class of CICY's comprises
several
thousand~\cite{\rHeC} topologically distinct manifolds corresponding to
some 250 pairs of values for the Hodge numbers ($h_{11},h_{21}$) and with Euler
numbers in the range $-200\leq\chi\leq 0$.  At the time of \cite{\rRolling}
this was the largest class of Calabi--Yau manifolds\ that could be systematically constructed.
Since then
another, perhaps larger, class has been studied. These are manifolds that
can be realized by a transverse polynomial (a polynomial $p$ such that $p$ and
$dp$ do not
simultaneously vanish) in \Footnote{Strictly speaking, the
Calabi--Yau manifolds\ are embedded in blow ups of $\relax{\rm I\kern-.18em P}_4^{\bf{k}}$} $\relax{\rm I\kern-.18em P}_4^{\bf{k}}$ , a
weighted $\relax{\rm I\kern-.18em P}_4$, with weights
${\bf k}=(k_1,k_2,k_3,k_4,k_5)$~\cite{{\rCLS,\rKS,\rMaxSkI}}.
The authors of
\cite{{\rKS,\rMaxSkI}} constructed a list ${\cal L}$ of 7555 weight vectors
$\bf{k}$ corresponding to these manifolds.  These weight vectors do not all
lead
to distinct CY manifolds though there are roughly 2,500 distinct pairs of Hodge
numbers. The Euler numbers of this class of manifolds lie in the range
$-960\leq\chi\leq 960$.  To our knowledge no
Calabi--Yau manifolds\ are known with Euler numbers outside this range. In virtue of this, we
considered it of interest to ask whether the moduli
spaces of all the manifolds of the list $\cal L$
are connected together. The result that we report in this paper is that they
are.

Reid conjectured that the moduli spaces of all Calabi--Yau manifolds\ are connected via
conifolds \cite{\rRolling}, though it would be in accord with the conjecture
for the
connection to be through non-K\"ahler manifolds with vanishing first
Chern class. We have found that, at least within our limited class,
it is not neccessary to leave the family of K\"ahler Calabi--Yau manifolds\ in order to
show that the moduli spaces form a connected web. Though the singular manifolds
that connect the moduli spaces are in many cases more singular than conifolds.
Since the list contains the quintic threefold, $\relax{\rm I\kern-.18em P}_4[5]$, which is also a
CICY, it follows that these moduli spaces are connected also to the web of
CICY's.

For the case of CICY's it was possible to show that the moduli spaces
of the manifolds were connected by means of an analytic argument.
Each CICY is specified by a degree matrix and the authors of \cite{\rPGTH} have
shown that a CICY exists for each such matrix.
A conifold transition corresponds to a certain operation on the degree
matrix and it is straightforward to see that one can transform any two
given degree matrices into one another by a sequence of such moves.
The analogous argument for weighted CICY's fails however, at least in
its naive form, since there are degree matrices for which the
corresponding weighted CICY has terminal singularities (which can not
be resolved to give a manifold with $c_1 = 0$).
Although there is no general proof that the moduli spaces are connected
for weighted CICY's it is clear \cite{\rLS} that many weighted CICY's
are connected to the web of CICY's and weighted~$\relax{\rm I\kern-.18em P}_4$'s.

Our investigation relies heavily on the toric construction of Calabi--Yau manifolds\ due to
Batyrev~\cite{{\rBB,\rB}}. For a manifold \ca{M}\ of a degree $d$, (defined as
the vanishing locus of a
polynomial $p$ of a degree $d$ in a weighted $\relax{\rm I\kern-.18em P}_4$) denote by $(x_1, \ldots,
x_5)$ the
homogeneous coordinates of the projective space and by ${\bf x}^{\bf m}$ the
monomial
$x_1^{m_1}\ldots x_5^{m_5}$. The set of all possible exponents ${\bf m}$
corresponding to a degree $d = k_1 + \ldots + k_5$ polynomial forms the Newton
polyhedron of $\ca{M}$. Batyrev observed that in many cases the Newton
polyhedron
of a manifold of the list $\ca{L}$ has a certain property termed reflexivity.
In ~\cite{\rCDK}
it was checked that in fact  all the manifolds of the list have this property.
Now a
converse obtains: if a polyhedron $\D$ is reflexive then a Calabi--Yau manifold\ $\ca{M}_\D$
may be constructed from $\D$. It may happen that a given reflexive polyhedron
contains a subsets of points that themselves form a reflexive polyhedron, $\delta}\def\D{\Delta$.
When this happens the moduli spaces of $\ca{M}_\D$ and
$\ca{M}_\delta}\def\D{\Delta$ intersect \cite{\rBKK}.
This is so because the polynomial of $\ca{M}_\delta}\def\D{\Delta$
contains a subset of the monomials of that for $\ca{M}_\D$ and so can be
deformed continuously by letting some of the coefficients in the polynomial for
$\ca{M}_\D$ tend to zero. It follows also from a recent theorem of
Hayakawa~\cite{\rHaya} that the distance from the generic smooth manifold
$\ca{M}_\D$ or
$\ca{M}_\delta}\def\D{\Delta$, to the singular manifolds $\ca{M}^\sharp$ (that correspond to the
intersection of the two moduli spaces) is~finite.

Recently, a possible explanation of a manner in which string theory unifies the
moduli spaces of many (possibly all) Calabi--Yau vacua was
discovered~\cite{{\rStro,\rBMS}}. It is argued that black hole condensation can
occur at conifold singularities in
the moduli space of type IIB  Calabi--Yau string vacua, and in some
cases this condensate
signals a smooth transition to a new, topologically different Calabi--Yau
vacuum. The extent to which it is possible to give a similar physical
interpretation to the more complicated
singularities that arise here is an interesting open question.

The paper is organized as follows. In \SS2 we try to motivate
intuitively  how we can deduce the connectivity of moduli
spaces from the nesting of polyhedra. In \SS3 we provide some of
the mathematical details to support this idea. After recalling some
facts about how reflexive polyhedra are constructed, we establish that
if one polyehedron contains another, then the moduli spaces of the
corresponding families of Calabi--Yau manifolds\ are connected. We give also a concrete
example of this procedure. In \SS4 we provide some of the
details about the algorithm used to show the connectedness of the
moduli spaces associated with $\cal L$. Finally, in the appendix we
provide a list of polyhedra that realize the
connectedness as described in this article.

We are grateful to D.~Morrison for initially suggesting this problem to us.

Just prior to the submission of this article, we were made aware of similar
results reported in ~\cite{\rCGGK}.

\vfill\eject

\section{p}{Nesting of Reflexive Polyhedra}
Newton polyhedra are important since they will allow us to describe how the
moduli spaces meet. Suppose that a Newton polyhedron $\D_1$ from $\cal L$
contains a reflexive polyhedron $\D_2$ as a subpolyhedron.
Then the rough idea
is that $\D_2$ is obtained from $\D_1$ by setting to zero certain coefficients
in the polynomial $p_1$ of $\D_1$. This is clearly a continuous operation. In
this way we see that the moduli space of the Calabi--Yau manifolds\ corresponding to $\D_1$
intersects the moduli space of Calabi--Yau manifolds\ corresponding to $\D_2$. If there is a
third polyhedron $\D_3$ that also contains $\D_2$
$$ \Delta_1 \supset \Delta_2 \subset \Delta_3
\eqlabel{inclusion}.$$
then the moduli space of the manifolds corresponding to $\Delta_3$ also
intersects the moduli space
of those corresponding to $\D_2$ and the three moduli spaces are connected.

The moduli spaces corresponding to all $\D$'s that contain a common
subpolyhedron are connected.
If we denote by $F_{\delta}$ the ``family'' of moduli spaces corresponding to
polyhedra that contain a given
subpolyheron $\delta$, and if any polyhedron of a
family $F_{\delta_1}$ has a subpolyhedron in common with any
polyhedron of a family $F_{\delta_2}$, then the two families
$F_{\delta_1}$ and $F_{\delta_2}$ are connected.

There are however some aspects of this process that need to be explained. The
apparent difficulty is that $\D_1$ corresponds to a family of hypersurfaces in
a weighted projective space, with weight vector ${\bf k}_1$ say, and $\D_2$ to
a hypersurface in a
 weighted projective space with a different weight ${\bf k}_2$, say. This
difficulty
is only apparent; the essential point is that, given a reflexive polyhedron
$\D$, a deformation class of Calabi--Yau manifolds\ may be constructed from $\D$, such that the
generic manifold in the class is smooth. It is important that the reflexivity
of $\D$ is sufficient
for this to be true. $\D$ does not have to correspond to a member of $\ca{L}$.
This is
fortunate, since some of the polyhedra that we use to prove the connectivity of
the moduli space, do not belong to $\ca{L}$. The resolution of the apparent
difficulty concerning the different weight vectors, is in essence just a point
made above about the
 smoothness of the generic Calabi--Yau manifold\ corresponding to a reflexive polyhedron. By
setting to
 zero the coefficients that take us from $\D_1$ to $\D_2$, we obtain a singular
manifold
$\ca{M}^\sharp_1$, in $\relax{\rm I\kern-.18em P}^{{\bf k}_1}_4$. This same singular space can be
realized as the limit
of a hypersurface in $\relax{\rm I\kern-.18em P}^{{\bf k}_2}_4$. It is important to note that a family
of smooth Calabi--Yau manifolds\
are in one--to--one correspondence with a dual\Footnote{The dual of a convex
polyhedron is defined in \SS3.1} pair of reflexive polyhedra, $(\D_1,
\nabla_1)$, say. There is, similarly, a family for $(\D_2, \nabla_2)$. Now,
$\D_1 \supset \D_2$ and it is important that  duality reverses the inclusion
for the dual polyhedra so that $\nabla_1 \subset \nabla_2$. The two families
meet in manifolds $\ca{M}^\sharp$, which may be considered to correspond to the
non--dual pair $(\D_2, \nabla_1)$.

We express this as a diagram:
$$
\matrix{\ca{M}_1:&~~~~~~\, (~\D_1,\nabla_1~)\cr
        ~~~~~~~~~&\downarrow\cr
        \ca{M}^\sharp:&~~~~~~\,(~\D_2,\nabla_1~)\cr
         ~~~~~~~~~&~~~~~~~~~~\, \uparrow\cr
         \ca{M}_2:&~~~~~~\,(~\D_2,\nabla_2~)\cr}~\eqlabel{*}$$
where the arrows denote specializations of either the polynomial corresponding
to $\ca{M}_1$ or the mirror polynomial corresponding to $\ca{M}_2$.

This diagram allows a nice interpretation. We may think of the singularization
$(\D_1, \nabla_1) \longrightarrow (\D_2, \nabla_1)$ as being due to the
specialization of the polynomial which forces the hypersurface to be singular.
Alternatively,
we can singularize via $(\D_2, \nabla_2) \longrightarrow (\D_2, \nabla_1)$.
We will see below that the process $\nabla_2 \longrightarrow \nabla_1$ can be
thought of
as a singularization of the embedding space. Speaking loosely, we can say that
we achieve the same effect by singularizing the hypersurface while leaving
alone the embedding space of one manifold, or by leaving the hypersurface alone
and
singularizing the embedding space of the other manifold.

The next section, which is rather technical, shows that the operations
that correspond to reversing the arrows in \eqref{*} render the
manifolds $\ca{M}_1$ and $\ca{M}_2$ smooth.
\vfill\eject
 %
\section{th}{Calabi-Yau Manifolds in Projective Varieties}
\vskip-40pt
\subsection{Generalities}
We consider Calabi--Yau manifolds\ as hypersurfaces in toric varieties that are toric
deformations  of weighted projective spaces. These varieties, as well
as the relations between them, will be described in terms of toric geometry.
To this end we briefly review the toric construction of projective
varieties~\cite{{\rAGM,\rBB,\rB,\rCDK}}.

We start with a weighted projective space $\relax{\rm I\kern-.18em P}_4^{\bf k}$,  and consider the
family of homogeneous polynomials $p = p(x_1, \ldots , x_5)$ of degree
$d = \sum_{i=1}^{5} k_{i}$. As in \SS1, we associate a vector of exponents to
each
monomial and write $\bf x^{\bf m}$ for
$x_1^{m_1}x_2^{m_2}x_3^{m_3}x_4^{m_4}x_5^{m_5}$.
Thus, a general homogeneous polynomial of degree $d$ has the form:
$$p = \sum_{{{\bf k} \cdot {\bf m} = d,\atop m_i\ge0}}c_{\bf m}{\bf x}^{\bf
m}~.$$
Each degree vector $\bf m $ can be regarded as a point in $\relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi^5 \otimes{\relax{\rm I\kern-.18em R}}$,
and the convex hull of these points forms the Newton polyhedron $
\Delta({\bf k})$ of $p$, though to avoid encumbering
the notation, we will largely suppress the dependence on {\bf k} in the
following.
Because of the relation between $d$ and $\bf k$
the only integral point inside $\Delta$ is ${\bf 1}\buildrel\rm def\over = (1,1,1,1,1)$.

The Newton polyhedron lives in the four dimensional sublattice of $\relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi^5$
defined by:
$$\Lambda = \lbrace {\bf m} \in \relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi^5 \mid {\bf k} \cdot {\bf m} = d \rbrace$$
or, after translating the origin to the interior point {\bf 1} ~and setting
${\bf m}' = {\bf m} - {\bf 1}$, by:
$$\Lambda = \lbrace {\bf m}' \in \relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi^5 \mid {\bf k} \cdot {\bf m}' = 0
 \rbrace~.$$
We denote by V the dual lattice to $\Lambda$.
The corresponding vector spaces in which these lattices are embedded
are $\Lambda_{\relax{\rm I\kern-.18em R}} = \Lambda \otimes \relax{\rm I\kern-.18em R}$ and
$\hbox{V}_{\relax{\rm I\kern-.18em R}} = \hbox{V} \otimes
\relax{\rm I\kern-.18em R}$.
Inside these vector spaces the Newton polyhedron and its dual are defined as:
$$\eqalign{\Delta({\bf k}) &= \hbox{ convex hull of} \quad \lbrace {\bf m}' \in
\Lambda({\bf k}) \mid  m'_i \geq -1,  i=1, \ldots ,5 \rbrace \cr
\nabla({\bf k}) &= \lbrace {\bf x} \mid \langle
  {\bf x}, {\bf y} \rangle \geq -1, \forall {\bf y} \in \Delta \rbrace\cr}$$
All the Newton polyhedra constructed this way from $\cal L$, have been checked
 to be reflexive \cite{\rCDK}.
 Reflexivity means (in geometrical terms) that:
 \item{ 1.}$\Delta$ has integer vertices
 \item{ 2.}There is one and only one interior point in  $ \Delta$
 \item{ 3.}The equation of any face of codimension 1, which we write
as $c_1 y_1 + \cdots + c_4 y_4 = 1$ , has coefficients $c_1, \ldots ,
c_4$ that are integers with no common factor.

\noindent The reflexivity condition is important since a polyhedron is
reflexive
if and only if it is the support of global sections of the anticanonical
sheaf on
a Gorenstein Fano variety $\ca{V}_{(\Sigma, \nabla)}$~\cite{\rB}. Any such
section is a linear combination of monomials that correspond to integral
points in
$\Delta$. This variety is a blow up of the $\relax{\rm I\kern-.18em P}_4^{\bf k}$ that
we started with.
A hypersurface in such a variety, the zero locus of homogeneous
polynomials of fixed degree, admits a Calabi--Yau\ resolution.
The fan of the embedding variety $\Sigma$ is the fan over a triangulation
of the
faces of the dual polyhedron $\nabla$.

There are two ways of constructing refinements of the fan
$\, \Sigma$. One is to take all rays supported by integral points
$x \in \nabla \cap \hbox{V}$
in the dual polyhedron and, given a triangulation, to construct the
respective fan (we may still be left with cones of volume \Footnote{If all
maximal cones of the fan have volume equal to 1, then the variety is smooth.}
greater than~1, but we are guaranteed that the Calabi--Yau\ hypersurfaces are going to
be
smooth). In this case we have a crepant \Footnote{A map
is crepant if it preserves the canonical class: in both varieties, Calabi--Yau\
hypersurfaces can be defined in terms of the same set of functions.}
morphism of toric varieties, $\phi :
\ca{V}_{(\Sigma ', \nabla )} \longrightarrow \ca{V}_{(\Sigma,
\nabla )}$. When we refine a fan we say that we blow up the associated
variety. The inverse procedure will be called a blow down. The variety
$\ca{V}_{(\Sigma ', \nabla )}$ is obtained from $\ca{V}_{(\Sigma, \nabla )}$ by
toric resolutions that do not affect the canonical class: the family of
functions
that may be used to define a
Calabi--Yau\ hypersurface is not changed in the process. The other way of refining the
fan
is to choose
additional rays that do not correspond to integral points in $
\nabla $. These rays intersect the facets of $
\nabla $ in non-integral points. In some cases the
primitive integer vectors along these rays may define a new polyhedron
$\hat{\nabla}$ which is also reflexive. Even
though the canonical class of the variety has been affected in the
process (there is a new set of functions that we can use in defining Calabi--Yau\
hypersurfaces) the reflexivity of the polyhedron $\hat{\nabla}$ associated with
the
new variety guarantees the existence of Calabi--Yau\ hypersurfaces. We claim that
whenever a variety $ \ca{V}_{(\Sigma,
\nabla )}$ admits such a
noncrepant blow up there is an isomorphism between the induced deformation
of  a subset of the Calabi--Yau\
family originally embedded in $ \ca{V}_{(\Sigma,\nabla )}$ and
the Calabi-Yau family of $\ca{V}_{(\hat{\Sigma} , \hat{\nabla} )}$.

\subsection{Connectivity of Moduli Spaces}
As mentioned in the introduction we aim to show the connectedness of the moduli
spaces for the 7555 families of Calabi--Yau manifolds\ that can be obtained as
transverse hypersurfaces in projective varieties~\cite{{\rKS,\rMaxSkI}}. The
constructive method outlined above tells us that each polyhedron $
\Delta $ lives in the hyperplane defined by the weight vector
$\bf k$. The sublattice $\Lambda $ generated by this hyperplane in
$\relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi^5$ has relative volume $\sqrt{\sum_{i=1}^5 k_i^2}$. Once we find an
appropriate basis for $\Lambda $ we express the coordinates of
all integral points with respect to it. Note that there are an infinite
number of ways to choose a basis for any lattice of dimension greater
then 1 and we will identify polyhedra $ \Delta(\bf k) $ and $
\Delta(\bf k')$ if there is a $GL(4,\relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi)$ bijection between
them. Because the group $GL(4,\relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi)$ is volume preserving we do not change the
structure of the fans supported by $ \Delta$. Each maximal cone
will preserve its volume as well as the number of integral points it
contains. This shows that the two varieties are indeed isomorphic, $
\ca{V}_{(\Sigma,\nabla({\bf k}))} \cong  \ca{V}_{(\Sigma ',\nabla({\bf k
'}))}$, and this is also
true for the respective families of Calabi--Yau\ hypersurfaces.
When we say that  $ \Delta_2 \subset  \Delta_1$,
we mean that there is a transformation that matches all the
points in  $ \Delta_2$ to a subset of points in $  \Delta_1$.
For example, if the point $a$ in $ \Delta_1$ corresponds to the
monomial ${\bf x}^{\bf m}$ and the point $b$ in $ \Delta_2$
corresponds to the monomial ${\bf y}^{\bf n}$, then identifying $a$
and $b$ implies ${\bf x}^{\bf m} = {\bf y}^{\bf n}$.

Let us now examine more closely the consequences of the inclusion
$\Delta_2 \subset \Delta_1$.\hfil\break
We have:
$$
\matrix{ \ca{V}_{(\Sigma_2,\nabla_2)}      & ~~~~
&\ca{V}_{(\Sigma_1,\nabla_1)}\cropen{3pt}
        \updownarrow   & ~~~~   &\updownarrow\cropen{3pt}
        \kern4pt \Delta_2   &\subset &\kern5pt \Delta_1\cropen{3pt}
        \updownarrow     & ~~~~   &\updownarrow\cropen{3pt}
        \kern4pt \nabla_2 &\supset &\kern5pt \nabla_1\cr}
$$
where $\nabla_2$ is the support of
a noncrepant toric deformation of $ \ca{V}_{(\Sigma_1,\nabla_1)}$ ($\Sigma_2$
is a refinement of $\Sigma_1$), such
that
$$ \ca{V}_{(\Sigma_2,\nabla_2)} \buildrel \phi \over
\longrightarrow  \ca{V}_{(\Sigma_1,\nabla_1)}$$
is a proper birational morphism of toric varieties.

Consider ${\cal M}_2
\subset \ca{V}_{(\Sigma_2,\nabla_2)}$ to be a generic
hypersurface. The points in $\nabla_2 \cap
\hbox{V}_2$ correspond to a subset of monomials
in $\nabla_1 \cap \hbox{V}_1$ that define
a certain hypersurface ${\cal M}_1^\sharp \subset
\ca{V}_{(\Sigma_1,\nabla_1)}$.
The crucial observation is
that the pullback of ${\cal M}_1^\sharp$ to $\ca{V}_{(\Sigma_2,\nabla_2)}$
under $\phi$ is isomorphic to ${\cal M}_2 \,$\cite{{\rCDK}}.
Otherwise said, if $\ca{V}_{(\Sigma_1,\nabla_1)}$ is
blown up to $\ca{V}_{(\Sigma_2,\nabla_2)}$, then the induced
deformation of ${\cal M}_1^\sharp$ is isomorphic to ${\cal M}_2$.

$$
\matrix{
 {\cal M}_2&\leftarrow{\rm birational~ map} \rightarrow &{\cal M}^\sharp_1\cr
\cap& ~~~~                                                   &\cap\cr
\ca{V}_{(\Sigma_2,\nabla_2)}& \rightarrow  {\rm
non~crepant ~blow up} \rightarrow&
\ca{V}_{(\Sigma_1,\nabla_1)} \cr}
$$

What do the inclusions  \eqref{inclusion}, $\Delta_1 \supset \Delta_2 \subset
\Delta_3$, tell us?
There is a special sub-family of hypersurfaces  ${\cal M}_1^\sharp \subset
\ca{V}_{(\Sigma_1,\nabla_1)}$ that can be ``deformed'' into the full family of
hypersurfaces  ${\cal M}_2 \subset \ca{V}_{(\Sigma_2,\nabla_2)}$, that
in turn can be ``deformed'' into a special sub-family  ${\cal M}_3^\sharp
\subset \ca{V}_{(\Sigma_3,\nabla_3)}$.
We remark that the resolution of either  ${\cal M}_1^\sharp$ or
${\cal M}_3^\sharp$
allows us to control all the polynomial deformations of~${\cal M}_2$.
\subsection{Illustration of the Method}
We want to show how the above analysis applies to a pair of varieties.
Consider the manifold
$\relax{\rm I\kern-.18em P}^{(24,51,133,416,624)}[1248]^{h_{11}=214}_{h_{21}=10}$.
There are 18 points in the
polyhedron and 12 of these are shared with the polyhedron associated
with $\relax{\rm I\kern-.18em P}^{(54,56,151,522,783)}[1566]^{h_{11}=251}_{h_{21}=5}$.
We are going to look at the
vertices of the interior polyhedron and see how they relate to points of
the exterior polyhedron.
A basis for the sublattice defined by ${\bf k}_1 = (24,51,133,416,624)$
is:
$$\eqalign{
{\bf v}_1 &= (-26,\-0,\-0,\-0,\-1)\cr
{\bf v}_2 &= (-25,\-1,\-1,\-1,\-0) \cr
{\bf v}_3 &= (-52,\-0,\-0,\-3,\-0) \cr
{\bf v}_4 &= (-17,\-8,\-0,\-0,\-0)}$$
The points that correspond to the vertices of the inside polyhedron are
$$\eqalign{
{\bf  a}_1 &= (\-1,-1,\-0,\-0) = (\-0,\-0,\-0,\-0,\-2) \cr
{\bf a}_2 &= (-1,-1,\-1,\-0) = (\-0,\-0,\-0,\-3,\-0) \cr
{\bf a}_3 &= (-1,\-8,-3,-1) = (\-0,\-1,\-9,\-0,\-0) \cr
{\bf a}_4 &= (-1,\-2,-1,\-0) = (\kern3pt 29,\-3,\-3,\-0,\-0) \cr
{\bf a}_5 &= (-1,-1,\-0,\-3) = (\-1,\kern3pt 24,\-0,\-0,\-0)}$$
The vectors on the extreme right with 5 components are the $\bf m$
vectors from which we can read the monomials to which they correspond.
Turning now to the second manifold, we find that
a basis for the sublattice defined by ${\bf k}_2 = (54,56,151,522,783)$
is:
$$\eqalign{{\bf w}_1 &= (\kern3pt 29,\-0,\-0,\-0,-2) \cr {\bf w}_2 &=
(\-1,\-1,\-1,\-1,-1) \cr
{\bf w}_3 &=
(\-0,\-0,\-0,\-3,-2) \cr {\bf w}_4 &= (\kern3pt 10,\-9,\-0,\-1,-2)}$$ The
vertices of the interior
polyhedron are:
$$\eqalign{{\bf b}_1 &= (\-0,-1,\-0,\-0) = (\-0,\-0,\-0,\-0,\-2) \cr
{\bf b}_2 &= (\-0,-1,\-1,\-0) = (\-0,\-0,\-0,\-3,\-0) \cr
{\bf b}_3 &= (\-0,\-9,-3,-1) = (\-0,\-1,\kern3pt 10,\-0,\-0) \cr
{\bf b}_4 &= (-1,-1,-1,\-3) = (\-1,\kern3pt 27,\-0,\-0,\-0) \cr
{\bf b}_5 &= (\-1,-1,\-0,\-0) = (\kern3pt 29,\-0,\-0,\-0,\-0)}$$
\vskip10pt
Define $ {\bf A}_i^j = ({\bf a}_i)^j$ and ${\bf B}_i^j = ({\bf b}_i)^j$. Then
$\bf A =\bf B \bf T$ where
$${\bf T} \, = \, \left(\matrix{-2&0&0&3\cr
                  -1&1&0&0\cr
                  -2&0&1&0\cr
                  -2&1&0&1\cr}\right)$$
If we take the space $\relax{\rm I\kern-.18em P}^{(24,51,133,416,624)}[1248]^{h_{11}=214}_{h_{21}=10}$
to have
homogeneous coordinates
${x_i}$ and the space $\relax{\rm I\kern-.18em P}^{(54,56,151,522,783)}[1566]^{h_{11}=251}_{h_{21}=5}$
to have
homogeneous coordinates ${y_i}$ then the correspondence ${\bf a}_i
\leftrightarrow {\bf b}_i$ gives the following identification of
monomials:
$$\eqalign{x_5^2 &= y_5^2\cr
           x_4^2 &= y_4^2\cr
           x_{2}x_3^9 &= y_{2} \, y_3^{10} \cr
           x_{1}^{29} \, x_{2}^{3} \, x_{3}^{3} &= y_{1} \, y_{2}^{27}\cr
           x_{1} \, x_2^{24} &= y_1^{29} \cr}$$
which gives in turn the birational map between the two varieties:
$$\eqalign{y_1 &= x_1^{1/29} \, x_2^{24/29}\cr
           y_2 &= x_1^{840/783} \, x_2^{63/783}x_3^{1/9}\cr
           y_3 &= x_1^{-84/783} \, x_2^{72/783}x_3^{8/9}\cr
           y_4 &= x_4 \cr
           y_5 &= x_5}$$
The map is one to one despite appearances. This must be the case in
virtue of our analysis in \mathhexbox278 3.2. and may be explicitly checked
by using the scaling properties of the two sets of coordinates.
To conclude, we have a sub-family of hypersurfaces in the variety
described by the exterior polyhedron given by the zero locus of the
polynomial
$$\eqalign{p^{\sharp} =& ~x_1x_2x_3x_4x_5 + x_5^5 + x_4^3 + x_2x_3^9 +
x_1^2x_2^2x_3^2x_4^2 + x_1^{29}x_2^3x_3^3\cr & + x_1^3x_2^3x_3^3x_5 +
x_1^4x_2^4x_3^4x_4 + x_1^6x_2^6x_3^6 + x_1^{10}x_2^9x_3x_4 +
x_1^{12}x_2^{11}x_3^3 + x_1x_2^{24}}$$
that can be blown up to a generic hypersurface in the variety
associated with the interior polyhedron given by the general
polynomial
$$\eqalign{p^{\flat} =& ~y_1y_2y_3y_4y_5 + y_5^2 + y_4^3 + y_2y_3^{10} +
y_1^2y_2^2y_3^2y_4^2 + y_1y_2^{27}\cr & + y_1^3y_2^3y_3^3y_5 +
y_1^4y_2^4y_3^4y_4 + y_1^6y_2^6y_3^6 + y_1^{10}y_2^9y_4 +
y_1^{12}y_2^{11}y_3^2 + y_1^{29}}.$$
\vfill\eject
 %
\section{comp}{The Computation}
 %
In principle, the task of determining all of the reflexive sub-polyhedra
(RSP's) of a given reflexive
polyhedron (RP) is straightforward. Suppose the RP in question has $N$ points.
One can then first look
for all RSP's with $N-1$ points. There are $N-1$ candidates (recall that the
interior point of the RP must
also be the interior point of the RSP).
For each candidate, one would have to determine if it is reflexive. This
requires
identifying the faces, and determining their equations. In general, there are
${N-1}\choose {M-1}$
candidates for RSPs with $M$ points. So, the total number of candidates one
would have to consider is:
$$\sum_{M=6}^{N-1} {{N-1}\choose {M-1}} \approx 2^{N-1}.$$ Since many RP's have
$N>200$, this is not feasible.
While this  method is complete, and would have to be followed in
order to establish that
a group of RP's are {\it not} directly connected, it is thankfully not
necessary if one is trying
to establish their connectedness. One might imagine that it might happen that
all of the
RP's under consideration contain a particular RSP, and connectedness would
follow immediately,
irrespective of any other RSP's that they might contain. This turns out not to
be the case, but is the spirit in which we have attacked the
problem.

There is no such RSP. To see this is simple. There do exist reflexive polyhedra
with 6 points. These are simplices in four dimensions (recall
that there is always one interior point), and they cannot contain any RSP's,
hence they themselves would
have to be the magical RSP's. However, there are three inequivalent RP's with 6
points. Although the simplest guess fails, nevertheless one might hope to
restrict one's search for RSP's to some
simple objects, such as simplices.

Our initial strategy was to identify the reflexive simplices within each of the
7555
RP's. In fact we should identify those reflexive simplices which
themselves contain no
reflexive simplices. We will refer to such objects as 5--vertex irreducible
simplices, that is they are simplices that contain no 5--vertex RSP's. In
general, any
reflexive polyhedron which does not contain any $n$-vertex reflexive polyhedra
(apart from itself) is an $n$-vertex irreducible polyhedron.
The combinatoric barrier is
now reduced from $2^{N-1}$ to ${N-1}\choose 5$, which is a good deal more
manageable.
In fact, it was a relatively quick matter to decompose all RP's with
$N\le 120$ into 5-vertex irreducible simplices. Very early in this procedure
(i.e. after decomposing a
small fraction of these RP's), a set of 41 5-vertex irreducible simplices was
generated,
ranging in size from $N=6$ to $N=26$ points. It turned
out that these were the only ones that were generated by this procedure, though
we have no proof that
others do not exist. Among them are 18 that are not on the list of 7555 that we
started
 from. Being simplices, these must correspond to Fermat polynomials, but since
they are not all in the list, they must in fact correspond to manifolds of the
form ${\lbrace p=0 \rbrace}/G$, where $p$ is Fermat and $G$ is a group of
automorphisms.

Establishing the connectedness of all of those RP's that contain one or more of
these simplices
is then a matter of showing that these 41 simplices are connected to each
other. Let us denote the set of 5-vertex irreducible simplices
contained in the $i$th RP as ${\bf V}^5_i$. We define ${\bf C}^5_1 ={\bf
V}^5_1$.
 In general:
$${\bf C}^5_i = {\bf C}^5_{i-1} \cup \delta_{i-1,i}~,\quad \hbox{where}
\quad \delta_{i,j} = \cases{\emptyset &if ${\bf C}_i^5\cap {\bf V}_j^5
=\emptyset$\cr
                            {\bf V}_j^5     &if ${\bf C}_i^5\cap {\bf
V}_j^5\ne\emptyset$\cr}
 $$
If for any $i$, ${\bf C}^5_i$ contains all 41 of the 5-vertex irreducible
simplices, then their
connectedness has been established. This procedure, which is a sufficient but
not necessary condition, can be described as follows.
For each polyhedra with $N\le 120$, we have a list of which of the 41 reflexive
simplices that it contains, ${\bf V}^5_i$. Clearly, all of the simplices that
are contained in ${\bf V}^5_1$ are connected, since they are all connected to
the first polyhedron. So this is our initial list of connected simplices.
Now, successively examine each of the ${\bf V}^5_i$. Whenever there is a
simplex in ${\bf V}^5_i$ that is also in our list of connected simplices, we
add {\it all} of the simplices in ${\bf V}^5_i$ to our list. As soon as this
list contains all 41 simplices, we have shown that they are connected. This is
indeed the case, hence all those
RP's with
$N\le 120$ and ${\bf V}^5_i \neq \emptyset$, which amounts to 6133 RP's, are
connected.

For those RP's with $N>120$, we take advantage of the apparent completeness of
the list of 5-vertex
irreducible simplices. So instead of finding {\it every} 5-vertex irreducible
simplex in these
larger objects, we stop our search as soon as we find {\it one}. This approach
succeeds, in that all
of these larger RP's  contain at least one of the 41  previously found 5-vertex
irreducible
simplices. This accounts for a further 1280 of the RP's.

We were left with 142 RP's that do not contain any reflexive simplices.
As they are relatively small ($N\le 28$), it was feasible to try and search for
RSPs that had
6 vertices, and were themselves 6-vertex irreducible. Applying this procedure,
we found that
132 of the remainder contained 6-vertex irreducible 6-vertex RSPs, and
 that 10 did not.

In order to see if these 132 RP's were connected to the 6133 RP's that were
already known
to be interconnected, it was necessary to make sure that all of the 6-vertex
irreducible 6-vertex
RSPs that were generated by the former list, were also generated by some subset
(hopefully
a small one) of the latter list. In practice, it was easy to establish this for
123 of
them. The remaining 9 are difficult cases.

We then attempted to decompose these 9, plus the 10
RP's that contained neither reflexive simplices, nor reflexive 6-vertex
polyhedra,
into 7-vertex irreducible 7-vertex RSPs. Seventeen of them yielded to this
procedure, and were
easily connected to the other 7536 RP's. This leaves us with two polyhedra.
They correspond
to $\relax{\rm I\kern-.18em P}^{(21,24,82,111,119)}_4[357]$ and $\relax{\rm I\kern-.18em P}^{(18,21,58,77,78)}_4[252]$. They
each contain the same
6-vertex irreducible 6-vertex RSP -- one which we could not easily locate in
another RP.
Let us call it $\Pi$.

In order to take care of these last two polyhedra, we make use of the
elementary observation made previously, that if $\D_2$ and $\D_1$ are reflexive
polyhedra, and  $\D_2\subset \D_1$, then $\nabla_1 \subset \nabla_2$, where
$\nabla_2$ and $\nabla_1$ are the dual polyhedra of $\D_2$ and $\D_1$.
First, we established that the dual of $\Pi$ was one of the RP's
from our original list. It corresponds to $\relax{\rm I\kern-.18em P}^{(1,1,4,6,6)}_4[18]$. Thus the
duals of whatever RSPs are contained within this RP, contain $\Pi$. In fact,
the RP corresponding
to $\relax{\rm I\kern-.18em P}^{(1,1,4,6,6)}_4[18]$ contains
a simplex whose dual polyhedron corresponds to $\relax{\rm I\kern-.18em P}^{(1,1,1,6,9)}_4[18]$. So,
we learn that
$\relax{\rm I\kern-.18em P}^{(21,24,82,111,119)}_4[357]$ and $\relax{\rm I\kern-.18em P}^{(18,21,58,77,78)}_4[252]$
each contain an RSP that is also contained in the RP corresponding to
$\relax{\rm I\kern-.18em P}^{(1,1,1,6,9)}_4[18]$.
Furthermore, the RP corresponding to $\relax{\rm I\kern-.18em P}^{(1,1,1,6,9)}_4[18]$ is one of the
RP's that had previously been
connected via 5-vertex irreducible simplices.

We have thus succeeded in establishing that the moduli spaces of all three
dimensional CICYs and all
transverse polynomials in four dimensional weighted projective spaces are
interconnected.
\vskip0.7in
\acknowledgements
We gratefully acknowledge fruitful conversations with S.~Katz, S.~Keel,
D.~Morrison and D.~Saltman. This work was supported by NSF Grants PHY95-11632
and PHY94-07194 and the Robert A.~Welch Foundation. P.~Candelas and
M.~Mandelberg would like to thank SBITP for
hospitality while part of this work was being done.
\vskip0.7in
\chapno=-1
\section{app}{Appendix: The $n$-Vertex Irreducible RP's that Connect the
Manifolds}
Here we give some information regarding those polyhedra which
we have used to connect the moduli spaces of all the manifolds of the list
\ca{L}.
For each polyhedron, the following information is given: the number of points
it
contains, its associated hodge numbers, and, in the case that it corresponds to
one
of the polyhedra corresponding to one or more of the spaces in
$\cal L$, the weights of those spaces, enclosed in square brackets. Instead, if
it
does not correspond to any entry on
$\cal L$, the vertices of the polyhedron are given, enclosed in parentheses.
There
are three tables, one for each $n$-vertex irreducible $n$-vertex RP's with
$n=5,6$
and $7$.
\vfill\eject
\centerline{{\bf Table 1:} 5-vertex irreducible 5-vertex reflexive polyhedra}
\vskip10pt
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\noalign{\hrule}
&
  &\hbox{\tenrm Pts}
    &&$h_{11}$\hfil
      &&$h_{21}$\hfil
        &&\hbox{\eightrm Vertices/Weights}&\cr
\noalign{\hrule\vskip3pt\hrule}
&   &6   &&103   &&1   && [52,60,63,75,125] &\cropen{-3pt}
&   &    &&      &&    && [48,50,60,63,79] &\cr
\noalign{\hrule}
&   &6   &&101   &&1   && [41,48,51,52,64] &\cropen{-3pt}
&   &    &&      &&    && [51,60,64,65,80] &\cr
\noalign{\hrule}
&   &6    &&21   &&1   && (0,1,-1,0) &\cropen{-3pt}
&   &     &&     &&    && (-2,1,1,-1) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,0,2) &\cropen{-3pt}
&   &     &&     &&    && (0,0,-1,0) &\cropen{-3pt}
&   &     &&     &&    && (3,-1,1,-1) &\cr
\noalign{\hrule}
&   &7    &&149  &&1   && [75,84,86,98,343] &\cropen{-3pt}
&   &     &&     &&    && [43,48,56,98,147] &\cropen{-3pt}
&   &     &&     &&    && [48,49,56,86,153] &\cropen{-3pt}
&   &     &&     &&    && [42,43,49,75,134] &\cr
\noalign{\hrule}
&   &7    &&145  &&1   && [73,80,90,162,405] &\cropen{-3pt}
&   &     &&     &&    && [64,72,73,115,324] &\cropen{-3pt}
&   &     &&     &&    && [40,45,73,81,166] &\cropen{-3pt}
&   &     &&     &&    && [40,45,63,64,148] &\cr
\noalign{\hrule}
&   &7    &&112  &&4   && [42,48,57,109,128] &\cr
\noalign{\hrule}
&   &7    &&86   &&2   && [27,36,49,56,84] &\cropen{-3pt}
&   &     &&     &&    && [27,36,41,52,60] &\cr
\noalign{\hrule}
&   &7    &&89   &&5   && [36,40,89,99,132] &\cr
\noalign{\hrule}
&   &7    &&38   &&2   && (1,-1,1,-1) &\cropen{-3pt}
&   &     &&     &&    && (-2,-1,1,1) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,-1,2) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (0,2,-1,0) &\cr
\noalign{\hrule}
&   &8    &&165  &&3   && [60,66,74,163,363] &\cropen{-3pt}
&   &     &&     &&    && [40,44,74,121,205] &\cropen{-3pt}
&   &     &&     &&    && [36,40,66,89,165] &\cr
\noalign{\hrule}
&   &8    &&128  &&2   && [25,30,66,88,121] &\cropen{-3pt}
&   &     &&     &&    && [25,30,54,82,109] &\cropen{-3pt}
&   &     &&     &&    && [30,41,50,54,125] &\cr
\noalign{\hrule}
&   &8    &&103  &&7   && [30,39,84,127,140] &\cr
\noalign{\hrule}
&   &8    &&101  &&5   && (-1,0,0,1) &\cr
&   &     &&     &&    && (-1,0,1,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,0,0) &\cropen{-3pt}
&   &     &&     &&    && (11,-1,-4,-2) &\cr
\noalign{\hrule}
&   &8    &&60   &&6   && (0,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (2,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-3,5,-1,-1) &\cr
\noalign{\hrule}}}
\hfill
\vbox{\offinterlineskip\halign{\strut#&\vrule#&
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\quad\sixrm#\quad\hfil&#\vrule\cr
\noalign{\hrule}
&
  &\hbox{\tenrm Pts}
    &&$h_{11}$\hfil
      &&$h_{21}$\hfil
        &&\hbox{\eightrm Vertices/Weights}&\cr
\noalign{\hrule\vskip3pt\hrule}
&   &8    &&63   &&3   && (0,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (-2,5,-1,-2) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (2,-1,0,1) &\cr
\noalign{\hrule}
&   &8    &&31   &&13  && (-3,2,0,1) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,3,-1) &\cropen{-3pt}
&   &     &&     &&    && (-1,1,-1,1) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (4,0,-2,-1) &\cr
\noalign{\hrule}
&   &9    &&148  &&4   && [24,28,83,90,135] &\cropen{-2pt}
&   &     &&     &&    && [24,28,77,78,129] &\cropen{-2pt}
&   &     &&     &&    && [24,42,52,71,147] &\cr
\noalign{\hrule}
&   &9    &&83   &&3   && (0,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (2,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (-1,7,-4,-1) &\cr
\noalign{\hrule}
&   &9    &&75   &&3   && (2,1,-1,-1) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,0,1) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,-1,2) &\cropen{-3pt}
&   &     &&     &&    && (1,-2,6,-1) &\cr
\noalign{\hrule}
&   &9    &&55   &&7   && (0,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,1,-1,1) &\cropen{-3pt}
&   &     &&     &&    && (-2,3,1,-2) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (2,-1,0,0) &\cr
\noalign{\hrule}
&   &9    &&43   &&3   && (3,0,0,-1) &\cropen{-3pt}
&   &     &&     &&    && (1,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (1,0,2,-1) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-5,5,-3,2) &\cr
\noalign{\hrule}
&   &9    &&29   &&5   && (-1,0,-1,2) &\cropen{-3pt}
&   &     &&     &&    && (-1,1,-1,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,-1,0) &\cropen{-3pt}
&   &     &&     &&    && (3,-1,1,-1) &\cropen{-3pt}
&   &     &&     &&    && (-1,1,3,-2) &\cr
\noalign{\hrule}
&   &10   &&272  &&2   && [91,96,102,578,867] &\cropen{-2.2pt}
&   &     &&     &&    && [64,68,91,355,578] &\cropen{-2.2pt}
&   &     &&     &&    && [48,51,91,289,388] &\cropen{-2.2pt}
&   &     &&     &&    && [36,51,64,187,274] &\cr
\noalign{\hrule}
&   &10   &&143  &&7   && [40,45,143,152,380] &\cr
\noalign{\hrule}
&   &10   &&105  &&3   && [33,36,40,89,198] &\cr
\noalign{\hrule}}}}
\vfill\eject
\centerline{{\bf Table 1(cont'd):} 5-vertex irreducible 5-vertex reflexive
polyhedra}
\vskip10pt
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&   &10   &&21   &&9   && (2,-1,0,0) &\cropen{-1.9pt}
&   &     &&     &&    && (-1,-1,-1,2) &\cropen{-1.9pt}
&   &     &&     &&    && (-1,-1,1,0) &\cropen{-1.9pt}
&   &     &&     &&    && (-1,-1,1,1) &\cropen{-1.9pt}
&   &     &&     &&    && (-1,5,-1,-3) &\cr
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&   &10   &&17   &&21  && (2,-1,-1,0) &\cropen{-1.9pt}
&   &     &&     &&    && (0,0,-1,0) &\cropen{-1.9pt}
&   &     &&     &&    && (-1,3,-1,0) &\cropen{-1.9pt}
&   &     &&     &&    && (-1,-1,-1,1) &\cropen{-1.9pt}
&   &     &&     &&    && (0,-1,4,-1) &\cr
\noalign{\hrule}
&   &11   &&164  &&8   && (1,-1,0,0) &\cropen{-1.9pt}
&   &     &&     &&    && (0,-1,1,0) &\cropen{-1.9pt}
&   &     &&     &&    && (0,-1,0,0) &\cropen{-1.9pt}
&   &     &&     &&    && (-1,5,-3,-1) &\cropen{-1.9pt}
&   &     &&     &&    && (-2,-1,0,4) &\cr
\noalign{\hrule}
&   &11   &&131  &&11  && [24,33,138,173,184] &\cr
\noalign{\hrule}
&   &11   &&69   &&9   && (-1,0,0,1) &\cropen{-1.9pt}
&   &     &&     &&    && (-1,1,0,0) &\cropen{-1.9pt}
&   &     &&     &&    && (-1,-2,4,0) &\cropen{-1.9pt}
&   &     &&     &&    && (-1,0,0,0) &\cropen{-1.9pt}
&   &     &&     &&    && (5,-2,-2,-1) &\cr
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&   &11   &&59   &&11  && (-1,0,1,0) &\cropen{-1.9pt}
&   &     &&     &&    && (-1,2,-1,0) &\cropen{-1.9pt}
&   &     &&     &&    && (0,-1,0,1) &\cropen{-1.9pt}
&   &     &&     &&    && (0,-1,0,0) &\cropen{-1.9pt}
&   &     &&     &&    && (6,-1,0,-1) &\cr
\noalign{\hrule}}}
\hfill
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&   &11   &&243  &&3   && [24,33,92,173,230] &\cropen{-3pt}
&   &     &&     &&    && [24,44,69,161,254] &\cropen{-3pt}
&   &     &&     &&    && [24,44,63,155,242] &\cr
\noalign{\hrule}
&   &12   &&251  &&5   && [54,56,151,522,783] &\cropen{-3pt}
&   &     &&     &&    && [30,56,87,290,407] &\cr
\noalign{\hrule}
&   &13   &&82   &&10  && (2,-1,1,-1) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,1,1) &\cropen{-3pt}
&   &     &&     &&    && (-1,1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (5,-1,-5,3) &\cr
\noalign{\hrule}
&   &13   &&35   &&19  && (1,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,3,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (-1,7,-4,-1) &\cr
\noalign{\hrule}
&   &14   &&271  &&7   && [48,51,181,560,840] &\cropen{-3pt}
&   &     &&     &&    && [36,51,140,403,630] &\cr
\noalign{\hrule}
&   &15   &&227  &&11  && [30,38,234,283,585] &\cropen{-3pt}
&   &     &&     &&    && [20,38,156,195,371] &\cr
\noalign{\hrule}
&   &15   &&103  &&7   && [18,20,57,85,180] &\cr
\noalign{\hrule}
&   &17   &&321  &&9   && [42,46,241,658,987] &\cropen{-3pt}
&   &     &&     &&    && [24,46,141,376,541] &\cr
\noalign{\hrule}
&   &21   &&131  &&11  && [15,16,93,116,240] &\cr
\noalign{\hrule}
&   &26   &&491  &&11  && [41,42,498,1162,1743] &\cropen{-3pt}
&   &     &&     &&    && [36,41,421,996,1494] &\cropen{-3pt}
&   &     &&     &&    && [28,41,332,761,1162] &\cropen{-3pt}
&   &     &&     &&    && [21,41,249,581,851] &\cr
\noalign{\hrule}}}}
\vfill\eject
\centerline{{\bf Table 2:} 6-vertex irreducible 6-vertex reflexive polyhedra}
\vskip10pt
\line{
\vbox{\offinterlineskip\halign{\strut#&\vrule#&
\hfil\quad\sixrm#\quad&#\vrule&
\hfil\quad\sixrm#\quad&#\vrule&
\hfil\quad\sixrm#\quad&#\vrule&
\quad\sixrm#\quad\hfil&#\vrule\cr
\noalign{\hrule}
&
  &\hbox{\tenrm Pts}
    &&$h_{11}$\hfil
      &&$h_{21}$\hfil
        &&\hbox{\eightrm Vertices/Weights}&\cr
\noalign{\hrule\vskip3pt\hrule}
&   &7    &&95   &&2   && [32,42,45,91,105] &\cropen{-3pt}
&   &     &&     &&    && [26,33,48,75,91] &\cropen{-3pt}
&   &     &&     &&    && [26,33,45,60,64] &\cropen{-3pt}
&   &     &&     &&    && [23,28,34,53,55] &\cr
\noalign{\hrule}
&   &7    &&86   &&2   && [27,29,64,72,96] &\cropen{-3pt}
&   &     &&     &&    && [22,29,49,50,75] &\cropen{-3pt}
&   &     &&     &&    && [29,36,75,84,112] &\cropen{-3pt}
&   &     &&     &&    && [27,28,59,63,75] &\cropen{-3pt}
&   &     &&     &&    && [21,29,53,56,65] &\cropen{-3pt}
&   &     &&     &&    && [24,29,50,56,65] &\cr
\noalign{\hrule}
&   &8    &&132  &&2   && [44,48,63,131,286] &\cropen{-3pt}
&   &     &&     &&    && [28,39,48,91,158] &\cr
\noalign{\hrule}
&   &8    &&105  &&3   && (0,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (5,1,-1,-2) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (2,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,1) &\cr
\noalign{\hrule}
&   &8    &&87   &&3   && [22,30,63,95,105] &\cr
\noalign{\hrule}
&   &8    &&43   &&7   && (-1,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (1,-1,-1,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (0,1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (1,4,-1,-1) &\cr
\noalign{\hrule}
&   &8    &&59   &&3   && (0,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (1,-1,-1,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (0,1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (1,4,-1,-1) &\cr
\noalign{\hrule}
&   &10   &&152  &&6   && [34,36,122,131,323] &\cropen{-3pt}
&   &     &&     &&    && [20,34,64,69,153] &\cr
\noalign{\hrule}
&   &10   &&131  &&3   && [33,40,122,130,325] &\cropen{-3pt}
&   &     &&     &&    && [20,33,61,65,146] &\cr
\noalign{\hrule}
&   &10   &&139  &&5   && (-1,1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (5,0,-1,-1) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,1,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,0,0) &\cropen{-3pt}
&   &     &&     &&    && (1,0,-1,1) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,-1,3) &\cr
\noalign{\hrule}
&   &10   &&111  &&9   && (-1,1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,1,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,0,1) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,0,0) &\cropen{-3pt}
&   &     &&     &&    && (3,-1,-1,-1) &\cropen{-3pt}
&   &     &&     &&    && (17,-6,-6,-4) &\cr
\noalign{\hrule}}}
\hfill
\vbox{\offinterlineskip\halign{\strut#&\vrule#&
\hfil\quad\sixrm#\quad&#\vrule&
\hfil\quad\sixrm#\quad&#\vrule&
\hfil\quad\sixrm#\quad&#\vrule&
\quad\sixrm#\quad\hfil&#\vrule\cr
\noalign{\hrule}
&
  &\hbox{\tenrm Pts}
    &&$h_{11}$\hfil
      &&$h_{21}$\hfil
        &&\hbox{\eightrm Vertices/Weights}&\cr
\noalign{\hrule\vskip3pt\hrule}
&   &11   &&114  &&6   && (-1,0,1,0) &\cropen{-2pt}
&   &     &&     &&    && (2,-1,1,-1) &\cropen{-2pt}
&   &     &&     &&    && (-1,1,0,0) &\cropen{-2pt}
&   &     &&     &&    && (0,-1,0,0) &\cropen{-2pt}
&   &     &&     &&    && (2,-1,-2,2) &\cropen{-2pt}
&   &     &&     &&    && (3,-1,-3,2) &\cr
\noalign{\hrule}
&   &11   &&101  &&5   && [17,24,99,124,132] &\cropen{-2pt}
&   &     &&     &&    && [17,18,70,93,99] &\cr
\noalign{\hrule}
&   &11   &&70   &&4   && (0,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (1,0,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,2,0) &\cropen{-3pt}
&   &     &&     &&    && (0,3,-2,0) &\cropen{-3pt}
&   &     &&     &&    && (0,6,-3,-1) &\cr
\noalign{\hrule}
&   &13   &&43   &&11  && (-1,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (3,-1,-2,0) &\cropen{-3pt}
&   &     &&     &&    && (3,6,-2,-1) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (1,3,-1,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,0,1) &\cr
\noalign{\hrule}
&   &14   &&194  &&10  && (1,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-5,9,-2,-2) &\cropen{-3pt}
&   &     &&     &&    && (-16,31,-8,-6) &\cr
\noalign{\hrule}
&   &15   &&183  &&7   && [23,30,182,220,455] &\cropen{-2.5pt}
&   &     &&     &&    && [23,24,141,176,364] &\cropen{-2.5pt}
&   &     &&     &&    && [15,23,91,110,216] &\cr
\noalign{\hrule}
&   &15   &&157  &&9   && (0,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (1,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (-1,19,-5,-4) &\cropen{-3pt}
&   &     &&     &&    && (-1,24,-6,-5) &\cr
\noalign{\hrule}
&   &15   &&166  &&8   && (1,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,19,-5,-4) &\cropen{-3pt}
&   &     &&     &&    && (-1,24,-6,-5) &\cr
\noalign{\hrule}
&   &15   &&140  &&10  && (-1,2,0,-1) &\cropen{-3pt}
&   &     &&     &&    && (0,0,1,-1) &\cropen{-3pt}
&   &     &&     &&    && (0,0,0,-1) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (5,0,0,-1) &\cropen{-3pt}
&   &     &&     &&    && (5,0,-1,0) &\cr
\noalign{\hrule}}}}
\vfill\eject
\centerline{{\bf Table 2(cont'd):} 6-vertex irreducible 6-vertex reflexive
polyhedra}
\bigskip
\centerline{
\vbox{\offinterlineskip\halign{\strut#&\vrule#&
\hfil\quad\sixrm#\quad&#\vrule&
\hfil\quad\sixrm#\quad&#\vrule&
\hfil\quad\sixrm#\quad&#\vrule&
\quad\sixrm#\quad\hfil&#\vrule\cr
\noalign{\hrule}
&
  &\hbox{\tenrm Pts}
    &&$h_{11}$\hfil
      &&$h_{21}$\hfil
        &&\hbox{\eightrm Vertices/Weights}&\cr
\noalign{\hrule\vskip3pt\hrule}
&   &15   &&149  &&9   && (-1,2,0,-1) &\cropen{-3pt}
&   &     &&     &&    && (0,0,1,-1) &\cropen{-3pt}
&   &     &&     &&    && (0,0,0,-1) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (5,0,0,-1) &\cropen{-3pt}
&   &     &&     &&    && (5,1,-1,-1) &\cr
\noalign{\hrule}
&   &15   &&131  &&11  && (-1,2,0,-1) &\cropen{-3pt}
&   &     &&     &&    && (0,0,0,-1) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (5,0,0,-1) &\cropen{-3pt}
&   &     &&     &&    && (5,0,-1,0) &\cr
\noalign{\hrule}
&   &17   &&295  &&7   && [37,42,216,590,885] &\cropen{-3pt}
&   &     &&     &&    && [28,37,144,381,590] &\cropen{-3pt}
&   &     &&     &&    && [21,37,108,295,424] &\cr
\noalign{\hrule}
&   &22   &&376  &&10  && (-1,0,1,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,0,1) &\cropen{-3pt}
&   &     &&     &&    && (-1,1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,0,0) &\cropen{-3pt}
&   &     &&     &&    && (13,0,-4,-2) &\cropen{-3pt}
&   &     &&     &&    && (59,-1,-20,-8) &\cr
\noalign{\hrule}}}}
\bigskip
\centerline{{\bf Table 3:} 7-vertex irreducible 7-vertex reflexive polyhedra}
\bigskip
\centerline{
\vbox{\offinterlineskip\halign{\strut#&\vrule#&
\hfil\quad\sixrm#\quad&#\vrule&
\hfil\quad\sixrm#\quad&#\vrule&
\hfil\quad\sixrm#\quad&#\vrule&
\quad\sixrm#\quad\hfil&#\vrule\cr
\noalign{\hrule}
&
  &\hbox{\tenrm Pts}
    &&$h_{11}$\hfil
      &&$h_{21}$\hfil
        &&\hbox{\eightrm Vertices/Weights}&\cr
\noalign{\hrule\vskip3pt\hrule}
&   &8    &&51   &&3   && [20,21,25,26,33] &\cr
\noalign{\hrule}
&   &10   &&92   &&5   && (0,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (-1,2,0,-1) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (1,0,0,0) &\cropen{-3pt}
&   &     &&     &&    && (3,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (4,0,-1,0) &\cr
\noalign{\hrule}
&   &13   &&152  &&6   && (1,-1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,1,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,-1,0,1) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,9,-2,-2) &\cropen{-3pt}
&   &     &&     &&    && (-1,11,-2,-3) &\cropen{-3pt}
&   &     &&     &&    && (-1,20,-4,-5) &\cr
\noalign{\hrule}
&   &14   &&167  &&7   && (-1,1,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,0,1) &\cropen{-3pt}
&   &     &&     &&    && (4,0,-1,-1) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,0,0) &\cropen{-3pt}
&   &     &&     &&    && (-1,0,2,0) &\cropen{-3pt}
&   &     &&     &&    && (0,0,3,-1) &\cropen{-3pt}
&   &     &&     &&    && (0,-1,5,-1) &\cr
\noalign{\hrule}}}}
\vfill\eject
\baselineskip=12pt plus 1pt minus 1pt
\listreferences
\bye


A.C. Avram(1), E. Derrick(2), D. Jancic(1) ((1) U. Texas at Austin,
  (2) Humboldt Uni. zu Berlin)

On Semi-Periods

18 pages, plain TeX. Revised derivation of $\Delta^*$ system of
  equations; version to appear in Nuclear Physics B

Abstract:
  The periods of the three-form on a Calabi-Yau manifold are found as solutions
of the Picard-Fuchs equations; however, the toric varietal method leads to a
generalized hypergeometric system of equations which has more solutions than
just the periods. This same extended set of equations can be derived from
symmetry considerations. Semi-periods are solutions of this extended system.
They are obtained by integration of the three-form over chains; these chains
can be used to construct cycles which, when integrated over, give periods. In
simple examples we are able to obtain the complete set of solutions for the
extended system. We also conjecture that a certain modification of the method
will generate the full space of solutions in general.

\section{intro}
{Preamble}
The moduli space of a Calabi--Yau manifold\ naturally splits into two
spaces:
the space $M_{21}$ of complex structure parameters, and
the space $M_{11}$ of (complexified) K\"ahler\ class parameters.
Calculations involving the complex structure parameters are
exact,
while those involving the K\"ahler\ class parameters are
corrected by instantons
(for a comprehensive review, see
  ~\Ref{\rBeast}{T.~H\"ubsch, {\it Calabi--Yau manifolds ---A Bestiary for
       Physicists}\hfill\break (World Scientific, Singapore, 1992).}).
Nevertheless,
under mirror symmetry, $M_{11}$ of one manifold is related
to $M_{21}$ of the mirror manifold.
Thus the parameters of the low energy effective action
of a string theory compactified on a Calabi--Yau manifold\
can be calculated exactly by studying the space $M_{21}$ of
the manifold, and the space $M_{21}$ of its mirror.

Since the discovery that Special Geometry applies to the
moduli space of Calabi--Yau manifolds
  \REF{\rRolling}{P.~Candelas, P.S.~Green and T.~H\"ubsch,
      \npb{330} (1990) 49.}
  \REF{\rAS}{A.~Strominger, \cmp{133} (1990) 163.}
  \REF{\rCd}{P.~Candelas and X.~C.~de~la~Ossa, \npb{355} (1991) 455.}
 \ \cite{{\rRolling,\rAS,\rCd}},
the calculation of the periods
of the three-form has become something of an industry.
The space of complex structures is fully described by the
set of periods, in the sense that by knowing the periods,
one can calculate the metric on the moduli space and hence the kinetic term
and the Yukawa couplings in the low energy effective action.

Begin with the case of a family of hypersurfaces
${\cal M}_{\phi}\def\Ph{\Phi}\def\vph{\varphi}$ defined as the zero-set of a defining polynomial $p_\phi}\def\Ph{\Phi}\def\vph{\varphi$.
The periods are defined as
 %
$$ \vp_j(\phi}\def\Ph{\Phi}\def\vph{\varphi_\alpha)~~ \buildrel\rm def\over = ~~\int_{\gamma}\def\G{\Gamma^j} \O(\phi}\def\Ph{\Phi}\def\vph{\varphi_\alpha)~, \eqlabel{ePerDef}$$
where $\O(\phi}\def\Ph{\Phi}\def\vph{\varphi_\alpha)$ is the nowhere vanishing holomorphic
three-form
{}~\REF{\rPC}{P.~Candelas, \npb{298} (1988) 458.}
\REF{\rMA}{M.~Atiyah, R.~Bott and L.~G{\aa}rding,
                       Acta Math. {\bf 131} (1973) 145.}
\cite{{\rPC,\rMA}}

$$ \O(\phi}\def\Ph{\Phi}\def\vph{\varphi_\alpha)~~ \buildrel\rm def\over =
           ~~{\rm Res}_{{\cal M}_\phi}\def\Ph{\Phi}\def\vph{\varphi}\bigg[{(x{\rm d}^4x)\over p_\phi}\def\Ph{\Phi}\def\vph{\varphi}\bigg] $$
 %
on the Calabi-Yau three-fold specified by the parameters $\phi}\def\Ph{\Phi}\def\vph{\varphi_\alpha$,
and $\gamma}\def\G{\Gamma^j$ is a three-cycle labelled by $j$.
The four-differential
$(x{\rm d}^4x)$ is the `natural' one: on a weighted projective $N$-space
 $\wcp{(k_1,\ldots,k_{N+1})}{N}$, we have
 %
$$ (x{\rm d}^Nx)~~ \buildrel\rm def\over =
             ~~{1\over (N{+}1)!} \epsilon}\def\ve{\varepsilon^{i_1 i_2\cdots i_{N+1}}
                k_{i_1} x_{i_1} {\rm d} x_{i_2} \cdots {\rm d} x_{i_{N+1}}~, $$
where $k_i$ are the weights of the coordinates. That is, in
 $\wcp{(k_1,\ldots,k_{N+1})}{N}$,
 %
$$ (x_1,\ldots,x_{N+1}) \cong (\lambda}\def\L{\Lambda^{k_1}x_1,\ldots,\lambda}\def\L{\Lambda^{k_{N+1}}x_{N+1})~,
             \qquad \lambda}\def\L{\Lambda \in \relax\hbox{\kern.25em$\inbar\kern-.3em{\rm C}$}^*~. $$
The period~\eqref{ePerDef}\ will here be calculated by choosing one of the
standard coordinate patches in $\wcp{(k_1,\ldots,k_5)}{4}$, ${\cal U}_m$,
where $x_m\neq0$ so that $x_m=1$ by projectivity. There
{}~\Ref{\rSpokes}{P.~Berglund, E.~Derrick, T.~H\"ubsch and
    D.~Jan\v{c}i\'{c}, \npb{420} (1994) 268, hep-th/9311143.}:
$$ \vp^{(m)}_j(\phi}\def\Ph{\Phi}\def\vph{\varphi_\alpha)~~ = ~~C \int_{\G^j}
          {\prod_{i\neq m}{\rm d} x_i\over p_\phi}\def\Ph{\Phi}\def\vph{\varphi\vert_m}~,
\eqlabel{eOM}$$
 %
with $C$ a convenient prefactor. This is easily seen to apply
for Calabi-Yau weighted hypersurfaces of arbitrary dimension.

We separate the polynomial into a reference polynomial $p_0$
independent of the moduli~$\phi}\def\Ph{\Phi}\def\vph{\varphi$, and a perturbative part $\D$
which does depend on the $\phi}\def\Ph{\Phi}\def\vph{\varphi$.
This amounts to choosing a reference point in moduli space,
and expanding around that point.
The basic idea here is,
while working in the patch ${\cal U}_m$ but suppressing the label,
to expand $1/p_\phi}\def\Ph{\Phi}\def\vph{\varphi$ around $1/p_0$, and utilize the Laplace transform
 %
$$ {1\over p_\phi}\def\Ph{\Phi}\def\vph{\varphi}~ = ~
{1\over p_0 + \D} ~=~
    \sum_{n=0}^\infty {(-\D)^n \over \- (p_0)^{n+1} }
{}~=~ {d\over k_m} \int_0^\infty {\rm d} x_m \sum_n {(-\D)^n \over n!} e^{- p_0}
{}~,\quad \mathop{\Re e}(p_0)>0~,\eqlabel{eLaplace}$$
 %
which produces a ``small-$\phi$''
expansion of the periods~\eqref{eOM}.
We have relabelled the Laplace transform parameter
to $x_m^{d/k_m}$, and so the expansion appears as if in homogeneous
coordinates.
With a choice of the poly-contours $\{\G^j\}$, Eq.~\eqref{eOM}\ may be
considered a definition of the periods.

The semi-period is a building block for a period.
In other words, it is the integral of the three--form
over a chain instead of a cycle, with the understanding that
the chain can be manipulated in some way to build cycles.
The semi-period construction we will discuss is
naturally related to the period construction in
  \cite{\rSpokes}.
The prototypical semi-period we write as
$$ F_0 = \int_V \, {\rm d}^{N+1} x\, \sum_{n=0}^\infty\, {(-\D)^n \over n!}\,
                              e^{-p_0}~,
\eqlabel{protosp}$$
where $V$ is the ${N+1}$-chain
$\{x_1,\ldots,x_{N+1}$ real and positive$\}$.

As was worked out in \cite{\rSpokes}, these chains are enough
to build cycles and thus to calculate periods.
The advantage of this approach is its ease of calculation of a
full set of periods in different regions of moduli space.
This method lends itself to many different constructions:
all the types of polynomial hypersurfaces discussed in
  \REF{\rCfC}{P.~Berglund and T.~H\"ubsch,
              \npb{411} (1994) 223, hep-th/9303158.}
  \REF{\rCLS}{P.~Candelas, M.~Lynker and R.~Schimmrigk,
                      \npb{341} (1990) 383.}\
  \cite{{\rCfC,\rCLS}},
(weighted) complete intersection spaces
  \REF{\rCYCI}{T.~H\"ubsch, \cmp{108} (1987) 291.}
  \REF{\rGH}{P.~Green and T.~H\"ubsch, \cmp{109} (1987) 99.}
  \REF{\rCDLS}{P.~Candelas, A.M.~Dale, C.~A.~L\"utken and R.~Schimmrigk,\hfill\break
       \npb{298} (1988) 493.}
  \REF{\rGRY}{B.~Greene, S.-S.~Roan and S.-T.~Yau, \cmp{142} (1991) 245.}\
\cite{{\rBeast, \rCYCI, \rGH, \rCDLS, \rGRY}},
generalized Calabi-Yau manifolds
 \REF{\rZ}{P.~Candelas, E.~Derrick and L.~Parkes,
        \npb{407} (1993) 115,\hfill\break hep-th/9304045.}
  \REF{\rRolf}{R.~Schimmrigk, \prl{70} (1993) 3688, hep-th/9210062.}\
\cite{{\rZ, \rRolf}},
Landau--Ginzburg vacua
  \REF{\rMaxSkII}{M.~Kreuzer and H.~Skarke, ``On the Classification
       of Quasihomogeneous Functions'', CERN preprint CERN-TH-6373/92.}
  \REF{\rMaxSkI}{M.~Kreuzer and H.~Skarke, \npb{388} (1992) 113,
               hep-th/9205004.}
  \REF{\rAR}{A.~Klemm and R.~Schimmrigk,
       \npb{411} (1994) 559, hep-th/9204060.}\
\cite{{\rMaxSkII,\rMaxSkI,\rAR}},
and toric varietal constructions related to these and other types
 ~\REF{\rBatyrev}{V.~Batyrev,
    Duke Math. Journal, Vol 69, No 2, (1993) 349.}
  \REF{\rHKTY}{S.~Hosono, A.~Klemm, S.~Theisen and S.-T.~Yau,
     \cmp{167} (1995) 301, hep-th/9308122.}
  \REF{\rHKTYII}{S.~Hosono, A.~Klemm, S.~Theisen and S.-T.~Yau,
      \npb{433} (1995) 501, hep-th/9406055.}
  \REF{\rCdK}{P.~Candelas, X. de la Ossa and S. Katz,
    \npb{450} (1995) 267, hep-th/9412117.}\
\cite{{\rBatyrev,\rHKTY,\rHKTYII,\rCdK}}.

A semi-period is also the solution of
a generalized hypergeometric system of equations.
The periods are solutions of the Picard-Fuchs (PF) equations.
In the toric varietal approach,
differential equations are constructed based on
the points in the dual polyhedron and the generators of Mori cone
  ~\REF{\rBKK}{P.~Berglund, S.~Katz and  A.~Klemm,
    \npb{456} (1995) 153, hep-th/9506091.}
\cite{{\rBatyrev,\rHKTY,\rBKK}}.
The system of equations obtained in this manner has extra solutions beside
periods, since the complete set of solutions is larger than
the set of linearly independent periods.
We show by construction that semi-periods such as \eqref{protosp}
are such extra solutions.
All of the periods can be obtained either as the solutions to PF equations,
or as linear combinations of semi-periods.

In many if not all cases the same generalized hypergeometric system that was
obtained from toric data can be constructed based on the symmetries of the
period, where the period is expressed as an integral over a cycle
 ~\REF{\rY}{S.-T. Yau, talk presented at Workshop on Mirror Symmetry
    and S-Duality, ICTP, Trieste, Italy, June 1995.}
\cite{{\rHKTY,\rY}}.
These equations are also satisfied by a semi-period,
which is expressed using the
same integrand, but integrated over a particular chain, not a cycle.
Not every chain yields a solution to the differential equations;
for those that do, we call the integral a semi-period.

Our aim in this paper is to construct, calculate and
investigate the semi-period solutions to the $\Delta^*$
hypergeometric system of equations.
The layout of the paper is as follows:
Section~\chapref{ghs} describes the construction of
the generalized hypergeometric system of equations using the
toric varietal approach and
Section~\chapref{sex} illustrates the procedure
by doing a simple example: the $\relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi_3$ torus.
An interesting set of examples are explored in
Sections~\chapref{example} and \chapref{exampleIII}.

\section{ghs}
{Construction of the $\Delta^*$ Hypergeometric System
 associated to Toric Manifolds}
In this section we briefly review the construction of the
$\Delta^*$
hypergeometric system for {Calabi--Yau manifolds } described as the zero loci of
homogeneous polynomials in weighted projective spaces.

We denote a weighted projective space as $\relax{\rm I\kern-.18em P}^r_{\bf k}$,
by which we mean the set of complex $x_i$ (not all zero)
identified under
$(x_1, x_2, \ldots, x_{r+1}) \sim
  (\lambda}\def\L{\Lambda^{k_1} x_1,\lambda}\def\L{\Lambda^{k_2} x_2, \ldots, \lambda}\def\L{\Lambda^{k_{r+1}} x_{r+1}) $
for all $\lambda}\def\L{\Lambda\neq 0$.
Consider the Calabi--Yau manifold\ $\relax{\rm I\kern-.18em P}^r_{\bf k}[d]$ with $\sum_{i=1}^{r+1}
k_{i} = d$, where  $d$ is the degree of the defining
polynomial whose general expression is :
$$
p = \sum_{\bf m} c_{\bf m} x^{\bf m}
$$
Here $\bf m$ is the degree vector such that ${\bf m} \cdot {\bf k} = d$.

We associate to each monomial a point with coordinates
${\bf m} =(m_{1}, \dots , m_{r+1})$ in the lattice $\relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi^{r+1}$.
The set of all monomials compatible with ${\bf m} \cdot {\bf k} =d$ will
define a convex polyhedron $\Delta$ in $\relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi^{r+1} \otimes \relax{\rm I\kern-.18em R}$, which is
moreover reflexive (in the sense of \cite{\rBatyrev}).
This polyhedron lies in a hyperplane defined by the weight vector normal to it
and will contain all the points of the sublattice that lie within its
boundaries.
By uniting the vertices of the polyhedron $\Delta$ with the origin
we obtain a
cone whose dual can be easily constructed by the method described in
\cite{\rBatyrev}.

The vertices of the dual polyhedron $\Delta^*$ are given by the first
intersection point between the generators of the dual
cone and the dual lattice.
Because none of the weights in the definition of
$\relax{\rm I\kern-.18em P}^r_{\bf k}$ is zero, we
can project the dual polyhedron on a
$\relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi^{r} \otimes \relax{\rm I\kern-.18em R}$ subspace and then find a basis for the sublattice
defined by the projected points of the dual polyhedron.
We take the unique interior point as the origin of the new coordinate system,
and then put back the dual polyhedron in
$\relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi^{r+1} \otimes \relax{\rm I\kern-.18em R}$ along the hyperplane $x_{0}=1$.

This leaves us with the following set of dual points:
$$
 \bar{\nu}_{i}^{*} = (1, \nu_{i}^{* 1}, \ldots , \nu_{i}^{* r})
\qquad i=1, \ldots ,p   ~~~,
\eqlabel{nu2}
$$
and the interior point
$$
 \bar{\nu}_{0}^{*} = (1,0, \ldots, 0)~.
$$

The space of linear dependence relations between the dual points,
$\sum_{i=0}^p \, l_{i}^{a} \, \bar{ \nu}_{i}^{*} = 0$,
will have dimension $p-r$. Nevertheless as Batyrev remarked in \cite{\rBatyrev}
we are interested only
in points that do not lie in
the interior of codimension $1$ faces. Assuming there are $n +1$
of them, we denote $A = \{ \bar{\nu}_{0}^{*},  \bar{\nu}_{i_1}^{*},
\dots, \bar{\nu}_{i_n}^{*} \}$ the subset of such points in the dual
polyhedron.
The linear dependences between points in $A$ are described by the lattice of
rank $n - r$:

$$
 L = \{ (l_{0}, \dots, l_{n}) \in \relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi^{n+1} \mid \sum_{i=0}^n\, l_{i}^{a} \,
\bar{ \nu}_{i}^{*} = 0, \bar{ \nu}_{i}^{*} \in A \}
\eqlabel{lindep}
$$
Considering the affine complex space $\relax\hbox{\kern.25em$\inbar\kern-.3em{\rm C}$}^{p+1}$ with coordinates $(a_{0},
\dots, a_{p})$ we define a consistent system of
differential operators:
$$\eqalign{
 Z_j \, &= \, \sum_{i=0}^{n} \, \bar{ \nu}^*_{i,j} \, a_i
{\partial\over\partial a_i} \, - \, \beta_j  \cr
 D_{{\bf l}^a} \, &= \, \prod_{l_i>0} \left({\partial \over \partial
a_i}\right)^{l_i^a} \, - \, \prod_{l_i<0} \left({\partial \over \partial
a_i}\right)^{-l_i^a} ; \quad  {\bf l}^a \in L \cr}
\eqlabel{sys}
$$
The exponent of $\bf\beta$ that appears in the definition of the differential
operators $Z_j$ will be taken
to be $(-1, 0, \ldots, 0)$.
With this choice the period integrals
vanish when acted upon by the operators  \eqref{sys}  \cite{\rBatyrev}.

The set of operators \eqref{sys} taken together form the
$\Delta^*$ hypergeometric system of equations.
This is an example of a class of generalized hypergeometric equations
described by
Gel'fand, Kapranov and Zelevinski \
  \Ref{\rgelfand}{I.~M.~Gelfand, A.~V.~Zelevinski and M.~M.~Kapranov,
   Func. Anal. Appl. Vol. 23 No. 2 (1989) 94.},
and hence is also called a GKZ system.
In general,
the $\Delta^*$ system is not enough to determine the Picard-Fuchs equations;
it must be extended by supplementing further differential
operators.  The methods for doing this are discussed in \
\REF{\rHLY}{S.~Hosono, B.~H.~Lian and S.-T.~Yau,
     ``GKZ-Generalized Hypergeometric System in Mirror Symmetry
      of Calabi-Yau Hypersurfaces'', alg-geom/9511001.}
\cite{{\rHKTY, \rBKK, \rHLY}}.
In other words, there are more solutions to this system of equations than
just the periods.
We conjecture that semi-periods form a complete set of solutions.

The periods and semi-periods
depend on  $a_i$ through $\zeta^a$ \cite{\rHKTY}, defined as:
$$
\zeta^a = (-1)^{{ \hat l}_0^a} \, {\bf a}^{{\bf{\hat l}}^a} ~.\eqlabel{zetadef}
$$
We used ${\bf {\hat l}}^a$ to denote the generators of the Mori cone which can
be found by standard
methods(see for example \cite{\rBKK}).
The variables $\zeta^a$
are not only the natural choices in order to
satisfy the linear constraints imposed by the $Z^j$,
but are also good coordinates on the
moduli space for describing the large complex structure limit of the mirror
manifold.

\section{sex}{Simple Example}
\subsection{The $\Delta^*$ Hypergeometric System for the $\relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi_3$ Torus}
The $\relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi_3$ torus can be described as a cubic hypersurface in $\relax{\rm I\kern-.18em P}^2$.
The  vector of weights is ${\bf k} = (1,1,1)$,
and so the vertices of the dual polyhedron become
$$
 \bar{\nu}^*_1 = (1,-1,-1) \quad \bar{\nu}^*_2 = (1,1,0) \quad
\bar{\nu}^*_3 = (1,0,1) ~, $$
and the interior point is $ \bar{\nu}^*_0 = (1,0,0)$.

There is one relation of the form
$ \sum_i l_i \bar{\nu}^*_i = 0$,
satisfied by ${\bf l} = (-3,1,1,1)$.
Taking this into the account, the differential operators defined by \eqref{sys}
will be of the following form:
$$ D = {\partial\over\partial a_1}\, {\partial\over\partial a_2}\,
       {\partial\over\partial a_3}
        - \left({\partial\over\partial a_0}\right)^{3}
\eqlabel{Dtorus}$$
and
$$\eqalign{
 Z_1  &= \left( \sum_{i=0}^3 a_i {\partial\over\partial a_i} +1
            \right)  \cr
 Z_2  &= \left(  a_2 {\partial\over\partial a_2} -
                    a_1 {\partial\over\partial a_1}    \right)  \cr
 Z_3  &= \left(  a_3 {\partial\over\partial a_3} -
                    a_1 {\partial\over\partial a_1}    \right) ~. \cr}
\eqlabel{Zitorus}$$
The equations $Z_i \Pi = 0$ are solved by $\Pi$ of the form
$$\Pi = {1\over a_0} \tilde\Pi\left(-{a_1 a_2 a_3 \over a_0^3}\right)~,$$
and the equation $D \Pi = 0 $ becomes
$$ \left[ \Theta^3 -\zeta (3\Theta + 3)(3\Theta +2)(3\Theta + 1)\right]
   \tilde\Pi = 0~,\eqlabel{Dtorus}$$
where $\Theta = \zeta {\partial\over\partial \zeta}$, and $\zeta$ is the argument
of $\tilde\Pi$, found from \eqref{zetadef}.

Solving the generalized hypergeometric equation one has
{}~\Ref\rErdelyi{A.~Erd\'elyi, F.~Oberhettinger, W.~Magnus and
       F.~G.~Tricomi, {\it Higher Transcendental Functions, 3 Vols.}
       (McGraw--Hill, New York, 1953).}\ :
$$\tilde \Pi = \Psymbol{0&1/3&0\cr 0&2/3&1\cr 0&1&0\cr}
                       {\cr \cr 3^3 \zeta \cr \cr}
{}~.$$
We will be interested in the region where $a_0$ is small.
Since $\zeta$ is related to $a_0^{-3}$, we should look at the solutions
near $\zeta\rightarrow\infty$.
Remembering that $\Pi = {1\over a_0} \tilde\Pi$, and setting
$a_1 = a_2 = a_3 = 1$, $\zeta = -1/a_0^3$, the solutions to the
$\Delta^*$-hypergeometric system are
$$\eqalign{
  \Pi_0 &=\hphantom{a_0^2}\
           {}_2F_1\left({1\over 3},{1\over 3};{2\over 3};
                        -\left({a_0\over 3}\right)^3\right) \cr
  \Pi_1 &= a_0\  {}_2F_1\left({2\over 3},{2\over 3};{4\over 3};
                        -\left({a_0\over 3}\right)^3\right) \cr
  \Pi_2 &= a^2_0\ {}_3F_2\left(1,1,1;{4\over 3},{5\over 3};
                        -\left({a_0\over 3}\right)^3\right)~. \cr}
\eqlabel{Pistorus}$$

\subsection{The Equation from Symmetry Considerations}
Suppose we look at the period defined as
$$ \Pi = \int_\G {x_1 {\rm d} x_2 {\rm d} x_3 \over
 a_1 x_1^3 +a_2 x_2^3 +a_3 x_3^3 + a_0 \ x_1 x_2 x_3 }~,
\eqlabel{perdef}$$
where $\G$ is some cycle.  Since $\Pi$ is
independent of $x_1$,
it is equivalent to
$$ \Pi = {1\over 2 \pi i} \int_{\G'} {{\rm d} x_1 {\rm d} x_2 {\rm d} x_3 \over
 a_1 x_1^3 +a_2 x_2^3 +a_3 x_3^3 + a_0 \ x_1 x_2 x_3 }~,
\eqlabel{perdefII}$$
where $\G' = C_1 \times \G$, and $C_1$ is a loop around $x_1=0$.
Now look at the symmetries of $\Pi$ as a function of the $a_i$:
$$
\Pi(a_0,\lambda}\def\L{\Lambda^3 a_1, \lambda}\def\L{\Lambda^{-3} a_2,a_3) = \Pi(a_0,a_1,a_2,a_3) ~,
 \eqlabel{symtorI}$$
and the same with $a_2\rightarrow a_3$, so
$$\eqalign{
 a_1 \pd{\Pi}{a_1}  &= a_2 \pd{\Pi}{a_2} \cr
 a_1 \pd{\Pi}{a_1}  &= a_3 \pd{\Pi}{a_3} ~,\cr}\eqlabel{ZsymtorI}$$
while
$$\Pi(\lambda}\def\L{\Lambda^3a_0,\lambda}\def\L{\Lambda^3 a_1,\lambda}\def\L{\Lambda^3 a_2,\lambda}\def\L{\Lambda^3 a_3) =\lambda}\def\L{\Lambda^{-3} \Pi(a_0,a_1,a_2,a_3)
\eqlabel{symtorII}$$
implies that
$$ a_0 \pd{\Pi}{a_0}  +\sum_{i=1}^3 a_i \pd{\Pi}{a_i} = -\Pi~.
\eqlabel{ZsymtorII}$$
These are the $Z_i$ equations \eqref{Zitorus}.
Finally, since $x_1^3 x_2^3 x_3^3 = (x_1 x_2 x_3)^3$,
the period satisfies the relation
$$ \pd{}{a_1} \pd{}{a_2} \pd{}{a_3} \Pi=
   \left( {\partial\over\partial a_0}\right)^3 \Pi ~,
\eqlabel{Dsymtor}$$
which is the $D$ equation \eqref{Dtorus}.

Thus we are able to find the equations from consideration
of the symmetries of the period,
without going through the toric variety construction.

\subsection{The Semi-Periods for the $\relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi_3$ Torus}
Consider the integrand in the definition of the period \eqref{perdef},
and note that we can take a Laplace transform as long as $\mathop{\Re e}(p)>0$:
$$ {x_1 {\rm d} x_2 {\rm d} x_3 \over p}  =
   \int_{x_1=0}^\infty {\rm d}^3 x\  e^{-p}~.$$
Taking the polynomial to be
$$p =  a_1 x_1^3 +a_2 x_2^3 +a_3 x_3^3 + a_0 x_1 x_2 x_3 ~,$$
we define the integration contour
$ V = \{x_1, x_2, x_3 \hbox{ real and positive} \} $
and construct
$$ F = \int_V {\rm d}^3 x\  e^{-p} ~.\eqlabel{FZ}$$
$F$ as defined also satisfies the relations \eqref{ZsymtorI}
\eqref{ZsymtorII} \eqref{Dsymtor} derived
from symmetry considerations above,
and hence is the extra solution besides the periods.
This is a semi-period.
By using the symmetries \eqref{symtorI} \eqref{symtorII},
we bring the polynomial in the integrand of $F$ to canonical form
$$p =   x_1^3 + x_2^3 + x_3^3 + \phi\  x_1 x_2 x_3 ~.$$
Now it is easy to calculate $F(\phi)$:
$$\eqalign{
F &= \sum_{n=0}^\infty {(-\phi)^n\over n!} \
    \prod_{i=1}^3 \int_0^\infty {\rm d} x_i \, x_i^n\, e^{-x_i^3}\cr
  &= {1\over 3^2} \sum_{n=0}^\infty
     {(-\phi)^n\over n!} \, \Gamma^3\left( {n+1\over 3}\right) \cr}
\eqlabel{FZcalc}$$
which is, of course, a linear combination of
$\Pi_0$, $\Pi_1$ and $\Pi_2$ of \eqref{Pistorus}, with $a_0 = \phi$.

But now, consider integrating over a slightly different contour.
Let $x_2$ run from 0 to $\omega}\def\O{\Omega \infty$, $\omega}\def\O{\Omega$ being a cube root of unity,
and call this new contour $A_2 V$.
This still satisfies the equations, but by a change of variables
we see that
$$ \int_{A_2 V} {\rm d}^3 x\  e^{-p(\phi)} = \omega}\def\O{\Omega F(\omega}\def\O{\Omega \phi) ~.$$
There are three semi-periods we can form in this way,
summarized by integrating
$x_2$ from 0 to $\omega}\def\O{\Omega^{\delta}\def\D{\Delta} \infty$, $\delta}\def\D{\Delta=0,1,2$.
This inserts a factor of $\omega}\def\O{\Omega^{n+1}$ in the summation of
\eqref{FZcalc}, and leads to
three independent linear combinations of \eqref{Pistorus}.

So what happens when we form a spoke in the $x_2$ plane,
that is, we integrate $x_2$ along a contour that comes in infinity
along one of these paths, and returns along the other?
The integration contour is $V - A_2 V$, so the result of
the integration is $F(\phi) - \omega}\def\O{\Omega F(\omega}\def\O{\Omega \phi)$,
which is a linear combination of $\Pi_0$ and $\Pi_1$.
Note that in generating periods for the torus by using spokes
\cite{\rSpokes},
we take exactly these sorts of contours, and
the periods are linear combinations of
$\Pi_0$ and $\Pi_1$.
Details of this example are in the Appendix in \cite{\rZ}.
A cycle could be generated, for example, by integrating over
$ (1-A_2) (1-A_3) V$, and the period that results is
$ F(\phi) - 2 \omega}\def\O{\Omega F(\omega}\def\O{\Omega \phi) + \omega}\def\O{\Omega^2 F(\omega}\def\O{\Omega^2 \phi)$.

Why do the semi-periods satisfy the $\Delta^*$ hypergeometric
system?
The differential equations can be generated by considering the
symmetries of the period  integral \eqref{perdefII}.
The symmetry in \eqref{symtorI} is demonstrated by a change of variables
in the integrand of \eqref{perdefII},
$x_1\rightarrow \lambda}\def\L{\Lambda x_1$ and $x_2\rightarrow \lambda}\def\L{\Lambda^{-1} x_2$.
Since the region of integration has no boundary,
the expression is the same.
Applying the same consideration to \eqref{FZ} or \eqref{FZcalc},
we see that (taking $\lambda}\def\L{\Lambda$ to be real), the boundary is unchanged
and again, the expression is the same.
Thus the same symmetries are exhibited by the semi-periods,
since the boundary of the chain has been chosen properly.

The $\Delta^*$ system contains the Picard-Fuchs equations, and is thus
solved by the periods;
but it is also solved by the semi-periods.
The methods for reducing the order of the $\Delta^*$ system
to the order of the PF equations are discussed in
\cite{{\rHKTY,\rBKK,\rHLY}}.
In~\cite{\rBKK}, use is made of further properties of
the periods that can be derived using partial integration.
These properties are not shared by the semi-periods,
and hence this method naturally selects periods over semi-periods.
References~\cite{\rHKTY} and~\cite{\rHLY} find other ways
of selecting periods, either by reducing the order of the
$\Delta^*$ system, or imposing additional differential equations.

There are, of course,  more semi-periods than periods.
Even though we have not been able to explicitly construct them in
general, we believe that all solutions of the $\Delta^*$ system can be
obtained by integrating the three--form over well chosen chains.
We have not found a general principle that guarantees that
all solutions to the $\Delta^*$ system can be found as
integrals over chains ---
it is, however, true in simple cases.
It seems likely that
the selection of these chains is highly model dependent.
For the types of chains we consider,
the number of semi-periods generated in a particular corner of
moduli space depends on the symmetries of $p_0$,
the reference polynomial.

\section{example}{A Known Example}
Here is an example for which the periods are discussed
in several places,
  \REF{\rGoryVII}{P.~Berglund, P.~Candelas, X.~de la Ossa, A.~Font,
       T.~H\"ubsch, D.~Jan\v{c}i\'{c} and F.~Quevedo,
        \npb{419} (1994) 352, hep-th/9308005.}\
namely Section 4.2 of \cite{\rGoryVII},
Section 3.1 of \cite{\rSpokes} and
Section 3.1 of \cite{\rBKK}.
Let us examine the semi-periods for a particular family of
hypersurfaces, $\cp{(2,2,3,7,7)}[21]^{50,11}$,
where the superscripts indicate the
Hodge numbers $b_{2,1}=50$ and $b_{1,1}=11$,
with the polynomial chosen as
$$ p = a_1 y_1^7 y_4 + a_2 y_2^7 y_5 + a_3 y_3^7 +
       a_4 y_4^3 + a_5 y_5^3 + a_0 y_1 y_2 y_3 y_4 y_5 +
       a_6 (y_1 y_2 y_3)^3 ~.$$
To calculate a semi-period $F$, we must
pick a corner of moduli space.
Supposing we treat the last two terms in $p$ as
perturbations, $F$ is  calculated to be
$$\eqalign{
 F
   &=
  \sum_{m,k=0}^{\infty} \,{(-a_0)^m \over m!}{(-a_6)^k \over k!}\cr
   &\quad\quad\quad\times
  \int_0^\infty {\rm d}^5 y \ (y_1 y_2 y_3)^{m+3k}\, (y_4 y_5)^m\
  e^{-\left(a_1 y_1^7 y_4 + a_2 y_2^7 y_5 + a_3 y_3^7 +
       a_4 y_4^3 + a_5 y_5^3 \right)} \cr
  &= {1\over 3^2 7^3 }\,
  \sum_{{m,k=0 \atop 2m+2>k}}^{\infty}
             \,{(-a_0)^m \over m!}{(-a_6)^k \over k!}\,
  {\Gamma^3\left({m+3k+1\over 7}\right)\over
         (a_1 a_2 a_3)^{m+3k+1\over 7} }\
  {\Gamma^2\left({2m-k+2\over 7}\right)\over
         (a_4 a_5)^{2m-k+2\over 7} }        ~.       \cr}
\eqlabel{F}$$

Things to notice about this in reference to the references
\cite{{\rGoryVII,\rSpokes,\rBKK}}:
\item{$\bullet$}This can be expressed as a function of the
two large complex structure parameters  as defined in
\cite{{\rGoryVII,\rBKK}},
$\zeta_1 = -{a_4 a_5 a_6\over a_0^3}$ and
$\zeta_2 = - {a_1 a_2 a_3 \over a_0 a_6^2}$, via
$$ \eqalign{
F &= {1\over a_0 }\tilde F(\zeta_1,\zeta_2)\cr
\tilde F &= \sum_{r=0}^1 \sum_{j,l=0}^\infty \,
             \left(-{1\over \zeta_1}\right)^{{2j+2-r\over 7}}\,
             \left(-{1\over \zeta_2}\right)^{l+{j+1+3r\over 7}}\,
        { (-1)^{l+j+r}\over \Gamma(l+j+1)\Gamma(2l+1+r) }\cr
&\quad\quad\quad\times
  \Gamma^3\left(l+{j+1+3r\over 7}\right)\,
  \Gamma^2\left({2j+2-r\over 7}\right)~.\cr}
\eqlabel{FII}$$
$F$ has been calculated in an expansion around large $\zeta_1$
and $\zeta_2$.
\item{$\bullet$}This obeys the equations found in \cite{\rBKK},
$$\eqalign{
  \partial^3_{a_0} F -
        \partial_{a_4}\partial_{a_5}\partial_{a_6} F &=0\cr
  \partial^2_{a_6} \partial_{a_0} F -
        \partial_{a_1}\partial_{a_2}\partial_{a_3} F &=0~,
\cr}\eqlabel{pfexample}$$
which become the extended Picard--Fuchs equations
$$\eqalign{
  \Big\{& \Theta_1^2 (\Theta_1 - 2 \Theta_2) -
   \zeta_1 (3 \Theta_1 + \Theta_2 + 3)\, (3 \Theta_1 + \Theta_2 + 2)\,
          (3 \Theta_1 + \Theta_2 + 1) \Big\} \tilde F = 0\cr
  \Big\{& \Theta_2^3 -
   \zeta_2 ( \Theta_1 -2 \Theta_2 -1)\, ( \Theta_1 -2 \Theta_2)\,
          (3 \Theta_1 + \Theta_2 + 1) \Big\} \tilde F = 0~,\cr}$$
using $\Theta_i =\zeta_i \partial_{\zeta_i}$.
\item{$\bullet$}
There are seven semi-periods that are easy to find in this expansion.
These were exploited in \cite{\rSpokes} to calculate the
periods.  The different semi-periods are obtained by
integrating over different chains in \eqref{F},
as detailed in \cite{\rSpokes}.
Consider changing the region of integration in \eqref{F} so that
$y_1$ runs from 0 to $\lambda}\def\L{\Lambda \infty$,
where $\lambda}\def\L{\Lambda$ is a seventh root of unity.
The net effect is to change $F$ by
$$ A : F(a_0,a_6) \rightarrow \lambda}\def\L{\Lambda F(\lambda}\def\L{\Lambda a_0, \lambda}\def\L{\Lambda^3 a_6) ~.$$
Explicitly, this leads to the insertion of $\lambda}\def\L{\Lambda^{m+3k+1}$
inside the summation in \eqref{F}, or an insertion of $\lambda}\def\L{\Lambda^{j+1+3r}$
inside the summation in \eqref{FII}.
This leads to seven semi-periods.
It has been shown in \cite{\rSpokes}
that the six independent periods can be calculated using spokes,
that is, using differences of semi-periods generated in
this way.
The action of $A$ is discussed in \cite{{\rGoryVII,\rSpokes}}.
\section{exampleIII}
{Nesting Example}
The following example is interesting because it
shows how $K3$ and $T^2$ can appear.
In this section
we extend an argument by Klemm, Lerche and Mayr \
  \Ref{\rKLM}{A.~Klemm, W.~Lerche and P.~Mayr,
    \plb{357} (1995) 313, hep-th/9506112.} \
to include the semi-periods;
see also \
   \Ref{\rLYII}{B.~H.~Lian and S.-T.~Yau, ``Mirror Maps, Modular Relations
      and Hypergeometric Series II'', hep-th/9507153.} \
and \cite{\rHKTY}.
Consider the manifold $\cp{(1,1,2,4,4)}[12]^{101,5}$,
and choose  a particular family of  hypersurfaces,
$$ p = a_1 x_1^{12} + a_2 x_2^{12} + a_3 x_3^{6} +
       a_4 x_4^3 + a_5 x_5^3 + a_0 x_1 x_2 x_3 x_4 x_5 +
       a_6 (x_1 x_2)^{6} + a_7 (x_1 x_2 x_3)^3 ~.$$
This is a $K3$ fibration, as can be seen by making
the substitutions $x_2 = \lambda}\def\L{\Lambda x_1$, $\tilde x_1 = x_1^2$,
which leads to
$$ p = \tilde x_1^{6} (a_1 + \lambda}\def\L{\Lambda^{12} a_2 + \lambda}\def\L{\Lambda^{6} a_6)
        +a_3 x_3^{6} +   a_4 x_4^3 + a_5 x_5^3 + \lambda}\def\L{\Lambda a_0 \tilde x_1 x_3 x_4 x_5
+
         + \lambda}\def\L{\Lambda^3 a_7 (\tilde x_1 x_3)^3 ~.$$
The hypersurface $p=0$ describes a  $K_3$ hypersurface
\cp{(1,1,2,2)}[6].
By a similar substitution,
this $K3$ is shown to be an elliptic fibration over $\relax{\rm I\kern-.18em P}^1$ with generic
fiber \cp{(1,1,1)}[3], otherwise known as the $\relax\ifmmode\hbox{\ss Z\kern-.4em Z}\else{\ss Z\kern-.4em Z}\fi_3$ torus, described in
Section~\chapref{sex}.

The dual polyhedron of $\cp{(1,1,2,4,4)}[12]$ contains 8 points:
$$\eqalign{
 \bar{\nu}_0^* &= (1,\- 0,\- 0,\- 0,\- 0)\cr
 \bar{\nu}_1^* &= (1,\- 4,\- 0,\- 2,\- 3)\cr
 \bar{\nu}_2^* &= (1,\- 0,\- 0,\- 0,\- 1)\cr
 \bar{\nu}_3^* &= (1,\- 0,\- 0,\- 1,\- 0)\cr
 \bar{\nu}_4^* &= (1,\- 0,\- 1,\- 0,\- 0)\cr
 \bar{\nu}_5^* &= (1, - 1, - 1, - 1, - 1)\cr
 \bar{\nu}_6^* &= (1,\- 2,\- 0,\- 1,\- 2)\cr
 \bar{\nu}_7^* &= (1,\- 1,\- 0,\- 1,\- 1)\cr} $$
We find $6$ vectors in the Mori cone, which can all be expressed with
positive integer coefficients in terms of the following basis:
$$\eqalign{
{\bf l}^{1} &= ( - 3,\- 0,\- 0,\- 0,\- 1,\- 1,\- 0,\- 1)\cr
{\bf l}^{2} &= (\- 0,\- 0,\- 0,\- 1,\- 0,\- 0,\- 1, - 2)\cr
{\bf l}^{3} &= (\- 0,\- 1,\- 1,\- 0,\- 0,\- 0, - 2,\- 0)\cr
}
$$
The large complex structure coordinates are:
$$\eqalign{
\zeta_1 &= - {a_4 a_5 a_7 \over a_0^3} \cr
\zeta_2 &= {a_3 a_6 \over a_7^2}  \cr
\zeta_3 &= {a_1 a_2 \over a_6^2} \cr}
$$
In this particular example
the constraint operators $Z_j$ have the expressions:
$$\eqalign{
Z_1 &= \sum_{i=0}^{7} \, a_i {\partial\over\partial a_i} +1 \cr
Z_2 &= 4 a_1 {\partial\over\partial a_1}
       - a_5 {\partial\over\partial a_5}
      +2 a_6 {\partial\over\partial a_6}
      +  a_7 {\partial\over\partial a_7} \cr
Z_3 &=  a_4 {\partial\over\partial a_4}
       - a_5 {\partial\over\partial a_5} \cr
Z_4 &= 2 a_1 {\partial\over\partial a_1}
       + a_3 {\partial\over\partial a_3}
      - a_5 {\partial\over\partial a_5}
      +  a_6 {\partial\over\partial a_6}
      +  a_7 {\partial\over\partial a_7} \cr
Z_5 &= 3 a_1 {\partial\over\partial a_1}
       + a_2 {\partial\over\partial a_2}
       - a_5 {\partial\over\partial a_5}
      +2 a_6 {\partial\over\partial a_6}
      +  a_7 {\partial\over\partial a_7} \cr}
$$
The $D_{\bf l}$ operators are:
$$\eqalign{
D_1 &= {\partial\over\partial a_1} {\partial\over\partial a_2}
      -\left({\partial\over\partial a_6} \right)^2 \cr
D_2 &= {\partial\over\partial a_4} {\partial\over\partial a_5}
       {\partial\over\partial a_7}
       -\left({\partial\over\partial a_0} \right)^3 \cr
D_3 &= {\partial\over\partial a_3} {\partial\over\partial a_6}
      -\left({\partial\over\partial a_7} \right)^2 \cr}
$$
If we parametrize the solutions of the above system as follows:
$$
 F({\bf a})= {1 \over a_0} {\tilde F}(\zeta_i)
$$
then $\tilde{F}( \zeta_i)$ is a solution of $D_i\, \tilde F = 0$, with
$$\eqalign{
{\cal D}_1 &= \theta_{3}^2 - \zeta_{3} \prod_{i=0}^{1} (\theta_{2} - 2
\theta_{3}-i) \cr
{\cal D}_2 &= \theta_{1}^2 (\theta_{1}-2 \theta_{2}) - 3^3 \zeta_{1}
\prod_{i=1}^{3} (\theta_{1} +i/3) \cr
{\cal D}_3 &= \theta_{2} (\theta_{2}- 2 \theta_{3}) - \zeta_{2}
\prod_{i=0}^{1} (\theta_{1} -2 \theta_{2} -i) \cr}
\eqlabel{d3}
$$
where $ \theta_i = \zeta_i {\partial \over \partial \zeta_i}$.

The dimension of the space of solutions for the $\Delta^*$-generalized
hypergeometric system is finite and can be calculated by two methods.

The first method uses the associated indicial equations.
In the corner of the moduli space where all the $\zeta_i$'s are large,
we obtain the following system of indicial equations (after the change
of variables $\zeta_i = {1 \over \tilde{\zeta}_i}$ ):
$$\eqalign{
\prod_{i=0}^{1} & (- \tilde{s}_2 +2 \tilde{s}_3 -i) = 0 \cr
\prod_{i=1}^{3} & (- \tilde{s}_1 +i/3) =0 \cr
\prod_{i=0}^{1} & (-\tilde{s}_1 +2 \tilde{s}_2 -i) =0~~. \cr}
\eqlabel{indeqsol}$$
The above system has $12$ solutions. It was obtained by acting with
the system of operators~\eqref{d3} on
$\tilde{F}(\tilde{\zeta}_i)$ given in the following form:
$$
 \tilde{F}(\tilde{\zeta}_i)~ =~
\tilde{\zeta}_1^{\tilde{s}_{1}}
\tilde{\zeta}_2^{\tilde{s}_{2}}
\tilde{\zeta}_3^{\tilde{s}_{3}}
\sum_{a,b,c=0}^{\infty}C_{abc}
\tilde{\zeta}_1^a\tilde{\zeta}_2^b
\tilde{\zeta}_3^c \quad \hbox{ for }~
\tilde{\zeta}_i \rightarrow 0 ~~.
\eqlabel{indeq}
$$

The other method of counting the number of solutions is by calculating
the volume of the dual cone \cite{\rgelfand}.
When the dual polyhedron is a simplex, the volume is the absolute value
of the determinant of the generators of the dual cone (the volume of
the $n$-dimensional cube is taken to be $n!$).
Otherwise, we can use the simplicial decomposition used to find the
generators of the Mori cone, and then add up the volumes of the
different simplices.

Taking the last two differential operators from \eqref{d3}, but dropping the
$\Theta_3$ term in the
second, we are left with the $\Delta^*$ hypergeometric system for
the $K3$ hypersurface
\cp{(1,1,2,2)}[6].
To see this we have to do the same type of analysis for
\cp{(1,1,2,2)}[6].
The dual polyhedron contains $6$ points:
$$\eqalign{
 \bar{\nu}_0^* &= (\- 1,\- 0,\- 0,\- 0) \cr
 \bar{\nu}_1^* &= (\- 1, - 1, - 1, - 1) \cr
 \bar{\nu}_2^* &= (\- 1,\- 1, - 1, - 1) \cr
 \bar{\nu}_3^* &= (\- 1,\- 0,\- 1,\- 0) \cr
 \bar{\nu}_4^* &= (\- 1,\- 0,\- 0,\- 1) \cr
 \bar{\nu}_5^* &= (\- 1,\- 0, - 1, - 1) \cr}
$$
out of which one  is interior
to a $1$-face, so we are going to study the family:
$$
 p = \sum_{i=1}^{4} a_i x_i^{6 \over k_i} + a_0 \prod_{i=1}^{4} x_i +
a_5 x_1^3 x_2^3
$$
The Mori cone has two generators:
$$\eqalign{
{\bf l}^1 &= ( - 3,\- 0,\- 0,\- 1,\- 1,\- 1)\cr
{\bf l}^2 &= (\- 0,\- 1,\- 1,\- 0,\- 0, -2)\cr}
$$
The large complex structure parameters are:
$$\eqalign{
\zeta_1 &= -{a_3 a_4 a_5 \over a_0^3} \cr
\zeta_2 &= {a_1 a_2 \over a_5^2} \cr}
\eqlabel{zeta2}
$$
In terms of $\theta_1 = \zeta_1 {\partial\over\partial \zeta_1}$ and
$\theta_2 = \zeta_2 {\partial\over\partial \zeta_2}$, the
differential operators are:
$$\eqalign{
{\cal D}_1 &= \theta_1^2 (\theta_1 -2 \theta_2)- 3^3 \zeta_1 \prod_{i=1}^3
(\theta_1 + i/3) \cr
{\cal D}_2 &= \theta_2^2 - \zeta_2 \prod_{i=0}^1 (\theta_1 -2 \theta_2
-i)\cr}
\eqlabel{d2}
$$
The system \eqref{d2} can be shown to have 6 linearly independent solutions.
We see that in the limit $\zeta_3 \rightarrow 0$ the system
\eqref{d3} reduces to \eqref{d2}, while if we also let $\zeta_2
\rightarrow  0$ the operator ${\cal D}_1$ reduces to the $\cal D$ operator in
\eqref{Dtorus}.

Obviously, one would expect the solutions for these hypergeometric systems to
obey the same rule, {\it i.e.,\ \/} taking the limit $\zeta_2\rightarrow 0$ in the
semi-period expression for $K3$, one would get the semi-period expresion for
$T^2$. Similarly, the limit $\zeta_3\rightarrow 0$ in the semi-periods for
\cp{(1,1,2,4,4)}[12] will yield $K3$ semi-periods. To show this, we
will analytically continue our expressions to the regions where the desired
parameter is small, and then take the zero limit of that parameter.

First, we need to calculate semi-periods.
Working in the region of the moduli space where $p_0$ represents the Fermat
part of the defining polynomial,
and dropping some numerical factors, one has the following:\hfil\break
--for the torus
$$ \eqalign{
  F_{T^2}
           &= {1\over a_0}\, \sum_{m=0}^\infty\,
         \lambda}\def\L{\Lambda^{(\delta}\def\D{\Delta_2+\delta}\def\D{\Delta_3)(m+1)}\,
         \left({1 \over \zeta_1}\right)^{{m+1\over 3}}
         {(-1)^m \over m!}\,
         \Gamma^3 \left({m+1\over 3}\right)~;\cr}
\eqlabel{spt2}$$
--for the $K3$
$$ \eqalign{
  F_{K3}
           &= {1\over a_0}\, \sum_{m=0}^\infty
         \lambda}\def\L{\Lambda^{(\delta}\def\D{\Delta_2+2\delta}\def\D{\Delta_3+2\delta}\def\D{\Delta_4)(m+1)}\,
         \left({1 \over \zeta_1}\right)^{{m+1\over 3}}
         {(-1)^m\over m!}\, \Gamma^2 \left({m+1\over 3}\right)\cr
&\hskip10pt \times
         \left({1 \over \zeta_2}\right)^{{m+1\over 6}}
        \sum_{n=0}^\infty
         \lambda}\def\L{\Lambda^{3 n \delta}\def\D{\Delta_2}\,
         \left({1 \over \zeta_2}\right)^{{n\over 2}}
         {(-1)^n\over n!}
       \Gamma^2 \left({m+3n+1\over 6}\right)~;\cr}
\eqlabel{spk3}$$
--for the Calabi--Yau
$$\eqalign{
  F_{CY}
           &= {1\over a_0}\, \sum_{m=0}^\infty
         \lambda}\def\L{\Lambda^{(\delta}\def\D{\Delta_2+2\delta}\def\D{\Delta_3+4\delta}\def\D{\Delta_4+4\delta}\def\D{\Delta_5)(m+1)}\,
         \left({1 \over \zeta_1}\right)^{{m+1\over 3}}
         {(-1)^m\over m!}\, \Gamma^2 \left({m+1\over 3}\right)\cr
&\hskip10pt \times
         \left({1 \over \zeta_2}\right)^{{m+1\over 6}}
      \sum_{n=0}^\infty
         \lambda}\def\L{\Lambda^{3 n (\delta}\def\D{\Delta_2+2\delta}\def\D{\Delta_3)}\,
         \left({1 \over \zeta_2}\right)^{{n\over 2}}
         {(-1)^n\over n!}
       \Gamma \left({m+3n+1\over 6}\right)\cr
&\hskip10pt \times
        \left({1 \over \zeta_3}\right)^{{m+3n+1\over 12}}
      \sum_{p=0}^\infty
         \lambda}\def\L{\Lambda^{6 p \delta}\def\D{\Delta_2}\,
        \left({1 \over \zeta_3}\right)^{{p\over 2}}
         {(-1)^p\over p!}
             \Gamma^2 \left({m+3n+6p+1\over 12}\right)~.\cr}
\eqlabel{spcy}$$
Here $\lambda}\def\L{\Lambda = e^{2\pi i \over d}$,
$d$ being the degree of the polynomial (3, 6 or 12 respectively),
and $\delta}\def\D{\Delta_i = 0,1,...{d \over k_i}-1$.
This means our contours connect the
points $0,{\lambda}\def\L{\Lambda^{k_i\delta}\def\D{\Delta_i}} \infty$ in the $x_i$ plane.
Not all different chains lead to different semi-periods;
{}from the above equations one can see the number of linearly independent
semi-periods that can be obtained is 3 for the
torus, 6 for $K3$ and 12 for the three-dimensional Calabi--Yau manifold .
These then are all the solutions of the respective hypergeometric equations.
To see that the solutions \eqref{spcy} are of the type
\eqref{indeq}, one can break up the summations in \eqref{spcy}
over $m,$ $n$ and $p$
such that
 $m = 3\tilde m + i$ for $i=0,1,2,$ $n = 2\tilde n + j$ for
$j=0,1,$ and $p = 2\tilde p + k$ for $k=0,1$.
Now the summation over $m$
becomes a summation from $0$ to $\infty$ over $\tilde m$
and a summation from $0$ to $2$ for $i$,
and likewise for the others.
Now it is clear that the semi-period separates into
12 summations with different leading order behaviours as
the $1/\zeta_i \rightarrow 0$.
These leading orders are those predicted by the
solutions of \eqref{indeqsol}.

{}For convenience in doing the analytic continuation
we set $\delta}\def\D{\Delta_2=0$ and focus on the $\zeta_3$ dependent part of \eqref{spcy}.
Its Barnes integral representation is as follows
{}~\Ref{\rSGI}{G.~Sansone and J.~Gerretsen, {\it Lectures on the Theory of
Functions of a Complex Variable} (Wolters-Noorhoff, Groningen 1969, the
Netherlands).}:
$$ \eqalign{
   {-1\over 2 \pi i} \int_0^{i \infty}
{\rm d} s \left({1\over \zeta_3}\right)^{{m+3n+6s+1\over 12}}\Gamma(-s)
\Gamma^2 \left({m+3n+6s+1\over 12}\right)~.\cr}
\eqlabel{Barnes}$$
The poles of $\Gamma(-s)$ lie along the positive axis. If the contour of
integration encloses these poles, then we obtain our starting expression,
since for $s\rightarrow -p$
$${\rm Res}\, \Gamma(s) ={(-1)^{p+1}\over p!}~. $$
The poles of $\Gamma^2 \left({m+3n+6s+1\over 12}\right)$ lie at
${m+3n+6s+1\over 12} = -k$, where $k = 0,1,2,3...$. If we enclose these poles
in the integrating contour ({\it i.e.,\ \/} close the contour to the left)
the value of the integral \eqref{Barnes} will be:
$${- \sum_{k=0}^\infty \Gamma\left({12k+m+3n+1\over 6}\right) \left({1 \over
\zeta_3}\right)^{-k} \left({1 \over k!}\right)^{2} 2 \psi}\def\Ps{\Psi_{1}(1+k)~.}
\eqlabel{ksum}$$
Here \cite{\rErdelyi} \
$$ \psi}\def\Ps{\Psi_{1}(k+1)=1+{1\over 2}+{1\over 3}+...+{1\over k}-\gamma}\def\G{\Gamma~,$$
and $\gamma}\def\G{\Gamma$ is the Euler-Mascheroni constant.
We have also used the fact that
$$\Gamma(-k+\epsilon}\def\ve{\varepsilon)={(-1)^k \over k!}{({1\over \epsilon}\def\ve{\varepsilon}+{\psi}\def\Ps{\Psi_{1}(1+k)}+{\it O} (\epsilon}\def\ve{\varepsilon))}$$
Thus we see that the limit $\zeta_3 \rightarrow 0$ in \eqref{spcy} will give
an expression proportional to \eqref{spk3}.

In a similar manner, one can show that
by analytically continuing \eqref{spk3} to small $\zeta_2$,
the limit $\zeta_2 \rightarrow 0$  is proportional to \eqref{spt2}.

 %
\vfill\noindent
{\bf Acknowledgments}:
The authors gratefully acknowledge much input and
guidance from Philip Candelas. We also want to thank Tristan H\"ubsch
and Mihaela Manoliu for interesting discussions.
A.C.A. and D.~J. were supported by
NSF grant PHY 9009850, PHY 9511632 and the Robert~A.~Welch Foundation.
E.~D. was supported by the Alexander von Humboldt Foundation.
E.~D. would like to thank  Howard University for hospitality while part of this
work was germinating.
Some of this work was carried out while E.~D. was a member of
the Theory Group at the University of Texas.

\vfill\eject

\listreferences

\bye