Chiral Rings Do Not Suffice: N=(2,2) Theories with Nonzero Fundamental
  Group

Abstract:
  The Kahler moduli space of a particular non-simply-connected Calabi-Yau
manifold is mapped out using mirror symmetry. It is found that, for the model
considered, the chiral ring may be identical for different associated conformal
field theories. This ambiguity is explained in terms of both A-model and
B-model language. It also provides an apparent counterexample to the global
Torelli problem for Calabi-Yau threefolds.

\section*{Abstract}
\else \small
\begin{center}
{\bf ABSTRACT}
\end{center}
\quotation
\fi}

\def\thebibliography#1{\section*{References\@mkboth
 {REFERENCES}{REFERENCES}}\small\list
 {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth
 \advance\leftmargin\labelsep
 \usecounter{enumi}}
 \def\hskip .11em plus .33em minus .07em{\hskip .11em plus .33em minus .07em}
 \sloppy\clubpenalty4000\widowpenalty4000
 \sfcode`\.=1000\relax}
\let\endthebibliography=\endlist

\newif\iffn\fnfalse

\@ifundefined{reset@font}{\let\reset@font\empty}{}
\long\def\@footnotetext#1{\insert\footins{\reset@font\footnotesize
    \interlinepenalty\interfootnotelinepenalty
    \splittopskip\footnotesep
    \splitmaxdepth \dp\strutbox \floatingpenalty \@MM
    \hsize\columnwidth \@parboxrestore
   \edef\@currentlabel{\csname p@footnote\endcsname\@thefnmark}\@makefntext
    {\rule{\z@}{\footnotesep}\ignorespaces
      \fntrue#1\fnfalse\strut}}}
\makeatother

\ifamsf
  \newfont{\bf}{msbm10 scaled\magstep2}
  \newfont{\bbbfont}{msbm10 scaled\magstep1}
  \newfont{\smallbbbfont}{msbm8}
  \newfont{\tinybbbfont}{msbm6}
  \newfont{\smallfootbbbfont}{msbm7}
  \newfont{\tinyfootbbbfont}{msbm5}
\fi

\ifscrf
  \newfont{\scrfont}{rsfs10 scaled\magstep1}
  \newfont{\smallscrfont}{rsfs7}
  \newfont{\tinyscrfont}{rsfs7}
  \newfont{\smallfootscrfont}{rsfs7}
  \newfont{\tinyfootscrfont}{rsfs7}
\fi

\ifamsf
  \newcommand{\bf}[1]{\iffn
      \mathchoice{\mbox{\footbbbfont #1}}{\mbox{\footbbbfont #1}}
      {\mbox{\smallfootbbbfont #1}}{\mbox{\tinyfootbbbfont #1}}\else
      \mathchoice{\mbox{\bbbfont #1}}{\mbox{\bbbfont #1}}
      {\mbox{\smallbbbfont #1}}{\mbox{\tinybbbfont #1}}\fi}
\else
  \def\bf{\bf}
  \def\bf{\bf}
\fi

\ifscrf
  \newcommand{\cal}[1]{\iffn
    \mathchoice{\mbox{\footscrfont #1}}{\mbox{\footscrfont #1}}
    {\mbox{\smallfootscrfont #1}}{\mbox{\tinyfootscrfont #1}}\else
    \mathchoice{\mbox{\scrfont #1}}{\mbox{\scrfont #1}}
    {\mbox{\smallscrfont #1}}{\mbox{\tinyscrfont #1}}\fi}
\else
  \def\cal{\cal}
\fi

\def\operatorname#1{\mathop{\rm #1}\nolimits}
\def{\Bbb C}{{\bf C}}
\def{\cal F}{{\cal F}}
\def{\cal O}{{\cal O}}
\def{\Bbb P}{{\bf P}}
\def{\Bbb Q}{{\bf Q}}
\def{\Bbb R}{{\bf R}}
\def{\Bbb Z}{{\bf Z}}
\def\operatorname{Aut}{\operatorname{Aut}}
\def\mathop{\widetilde{\rm Aut}}\nolimits{\mathop{\widetilde{\rm Aut}}\nolimits}
\def\operatorname{Hom}{\operatorname{Hom}}
\def\operatorname{Ext}{\operatorname{Ext}}
\def\operatorname{Ker}{\operatorname{Ker}}
\def\ |\ {\ |\ }
\def\operatorname{Spec}{\operatorname{Spec}}
\def\operatorname{Area}{\operatorname{Area}}
\def\operatorname{Vol}{\operatorname{Vol}}
\def\operatorname{gen}{\operatorname{gen}}
\def\operatorname{div}{\operatorname{div}}
\def\operatorname{Div}{\operatorname{Div}}
\def\operatorname{WDiv}{\operatorname{WDiv}}
\def\operatorname{ad}{\operatorname{ad}}
\def\operatorname{tr}{\operatorname{tr}}
\def\operatorname{CPL}{\operatorname{CPL}}
\def\operatorname{cpl}{\operatorname{cpl}}
\def\operatorname{Im}{\operatorname{Im}}

\def\opeq#1{\advance\lineskip#1 \advance\baselineskip#1
	\advance\lineskiplimit#1}
\def\eqalign#1{\null\,\vcenter{\opeq{2.5\jot}\mathsurround=0pt
	\everycr={}\tabskip=0pt
	\halign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
	\crcr#1\crcr}}\,\null}

\def$\sigma$-model{$\sigma$-model}
\defnon-linear \sm{non-linear $\sigma$-model}
\def\sm\ measure{$\sigma$-model\ measure}
\defCalabi-Yau{Calabi-Yau}
\defK\"ahler form{K\"ahler form}
\defLandau-Ginzburg{Landau-Ginzburg}
\def{\Scr R}{{\cal R}}
\def{\Scr M}{{\cal M}}
\def{\Scr A}{{\cal A}}
\def{\Scr K}{{\cal K}}
\def{\Scr D}{{\cal D}}
\def{\Scr Z}{{\cal Z}}
\defS_{\hbox{\scriptsize LG}}{S_{\hbox{\scriptsize LG}}}
\def\cM{{\Scr M}}
\def{\hfuzz=100cm\hbox to 0pt{$\;\overline{\phantom{X}}$}\cM}{{\hfuzz=100cm\hbox to 0pt{$\;\overline{\phantom{X}}$}{\Scr M}}}
\defalgebraic measure{algebraic measure}
\def\ff#1#2{{\textstyle\frac{#1}{#2}}}
\def{\cal F}#1#2{{}_{#1}F_{#2}}

\newcommand{\text}[1]{\mathchoice{\mbox{\rm #1}}{\mbox{\rm #1}}
    {\mbox{\scriptsize\rm #1}}{\mbox{\tiny\rm #1}}}

\ifscrf
  \def\Tch#1#2{{\cal T}^{\;\,#1,#2}}
\else
  \def\Tch#1#2{{\cal T}^{#1,#2}}
\fi

\ifamsf
  \def.{\mathbin{\mbox{\bbbfont\char"6E}}}
  \def\hbox{does not divide}{\mathbin{\mbox{\bbbfont\char"2D}}}
\else
  \def.{.}
  \def\hbox{does not divide}{\hbox{does not divide}}
\fi

\nofiles
\makeatletter
\relax
\citation{CHSW:}
\citation{CHSW:}
\citation{LVW:}
\citation{W:AB}
\@writefile{toc}{\string\contentsline\space {section}{\string\numberline\space
{1}Introduction}{1}}
\newlabel{s:intro}{{1}{1}}
\newlabel{eq:Fermat}{{1}{1}}
\newlabel{eq:Ggen}{{2}{1}}
\citation{Vafa:vacua}
\citation{N:torus,AM:K3p}
\citation{DW:gp}
\citation{DW:gp,SW:spin,Vafa:tor}
\@writefile{toc}{\string\contentsline\space {section}{\string\numberline\space
{2}A-model Analysis}{2}}
\newlabel{s:A}{{2}{2}}
\citation{W:phase,AGM:II}
\citation{BT:}
\citation{EM:hom}
\citation{EM:hom}
\newlabel{eq:ES2}{{5}{3}}
\newlabel{eq:ES3}{{6}{3}}
\citation{Vafa:tor}
\citation{GP:orb}
\citation{ALR:coup}
\citation{GP:orb}
\citation{CDGP:}
\citation{CDGP:}
\citation{CDGP:}
\citation{CDGP:}
\newlabel{eq:ES1}{{7}{4}}
\@writefile{toc}{\string\contentsline\space {section}{\string\numberline\space
{3}The B-model of the Mirror}{4}}
\newlabel{s:B}{{3}{4}}
\newlabel{eq:Bmod}{{9}{4}}
\citation{CDGP:}
\citation{CDGP:}
\citation{BCOV:big}
\citation{BCOV:big}
\citation{Mor:cp}
\newlabel{eq:Bmodnew}{{10}{5}}
\citation{CDGP:}
\citation{CDGP:,AM:rat}
\@writefile{toc}{\string\contentsline\space {section}{\string\numberline\space
{4}Some curve counting}{6}}
\newlabel{s:count}{{4}{6}}
\newlabel{eq:rcX}{{14}{6}}
\citation{BCOV:ell,BCOV:big}
\citation{BCOV:ell}
\citation{BCOV:ell,CDFKM:I,CFKM:II}
\citation{Katz:m}
\newlabel{eq:F1q}{{17}{7}}
\citation{Hulek:}
\citation{GH:alg}
\citation{Hulek:}
\newlabel{eq:cvs}{{21}{8}}
\newlabel{eq:param}{{22}{8}}
\newlabel{eq:condition}{{23}{8}}
\citation{BCOV:ell,CDFKM:I,CFKM:II}
\citation{Tod:WP}
\newlabel{eq:ecX}{{26}{9}}
\newlabel{eq:dcount}{{27}{9}}
\@writefile{toc}{\string\contentsline\space {section}{\string\numberline\space
{5}Conclusions}{9}}
\newlabel{s:conc}{{5}{9}}
\citation{Vafa:vacua}
\citation{Vafa:tor}
\citation{me:orb2}
\bibcite{CHSW:}{1}
\bibcite{LVW:}{2}
\bibcite{W:AB}{3}
\bibcite{Vafa:vacua}{4}
\bibcite{N:torus}{5}
\bibcite{AM:K3p}{6}
\bibcite{DW:gp}{7}
\bibcite{SW:spin}{8}
\bibcite{Vafa:tor}{9}
\bibcite{W:phase}{10}
\bibcite{AGM:II}{11}
\bibcite{BT:}{12}
\bibcite{EM:hom}{13}
\bibcite{GP:orb}{14}
\bibcite{ALR:coup}{15}
\bibcite{CDGP:}{16}
\bibcite{BCOV:big}{17}
\bibcite{Mor:cp}{18}
\bibcite{AM:rat}{19}
\bibcite{BCOV:ell}{20}
\bibcite{CDFKM:I}{21}
\bibcite{CFKM:II}{22}
\bibcite{Katz:m}{23}
\bibcite{Hulek:}{24}
\bibcite{GH:alg}{25}
\bibcite{Tod:WP}{26}
\bibcite{me:orb2}{27}
\makeatother

\begin{document}
\setcounter{page}0
\title{\large\bf Chiral Rings Do Not Suffice:\\
$N$=(2,2) Theories with Nonzero\\
Fundamental Group\\[10mm]
}
\author{
Paul S. Aspinwall,\\[0.7cm]
\normalsize School of Natural Sciences,\\
\normalsize Institute for Advanced Study,\\
\normalsize Princeton, NJ  08540\\[10mm]
David R. Morrison \\[0.7cm]
\normalsize  Department of Mathematics, \\
\normalsize  Box 90320, \\
\normalsize  Duke University, \\
\normalsize  Durham, NC 27708-0320}

{\hfuzz=10cm\maketitle}

\def\large{\large}
\def\large\bf{\large\bf}

\begin{abstract}

The K\"ahler moduli space of a particular
non-simply-connected Calabi-Yau
manifold is mapped out using mirror symmetry. It is found that, for
the model considered, the chiral ring may be identical for different
associated conformal field theories. This ambiguity is explained in
terms of both A-model and B-model language.
It also provides an apparent counterexample to the global Torelli problem
for Calabi-Yau threefolds.

\end{abstract}

\vfil\break

\section{Introduction}		\label{s:intro}

One of the very first Calabi-Yau\ spaces to be considered for a superstring
compactification was defined as follows \cite{CHSW:}. Take the complex
projective space ${\Bbb P}^4$ with homogeneous coordinates
$[z_0,z_1,\ldots,z_4]$ and construct a quintic hypersurface within it
{}from the condition
\begin{equation}
  p=z_0^5+z_1^5+z_2^5+z_3^5+z_4^5=0.	\label{eq:Fermat}
\end{equation}
Let us denote this space by $Q$. Now define a Calabi-Yau\ space as
 $X=Q/G$ where $G$ is the freely acting group of identifications
isomorphic to ${\Bbb Z}_5\times{\Bbb Z}_5$ generated by
\begin{equation}
  \eqalign{g_1:[z_0,z_1,z_2,z_3,z_4]&\mapsto
	[z_0,\zeta z_1,\zeta^2z_2,\zeta^3z_3,\zeta^4z_4],
	\qquad \zeta=e^{2\pi i/5},\cr
	g_2:[z_0,z_1,z_2,z_3,z_4]&\mapsto
	[z_1,z_2,z_3,z_4,z_0].}	\label{eq:Ggen}
\end{equation}
The smooth Calabi-Yau\ manifold $X$ has Euler characteristic $-8$ and can
thus be used to build a four generation model. Although these days,
four generations is not considered an attractive feature of a model, it
will still prove interesting to study the space $X$ and, in
particular, the moduli space of $N$=(2,2) superconformal field theories
containing a point corresponding to this manifold.

$N$=(2,2) superconformal field theories are used to represent a Calabi-Yau\
compactification with the ``spin connection embedded in the gauge
group'' \cite{CHSW:}. The moduli space of such conformal field
theories, i.e., the moduli space of string vacua of this type is
actually quite simple to analyze in many cases. One of the reasons for
this is the existence of the ``chiral ring'' (see, for
example, \cite{LVW:}). To analyze a generic conformal field theory it
is usual to concentrate on the set of primary fields, which unfortunately are
infinite in number.\footnote{If the conformal
field theory is rational, the number of primary fields
will be finite and the analysis
may not be so difficult. However, it is believed that such rational conformal
field theories do not deform and thus can, at best, form only a
countable dense subset of the space of conformal field theories
(like the set of rational numbers in the space of
reals).} However, in the case of $N$=(2,2)
superconformal field theories we may look at two distinguished
 subsets of the primary
fields, the {\em chiral\/} primary fields and the {\em antichiral\/}
primary fields.  (The difference between the definitions
is simply a change in the sign of the $U(1)$ charge). The advantage of this is
that the number of (anti)chiral primary fields is finite, and remains constant
over the moduli space. This makes analysis of the set of such fields much
easier.  Furthermore, the ``na\"{\i}ve product''
\begin{equation}
(\phi\chi)(z)=\lim_{z'\to z}\phi(z')\chi(z)
\end{equation}
of two chiral
primary fields is again a chiral primary field, giving the set of such
fields the structure of a ring.  The structure constants of this ring
can be found by calculating
 two point and  three point functions.

An $N$=(2,2) superconformal field theory may be twisted to form a
topological field theory. This may be done in two inequivalent ways
\cite{W:AB} yielding the ``A-model'' or ``B-model''. The observables
in each of these models correspond to the (anti)chiral primary fields of the
original $N$=(2,2) theory, the difference amounting to a choice of
sign of one of the $U(1)$ charges in the $N$=(2,2) superconformal
algebra. Deformations of the $N$=(2,2) superconformal field theory can
generated by (anti)chiral primary fields and thus map into
deformations of the A-model
and B-model, where they are detected by the ring structures.
It is tempting to speculate that analyzing the
two point and three
point functions of the observables in the two topological field
theories gives sufficient information to classify completely the
$N$=(2,2) theory. That is, the two chiral rings of the conformal
field theory might contain sufficient information to obtain all other
correlation functions from them.

A simple counterexample to this proposition is the
complex 3-torus \cite{Vafa:vacua}. In
this case the A-model (having no instantons at tree-level) is too
trivial and contains no local
information. One possible explanation for this is that
in the case of the torus,
 the local supersymmetry is actually $N$=(4,4). Whenever this
happens the moduli space takes on a significantly different form which
no longer splits naturally into A-model and B-model part (see, for
example, \cite{N:torus,AM:K3p}). One might thus modify the proposal to
apply to theories with only $N$=(2,2) symmetry. We shall see from our
example that this too fails---for global rather than local reasons.

In section \ref{s:A} we will discuss the A-model on $X$. In section
\ref{s:B} the moduli space will be formulated by considering the
B-model on the mirror of $X$. To help understand the form of the
moduli space we will consider rational and elliptic curves on $X$ in
section \ref{s:count} and finally in section \ref{s:conc} we will
present concluding remarks.

\section{A-model Analysis}   	\label{s:A}

In terms of the geometric data of the target space, the B-model
captures the information concerned with the complex structure. The
A-model moduli however are concerned with variation of the
complexified K\"ahler form. By expressing the constraint in the form
(\ref{eq:Fermat}) we have effectively fixed the complex structure and
thus frozen the B-model data. Thus we will concentrate only on the
A-model.

In the conventional approach to the A-model one describes a non-linear \sm\
whose target space is equipped with a K\"ahler metric
given by a K\"ahler form $J$ and
a real 2-form, $B$. These may be combined to form a complex 2-form
$B+iJ$ upon whose cohomology class the A-model depends. This
description does not capture all the possibilities however.

For a map $\phi:\Sigma\to X$ from the world-sheet into the target
space, the A-model correlation functions vary as a function of
\begin{equation}
  \xi = \exp(2\pi i\int_\Sigma \phi^*(B+iJ)),
\end{equation}
where $\phi^*$ is the pull-back. Thus the degree of freedom
represented by $B+iJ$ can be thought of as an element of
$\operatorname{Hom}(H_2(X),{\Bbb C}^*)$. That is, it associates some non-zero complex
number, $\xi$, to each homology class of the image of $\Sigma$ in
$X$. The group $\operatorname{Hom}(H_2(X),{\Bbb C}^*)$ may contain more freedom than that
described by the complex 2-form
$B+iJ$. It is possible that the singular homology group
$H_2(X)$ contains {\em torsion}, i.e., there is an element $\tau\in
H_2(X)$ such that $N\tau\sim0$ for some integer $N$. Using only de
Rham cohomology in the form of $B+iJ$ will then miss the corresponding
torsion elements of $\operatorname{Hom}(H_2(X),{\Bbb C}^*)$. Since it would seem natural
to allow for torsion elements of $H_2(X)$ to be associated with a
non-trivial $\xi$ it would appear that the A-model moduli space is
better described as $\operatorname{Hom}(H_2(X),{\Bbb C}^*)$ rather than the potentially
smaller space of $B+iJ$'s.
This is closely related to issues studied in \cite{DW:gp}.
This form of the moduli space can also be justified by looking at
fundamental properties of maps of the world-sheet into the target space
\cite{DW:gp,SW:spin,Vafa:tor}.

It is important to note that the A-model picture presented here
depends on which ``phase'' of moduli space we are in in the sense of
\cite{W:phase,AGM:II}. We will assume that we are in some
neighbourhood of the large radius limit of $X$ and thus in the Calabi-Yau\
phase. When one leaves this phase, the moduli space will no longer
appear to be in the form $\operatorname{Hom}(H_2(X),{\Bbb C}^*)$.

For the quintic threefold, $Q$, $\dim H^2_{\rm DR}(Q)=1$ and there is
no torsion in $H_2(Q)$. Thus, in the neighbourhood of the large radius
limit (where the A-model is well-defined) the moduli space is locally
isomorphic to $\operatorname{Hom}(H_2(Q),{\Bbb C}^*)\cong {\Bbb C}^*$. Since $Q$ is a simply-connected
non-ramified cover of $X$, we have that
$\pi_2(X)\cong\pi_2(Q)\cong H_2(Q)\cong{\Bbb Z}$ \cite{BT:}.
Since $X=Q/G$ we also have that $\pi_1(X) \cong G\cong {\Bbb Z}_5\times{\Bbb Z}_5$.
Given $\pi_1(X)$ and $\pi_2(X)$ we may calculate $H_2(X)$ and consequently
$\operatorname{Hom}(H_2(X),{\Bbb C}^*)\cong H^2(X,{\Bbb C}^*)$ by the method of Eilenberg and MacLane
\cite{EM:hom}.
According to \cite{EM:hom} there is an exact sequence
\begin{equation}
  0\to \pi_2(X) \to H_2(X) \to H_2(G) \to 0
\label{eq:ES2}
\end{equation}
which relates the homotopy and homology groups of $X$ to the group
homology $H_2(G)$.  In the present case, the group homology can
be calculated as
$H_2({\Bbb Z}_5\times{\Bbb Z}_5)\cong{\Bbb Z}_5$, and the exact sequence (\ref{eq:ES2})
becomes
\begin{equation}
  0\to{\Bbb Z}\to H_2(X)\to{\Bbb Z}_5\to 0.	\label{eq:ES3}
\end{equation}
There are two possibilities for $H_2(X)$ compatible with (\ref{eq:ES3}),
depending on whether the exact sequence splits:
either $H_2(X)\cong{\Bbb Z}\times{\Bbb Z}_5$, or $H_2(X)\cong{\Bbb Z}$.  In either case
there will be a homology class $e$ which is not represented by
a sphere; in the former case, $e$ may be chosen so that $5e\sim0$
and in the latter so that $5e$ generates $\pi_2(X)\cong H_2(Q)$.
(These two possibilities are mutually exclusive.)

In fact, it is the second possibility $H_2(X)\cong{\Bbb Z}$ which occurs for
our example.  We will show this in section 4 by exhibiting an elliptic
curve $E$ on $X$ whose inverse image $\pi^{-1}(E)$ on $Q$ is an irreducible
elliptic curve of degree $5$.  The homology class $e$ of $E$ cannot
lie in $\pi_2(X)$, since for every rational curve $\Gamma$ on $X$,
the inverse image $\pi^{-1}(\Gamma)$ consists of 25 disjoint rational
curves, all of the same degree $d$, so that the degree of $\pi^{-1}(\Gamma)$
is a multiple of 25.

The group $H^2(X,{\Bbb C}^*)\cong\operatorname{Hom}(H_2(X),{\Bbb C}^*)$
which describes the degree of freedom
represented by $B+iJ$ fits in an exact sequence of its own
\begin{equation}
  0\to{\Bbb Z}_5\to H^2(X,{\Bbb C}^*)\mathrel{\mathop\to^\alpha}{\Bbb C}^*\to 0,	\label{eq:ES1}
\end{equation}
whose structure is easily deduced from that of (\ref{eq:ES3}).
In particular, $H^2(X,{\Bbb C}^*)\cong{\Bbb C}^*$ and
the map $\alpha$ is a five-fold cover.

This
occurrence of $H^2(G,{\Bbb C}^*)$ when modding out a space by a group action
$G$ was first discussed in the context of string theory in
\cite{Vafa:tor}\footnote{In that paper only the
$B$-field is discussed leading to the equivalent group $H^2(G,U(1))$.}
where it was given the name ``discrete torsion''. Note that in our
example this name is somewhat misleading since $H^2(X,{\Bbb C}^*)$ is torsion
free.

\section{The B-model of the Mirror}	\label{s:B}

Given $X$ we now hope to find another space $Y$, which is the ``mirror'' of
$X$, such that the A-model on $X$ is equivalent to the B-model on $Y$.
To be more precise, $Y$ will actually be in the Landau-Ginzburg phase
rather than Calabi-Yau\ phase but the phase picture is not important for the
B-model and we may imagine $Y$ to be a Calabi-Yau\ manifold for all practical
purposes.
In the case of the quintic and many of its quotients the mirror is
given by the method of \cite{GP:orb}. $X$ does not quite
fall into this class but we may use an extension of this
method \cite{ALR:coup}
to find the mirror. Define  $G_1$, isomorphic to
${\Bbb Z}_5$, to be the group generated by the element $g_1$  defined in
eq.~(\ref{eq:Ggen}). By the
method of \cite{GP:orb},
the manifold $Q/G_1$ is
known to be mirror to $Q/\widetilde{G_1}$, where
$\widetilde{G_1}\cong{\Bbb Z}_5\times{\Bbb Z}_5$ is
generated by $g_1$ and another element $g_3$ defined by
\begin{equation}
  g_3:[z_0,z_1,z_2,z_3,z_4]\mapsto
	[z_0,\zeta z_1,\zeta^3z_2,\zeta z_3,z_4],
	\qquad \zeta=e^{2\pi i/5}.
\end{equation}
$g_2$ now acts on the mirror pair of theories corresponding to the
spaces $Q/G_1$ and
$Q/\widetilde{G_1}$ in precisely the same manner
 (once the mirror map is taken into
account). Thus we may divide both spaces (i.e., orbifold both
conformal field theories) by the group generated by this action to
yield another mirror pair. This pair consists of $X$ and $Y\cong
Q/\widetilde{G}$ where $\widetilde{G}\cong{\Bbb Z}_5.({\Bbb Z}_5)^2$ and is
generated by $g_1$, $g_2$ and $g_3$.

The moduli space of the B-model consists of varying the complex
structure of $Y$ which may be done by varying the defining equation
(\ref{eq:Fermat}) of $Q$.
There is only one deformation compatible with
the group $\widetilde{G}$ and we follow
\cite{CDGP:} by using the following parametrization
\begin{equation}
  p_\psi=z_0^5+z_1^5+z_2^5+z_3^5+z_4^5-5\psi z_0z_1z_2z_3z_4=0.
		\label{eq:Bmod}
\end{equation}

Further analysis of the moduli space and the mirror map between this
B-model and the A-model of section \ref{s:A} is now very close to the
analysis of \cite{CDGP:} where the A-model considered was that
associated to $Q$. The only difference between the B-model considered
here and that of \cite{CDGP:} is that a different 125 element group of
symmetries (call it $\widehat G$)
is used to divide $Q$, although both lead to the same
general form of defining equation (\ref{eq:Bmod}). Little of the
analysis of \cite{CDGP:} depends on the exact form of the group
dividing $Q$ and so can be copied over to the case considered here.

In \cite{CDGP:} it was observed that the family of Calabi-Yau\ manifolds
described by (\ref{eq:Bmod}) admits a symmetry $R$ defined by
$z_0\to\zeta z_0$, $z_i\to z_i$ for $i>0$, and $\psi\to\zeta^{-1}\psi$.
This symmetry establishes an isomorphism between the Calabi-Yau\ manifolds
at $\psi$ and at $\zeta\psi$, and shows that the correct parameter
for this family is in fact $\psi^{-5}$ rather than $\psi$.
(This parameter can be seen directly by changing coordinates via
$z_0=\psi^{-1}\tilde z$ to give a new defining equation
\begin{equation}
  \psi^{-5}\tilde z^5+z_1^5+z_2^5+z_3^5+z_4^5-5\tilde z z_1z_2z_3z_4=0
		\label{eq:Bmodnew}
\end{equation}
in which the parameter is visibly $\psi^{-5}$.)
When forming a quotient of this family by $\widehat G$ to obtain
the mirror family of $Q$, $R$ remains a symmetry of the quotient---in
fact, $\widehat G$ acts on (\ref{eq:Bmodnew}) equally well as
on (\ref{eq:Bmod}).  However, when forming the quotient by our
group $\widetilde G$ which includes a permutation, $R$ is no
longer a symmetry---it does not normalize the group $\widetilde G$,
nor does it preserve the alternate form (\ref{eq:Bmodnew}) of
the defining equation.

This immediately tells us that the chiral ring of $X$ has ``lost'' some
information concerning the conformal field theory. The chiral ring as
calculated in \cite{CDGP:} is a function of $\psi^5$. However since
$R$ is not a symmetry of $Y$ we expect the points given by
$\psi,\zeta\psi,\zeta^2\psi,\ldots$ to correspond to different
conformal field theories.\footnote{It might be objected that there
could be some other symmetry, not manifest in the present description,
which produces an isomorphism between the theory at $\psi$ and
the theory at $\zeta\psi$.  The calculations in the next section will
demonstrate that this is not the case.}

In order to construct the mirror map between the A- and B-models we
first require a set of ``flat'' coordinates on the B-model moduli
space (see \cite{BCOV:big} for a full discussion of this issue). This
is obtained by considering the variation of Hodge structure on $Y$.
That is, we consider the periods $\varpi_i=\int_{\gamma_i}\Omega$
where $\Omega$ is a (3,0)-form and $\gamma_i$ are elements of
$H_3(Y)$.

Let us use $q$ to denote the image of $e$, the fundamental generator
of $H_2(X)$, under the action of an element of $\operatorname{Hom}(H_2(X),{\Bbb C}^*)$.
The mirror map then relates the A- and B-model moduli spaces by
relating $q$ to $\psi$. The local geometry of the $N$=(2,2) moduli
space tells us that \cite{BCOV:big}
\begin{equation}
  q = \exp\left(\frac{\varpi_1(\psi)}{\varpi_0(\psi)}\right),
\end{equation}
for two suitably chosen periods $\varpi_0$ and $\varpi_1$. To find
exactly which periods to use one must look at global considerations of
$H_3(Y)$ as discussed in \cite{Mor:cp}. Rather than use this method
directly we may use a simple argument as follows. By mapping the
correlation functions of the A-models of $X$ and $Q$ to each other we
obtain a five-fold cover of the moduli space of $Q$ by the moduli
space of $X$. This cover is branched at $q=0$, which represents the
large radius limit.  A path once around this
point in the $X$ moduli space corresponds to changing $B$ by the generator
of $H^2(X,{\Bbb Z})$.   But since $e_Q$, the generator of $H_2(Q)$, descends
to $5e$ on $X$, a path which winds once around the large radius
limit of the $Q$ moduli space
(changing that space's $B$ by the generator of $H^2(Q,{\Bbb Z})$)
 will wind 5 times around
$q=0$ in the $X$ moduli space.
Thus, denoting by
$q_Q$ the image of $e_Q$ for the $Q$ moduli space, we obtain
\begin{equation}
  q^5 = q_Q.
\end{equation}
{}From \cite{CDGP:} it then follows that
\begin{equation}
  \eqalign{\varpi_0&=\sum_{N=0}^\infty\frac{(5N)!}{(N!)^5}(5\psi)^{-5N}\cr
  \varpi_1&=-\varpi_0\log(5\psi)+\sum_{N=1}^\infty\frac{(5N)!}{(N!)^5}
  \left[\Psi(5N+1)-\Psi(N+1)\right](5\psi)^{-5N}.\cr}
\end{equation}

\section{Some curve counting} 	\label{s:count}

We may now proceed and count rational curves on $X$. The suitably
normalized three-point function for the A-model is
\begin{equation}
  \langle{\cal O}^3\rangle=25+14375q^5+24384375q^{10}+\ldots,
		\label{eq:rcX}
\end{equation}
It follows \cite{CDGP:,AM:rat} that, for $n_i$ the number of rational
curves of degree $i$ on a generic manifold diffeomorphic to $X$ and
$n_i(Q)$ the same quantity for $Q$, we have
\begin{equation}
  \eqalign{n_i&=0, \qquad\qquad{\rm when}\;5\hbox{does not divide} i,\cr
  n_{5i}&=n_i(Q)/25.\cr}
\end{equation}
This is exactly what we would expect from geometry. The group $G$ is
of order 25 and acts freely on $Q$. Since rational curves do not admit
a freely acting symmetry, $G$ must identify the rational curves on $Q$
in groups of 25 (as observed earlier).
A curve of degree $i$ on $Q$ maps into a curve of
degree $5i$ on $X$ because of the relationship between $H_2$ of the
two spaces.\footnote{The {\em degree}\/ in both cases refers to
the number of intersection points with a generator of $H^2$.}

The form of the expression (\ref{eq:rcX}) shows that the chiral ring of
$X$ does not contain enough information to classify the conformal
field theory. This series may be considered as an instanton expansion.
Since this is a tree-level computation, the instantons are spheres.
Spheres correspond to elements of $\pi_2(X)$ and thus can only
represent homology classes which are a multiple of $5e$. Thus
(\ref{eq:rcX}) is a power series in $q^5$ and cannot fully distinguish
between conformal field theories.

In order to measure any quantity which depends properly on $q$ rather
than $q^5$ we must therefore go beyond tree level. Non trivial
information is obtained beyond genus 0 when the topological A-model is
coupled to gravity \cite{BCOV:ell,BCOV:big}. In this case one may
consider a partition function $F_1$ defined for one-loop world-sheets.
This partition function contains information concerning elliptic
curves on the target space and, with luck, may be used to count them
as follows \cite{BCOV:ell}.
The holomorphic anomaly dictates that $F_1$ is of
the form
\begin{equation}
  F_1 = \log\left[\left(\frac\psi{\varpi_0}\right)^w\!\!
	f(\psi)\,q\frac{d\psi}{dq}\right] + {\rm const},
\end{equation}
where $w=3+h^{1,1}-\chi/12$ (which is
$\ff{14}3$ for $X$), and $f(\psi)$ is an unknown
holomorphic
function of $\psi$. The relationship between $F_1$ and the number,
$n_i$, of rational curves of degree $i$ and elliptic curves, $d_i$, of degree
$i$ on $X$ is given by
\begin{equation}
  F_1 = -\ff1{12}(c_2.e)\log q -\sum_i\left\{2d_i\log\eta(q^i)
   +\ff16n_i\log(1-q^i)\right\}
   + {\rm const},	\label{eq:F1q}
\end{equation}
where
\begin{equation}
  \eta(q) = \prod_{n=1}^\infty(1-q^n),
\end{equation}
and $(c_2.e)$ is obtained by wedging the second Chern class of $X$
with the 2-form dual to $e$ and integrating over $X$.

Knowing $c_2(X)$ and the fact that $F_1$ should be finite for a good
conformal field theory is often sufficient to determine $f(\psi)$.
This was the case for all the examples studied in
\cite{BCOV:ell,CDFKM:I,CFKM:II} but fails for our example. However, we
may find the solution by working a little harder.
The function $f(\psi)$
is generally of the form
\begin{equation}
  f(\psi) = \prod_{s}(\psi_s-\psi)^{a_s},
\end{equation}
where $s$ runs over the points (where $\psi=\psi_s$) in the moduli
space where the conformal field theory is ``bad''. The constants $a_s$
are to be determined. In our example, $Y$ is singular (and thus the
associated conformal field theory is bad) whenever $\psi^5=1$. The
fact that the series part of the the expansion in (\ref{eq:F1q})
has rational coefficients means that $F_1$ must be of the form
\begin{equation}
  f(\psi) = (1-\psi)^{a_0}(1+\psi+\psi^2+\psi^3+\psi^4)^{a_1}.
\end{equation}
The fact that $(c_2.e)=10$ for $X$ the tells us that $a_0+4a_1=-\ff{29}6$.

We can also directly count the number of
degree one elliptic curves on $X$ as follows. In contrast to the usual
cases (see for example \cite{Katz:m}) this may be done very explicitly.
The inverse image of such a curve has total degree $5$ on $Q$.
In principle this might split as $5$ curves of degree 1, permuted by
$G$ (and each invariant under some ${\Bbb Z}_5$ subgroup), but since there
are no elliptic curves of degree 1 on $Q$ this is impossible.
Thus, the inverse image is an irreducible curve of degree 5, preserved
by $G$.  Our task is to count those curves.

Any elliptic curve of degree 5 in ${\Bbb P}^4$ is defined by an ideal generated by
 $5$ quadrics.  When the curve is preserved by $G$, this space of quadrics
must form a projective
representation of $G$, and in fact can be generated by  $5$
quadrics of the form
\begin{equation}
  \eqalign{\alpha z_0^2 + &\beta z_1z_4 + \gamma z_2z_3 \cr
      \alpha z_1^2 + &\beta z_2z_0 + \gamma z_3z_4 \cr
      &\vdots\cr
      \alpha z_4^2 + &\beta z_0z_3 + \gamma z_1z_2\cr}	\label{eq:cvs}
\end{equation}
for some constants $\alpha$, $\beta$, and $\gamma$ (cf.\ \cite{Hulek:}).
Generic values of
those constants lead to 5 quadrics which do not intersect; however,
for any constants satisfying
\begin{equation}
\alpha^2+\beta\gamma=0,
\label{eq:param}
\end{equation}
the intersection is a curve of degree 5.\footnote{This is most easily
seen by intersecting with $z_0=0$ and explicitly solving for the
points of intersection.}  A result from classical projective geometry
(cf.\ \cite{GH:alg}) says that the genus of this curve is at most 1.
On the other hand, if the curve does not pass through the fixed points
of $G$ on ${\Bbb P}^4$, i.e.\ if
\begin{equation}
\alpha\ne0, \text{ and }
\alpha + \zeta\beta + \zeta^{-1}\gamma\ne0 \text{ for all }\zeta^5=1,
\label{eq:condition}
\end{equation}
then $G$ acts without fixed points so the genus cannot be 0 and must
be 1.

When (\ref{eq:param}) is satisfied and $\alpha\ne0$, we can take $\alpha=1$,
$\beta=-1/a$, $\gamma=a$ (as in \cite{Hulek:}).  The corresponding
quadrics (\ref{eq:cvs}) form a Gr\"obner basis for the ideal of
the curve, with leading monomials $z_2z_3$, $z_3z_4$, $z_2^2$, $z_3^2$,
$z_4^2$.  Since this is so, it is easy to find the condition that
the quintic $Q$ defined by (\ref{eq:Fermat}) contain the curve---it is
simply
\begin{equation}
  a^{10}+6a^5-1=0.
\end{equation}
Thus there are precisely 10 such curves.   (Note that all solutions
also satisfy (\ref{eq:condition}).)  These descend to 10 elliptic
curves on $X$, and we conclude that
 $d_1=\ff52(a_1-a_0)=10$. This tells us that $a_0=-\ff{25}6$
and $a_1=-\ff16$.  It also shows, as asserted
earlier, that (\ref{eq:ES3}) does not split.

It is interesting to note that it appears that the exponents $a_s$
might be determined purely by the type of singularities of $Y$ when
$\psi=\psi_s$.
As noted above, $Y$ is singular when $\psi^5=1$. If $\psi\neq1$ then
the form of the singularity is that of a single isolated ``simple''
singularity, i.e., it is locally of the form of the hypersurface
\begin{equation}
  x_1x_2=x_3x_4
\end{equation}
in ${\Bbb C}^4$ near the origin. For $\psi=1$, there are 5 singularities
locally of the form of an orbifold of a simple singularity by ${\Bbb Z}_5$.
In all the cases considered in \cite{BCOV:ell,CDFKM:I,CFKM:II} the
value of $a_s$ for a simple singularity is $-\ff16$ and indeed we have
found the same value in this case.

We now have
\begin{equation}
  F_1 = -\ff56\log q+{\rm const}+20q+50q^2+\ff{500}3q^3+\ldots,
	\label{eq:ecX}
\end{equation}
and we determine
\begin{equation}
  \eqalign{d_1&=10\cr d_2&=10\cr d_3&=70\cr d_4&=280\cr
	d_5&=888\cr &\quad\vdots\cr}	\label{eq:dcount}
\end{equation}
We have extended this calculation through $d_{125}$ and find positive
integers for every degree.

\section{Conclusions} 	\label{s:conc}

We have shown that the space of correlation functions for the A-model
on $X$
does not faithfully represent the moduli space of conformal field
theories on $X$. That is, the chiral ring is not sufficient to
determine the conformal field theory. In this case for generic theories,
there are 5 conformal field theories for each chiral ring. The
exception the this occurs at $\psi=0$ or $\infty$ where the chiral ring
does uniquely determine the theory. This ambiguity is clearly shown in
(\ref{eq:rcX}), valid for small $q$, where the correlation functions are
a function of $q^5$. This occurs because the correlation function is at
tree level and spheres do not span $H_2(X)$. One needs to go to a loop
effect such as (\ref{eq:ecX}) to observe a faithful $q$ dependence. In
our example, $H_2(X)$ is generated by tori and so a one-loop partition
function suffices.

Our results can also be interpreted as providing a
counterexample\footnote{Further analysis of the B-model is likely
needed to make this into a mathematically rigorous
counterexample.}
to the global Torelli problem for Calabi-Yau\ manifolds (contrary to
assertions in \cite{Tod:WP}).
The Torelli problem asks whether the variation of Hodge structure
determines the complex structure on the manifold.  The variation
of Hodge structure is completely determined by the chiral ring,
and so depends only on $\psi^{-5}$, but we have seen that the conformal
field theory on the manifold (and so presumably the complex structure
itself) actually depends on $\psi$.

If a manifold is simply connected, we have $\pi_2=H_2$ and so $H_2$ is
generated by spheres. This suggests that the failure of the chiral
ring to determine the conformal field theory is caused by non-trivial
$\pi_1$. Indeed it is precisely the (torsion)
group $H^2(\pi_1(X),{\Bbb C}^*)$ which
led to the ${\Bbb Z}_5$ ambiguity in the identification of the conformal
field theory from the chiral ring. One is thus led to hypothesize
that this is precisely the data required to supplement the chiral
ring (cf.~question 6 in \cite{Vafa:vacua}). That is, given a theory
corresponding to target space $X$ with a
mirror $Y$, in the Calabi-Yau\ phase of both $X$ and $Y$ (i.e., near the large
radius limits) we might have\footnote{Modulo the ${\Bbb Z}_2$ symmetry
coming from complex conjugation.}
\begin{equation}
  \vcenter{\hbox{Moduli Space of}\break\hbox{$N$=(2,2) models}}
	\;\cong\;
  \vcenter{\hbox{Moduli Space}\break\hbox{\hfill of A-ring\hfil}}
  	\;\times\;
  \vcenter{\hbox{Moduli Space}\break\hbox{\hfill of B-ring\hfil}}
  \;\times H^2(\pi_1(X),{\Bbb C}^*)\times H^2(\pi_1(Y),{\Bbb C}^*),
\end{equation}
where by A-ring we mean the chiral ring of the $N$=(2,2) theory as
determined by the A-model and similarly for the B-ring.

One of the motivations for studying this model was to hope to gain
some understanding of the form of the moduli space when there is
torsion in $H_2(X)$ although this did not actually happen for the case
we considered. This might have provided some insight into the
situation when there is ``discrete torsion'' in orbifolds
\cite{Vafa:tor}. Although mirror symmetry has managed to give a
complete picture of the blowing-up of an orbifold without discrete
torsion, at least in the case where points are fixed by abelian groups
(see, for example, \cite{me:orb2}), little is known about this same
process when some discrete torsion is present. Therefore, although
considerable progress has been made in recent years on the form of the
moduli space of $N$=(2,2) theories there is still much to be understood.

\section*{Acknowledgements}

It is a pleasure to thank S. Katz, R. Plesser, C. Vafa, and E. Witten for
useful
conversations.
D.R.M. thanks the School of Natural Sciences, Institute for Advanced
Study for hospitality during the preparation of this paper.
The work of P.S.A. was supported by NSF grant PHYS92-45317,
and the work of D.R.M. was supported by an American Mathematical
Society Centennial Fellowship.


Doron Gepner (Caltech and Weizmann Institute)

11 pages, Caltech preprint, May, 1993, phyzzx macropackage

Gepner
Doron
Caltech and Weizmann Institute

\chapter{#1}}
\def{\rm \chapterlabel.\the\equanumber}{{\rm \chapterlabel.\the\equanumber}}
\def\par\noindent{\bf Proof}{\par\noindent{\bf Proof}}
\def\Lemma{\par\noindent\global\advance\lemnumber by 1
           {\bf Lemma\ (\chapterlabel\the\lemnumber)}}
\def\Corollary{\par\noindent\global\advance\conumber by 1
           {\bf Corollary\ (\chapterlabel\the\conumber)}}
\def\Theorem{\par\noindent\global\advance\thnumber by 1
           {\bf Theorem\ (\chapterlabel\the\thnumber)}}
\def\par\noindent{\bf Conjecture\ }{\par\noindent{\bf Conjecture\ }}
\def\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}{\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}}
\def\global\advance\equanumber by 1{\global\advance\equanumber by 1}
\def\twoline#1#2{\displaylines{\qquad#1\hfill(\global\advance\equanumber by 1{\rm \chapterlabel.\the\equanumber})\cr\hfill#2
\qquad\cr}}
\def\llap(\adveq\myeq){\llap(\global\advance\equanumber by 1{\rm \chapterlabel.\the\equanumber})}
\def\displaytwo#1#2{\displaylines{\hfill#1\hfill\cr\hfill#2\hfill
\llap(\adveq\myeq)\cr}}
\def\manyeq#1{\eqalign{#1}\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}}
\def\threeline#1#2#3{\displaylines{\qquad#1\hfill\cr\hfill#2\hfill\llap{(\global\advance\equanumber by 1{\rm \chapterlabel.\the\equanumber})}\cr
\hfill#3\qquad\cr}}
\font\tensl=cmsl10
\font\tenss=cmssq8 scaled\magstep1
\outer\def\quote{
   \begingroup\bigskip\vfill
   \def\endgroup\eject{\endgroup\eject}
    \def\par{\ifhmode\/\endgraf\fi}\obeylines
    \tenrm \let\tt=\twelvett
    \baselineskip=10pt \interlinepenalty=1000
    \leftskip=0pt plus 60pc minus \parindent \parfillskip=0pt
     \let\rm=\tenss \let\sl=\tensl \everypar{\sl}}
\def\from#1(#2){\smallskip\noindent\rm--- #1\unskip\enspace(#2)\bigskip}
\def\fro#1{\smallskip\noindent\rm--- #1\bigskip}

\def\WIS{\address{Department of Physics\break
      Weizmann Institute of Science\break
      Rehovot 76100, Israel}}

\def\CALT{\address{Division of Physics, Mathematics
and Astronomy\break
Mail Code 452--48\break
California Institute of Technology\break
Pasadena, CA 91125}}

\def\NIKHEF{\address{NIKHEF-H\break
Kruislaan 409\break
NL-1098, SJ Amsterdam\break
Netherlands\break}}
\def\ITP{\address{Institute for Theoretical Physics\break
         University of California\break
         Santa Barbara, California 93106}}

\def\r#1{$\lbrack \rm#1 \rbrack$}
\defDepartment of Physics{Department of Physics}
\def\rightarrow{\rightarrow}
\def\d#1{{\rm d}#1\,}
\def{\cal L}{{\cal L}}
\def{\cal H}{{\cal H}}
\def{\rm mode}{{\rm mode}}
\def{\rm tach}{{\rm tach}}
\def{\rm phys}{{\rm phys}}
\def{\rm ch}{{\rm ch}}
\def\mathop{\rm dim}\nolimits{\mathop{\rm dim}\nolimits}
\def\mathop{\rm mult}\nolimits{\mathop{\rm mult}\nolimits}
\def\semidirect{\mathrel{\raise0.04cm\hbox{${\scriptscriptstyle |\!}$
\hskip-0.175cm}\times}}
\def{\vee}{{\vee}}
\def\dot W{\dot W}
\def\mathop{\rm mod}\nolimits{\mathop{\rm mod}\nolimits}
\def{\cal Z}{{\cal Z}}
\def\mathop{\rm lcd}\nolimits{\mathop{\rm lcd}\nolimits}
\def\mathopen{:}{\mathopen{:}}
\def\mathclose{:}{\mathclose{:}}
\def\ref#1{$^{#1}$}
\def\dagger{\dagger}
\def\prime#1{{#1^\prime}}
\def\prb#1{{\bar #1}^\prime}
\def\tilde{\tilde}
\def\tilde{\tilde}
\def\mathop{\rm sign}\nolimits{\mathop{\rm sign}\nolimits}
\def\bar J{\bar J}
\def\mathop\oplus\limits{\mathop\oplus\limits}
\def{\rm d}z{{\rm d}z}
\def\mathop{\rm Tr}\limits{\mathop{\rm Tr}\limits}
\def{\vee}{{\vee}}
\def\mathop{\rm Im}\nolimits{\mathop{\rm Im}\nolimits}
\def{1\over2}{{1\over2}}
\def\lbrack{\lbrack}
\def\rbrack{\rbrack}
\def\mathop{\rm Dim}\nolimits{\mathop{\rm Dim}\nolimits}
\def\mathop{\rm Ker}\nolimits{\mathop{\rm Ker}\nolimits}
\def\mathop{\rm rank}\nolimits{\mathop{\rm rank}\nolimits}
\def\prime{\prime}
\def\diam{{\hbox{\hskip-0.02in
\raise-0.126in\hbox{$\displaystyle\bigvee$}\hskip-0.241in
\raise0.099in\hbox{ $\displaystyle{\bigwedge}$}}}}
\def\mathop{{\diam}}\limits{\mathop{{\diam}}\limits}
\def\dw#1#2#3#4{
{\scriptstyle{#4}}\,
{\mathop{{\diam}}\limits_{#3}^{#1}}
{\scriptstyle{#2}}  }
\def\bw#1#2#3#4#5{{w\left(\matrix{#1&#2\cr#3&#4\cr}\bigg\vert #5\right)}}
\def\sw#1#2#3#4#5{{S\left(\matrix{#1&#2\cr#3&#4\cr}\bigg\vert #5\right)}}

\def\sim{\sim}
\date{May, 1993}
\titlepage
\title{On Solvable Lattice Models and Knot Invariants}
\author{Doron Gepner}
\CALT\foot{On leave from the Weizmann Institute, Israel. Incumbent
of the Sorrela and Henry Shapiro Chair.}
\vskip15pt
\abstract
Recently, a class of solvable interaction round the face lattice models
(IRF) were constructed for an arbitrary rational conformal field theory (RCFT)
and an arbitrary field in it. The Boltzmann weights of the lattice models
are related in the extreme ultra violet limit to the braiding matrices
of the rational conformal field theory. In this note we use these new
lattice models to construct a link invariant for any such pair of an RCFT
and a field
in it. Using the properties of RCFT and the IRF lattice models, we prove
that the invariants so constructed always obey the Markov properties, and
thus are true link invariants. Further, all the known link invariants,
such as the Jones, HOMFLY and Kauffman polynomials arise in this way,
along with giving a host of new invariants, and thus also a unified approach
to link polynomials. It is speculated that all link invariants arise
from some RCFT, and thus the problem of classifying link and knot
invariants is equivalent to that of classifying two dimensional conformal field
theory.
\endpage
The intriguing interplay between knot theory and two dimensional
physics has benefited considerably both fields (for a review, see, e.g.,
\REF\Wadati{M. Wadati, T. Deguchi and Y. Akutsu, Phys. Rep. 180 (1989) 247}
\r\Wadati). The purpose of this note is to put forwards a general framework
for link invariants stemming from solvable lattice models. It was recently
shown that solvable fusion interaction round the face (IRF) lattice models
are in a one--to--one correspondence with a pair of a rational conformal field
theory
and a field in it \REF\One{D. Gepner, Foundations of rational conformal field
theory, I, Caltech preprint, CALT 68--1825, November (1992)}\r\One.
It follows as we shall see that for each such pair
one can form a link invariant, and that this class of link invariants is
in a one to one correspondence with such pairs.

Let us review the construction of the Boltzmann weights described in ref.
\r\One.
Consider a rational conformal field theory (RCFT) $\cal O$, and a field in
it, $x$, which for simplicity we shall assume to be a primary field.
We than construct a solvable IRF model, denoted by IRF$({\cal O},x)$ following
\r\One, whose admissibility conditions are given by fusion with respect for
$x$ and whose Boltzmann weights reduce in the extreme ultra violet limit
to a specialization of the braiding matrix of the RCFT (see \r\One\ for more
detail). We put on the vertices of the lattice, which is a square two
dimensional one, state variables which
are the primary fields of $\cal O$ and are labeled by $a,b,c,\ldots$. The
pair $a$ and $b$ is allowed to be on the same link, $a\sim b$, if and
only if, the fusion coefficient $N_{ax}^b>0$. The partition function
of the model is
$$Z=\sum_{\rm configurations} \prod_{\rm faces} \bw a b c d u,\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where $a,b,c$ and $d$ are the four states (primary fields) on the vertices of
the
face, $\bw a b c d u$ is the Boltzmann weight associated to the
face, and $u$ is a spectral parameter which labels a family of models.
The Boltzmann weights obey the star triangle equation (STE), from which
it follows that the transfer matrices for different values of the
spectral parameter $u$ commute, and thus the model is solvable.

The Boltzmann weights of the model IRF$({\cal O}, x)$ were given in ref.
\r\One, and are conveniently described in an operator form.
To do so define the operator $X_s(u)$, the face transfer matrix, by
$$X_s(u)_{l_1,l_2,\ldots,l_n}^{m_1,m_2,\ldots,m_n}=
\prod_{i\neq s}\delta(l_i,m_i) \bw {l_{i-1}} {m_i} {l_i} {l_{i+1}} u,\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where $l_i$ and $m_i$ are the states on two adjacent diagonals of the
lattice. The face transfer matrix of the model IRF$({\cal O},x)$ is \r\One
$$X_s(u)=\sum_{a=1}^N P^a_s f^a(u),\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where $a=1,2,\ldots,N$ labels the fields appearing in the operator product
$x\cdot x$,
$P^a$ is a projection operator of the braiding matrix on the $a$ field
in the operator product defined by
$$P^a=\prod_{j=1\atop j\neq a}^N {B_s-\lambda_j\over \lambda_a-\lambda_j},\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
and where $B_s$ is the braiding matrix of the RCFT at the face $s$, and
$\lambda_j$ are its eigenvalues, which are given by
$$\lambda_j=e^{i\pi (2\Delta_x-\Delta_j)},\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
and $\Delta_x$ and $\Delta_j$ are the conformal dimensions of the field
$x$ and the $j$ field in the operator product $x\cdot x$, respectively.

The functions $f^a(u)$ are defined by,
$$f^a(u)=\prod_{j=1}^{a-1} \sin (\zeta_j+u) \prod_{j=a}^{N-1}
\sin(\zeta_i-u),\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where
$$\zeta_i=\pi(\Delta_{i+1}-\Delta_i)/2,\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
and $\lambda=\zeta_1$ is the
crossing parameter of the model.
The projection operators obey,
$$\eqalign{P^a_s P^b_s&=\delta_{ab} P_s^a,\cr
           1&=\sum_{a=1}^N P_a,\cr
          B_s&=\sum_{a=1}^N P^a_s \lambda_a,\cr}\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
from which it follows that the face transfer matrix obeys the unitarity
condition,
$$X_s(u) X_s (-u)=\rho(u)\rho(-u),\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where the unitarity factor is
$$\rho(u)=f^N(u)=\prod_{i=1}^{N-1} \sin(\zeta_i+u).\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
Also, this implies the regularity condition,
$$X_s(0)=\rho(0)\cdot 1.\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
An important, and highly non trivial, property of the Boltzmann weights
is the crossing symmetry,
$$\bw a b c d {\lambda-u}=\left({\psi_b \psi_c \over \psi_a \psi_d}
\right)^{1\over2} \bw c a d b u,\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where the crossing multiplier $\psi_a$ is given in terms of the torus
modular function $S_{ab}$,
$$\psi_a={S_{a,0}\over S_{0,0}},\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where `$0$' denotes the unit field.
Repeating the crossing transformation twice implies the charge conjugation
symmetry:
$$\bw a b c d u=\bw d c b a u.\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$

It is convenient to define the two braiding operators,
$$G_i^\pm=\lim_{u\rightarrow \pm \infty} X_i(u)/\rho(u),\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where $G_i^+$ (denoted also for simplicity by $G_i$)
differs from the conformal braiding matrix $B_i$ by an
irrelevant phase. In terms of the Boltzmann weights, this is
\def\bs#1#2#3#4#5{\sigma\left(\matrix{ #1 & #2 \cr #4& #3\cr}\bigg\vert
#5\right)}
$$\bs a b c d {\pm}=\lim_{u\rightarrow\pm\infty} \bw a b c d u/\rho(u),\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
from which it follows that $G_i^+=(G_i^-)^\dagger$, i.e., they are complex
conjugates of each other, and that $G_i^\pm$ obey the Braid group relationships
which are
$$\eqalign{G_i G_{i+1} G_i&=G_{i+1} G_i G_{i+1},\cr
           G_i G_j&=G_j G_i \qquad {\rm for\ } |i-j|>1,\cr }\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
which is the relation obeyed by the generators of the braiding group,
i.e., $G_i$
can be considered as the generator of the braiding of the $i$ and
$i+1$ strands in a braid. By Artin theorem these are the generating
relations for the braid group.

A link is formed by connecting the end points of a braid. Labeling
the end points $l_1,l_2,\ldots ,l_n$ and $m_1,m_2,\ldots,m_n$, as
before, we connect with a strand the $l_i$ and $m_i$ end points, for
all $i$. This procedure is ambiguous as different braids may give
the same (topologically) link. We call such braids equivalent. It
was shown by Markov \REF\Markov{A.A. Markov, Recueil Math. Moscou 1 (1935) 73}
\r\Markov, that two braids are equivalent if an only if they
can be related by the sequence of moves of the two types,
$$({\rm I})\qquad AB\rightarrow BA \qquad {\rm for\ } A,B\in B_n,\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
$$({\rm II})\qquad A\rightarrow A G_n^{\pm 1} \qquad {\rm for\ }
A\in B_n,\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where $B_n$ denotes the braid group on $n$ elements, defined by the relations
eq. (17).

In order to classify links we wish to form a functional $\alpha$ which
assigns a complex number for each link, in such a way that topologically
equivalent links will have the same value of $\alpha$,
$\alpha(A)=\alpha(B)$ if $A$ and $B$ are equivalent topologically.
To do so, it is thus sufficient to demand that $\alpha$ is invariant
under the Markov moves. We define a Markov trace on a braid, $\phi(A)$,
for $A\in B_n$, to be a complex number obeying the
properties,
$$\eqalign{({\rm (I)}\qquad \phi(AB)&=\phi(BA),\qquad A,B\in B_n,\cr
({\rm II})\qquad \phi(A G_n)&=\tau \phi(A), \qquad \phi(A G_n^{-1})=
\bar\tau \phi(A), \qquad A\in B_n,\cr}\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
and where the parameters $\tau$ and $\bar \tau$ are
$$\tau=\phi(G_i),\qquad \bar \tau=\phi(G_i^{-1}).\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
The link invariant $\alpha(A)$ is formed in terms of the Markov trace
$\phi(A)$, by
$$\alpha(A)=(\tau\bar\tau)^{-(n-1)/2} (\tau/\bar \tau)^{e(A)/2} \phi(A),\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where $e(A)$ is the exponent sum of the braid, i.e.,
$$e(\prod_{i=1}^n G_i^{a_i})=\sum_{i=1}^n a_i, \global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
which is evidently a well defined grading, since it is preserved by the braid
group relationships, eqs. (17).

We next proceed to describe a Markov trace based on the lattice model
IRF$({\cal O},x)$. Note that any element of the braid group, $A\in B_n$
is represented by some diagonal to diagonal transfer matrix,
$A_{l_1,l_2,\ldots,l_n}^{m_1,m_2,\ldots, m_n}$, where the generators
are represented by the conformal braiding matrix $G_i$. Now, define the
diagonal matrix,
$$(H^n)_{l_1,l_2,\ldots l_n}^{m_1,m_2,\ldots, m_n}=
\prod_{i=1}^n \delta(l_i,m_i) {S_{l_n,0}\over S_{l_1,0}},\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where $S$ is, as before, the torus modular matrix, which gives the
crossing multiplier. Define also a constrained trace by,
\def\mathop{\hat{\rm Tr}}\nolimits{\mathop{\hat{\rm Tr}}\nolimits}
$$\mathop{\hat{\rm Tr}}\nolimits(A)=\sum_{l_2,l_3,\ldots,l_n} A_{l_1,l_2,\ldots,l_n}^{l_1,l_2,
\ldots,l_n}.\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
Then the Markov trace is defined by
$$\phi(A)={\mathop{\hat{\rm Tr}}\nolimits(H^n A) \over \mathop{\hat{\rm Tr}}\nolimits(H^n)},\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
for any element of the braid group $A$.
It remains to show that the Markov trace so defined, $\phi(A)$ obeys
the properties (I) and (II), eqs. (18--19). Property (I) follows trivially
from the definition, while property (II) follows from a straight forwards
calculation, provided that the Boltzmann weights obey the Markov property,
$$\sum_{b\sim a} \bw b a a c u {S_{b,0}\over S_{a,0}}=H(u)\rho(u),\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where $H(u)$ is some function independent of $a$ and $c$.
The parameters $\tau$ and $\bar \tau$ are given by
$$\tau,\bar\tau=\lim_{u\rightarrow\pm\infty} H(u)/H(0),\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where $\tau$ ($\bar\tau$) corresponds to the plus (minus) sign in the limit.

Using the crossing property, eq. (12), it is straight forwards to show that the
extended Markov property holds provided that the following relation
is valid,
$$X_i(\lambda) X_i(u)=\beta(u) X_i(\lambda),\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where $\beta(\lambda-u)=H(u)\rho(u)$.
We shall now show that for the models IRF(${\cal O},x)$ the property eq. (29)
holds and that thus $\phi$ is always a good Markov trace.
This is a simple calculation using eqs. (8). We note that
$X_i(\lambda)=P^N_i f^N(\lambda)$, since $f^a(\lambda)$ vanishes for $a\neq N$.
Thus $X_i(\lambda)$ is indeed a projection operator and so
$$X_i(\lambda) X_i(u)=f^N(\lambda) \sum_{a=1}^N P^N_i P^a_i f^a(u)=
\beta(u) X_i(\lambda),\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where we used eqs. (3,8), and
$$\beta(u)=f^N(u)=\prod_{a=1}^N \sin(\zeta_i+u).\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
It follows that the parameters are
$$H(u)=\prod_{i=1}^N {\sin(\lambda+\zeta_i-u)\over \sin(\zeta_i+u) },\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
and
$$\tau=e^{iN\lambda}\prod_{i=1}^N {\sin(\zeta_i)\over \sin(\lambda+\zeta_i)},
\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
and $\bar\tau=\tau^\dagger$.
It follows that the invariant we defined, eq. (22), indeed assumes the
same values for topologically equivalent links, and thus can be used
to classify knots and links.

For a number of examples of IRF models, the link invariants we defined
here were
previously calculated (for a review, see \r\Wadati, and references therein).
For example, the unrestricted Lie algebra model $A_{m-1}$ give rise
to the HOMFLY polynomial \REF\HOMFLY{P. Freyd, D. Yetter, J. Hoste, W.B.R.
Lickorish, K. Millet and A. Ocneanu, Bull. Am. Math. Soc. 12 (1985) 239}
\r\HOMFLY\ (as a polynomial in $m$ and the crossing parameter), which
is a two variable generalization of the original Alexander
polynomial \REF\Alex{J.W. Alexander, Trans. Am. Math. Soc. 30 (1928) 275}
\r\Alex\ (at the limit $m\rightarrow 0$) and the more recent Jones polynomial
\REF\Jones{V.F.R. Jones, Bull. Am. Math. Soc. 12 (1985) 103}\r\Jones\
($m=2$ case). The unrestricted $B_m$, $C_m$ and $D_m$ IRF models
give the Kauffman polynomial \REF\Kauf{L.H. Kauffman, Trans. Am. Math. Soc.,
to be published.}\r\Kauf.
These models correspond to the current algebra RCFT based on the Lie algebras
$A$, $B$, $C$, $D$, with the field which is the fundamental representation
for $A_n$, and the vector representation for the other algebras.

It is noteworthy that the construction presented
here, while encompassing all the known link invariants, provides for a
very far reaching generalization of these, along with a unified framework
for their construction. Such new invariants are indeed needed in the
problem of classifying links as it is well known that two topological distinct
links may certainly have identical classifying polynomials (see for example
Birman's example \REF\Birman{J.S. Birman, Invent. Math. 81 (1985) 287}
\r\Birman\ of two different knots that have the same Jones polynomial).

The link invariants we defined eq. (22) may be calculated directly for each
IRF model by substituting the Boltzmann weights and preforming
the traces. This is however rather cumbersome for big links. A considerable
simplification
is provided by the skein relations which relate the invariants of different
links
\REF\Conway{J.H. Conway, in Computational Problems in Abstract Algebra, ed. J.
Leech (Pargamon, New--York, 1970) p. 329}\r{\Alex,\Conway}.
To derive skein relations for the invariants described here, first
note that the Braiding matrix $G_i$ obeys a fixed $N$th order polynomial
equation,
$$\sum_{m=0}^N a_m G_i^m=\prod_{m=1}^N (G_i^m-\lambda_m)=0,\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where we used eq. (5).
Define the link $L_{m}$ to be the link obtained with the insertion
of the braid element $G_i^m$, i.e., if $L$ described by the braid
$A$, then $L_m$ is described by the braid $AG_i^m$. Using the polynomial
relation, eq. (34), we find immediately the relation for the Markov trace,
$$\sum_{m=-k}^{N-k} a_m \phi(L_m)=0,\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
for any $k$. Substituting this into the definition of the invariant, eq. (22),
we find the skein relation,
$$\sum_{-k}^{N-k} b_m \alpha(L_m)=0,\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
where
$$b_m=a_m (\tau/\bar \tau)^{-m/2}.\global\advance\equanumber by 1\eqno{\rm (\chapterlabel\the\equanumber)}$$
The skein relation, eq. (36), is a very effective tool for the calculation of
link invariants.

We thus describe in this note a whole wealth of link invariants which are
in a one to one correspondence with a pair of a rational conformal field
theory and a field in this theory. The RCFT and the field chosen
are arbitrary, and every RCFT gives rise to different invariants.
It is tantalizing to speculate on this in a number of directions.
First, since all known link polynomials arise in this fashion, one might
conjecture that the category of link invariants and the category
of pairs of conformal field theory and a field in it are in fact equivalent
ones, and that the problem of classifying link invariants is thus the same
as that of classifying conformal field theory. Second, one might
ponder the generalization of these ideas to all conformal
field theories, not necessarily rational. There does not seem to be any
obstacle in doing so, and the entire construction might be carried,
mutatis mutandis.
This will also open up an entire
different type of invariants, so called irrational, which, in particular,
obey an infinite order skein relations, i.e., a Laurent series type rather
than polynomial. Such invariants appear not to have been studied before.

Finally, it is hoped that the results described here will be of help
in the further understanding of both knot theory and two dimensional
physics, along with the fascinating interrelationship between them.
\refout
\bye


Liouville Equation And Schottky Problem

12 pages, LaTex file. Expanded version and misprints corrected

Abstract:
  An Ansatz for the Poincar\'e metric on compact Riemann surfaces is proposed.
This implies that the Liouville equation reduces to an equation resembling a
non chiral analogous of the higher genus relationships (KP equation) arising in
the framework of Schottky's problem solution. This approach connects
uniformization (Fuchsian groups) and moduli space theories with KP hierarchy.
Besides its mathematical interest, the Ansatz has some applications in the
framework of quantum Riemann surfaces arising in 2D gravity.

\section{#1}}

\mysection{Schottky Problem And KP Hierarchy}

 Let us consider a genus $h$ compact Riemann surface
$\Sigma$. A fundamental object defining the complex structure
of $\Sigma$ is the Riemann period matrix
\begin{equation}
\Omega_{ij}\equiv\oint_{\beta_i}\omega_j,
\label{aa1}\end{equation}
where the $\omega_k$'s denote the $h$ holomorphic differentials with the
standard normalization $\oint_{\alpha_i}\omega_j=\delta_{ij}$. By means
of the Riemann bilinear relations it can be proved that
$\Omega_{ij}$ is symmetric and has positive definite imaginary part
(see for example \cite{fk1}).
Let us consider the Siegel space
\begin{equation}
{\cal A}_h={\cal H}_h/Sp(h,{\bf Z}),
\label{aa2}\end{equation}
where ${\cal H}_h$ denotes the Siegel upper-half plane, that is the
space of symmetric $h\times h$ matrices with
positive definite imaginary part.
To recognize  the locus in ${\cal A}_h$ of the Riemann
period matrices is the famous Schottky problem. This problem has been
 solved essentially by Dubrovin, Mulase and Shiota
\cite{dubrovin,mulase,shiota}. The solution is based on the proof of the
Novikov conjecture stating that
\begin{equation}
u(x,y,t)=2\partial_x^2 \log \Theta (Ux+Vy+Wt+z_0|\Omega),
\label{S1}\end{equation}
satisfies the KP equation if and only if
$\Omega$ is the period matrix of some $\Sigma$.
The corresponding equations
on $\Omega$ (see eq.(\ref{Boris})) were derived in \cite{dubrovin} where it
was proved that they determine an algebraic variety
with a component given by the matrices of the $\beta$-periods.
In \cite{shiota} Shiota pointed out that if $u$
in eq.(\ref{S1}) satisfies the KP equation,
then there are vectors $U^k$, such that the function
\begin{equation}
u(t_1,t_2,\ldots)=2\partial_t^2\log \Theta
\left(\sum_{k=1}^\infty U^kt_k|\Omega\right),
\qquad t_1=x,\, t_2=y,\, t_3=t,
\label{S2}\end{equation}
determines solutions of the KP hierarchy
\begin{equation}
\left[{\partial\over
 \partial t_j}-L_j,{\partial\over \partial t_k}-L_k\right]
=0,\label{S3}\end{equation}
where the order $k$ differential operators $L_k$
have coefficients depending on $\vec{t}\equiv (t_1,t_2,\ldots)$
  and are determined by the equation $(\partial_{t_k}-L_k)
\psi(\vec{t},z)=0$, $\psi$ being the
Baker-Akhiezer function on $\Sigma$.
Since the space of vectors $U^k$ is $h$-dimensional,
 there are two commuting operators of coprime order
which are linear combinations of the $L_k$'s.
Therefore one can apply the results in \cite{krichever1}
to show that $\Omega$ is the Riemann matrix of the surface
defined by these operators.

\mysection{The Ansatz}

 Let $\Sigma$ be a compact Riemann surface of genus $h>1$.
It is well-known that the Liouville equation on $\Sigma$
\begin{equation}\partial_z\partial_{\bar z}\varphi(z,\bar z)={1\over 2}e^{
\varphi(z,\bar z)},\label{1l}\end{equation}
is uniquely satisfied by the Poincar\'e metric (with Gaussian curvature
$-1$).
This metric can be written in terms of the
inverse map of uniformization
\begin{equation}
J^{-1}_H:\Sigma\longrightarrow H,
\label{e1}\end{equation}
where $H=\{w|{\rm
Im} \, w > 0\}$ denotes the upper half plane.
The Poincar\'e metric on $H$ is
\begin{equation}
ds^2={|dw|^2\over({\rm Im} \, w)^2},
\label{e2}\end{equation}
so that on $\Sigma\cong H/\Gamma$ (here $\Gamma$ is a hyperbolic
Fuchsian group)
\begin{equation}
e^{ \varphi(z,\bar {z})}={|{J_H^{-1}(z)}'|^2\over({\rm Im}\,
J_H^{-1}(z))^2},\label{e3}\end{equation}
which is invariant under $SL(2,{\bf R})$ fractional transformations of
$J^{-1}_H$.
Unfortunately no one has succeeded in writing down
$J^{-1}_H$ in terms of the moduli of $\Sigma$.

Here we consider
the following Ansatz for the Poincar\'e metric\footnote{Notice
that a possible choice for the matrix to be positive definite is to set $A_{ij}
={\Omega_{ij}^{(2)}}^{-1}$, in this case (\ref{prova11}) coincides
with the Bergman metric.}
\begin{equation}e^\varphi=\sum_{i,j=1}^h\omega_iA_{ij}\overline\omega_j.
\label{prova11}\end{equation}
To get the inverse map one has to solve the Schwarzian equation
\begin{equation}\{J_H^{-1},z\}=
T^F(z),\qquad \label{4}\end{equation}
where
\begin{equation}
T^F(z)=\varphi_{zz}-{1\over 2}\varphi_z^2,
\label{stress1}\end{equation}
is the classical Liouville stress tensor
(or Fuchsian projective connection).
By (\ref{prova11}) we have
\begin{equation}
T^F(z)={2\sum_{i,j=1}^h\omega_i''A_{ij}\overline \omega_j-
3\left(\sum_{i,j=1}^h\omega_i'A_{ij}
\overline\omega_j\right)^2\over 2\left(\sum_{i,j=1}^h\omega_iA_{ij}
\overline\omega_j\right)^2}.
\label{stress2}\end{equation}
Observe that eq.(\ref{1l}) implies that
\begin{equation}
\partial_{\bar z}T^F(z)=0.
\label{stress2er}\end{equation}

Eq.(\ref{4}) can be reduced to the linear equation
\begin{equation}
\left(2\partial^2_z+T(z)\right)\psi=0.
\label{ficxk}\end{equation}
Actually it turns out that, up to $SL(2,{\bf C})$ linear fractional
transformations,
\begin{equation}
J^{-1}=\psi_1/\psi_2,\label{g1000}\end{equation}
with $\psi_1$ and $\psi_2$ two linearly independent solutions
of (\ref{ficxk}) (see \cite{mma} for a discussion on this point).

Inserting (\ref{prova11}) in (\ref{1l}), the
Liouville equation becomes\begin{equation}{\sum_{i,j,k,l=1}^h\omega_l^2
\partial_z\left({\omega_{i}/\omega_l}\right)A_{ij}A_{lk}\overline\omega_k^2
\partial_{\bar z}\left({\overline\omega_j/\overline\omega_k}\right)}=\left(
\sum_{i,j=1}^h\omega_iA_{ij}\overline\omega_j\right)^3.\label{2l}\end{equation}

This equation has a strict similarity with the relations between the
periods of holomorphic differentials on Riemann surfaces \cite{dubrovin}.
Thus one should expect that $A_{ij}$ depends on the moduli
through the Riemann period matrix.
To show this similarity, we write down the fundamental relations
given in \cite{dubrovin}. Let us introduce the following notation
$$
U_k=-\omega_k(P),
$$
$$
V_k=-\omega'_k(P),
$$
\begin{equation}
W_k=-{1\over 2}\omega''_k(P)-{1\over 2}c(P)U_k,
\label{w2}\end{equation}
where $c(P)$ is a projective connection \cite{dubrovin} and $P$ is an
arbitrary point on $\Sigma$.
In \cite{dubrovin} Dubrovin proved that the function (\ref{S1})
is a solution of the KP equation
\begin{equation}
u_{yy}=(4u_t-6uu_x-u_{xxx})_x,
\label{dgpl}\end{equation}
if and only if the following relations between $U,V,W,\Omega$ and
an additional constant $d$ are satisfied
(see \cite{dubrovin} for
notation)
\begin{equation}
\sum_{i,j,k,l=1}^hU_iU_jU_kU_l\widehat \Theta_{ijkl}[n]+
\sum_{i,j=1}^h\left({3\over 4}V_iV_j-U_iW_j\right)
\widehat \Theta_{ij}[n]+d \widehat \Theta [n]=0, \qquad n\in{\bf Z}^h_2.
\label{Boris}\end{equation}
We emphasize that this result is a fundamental step
to solve Schottky's problem.

Our remark is that eq.(\ref{2l}) looks like a non chiral generalization
of (\ref{Boris}). In the notation introduced above eq.(\ref{2l}) reads
\begin{equation}\sum_{i,j,k,l=1}^h
\left(U_lV_i-U_iV_l\right)A_{ij}A_{lk}
\left(\overline U_k\overline V_j-\overline U_j \overline V_k\right)
=\left(
\sum_{i,j=1}^hU_iA_{ij}\overline U_j\right)^3.
\label{2lh}\end{equation}
We stress that solving this equation is equivalent to solving crucial
questions arising in uniformization theory, Fuchsian groups and related
subjects. In particular, Weil-Petersson's 2-form $\omega_{WP}$
can be recovered using
the fact that the classical Liouville action evaluated at the classical
solution is the K\"ahler potential of $\omega_{WP}$ \cite{0}.

Another aspect that should be investigate
is whether eq.(\ref{2lh}) furnishes conditions on the period
matrix in a more manageable form than KP equation (\ref{dgpl})-(\ref{Boris}).

A possible approach to study eq.(\ref{2l}) is using
Krichever-Novikov's differentials $\psi_j^{(n)}$ \cite{kn}.
These differentials are holomorphic on $\Sigma\backslash\{P_+,P_-\}$
with prescribed behaviour at $P_\pm$.
In particular, in terms of local coordinates
$z_\pm$ vanishing at $P_\pm\in \Sigma$, we have
\begin{equation}
\psi_j^{(n)}(z_\pm)(dz_\pm)^n=a_j^{(n)\pm}
z_\pm^{\pm j -s(n)}\left(1+{\cal
O}(z_\pm)\right)\left(dz_\pm\right)^n,
\quad s(n)={h\over 2}-n(h-1),\label{onehalf}\end{equation}
where $j\in{\bf Z}+h/2$ and $n\in{\bf Z}$.
By the Riemann-Roch theorem, $\psi_j^{(n)}$ is uniquely fixed
by choosing the value of $a_j^{(n)+}$ or $a_j^{(n)-}$. In the
following we set $a_j^{(n)+}=1$.

These differentials can be written in terms of theta
functions\footnote{In the appendix we illustrate the method to construct
differentials in higher genus Riemann surfaces.} \cite{cmp}
\begin{equation}
\psi_j^{(n)}(z)=C_j^{(n)}
\Theta\left(I(z)+{\cal D}^{j;n}|\Omega\right)
 {\sigma(z)^{2n-1} E(z, P_+)^{j-s(n)}\over
 E(z,P_-)^{j+s(n)}},
\label{psij}\end{equation}
where
$$
{\cal D}^{j;n}=\left(j-s(n)\right)I(P_+)-
\left(j+s(n)\right)I(P_-)+(1-2n)\Delta,
$$
and constant $C_j^{(n)}$ is fixed by
the condition $a_j^{(n)+}=1$.

Let  ${\cal C}$ be a homologically trivial
contour separating $P_+$ and $P_-$.
The dual of $\psi_j^{(n)}$  is defined by
\begin{equation}
{1\over 2\pi i}\oint_{\cal C} \psi_j^{(n)}
\psi^k_{(n)}=\delta_j^k,\label{dualpsi}\end{equation}
which implies
\begin{equation}
\psi^j_{(n)}=\psi_{-j}^{(1-n)}.\label{easy1}\end{equation}
Note that (\ref{onehalf}) provides a basis for the
$1-2s(n)=(2n-1)(h-1)$ holomorphic $n$-differentials
on $\Sigma$ ($h\ge 2$)
\begin{equation}
{\cal H}^{(n)}=\left\{\psi_k^{(n)}\big |s(n)\le k\le -s(n)\right\},
\qquad n\ge 2.
\label{qdrtcdfab}\end{equation}
Furthermore, from
\begin{equation}
\widetilde{\cal H}^{(m)}=\left\{\psi_k^{(m)}
\big |1-s(m)\le k\le s(m)-1\right\},\qquad m\le -1,
\label{qdrtcdfabc}\end{equation}
one can define the space of generalized Beltrami differentials.
They are vanishing everywhere on $\Sigma$ except in a disk
where coincide with \cite{mma}
\begin{equation}
\widetilde{\cal B}^{(m)}=\left\{\partial_{\bar z}\psi_k^{(m)}
\big |1-s(m)\le k\le s(m)-1\right\},\qquad m\le -1,
\label{qdrtcdfabcdd}\end{equation}
 (for $m=-1$ one gets the Beltrami differentials considered in
\cite{cmp}). Observe that the differentials in
(\ref{qdrtcdfabc}) have poles both in $P_+$ and $P_-$.
In particular, $\widetilde{\cal H}^{(1-n)}$ is the dual space of
${\cal H}^{(n)}$.

We now  expand the holomorphic 3-differentials in
(\ref{2l}) in terms of the basis introduced above. We have
\begin{equation}
\omega_i^2\partial_z(\omega_j/\omega_i)=\sum_{p=1}^{5h-5}a_{ij}^p\psi_{p+s(3)-1
}^{(3)},\qquad a_{ij}^p={1\over 2\pi i}\oint_{{\cal C}}\psi_{-p-s(3)+1}^{(-2)}
\omega_i^2\partial_z(\omega_j/\omega_i),\label{i1}\end{equation}\begin{equation}
\omega_i\omega_j\omega_k=\sum_{p=1}^{5h-5}b_{ijk}^p\psi_{p+s(3)-1}^{(3)},\qquad
b_{ijk}^p={1\over2\pi i}\oint_{{\cal C}}\psi_{-p-s(3)+1}^{(-2)}\omega_i\omega_j
\omega_k.\label{i2}\end{equation}
Inserting these expansions in (\ref{2l}) we
get the `Liouville relations'
\begin{equation}\sum_{i,j,k,l=1}^ha_{ij}^pA_{ik}A_{jl}\overline a^q_{kl}=
\sum_{i,j,k,l,m,n=1}^hb_{ijm}^pA_{ik}A_{jl}A_{mn}\overline b_{kln}^q.\label{i3}
\end{equation}\\Let us notice that the coefficients $a^q_{kl}$ and $b_{kln}^q$
are functionals of the holomorphic differentials and their derivatives computed
at $P_+$ and coincide with the vectors of $\beta$-periods of second-kind
differentials.

The above expansions provide relations involving the holomorphic differentials,
theta functions and their derivatives. To see this it is
sufficient to notice that the coefficients $a^p_{ij}$ and
$b^p_{ijk}$ are vanishing for $p<1$ and $p>5h-5$. The reason is that
in this range the $\psi_{-p-s(3)+1}^{(-2)}$'s
are holomorphic in $P_-$ or $P_+$.
This implies that for $p<1$ and $p>5h-5$, the contribution to $a^p_{ij}$ and
$b^p_{ijk}$ coming from the poles at $P_-$ or $P_+$
add to zero. Notice that this `residue formula' is crucial to
get important relations such as Hirota's formulation of the KP
hierarchy (see for example \cite{johnfay}).

\mysection{The Accessory Parameters}

 Here we consider some aspects concerning the Fuchsian accessory
parameters. First of all we introduce the projective connection
 \begin{equation}
T^S(z)=\left\{J^{-1}_\Omega,z\right\},
\label{proOmega}\end{equation}
where $J_\Omega:\Omega\to\Sigma$ denote the Schottkian uniformization map.
Here $\Omega$ denotes the region of discontinuity in
$\widehat {\bf C}={\bf C}\cup\{\infty\}$
of the Schottky group ${\cal S}$ and $\Sigma \cong\Omega/{\cal S}$.
Let us introduce the following notation for the Krichever-Novikov vector
fields and quadratic differentials
\begin{equation}
e_k\equiv \psi_k^{(-1)},\qquad\quad
\Omega^k\equiv \psi_{-k}^{(2)}.
\label{fdgs}\end{equation}
Let ${\cal T}_\Sigma$ be the holomorphic projective connection on
$\Sigma$ obtained from the symmetric bidifferential of the second-kind with
bi-residue 1 and zero $\alpha$-periods.
The Fuchsian accessory parameters $\lambda_1^{(F)},...,
\lambda_{3h-3}^{(F)}$ and the Schottkian accessory parameters
$\lambda_1^{(S)},...,\lambda_{3h-3}^{(S)}$ are defined by
\begin{equation}
T^F={\cal T}_\Sigma+\sum_{k=1}^{3h-3}
\lambda_k^{(F)}\Omega^{k+1-h_0}, \qquad
T^S={\cal T}_\Sigma+\sum_{k=1}^{3h-3}
\lambda_k^{(S)}\Omega^{k+1-h_0},\quad h_0\equiv
{3\over 2}h. \label{m2}\end{equation}

In order to write ${\cal T}_\Sigma$ explicitly we consider
 an arbitrary nonsingular point $f$ of the theta divisor,
that is $\Theta(f)=0$ and grad$\,\Theta(f)\ne 0$.
 We define
\begin{equation}
H_f(z)=\sum_{k=1}^h\Theta_{k}(f)\omega_k(z),
\label{m34}\end{equation}
\begin{equation}
Q_f(z)=\sum_{j,k=1}^h\Theta_{jk}(f)\omega_j(z)
\omega_k(z),\label{m35}\end{equation}
\begin{equation}
T_f(z)=\sum_{i,j,k=1}^h\Theta_{ijk}(f)
\omega_i(z)\omega_j(z)\omega_k(z).\label{m36}\end{equation}
The holomorphic projective connection is \cite{fay}
\begin{equation}
{\cal T}_\Sigma(z)=\left\{\int^z_{P_0} H_f,z\right\}+
 {3\over 2} \left(Q_f(z)\over H_f(z)\right)^2-2{T_f(z)\over H_f(z)}.
 \label{m37}\end{equation}
At a zero of $H_f$ we have
\begin{equation}
Q_f(z_0)=\pm H_f'(z_0),
\qquad T_f(z_0)=-H_f''(z_0)\pm {3\over 2}Q_f'(z_0),
\label{m38}\end{equation}
with the sign $\pm$ chosen accordingly as
 $\Theta(z-z_0\mp f)\equiv 0,\; \forall z\in\Sigma$.

Besides $T^F$ and $T^S$, also ${\cal T}_\Sigma$ can be expressed
as a Schwarzian derivative. To do this we simply note that
according to the general rule described above the equation
\begin{equation}
\left({\partial^2\over \partial z^2}+{1\over2}{\cal T}_\Sigma(z)
\right)\phi(z)=0,
\label{new1aqw}\end{equation}
has as solutions two linearly independent
$-{1\over 2}$-differentials $\phi_1,\,\phi_2$, satisfying
the equation
\begin{equation}
{\cal T}_\Sigma(z)=\left\{{\phi_2/ \phi_1},z\right\}.
\label{dfsfd}\end{equation}

Note that the Fuchsian accessory parameters are given by
\begin{equation}
\lambda_k^{(F)}={1\over 2\pi i}\oint_{{\cal C}}\left(
 \left\{J_H^{-1}(z),z\right\}
-\left\{J_\Sigma^{-1}(z),z\right\}
\right)e^{k+1-h_0},
\label{m39}\end{equation}
where
\begin{equation}
J_\Sigma^{-1}(z)={\phi_2/ \phi_1}.
\label{m391}\end{equation}
It is interesting to note that the integrand resembles
the chain rule for the Schwarzian derivative
\begin{equation}
\{w(t(z)),z\}(dz)^2-\{t(z),z\}(dz)^2=
\{w(t),t\}(dt)^2,
\label{oiioj}\end{equation}
in particular
\begin{equation}
\left\{J_H^{-1}\left(J_\Sigma^{-1}(z)\right),z\right\}
-\left\{J_\Sigma^{-1}(z),z\right\}=
\left\{J_H^{-1}\left(J_\Sigma^{-1}\right),
J_\Sigma^{-1}\right\}\left(\partial_zJ_\Sigma^{-1}(z)\right)^2.
\label{oiiojb}\end{equation}

We stress that the accessory parameters can be written
as a line integral of a one-form written in terms of theta
functions and holomorphic differentials.
In particular for the Fuchsian accessory parameters we have
$$
\lambda_k^{(F)}={1\over 2\pi i}\oint_{{\cal C}}\left(
{2\sum_{i,j=1}^h\omega_i''A_{ij}\overline \omega_j-
3\left(\sum_{i,j=1}^h\omega_i'A_{ij}
\overline\omega_j\right)^2\over 2\left(\sum_{i,j=1}^h\omega_iA_{ij}
\overline\omega_j\right)^2}
-\left\{\int^z_{P_0} H_f,z\right\} \right.
$$
\begin{equation}
\left. -{3\over 2} \left(Q_f(z)\over H_f(z)\right)^2
+2{T_f(z)\over H_f(z)}\right)e^{k+1-h_0}.\label{maa39}\end{equation}

In the second reference in \cite{0},
where the results for the punctured Riemann sphere are generalized
to higher genus Riemann surfaces, a relationship has been established between
$c_k^{(h)}=\lambda_k^{(F)}-\lambda_k^{(S)}$, the Liouville action
evaluated on the classical solution
and the Weil-Petersson metric.
In particular it turns out that
\begin{equation}
{1\over 2}{\partial S^{(h)}_{cl}\over \partial z_i}=c_i^{(h)},
\qquad {\partial c_i^{(h)}\over \partial \bar z_j}=-{1\over 2}
\left \langle
 {\partial\over \partial z_i}\,,{\partial\over \partial z_j
}\right \rangle_{WP},
\label{addendum1}\end{equation}
where the brackets denote the Weil-Petersson metric on the
Teichm\"uller space $T_h$
projected onto the Schottky space whose coordinate are
 $z_1,...,z_{3h-3}$.
Since the difference
\begin{equation}
\Theta(z)=T^F(z)-T^S(z)=
\sum_{k=1}^{3h-3}c_k^{(h)}\Omega^{k+1-h_0}(z),\label{difference}\end{equation}
 is a holomorphic quadratic differential
(i.e. a section of $T^\star T_h$),
the formulas in eq.(\ref{addendum1}) are equivalent to
 \begin{equation}
\partial S^{(h)}_{cl}=2\Theta,
\qquad {\overline\partial}\partial S^{(h)}_{cl}=-2i\omega_{WP},
\label{41}\end{equation}
where $d=\partial+\overline\partial$ is the exterior differentiation
on the Schottky space and $\omega_{WP}$ is the Weil-Petersson 2-form
on this space. Because the Schottky projective connection
depends holomorphically on the moduli we have
\begin{equation}
\overline\partial
T^F= - i\omega_{WP},\label{wppc}\end{equation}
that by (\ref{stress2}) gives
\begin{equation}
\omega_{WP}=i\overline
\partial{2\sum_{i,j=1}^h\omega_i''A_{ij}\overline \omega_j-
3\left(\sum_{i,j=1}^h\omega_i'A_{ij}
\overline\omega_j\right)^2\over 2\left(\sum_{i,j=1}^h\omega_iA_{ij}
\overline\omega_j\right)^2}.
\label{stress2ba}\end{equation}

Similar results have been derived by Fay \cite{fay2}. In particular it
turns out that
\begin{equation}
\left\{J_H^{-1},z\right\}= {\cal T}_\Sigma-24\pi i
\sum_{j,k=1}^h \left( {\partial\over \partial \Omega_{jk}}
\log c_0\right) \omega_j(z)\omega_k(z),\label{fay11}\end{equation}
where
\begin{equation}
c_0=\left[{8\pi^2 {\rm det}' \Delta\over {\rm det}\, {\rm
Im}\,\Omega}\right]^{-1/2},\label{anomalia}\end{equation}
is the anomaly in the spin-1/2 bosonization formula computed
with respect to the Poincar\'e metric $e^\varphi$.

The connection with the Weil-Petersson metric
on $T_h$  arises if we consider the quasi-conformal mapping
\begin{equation}
\partial_{\bar z} f^\rho =\rho \partial_z f^\rho,
\qquad \rho=t_1\nu_1+t_2\nu_2.\label{quasiabbastanza}\end{equation}
It turns out that
\begin{equation}
 -24\pi \partial\overline\partial
\log c_0 =\big<\nu_1,\nu_2\big>_{WP},\label{fay12}\end{equation}
where $\big<\nu_1,\nu_2\big>_{WP}=\int_\Sigma e^\varphi
\nu_1\overline\nu_2$ and
\begin{equation}
\partial=\partial_{t(p)}=\sum_{j,k=1}^h {\partial\over \partial \Omega_{jk}}
\delta \Omega_{jk},\label{schiffer1}\end{equation}
is the Schiffer variation (see \cite{fay2} for details).

Another possible way to investigate eq.(\ref{prova11}) is
by noticing that both the first and second variations vanish for
the deformation of the
complex structure induced by the harmonic Beltrami differentials
\cite{ahlfors,wolpert3}.
Applying this condition to (\ref{prova11}) should give further informations on
the form of matrix $A_{ij}$.

As a final remark, we observe that, besides any mathematical interest, the
solution for the Poincar\'e metric is crucial to get
explicit expressions for correlators in string theory. In particular in
the `uniformization approach' to 2D quantum gravity
\cite{tak1,mma} one needs the explicit expression for the Liouville action
evaluated at the classical solution to compute the `VEV of quantum Riemann
surfaces' $\langle\Sigma\rangle$ (see \cite{tak2,mmb}).