String Theory on K3 Surfaces

Mirror Symmetry II (B. Greene and S.-T. Yau, eds.), International
  Press, Cambridge, 1997, pp. 703-716

Abstract:
  The moduli space of N=(4,4) string theories with a K3 target space is
determined, establishing in particular that the discrete symmetry group is the
full integral orthogonal group of an even unimodular lattice of signature
(4,20). The method combines an analysis of the classical theory of K3 moduli
spaces with mirror symmetry. A description of the moduli space is also
presented from the viewpoint of quantum geometry, and consequences are drawn
concerning mirror symmetry for algebraic K3 surfaces.

\section*{Abstract}
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{\bf ABSTRACT}
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\begin{document}
\setcounter{page}0
\title{\large\bf String Theory\\
on K3 Surfaces\\[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 moduli space of $N$=(4,4) string theories with a K3 target space
is determined, establishing in particular that the discrete symmetry
group is the full integral orthogonal group of an even unimodular
lattice of signature (4,20).
The method combines an analysis of the classical theory of K3
moduli spaces with mirror symmetry. A description of the moduli
space is also presented from the viewpoint of quantum geometry,
and consequences are drawn concerning mirror symmetry for algebraic
K3 surfaces.

\end{abstract}

\vfil\break

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

In recent years, Calabi-Yau\ manifolds have
received great attention in the string literature. This is mainly because
compactification on
such spaces may be used to reduce the number of dimensions in models built
 from the intrinsically ten dimensional critical superstring \cite{CHSW:}.
The focus has largely been on the case of complex dimension $d=3$,
since the corresponding compactification is to four space-time dimensions.
(It is also the case that, in many respects, within topological
field theory on Calabi-Yau\ target spaces
the case $d=3$ plays a special r\^ole
\cite{W:aspects,BCOV:big}.)
However,
one might expect interesting properties for other values of $d$ as well.

In this paper we analyze the
case $d=2$ whose special features derive from the fact that the topology
of the target must be either
a torus or a K3 surface. The classification of
toroidal target spaces has been known for some time \cite{N:torus,HNW:torus}.
The local geometry of the moduli space of conformal field
theories with K3 target space topology
was understood shortly afterwards \cite{Sei:K3} and although some
aspects of the global form of the moduli space have been conjectured
and studied \cite{Sei:K3,Vafa:qu,GS:sym}, more precise statements
concerning this question have remained elusive.

We will use mirror symmetry to address this question of
the form of the moduli space. We find that, given a couple of minor
assumptions, combining a $\sigma$-model\ analysis near the field theory
limit with the study of a mirror symmetry transformation
is sufficient to give the precise form of the moduli
space. This paper is not intended to provide the full mathematical
exposition of the analysis required --- for that the reader is
referred to \cite{AM:K3m}. Here we only give a brief summary of the
methods used and conclusions reached by the analysis.

The analysis of the moduli space of string theories on a K3 surface
differs markedly from that of
the more familiar Calabi-Yau\ moduli spaces related to string compactifications
to four dimensions.
Firstly, in studying the case where the target space $X$ is a Calabi-Yau\ threefold,
the assumption $h^{2,0}=0$ is usually made.
Under this assumption,  the
deformations of complex structure of $X$ and of the (complexified)
K\"ahler form\ ``decouple'' to give the complete moduli space a local product structure
\cite{Cand:mir} (at least over generic points in the moduli space).
These two types of deformations may then be studied independently.
For a K3 surface we have $h^{2,0}=1$ and the above structure is lost.
We are thus forced to analyze deformations of complex structure and
deformations of K\"ahler form\ together.

In another respect however the K3 case
is simpler --- the $N$=(2,2) superconformal invariance of a string
with Calabi-Yau\ target space is extended to $N$=(4,4)
supersymmetry in the case $d=2$
\cite{Eg:K3}.
This is equivalent to the geometric statement that a Calabi-Yau\ manifold in
complex dimension two admits a hyperk\"ahler metric. This $N$=(4,4)
structure serves to fix the local form of the moduli space
completely. This can be contrasted with the case $d=3$ where the
extended chiral algebra
\cite{Odake:} has yet to provide much insight into the classification
of string theories.

As in the case $d=3$, the local dimension of the moduli space of
conformal field theories can be matched to the dimension of the moduli
space of the geometrical objects representing the target space if we
regard the ``geometrical object'' as including, in
addition to the target space metric, the specification of
a $B$-field, i.e., an
element of the second real (de Rham) cohomology group of the target. In the
case
$d=3$, the $B$-field naturally combined with the K\"ahler form to
provide a complexified K\"ahler form. In the case of K3 this is
clearly impossible --- $B$ lives in a 22-dimensional space whereas the
dimension of the space of K\"ahler forms varies with the complex
structure but is at most 20. Despite this fact we will still find that
a beautiful structure arises when the $B$-field is included in the
moduli space.

At the topological level, mirror symmetry might appear rather trivial
for a K3 surface since the mirror of a K3 surface (equipped with metric
and $B$-field) is another such K3
surface. However, the geometric data of the
mirror K3 surface thus obtained is not, in
general, isomorphic to the original. Thus the mirror map acts as a
non-trivial automorphism on the ``classical'' moduli space of K3 surfaces
(i.e., the $\sigma$-model\ moduli space of metrics and $B$-fields).
The mirror map in this context was first studied long ago in
\cite{Mart:CCC} and some of the observations in that paper lead to
some of the methods used here.

Although the purpose of this paper is to concentrate on the case where the
target space is a K3 surface, it should be noted that much of
what follows applies to any target space with $h^{2,0}=1$. The only
steps in the following argument which will not be directly applicable
to the determination of the moduli space of this more general case are the
explicit form of the moduli space of
Einstein metrics, and the particular self-mirror K3 theory used.

In section \ref{s:mod} we will present the outline of the construction
of the moduli space of $N$=(4,4) and $N$=(2,2) theories. In section
\ref{s:coh} we discuss the interpretation of the moduli space in terms
of the space of total cohomology of K3. In section \ref{s:alg},
some  aspects of mirror
symmetry on algebraic K3 surfaces are  discussed.

\section{The Moduli Space}	\label{s:mod}

Let us first fix some notations. $\Gamma\backslash G/H$ denotes the double
coset
space resulting from dividing the group $G$ by the right-action of
$H$ and left-action of $\Gamma$. ($G$ and $H$ will always be continuous Lie
groups and $\Gamma$ will be discrete.) $\Lambda^{a,b}$ denotes the unique
(up to isomorphism) even
self-dual lattice in $(a+b)$-dimensional space with signature $(a,b)$,
when $ab\ne0$.
Basis vectors may be taken such that the inner product has the
following form:
\begin{equation}
  \Lambda^{a,b}:\langle,\rangle\cong
\underbrace{(-E_8)\oplus(-E_8)\oplus\ldots}_{n\rm\;times}
\oplus H\oplus H\oplus\ldots,
\end{equation}
where $b-a=8n$
(one uses $E_8$ in place of $-E_8$ when $b-a<0$),
$E_8$ is the Cartan matrix of the Lie algebra $E_8$,
and $H$ is the hyperbolic plane:
\begin{equation}
 H = \left(\begin{array}{cc}0&1\\1&0\end{array}\right). \label{eq:H}
\end{equation}
We will use $X$ to denote a specific smooth K3 surface. One can show that
the intersection pairing on
$H^2(X,{\Bbb Z})$
gives it the structure of a lattice isomorphic to
$\Lambda^{3,19}$ (see for example \cite{BPV:}).
Let ${\Bbb R}^{a,b}$ be a $(a+b)$-dimensional space with an inner product
of signature $(a,b)$.
$O(a,b)$ is the orthogonal group on ${\Bbb R}^{a,b}$ and $O(\Lambda^{a,b})$
is the subgroup preserving $\Lambda^{a,b}\subset {\Bbb R}^{a,b}$. Let
\begin{equation}
 \Tch ab=O(a,b)/\left(O(a)\times O(b)\right).
\end{equation}
We can identify $\Tch ab$ as the set of space-like $a$-planes in
${\Bbb R}^{a,b}$ (i.e., $a$-planes on which the inner product is
positive-definite).  $\Tch ab$ can also be regarded as one of the
two connected components of the set of {\em oriented}\/ space-like
$a$-planes.  Under this latter interpretation, we can write
\begin{equation}
\Tch ab=O^+(a,b)/\left(SO(a)\times O(b)\right),
\end{equation}
where $O^+(a,b)$ is the index 2 subgroup of $O(a,b)$ with the
same connected components as $SO(a)\times O(b)$.

It was shown in \cite{Sei:K3} using arguments from supergravity
\cite{deRoo:} that the required moduli space, ${\Scr M}$, of conformal field
theories on a K3 surface is locally of the
form $\Tch 4{20}$. This is probably best understood  from
the argument
presented in \cite{Cec:nonp}, which we now review.
The $N$=(4,4) superconformal algebra
contains affine $SU(2)$ algebras in both the left and right sectors.
This symmetry acts on the marginal operators spanning the tangent
spaces of ${\Scr M}$. The existence of such a symmetry
 restricts the form of the holonomy group of the
Zamolodchikov metric of ${\Scr M}$.
The restriction is so severe, in fact, that
it then follows from Berger's classification \cite{Berg:}
of holonomy groups that either ${\Scr M}$ is locally isomorphic to a
quaternionic symmetric space, or that the holonomy is reducible.
Moreover, the
non-flatness of the Zamolodchikov metric is enough to rule out the
reducible holonomy case.
A simple analysis of any conformal field theory giving rise
to a K3 target space tells us that ${\Scr M}$ has  real dimension 80
\cite{Sei:K3,Eg:K3}.\footnote{One way to do this count is as
follows:  the space of complex structures has complex dimension 20
(real dimension 40), the space of K\"ahler forms has real dimension
20, and the space of $B$-fields has real dimension 22.  However, for each
Ricci-flat metric on the K3 surface there is an $S^2$ of complex
structures, so the overall dimension count is $40+20+22-2=80$.}
This,
together with the known classification of quaternionic symmetric
spaces (cf.~\cite{Wolf:}), completes the proof.

Let us now make two assumptions about the form of ${\Scr M}$, both of which
we consider to be quite reasonable. Firstly we assume ${\Scr M}$ is
geodesically complete\footnote{As we will point out below, we must
include orbifold points in our moduli space in order to ensure this
geodesic completeness.}
 and secondly we assume ${\Scr M}$ to be Hausdorff.
While some non-Hausdorff moduli spaces have appeared in string theory
\cite{Moore:nH}, this happened in the context of a target space of
indefinite signature. There is no reason to believe that something as
unpleasant as this should happen in the case of a K3 surface.
It then follows that
\begin{equation}
  {\Scr M}\cong \Gamma\backslash \Tch 4{20},
\end{equation}
for some group $\Gamma$ acting discretely on $\Tch 4{20}$.
Thus to complete the description of ${\Scr M}$, we only need to find $\Gamma$.

We begin our determination of $\Gamma$ by analyzing the
classical form of the moduli
space of K3 surfaces. Using the techniques of \cite{Borel-Serre}, one may
decompose $\Tch 4{20}$ as
\begin{equation}
  \Tch4{20} \cong {\Bbb R}_+\times\Tch3{19}\times {\Bbb R}^{3,19},
		\label{eq:decomp}
\end{equation}
where ${\Bbb R}_+$ is the half-line of positive real numbers which we
parameterize by $\lambda$.
The decomposition depends on a choice of null vector $v\in{\Bbb R}^{4,20}$,
and the space $v^\perp/{\Bbb R} v$ provides the ${\Bbb R}^{3,19}$ on which
$O(3,19)$ acts and on which $\Tch 3{19}$ is based.
The space $\Tch 3{19}$ is known to be isomorphic to
the Teichm\"uller space for Einstein metrics of volume one on a K3 surface
(and their orbifold limits)
\cite{Mor:Katata,Kobayashi-Todorov,Anderson}. It is natural then to
identify $\lambda$ as giving the size of the K3 surface and ${\Bbb R}^{3,19}$
as the moduli space of $B$-fields on the target space. These
identifications may be established by looking at the metrics on the
above mentioned spaces. The Zamolodchikov metric on $\Tch 4{20}$ is
known to be the left-invariant metric. From \cite{Borel:stable} this induces
a metric on each of the terms on the right hand of (\ref{eq:decomp})
where this should now be viewed as a ``warped product'', i.e., the
metric does not respect the product structure. One may now show
(using \cite{Anderson}) that
the metrics induced are those given precisely by the extended
Weyl-Petersson metric in the sense of \cite{CGH:WP}. Such an isometric
identification allows us to
identify every point in $\Tch4{20}$ with an Einstein metric on a K3
surface (or orbifold metric)
together with a $B$-field. This can be taken as another
version of the statement that the non-linear \sm\ is exactly conformally
invariant on a K3 surface with Ricci-flat metric
\cite{AGG:N=4,Hull:N=4,BS:N=4}. It also shows that the Zamolodchikov
metric and Weyl-Petersson metric coincide exactly on the moduli space
--- a fact which in general $N$=(2,2) theories holds only to leading
order in the large
radius limit \cite{CHS:metric}.

The moduli space of smooth Einstein metrics of volume one on a K3
surface $X$ is determined in \cite{Anderson} to be
$\Gamma_0\backslash\Tch3{19}-{\Scr Z}$, where $\Gamma_0=\operatorname{Diff}(X)/
\operatorname{Diff}^0(X)$ is the group of
components of the diffeomorphism group of $X$, and ${\Scr Z}$ is the space of
orbifold metrics. (It is
generally believed that such
orbifolds should be included when considering a string target space
\cite{DHVW:}; if we include them, we get the geodesically complete
space $\Gamma_0\backslash\Tch3{19}$.)
The discrete group $\Gamma_0$ is determined in
 \cite{Mat:d,Borcea:d,Don:inv} to coincide with the group
\begin{equation}
O^+(\Lambda^{3,19}):=O(\Lambda^{3,19})\cap O^+(3,19),
\end{equation}
which has index 2 in $O(\Lambda^{3,19})$. The
``missing'' ${\Bbb Z}_2$ in $O(\Lambda^{3,19})$
may be generated by $-I$ acting on ${\Bbb R}^{3,19}$.

We also know that the non-linear \sm\
on a Calabi-Yau\ manifold is invariant under all translations $B\to B+v$, where $v\in
H^2(X,{\Bbb Z})$. Thus we should divide the space ${\Bbb R}^{3,19}$ of $B$-fields
by additive translations by
$\Lambda^{3,19}$.   (As an abstract group, these additive translations
simply form a ${\Bbb Z}^{22}$.)
Furthermore one may consider complex conjugation of the target space.
If one considers such a complex conjugation for a $\sigma$-model\ one sees that
the transformation $B\to-B$ is required in addition to the conjugation
of the complex structure of the target. Such a transformation may be
represented by $-I$ and thus generates the missing ${\Bbb Z}_2$ from above.
Therefore, in terms of the decomposition (\ref{eq:decomp})
one sees a group $O(\Lambda^{3,19}).\Lambda^{3,19}$ (which is the full
space-group of $\Lambda^{3,19}$) of identifications
that should be made on the right-hand side and thus on $\Tch4{20}$.
Thus we obtain
\begin{equation}
  \Gamma\supseteq O(\Lambda^{3,19}).\Lambda^{3,19}. \label{eq:clsym}
\end{equation}

This is the maximal set of identifications that can be made from
classical geometry. Any further statement requires some quantum
geometry. The mirror construction of \cite{GP:orb} provides us with
such a tool. First we need the clarify the meaning of the mirror map
in the context of $N$=(4,4) theories.
Mirror symmetry reverses the sign of a $U(1)$ charge derived from an
$N$=2 chiral algebra.  Now
an $N$=4 chiral algebra
contains an $SU(2)$ affine subalgebra, which is larger than the
$U(1)$ affine subalgebra of an $N$=2 theory.  Given an $N$=4 theory,
though, we may choose an $N$=2
subalgebra of the $N$=4 algebra by specifying the corresponding
$U(1)\subset SU(2)$. Each $N$=4 theory thus gives rise to an
$S^2\cong SU(2)/U(1)$ of $N$=2 theories. This leads to construction of
the moduli space of $N$=(2,2) theories of string on a K3,
which we denote by ${\Scr M}_{q\bar q}$, as a fibre bundle
\begin{equation}
  \pi:\;{\Scr M}_{q\bar q}\to{\Scr M},
\end{equation}
with fibre $S^2\times S^2$.
Each point in ${\Scr M}_{q\bar q}$ corresponds to an $N$=(4,4) theory
on which a particular $N$=(2,2) structure has been chosen;
the subscripts $q$ and $\bar q$ denote
charges with respect to the left and right $U(1)$ currents respectively
within the
$N$=(2,2) theory.

A {\em (left) mirror map}\/ on the corresponding Teichm\"uller
space $\Tch4{20}_{q\bar q}$ of $N$=(2,2) theories is a map
$\mu:\Tch4{20}_{q\bar q}\to\Tch4{20}_{q\bar q}$ with the property
that, for all $S\in\Tch4{20}_{q\bar q}$, the $N$=(2,2) theories
at $S$ and $\mu(S)$ are isomorphic, but with a switch of charge
assignments $(q,\bar
q)\leftrightarrow(-q,\bar q)$. We also define {\em right mirror maps}\/
which switch $(q,\bar q)\leftrightarrow(q,-\bar q)$. Such a map
on $\Tch4{20}_{q\bar q}$  can be pushed down to a map $\bar\mu$ on
the Teichm\"uller space $\Tch4{20}$ ---
since two theories which are mirrors as $N$=(2,2) theories
are isomorphic as $N$=(4,4) theories, any mirror map within $\Tch
4{20}$ should give rise to an element of $\Gamma$. Note that some
mirror maps on $\Tch4{20}_{q\bar q}$ may be trivial on $\Tch4{20}$,
in the sense that they induce the identity element of $\Gamma$.
Such maps relate pairs of points in $\Tch4{20}_{q\bar q}$ which
lie over the same point in the base space $\Tch4{20}$.
What will be of more interest are
the nontrivial mirror maps which map to a nontrivial element of
$\Gamma$. The difference between trivial and nontrivial maps is shown
schematically in figure \ref{fig:f1}.

\iffigs
\begin{figure}
  \centerline{\epsfxsize=14cm\epsfbox{k3-f1.ps}}
\unitlength=1mm \begin{picture}(0,0)
\put(3,45){\makebox{$\Tch4{20}_{q\bar q}$}}
\put(3,12){\makebox{$\Tch4{20}$}}
\end{picture}
  \caption[f1]{Trivial and nontrivial mirror maps. $\Tch4{20}_{q\bar q}$ is
an $(S^2\times S^2)$-bundle over $\Tch4{20}$ as shown. A theory (shown as
an open
circle) is associated with a pair of points, one on each sphere. In
this figure $\mu_1$ is trivial and $\mu_2$ is nontrivial.}
  \label{fig:f1}
\end{figure}
\fi

We now consider an example of a generically nontrivial mirror map with a fixed
point on $\Tch4{20}$. Given the
Gepner model \cite{Gep:} associated via \cite{GVW:} to the K3
hypersurface
\begin{equation}
  X_0^2+X_1^3+X_2^7+X_3^{42}=0 \label{eq:K12Y}
\end{equation}
in the weighted projective space ${\Bbb P}^3_{\{21,14,6,1\}}$ we find the
mirror as an orbifold of the original space by the method of \cite{GP:orb}. The
orbifolding group thus found is trivial and so this theory is its own
mirror.
The fact that this theory is self-mirror is also related to some
issues regarding Arnold's ``strange duality'' observed in \cite{Mart:CCC}.
This mirror map acts non-trivially on the marginal operators of
this theory and thus on the tangent bundle of $\Tch4{20}$.
By analyzing this action, it is possible to show that there is a lattice
$\Lambda^{4,20}\subset{\Bbb R}^{4,20}$ such that the induced automorphism $\bar\mu$
of $\Tch4{20}$ lies in $O(\Lambda^{4,20})$.  In
fact, there
is a decomposition
\begin{equation}
  \Lambda^{4,20}\cong \Lambda^{2,10}\oplus\Lambda^{2,10},
		\label{eq:twobits}
\end{equation}
such that $\bar\mu$ acts by simply exchanging the two terms on
the right-hand side. The decomposition can be chosen so that one of the
$\Lambda^{2,10}$ lattices in (\ref{eq:twobits}) is a sublattice of the
$\Lambda^{3,19}$ lattice on which the classical $O(\Lambda^{3,19})$
symmetry of (\ref{eq:clsym}) acts.

We have thus found the explicit form of another generator of $\Gamma$. A
straight-forward but somewhat involved calculation then shows that the
classical symmetries, $O(\Lambda^{3,19})
.\Lambda^{3,19}$, and the above ${\Bbb Z}_2$ mirror map
generate the whole of $O(\Lambda^{4,20})$. That is, we have
\begin{equation}
  \Gamma\supseteq O(\Lambda^{4,20}).  \label{eq:Gmax}
\end{equation}
But now from \cite{Allan:}, which establishes that $O(\Lambda^{4,20})$
is a {\em maximal}\/ discrete group acting on $\Tch4{20}$ to yield a
Hausdorff quotient, it follows that
\begin{equation}
  \Gamma\cong O(\Lambda^{4,20}),
\end{equation}
completing the proof that the moduli space has the precise form
\begin{equation}
{\Scr M}=O(\Lambda^{4,20})\backslash O(4,20)/\left(O(4)\times O(20)\right)
\end{equation}
(as had been speculated by Seiberg \cite{Sei:K3}).

To summarize, we have shown that the full group of identifications to
be made on the Teichm\"uller space is generated by the following:
\begin{enumerate}
  \item Classical
identifications made for the moduli space of complex structures.
  \item The integral $B$-field shifts.
  \item Complex conjugation (together with a change
in sign of the $B$-field).
  \item The mirror map.
\end{enumerate}
One should note that
this is	 also true for the case $d=1$ where the target space must be a
torus~\cite{DVV:torus}.

It is interesting to note that this moduli space ${\Scr M}\cong\Gamma
\backslash\Tch4{20}$ is precisely that which one would obtain for
toroidal compactifications of the heterotic string down to
six-dimensional spacetime using the methods of \cite{N:torus}. This
equivalence between the moduli space of K3 compactifications and
toroidal compactifications (for the relevant strings) was established
locally in \cite{Sei:K3} but it is unclear why they turn out to be
globally isomorphic.

\section{The Space of Total Cohomology}   \label{s:coh}

The derivation of the moduli space ${\Scr M}$ in the previous section and
\cite{AM:K3m} is somewhat unpleasant and yet produces a beautifully
symmetric result. This is a consequence of having derived the moduli space
using
the ideas of classical geometry necessitating the intermediate step of
equation (\ref{eq:clsym}). Now that we have the moduli space we may
reinterpret it from the standpoint of quantum geometry in a more
symmetric way.

First let us quickly review the form of the moduli space of complex
structures on a classical K3 surface. Consider a non-vanishing
holomorphic 2-form $\Omega$
on $X$ representing an element of $H^{2,0}(X)$. Let
us specify a ``marking'' of $X$, i.e., a basis $\gamma_i$,
$i=1,\ldots,22$ of $H_2(X,{\Bbb Z})$ determining an isomorphism of that lattice
with $\Lambda^{3,19}$. The global
Torelli theorem \cite{P-SS:,BR:,Mor:Katata} essentially says
that the complex structure of $X$ is
expressed uniquely in terms of the periods
\begin{equation}
  \int_{\gamma_i}\Omega,
\end{equation}
up to a change of basis of $H_2(X,{\Bbb Z})$. To state this more
precisely, we write $\Omega=\xi+i\eta$,
where $\xi,\eta\in H^2(X,{\Bbb R})\cong \Lambda^{3,19}\otimes_{{\Bbb Z}}{\Bbb R}$. The
Hodge-Riemann bilinear relations assert that
\begin{equation}
  \eqalign{
  \langle\Omega,\Omega\rangle&=\int_X\Omega\wedge\Omega=0\cr
  \langle\Omega,\bar\Omega\rangle&=\int_XdV\,\|\Omega\|^2>0.\cr}
\end{equation}
This implies that the oriented 2-plane $\xi\wedge\eta$
in $H^2(X,{\Bbb R})$ spanned by $\xi$ and
$\eta$ is space-like, i.e., has positive definite metric.
In fact, specifying $\Omega$ up to multiplication by ${\Bbb C}^*$ (which
determines the Hodge structure) is equivalent to specifying the
oriented 2-plane $\xi\wedge\eta$.  Now
the moduli space of oriented space-like 2-planes
in $H^2(X,{\Bbb R})\cong{\Bbb R}^{3,19}$ is given by
$O(3,19)/\left(SO(2)\times O(1,19)\right)$.
When we mod out by diffeomorphisms, we would expect to obtain a
description of the moduli space of complex structures on a K3 surface
as an open subset of
\begin{equation} \label{eq:cxmod}
  O^+(\Lambda^{3,19})\backslash O(3,19)/\left(SO(2)\times O(1,19)\right)
\end{equation}
(and we would expect to get the entire space (\ref{eq:cxmod}) if we
include orbifold complex structures).  Actually, there are some technical
difficulties in interpreting (\ref{eq:cxmod}) as a moduli space,
but the interpretation is essentially correct (see \cite{Mor:Katata} for
more precise statements).

The extension of this to string theory follows immediately. The
space $\Tch4{20}$ is one connected component of
the set of oriented space-like 4-planes in ${\Bbb R}^{4,20}$.
The space ${\Bbb R}^{4,20}$ can be identified with the space
$H^*(X,{\Bbb R})$ of all real cohomology groups on $X$ --- the inner product
is obtained by supplementing the intersection pairing on $H^2(X,{\Bbb R})$
 by a pairing on $H^0(X,{\Bbb R})\oplus H^4(X,{\Bbb R})$ which has matrix
(\ref{eq:H}) on the standard generators of those two spaces (i.e., the class
of a point, and the class of the entire space $X$).
The natural lattice $H^*(X,{\Bbb Z})$ inside of $H^*(X,{\Bbb R})\cong{\Bbb R}^{4,20}$
inherits an integer-valued pairing from this inner product,
giving it the structure
$\Lambda^{4,20}$.
 An $N$=(4,4) theory
on a K3 surface is thus specified uniquely by an oriented
 space-like 4-plane in
$H^*(X,{\Bbb R})$ (belonging to the correct component)
where the 4-plane is located relative to the $H^*(X,{\Bbb Z})$
lattice.

This picture becomes clearer when we go to the $N$=(2,2) moduli space.
Now the moduli space, ${\Scr M}_{q\bar q}$, is an $(S^2\times S^2)$-bundle over
${\Scr M}$. In fact one may show that the Teichm\"uller space takes
the form
\begin{equation}
  \Tch4{20}_{q\bar q}=O^+(4,20)/\left(SO(2)\times SO(2)\times O(20)\right).
\end{equation}
In the language of the previous paragraph, this is one component of the space
of
orthogonal pairs of oriented
space-like 2-planes in ${\Bbb R}^{4,20}$. That is, we not only
specify an oriented
space-like 4-plane but we give it an internal structure of being
spanned by two orthogonal oriented
2-planes. We may now specify these oriented 2-planes
in terms of complex vectors as in the classical case. Namely, define
$\Omega$ and $\mho$ as arbitrary elements of $H^*(X,{\Bbb C})$ (mod ${\Bbb C}^*$)
such that
\begin{equation}
  \eqalign{
  \langle\Omega,\Omega\rangle&=\langle\Omega,\mho\rangle=
  \langle\mho,\mho\rangle=\langle\bar\Omega,\mho\rangle=0,\cr
  \langle\Omega,\bar\Omega\rangle&>0,\quad\langle\mho,\bar\mho\rangle>0.
  \cr} \label{eq:Om}
\end{equation}
The space of such vectors gives precisely the Teichm\"uller space
required. Thus we may identify the moduli space of $N$=(2,2) theories
on a K3 surface as being the space of all possible choices of $\Omega$
and $\mho$ subject to (\ref{eq:Om}) where the positions of these
vectors are located relative to $H^*(X,{\Bbb Z})$. This statement may be
considered to be the quantum version of the global Torelli theorem.

As explained earlier,  mirror maps will act on the Teichm\"uller space
 $\Tch4{20}_{q\bar
q}$. In fact a nontrivial
mirror map may be identified with the interchange $\Omega
\leftrightarrow\mho$. Our picture now begins to look very similar to
that pursued in \cite{Cand:mir} for the case $d=3$. There is a slight
difference since in the latter case one had
\begin{equation}
  \eqalign{\Omega&\in H^{3,0}\oplus H^{2,1}\oplus H^{1,2}
	\oplus H^{0,3},\cr
   \mho&\in H^{0,0}\oplus H^{1,1}\oplus H^{2,2}
	\oplus H^{3,3}.\cr}
\end{equation}
For the K3 case however, we may only specify the general statement
that both $\Omega$ and $\mho$ lie in the full cohomology
$H^*$ at a generic point in moduli space. The
following construction may be used to reconcile the
approaches. Let us choose two null vectors $v$ and $w$
in $H^*(X,{\Bbb Z})\cong \Lambda^{4,20}$ such
that $\langle v,w\rangle=1$, so that $v$ and $w$ span a hyperbolic plane.
Let $F$ be
the subspace of $H^*(X,{\Bbb R})$ perpendicular to $v$ and $w$. For each
oriented space-like 4-plane $\Pi$, we define
\begin{equation}
\eqalign{\Omega&:=\ \mbox{complex vector associated to }\Pi\cap F,\cr
\mho&:=\ \mbox{complex vector associated to }\Omega^\perp\cap\Pi.\cr}
\end{equation}
This
amounts to choosing a specific $N$=(2,2) theory to represent an
$N$=(4,4) theory. We may now identify $F$ with $H^2(X,{\Bbb R})$. If
$\Omega$ specifies the $H^{2,0}$ direction, (\ref{eq:Om}) then implies
\begin{equation}
  \eqalign{\Omega&\in H^{2,0}\oplus H^{1,1}\oplus H^{0,2},\cr
   \mho&\in H^{0,0}\oplus H^{1,1}\oplus H^{2,2}.\cr}
\end{equation}
The complex structure of $X$ can thus be determined from $\Omega$ in
the usual way. One can also obtain the K\"ahler form and $B$-field
information from $\mho$.  The mirror map interchanges the two.

Note that we may equally specify $\Omega\to\bar\mho,\mho\to\bar\Omega$
as a mirror map. We may compose this with the previous mirror map to
produce the map $\Omega\to\bar\Omega,\mho\to\bar\mho$. This map may be
identified as the map taking $(q,\bar q)\to(-q,-\bar q)$ in the
$N$=(2,2) theory or equivalently as complex conjugation of the target
space. Note that such a map is usually trivial in the moduli space of
$N$=(2,2) theories but that our construction, through specific
labeling of the $U(1)$ charges, leads to a non-trivial map.

\section{Mirror Symmetry for Algebraic K3 Surfaces}      \label{s:alg}

Now that we have identified how mirror maps
act on the moduli space of conformal field theories on a K3 surface,
we can explain several of the mirror-type phenomena which have been
observed for algebraic K3 surfaces.

Given any subgroup $M$ of $H^2(X,{\Bbb Z})$ such that the cokernel
$H^2(X,{\Bbb Z})/M$ has no torsion and the intersection form on
$M\otimes{\Bbb R}$ has signature $(1,\rho-1)$, we can associate the
family of ``algebraic K3 surfaces of type $M$'' which consists of
all complex structures on the K3 for which the classes on $M$ are
all of type $(1,1)$.  This construction was introduced into the
mathematics literature \cite{Pinkham} as part of the explanation
of Arnold's ``strange duality'' put forward by Pinkham and Dolgachev,
and moduli spaces of these structures were studied in detail in
\cite{Mor:Tod,MS:}.
The moduli space of algebraic K3 surfaces of type $M$ has complex dimension
$20-\rho$ when $M\otimes{\Bbb R}$ has signature $(1,\rho-1)$.

We can extend this idea, and
define the set of ``CFT's of type $M$'' on a K3 surface
to be the image in the conformal field theory moduli space of the
set of $\sigma$-model s  whose complex structure is an algebraic K3 surface
of type $M$, and whose K\"ahler class and $B$-field are taken from
the space $M\otimes{\Bbb R}$.
The complexified K\"ahler moduli space
has complex dimension $\rho$, so the entire CFT moduli space of type
$M$ has complex dimension $(20-\rho)+\rho=20$.  Since the construction
segregates complex and K\"ahler deformations, the CFT moduli space of
type $M$ embeds naturally into the $N$=(2,2) moduli space ${\Scr M}_{q\bar{q}}$.

Suppose that the orthogonal complement $M^\perp$ can be written in
the form $M^\perp=H\oplus N$ for some lattice $N$ (orthogonal direct
sum).  Then a mirror map $\mu$ can be defined which exchanges the
given hyperbolic plane $H$ with the hyperbolic plane $H^0(X)\oplus H^4(X)$.
When $\mu$ is applied to the CFT moduli space of type $M$, it reverses
the r\^oles of K\"ahler and complex structure deformations.  It is
not hard to see that the resulting moduli space is in fact the CFT
moduli space of type $N$ --- the complex structure is one for which $N$
(rather than $M$) consists of $(1,1)$ classes, and the K\"ahler structure
is now taken from $N\otimes{\Bbb R}$.
This gives a precise CFT interpretation of Arnold's strange duality
and other related phenomena for K3 surfaces (an earlier version of
which was given by Martinec \cite{Mart:CCC}).

Finally we note a link between our construction of the mirror map for
K3 surfaces and Voisin's construction of mirror pairs of Calabi-Yau\
threefolds \cite{Voi:K3} (see also \cite{Bor:K3}). The latter
construction consists of taking an orbifold $Y=(X\times E)/{\Bbb Z}_2$, where
$E$ is a torus of complex dimension one. It was shown in \cite{Voi:K3}
that for suitable pairs $X_1,X_2$ of $X$, $Y_1$ and $Y_2$ are mirror
pairs at the level of the Hodge numbers, $H^{p,q}(Y_1)=H^{3-p,q}(Y_2)$;
and ``A-model'' and ``B-model'' \cite{W:AB} correlation functions.
The K3 surfaces involved must be of type $M$, with $M$ the lattice
invariant under the ${\Bbb Z}_2$-action.
One may show
that our mirror map for K3 surfaces directly establishes $X_1$ and
$X_2$ as a mirror pair and thus it follows that $Y_1$ and $Y_2$ are
truly mirror manifolds at the level of conformal field theory.

\section*{Acknowledgements}

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.


Resolution of simple singularities yielding particle symmetries in a
  space-time

Abstract:
  A finite subgroup of the conformal group SL(2,C) can be related to invariant
polynomials on a hypersurface in C^3. The latter then carries a simple
singularity, which resolves by a finite iteration of basic cycles of
deprojections. The homological intersection graph of this cycles is the Dynkin
graph of an ADE Lie group. The deformation of the simple singularity
corresponds to ADE symmetry breaking. A 3+1-dimensional topological model of
observation is constructed, transforming consistently under SL(2,C), as an
evolving 3-dimensional system of world tubes, which connect ``possible points
of observation". The existence of an initial singularity for the 4-dimensional
space-time is related to its global topological structure. Associating the
geometry of ADE singularities to the vertex structure of the topological model
puts forward the conjecture on a likewise relation of inner symmetries of
elementary particles to local space-time structure.

\section{\bf Introduction}
\setcounter{equation}{0}
Relations between finite simple groups, simple singularities and simply
laced Lie groups, denoted by the Cartan series A, D and E,
are wellknown$^{{1}-\Ur}$, but have not been used so far
to tie up a particle's inner symmetry, expressed by an ADE Lie group,
explicitly to local space-time geometry near the particle.
This geometry is assumed to be given as a vertex structure in a
$3+1$-dimensional topological model of observation,
which is based on a network of spatial world tubes evolving in time.
The present approach, inspired by string theory$^\Gr$, differs
from that by working a priori in $3+1$ dimensions. Breaking of
conformal symmetry is considered here as naturally given by the global
topology of space-time and the structure of observation.
The following section defines the topological model and discusses
democracy between ``possible points of observation" (PPOs) in its
relation to a possible initial singularity of the space-time and its global
topology.

Sec. 3 presents an overview over the conformal action of SL(2,C) and its
relatives on the structures of the model.

Sec. 4 resumes known results$^{{1}-\Kl}$ on the finite subgroups
of SL(2,C), associated simple singularities, their resolutions and
homological relation of the latter to the ADE Lie groups.

Sect. 5 describes the singularities of the A and D series explicitly on
characteristic algebraic hypersurfaces in $C^3$, following essentially
the original work of F. Klein$^\Kl$.

Sec. 6 reviews deformations of simple
singularities$^{\Du,\Hu,\Ur}$ and presents a theorem$^\Ur$,
wellknown to algebraic topologists rather than physicists,
which allows to decide
the existence of certain deformations of the singularities by a graph
theoretical method. By the latter the relation to symmetry
breaking transitions between the associated ADE Lie groups becomes evident.
This together with Sec. 3 has recently encouraged  the author to
conjecture$^\Ra$ a close relation of ADE symmetries and local structure of
observation in space-time.

In Sec. 7 the geometric structure of the resolution of
singularities of series A is described explicitly. A properly chosen real
section can be identified  with some vertex of the model presented in
Sec. 2.
This confirms the conjecture above.

Finally we discuss the perspective of results for this model.

\section{\bf Topological model of space-time observation}
\setcounter{equation}{0}
Starting with $n+1$ vertices and demanding
any of them being connected to any other one by a stringy world tube,
the lowest dimension allowing the connections to be nonintersecting
is $d=3$, which shall be the case. Up to Sec. 6 the vertices are assumed
to contain a ``possible point of observation" (PPO) each.
{}From any vertex there are $n$ tubes going out, as the
only possibility to exchange information with other vertices.
The whole worldsheet shall be closed.
This structure is assumed to be embedded in some spacelike section $M^3$ of
the Lorentzian part $M^4\cong M^3\times  \mbox{$\rm R $} $ of a
classical space-time and to evolve in $M^4$ according to a synchronized
time of one of the PPOs, which will be called ``the observer" O.
Since all $n+1$ PPOs are equal, anyone of them could
observe a likewise system of $n$ vertices. This fact is called ``democracy".
Fig. 1 shows an example of 5 vertices connected to each
other and all other ones.

Unification/separation of a single pair of vertices yields
separation/ unification of $n$ pairs of points on the worldsheet
after pinching/before blowing up n (closed) strings respectively:
\medskip  {\hfill \break}
{\normalsize
\begin{tabular}{ll} \quad
\begin{tabular}{llll}
{unify 2 PPOs}& $ {\Rightarrow } $& n-times pinch \& split &{$ S^1  {\longrightarrow }  \cdot  {\longrightarrow }  : $}
\\
{split 1 PPO} & $ {\Rightarrow } $& n-times glue \&blow up
&{${\,}:{\,} {\longrightarrow } \cdot {\longrightarrow }  S^1 $}
\end{tabular} &\quad  (2.1)
\end{tabular} }
In (2.1) the topological transformations of pinching
or blowing up of a closed string $S^1$ are denoted resp. by
$S^1 {\longrightarrow }  \cdot$ and $\cdot {\longrightarrow }  S^1$,
while $\cdot  {\longrightarrow }  :$ denotes the separation of a pair of points $(:)$ by
splitting a single point $(\cdot)$ and $:  {\longrightarrow }  \cdot$ is the inverse
operation, the unification of the pair, gluing it into a single point.
\vspace{5.0truecm}
\begin{center}
{\normalsize Fig. 1: Net of vertices of PPOs connected by stringy tubes}
\end{center}
After O has been chosen and the
connecting tubes between all observed vertices removed, the worldsheet
connected to the observer can be deformed smoothly into a sphere
 \mbox{$ S^2 $} , centered at O, with $n$ punctures, corresponding to the observed
vertices.
This can be achieved by conformal rescaling of the original worldsheet
metric $h_0$ to $h=e^{\Phi} h_0$ by an appropriate choice$^\Gr$ of some
scalar field $\Phi$ on the tubular world sheet. The removed
connections between the observed vertices would correspond to
$n(n-1)/2$ handles attached to \mbox{ \mbox{$ S^2 $} .}

In a continuum limit $n \to \infty$ the punctures on  \mbox{$ S^2 $}  are assumed to
become dense on the whole  \mbox{$ S^2 $} . Assuming a homogeneous distribution
of punctures implies local rotation symmetry$^{9}$ on $M^3$.

Since O is at the center and all others PPOs are on  \mbox{$ S^2 $} , one might  think
that the original democracy is broken by observation. Whether this is so
depends however on the global topology of $M^3$.  {\hfill \break}
If e.g. $M^3\cong S^3$ (Fig. 2 a),
it is possible to embed O together with  \mbox{$ S^2 $}  in $M^3$.
By shrinking  \mbox{$ S^2 $}  , O can be exchanged with an other PPO
by an arbitrarily small deformation on $M^3$. Such a
situation is given in Bianchi IX models, e.g. de Sitter space-time,
which is free of an initial singularity.  {\hfill \break}
If on the other hand e.g. $M^3\cong  \mbox{$ S^2 $} \times  \mbox{$\rm R $} $ (Fig. 2 b),
embedding  \mbox{$ S^2 $}  symmetrically in $M^3$ excludes O from $M^3$.
Thus it cannot be exchanged with any other PPO.
Such a situation appears in Kantowsky Sachs (KS) models,
which have an essential singularity (i.e. one which cannot be removed
by coordinate transformations) at a point or even an interval of
time$^\Qa$.
\vspace{5.0truecm}
\begin{center}
{\normalsize Fig. 2: Democracy a) conserved in $M^3\cong S^3$,
b) broken in $M^3\cong \mbox{$ S^2 $} \times  \mbox{$\rm R $} $}
\end{center}
Singularities of spherical space-times like those mentioned above have been
classified recently in Ref. \Qa. One is led
to the conjecture that singularities in space-time are essentially effected
by a breaking of the permutation symmetry between all PPOs (vertices).

Note that the affliction of classical space-times by singularities mentioned
above refered to the case of Einstein-Hilbert (EH) theory, given by a
Lagrangian $L\!=\!R$.
For theories with additional higher order curvature invariants in $L$
a singularity which appears for a certain model in EH theory may be
evitable$^\Sa$, since the strong energy condition may not be valid and then
the Hawking-Penrose  theorem$^{12}$ does not apply.

A resolution of scale factor singularities$^\Qa$ of the
classical space-time in the quantum model$^{\An,{14}}$
corresponds$^\Ra$ to blowing up O to $S^3$.
Note that quantum theory even more than
special relativity (SR) does not admit a separation of O from the
space-time.
The usual projection  \mbox{$ S^2 $}  $\to$ C used to evaluate string theory as an
effective QFT in flat 2-space yields a conformal anomaly$^\Gr$ for
dimensions $D \ne 26$. Note that this projection simultaneously destroys
the symmetry between O and other PPOs.

Finally the alternative approach of implementing the principle of
SR at the quantum level by using Quantum Frames of Reference (QFR),
yields a quantized space-time$^{15}$. This should be consistent with
the present multi-vertex approach by attaching a QFR to each vertex.

\section{\bf Action of SL(2,C) on the model structures}
\setcounter{equation}{0}
Consider now the group of invertible complex $2\times 2$ matrices GL(2,C).
The elements of determinant 1 form a normal subgroup SL(2,C). Division
of GL(2,C) resp. SL(2,C) by its center $\rm C^*$ resp. $\{\pm E\}$
yields the group PGL(2,C) resp. PSL(2,C), which is the analogue in
projective space, and can be identified with the automorphisms of  \mbox{$ S^2 $}
(or the M\"obius tranformations Aut $\rm\hat C$) resp. the proper
ortochronous Lorentz group $\rm L_{+}^\uparrow$
(or the 3-dim. complex orthogonal group). One gets the
following diagram with exact vertical sequences and an exact horizontal
splitting one: {\hfill \break}
\medskip
{\normalsize
\begin{tabular}{cccccccccl}
&& &1& &1& & & & \\
&& &$ {\uparrow } $& &$ {\uparrow } $& & & & \\
SO(3,C)&=&$\rm L_{+}^ {\uparrow }  =$&PSL(2,C)&&PGL(2,C)&= Aut $\rm\hat C$&=&Aut \mbox{$ S^2 $} &\\
&& &$ {\uparrow } $& &$ {\uparrow } $& & & & \\
&1&$ {\hookrightarrow } $&SL(2,C)&$\to$&GL(2,C)&
${{ det\atop {\longrightarrow } } \atop \hookleftarrow}$&$\rm C^* $&$ \to\ 1$&  \\
&& &$ {\uparrow } $& &$ {\uparrow } $& & & & \\
&& &$\{\pm E\}$& &$\rm C^*$& & & & \\
&& &$ {\uparrow } $& &$ {\uparrow } $& & & & \\
&& &1& &1& & & (3.1)&
\end{tabular}
\medskip }
By (3.1) {\it transformations of  \mbox{$\rm C^2 $} -spinors}, SL(2,C),
project to {\it transformations of bivectors}, SO(3,C),
and {\it proper orthochronous Lorentz transformations},  \mbox{$\rm L_{+}^\up $} ,
and they essentially constitute (modulo dilatations and
phase factors){\it conformal transformations}, PGL(2,C),
on the Riemann sphere or
projective plane, which correspond conformally to transformations of a
tubular worldsheet$^\Gr$. Thus the transformation properties of the model of
Sec. 2 are fixed, since all its structures are covered by the
transformations above. In this paper only the worldsheet as observational
structure and the 4-dimensional space-time are considered.
(For a transcription to the language of spinors and their relation to the
light cone see Ref. \Pe.)

In the following section we will deal with another
very important structural property of SL(2,C), namely its finite subgroups,
which are related uniquely to the simple singularities.

\section{\bf SL(2,C) subgroups and simple singularities}
The finite subgroups F of SL(2,C), together with the properties of the
related simple singularities$^{{1},\Du,\Ki}$, are listed
in the following table:
\medskip {\hfill \break}
{\normalsize
\begin{tabular}{llrrcl}
F    & name           &order&$R$(X,Y,Z) \qquad &{\ }basic graph\qquad& Lie G
\medskip\\
$C_n$&cyclic      &$n${\ } &$ {\rm X} ^n +{\ } {\rm Y} ^2{\,}+ {\rm Z} ^2  $&& ${\rm A}_{n-1} $\\
$D_n$&binary dihedral &4$n${\ } &$ {\rm X} ^{n+1}\!+\! {\rm X}  {\rm Y} ^2\!+ {\rm Z} ^2$&&
${\rm D}_{n+2}$\\
$T$&binary tetrahedral&24{\ } &$\rm X^4 +{\ }Y^3{\,}+Z^2  $&& $\rm E_6  $\\
$O  $&binary octahedral &48 {\ }&$\rm X^3Y +{\ }Y^3{\,}+Z^2 $&& $\rm E_7$\\
$J  $&binary icosahedral&120{\ } &$\rm X^5  +{\ }Y^3{\,}+Z^2 $&& $\rm E_8$
\end{tabular}
\begin{center}
Table: Properties of finite SL(2,C)-subgroups and simple singularities
\end{center}
}\hspace*{-0.75truecm}
The A and D series are running over integers $n\ge 2$.
Any F is
located in SU(2), the maximal compact subgroup of SL(2,C) and the double
cover of SO(3,R). That yields (for F$\neq C_{2j+1}$) the fibration
\begin{center}
{\normalsize
\begin{tabular}{cr}\qquad\quad
\begin{tabular}{ccccccccc}
1& $ \to $ &$\{\pm E\}$ & $ {\hookrightarrow } $ & SL(2,C) & $\to$ & SO(3,C) &$ \to $&1\\
&&  & & $\cup$ & & $\cup$ && \\
1& $ \to $ & $\{\pm E\}$ &$ {\hookrightarrow }  $&SU(2) &$\to$ & SO(3,R) &$ \to $&1 \\
&&  & & $\cup$ & & $\cup$ && \\
1& $ \to $ &$\{\pm E\}$ & $ {\hookrightarrow } $ & F & $ \to $& PF &$ \to $&1
\end{tabular} & \qquad  (4.1)
\end{tabular} }
\end{center}
\medskip
(4.1) explains the names of the F, since PF is the symmetry group of a
regular polyhedron.

Let us now consider the  F-invariant polynomials on $\rm C^2$.
They are generated by 3 fundamental F-invariants  X, Y, Z  subject to 1
constraint  {$R$(X,Y,Z)=0}. The normal forms of  $R$(X,Y,Z) are listed in
the table above. Thus we have a quotient map
$$ \qquad \qquad \quad \qquad
q :\left\{ \begin{array}{ccc}
		\rm C^2 & \to &\rm C^3 \\
   {z_1 \choose z_2}\equiv z & \mapsto & \left( \begin{array}{c}
						{\rm X} ( z )\\
						{\rm Y} ( z )\\
						{\rm Z} ( z )
					       \end{array} \right)
	      \end{array} \right. \qquad \qquad \qquad (4.2)
$$
The image $q(\rm C^2) = C^2 / F$ is a hypersurface in $\rm C^3$ given by
the constraint R. For any F a simple singularity (at 0) is just given by
(the complex analytic germ of)
$\rm C^2 / F$. For a complete classification list of all hypersurface
singularities see Ref. \Ar. Here we restrict to the simple ones.
Their properties have already been examined by F. Klein$^\Kl$. A modern
overview is given by Ref. \Du.

In the following we need the notion of a resolution of a singular variety
$X$ (here $X = \rm C^2 / F$). This is given by a surjective proper map
$\pi : Y \to X$
from a regular variety $Y$ to the singular $X$, such that the regular subset
$ X_{reg}$ (=$X\!-0$ in the present case) is diffeomorphic
with its pre-image $\pi^{-1} \ X_{reg}$, which is required to be dense
in $Y$.
Normally we want $Y$ to be a minimal resolution, demanding
$\forall \pi',Y'\ \exists j : Y  {\hookrightarrow }  Y'$ such that $ \pi' \circ  j = \pi$.

It is now a very special property of the simple singularities that they can
be resolved by a finite iteration of a deprojection proceedure called
blow up or $\sigma$-process$^{{1},\Ki}$.
For $\rm C^3$ the blow up at 0 is given by the canonical projection
$\beta$ of the tautological line bundle
$\tau  = \{(x,l) \in {\rm C^3 \times P^2} : x \in l \}$
onto $\rm C^3$. The singular fiber of $\beta$ is given by the complex
projective space $\rm P^2$.
Then the blow up $\hat X$ of a subvariety $X \subset  \mbox{$\rm C^3 $} $ at 0 is defined
by the closure of the pre-image of $X_{reg}=X-0$ under $\beta$
$$       \qquad \qquad     \qquad \qquad
\begin{array}{ccl}
 \mbox{$\rm C^3 $}  & {\beta\atop\gets} & \tau \\
\cup & &\cup \\
X &\gets&\hat X =  \overline{\beta^{-1}(X-0)} \\
\cup & &\cup \ \mbox{\small\it dense} \\
X-0 &\gets&\beta^{-1}(X-0) \\
\end{array}  \qquad \qquad \qquad (4.3)
$$
Assume $\pi = \beta^n : Y \to X$ is a resolution of a simple singularity.
The exeptional set, defined as
$E = { Y} / \pi^{-1} X_{reg}$,
is a finite union of
homological cycles $\rm C_1, ..., C_r$ with each $\rm C_k$ isomorphic to
${\rm P^1}  {\cong }  \frac{S^3}{S^1}  {\cong }  S^2$. So the cycles are actually
2-spheres. The first Chern class of the normal bundle of $\rm C_k$ in $Y$ is
${\rm c_1} ({\rm C_k},Y) = -2$,
which is also known as the selfintersection number$^\Hu$ of $\rm C_k$ in $Y$.

In fact the cycles $\rm C_k$ here either intersect in 1 point or do not
intersect at all. So we can assign a basic graph to a simple singularity
representing each cycle of its resolution as a dot and connecting 2 dots
by an edge if the cycles intersect. It turns out that any basic graph of a
simple singularity corresponds to the Dynkin graph of a simply laced
Lie group.
The last two columns of the table above show the basic graph and the
corresponding Lie group in Cartan's notation.

\section{\bf Explicit Description of the A and D Series}
\setcounter{equation}{0}
Let us now consider some instructive examples of the
singularities explicitly. First the A series:  {\hfill \break}
The constraint $ R =  {\rm X} ^n +  {\rm Y} ^2 +  {\rm Z} ^2 = 0$ for the
cyclic singularities
is equivalent to
$Q =  {\rm X} ^n+\rm U V = 0$ by the substitutions
\begin{center}
\begin{tabular}{l}
U = {Y + $i$ Z} \vspace{0.15cm}\\
V = {Y $-$ $i$ Z} \\
\end{tabular}
respectively
\begin{tabular}{l}
Y = {${1\over 2}\,$}\,{(U + V)} \vspace{0.15cm}\\
Z = {${1\over 2i}$}\,{(U $-$ V)} \\
\end{tabular}
\end{center}
In SL(2,C) the cyclic group of order $n$ is represented as  {\hfill \break}
$C_n=\{1,E_n,..,E_n^{n-1}\}$ with
$$
E_n = \left(
\begin{array}{cc}
e^{i\frac{2\pi}{n} } & 0       \\
  0                & e^{-i\frac{2\pi}{n} }
\end{array}
\right).
$$
Obviously X=\,$z_1 z_2$, U=\,$i z_1^n$, V=\,$i z_2^n$ are 3 independent
$C_n$-invariants, satisfying Q=0. Thus the 3 generic
$C_n$-invariant polynomials with R=0 are
$$ \qquad \qquad \qquad\qquad\qquad
\begin{array}{l}
 {\rm X}  = z_1 z_2 \vspace{0.15cm}\\
 {\rm Y}  = \frac{i}{2}\, (z_1^n + z_2^n) \vspace{0.15cm}\\
 {\rm Z}  = \frac{1}{2}\, (z_1^n - z_2^n)
\end{array} \qquad\qquad\qquad\qquad \quad (5.1)
$$
Now let us go over to the D series: {\hfill \break}
The binary dihedral group $D_n$ is represented in SL(2,C) by combining
$C_{2n}$ with
$$
B = \left(
\begin{array}{cc}
0 & 1 \\
  -1  & 0
\end{array}
\right).
$$
Note that $B^2 = E_{2n}^n = \{-E\}\in C_{2n}$.
It is now easy to verify that
$$ \qquad \qquad\qquad\qquad\quad
\begin{array}{l}
 {\rm X}  = (z_1 z_2)^2 \vspace{0.15cm}\\
 {\rm Y}  = \frac{i}{2}\, (z_1^{2n} + z_2^{2n}) \vspace{0.15cm}\\
 {\rm Z}  = \frac{1}{2}\, z_1 z_2 \,(z_1^{2n} - z_2^{2n})
\end{array}  \qquad\qquad\qquad \qquad (5.2)
$$
are 3 independent $D_n$-invariant polynomials, which satisfy just the
constraint $ R =  {\rm X} ^{n+1} +  {\rm X}  {\rm Y} ^2 +  {\rm Z} ^2 = 0$ given above.  {\hfill \break}
The lowest order nontrivial group of this series is $D_2$.
Its projection onto
$L_+^{ {\uparrow } }$ yields ${\rm P}D_2 = V_4 \equiv \{E,C,P,T\}$, the
Kleinian 4-group, which is the unique abelian group of order 4 with
$C^2=P^2=T^2=E$ and $CPT=E$.

Similar like the invariants (5.1) and (5.2) for the A resp. D series
one can construct the invariants of type E.
The results are already contained in the original work$^\Kl$ of F. Klein.
The following section is devoted to the behavior of the ADE singularities
under algebraic perturbations.

\section{\bf Deformations and symmetry breaking}
\setcounter{equation}{0}
For the singularity given by $R$(X,Y,Z) = 0
let us define (following Refs. \Du, \Hu\ and \Ur)
small deformation (fiber)s
in an $ \mbox{$\varepsilon$} $-neighbourhood of 0 by
$$
M_{S,t}\!=\!\{( {\rm X} , {\rm Y} , {\rm Z} )\!\in\!B^3( \mbox{$\varepsilon$} )\!\subset\! \mbox{$\rm C^3 $}  \mid
	      R( {\rm X} , {\rm Y} , {\rm Z} ) + t S( {\rm X} , {\rm Y} , {\rm Z} )\!=\!0 \} \quad (6.1)
$$
with $t$ small w.r.t. \mbox{$\varepsilon$} . {\hfill \break}
Clearly $M_{S,0}$ is the singularity itself. $M_{1,t}$ is the Milnor fiber,
which characterizes the resolution, since $H_2(M_{1,t}, {\rm Z} )$ is just
generated by the cycles $\rm C_k$ introduced in sec. 4.
Following theorem$^{\Ur}$ holds: {\hfill \break}
{\it
Given a simple singularity and independently a Dynkin graph G of ADE type
(i.e.\,each  connected component of G is of ADE type).
 {\hfill \break}
Then the following properties are equivalent:
 {\hfill \break}
I. There exists a deformation $M_{S,t}$ with only simple singularities,
\hspace*{0.6cm} such that the combination of these corresponds exactly to G.
 {\hfill \break}
II. G is a subgraph of the basic graph of the singularity.
}

Similar theorems hold also for larger classes of singularities$^\Ur$.
The powerful implication of the theorem above is that it allows to decide
just graph theoretically the existence of both, deformations of simple
singularities and corresponding symmetry breaking transitions$^\Ra$
between the associated ADE Lie groups (see example in Fig. 3).
\vspace{3.5truecm}
\begin{center}
{\normalsize Fig. 3: Example of possible symmetry breaking}
\end{center}
Considering also the actions of
SL(2,C) on space-time and tubular worldsheet according to Sec. 3,
the author has conjectured$^\Ra$ an intimate relation between space-time and
observational structure on the on hand and internal (Yang Mills like)
symmetries on the other.
 {\hfill \break}

In the following section we describe the geometric structure of some
simple singularities explicitly and relate them to the structure of the
model of Sec. 2, thus confirming the conjecture above.

\section{\bf Geometry of the singularities in the model}
\setcounter{equation}{0}
Let us consider the geometry of the most simple singularity
$\rm A_1$ given by the constraint
$$    \qquad  \qquad \qquad\qquad\qquad
-  {\rm X} ^2 + \underbrace{(i  {\rm Y} )^2 + (i  {\rm Z} )^2}_{R^2} = 0 .
\qquad \qquad   \qquad \ (7.1)
$$
This surface is a double cone with double point singularity at 0,
the 2 cones corresponding to $X=\pm R$. It is resolved by a single
blow up (4.3)
yielding a surface isomorphic to $T^*\rm P^1$ and the singular fiber is
given by the zerosection $P^1$ (see Fig. 4).
$T^*\rm P^1$ is the unique line bundle of Chern class -2 and thus it is
the standard building block for the resolutions of higher ADE singularities
involving more fundamental cycles.
\vspace{8.0truecm}
\begin{center}
{\normalsize Fig. 4: Resolution of $A_1$}
\end{center}
 {\hfill \break}
More generally an ${\rm A}_n$ singularity is given by a surface
$ {\rm X} ^{n+1}=R^2$,
with $R=i( {\rm Y} ^2+ {\rm Z} ^2)^{\frac{1}{2}}$ like above.
This surface is built by $n+1$ cones given by
$ {\rm X} =e^{\frac{2\pi i k}{n+1}} R^{\frac{2}{n+1}}$ with $k=1,..,n+1$,
their tops joining in the singularity 0. The
resolution can be found by $n$ blow ups (4.3), with $n-i$ cones left after
the $i$th blow up. The resolution is given by a vertex of $n+1$ joining
tubes  \mbox{$T^*\rm P^1 $}  rather than cones. The singular fiber of the resolution
is given by a joining chain of $n$ spheres $ \mbox{$\rm P^1 $} \vee  \mbox{$\rm P^1 $} ..\vee  \mbox{$\rm P^1 $} $.

This is now demonstrated in detail for $n=2$ (see Fig. 5).
The surface is given by
3 cones joining in a point. The first
blow up yields an $\hat X$ like a $T^* \mbox{$\rm P^1 $} $ tube with a cone
attached to the image of the
zerosection  \mbox{$\rm P^1 $}  in a point $\hat 0$. This is however not yet a resolution,
since $ \overline{\beta^{-1}  X_{reg}}$ is singular in $\hat 0$.
Applying (4.3) once more
$\hat 0$ is blown up to a further  \mbox{$\rm P^1 $}  intersecting the first one just in a
point. This yields the resolution,
given by 3 regularly joining  \mbox{$T^*\rm P^1 $}  tubes.
The singular fiber is the join $ \mbox{$\rm P^1 $} \vee \mbox{$\rm P^1 $} $ of 2 spheres. {\hfill \break}
\vspace*{8.8 truecm}
\begin{center}
{\normalsize Fig. 5: Resolution of $A_2$}
\end{center}
 {\hfill \break}
In order to relate these complex structures to the real structures of our
topological model, let us take a proper real section, such that $ \mbox{$\rm P^1 $}   {\cong }
S^2$ becomes restricted to $S^1$ and the complex cotangent bundle  \mbox{$T^*\rm P^1 $}
restricts to the real cotangent bundle $ \mbox{$T^* S^1 $}  {\cong }  S^1\!\times\!\rm R^1$.
Thus we can identify the tubular strings between PPOs as real sections
of resolutions of $\rm A_1$ singularities, which themselves might have
been created each
by 2 points of the worldsheet identifying to a double point. So
$\rm A_1$ singularities and resolution appear naturally in processes
like (2.1).

Until now we have considered any single PPO as vertex with $n$ tubes
attached.
Note however that this point of view can be transformed into a different
one
by a smooth deformation of the worldsheet replacing any $n$-tube vertex
replaced by $n-k+1$ vertices with maximally $k$ external tubes
($3\le k\le n$).
Taking such a vertex as PPO, the associated 2-sphere would have only $k$
punctures. The most refined decomposition
is the tree level with $k=3$.
At this level any vertex corresponds to a real
section of an $\rm A_2$ resolution.
Generally at level $k$ any maximal vertex corresponds to an
${\rm A}_{k-1}$ resolution.

However at any refined level  democracy among vertices is broken, since no
longer each of them is connected to any other one. Democracy of PPOs can
be retained if we consider only a whole cluster of $n-k+1$ ``confined"
vertices, rather than a simple vertex, as PPO.

Note that the resolution by passing in (6.1) from $M_{1,0}$ to
$M_{1,t}$ with
$t\ne 0$ is similar like in QFT shifting particles off the singular light
cone by giving them a (small) nonzero mass. In fact the geometry of the
$\rm A_1$-resolution corresponds to a complex mass shell.

We had chosen real sections, like e.g.\,in (7.1), such that they
correspond to the
worldsheet structure in a syncronized spacelike section $M^3$.
Note however that the complex structure of the resolution contains various
possibilities for real sections and choices of signatures. Recently it was
pointed out, how a change of space-time signature may be posssible$^{18}$
and interpreted$^\An$,
where instantons play a crucial role. A full space-time
analysis of the model of Sec. 2 provides an instanton interpretation$^\Hi$
of geometric structures related to ADE singularities.

\section{\bf Discussion}
\setcounter{equation}{0}
Using a specific model of observation with special emphasis to microscopic
democracy of PPOs (``possible points of observation") in space-time,
conjectures about an intimate relation
between inner symmetries of the Yang Mills type and local structure
of observation in space-time near a particle are confirmed. {\hfill \break}
Explicitly the resolution of singularities of the A series appear naturally
as vertices of the tubular network which is transporting the information
between PPOs. The $\rm A_1$ singularity and its resolution (corresponding to
the SU(2) symmetry  of electroweak interaction) appears in any
unification/separation process of (PPO) vertices as pinch/blow up
of specific strings on the tubular worldsheet.

Abelian U(1) factors
are not considered in the ADE theory but might be represented
by (counter)clockwise twistings of the stringtubes around
their axis according to the polarisation of a lightwave propagating along
some of them.

The resolutions of the ${\rm A}_{k-1}$ singularities for $k\ge 3$
(corresponding to a SU($k$) Yang Mills symmetry) have real sections
isomorphic to a $k$-tube vertex. Confinement might be related to the
necessity to consider clusters rather than single vertices as
PPOs in order to restore democracy at the most refined level $k=3$.

We have coined the term PPO for an undetermined location somewhere in the
interior at a vertex of an extended observer carrying the frame of
reference. In contrast to a
quantization of space-time into disconnected parts$^{15}$, our model
assumes
(at least in the initial phase considered here) all PPOs to be located
in a common connected region, namely the interior of the closed tubular
worldsheet. Regions connected only by stringtubes with
diameter very much below Plancklength $L_{Pl}$ should be considered as
quasidisconnected, which yields
the correspondence to the QFR (Quantum Frame of Reference) approach$^{15}$.

String pinchings according to (2.1) can be performed equivalently around
interior and exterior of the tubular worldsheet w.r.t. the embedding space,
both yielding the same unification of vertex tubes.
Note the
resemblance of the present approach to knot theory and neural networks.
 {\newpage }
Assuming a mean extension of the vertices
of only few orders of magnitude above $L_{Pl}$ and
a mean stringtube diameter of order $L_{Pl}$ or below, the PPOs
become sharp only for
$L_{Pl} \to 0$ in the macroscopic continuum limit
$n\to \infty$. In this limit
there should appear a connection to the gauge bundle framework.

By the theorem$^\Ur$ of Sec. 6, the Dynkin graphs describe
the possible ADE symmetry breaking transitions and deformations
of singularities. Notably in our model all this happens a priori
in a space-time of $D=4$ rather than decending from $D=26$.

For an evolutionary process there remains therefore the crucial question,
whether a classical space-time (initial) singularity appears.
This question
is closely related to the question of PPO democracy via the global
topology of $M^3$.
Furthermore a resolution
of a scalefactor singularity corresponds to a deprojection process,
similar to those involved in the ADE resolutions.
All that indicates a unified description of both the
singularities {\it in} and {\it of} space-time. This and a more detailed
evaluation of the physical predictions of the model will be topic of a
forthcoming paper.
 {\hfill \break} \nl {\hfill \break}
{\Large {\bf Acknowledgements}}
 {\hfill \break} \nl
The author would like to thank the unknown referee for his useful comments
on the paper. Furthermore he is grateful to
H.-J. Schmidt for critical remarks on the
topic and thanks the Cosmology Group of the Mathematics Department at
Potsdam University for their hospitality.
 {\newpage } \hspace*{-0.75truecm}

{\Large {\bf References}}
 {\hfill \break} \nl
$^{1}$  E. Brieskorn, Math. Ann. 166 (1966) 76
 {\hfill \break}
$^\Du$  A. Durfee, Enseign. Math. 25 (1979) 131
 {\hfill \break}
$^\Hu$  S.M. Husein-Zade, Russian Math. Surveys 32:2 (1977) 23
 {\hfill \break}
$^\Ki$ D. Kirby, Proc.\,London Math.\,Soc.\,(3), 6\,(1956) 597,
{7\,(1957)\,1}
 {\hfill \break}
$^\Kl$ F. Klein, Vorlesungen \"uber das Ikosaeder und die Aufl\"osung der
\hspace*{0.12cm} Gleichungen vom f\"unften Grade (Teubner, Leipzig, 1884)
 {\hfill \break}
$^\Ur$ T. Urabe, ICTP Lecture Notes SMR.567/4 (1991)
 {\hfill \break}
$^\Gr$  M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory,
\hspace*{0.12cm}        Cambridge Univ. Press (1987)
 {\hfill \break}
$^\Ra$ M. Rainer, Projective Geometry for Relativistic Quantum
 {\hfill \break}
\hspace*{0.12cm} Physics, preprint 92/05, Univ. Potsdam (1992)
 {\hfill \break}
$^{9}$  G.F.R. Ellis, J. Math. Phys. 8 (1967) 1171
 {\hfill \break}
$^\Qa$  A. Qadir, J. Math. Phys. 33 (1992) 2262
 {\hfill \break}
$^\Sa$ A.D. Sacharov, Dokl. Akad. Nauk SSSR 177 (1967) 70, cf.
 {\hfill \break}
\hspace*{0.12cm} the review: H. Fuchs, U. Kasper, D.-E. Liebscher,
V. M\"uller,
 {\hfill \break}
\hspace*{0.12cm} H.-J. Schmidt, Fortschr. Phys. 36,6 (1988) 427
 {\hfill \break}
$^{12}$  S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of
\hspace*{0.12cm} Space-Time (Cambridge Univ. Press, Cambridge, 1973)
 {\hfill \break}
$^\An$ Y. Anini, Quantum Tunneling and the Change of Signature of
 {\hfill \break}
\hspace*{0.12cm}  the Spacetime Metric, ICTP preprint 2435/92 (1992)
 {\hfill \break}
$^{14}$  J.B. Hartle and S.W. Hawking, Phys. Rev. D 28 (1983) 2960
 {\hfill \break}
$^{15}$  V.P. Bransky,
	Quantum Theory of Relativity as Quantum Clep-
 {\hfill \break}
\hspace*{0.12cm} sodynamics,
	Proc. London Philosophical Conference (1992) 72
 {\hfill \break}
$^\Pe$ R. Penrose, W. Rindler, Spinors and Spacetime, Cambridge Univ.
\hspace*{0.12cm} Press (1986)
 {\hfill \break}
$^\Ar$  V. Arnold, Invent. Math. 35 (1976) 87
 {\hfill \break}
$^{18}$  G.F.R. Ellis, Change of Signature in Classical Relativity,
 {\hfill \break}
\hspace*{0.12cm}  preprint SISSA 170/91/A (1991)
 {\hfill \break}
$^\Hi$  N. J. Hitchin, Math. Proc. Camb. Phil. Soc. 85 (1979) 465
 {\newpage }
Figure captions:
\begin{center}
{\normalsize Fig. 1: Net of vertices of PPOs connected by stringy tubes}
\end{center}
\begin{center}
{\normalsize Fig. 2: Democracy a) conserved in $M^3\cong S^3$,
b) broken in $M^3\cong \mbox{$ S^2 $} \times  \mbox{$\rm R $} $}
\end{center}
\begin{center}
{\normalsize Fig. 3: Example of possible symmetry breaking}
\end{center}
\begin{center}
{\normalsize Fig. 4: Resolution of $A_1$}
\end{center}
\begin{center}
{\normalsize Fig. 5: Resolution of $A_2$}
\end{center}
 {\hfill \break}
Table caption:
\begin{center}
{\normalsize Table: Properties of finite SL(2,C)-subgroups and simple
singularities}
\end{center}
\end{document}