We continue the investigation of $Spin(7)$ holonomy metric of cohomogeneity one with the principal orbit $SU(3)/U(1)$. A special choice of $U(1)$ embedding in $SU(3)$ allows more general metric ansatz with five metric functions. There are two possible singular orbits in the first order system of $Spin(7)$ instanton equation. One is the flag manifold $SU(3)/T^2$ also known as the twister space of ${CP}(2)$ and the other is ${CP}(2)$ itself. Imposing a set of algebraic constraints, we find a two-parameter family of exact solutions which have $SU(4)$ holonomy and are asymptotically conical. There are two types of asymptotically locally conical (ALC) metrics in our model, which are distingushed by the choice of $S^1$ circle whose radius stabilizes at infinity. We show that this choice of $M$ theory circle selects one of possible singular orbits mentioned above. Numerical analyses of solutions near the singular orbit and in the asymptotic region support the existence of two families of ALC $Spin(7)$ metrics: one family consists of deformations of the Calabi hyperK\"ahler metric, the other is a new family of metrics on a line bundle over the twister space of ${CP}(2)$. %\vspace{-3.0mm} %\begin{flushleft} %MSC 1991: 53C07, 53C25, 81T13, 81C60 \\ %Keywords: Instanton, Special holonomy, Supersymmetric Yang-Mills theory %\end{flushleft}
hep-th0111198.tex

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\begin{titlepage}
\begin{flushright}
{November, 2001} \\
{\tt hep-th/0111198}
\end{flushright}
\vspace{0.5cm}
\begin{center}
{\Large \bf
On $Spin(7)$ holonomy metric based on $SU(3)/U(1)$ : II
}%
\lineskip .75em
\vskip1.0cm
{\large Hiroaki Kanno\footnote{e-mail: kanno@math.nagoya-u.ac.jp}}
\vskip 1.0em
{\large\it Graduate School of Mathematics \\
Nagoya University, Nagoya, 464-8602, Japan}
\vskip 0.8cm
{\large Yukinori Yasui\footnote{e-mail: yasui@sci.osaka-cu.ac.jp}}
\vskip 1.0em
{\large\it Department of Physics, Osaka City University \\
Sumiyoshi-ku, Osaka, 558-8585, Japan}
\end{center}
\vskip0.5cm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{abstract}

We continue the investigation of $Spin(7)$ holonomy metric of
cohomogeneity one with the principal orbit $SU(3)/U(1)$.
A special choice of $U(1)$ embedding in $SU(3)$ allows more
general metric ansatz with five metric functions. There are two 
possible singular orbits in the first order system of $Spin(7)$ 
instanton equation. One is the flag manifold $SU(3)/T^2$ also 
known as the twister space of ${\bf CP}(2)$ and the other is 
${\bf CP}(2)$ itself. Imposing a set of algebraic constraints,
we find a two-parameter family of exact solutions which have 
$SU(4)$ holonomy and are asymptotically conical. There are 
two types of asymptotically locally conical (ALC) metrics in our model,
which are distingushed by the choice of $S^1$ circle whose radius
stabilizes at infinity. We show that this choice of $M$ theory circle
selects one of possible singular orbits mentioned above.
Numerical analyses of solutions near the
singular orbit and in the asymptotic region support
the existence of two families of ALC $Spin(7)$ metrics:
one family consists of deformations of the Calabi hyperK\"ahler metric, 
the other is a new family of metrics on
a line bundle over the twister space of ${\bf CP}(2)$.

%\vspace{-3.0mm}
%\begin{flushleft}
%MSC 1991: 53C07, 53C25, 81T13, 81C60 \\
%Keywords: Instanton, Special holonomy, Supersymmetric Yang-Mills theory
%\end{flushleft}

\end{abstract}
\end{titlepage}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

By string dualities intriguing dynamics in supersymmetric
compactification of superstrings and $M$ theory are often
associated with singularities in manifolds of special
holonomy which appear at finite distance in the moduli space.
If the singularity is isolated and conical, we may expect
that the details of the metric far from the singularity are
irrelevant and as an approximation of the singular geometry
take a simple Ricci-flat cone metric over an $n$-dimensional
Einstein manifold $M$;
%
\beq
ds^2 = dr^2 + r^2 d\Omega^2~,  \qquad (0 \leq r < +\infty)
\eeq
%
where the Einstein metric $d\Omega^2$ satisfies $R_{ab}=(n-1)g_{ab}$.
For supersymmetry a parallel spinor should exist on the cone
and it comes from a Killing spinor on $M$ \cite{Bar}, \cite{AFHS}, \cite{MP}.
Unless the Einstein manifold is the $n$-dimensional
sphere $S^n = SO(n+1)/SO(n)$, there is a conical singularity at
$r=0$. When the manifold $M$ is a homogeneous space $G/K$,
a resolution of the singularity may be provided by
the following metric of cohomogeneity one \cite{DW}, \cite{Wang}, \cite{EW};
%
\beq
d\widetilde{s}^2 = dt^2 + g_{G/K} (t)~,  \qquad  ( t_0 \leq t < +\infty)
\eeq
%
where  $g_{G/K}(t)$ is a $t$-dependent homogeneous metric
on the principal orbit $G/K$. This sort of resolution of
an isolated conical singularity has been employed recently
in the discussion of IR strong coupling dynamics of supersymmetric
gauge theories based on gauge theory/gravity
correspondence in large $N$ limit \cite{KS}, \cite{MN}, \cite{Vafa},
\cite{Ach}, \cite{AMV}, \cite{PZT}, \cite{EN}, \cite{AW}.


The requirement of special holonomy, which can be expressed
as linear constraints on the spin connection $\omega_{ab}$, gives
a first order system of flow equations for one-parameter family
of homogeneous metrics $g_{G/K}(t)$.
The boundary condition should be specified in solving the flow
equation. At the boundary $t=t_0$ there appears a singular
orbit $G/H$ with $K \subset H \subset G$. This singular orbit has a finite
volume and the original conical singularity is developed when the volume
of $G/H$ tends to vanishing. The coset $H/K$ has to be
a sphere $S^k$ for the principal orbit $G/K$ to degenerate
smoothly to the singular orbit \cite{Mos}. When there are several choices of
the subgroup $H$ such that $H/K \simeq S^k$,
there may be more than one way of
resolving the conical singularity. A famous example is given by
the conifold that is a cone over the five dimensional coset
space $T^{1,1} = SU(2) \times SU(2) / U(1)$.
There are three possible singular orbits \cite{CGLP2};
%
\begin{enumerate}
\item
$H= U(2)~, \quad G/H \simeq S^2~, \quad  H/K \simeq S^3~,$
\item
$H= SU(2)~, \quad G/H \simeq S^3~, \quad  H/K \simeq S^2~,$
\item
$H= U(1) \times U(1)~, \quad G/H \simeq S^2 \times S^2~, \quad
H/K \simeq S^1~.$
\end{enumerate}
%
In this paper we will see a similar example of this kind, when the principal
orbit is the seven dimensional coset space $N (1,1) = SU(3)/ U(1)$.



The other side of the boundary is specified by the asymptotic behavior
of the solution.
A standard behavior is that the homogeneous metric $g_{G/K}(t)$ asymptotically
approaches to the original Einstein metric $d\Omega^2$.
Such metrics are called asymptotically conical (AC). From the viewpoint of
compactifications of $M$ theory we are also interested in
the asymptotic behavior called asymptotically locally conical (ALC) 
\cite{CGLP3},
where there is a circle whose radius remains finite at infinity .
In \cite{Gom}, by considering the geometry of ALE fibration over
a supersymmetric cycle, it has been argued that an $M$ theoretic lift
of a type IIA geometry with $D6$ branes wrapping on the SUSY cycle
is given by purely gravitational configuration. (See also \cite{GS}
for a relation of the $M$ theoretic lift to $Spin_c$ structure.)
Such $D6$ brane configurations have been discussed from the
dual picture of eight dimensional supergravity in \cite{Her}, \cite{GM2}.
Since the $M$ theory circle which is related to
the string coupling of IIA theory should remain finite asymptotically,
the corresponding metric is expected to be ALC. In fact
when the SUSY cycle is $S^4$ in $Spin(7)$ manifold and $S^3$ in $G_2$ manifold,
such ALC metrics were constructed in \cite{CGLP3} and \cite{BGGG}, 
respectively.
More recently a similar ALC metric has been found for a SUSY cycle
${\bf CP} (2)$ in \cite{KY}, \cite{CGLP5}, \cite{GS}.
Even if we assume that the metric is ALC, the choice of $M$ theory circle
in the principal orbit may not be unique, when there are more than one 
irreducible
modules of dimension one in the isotropy representation on the tangent space
of the principal orbit.
Due to the special choice of $U(1)$ subgroup to be introduced shortly,
the isotropy representation of the coset space $N(1,1) = SU(3)/U(1)$ has
three one dimensional irreducible components.
Recently $M$ theory on ALC $Spin(7)$ manifolds has been discussed in 
\cite{GS}, \cite{CKL}.



In this article taking the homogeneous space $SU(3)/U(1)$ (also known as
the Aloff-Wallach space), we investigate aspects of the special holonomy
metrics of cohomogeneity one.
In our previous work \cite{KY} we left a choice of $U(1)$ subgroup in
$SU(3)$ free
so that the triality $W(SU(3))$(=the Weyl group) symmetry was manifest.
In the following we will fix the embedding so that the $U(1)$ subgroup is
${\rm diag} (e^{i\theta}, e^{i\theta}, e^{-2i\theta})$. In this
case we can make a more general metric ansatz with five functions,
while in general the number is four. In section two we derive
a first order system for $Spin(7)$ holonomy and classify a possible singular
orbit appearing at the boundary.  In our metric ansatz there is a natural
candidate for a K\"ahler two form. The closedness (or the integrability) of
the candidate two form gives a set of algebraic constraints
that allows us to solve the flow equation exactly.
In section three we present a two-parameter family of exact solutions
which is asymptotically conical.
They are $SU(4)$ holonomy metrics
on the line bundle over the flag manifold $SU(3)/T^2$, which is a
two-sphere bundle
over ${\bf CP}(2)$. When one of the parameters vanishes,
then the $S^2$ fiber collapses
and the metric reduces to the Calabi hyperK\"ahler metric on $T^* {\bf CP}(2)$.
An analysis of ALC solutions is carried out in section four.
Due to the generalized metric ansatz with five metric
functions there are two choices of a circle whose radius remains finite 
asymptotically.
We find that if we assume the metric is non-singular,
the ALC asymptotic behavior requires a reduction of
the number of metric functions from five to four, but the way of reduction
depends on the choice of $S^1$ factor in the principal orbit.
 From the perturbative analysis around the singular orbit
we see one of possible singular orbits is selected by each reduction
and thus there are two types of ALC $Spin(7)$ metrics.
The topology of the singular orbit and the choice of
$M$ theory circle cannot be independent and the singular orbit is
either ${\bf CP}(2)$ or $Flag_6 = SU(3)/T^2$
depending on the chioce of asymptotic $M$ theory circle.


Since we could not find explicit solutions in general, we have numerically
worked out perturbative series expansions both
around the singular orbit and in the asymptotic region.
In section five, based on this numerical analysis
we propose the \lq\lq moduli\rq\rq\ space
of $Spin(7)$ metrics of cohomogeneity one with the principal orbit
$SU(3)/U(1)$ for the special choice of $U(1)$ subgroup.
Especially we observe
that two types of ALC metrics in section four are in fact interpolated
by the exactly known Ricci-flat K\"ahler metrics obtained in section three.
The existence of ALC metrics whose singular orbit is $Flag_6$ is
shown only numerically. But their qualitative behavior is much like
the Atiyah-Hitchin metric in four dimensions. Hence we believe that
this is an analogue of $Spin(7)$ metric called ${\bf C}_8$ in \cite{CGLP5},
\cite{CGLP6}, whose singular orbit is ${\bf CP}(3)$, the twister space
of $S^4$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Instanton equation with five metric functions}
\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The maximal torus $T^2$ of the Lie group $SU(3)$ is two dimensional
and its $U(1)$ subgroup is specified by
integers $\overrightarrow n = (n_1, n_2, n_3)$ with $n_1 + n_2 + n_3=0$.
Without loss of generality we can assume that there is no
common divisor of $n_i$. Explicitly the $U(1)$ subgroup is given by
${\rm diag} (e^{in_1\theta}, e^{in_2\theta}, e^{in_3\theta})$.
In previous paper on $Spin(7)$ metric of cohomogeneity one
with the principal orbit $SU(3)/U(1)$, we took the following
metric ansatz \cite{KY};
%
\beq
g = dt^2 + a(t)^2 (\sigma_1^2 + \sigma_2^2) + b(t)^2
              (\Sigma_1^2 + \Sigma_2^2) + c(t)^2 (\tau_1^2 + \tau_2^2)
+ f(t)^2 T_A^2~,
\eeq
%
which is consistent with any choice of the embedding parameters
$\overrightarrow n$ and consequently gives a formulation
which has manifest symmetry under $\Sigma_3 = W(SU(3))$;
the Weyl group of $SU(3)$. Our convention of $SU(3)$ left invariant
one forms is summarized in Appendix A. The components of
invariant one form for the maximal torus $T^2$
are denoted as $T_A$ and $T_B$. The corresponding generators are given by
%
\beq
E_A = -\frac{1}{\Delta} \left(
\begin{array}{ccc}
\alpha_B & 0 & 0 \\
0 & \beta_B & 0 \\
0 & 0 & \gamma_B
\end{array}
\right),  \qquad
E_B=\frac{1}{\Delta} \left(
\begin{array}{ccc}
\alpha_A & 0 & 0 \\
0 & \beta_A & 0 \\
0 & 0 & \gamma_A
\end{array}
\right),  \label{base}
\eeq
%
with $\alpha_A + \beta_A + \gamma_A = \alpha_B + \beta_B + \gamma_B =0$
and $\Delta = \beta_A \alpha_B - \alpha_A \beta_B$.
The generator of the $U(1)$ subgroup is $E_B$.
Note that the bases of the Lie algebra $su(3)$ and the components
of the left invariant one form are in dual relation and hence
the role of parameters $\alpha_A, \beta_A, \gamma_A$ and
$\alpha_B, \beta_B, \gamma_B$ is exchanged.


When the $U(1)$ subgroup generated by $E_B$ decouples from
one of $\sigma$, $\Sigma$ and $\tau$, more general metric
ansatz is allowed since in this case the isotropy representation
of $SU(3)/U(1)$ becomes
%
\beq
su(3)/u(1)={\bf p}_1 \oplus {\bf p}_2 \oplus {\bf p}_3 \oplus
\widetilde{{\bf p}_3} \oplus
{\bf p}_4,
\eeq
%
where $\mbox{dim}~{\bf p}_1=
\mbox{dim}~{\bf p}_2 =2$
and $\mbox{dim}~{\bf p}_3 =
\mbox{dim}~\widetilde{{\bf p}_3} =
\mbox{dim}~{\bf p}_4=1$.
Then the metric ansatz becomes
%
\beq
g = dt^2 + a(t)^2 (\sigma_1^2 + \sigma_2^2) + b(t)^2
              (\Sigma_1^2 + \Sigma_2^2) + c_1(t)^2 \tau_1^2 +  c_2(t)^2 \tau_2^2
+ f(t)^2 T_A^2~, \label{ansatz}
\eeq
where we assume that $\tau_1$ and $\tau_2$ are singlets.
The reduction of the holonomy group from $SO(8)$ to $Spin(7)$ is represented by
the octonionic instanton equation \cite{CDFN}, \cite{AL}, \cite{BFK},
(See also Appendix B).
We can see the octonionic instanton equation derived
from the above ansatz does not have $T_B$ component,
if and only if $d\tau_i$ has no $T_B$ component.
We have $\nu_B =0$ and hence $\alpha_A=\beta_A$ (see Appendix A).
Then the generator of the $U(1)$ subgroup is
$E_B ={\rm diag} (1, 1, -2)$ and
the charge vector $\overrightarrow n$ in the
Maurer-Cartan equation of $dT_A$
is fixed to be $\overrightarrow n = (1,1,-2)$.
Note that this is the case where the action of the Weyl group degenerates.
We obtain the following system of first order differential equations
as the octonionic instanton equation on the spin connection $\omega_{ab}$
derived from the metric ansatz (\ref{ansatz});
%
\beqa
\frac{\dot a}{~a~} &=& \frac{b^2 + c_1^2 - a^2}{2abc_1}
+ \frac{b^2 + c_2^2 - a^2}{2abc_2} - \frac{f}{a^2}~, \CR
\frac{\dot b}{~b~} &=& \frac{c_1^2 + a^2 - b^2}{2abc_1}
+ \frac{c_2^2 + a^2 - b^2}{2abc_2}- \frac{f}{b^2}~, \CR
\frac{\dot c_1}{~c_1~} &=& \frac{a^2 + b^2 - c_1^2}{abc_1} +
\frac{2f}{c_1 c_2} + \frac{c_2^2 -c_1^2}{2c_1 c_2 f}~, \label{five} \\
\frac{\dot c_2}{~c_2~} &=& \frac{a^2 + b^2 - c_2^2}{abc_2} +
\frac{2f}{c_1 c_2} + \frac{c_1^2 -c_2^2}{2c_1 c_2 f}~, \CR
\frac{\dot f}{~f~} &=& \frac{f}{a^2}
+ \frac{f}{b^2} - \frac{2f}{c_1 c_2} + \frac{(c_1- c_2)^2}{2c_1 c_2 f}~.
\nonumber
\eeqa
%
This first order system has a discrete ${\bf Z}_2 \times {\bf Z}_2$ symmetry
generated by $(a,b,c_1,c_2) \to (\pm a, \mp b, -c_1, -c_2)$ and two exchange
symmetries $a \leftrightarrow b$ and $c_1 \leftrightarrow c_2$.
We note that though any independent sign flip of metric functions that
is not necessarily included above has no effect on the metric itself
or at the level of Ricci-flatness, but do {\it not} keep
the instanton equation invariant.
The first order system (\ref{five}) is an integral of
the second order Einstein equation
and the change in the instanton equation
means the different ways of integration.


Let us classify possibilities of the singular orbit compatible with
the evolution equation (\ref{five}). Group theoretically the singular
orbit is in one to one correspondence to a subgroup $H$ which
satisfy $U(1) \subset H \subset SU(3)$ and $H/U(1)$ should be
a sphere which is collapsing at the singular orbit.
Thus we find three possibilities;
%
\begin{enumerate}
\item
$H = U(1)\times U(1)$ ; In this case $H/U(1) \simeq S^1$ is collapsing and
the singular orbit is the twister space of ${\bf CP}(2)$; $SU(3)/H \simeq
Flag_6$, which is topologically a two sphere bundle over ${\bf CP}(2)$.
\item
$H = SU(2)$ ; In this case $H/U(1) \simeq S^2$ is collapsing and
the singular orbit is $SU(3)/H \simeq S^5$, which is the
Hopf bundle over ${\bf CP}(2)$.
\item
$H= S(U(2) \times U(1))$ ; In this case $H/U(1) \simeq S^3$ is collapsing and
the singular orbit is $SU(3)/H \simeq {\bf CP}(2)$ itself.
\end{enumerate}
%
We assume that the singular orbit is at $t=0$ and
make the following series expansion for small $t$;
%
\beqa
a(t) &=& p + \sum_{k \geq 1} a_k t^k~, \qquad
b(t) = q + \sum_{k \geq 1} b_k t^k~, \CR
c_1(t) &=& m + \sum_{k \geq 1} c_{1k} t^k~, \qquad
c_2(t) = n + \sum_{k \geq 1} c_{2k} t^k~, \CR
f(t) &=& r + \sum_{k \geq 1} f_k t^k~.
\eeqa
%
The parameters $p,q,m,n,r$ are regarded as the \lq\lq initial
conditions\rq\rq\ at
the singular orbit.
Substituting the series expansion to (\ref{five}) and looking at
the leading order, we obtain
%
\beqa
a_1 &=& \frac {1}{2}
\left( \frac{q}{m} + \frac{m}{q} - \frac{p^2}{mq} +
\frac{q}{n} + \frac{n}{q} - \frac{p^2}{nq} -\frac{2r}{p} \right)~, \CR
b_1 &=& \frac{1}{2}
\left(  \frac{p}{m} + \frac{m}{p} - \frac{q^2}{mp} +
\frac{p}{n} + \frac{n}{p} - \frac{q^2}{np} -\frac{2r}{q} \right)~, \CR
c_{11} &=& \frac{p}{q} + \frac{q}{p} - \frac{m^2}{pq} +\frac{2r}{n}
+\frac{n}{2r} - \frac{m^2}{2nr}~, \\
c_{21} &=& \frac{p}{q} + \frac{q}{p} - \frac{n^2}{pq} +\frac{2r}{m}
+\frac{m}{2r} - \frac{n^2}{2mr}~, \CR
f_1 &=& \frac{r^2}{p^2} + \frac{r^2}{q^2} -\frac{2r^2}{mn}
+ \frac{n}{2m} + \frac{m}{2n} -1~. \nonumber
\eeqa
%
Now the above three possibilities of the singular orbit correspond
respectively to the following initial conditions;
%
\begin{enumerate}
\item
$S^1$ is collapsing; $r=0$~,
\item
$S^2$ is collapsing; $p=0,~{\rm or}~~q=0$~,
\item
$S^3$ is collapsing; $p=r=0,~{\rm or}~~q=r=0$~.
\end{enumerate}
%
Firstly in case 2 there is no
regular solution at the singular orbit, since
there is no way to make $f_1$ regular\footnote{However,
another special choice of $U(1)$ embedding seems to allow
$S^5$ as a singular orbit \cite{CGLP5}. It might be very interesting
to see why it is the case,
since there is no odd dimensional SUSY cycle
in eight dimensions.}.
On the other hand in case 1
we see that the regularity of $c_{11}$ and $c_{21}$ requires
$m^2=n^2$. But $m=n$ implies $f_1=0$, which means that
the $S^1$ is collapsing \lq\lq too\rq\rq\ fast near $t=0$.
Thus only the case $m=-n$ can give non-singular
solutions. This also means that this type of singular orbit is
not allowed in generic cases where we have $c_1=c_2$.
In this case $p$ and $q$ are free parameters and $f_1=-2$.
Finally in case 3 there are non-singular
solutions if $m^2=n^2=q^2$ or $m^2=n^2=p^2$. Thus near $t=0$ we have
two types of
boundary conditions which correspond to case 1 and case 3, respectively~;
%
\beqa
g & \longrightarrow & dt^2+ 4t^2 T_{A}^2 + p^2(\sigma_{1}^2+\sigma_{2}^2)
+q^2(\Sigma_{1}^2+\Sigma_{2}^2)+ m^2(\tau_{1}^2+\tau_{2}^2)~, \label{ab} \\
g & \longrightarrow & dt^2+ t^2(T_{A}^2 + \sigma_{1}^2 + \sigma_{2}^2)
+m^2(\Sigma_{1}^2 + \Sigma_{2}^2 + \tau_{1}^2 + \tau_{2}^2)~. \label{bb}
\eeqa
%
Note that the terms with $\sigma_i^2$, $\Sigma_i^2$ and $\tau_i^2$ in 
(\ref{ab})
describe the singular orbit $Flag_6$ squashed by the parameters $p,q$ and $m$,
while the term with $\Sigma_i^2+\tau_i^2$ in (\ref{bb}) represents
the singular orbit ${\bf CP}(2)$ with the Fubini-Study metric.
In the following we call the first case
A-type boundary and the second case B-type boundary.
If we regard the homogeneous space $Flag_6$ as an $S^2$-bundle over
${\bf CP}(2)$, then B-type boundary may be reduced from A-type one
by making the fibre $S^2$ collapse. The higher order terms of the series
expansion are summarized in Appendix C.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Explicit AC solutions of $SU(4)$ holonomy}
\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In terms of the vielbeins (the orthonormal frames) of our metric ansatz,
we can write down the following non-degenerate two form
%
\beq
\omega=fdt \wedge T_{A} - c_1 c_2 \tau_1 \wedge \tau_2
-a^2 \sigma_1 \wedge \sigma_2 -b^2 \Sigma_1 \wedge \Sigma_2~,
\eeq
%
which is a natural candidate for a K\"ahler form.
Using the first order equation for functions  $a,b$ and $c_i$,
we see that the condition $d\omega =0$ is equivalent to
the constraints
%
\beq
a^2 + b^2 + c_1 c_2 = 0~, \quad c_1+c_2=0~.
\eeq
%
They are compatible with the first order system (\ref{five}) and
we obtain the following reduction with $c:= c_1 = - c_2$~;
%
\beqa
\dot a &=& -\frac{f}{a}~, \quad \dot b =-\frac{f}{b}~, \quad
\dot c =-\frac{2f}{c}~, \label{first} \\
\dot f &=& \frac{f^2}{a^2}
+ \frac{f^2}{b^2} + \frac{2f^2}{c^2} - 2~.
\label{second}
\eeqa
%
We can solve this reduced first order system exactly.
The first two equations of (\ref{first}) implies
%
\beq
b^2 - a^2 = \ell^2~,
\eeq
where $\ell^2$ is an integration constant. Due to the exchange
symmetry of $a$ and $b$, we may assume $b^2 - a^2 \geq 0$.
%
Furthermore by a change of variables $dt = c/(2f) dr$ we can integrate
$a,b,c$ to obtain
%
\beq
a^2 = \frac{1}{2} (r^2 - \ell^2)~, \quad  b^2 = \frac{1}{2} ( r^2 + \ell^2 )~,
\quad c^2 = r^2 ~,
\eeq
%
where we have fixed an integration constant by requiring $c^2=r^2$.
Substituting the above solution into the equation (\ref{second})
we have
%
\beq
\frac{d}{dr} f^2 = 2r - 2 f^2 \left( \frac{r}{r^2 - \ell^2}
+ \frac{r}{r^2 + \ell^2}  + \frac{1}{r} \right)~.
\eeq
%
It is possible to integrate this differential equation;
\beq
f^2 = \frac{r^2}{4} \left( 1 - \frac{\ell^4}{r^4} \right) U(r)~,
\quad U(r) = 1 - \frac{k^8}{(r^4 - \ell^4)^2} ~.
\eeq
%
We thus find the following metric of $SU(4)$ holonomy;
%
\beqa
ds^2 &=& \left( 1 - \frac{\ell^4}{r^4} \right)^{-1} U(r)^{-1} dr^2
+ \frac{1}{2} (r^2 - \ell^2) (\sigma_1^2 + \sigma_2^2)
+ \frac{1}{2} (r^2 + \ell^2) (\Sigma_1^2 + \Sigma_2^2)  \CR
& &~~~~~+ r^2 (\tau_1^2 + \tau_2^2)
+ \frac{1}{4} r^2  \left( 1 - \frac{\ell^4}{r^4} \right) U(r) T_A^2~, \quad
( (k^4+\ell^4)^{1/4} \leq r)  \label{SU4}
\eeqa
%
which is asymptotically conical. The singular orbit at
$r=(k^4+\ell^4)^{1/4} \; (k \neq 0)$
is the flag manifold $SU(3)/T^2$, or the twister space of ${\bf CP}(2)$.
Hence this metric is a Ricci-flat K\"ahler metric on the canonical
line bundle over the flag manifold and it is in the class discussed
in \cite{BB}, \cite{PP}. (See also \cite{HKN} on the construction
of Ricci-flat metrics on the canonical line bundle over Hermitian
symmetric spaces based on the K\"ahler potential
of supersymmetric gauge theory.) We also note that
when $\ell=0, k \neq 0$, the metric constructed in \cite{CGLP2} is reproduced.
The first order system in \cite{CGLP2} corresponds to the case $a=b,
c_1=-c_2$ in
this paper and cannot cover the most general case. This is the reason why
we can obtain more general solutions with two parameters.
On the other hand, when $\ell \neq 0, k =0$ then $U(r) \equiv 1$ and
the solution reduces to the Calabi
hyperK\"ahler metric over $T^{*}{\bf CP}(2)$ of $Sp(2)$ holonomy \cite{DS}, 
\cite{CGLP2}~;
%
\beqa
ds^2 &=& \left( 1 - \frac{\ell^4}{r^4} \right)^{-1} dr^2
+ \frac{1}{2} (r^2 - \ell^2) (\sigma_1^2 + \sigma_2^2)
+ \frac{1}{2} (r^2 + \ell^2) (\Sigma_1^2 + \Sigma_2^2)  \CR
& &~~~~~+ r^2 (\tau_1^2 + \tau_2^2)
+ \frac{1}{4} r^2  \left( 1 - \frac{\ell^4}{r^4} \right) T_A^2~, \quad
(\ell \leq r ) \label{cara}
\eeqa
%
with three K\"ahler forms~,
%
\beqa
\omega^1 &=& \omega~, \CR
\omega^2 &=& cdt \wedge \tau_1 - cfT_{A} \wedge \tau_2 - ab(
\sigma_1 \wedge \Sigma_1 - \sigma_2 \wedge \Sigma_2)~, \\
\omega^3 &=& cdt \wedge \tau_2 + cfT_{A} \wedge \tau_1 + ab(
\sigma_1 \wedge \Sigma_2 + \sigma_2 \wedge \Sigma_1)~.
\nonumber
\eeqa
%
The flag manifold $SU(3)/T^2$ is a two-sphere bundle over ${\bf CP} (2)$ and
in the limit $k \to 0$ the $S^2$-fiber collapses to develop the singular orbit
${\bf CP} (2)$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two types of ALC $Spin(7)$ metric}
\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


 From the view point of compactification of $M$ theory
it is of great interest to classify possible asymptotically
locally conical (ALC) metric. When $c_1 = c_2$,
an example of such ALC $Spin(7)$ metric is
given by \cite{KY}, \cite{CGLP5};
%
\beqa
ds^2 &=&  \frac {(r - \ell)^2}{(r + \ell)(r - 3\ell)} dr^2
+ (r - \ell)(r + \ell) (\sigma_1^2 + \sigma_2^2 +
\Sigma_1^2 + \Sigma_2^2) \CR
& &~~~~
+ (r - 3\ell)(r + \ell) (\tau_1^2 + \tau_2^2)
+ \ell^2\frac{(r + \ell)(r - 3\ell)}
{(r - \ell)^2}T_A^2~,  \quad (3\ell \leq r)
\eeqa
%
where the fiber over the base ${\bf CP}(2)$ is not ${\bf R}^4$ but
${\bf R}^4/{\bf Z}_2$. Note that this is different from the case of the Calabi
metric.
This is due to the fact that for the Calabi metric
it is $\sigma_1^2 + \sigma_2^2$ part that is collapsing at the singular
orbit, but it is $\tau_1^2 + \tau_2^2$ in the above ALC metric.


Let us consider the ALC $Spin(7)$ solutions that
interpolate between the short distance geometry of the form
${\bf R}^2 \times Flag_6$ (A-type boundary) or ${\bf R}^4 \times {\bf CP}(2)$
(B-type boundary) and the large distance geometry of the form
$S^1 \times C(Flag_6)$, where $C(Flag_6)$
is the 7-dimensional cone over $Flag_6$.
We then assume that the metric functions take the following form
for the large distance $t$~:
\begin{enumerate}
\item
$a(t)=ta_0+\alpha(t)$, $\;$
$b(t)=tb_0+\beta(t)$, $\;$
$c_1(t)=\gamma_1(t)$, \\
$c_2(t)=tc_{20}+\gamma_2(t)$, $\;$
$f(t)=tf_0+\zeta(t)$,
\item
$a(t)=ta_0+\alpha(t)$, $\;$
$b(t)=tb_0+\beta(t)$, \\
$c_1(t)=tc_{10}+\gamma_1(t)$, $\;$
$c_2(t)=tc_{20}+\gamma_2(t)$, $\;$
$f(t)=\zeta(t)$,
\end{enumerate}
where $a_0$, $b_0$, $c_{i0}$, $f_0$ are constants and
$\alpha$, $\beta$, $\gamma_i$, $\zeta$
are smooth functions tending to finite value for $t \rightarrow \infty$.
In case 1 the $S^1$ direction is $\tau_1$ and at large $t$ the function $c_1$
approaches a constant $M_1=\gamma_1(\infty)$, while in case 2 the
function $f$ of the $S^1$ direction $T_A$ approaches a constant
$M_2=\zeta(\infty)$. The octonionic instanton equation (\ref{five}) requires
the following conditions on the leading coefficients;
%
\beq
a_{0}^2=b_{0}^2=1, \;  a_0 b_0 c_{20}=1, \;  f_0=-1/2
\quad  \mbox{for case 1}~, \label{as1}
\eeq
%
\beq
a_{0}^2=b_{0}^2=1, \;  a_0 b_0 c_{20}=1, \;  c_{10}=c_{20}
\quad \mbox{for case 2}~. \label{as2}
\eeq
%
Note that they are different only in the last condition, but this 
difference produces
significant change as we will see shortly.
Thus the possible cone metrics on $C(Flag_6)$
consistent with the instanton equation are given by
%
\beqa
g_c &=& dt^2 + t^2(\sigma_1^2 + \sigma_2^2 + \Sigma_1^2 + \Sigma_2^2
+ \tau_2^2 + T_A^2/4)~, \\
g'_{c} &=& dt^2 + t^2(\sigma_1^2 + \sigma_2^2 + \Sigma_1^2 + \Sigma_2^2
+ \tau_1^2 + \tau_2^2)~,
\eeqa
%
corresponding to (\ref{as1}) and (\ref{as2}), respectively. It follows that
the boundary condition for the ALC metric is
%
\beq
g \rightarrow M_{1}^2 \tau_{1}^2+g_{c} \quad  \mbox{or} \quad
M_{2}^2 T_{A}^2+g'_{c}
\quad \mbox{for} \; t \rightarrow \infty.
\eeq
%
We call the first case $A_{\infty}^{\pm}$ and the second case $B_{\infty}$, and
prove the following proposition. The sign $\pm$ corresponds to the choice
$a_{0}=\pm b_{0}$ in (\ref{as1})\footnote{In (\ref{as2}) the choice of the sign
does not make difference due to the ${\bf Z}_2$ symmetry of the instanton 
equation.}~;
this difference does not appear at the
level of cone metrics, but it must be distinguished at the level of instanton
solutions.


\vskip0.8mm

\begin{flushleft}
{\bf Proposition} \quad If there exists a regular ALC solution interpolating
between $S^1 \times C(Flag_6)$ and ${\bf R}^2 \times Flag_6$ or
${\bf R}^4 \times {\bf CP}(2)$, then the following holds~:
\begin{enumerate}
\item
For the boundary $A_{\infty}^{\pm}$, $a(t)=\pm b(t)$ in the whole region
$0 \le t \le \infty$ and
the solution approaches ${\bf R}^2 \times Flag_6$ for $t \rightarrow 0$.
\item
For the boundary $B_{\infty}$, $c_1(t)=c_2(t)$ in the whole region
$0 \le t \le \infty$ and
the solution approaches
${\bf R}^4 \times {\bf CP}(2)$ for $t \rightarrow 0$.
\end{enumerate}
\end{flushleft}

\vskip0.8mm

\begin{flushleft}
{\bf Remark} \quad The part $dt^2+4t^2 T_A^2$ in the metric (\ref{ab})
looks like
\beq
dt^2+t^2 d\psi^2, \quad (0 \le \psi < 4\pi)
\eeq
when we fix the coordinates on $Flag_6$ in A-type boundary \cite{KY}.
Therefore, the range of $\psi$
must be adjusted to be that of usual polar coordinates on ${\bf R}^2$,
$0 \le \psi < 2\pi$. This means the manifold in the boundary
$A_{\infty}^{\pm}$ is $Flag_6/{\bf Z}_2$ rather than $Flag_6$, which
would have $0 \le \psi < 4\pi$. While in the case of $B_{\infty}$
it is not necessary to do such a modification since
\beq
dt^2+t^2(T_A^2+\sigma_1^2+\sigma_2^2)
\eeq
in (\ref{bb}) is the
standard metric on ${\bf R}^4$ written by the polar coordinates when we
fix the coordinates on ${\bf CP}(2)$.
\end{flushleft}

\vskip0.8mm

(Proof.) \quad We first consider the case $A_{\infty}^{+}$. From the
instanton equation
we have
%
\beqa
a(t) - b(t) &=& N \exp\left(\int^t u(t')dt' \right)~, \label{cc1}
\eeqa
%
where $N$ is an  integration constant and
%
\beq
u(t) = \frac{1}{2ab}\left(\frac{c_1^2-(a+b)^2}{c_1}+
\frac{c_2^2-(a+b)^2}{c_2} + 2f \right)~.
\eeq
%
Suppose that a regular solution exists
in the form (\ref{cc1}) with $N \neq 0$.
By using (\ref{as1}) with $a_0 = b_0$, it is easy to see the asymptotic
behavior
$a - b \simeq e^{-2t/M_{1}}$ for $t \rightarrow \infty$.
Note that the constant $M_1$ is required to be positive for
the exponentially small suppression.
If the solution approaches the singular
orbit $Flag_6$, then the product $c_1 c_2$ must be negative
by the result of section 2 (see also (\ref{sin1})). On the other hand,
$c_1 c_2$ is positive in the asymptotic
region since $c_1 c_2 \rightarrow M_1$ for $t \rightarrow \infty$
and hence $c_1$ or $c_2$ becomes zero at a certain time $t_0$,
which contradicts the regularity condition. If the solution approaches
the singular orbit ${\bf CP}(2)$, $f$ is positive as seen in
(\ref{sin2}), so the negative $f$ in the asymptotic region leads to
a contradiction. In the case of (\ref{sin3}), the product $c_1 c_2$
is negative for $t \rightarrow 0$,
which contradicts the sign in the asymptotic region.
Thus $a = b$ in the whole region, and if there exists a regular solution
with the boundary $A_{\infty}^{+}$, then it  approaches
${\bf R}^2 \times Flag_6$ for
$t \rightarrow 0$ since this boundary is only one consistent with $a=b$.
Furthermore, by the discrete symmetry of the instanton equation
there exists a regular solution with $a=-b$ for the boundary
$A_{\infty}^{-}$.

Next let us consider the case $B_{\infty}$. By the instanton equation
we have
%
\beq
c_{1}(t)-c_{2}(t)=N \exp\left(\int^t v(t')dt' \right)~,  \label{cc2}
\eeq
%
with
\beq
v(t)=\frac{4f^2-(c_1+c_2)^2}{2c_1 c_2 f}-\frac{c_1+c_2}{ab}.
\eeq
If we assume a regular solution (\ref{cc2}) with $N \neq 0$, then
similar arguments lead to a contradiction.
Thus $c_1=c_2$ in the whole region and the solution
approaches ${\bf R}^4 \times {\bf CP}(2)$ given by (\ref{sin2}) for
$t \rightarrow 0$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ Evidence for new global solutions}
\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

It is not easy to find exact solutions in general and we turn to numerical
computations to examine the existence of global solutions to
the octonionic instanton equation (\ref{five}).
The result of our analysis is summarized in Figure 1\footnote{The figure is attached
at the end of the section.}, which shows
possible lines for existence of global solutions in the two dimensional
parameter space of initial conditions at the singular orbit.
The circle $p^2+q^2=m^2$ represents the
Ricci-flat K\"ahler metrics of $SU(4)$ holonomy obtained in section 3.
As one can see from (\ref{SU4}), near the singular orbit at
$r_0=(k^4+\ell^4)^{1/4}$
we have
%
\beq
g \longrightarrow d\rho^2+4\rho^2 T_{A}^2 + \frac{1}{2}(r_0^2+\ell^2)
(\sigma_1^2+\sigma_2^2)
+\frac{1}{2}(r_0^2-\ell^2)(\Sigma_1^2+\Sigma_2^2)+
r_0^2(\tau_1^2+\tau_2^2)~,
\eeq
%
where $\rho^2=r_0(r-r_0)/2$. By comparing with the expansion (\ref{sin1}),
we see that they are
parametrized by the circle with radius $m=r_0$ in the $(p,q)$-space of
A-type boundary. The four points on the circle,
$(p,q)=(\pm m, 0)$ and $(0, \pm m)$,
correspond the Calabi hyperK\"ahler  metric given by (\ref{cara}),
where the holonomy group is
further reduced to $Sp(2)$. Note that the Calabi metric satisfies B-type
boundary and hence the singular orbit changes from $Flag_6$ to ${\bf CP}(2)$
at these points.
The wavy lines attaching to the Calabi metric are the ALC metrics
of $Spin(7)$ holonomy whose existence was expected by the second statement
of proposition ($B_{\infty}$ boundary). Indeed, from the numerical analysis
we can find non-singular solutions interpolating between $S^1 \times C(Flag_6)$
and ${\bf R}^4 \times {\bf CP}(2)$ for the parameter region $q_1 < -2/3$
of B-type boundary (\ref{sin2}), with $q_1=-2/3$ giving the Calabi metric
\cite{KY} \cite{CGLP5}.

Finally we discuss the new metrics of $Spin(7)$ holonomy depicted by the
lines $p=\pm q, p^2+q^2>m^2$ in Figure 1, which we shall denote by
${\bf C}_{8}^{*}$.
They are an analogue of $Spin(7)$
metrics ${\bf C}_8$ on the line bundle over ${\bf CP}(3)$
discussed in \cite{CGLP5} \cite{CGLP6}.
Although we have not been able to find the solutions in closed form,
the following arguments indicate they must exist.
The solutions on the two lines $p=q$ and $p=-q$ are related to each other
by the action of the discrete symmetry of the instanton equation
(\ref{five}), and so we will consider the case $p=q$. By rescaling the
parameter $p \rightarrow mp$, the perturtative expansion for
A-type boundary becomes
%
\beqa
a(t) &=& b(t)=m \left(p+\frac{6p^2-1}{4p^3}(t/m)^2+
\cdot \cdot \cdot \right)~, \CR
c_1(t) &=& m
\left(1+\frac{2p^2-1}{2p^2}(t/m)+\frac{12p^4-4p^2+3}{8p^4}(t/m)^2
+\cdot \cdot \cdot \right)~, \CR
c_2(t) &=& -m
\left(1-\frac{2p^2-1}{2p^2}(t/m)+\frac{12p^4-4p^2+3}{8p^4}(t/m)^2
+\cdot \cdot \cdot \right)~, \label{atype} \\
f(t) &=& -2t \left(1-\frac{12p^4+20p^2-1}{12p^4}(t/m)^2+
\cdot \cdot \cdot \right)~, \nonumber
\eeqa
%
which shows the reduction $a=b$ of the instanton equation.
If we put $c_3 \equiv -2 f$, the first order system with $a=b$ reduction 
is described by
\beqa
\frac{\dot a}{~a~} &=& \frac{c_1}{2a^2}
+ \frac{c_2}{2a^2} + \frac{c_3}{2a^2}~, \CR
\frac{\dot c_1}{~c_1~} &=& -\frac{c_1}{a^2}
+ \frac{c_1^2 - (c_2 -c_3)^2}{c_1 c_2 c_3}~, \label{ah} \\
\frac{\dot c_2}{~c_2~} &=& -\frac{c_2}{a^2}
+ \frac{c_2^2 - (c_3 -c_1)^2}{c_1 c_2 c_3}, \CR
\frac{\dot c_3}{~c_3~} &=& -\frac{c_3}{a^2}
+ \frac{c_3^2 - (c_1 -c_2)^2}{c_1 c_2 c_3}~.
\nonumber
\eeqa
%
After the rescaling $a \to \sqrt{2} a$ we obtain exactly the same first
order system as eq.(8) in \cite{CGLP5}, where
it was shown numerically that the solution with the boundary (\ref{atype})
is regular and ALC provided that the parameter $p$ is chosen to satisfy
$p^2 > 1/2$. It is easy to check the boundary $p^2=1/2$ corresponds to the 
AC solution
(\ref{SU4}) with the special value $\ell=0$. Thus two-parameter family of
ALC metrics
${\bf C}_{8}^{*}$ has the same topology as the canonical
line bundle over $Flag_6$. The large distance geometry of ${\bf C}_{8}^{*}$
can be worked out as follows. By the proposition in section 4,
${\bf C}_{8}^{*}$ approaches
the boundary $A_{\infty}^{+}$ for $t \rightarrow \infty$. After some
calculation we find that the asymptotic expansion up to order $t^{-3}$
is given by
%
\beqa
a(t) &=& t \left(1+\frac{3}{8}(M/t)^2+\frac{1}{4}(M/t)^3+
\frac{1}{2}\left(\frac{7}{64}-P \right)(M/t)^4+\cdot \cdot \cdot \right)~, \CR
c_1(t) &=& M\left(1-\frac{1}{2}(M/t)^2-\frac{1}{2}(M/t)^3+
\cdot \cdot \cdot \right)~, \label{series} \\
c_2(t) &=& c_3(t)=t\left(1-\frac{1}{2}(M/t)+
P(M/t)^4+ \cdot \cdot \cdot \right)~.
\nonumber
\eeqa
It should be noticed that the equality $c_2=c_3$ is valid for all orders,
if we assume that they can be expanded as power series in $t^{-1}$.
Hence, the series coincides with the asymptotic form of the ALC solutions
found in \cite{CGLP3}. The parameters $M,P$ correspond to $m,p$
in the expansion around the singular orbit.
Since the product $c_1 c_2=-m^2$ for $t \rightarrow 0$, we
must have $M < 0$. This sign is consistent with the exponentially small
correction of the asymptotic expansion. Indeed, we have
%
\beq
\dot c_2 - \dot c_3 =(c_2-c_3)\left(\frac{(c_2+c_3)^2-c_1^2}{c_1 c_2
c_3}-\frac{c_2+c_3}{a^2} \right)~,
\eeq
which leads to the asymptotic behavior $c_2-c_3 \simeq e^{4t/M}$ using the
expansion (\ref{series}),
and the metric functions behave similarly to those
in the Atiyah-Hitchin metric \cite{GM}, \cite{CGLP6}.


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

\newpage

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% Figure: Octonionic instantons
\put(10.6000,-40.4000){\makebox(0,0)[lb]{Figure 1: 
Possible lines for the existence of global metric of special holonomy.}}%
\end{picture}%

\newpage


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

\vskip10mm

\begin{center}
{\bf Acknowledgements}
\end{center}

We would like to thank G.W. Gibbons and C.N. Pope for correspondence.
This work is supported in part by the fund
for special priority area 707
"Supersymmetry and Unified Theory of Elementary Particles" and
the Grant-in-Aid for Scientific Research No. 12640074.


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


%\newpage
%\input{newspin(7).tex}
%\input{testspin.tex}
%\newpage


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Appendix A}
\renewcommand{\theequation}{A.\arabic{equation}}\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{flushleft}
{\bf Convention of $SU(3)$ Maurer-Cartan forms}
\end{flushleft}

We use the following
$SU(3)$ Maurer-Cartan equation that is $\Sigma_3$ symmetric;
%
\beqa
d\sigma_1 &=& \Sigma_1 \wedge \tau_1 - \Sigma_2 \wedge \tau_2 + \kappa_A
T_A \wedge \sigma_2
+ \kappa_B T_B \wedge \sigma_2~, \CR
d\sigma_2 &=& - \Sigma_1 \wedge \tau_2 - \Sigma_2 \wedge \tau_1 - \kappa_A
T_A \wedge \sigma_1
- \kappa_B T_B \wedge \sigma_1~, \CR
d\Sigma_1 &=& \tau_1 \wedge \sigma_1 - \tau_2 \wedge \sigma_2 + \mu_A T_A
\wedge \Sigma_2
+ \mu_B T_B \wedge \Sigma_2~, \CR
d\Sigma_2 &=& - \tau_1 \wedge \sigma_2 - \tau_2 \wedge \sigma_1 - \mu_A T_A
\wedge \Sigma_1
- \mu_B T_B \wedge \Sigma_1~, \\
d\tau_1 &=& \sigma_1 \wedge \Sigma_1 - \sigma_2 \wedge \Sigma_2 + \nu_A T_A
\wedge \tau_2
+ \nu_B T_B \wedge \tau_2~, \CR
d\tau_2 &=& - \sigma_1 \wedge \Sigma_2 - \sigma_2 \wedge \Sigma_1 - \nu_A
T_A \wedge \tau_1
- \nu_B T_B \wedge \tau_1~, \CR
dT_A &=& 2\alpha_A \sigma_1 \wedge \sigma_2  + 2\beta_A \Sigma_1 \wedge
\Sigma_2
+ 2\gamma_A \tau_1 \wedge \tau_2~, \CR
dT_B &=& 2\alpha_B \sigma_1 \wedge \sigma_2  + 2\beta_B \Sigma_1 \wedge
\Sigma_2
+ 2\gamma_B \tau_1 \wedge \tau_2~. \nonumber
\eeqa
%
This form of the Maurer-Cartan equation is symmetric under the (cyclic)
permutation of $(\sigma_i, \Sigma_i,
\tau_i)$.
 From the Jacobi identity we see that
the parameters
$\alpha, \beta, \gamma, \kappa, \mu,\nu$, which describe the "coupling" of
the Cartan generators $\{ T_A, T_B\}$ satisfy
%
\beqa
& & \alpha_A + \beta_A + \gamma_A = 0~, \;
\alpha_B + \beta_B + \gamma_B = 0~,  \CR
\kappa_A &=& \frac{1}{\Delta}(\beta_B - \gamma_B), \;
\kappa_B = -\frac{1}{\Delta}(\beta_A - \gamma_A), \;
\mu_A = -\frac{1}{\Delta}(\alpha_B-\gamma_B), \\
\mu_B &=& \frac{1}{\Delta}(\alpha_A - \gamma_A), \;
\nu_A = \frac{1}{\Delta}(\alpha_B - \beta_B), \;
\nu_B = -\frac{1}{\Delta}(\alpha_A - \beta_A) \nonumber
\eeqa
%
with $\Delta=\beta_A \alpha_B -\alpha_A \beta_B$
leaving four free parameters $(\alpha_{A,B}, \beta_{A,B})$.
We may further put the "orthogonality" conditions;
%
\beqa
\alpha_A \alpha_B + \beta_A \beta_B + \gamma_A \gamma_B &=& 0~, \CR
\kappa_A \kappa_B + \mu_A \mu_B + \nu_A \nu_B &=& 0~,
\eeqa
%
which reduces one parameter.

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Appendix B}
\renewcommand{\theequation}{B.\arabic{equation}}\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{flushleft}
{\bf Reduction of holonomy group and the octonionic instanton equation}
\end{flushleft}

One of ways to realize the reduction of holonomy group is to impose
appropriate linear relations on the $so(n)$ valued spin connection one form
$\omega_{ab} = - \omega_{ba}$. It is rather amusing that in all the three cases
which are most relevant from the viewpoint of $M$ theory conpactifications
the expected number of linear relations is always seven, since
${\rm dim}~SO(8) - {\dim}~Spin(7) = {\rm dim}~SO(7) - {\dim}~G_2=
{\rm dim}~SO(6) - {\dim}~SU(3) = 7$.  There are topological relations
behind this dimension counting; $Spin(7)/G_2 \simeq SO(8)/SO(7) \simeq S^7$ and
$G_2/SU(3) \simeq SO(7)/SO(6) \simeq S^6$.
In fact the following octonionic instanton equation gives
a \lq\lq master\rq\rq\ equation for seven conditions required for the 
reduction \cite{CDFN}, \cite{AL}.
%
\beq
\omega_{ab} = \frac{1}{2} \Psi_{abcd} \omega_{cd}~, \label{octduality}
\eeq
where totally anti-symmetric tensor $\Psi_{abcd}$ is defined by
the structure constants of octonions $\psi_{abc}$ as follows;
%
\beqa
\Psi_{abc0} &=& \psi_{abc}~,  \quad (1 \leq a,b,c, \cdots \leq 7) \CR
\Psi_{abcd} &=& -\frac{1}{3!} \epsilon_{abcdefg} \psi_{efg}~.
\eeqa
%
A conventional choice of the structure constants is
\beqa
\psi_{abc}= +1~, ~~~{\rm for}~~~(abc)=(123), (516), (624), (435), (471),
(572), (673)~.
\eeqa
It can be shown that (\ref{octduality}) implies the four form defined by
%
\beq
\Omega = \frac{1}{4!}
\Psi_{abcd} e^a \wedge e^b \wedge e^c \wedge
e^d~. \label {calib}
\eeq
%
is closed and the metric has $Spin(7)$ holonomy \cite{BFK}.
In the above convention of the structure constants of octonions
the explicit form of the octonionic instaton equation is
%
\beqa
\omega_{14} + \omega_{25} + \omega_{36}  + \omega_{07} &=& 0~, \CR
\omega_{71} + \omega_{62} + \omega_{35}  + \omega_{04} &=& 0~, \CR
\omega_{47} + \omega_{65} + \omega_{23}  + \omega_{01} &=& 0~, \CR
\omega_{67} + \omega_{12} + \omega_{54}  + \omega_{03} &=& 0~, 
\label{explicit} \\
\omega_{73} + \omega_{51} + \omega_{24}  + \omega_{06} &=& 0~, \CR
\omega_{57} + \omega_{46} + \omega_{31}  + \omega_{02} &=& 0~, \CR
\omega_{72} + \omega_{16} + \omega_{43}  + \omega_{05} &=& 0~. \nonumber
\eeqa
If we simply substitute $\omega_{0k},(1\leq k \leq 7)$, then 
(\ref{explicit}) gives
the seven conditions for $G_2$ holonomy.
Further putting $\omega_{7j},(1\leq j \leq 6)$ gives the seven conditions for
$SU(3)$ holonomy. We should emphasize that compared with the condition on
the Riemann curvature, the condition on the spin connection depends on the gauge
or the choice of coordinate system and therefore it is only a sufficient 
but not necessary condition for special holonomy.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Appendix C}
\renewcommand{\theequation} {C.\arabic{equation}}\setcounter {equation} {0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{flushleft}
{\bf Perturbative expansion around singular orbits}
\end{flushleft}

For A-type boundary the instanton equation is perturbatively solved
in the form,
\beqa
a(t) &=& p +\left(\frac{1}{p}+\frac{(p^2+q^2-m^2)(p^2-q^2+m^2)}
{4 p q^2 m^2} \right)t^2+ \cdot \cdot \cdot , \CR
b(t) &=& q +\left(\frac{1}{q}+\frac{(p^2+q^2-m^2)(-p^2+q^2+m^2)}
{4 p^2 q m^2} \right)t^2+ \cdot \cdot \cdot , \CR
c_{1}(t) &=& m+ \left(\frac{p^2+q^2-m^2}{2pq} \right)t
+\left(\frac{2}{m}-\frac{m(p^2+q^2-m^2)}{2p^2 q^2}
-\frac{(p^2+q^2-m^2)^2}{8p^2 q^2 m} \right)t^2+ \cdot \cdot \cdot ,
\label{sin1} \CR
c_{2}(t) &=& -m + \left(\frac{p^2+q^2-m^2}{2pq} \right)t
-\left(\frac{2}{m}-\frac{m(p^2+q^2-m^2)}{2p^2 q^2}
-\frac{(p^2+q^2-m^2)^2}{8p^2 q^2 m} \right)t^2+ \cdot \cdot \cdot , \CR
f(t) &=& -2t\left(1+
\left(\frac{p^4+q^4+m^4-10 p^2 m^2-10 q^2 m^2-14 p^2 q^2}{12 p^2 q^2 m^2}
\right)t^2+ \cdot \cdot \cdot  \right)~.
\eeqa
%
We note that the power series solution are completely fixed by
the "initial conditions" $p,q,m$. (This should be compared with
the case of B type boundary condition in the following.)
The reduction $c_1=-c_2$ is
reproduced by imposing $p^2 + q^2 = m^2$
and the reduction $a=\pm b$ by $p=\pm q$.

There are two possible solutions for B-type boundary. One of these is given by
%
\beqa
a(t) &=& t \left(1-\frac{1}{2}(q_{1}+1)(t/m)^2+ \cdot \cdot \cdot \right),
\CR
b(t) &=& m \left(1+\frac{1}{2}(t/m)^2+ \cdot \cdot \cdot \right),
\CR
c_{1}(t) &=& c_{2}(t)=m\left(1+(t/m)^2+ \cdot \cdot \cdot \right),
\label{sin2} \\
f(t) &=& t \left(1+q_{1}(t/m)^2+ \cdot \cdot \cdot \right).
\nonumber
\eeqa
%
The other solution has the following expansion~,
\beqa
a(t) &=& t \left(1-\frac{1}{6}(t/m)^2+ \cdot \cdot \cdot \right), \CR
b(t) &=& m \left(1+q_{2}(t/m)^2+ \cdot \cdot \cdot \right), \CR
c_{1}(t) &=& m \left(1+(t/m)^2+ \cdot \cdot \cdot \right), \label{sin3} \\
c_{2}(t) &=& -m\left(1+2(1-q_{2})(t/m)^2+ \cdot \cdot \cdot \right), \CR
f(t) &=& t \left(-1+ \frac{2}{3} (t/m)^2 + \cdot \cdot \cdot  \right).
\nonumber
\eeqa
%
These solutions include the free parameters $q_1$ and $q_2$ in addition to
the scaling parameter $m$. In particular,
both solutions with $q_{1}=-2/3$ and $q_{2}=1/2$ lead to a same metric
and this is in fact precisely the Calabi hyperK\"ahler metric on
$T^{*}{\bf CP}(2)$.

%\begin{flushleft}
%{\bf Asymptotic expansion}
%\end{flushleft}


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

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Greater Than Four, Nucl. Phys. {\bf B214} (1983) 452.

\bibitem{AL} B.S. Acharya and M. O'Loughlin,
Self-Duality in $D \leq 8$-Dimensional Euclidean Gravity,
Phys. Rev. {\bf D55} (1997) 4521, {\tt hep-th/9612182}.

\bibitem{BFK} I. Bakas, E.G. Floratos and A. Kehagias,
Octonionic Gravitational Instanton,
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Quelques Exemples de Varietes Riemanniennes Completes Non
Compactes a Courbure de Ricci Positive,
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\bibitem{PP} D.N. Page and C.N. Pope,
Inhomogeneous Einstein Metrics on Complex Line Bundles,
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\bibitem{HKN} K. Higashijima, T. Kimura and M. Nitta,
Gauge Theoretical Construction of Non-compact Calabi-Yau Manifolds,
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Ricci-flat Kahler Manifolds from Supersymmetric Gauge Theories,
{\tt hep-th/0108084}.

\bibitem{DS} A. Dancer and A. Swann,
Hyperk\"ahler Metrics of Cohomogeneity One
J. Geom. Phys. {\bf 21} (1997) 218.

\bibitem{GM} G.W. Gibbons and N.S. Manton,
Classical and Quantum Dynamics of BPS Monopoles, 
Nucl. Phys. {\bf B274} (1986) 183.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\bibitem{KN} Y. Konishi and M. Naka,
%Coset construction of $Spin(7),G_2$ gravitational instanton,
%{\tt hep-th/0104208}.
%
%\bibitem{BS} R. Bryant and S. Salamon,
%On the construction of some complete metrics with
%exceptional holonomy, Duke Math. J. . {\bf 58} (1989) 829.
%
%\bibitem{GPP} G.W. Gibbons, D.N. Page and C.N. Pope,
%Einstein metrics on $S^3, R^3$ and $R^4$ Bundles
%Commun. Math. Phys. {\bf 127} (1990) 529.
%
%\bibitem{CGLP1} M. Cveti\v c, G.W. Gibbons, H. L\"u and C.N. Pope,
%Ricci-flat metrics, Harmonic forms and Brane resolutions, {\tt 
%hep-th/0012011}.
%
%
%\bibitem{CGLP4} M. Cveti\v c, G.W. Gibbons, H. L\"u and C.N. Pope,
%Supersymmetric M3-branes and $G_2$ manifolds, {\tt hep-th/0106026}.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{thebibliography}



\end{document}














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First-Order Equations for Gauge Fields in Spaces of Dimension
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Self-Duality in $D \leq 8$-Dimensional Euclidean Gravity,
Phys. Rev. {\bf D55} (1997) 4521, {\tt hep-th/9612182}.

\bibitem{BFK} I. Bakas, E.G. Floratos and A. Kehagias,
Octonionic Gravitational Instanton,
Phys. Lett. {\bf B445} (1998) 69, {\tt hep-th/9810042}.

\bibitem{BB} L. Berard-Bergery,
Quelques Exemples de Varietes Riemanniennes Completes Non
Compactes a Courbure de Ricci Positive,
C. R. Acad. Sci., Paris, Ser. I302 (1986) 159.

\bibitem{PP} D.N. Page and C.N. Pope,
Inhomogeneous Einstein Metrics on Complex Line Bundles,
Class. Quantum Grav. {\bf 4} (1987) 213.

\bibitem{HKN} K. Higashijima, T. Kimura and M. Nitta,
Gauge Theoretical Construction of Non-compact Calabi-Yau Manifolds,
{\tt hep-th/0110216};
Ricci-flat Kahler Manifolds from Supersymmetric Gauge Theories,
{\tt hep-th/0108084}.

\bibitem{DS} A. Dancer and A. Swann,
Hyperk\"ahler Metrics of Cohomogeneity One
J. Geom. Phys. {\bf 21} (1997) 218.

\bibitem{GM} G.W. Gibbons and N.S. Manton,
Classical and Quantum Dynamics of BPS Monopoles, 
Nucl. Phys. {\bf B274} (1986) 183.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\bibitem{KN} Y. Konishi and M. Naka,
%Coset construction of $Spin(7),G_2$ gravitational instanton,
%{\tt hep-th/0104208}.
%
%\bibitem{BS} R. Bryant and S. Salamon,
%On the construction of some complete metrics with
%exceptional holonomy, Duke Math. J. . {\bf 58} (1989) 829.
%
%\bibitem{GPP} G.W. Gibbons, D.N. Page and C.N. Pope,
%Einstein metrics on $S^3, R^3$ and $R^4$ Bundles
%Commun. Math. Phys. {\bf 127} (1990) 529.
%
%\bibitem{CGLP1} M. Cveti\v c, G.W. Gibbons, H. L\"u and C.N. Pope,
%Ricci-flat metrics, Harmonic forms and Brane resolutions, {\tt 
%hep-th/0012011}.
%
%
%\bibitem{CGLP4} M. Cveti\v c, G.W. Gibbons, H. L\"u and C.N. Pope,
%Supersymmetric M3-branes and $G_2$ manifolds, {\tt hep-th/0106026}.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{thebibliography}