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\begin{flushright}
OU-HET 262 \\
April 1997
\end{flushright}

\begin{center}
\begin{Large}
{\bf Toward Second-Quantization of $D5$-Brane}\\
\end{Large}

\vspace{25pt}

\noindent
Toshio NAKATSU

\vspace{18pt}

\begin{small}
{\it Department of Physics,}\\
{\it Graduate School of Science, Osaka University,}\\
{\it Toyonaka, Osaka 560, JAPAN}
\end{small}

\vspace{25pt}

\underline{Abstract}
\end{center}

\vspace{10pt}

\begin{small}
A framework of second-quantization of
$D5$-branes is proposed.
It is based on the study of topology of the
moduli space of their low energy effective
worldvolume theory.
Among the topological cycles which resolve
singularities caused by overlapping $D5$-branes,
it is introduced those cycles which duals,
constituting a subspace of cohomology group
of the moduli space,
turn out to define the Fock space of
the second-quantized $D5$-branes.
The second-quantized operators are given by
creation and annihilation operators of those
cycles or their duals.
\end{small}

\newpage
\sect{Introduction and Summary}

               The discovery of the Dirichlet branes
($D$-branes) \cite{Polchinski} provides us a route to
the quantization of solitonic objects \cite{Strominger}
in string theory.
The quantum fluctuations of these solitonic objects are
described by open strings with one or both of their
boundaries constrained on them, for which they are named
Dirichlet branes.

                  Consider the Dirichlet five-branes
($D5$-branes) in Type IIB theory
with open strings having the $U(n)$ Chan-Paton factors
or Type IIB theory under the background of $n$ coincident
$D9$-branes. When there are $k$ coincident $D5$-branes at
$x^6= \cdots =x^9=0$, their low energy effective worldvolume
theory is given by a six-dimensional supersymmetric $U(k)$
gauge theory with global $U(n)$-symmetry
\cite{Polchinski-Witten},\cite{Witten1}.
In this article we investigate this six-dimensional theory
including the Fayet-Iliopoulos $D$-terms,
$\zeta_{\dot{A}\dot{B}}\int\!d^6\!x
~Tr~D^{\dot{A}\dot{B}}$.
The degenerate vacua ($or$ moduli space),
which is denoted by ${\cal M}(k)$, are
determined by the $D$-flat conditions.
${\cal M}(k)$ is the moduli space of $k$ $D5$-branes
with open strings having the $U(n)$ Chan-Paton factors.
Regarding auxiliary $D$-fields as a hyperk\"ahler
momentum map,
the moduli space is a $4nk$-dimensional
hyperk\"ahler manifold obtained by the
hyperk\"ahler quotient construction \cite{HKLR},
\begin{eqnarray}
\left.
{\cal M}(k)=
\mu_{\dot{A}\dot{B}}^{-1}
(\zeta_{\dot{A}\dot{B}})
\right/ U(k)~~~,
\nonumber
\end{eqnarray}
where
$\mu_{\dot{A}\dot{B}}$ is the hyperk\"ahler momentum
map of $U(k)$-symmetry
(a global part of the $U(k)$ gauge symmetry).

     Due to the coupling with open string
$D$-branes are identified with the BPS states which
have the Ramond-Ramond charges. These BPS states are
allowed to have the bound states which are marginally
stable \cite{Witten2},\cite{Sen}.
For the case of $D5$-branes
it is discussed in\cite{Harvey-Moore},\cite{FKN} the possibility of
identifying these BPS states with cohomology
elements of the moduli space of $D5$-branes and
thereby the second-quantized five-branes are proposed.
The purpose of this article is to provide a general
framework of the second-quantization of $D5$-brane
argued in \cite{FKN}.

     In the next section we study topology of the
moduli space utilizing the techniques developed by
Nakajima \cite{Nakajima}.
Due to $D5$-branes the Lorentz group $SO(9,1)$ reduces,
at least, to $SO(5,1)\times SO(4)$.
$SO(4)$, which originates from the rotations in the
four-dimensions $(x^6,\cdots,x^9)$, is a global symmetry
of the worldvolume theory.
To study the vacua it turns out useful to fix a complex
structure of ${\cal M}(k)$. One can fix it by giving a
complex structure of the four-dimensions.
The hyperk\"ahler quotient acquires the form,
\begin{eqnarray}
{\cal M}(k)=
\frac{\mu_{{\bf C}}^{-1}(0) \cap
           \mu_{{\bf R}}^{-1}(\eta)}
     {U(k)}~~~,
\nonumber
\end{eqnarray}
where the hyperk\"ahler momentum map
$\mu_{\dot{A}\dot{B}}$ is decomposed into
$\mu_{{\bf C}}$, the complex (holomorphic) part,
and
$\mu_{{\bf R}}$, the real part.
The three constants $\zeta_{\dot{A}\dot{B}}$
(coupled with the Fayet-Iliopoulos terms)
are rotated equal to zero except the only
one component, which is denoted by $\eta$.
$\eta$ is assumed to be positive.
In the case of $\eta$ being positive, the moduli space
${\cal M}(k)$ turns out to be a smooth hyperk\"ahler
manifold.

           In order to preserve this complex structure,
the structure group $SO(4)$
of the four-dimensions reduces to $U(2)$.
An abelian part of $U(2)$ rotates $(z_1,z_2)$,
the holomorphic coordinates of ${\bf C}^2$, by
phases and also acts on the moduli space ${\cal M}(k)$.
Its fixed points in ${\cal M}(k)$ are related with
overlapping $D5$-branes.
Any fixed point turns out to become zero as
$\eta \rightarrow 0$.
``zero" is a singularity of the moduli space obtained by
setting all the components $\zeta_{\dot{A}\dot{B}}$
being zero.
This singularity is caused by $k$ $D5$-branes
overlapping at the origin $(z_1,z_2)=(0,0)$.
From the perspective of four-dimensional gauge theory one
can say it corresponds to the small size limit of
$k$ $SU(n)$-instantons sitting at the origin.

                It is shown that one can associate an
appropriate set of $n$ Young tableaux with each fixed
point.
This correspondence is not one-to-one because of
degeneracies of the fixed points. Strictly speaking,
each set of $n$ Young tableaux satisfying conditions
(\ref{Young tableau}) corresponds to a fixed submanifold
of the moduli space.
Taking the Morse theoretical viewpoint
\cite{Nakajima} topology of ${\cal M}(k)$ is described
from considerations on these fixed submanifolds.
The action of the abelian part of $U(2)$ is hamiltonian
and its momentum map can be regarded,
with an appropriate combination,
as a perfect Morse function on the moduli space.
The Poincar\'e polynomial of the moduli space
turns out to have form (\ref{Poincare Poly})
\begin{eqnarray}
&&P_t({\cal M}(k))=
\sum_{(\Gamma_1,\cdots,\Gamma_n)}
t^{2 \left\{
n(k-\sum_{j=1}^{n}l(\Gamma_j))
    +\sum_{i<j}l(\Gamma_i \backslash \Gamma_j)
       -\sum_{i<j}\nu(\Gamma_i \backslash \Gamma_j)
\right\} }
P_t({\cal F}_{(\Gamma_1,\cdots, \Gamma_n)})
\nonumber
\end{eqnarray}
where the summation is performed with respect to
those set of $n$ Young tableaux satisfying conditions
(\ref{Young tableau}).
To each set of $n$ Young tableaux
$(\Gamma_1,\cdots,\Gamma_n)$,
the corresponding fixed submanifold is denoted by
${\cal F}_{(\Gamma_1,\cdots,\Gamma_n)}$.
Note that $l(\Gamma_i),l(\Gamma_i/\Gamma_j)$ and
 $\nu(\Gamma_i/\Gamma_j)$ are positive integers
introduced in the text.
In this expression of the Poincar\'e polynomial,
terms related with a given set of $n$
Young tableaux $(\Gamma_1,\cdots,\Gamma_n)$
can be read as
\begin{eqnarray}
&&
t^{2 \left\{
       n(k-\sum_{j=1}^{n}l(\Gamma_j))
             +\sum_{i<j}l(\Gamma_i \backslash \Gamma_j)
       -\sum_{i<j}\nu(\Gamma_i \backslash \Gamma_j)
\right\} }
P_t({\cal F}_{(\Gamma_1,\cdots, \Gamma_n)})
\nonumber \\
&& =
t^{2 \left\{
       n(k-\sum_{j=1}^{n}l(\Gamma_j))
             +\sum_{i<j}l(\Gamma_i \backslash \Gamma_j)
       -\sum_{i<j}\nu(\Gamma_i \backslash \Gamma_j)
\right\}
+~dim~{\cal F}_{(\Gamma_1,\cdots,\Gamma_n)} }
\left( 1 +O(t^{-2}) \right)~~.
\nonumber
\end{eqnarray}
A cycle which gives the leading will be called the
maximal dimensional cycle labelled by
$(\Gamma_1,\cdots,\Gamma_n)$ and denoted by
$C_{(\Gamma_1,\cdots,\Gamma_n)}$.
Among these topological cycles,
those labelled by $(\Gamma,\emptyset,\cdots,\emptyset)$
are studied in Section 3 from the viewpoint of $D5$-branes.
$\Gamma$ is an arbitrary Young tableau of $k$ boxes.
One can write it explicitly by
$\Gamma=\mbox{$[k_1,\cdots,k_l]$}$ where
$k_i$ $(1 \leq i \leq l)$ are non-decreasing
positive integers satisfying $k_1+\cdots +k_l=k$.
We begin Section 3 by examining the case of
$\Gamma=\mbox{$[k]$}$.
An realization of the cycle
$C_{({\scriptsize \mbox{$[k]$}},
\emptyset,\cdots,\emptyset)}$ is presented.
Any point of this cycle is shown to
describe the vacuum of overlapping $k$ $D5$-branes
which admits $2(nk-1)$ additional degrees of freedom.
(For the explicit form presented in the text,
$k$ $D5$-branes are degenerate at the origin.)
These parameters give a parametrization of the cycle
$C_{({\scriptsize \mbox{$[k]$}},
\emptyset,\cdots,\emptyset)}$.
They turn out to disappear
when $\eta$ goes to zero.
It means that this topological cycle is added
to resolve the aforementioned singularity of
the moduli space at
$\zeta_{\dot{A}\dot{B}}=0$.
By changing the center-of-mass of $k$ $D5$-branes
from the origin to an arbitrary point, say, $P$
in the four-dimensions
one can construct a cycle which is same as
$C_{({\scriptsize \mbox{$[k]$}},
\emptyset,\cdots,\emptyset)}$
except that $k$ $D5$-branes are degenerate
at $P$, not at the origin.
In order to make the position of $k$ coincident
$D5$-branes, this cycle will be denoted by
$C_{{\scriptsize \mbox{$[k]$}}}(P)$.
It is isomorphic to
$C_{({\scriptsize \mbox{$[k]$}},
\emptyset,\cdots,\emptyset)}$.

      Nextly we consider vacua of $k=k_1+\cdots+k_l$
$D5$-branes ($k_1 \geq \cdots \geq k_l$) in which each
$k_i$ pieces are degenerate at $P_i$.
In particular we concentrate on the vacuum obtained by
superposing the vacua of $k_i$ $D5$-branes which belong
to $C_{{\scriptsize \mbox{$[k_i]$}}}(P_i)$.
The additional degrees of freedom of each constituent
are now regarded as parameters of the superposed vacuum.
They constitute a $2(nk-l)$-dimensional cycle of
${\cal M}(k)$, which is shown to be identified with
the maximal cycle
$C_{({\scriptsize \mbox{$[k_1,\cdots,k_l]$}},
\emptyset,\cdots,\emptyset)}$.
In order to make the positions of overlapping
$k$ $D5$-branes manifest
this $2(nk-l)$-dimensional cycle will be denoted by
$C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,\cdots,P_l)$.

              In Section 4 we investigate on
superposition of two vacua one of which is the
vacuum of $k$ $D5$-branes belonging to
$C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,\cdots,P_l)$
while the other is a generic vacuum of
$\tilde{k}$ $D5$-branes.
This superposition defines an inclusion
of
$C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,\cdots,P_l) \times {\cal M}(\tilde{k})$
to ${\cal M}(k+\tilde{k})$.
Its image consists of vacua of $k+\tilde{k}$
$D5$-branes in which the configurations of $k$ pieces,
regarded as vacua of $k$ $D5$-branes, belong to
$C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,\cdots,P_l)$.
By letting the $l$-positions of the overlapping
$k$ $D5$-branes free in the four-dimensions
we obtain a noncompact submanifold of
${\cal M}(k+\tilde{k})$. It is denoted by
${\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}$.
For each $i$, considering the inclusion of
$C_{{\scriptsize \mbox{$[k_i]$}}}(P_i)$
$\times {\cal M}(\tilde{k}+\sum_{j \neq i}^lk_j)$
to ${\cal M}(k+\tilde{k})$ and
letting the position of the overlapping
$k_i$ $D5$-branes free, we also obtain the
noncompact submanifold
${\cal C}_{{\scriptsize \mbox{$[k_i]$}}}$.
The intersection of these submanifolds turns out to be
${\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}$ :
\begin{eqnarray}
\cap_{i=1}^l
{\cal C}_{{\scriptsize \mbox{$[k_i]$}}}
=
{\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
~~~.
\nonumber
\end{eqnarray}

                 The Poincar\'e duals of
the noncompact submanifolds
${\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}$
and
${\cal C}_{{\scriptsize \mbox{$[k_i]$}}}$
($1 \leq i \leq l$) are introduced,
which are denoted respectively by
${\cal O}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}$
and
${\cal O}_{{\scriptsize \mbox{$[k_i]$}}}$.
Their degrees ($or$ ghost numbers \cite{FKN})
are $2(nk-l)$ and $2(nk_i-1)$.
It is argued in the text that they can be made
depend only on the local data of the resolutions
of the singularities caused by overlapping $D5$-branes.
The above intersection formula can be rephrased as
\begin{eqnarray}
{\cal O}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
=
\wedge_{i=1}^l
{\cal O}_{{\scriptsize \mbox{$[k_i]$}}}~~~.
\nonumber
\end{eqnarray}

                It is also possible to interpret
${\cal O}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}$
as an element of the cohomology group
$H^*({\cal M}(k))$.
It is dual to the cycle
$C_{({\scriptsize \mbox{$[k_1,\cdots,k_l]$}},
\emptyset,\cdots,\emptyset)}$.
With this identification we define the subspace of
$H^*({\cal M}(k))$,
\begin{eqnarray}
\bigoplus_{k_1+\cdots+k_l=k}
{\bf C}
{\cal O}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}~~~.
\nonumber
\end{eqnarray}
Due to the intersection formula it can be written as
\begin{eqnarray}
\bigoplus_{k_1+\cdots +k_l=k}{\bf C}
{\cal O}_{k_1}\wedge \cdots \wedge
{\cal O}_{k_l}~~~~~~~~~~
\left( {\cal O}_{k_i}
\equiv {\cal O}_{{\scriptsize \mbox{$[k_i]$}}}
\right)~~~~~.
\nonumber
\end{eqnarray}
Therefore,
as a physical Hilbert space
of the topological field theory \cite{FKN},
this subspace of $H^*({\cal M}(k))$ admits
to have a structure analogous to the Fock space.
By considering the direct sum of the
moduli spaces the Fock space structure is fully recovered.
This extension will be reasonable from the viewpoint of
$D5$-branes since the number of $D5$-branes does not suffer
apriori any restriction.
(The four-dimensions is not compactified.)
So, finally we obtain the following subspace
of $\oplus_k H^*({\cal M}(k))$ :
\begin{eqnarray}
{\cal H}_{total}\equiv
\bigoplus_l \bigoplus_{k_1,\cdots,k_l}
{\bf C}{\cal O}_{k_1}\wedge \cdots \wedge
{\cal O}_{k_l}~~~~~.
\nonumber
\end{eqnarray}

                  Our proposal can be summarized
as follows :
${\cal H}_{total}$ is the Fock space of the
second-quantized $D5$-branes which allow the
$U(n)$ Chan-Paton factors, in which
${\cal O}_m$ can be identified with a marginally
stable bound state of $m$ $D5$-branes.
The second-quantized operators are introduced
as the creation and annihilation operators of
these bound states.

\sect{Moduli Space of D5-Branes}

                  The goal of this section is
to describe topology of the moduli space of
D5-branes which is introduced as the degenerate vacua
of their effective worldvolume theory.
We study it utilizing the techniques developed
by Nakajima \cite{Nakajima},\cite{Nakajima-Lec}.

              We start by giving a brief review on
the effective worldvolume theory of D5-brane.
Let us consider Type IIB theory with
open strings having the $U(n)$ Chan-Paton factor
($or$ Type IIB theory under the background
of $n$ coincident D9-branes).
When there is a D5-brane
an open string can have either
Neumann boundary with the (anti-)
fundamental representation
of $U(n)$ or
Dirichlet boundary on the D5-brane.
The Dirichlet boundary has the index of the fundamental
representation of $U(k)$ when $k$ D5-branes overlap
\cite{Witten2}.
The combinations of these boundary conditions
correspond to three different open string sectors :
Neumann-Neumann (NN),
Dirichlet-Dirichlet (DD)
and Dirichlet-Neumann (DN).
The quantization of DD and DN strings leads to the
(first) quantization of five-brane
\cite{Polchinski},\cite{Polchinski-Witten},\cite{Witten1}.

                 Suppose that there are $k$
coincident D5-branes at $x^6=\cdots =x^9=0$.
Their low energy effective worldvolume theory
can be described by massless modes of
the DD and DN strings.
It is a six-dimensional supersymmetric
$U(k)$ gauge theory
with global $U(n)$-symmetry
\cite{Polchinski-Witten},\cite{Witten1}.

              There are two kinds of
massless bosonic modes.
$A_{\mu}$ $(\mu =0,\cdots,5)$
give a $U(k)$ gauge field on the worldvolume.
$X^i$ $(i=6,\cdots,9)$ are scalar fields
which belong to the adjoint representation
of $U(k)$.
Their $U(k)$ gauge transformations are
\begin{eqnarray}
A_{\mu}(x) &\mapsto&
g(x)A_{\mu}(x)g(x)^{-1}-i \partial_{\mu}g(x)g(x)^{-1}~~,
\nonumber \\
X^i(x) &\mapsto&
g(x)X^i(x)g(x)^{-1}~~,
\end{eqnarray}
where $g(x)\in U(k)$.
These fileds are invariant under the global
$U(n)$-rotation
since the DD string has no Neumann  boundary.
The vacuum expectation values of $X^i$ become
the collective coordinates of D5-branes.
The $SO(4) \simeq SU(2)\times SU(2)_R$ which originates
from the rotations in the four-dimensions
$(x^6,\cdots,x^9)$ is a global symmetry group
of the worldvolume theory.
$SU(2)_R$ will be identified with
the $SU(2)$ R-symmetry.
In order to make it clear,
it is convenient to rewrite $X^i$ as
\begin{eqnarray}
X_{A \dot{A}}=X^i\sigma^i_{A\dot{A}}~~,
\end{eqnarray}
where
$\sigma^i_{A\dot{A}} =
(i\tau^1,i\tau^2,i\tau^3,{\bf 1}_2)$.
$\tau^{1,2,3}$ are the Pauli matrices and
${\bf 1}_2$ is a $2 \times 2$ identity matrix.
$A$ $(=1,2)$ and
$\dot{A}$ $(=\dot{1},\dot{2})$ are respectively
the $SU(2)$ and $SU(2)_R$ indices.
In the DN sector,
there is a $SU(2)_R$ doublet complex scalar
$H_{\dot{A}}$.
Since the DN string has both Neumann and Dirichlet
boundaries, each $H_{\dot{A}}$ transforms as
$({\bf k},{\bf \bar{n}})$ representation
under the action of $U(k) \times U(n)$ :
\begin{eqnarray}
H_{\dot{A}}(x)
\mapsto
g(x)H_{\dot{A}}(x)h^{-1}~~,
\end{eqnarray}
where $(g(x),h) \in U(k) \times U(n)$.
$H_{\dot{A}}$ is represented by a
$k \times n$ complex matrix
while $A_{\mu}$ and $X^i$ are represented by
$k \times k$ hermitian matrices.

                   The bosonic part of
the effective action will be given by
\begin{eqnarray}
S_{boson} &=&
\int \!d^6\!x~
Tr \left\{
\frac{1}{4}F^{\mu \nu}F_{ \mu \nu}
+\frac{1}{2}D^{\mu}X^{A\dot{A}}D_{\mu}X_{A\dot{A}}
+D^{\mu}H_{\dot{A}}D_{\mu}\bar{H}^{\dot{A}} \right.
\nonumber \\
&&
~~~~~~~~~~~~~~~~~~
\left.
-\frac{1}{2}D^{\dot{A}\dot{B}}D_{\dot{A}\dot{B}}
+D^{\dot{A}\dot{B}}(X_{A \dot{A}}X^A_{~\dot{B}}
+H_{\dot{A}}\bar{H}_{\dot{B}}) \right\} ~~~~,
\label{effective action}
\end{eqnarray}
where $D_{\mu}= \partial_{\mu}+iA_{\mu}$ and
$\bar{H}_{\dot{A}}
= \epsilon_{\dot{A}\dot{B}}(H_{\dot{B}})^{\dagger}$.
The degenerate vacua ($or$ moduli space) are determined
by the $D$-flat conditions obtained from
(\ref{effective action}),
\begin{eqnarray}
X_{A (\dot{A}}X^A_{~\dot{B})}
+H_{(\dot{A}}\bar{H}_{\dot{B})}=0~~,
\end{eqnarray}
which coincides with the ADHM equation \cite{ADHM}
of $SU(n)$-instantons on ${\bf R}^4$.

           One may modify effective action
(\ref{effective action}) by adding
the Fayet-Iliopoulos $D$-terms,
\begin{eqnarray}
S_{F.I.}=
-\zeta_{\dot{A}\dot{B}}\int \!d^6\!x
~Tr~D^{\dot{A}\dot{B}}~~~,
\end{eqnarray}
which change the $D$-flat conditions to
\begin{eqnarray}
X_{A (\dot{A}}X^A_{~\dot{B})}
+H_{(\dot{A}}\bar{H}_{\dot{B})}
=\zeta_{\dot{A}\dot{B}}{\bf 1}_k~~,
\label{ADHM}
\end{eqnarray}
where ${\bf 1}_k$ is a $k \times k$ identity matrix.
Notice that equation (\ref{ADHM}) can be regarded
as the ADHM equation \cite{Kronheimer-Nakajima}
of $SU(n)$-instantons
in the gravitational instanson background
\footnote{It is discussed in
\cite{Douglas-Moore}
from the viewpoint of D-brane.}.

\subsection*{D5-Brane Vacua as Hyperk\"ahler Quotient}

             Let us introduce the hyperk\"{a}hler structure
on the space of $X_{A\dot{A}}$ and $H_{\dot{A}}$.
The triplet of symplectic forms are given by
\begin{eqnarray}
\omega_{\dot{A}\dot{B}}=~
Tr~ \left\{\frac{1}{2}
dX_{A(\dot{A}}\wedge dX^A_{~\dot{B})}
+dH_{(\dot{A}}\wedge d\bar{H}_{\dot{B})} \right\},
\label{symplectic forms}
\end{eqnarray}
where,
since we are now considering the classical vacua,
the field variables
$X_{A \dot{A}}$ and $H_{\dot{A}}$ are
$c$-number matrices.
The hyperk\"{a}hler structure may be extractable
from the expansion
\begin{eqnarray}
\omega_{\dot{A}\dot{B}}
&=&
\omega_II_{\dot{A}\dot{B}}
+\omega_JJ_{\dot{A}\dot{B}}
+\omega_KK_{\dot{A}\dot{B}},
\label{omega-IJK}
\end{eqnarray}
where
\footnote{
$\bar{\sigma}_{ij}^{~\dot{A}}~_{\dot{B}}
=\frac{1}{4}
(\bar{\sigma}_i\sigma_j
      -\bar{\sigma}_j\sigma_i)^{\dot{A}}~_{\dot{B}}$,
where  $\bar{\sigma}_i=\sigma_i^{\dagger}$.
}
$I^{\dot{A}}_{~\dot{B}}
\equiv 2\bar{\sigma}_{12}^{~\dot{A}}~_{\dot{B}}$ ,
$J^{\dot{A}}_{~\dot{B}}
\equiv 2\bar{\sigma}_{13}^{~\dot{A}}~_{\dot{B}}$
and
$K^{\dot{A}}_{~\dot{B}}
\equiv 2\bar{\sigma}_{14}^{~\dot{A}}~_{\dot{B}}$
are the bases of $SU(2)_R$ which satisfy
$I^2=J^2=K^2=-1$ and $IJ=-JI=K$.

             Under the global $U(k)$-symmetry (the global
part of the $U(k)$ gauge symmetry of the worldvolume theory)
$X_{A\dot{A}}$ and $H_{\dot{A}}$ transform adjointly
and vectorialy. Their infinitesimal transforms are
\begin{eqnarray}
\delta X_{A\dot{A}}&=&
\mbox{$[$}\Omega, X_{A\dot{A}} \mbox{$]$},
\nonumber \\
\delta H_{\dot{A}}&=&
\Omega H_{\dot{A}},~~~~
\nonumber \\
\left(~ \delta \bar{H}_{\dot{A}} \right.
&=&
\left.
-\bar{H}_{\dot{A}}\Omega
~\right),
\label{u(k)-action}
\end{eqnarray}
where $\Omega \in u(k)$.
($\Omega^{\dagger}=-\Omega$).
These infinitesimal transforms define a vector field
$\xi_{\Omega}$
on the space of the field variables.
Notice that symplectic forms (\ref{symplectic forms})
are invariant
under $U(k)$-action (\ref{u(k)-action}).
Moreover they satisfy
\begin{eqnarray}
i_{\xi_{\Omega}}\omega_{\dot{A}\dot{B}}=
~Tr~ \left\{ \Omega
d \mu_{\dot{A}\dot{B}} \right\} ,
\label{inner-product1}
\end{eqnarray}
where
\begin{eqnarray}
\mu_{\dot{A}\dot{B}}
\equiv
X_{A(\dot{A}}X^A_{~\dot{B})}+
H_{(\dot{A}}\bar{H}_{\dot{B})}.
\label{momentum map}
\end{eqnarray}
``$i_{\xi_{\Omega}}$" in (\ref{inner-product1})
means taking an inner product
by the vector field
$\xi_{\Omega}$.
Equation (\ref{inner-product1}) besides
the invariance of $\omega_{\dot{A}\dot{B}}$
show that the $U(k)$-action is hamiltonian
with respect to the symplectic structures
$\omega_{I,J,K}$ and
that its momentum map is given
by
$\mu_{\dot{A}\dot{B}}
\equiv \mu_II_{\dot{A}\dot{B}}+\mu_JJ_{\dot{A}\dot{B}}
+\mu_KK_{\dot{A}\dot{B}}$, where $\mu_{I,J,K}$
are $u(k)$ (or $u(k)^{*}$)-valued.

              The hyperk\"{a}hler momentum map
$\mu_{\dot{A}\dot{B}}$ is directly related
with the $D$-flat conditions.
Adding the Fayet-Iliopoulos $D$-terms,
$\zeta_{\dot{A}\dot{B}}\int d^6x Tr D^{\dot{A}\dot{B}}$
$(\zeta^{\dot{A}}_{~\dot{B}} \in su(2)_R)$,
$ D$-flat conditions (\ref{ADHM}) can be written as
\begin{eqnarray}
\mu_{\dot{A}\dot{B}}=\zeta_{\dot{A}\dot{B}}{\bf 1}_k~~.
\label{Dflat}
\end{eqnarray}
The classical moduli space
of the low energy effective worldvolume theory
is now given by
the hyperk\"ahler quotient \cite{HKLR}
\begin{eqnarray}
{\cal M}(k)=
\left. \left. \left\{
~(X_{A\dot{A}},H_{\dot{A}})~ \right|~
\mu_{\dot{A}\dot{B}}=\zeta_{\dot{A}\dot{B}}{\bf 1}_k ~
\right\} \right/ U(k).
\label{moduli}
\end{eqnarray}
(\ref{moduli}) can be also regarded as the moduli
space of $k$ D5-branes.
As for the dimensionality, by simply counting up
the degrees of freedom, it turns out to be
\begin{eqnarray}
dim~{\cal M}(k) = 4nk~~.
\end{eqnarray}

\subsection*{Complex Structue of D5-Brane Vacua}

              Due to D5-branes the Lorentz group
$SO(9,1)$ reduces to $SO(5,1)\times SO(4)$.
The $SO(4)~(\simeq SU(2) \times SU(2)_R)$
which originates from the rotations in the four-dimensions is a
global symmetry group of the worldvolume theory.
($SU(2)_R$ is identified with the $SU(2)~ R$-symmetry.)
To investigate moduli space (\ref{moduli}) it turns out useful
to fix a complex structure in the four-dimensions.
Let us introduce the complex structure by
\begin{eqnarray}
x_{A\dot{A}}=
\left(
\begin{array}{cc}
z_1 & \bar{z}_2 \\
-z_2 & \bar{z}_1
\end{array} \right).
\label{cpx structure}
\end{eqnarray}
To preserve it the structure group
$SO(4) \simeq SU(2)\times SU(2)_R$
must be restricted to
$U(2) \simeq SU(2) \times U(1)_R$.
(\ref{cpx structure}) also fixes
a complex structure of the space of field variables.
One may introduce the complex $k \times k$ matrices
$B_a (a=1,2)$ instead of the hermitian matrices $X^i$
\begin{eqnarray}
X_{A\dot{A}}=
\left(
\begin{array}{cc}
B_1 & B_2^{\dagger} \\
-B_2 & B_1^{\dagger}
\end{array} \right).
\label{cpx structure2}
\end{eqnarray}

           With these complex matrices one can write down
the symplectic forms $\omega^{\dot{A}}_{~\dot{B}}$
as the combination  of $(1,1)$ and $(2,0)~((0,2))$ forms
\begin{eqnarray}
\omega^{\dot{A}}_{~\dot{B}}=
\omega_{{\bf R}}
\left( \begin{array}{cc}
\frac{1}{2} & ~ \\ ~ & -\frac{1}{2} \end{array} \right)
+
\omega_{{\bf C}}
\left( \begin{array}{cc}
~ & 1 \\ 0 & ~ \end{array} \right)
+
\bar{\omega}_{{\bf C}}
\left( \begin{array}{cc}
~ & 0 \\ 1 & ~ \end{array} \right),
\label{symplectic forms2}
\end{eqnarray}
where
$\omega_{{\bf R}}$ and $\omega_{{\bf C}}$
($\bar{\omega}_{{\bf C}}$) are
respectively $(1,1)$ and $(2,0)~((0,2))$ forms
on the space of field variables.
As regards $D$-flat conditions (\ref{Dflat})
they acquire the following form :
\begin{eqnarray}
&&
\mu_{{\bf C}}=0 ~,
\label{ADHM1} \\
&&
\mu_{{\bf R}}=\eta {\bf 1}_k~~,
\label{ADHM2}
\end{eqnarray}
where $\mu_{{\bf C},{\bf R}}$ are given by
\begin{eqnarray}
\mu_{{\bf C}} &=&
\mbox{$[$}B_1,B_2 \mbox{$]$}+
2H_{\dot{1}}H_{\dot{2}}^{\dagger} ,
\nonumber
\\
\mu_{{\bf R}}&=&
\mbox{$[$}B_1,B_1^{\dagger}\mbox{$]$}
+\mbox{$[$}B_2,B_2^{\dagger}\mbox{$]$}
+2H_{\dot{1}}H_{\dot{1}}^{\dagger}
-2H_{\dot{2}}H_{\dot{2}}^{\dagger}.
\end{eqnarray}
$\mu_{{\bf R}}$ and $\mu_{{\bf C}}$ are
the momentum maps of the $U(k)$-action
constructed respectively
from $\omega_{{\bf R}}$ and $\omega_{{\bf C}}$.
Notice that, by using the $SU(2)_R$-rotations,
we have set in (\ref{ADHM1}) and (\ref{ADHM2})
the three constants $\zeta^{\dot{A}}_{~~\dot{B}}$
equal to zero except the only one component
which is denoted by $\eta$.
$\eta$ is assumed to be positive.

              Throughout this section
it is convenient to regard the $k \times k$ matrices
$B_a$ and the $k \times n$ matrices $H_{\dot{A}}$
as elements of $Hom (V,V)$ and $Hom (W,V)$ where
$V$ and $W$ respectively denote
${\bf C}^k$ and ${\bf C}^n$.

                            The positivity of $\eta$
ensures the following stability \cite{Nakajima}
for any solution of $D$-flat conditions
(\ref{ADHM1},\ref{ADHM2}) :
\begin{eqnarray}
\bullet~~&&
\mbox{Any subspace $S$ of $V$ which
satisfies $B_a(S) \subset S$ and
$H_{\dot{1}}(W) \subset S$}
\nonumber \\
&&
\mbox{is equal to $V$.}
\label{stability condition}
\end{eqnarray}
Let us derive stability condition
(\ref{stability condition}) :
Suppose (\ref{stability condition}) does not hold.
Then there exists a subspace $S$
$(\neq \emptyset)$ of $V$ such that
$B_a(S)\subset S$ and $H_{\dot{1}}(W) \subset S$.
Let $S_{\perp}$ be the subspace of $V$ which
is orthogonal to $S$.
Notice that the actions of
$B_a$ and $B_a^{\dagger}$ are
closed respectively on $S$ and $S_{\perp}$.
Let $B_a|_S$ and $B_a^{\dagger}|_{S_{\perp}}$
denote their restrictions on $S$ and $S_{\perp}$.
$B_a$ will acquire the form
\begin{eqnarray}
B_a=
\left(
\begin{array}{cc}
B_a|_S &
D_a \\
0 &
\left(
B_a^{\dagger}|_{S_{\perp}}
\right)^{\dagger}
\end{array}
\right)~~~~
\left(
B_a^{\dagger}=
\left(
\begin{array}{cc}
\left(
B_a|_S
\right)^{\dagger} &
0 \\
D_a^{\dagger} &
B_a^{\dagger}|_{S_{\perp}}
\end{array}
\right) \right)~~.
\end{eqnarray}
Equation (\ref{ADHM2}), if one restrict it
on the subspace $S_{\perp}$,
can be written as
\begin{eqnarray}
\sum_{a} \mbox{$[
\left(B_a^{\dagger}|_{S_{\perp}}\right)^{\dagger},
B_a^{\dagger}|_{S_{\perp}}]$}
-\sum_a D_a^{\dagger}D_a
-2H_{\dot{2}}|_{S_{\perp}}
\left(H_{\dot{2}}|_{S_{\perp}}\right)^{\dagger}
=
\eta {\bf 1}_{S_{\perp}}~~,
\end{eqnarray}
where $H_{\dot{A}}|_{S_{\perp}}$ are
the projections of $H_{\dot{A}}$ onto
$S_{\perp}$.
($H_{\dot{1}}|_{S_{\perp}}=0$.)
Taking the trace of this equation leads
a cotradiction
because we set $\eta >0$.
Therefore $S=\emptyset$.

      At this stage it might be convenient to remark on
the $U(k)$-quotient which appears in (\ref{moduli}).
It originates in the $U(k)$ gauge symmetry of
the worldvolume theory.
Any fixed point of the $U(k)$-action, if it exists,
cause a singularity in moduli space (\ref{moduli}).
This singularity relates to symmetry enhancement
of the theory.
So it is important to ask whether such a fixed point
does appear or not.
Let $(B_a,H_{\dot{A}})$ be a fixed point of the
$U(k)$-action.
There exists $g$ $\in U(k)$ which satisfies
$gB_ag^{-1}=B_a$ and $gH_{\dot{A}}=H_{\dot{A}}$.
Notice that $(B_a,H_{\dot{A}})$
satisfies the stability condition.
Namely any vector $v$ of $V$ can be written
in the form,
$v=f(B)H_{\dot{1}}(w)$,
where $f(B)$ is an appropriate polynomial
of $B_a$ and $w$ is an element of $W$.
The action of $g$ on this vector can be evaluated as
\begin{eqnarray}
gv &=& gf(B)H_{\dot{1}}(w)
\nonumber \\
   &=& f(gBg^{-1})gH_{\dot{1}}(w)~~,
\end{eqnarray}
which turns out equal to $v$.
$gv=v$.
Since $v$ is an arbitrary vector of $V$,
it shows $g=1$.
Therefore, owing to stability condition
(\ref{stability condition}), the $U(k)$-action
is transitive. The $U(k)$-quotient gives
a smooth hyperk\"ahler manifold.

\subsection*{Fixed Points of $T^2$-Action}

                         An abelian subgroup of
the residual $U(2) \simeq SU(2)\times U(1)_R$
is interesting in the sense
that its fixed points will be related with
overlapping D5-branes.
This $U(1) \times U(1)_R$ $(or~T^2)$ symmetry
will rotate the complex coordinates $z_a$
of the four-dimensions by the phases
\begin{eqnarray}
&&
z_1 \mapsto e^{i \phi}z_1, ~~~
\nonumber \\
&&
z_2 \mapsto e^{i \theta}z_2.
\label{torus action}
\end{eqnarray}
As regards $B_a$ and $H_{\dot{A}}$
the $T^2$-action has the
forms
\begin{eqnarray}
&& B_1 \mapsto e^{i \phi}B_1~,~~~~
    H_{\dot{1}} \mapsto e^{i \phi}H_{\dot{1}},
\nonumber \\
&& B_2 \mapsto e^{i \theta}B_2~,~~~~
     H_{\dot{2}} \mapsto e^{-i \theta}H_{\dot{2}}.
\label{torus action BH}
\end{eqnarray}
Notice that the transforms of
$H=(H_{\dot{1}},H_{\dot{2}})$ are modified
in (\ref{torus action BH})
by an right $U(n)$-action of
the Chan-Paton symmetry.
The $U(1)_R$-symmetry rotates $H$ into
$(e^{i (\theta+\phi)/2}H_{\dot{1}}$,
$e^{-i(\theta+\phi)/2}H_{\dot{2}})$.
Multiplying it by $e^{-i(\theta-\phi)/2}$
$\in U(n)$,
we can identify the transform with
$(e^{i \phi}H_{\dot{1}},e^{-i \theta}H_{\dot{2}})$.

        From the adjoint action of $SU(2)_R$ on
$\omega^{\dot{A}}_{~\dot{B}}$ one can find that
$U(1)_R$, the abelian part,
rotates the symplectic forms as follows :
\begin{eqnarray}
\omega_{{\bf R}} \mapsto
\omega_{{\bf R}}, ~~~~~
\omega_{{\bf C}} \mapsto
e^{i(\theta+\phi)}\omega_{{\bf C}}.
\end{eqnarray}
Notice that $\omega_{{\bf R}}$ is invariant under
the $U(1)_R$-action.
In fact the $T^2$-action is hamiltonian
with respect to $\omega_{{\bf R}}$.
The corresponding k\"ahler momentum map
$\mu_{\phi,\theta}$ are given by
\begin{eqnarray}
\mu_{\phi}&=&
Tr \{ B_1B_1^{\dagger}
       +2H_{\dot{1}}H_{\dot{1}}^{\dagger} \},
\nonumber \\
\mu_{\theta}&=&
Tr \{ B_2B_2^{\dagger}
       +2H_{\dot{2}}H_{\dot{2}}^{\dagger} \}.
\label{momentum map of T}
\end{eqnarray}

             The action of $U(1) \times U(1)_R$
preserves $D$-flat conditions (\ref{Dflat}) and
also commutes with that of $U(k)$.
It means that
$U(1) \times U(1)_R$ can act on the moduli space
${\cal M}(k)$.
Let us give a closer look on the fixed points of
this $T^2$-action.
Notice that any two solutions of
$D$-flat conditions (\ref{Dflat})
which differ from each other
by the $U(k)$-action should be identified in
moduli space (\ref{moduli}).
Therefore,
for any fixed point $(B_a,H_{\dot{A}})$
$ \in {\cal M}(k) $ there exists a homomorphism
$\gamma~$ $U(1) \times U(1)_R \longrightarrow U(k)$
which satisfies
\begin{eqnarray}
e^{i \phi}B_1 &=&
\gamma(\phi,\theta)B_1\gamma(\phi,\theta)^{-1},
\nonumber \\
e^{i \theta}B_2 &=&
\gamma(\phi,\theta)B_2\gamma(\phi,\theta)^{-1},
\nonumber \\
e^{i \phi}H_{\dot{1}} &=&
\gamma(\phi,\theta)H_{\dot{1}},
\nonumber \\
e^{-i \theta}H_{\dot{2}} &=&
\gamma(\phi,\theta)H_{\dot{2}}.
\label{fixed point}
\end{eqnarray}
Since $\gamma(\phi, \theta) $ is diagonalizable,
we may decompose $V$ into the sum of the eigenspaces
of $\gamma(\phi,\theta)$ :
\begin{eqnarray}
V=\bigoplus_{p,q \in {\bf Z}}V(p,q)~~~~,
\label{decompo}
\end{eqnarray}
where $V(p,q)$ is an eigenspace of $\gamma$ with
its eigenvalue equal to $e^{i(p \phi+q \theta)}$,
that is,
$\left. \gamma(\phi,\theta) \right|_{V(p,q)}$
$=e^{i(p \phi+q \theta)}{\bf 1}_{V(p,q)}$.

      It is important to note that
these eigenvalues $e^{-i(p \phi+q \theta)}$
besides eigenspaces $V(p,q)$
are restricted by $D$-flat conditions (\ref{Dflat}),
especially by the positivity of $\eta$.
$B_a$ and $H_{\dot{A}}$ will be shown to satisfy
the following properties :
\begin{eqnarray}
\bullet
&&~
B_1(V(p,q))=V(p+1,q) ~~,~~B_2(V(p,q))=V(p,q+1)~~.
\label{surjectivity of B}
\\
\bullet
&&~
Im~H_{\dot{1}}=V(1,0)~~,~~H_{\dot{2}}=0~~.
\label{surjectivity of H}
\end{eqnarray}
And the allowed eigenvalues
which appear in (\ref{decompo})
will be restricted to
\begin{eqnarray}
      p \geq 1~,~~ q \geq 0  .
\label{allowed eigenvalues}
\end{eqnarray}
Therefore the decomposition has the form
\begin{eqnarray}
V=\bigoplus_{p \geq 1, q \geq 0}V(p,q)~~~~.
\label{decompo2}
\end{eqnarray}

            Let us derive these properties :
i)
We remark that relations (\ref{fixed point}) imply
$B_1(V(p,q))\subseteq V(p+1,q)$,
$B_2(V(p,q))\subseteq V(p,q+1)$
as for $B_a$ and
$H_{\dot{1}}(W)\subseteq V(1,0)$,
$H_{\dot{2}}(W)\subseteq V(0,-1)$
as for $H_{\dot{A}}$.
Because $(B_a, H_{\dot{A}})$ satisfies
stability condition (\ref{stability condition}),
eigenspaces $V(p,q)$ with
$p \leq 0$ or $q \leq -1$ can not appear
in (\ref{decompo}),
which means (\ref{allowed eigenvalues}).
In particular it implies $V(0,-1)=\emptyset$.
Thus we obtain $H_{\dot{2}}=0$.
ii)
Nextly we will derive surjectivity
(\ref{surjectivity of H}) of $H_{\dot{1}}$.
Suppose there exists a vector $v$ of $V(1,0)$
such that $H_{\dot{1}}^{\dagger}v=0$.
Notice that condition (\ref{allowed eigenvalues})
implies $V(0,0)=V(1,-1)=\emptyset$,
which tells us $B_a^{\dagger}v=0$.
The mulitiplication of $v$ by $\eta$
may be evaluated using $D$-flat condition
(\ref{ADHM2}),
\begin{eqnarray}
\eta v
&=&
\left(
\sum_a
\mbox{$[$}B_a,B_a^{\dagger}\mbox{$]$}
+2H_{\dot{1}}H_{\dot{1}}^{\dagger}
-2H_{\dot{2}}H_{\dot{2}}^{\dagger}
\right)v
\nonumber \\
&=&
-\sum_{a}B_a^{\dagger}B_av ,
\end{eqnarray}
which gives
\begin{eqnarray}
\eta |v|^2=-\sum_a|B_av|^2.
\end{eqnarray}
This cause a contradiction
except for the case of $v=0$
since the L.H.S. of the equation is non-negative
while the R.H.S. is non-positive. Thus we obtain
$v=0$. It means $Im H_{\dot{1}}=V(1,0)$.
iii)
As regards (\ref{surjectivity of B})
we shall derive the first equation.
Let us begin by considering the case of
$q=0$.
Suppose there exists a vector $v$ of $V(p+1,0)$
such that $B_1^{\dagger}v=0$.
Since we can now assume $p \geq 1$,
both $B_2^{\dagger}v$ and $H_{\dot{1}}^{\dagger}v$
vanish.
The multiplication of $v$ by $\eta$ can be also
evaluated using $D$-flat condition
(\ref{ADHM2}).
It gives us the equality,
\begin{eqnarray}
\eta |v|^2 = -\sum_a|B_av|^2 ,
\end{eqnarray}
which means $v=0$.
Thus we obtain $B_1(V(p,0))=V(p+1,0)$.
For the case of $q > 0$,
by using the induction on $q$,
one can derive the equation,
$B_1(V(p,q))=V(p+1,q)$.
One can also prove the second equation of
(\ref{surjectivity of B})
by repeating the same argument as above.

           What is the physical or geometrical meaning
of these fixed points of the $U(1)\times U(1)_R$-symmetry?
To answer this question it is convenient to consider
their behavior under the limit of $\eta$ being zero.
Let $(B_a,H_{\dot{A}})$ be a fixed point of
the $T^2$-action.
Owing to properties (\ref{surjectivity of B})
we may regard  $B_a$ as upper trianglular matrices.
$B_a=(b_{a~ij})$ where $b_{a~ij}=0$ for $i \geq j$.
The $(i,i)$-component of equation (\ref{ADHM2}) can be
written as follows :
\begin{eqnarray}
\sum_a
\left\{
\sum_{j=i+1}^k
|b_{a~ij}|^2
-\sum_{j=1}^{i-1}
|b_{a~ji}|^2
\right\}
+2
\left(
H_{\dot{1}}H_{\dot{1}}^{\dagger}
\right)_{ii}
=\eta.
\label{(i,i)-component}
\end{eqnarray}
Let us consider an implication of equation
(\ref{(i,i)-component}).
For the case of $i=1$,
setting $\eta=0$ in (\ref{(i,i)-component}),
we obtain
$b_{a~1j}=0$ $( 1 \leq ^{\forall} j \leq k)$ and
$\left(
H_{\dot{1}}H_{\dot{1}}^{\dagger}
\right)_{11}=0$.
For the case of $i \geq 2$,
it is shown recursively that
equations (\ref{(i,i)-component})
with $\eta$ being zero
gives the conditions,
$b_{a~ij}=0$ $( 1 \leq ^{\forall} j \leq k)$ and
$\left(
H_{\dot{1}}H_{\dot{1}}^{\dagger}
\right)_{ii}=0$.
Therefore we can conclude that
{\it any fixed point $(B_a, H_{\dot{A}})$
goes to zero as $\eta \rightarrow +0$}.
Notice that
``zero" is a singularity of the moduli space
at $\eta=0$.
It is caused by $k$ D5-branes
overlapping at the origin.
From the perspective of four-dimensional gauge theory
one can also say that it corresponds
to the small size limit of $k$ $SU(n)$-instantons
sitting at the origin.
The above behavior of the fixed points shows
that these fixed points and their associated cycles
(which we will discuss in the next section) are those
appearing by the resolution of the singularity which exists
in the case of $\eta$ being zero.

           With each fixed point of the $T^2$-action
it is possible to associate a set of $n$ Young tableaux
by taking the following procedure.
We first notice that,
due to relation (\ref{surjectivity of B}),
we can draw the following flow diagram
among the eigenspaces
$V(p,q)$ in (\ref{decompo2}) :
\begin{eqnarray}
\begin{array}{ccccc}
\vdots   & ~           & \vdots     & ~  & ~
\\
\uparrow & ~           & \uparrow   & ~  & ~
\\
V(2,0)   & \rightarrow & V(2,1)     & \rightarrow & \cdots
\\
\uparrow & ~           & \uparrow   & ~  & ~
\\
V(1,0)   & \rightarrow & V(1,1)     & \rightarrow & \cdots
\end{array}
\end{eqnarray}
In this flow diagram the up- and right-arrows
denote respectively the actions of $B_1$ and $B_2$.
We also note that relations (\ref{surjectivity of B})
and (\ref{surjectivity of H}) give the inequalities
on the dimensions of the eigenspaces in (\ref{decompo2})
\begin{eqnarray}
\bullet
&&~
dim~V(p,q) \geq dim~V(p\!+\!1,q)~~,~~
dim~V(p,q) \geq dim~V(p,q\!+\!1)~~.
\nonumber \\
\bullet
&&~
dim~V(p,q) \leq  n ~~(= dim~W)~~~.
\label{dim inequality}
\end{eqnarray}
Due to these inequalities
we may introduce $n$ subdiagrams among the eigenspaces.
Let us draw a flow diagram among the eigenspaces
which dimensions are not less than $d$
($1 \leq d \leq n$).
(Note that we obtain the original diagram
in the case of $d=1$.)
For each $d$,
replacing each eigenspace in the diagram by a box,
we will obtain a Young tableau $\Gamma_d$.
Dimensional inequalities (\ref{dim inequality})
can be rephrased as the conditions on
these $n$ Young tableaux
$\Gamma_1,\Gamma_2,\cdots,\Gamma_n$ :
\begin{eqnarray}
\bullet&&~
\Gamma_1 ~\supseteq~ \Gamma_2 ~\supseteq~
 \cdots ~\supseteq~ \Gamma_n~~.
\nonumber \\
\bullet&&~
|\Gamma_1|+|\Gamma_2|+\cdots+|\Gamma_n|=k~~.
\label{Young tableau}
\end{eqnarray}
Here,
for given two Young tableaux
$\Gamma$ and $\tilde{\Gamma}$,
we say
$\Gamma \supseteq \tilde{\Gamma}$
when the Young tableau $\tilde{\Gamma}$
can be obtainable from
the Young tableau $\Gamma$
by removing some boxes of $\Gamma$.
The total number of boxes of $\Gamma$
is denoted by $|\Gamma|$.

         The correspondence between the fixed points
of the $T^2$-action and
the sets of the $n$ Young tableaux is not one-to-one.
The fixed points will be degenerate.
This degeneracy is due to the commutativity
of the $U(1)\times U(1)_R$-symmetry and the
$U(n)$ Chan-Paton symmetry.
One can say that
each set of $n$ Young tableaux which satisfy conditions
(\ref{Young tableau}) corresponds to
a fixed submanifold of the $T^2$-action.

\subsection*{Topology of D5-Brane Vacua}

        The fixed points of the $T^2$-action may be
considered as the critical points of the following
Morse function on the moduli space \cite{Nakajima},
\cite{Nakajima-Lec}
\begin{eqnarray}
\mu_{T^2}\equiv\mu_{\phi}+\epsilon \mu_{\theta},
\label{Morse function}
\end{eqnarray}
where $\epsilon (>0)$ is a perturbation parameter.
Notice that the $T^2$-action is hamiltonian with respect to
$\omega_{{\bf R}}$  which is regarded as
the k\"ahler form of the moduli space.
$\mu_{\phi,\theta}$
(\ref{momentum map of T}) are now the k\"ahler
momentum map on the moduli space.

           Since the critical submanifolds of
Morse function (\ref{Morse function}) are classified
by the sets of $n$ Young tableaux
$(\Gamma_1,\cdots,\Gamma_n)$ which satisfy
conditions (\ref{Young tableau}),
let us write the corresponding critical submanifold by
${\cal F}_{(\Gamma_1,\cdots, \Gamma_n)}$.
To describe the Morse indices besides the dimensions
of these critical submanifolds,
we shall prepare a few notations on Young tableaux :
One can realize a Young tableau $\Gamma$
by a set of non-increasing positive integers,
$\Gamma=\mbox{$[k_1,k_2,\cdots,k_l]$}$.
($k_1 \geq k_2 \geq \cdots \geq k_l \geq 1$.)
We introduce $l(\Gamma)$, the length of $\Gamma$,
by $l(\Gamma)=l$.
(Notice that the total number of boxes of $\Gamma$,
which is denoted by $|\Gamma|$,
is $k_1+k_2+\cdots +k_l$.)
Let
$\Gamma=\mbox{$[k_1,k_2,\cdots,k_l]$}$
and
$\tilde{\Gamma}=
\mbox{$[\tilde{k}_1,\tilde{k}_2,\cdots,
\tilde{k}_{\tilde{l}}]$}$
be two Young tableaux which satisfy
$\Gamma \supseteq \tilde{\Gamma}$.
It means that $l \geq \tilde{l}$ and
$k_i \geq \tilde{k}_i$ for $1 \leq i \leq \tilde{l}$.
In such a situation it turns out useful
to extend
$\tilde{\Gamma}
=\mbox{$[\tilde{k}_1,\tilde{k}_2,
\cdots,\tilde{k}_{\tilde{l}}]$}$ by
writting
$\mbox{$[\tilde{k}_1,\tilde{k}_2,
\cdots,\tilde{k}_l]$}$
setting
$\tilde{k}_i=0$ for $\tilde{l}+1 \leq i \leq l$.
With this understanding
let us introduce
$l(\Gamma \setminus \tilde{\Gamma})$ as
the number of $k_i$
$(1 \leq i \leq l)$
which satisfy $k_i > \tilde{k}_i$.
And also we introduce
$\nu(\Gamma \setminus \tilde{\Gamma})$
as the number of $\tilde{k}_i$
$(1 \leq i \leq l)$
which satisfy
$\tilde{k}_i < k_i$ and
$\tilde{k}_i=k_{i+1}$.
($k_{l+1}\equiv 0$.)

           The dimensions of the critical submanifold
${\cal F}_{(\Gamma_1,\cdots, \Gamma_n)}$
turn out to be
\begin{eqnarray}
dim~ {\cal F}_{(\Gamma_1,\cdots, \Gamma_n)}
=
2 \sum_{i<j}\nu (\Gamma_i \backslash \Gamma_j),
\label{dim of F}
\end{eqnarray}
and the Morse index at
${\cal F}_{(\Gamma_1,\cdots,\Gamma_n)}$
is given by
\begin{eqnarray}
2 \left\{
n(k-\sum_{j=1}^{n}l(\Gamma_j))
    +\sum_{i<j}l(\Gamma_i \backslash \Gamma_j)
       -\sum_{i<j}\nu(\Gamma_i \backslash \Gamma_j) \right\}.
\label{Morse index}
\end{eqnarray}
Since the critical submanifolds have no odd dimensional
cycles the Morse function is perfect and the Poincar\'e
polynomial of the moduli space has the form
\begin{eqnarray}
&&P_t({\cal M}(k))=
\sum_{(\Gamma_1,\cdots,\Gamma_n)}
t^{2 \left\{
n(k-\sum_{j=1}^{n}l(\Gamma_j))
    +\sum_{i<j}l(\Gamma_i \backslash \Gamma_j)
       -\sum_{i<j}\nu(\Gamma_i \backslash \Gamma_j)
\right\} }
P_t({\cal F}_{(\Gamma_1,\cdots, \Gamma_n)})
\nonumber \\
&&~~~~~
\label{Poincare Poly}
\end{eqnarray}

                 Formulae (\ref{dim of F}) and
(\ref{Morse index}) can be derived by using the
techniques developed by Nakajima \cite{Nakajima},
\cite{Nakajima-Lec}.
It is important to remark that
one can also introduce \cite{Nakajima} moduli space
(\ref{moduli}) by the complex symplectic quotient
\begin{eqnarray}
{\cal M}(k)=
\mu_{{\bf C}}^{-1}(0)^s / GL(k:{\bf C}),
\label{moduli2}
\end{eqnarray}
where
\begin{eqnarray}
\mu_{{\bf C}}^{-1}(0)^s =
\left\{
  (B_a,H_{\dot{A}})
\left|
  \begin{array}{l}
\cdot ~~
\mu_{{\bf C}}=0 \\
\cdot ~~
\mbox{stability condition (\ref{stability condition})}
\end{array}
\right. \right\}.
\end{eqnarray}
The action of $GL(k:{\bf C})$ in (\ref{moduli2})
is given by
\begin{eqnarray}
&&
B_a ~\mapsto~~gB_ag^{-1}~,
\nonumber \\
&&
H_{\dot{1}}~\mapsto~~gH_{\dot{1}}~,
\nonumber \\
&&
H_{\dot{2}}^{\dagger}~\mapsto
~~H_{\dot{2}}^{\dagger}g^{-1}~~,
\label{GL(k:C)action}
\end{eqnarray}
where $g \in GL(k:{\bf C})$.
Roughly speaking, the quotient by $GL(k:{\bf C})$
means that we are considering not the unitary group but
its complexfication as the gauge symmetry.

           Taking complex symplectic description
(\ref{moduli2}) of the moduli space
let us consider an infinitesimal deformation
($B_a+\delta B_a,H_{\dot{A}}+\delta H_{\dot{A}}$)
from a given vacuum ($B_a,H_{\dot{A}}$).
In order that the infinitesimal deformation
still describes a vacuum configuration,
$(\delta B_a,\delta H_{\dot{A}})$ necessarily
satisfies the equation,
\begin{eqnarray}
\mbox{$[\delta B_1,B_2]$}+
\mbox{$[B_1,\delta B_2]$}+
2\delta H_{\dot{1}} H_{\dot{2}}^{\dagger}+
2 H_{\dot{1}} \delta H_{\dot{2}}^{\dagger}
=0 ~~.
\label{infinitesimal muC}
\end{eqnarray}
The L.H.S. of (\ref{infinitesimal muC}) is nothing but
the infinitesimal deviation of $\mu_{\bf C}$,
which will be denoted by
$\alpha(\delta B_a,\delta H_{\dot{1}},
\delta H_{\dot{2}}^{\dagger})$.
Not every solution of equation
(\ref{infinitesimal muC})
gives an independent vacuum.
One should take account of gauge symmetry
(\ref{GL(k:C)action}). The infinitesimal
$GL(k:{\bf C})$-transform can be read as
\begin{eqnarray}
\delta_Y
\left( \begin{array}{c}
B_a \\
H_{\dot{1}} \\
H_{\dot{2}}^{\dagger}
\end{array} \right)
=
\left( \begin{array}{c}
\mbox{$[Y,B_a]$} \\
YH_{\dot{1}} \\
-H_{\dot{2}}^{\dagger}Y
\end{array}\right)~~~,
\label{gl(k:C)action}
\end{eqnarray}
where $Y \in gl(k:{\bf C}) (\equiv Hom (V,V))$.
The R.H.S. of equations (\ref{gl(k:C)action}) gives
a matrix-valued function of $Y$,
which will be denoted by $\beta(Y)$.
Now the tangent space of the moduli space
at a given vacuum
$(B_a,H_{\dot{A}})$ can be realized
using these $\alpha$ and $\beta$.
Namely let us introduce the complex :
\begin{eqnarray}
Hom(V,V)
\stackrel{\beta}{\longrightarrow}
\begin{array}{c}
Hom(V,V)^{\oplus 2} \\
\oplus \\
Hom(W,V) \\
\oplus \\
Hom(V,W)
\end{array}
\stackrel{\alpha}{\longrightarrow}
Hom(V,V)~~~,
\label{tangent space}
\end{eqnarray}
where $\alpha \cdot \beta =0$ holds.
Then we can identify
the (holomorphic) tangent space at the vacuum
($B_a,H_{\dot{A}}$) with $Ker~\alpha / Im~\beta$.
Notice that,
owing to the stability of $(B_a,H_{\dot{A}})$,
$\alpha$ and $\beta$ in (\ref{tangent space})
turn out to be respectively
surjective and injective.

        Let $(B_a,H_{\dot{A}})$ be a fixed point of
the $U(1)\times U(1)_R$-symmetry.
In such a case there exists a homomorphism
$\gamma$ (cf.(\ref{fixed point}))
and we can decompose $V$
into the direct sum of the eigenspaces of $\gamma$.
(These eigenspaces have the corresponding
$U(1)\times U(1)_R$-charges (cf.(\ref{decompo2})).)
The spaces of matrices appearing in complex
(\ref{tangent space}) can be also decomposed into
the eigenspaces of $\gamma$ with the definite
$U(1)\times U(1)_R$-charges
\footnote{
$Hom (V,V)\simeq V^* \otimes V$ and
$Hom (W,V)\simeq W^* \otimes V$.
In these identifications
we should count the $U(1)\times U(1)_R$-charges
of the eigenspace $V(p,q)$ as $(-p,-q)$
(cf.(\ref{torus action tangent})).}.
Since $\alpha$ and $\beta$ in (\ref{tangent space})
are respectively surjective and injective,
it is possible  to obtain
the character of $U(1)\times U(1)_R$
on the tangent space at the fixed point,
by identifying it with
$Ker~\alpha/Im~\beta$,
from the characters on the spaces of matrices
appearing in (\ref{tangent space})
\footnote{A little modification is needed since
$\alpha$ and $\beta$ in (\ref{tangent space}) have
the $U(1)\times U(1)_R$-charges.
For the exact treatment we refer \cite{Nakajima},
in which $n=1$ case is studied.}.
Counting the non-positive charges in the character
on the tangent space we obtain formulae
(\ref{dim of F}) and (\ref{Morse index}).

            To end this section
it may be convenient to comment
on the fixed submanifolds,
especially those of the type
$(\Gamma_1,\Gamma_2,\cdots,\Gamma_n)
=(\Gamma,\emptyset,\cdots,\emptyset)$.
$\Gamma$ is a Young tableau of $k$ boxes.
In these cases
all the eigenspaces of $\gamma$
is one-dimensional, $dim V(p,q)=1$,
and the rank of $H_{\dot{1}}$ is equal to one.
From the corresponding flow diagrams one can normalize
the $k \times n$ complex matrices $H_{\dot{A}}$ as
\footnote{Here we use hyperk\"ahler description
(\ref{moduli}) of the moduli space.}
\begin{eqnarray}
H_{\dot{1}}=
\left(
\begin{array}{cccc}
0      & 0      & \ldots & 0 \\
\vdots & \vdots &        & \vdots \\
\vdots & \vdots &        & \vdots \\
0      & 0      & \ldots & 0 \\
\sqrt{k\eta/2}      & 0      & \ldots & 0
\end{array}
\right)~~,~~~~~H_{\dot{2}}=0.
\label{H0}
\end{eqnarray}
Notice that the $U(n)$ Chan-Paton symmetry
which commutes with the $U(1)\times U(1)_R$-symmetry
can rotates $H_{\dot{A}}$ to $H_{\dot{A}}h$
where $h \in U(n)$. The critical submanifolds
${\cal F}_{(\Gamma, \emptyset,\cdots, \emptyset)}$
will be generated by this $U(n)$-action.
From the matrix forms of (\ref{H0}) one can see
that the action of $U(n-1) \times U(1)$ (of $U(n)$)
is irrelevant. (The action of $U(1)$, which
gives the phase of $H_{\dot{1}}$,
can be absorbed into the $U(k)$ gauge symmetry.)
Therefore the critical submanifolds are
\begin{eqnarray}
{\cal F}_{(\Gamma, \emptyset,\cdots, \emptyset)} &\simeq&
\frac{U(n)}{U(n-1)\times U(1)}
\nonumber \\
&=& {\bf C}P_{n-1}.
\label{Pn-1}
\end{eqnarray}

\sect{Cycles of D5-Brane Vacua}

                 There exist several nontrivial
cycles in the moduli space ${\cal M}(k)$.
Apart from topologies of the critical submanifolds
of Morse function (\ref{Morse function})
these nontrivial cycles will be labelled,
first of all, by the sets of $n$ Young tableaux
which satisfy condition (\ref{Young tableau}).
In the Poincar\'{e} polynomial (\ref{Poincare Poly}),
terms related with a given set of
$n$ Young tableaux
$(\Gamma_1,\cdots,\Gamma_n)$ are
\begin{eqnarray}
&&
t^{2 \left\{
       n(k-\sum_{j=1}^{n}l(\Gamma_j))
             +\sum_{i<j}l(\Gamma_i \backslash \Gamma_j)
       -\sum_{i<j}\nu(\Gamma_i \backslash \Gamma_j)
\right\} }
P_t({\cal F}_{(\Gamma_1,\cdots, \Gamma_n)})
\nonumber \\
&& =
t^{2 \left\{
       n(k-\sum_{j=1}^{n}l(\Gamma_j))
             +\sum_{i<j}l(\Gamma_i \backslash \Gamma_j)
       -\sum_{i<j}\nu(\Gamma_i \backslash \Gamma_j)
\right\}
+~dim~{\cal F}_{(\Gamma_1,\cdots,\Gamma_n)} }
\left( 1 +O(t^{-2}) \right)~~.
\nonumber \\
&&~~~
\label{expansion}
\end{eqnarray}
A cycle which gives the leading of
(\ref{expansion}) will be called the maximal
dimensional cycle labelled by
$(\Gamma_1,\cdots,\Gamma_n)$.
It will be denoted by
$C_{(\Gamma_1,\cdots,\Gamma_n)}$.
The goal of this section is the interpretation of
these topological cycles in terms of D5-branes.
In particular,
we will investigate the maximal dimensional cycles
which are labelled by
$(\Gamma_1,\Gamma_2,\cdots,\Gamma_n)=$
$(\Gamma,\emptyset,\cdots,\emptyset)$.

             In order to proceed further we will need
explicit forms of these cycles.
These explicit forms may be handled by using
another equivalent description (\ref{moduli2})
of the moduli space.

            Let us start by giving a simple
example of these maximal cycles.
Consider the case of $l(\Gamma)=1$.
Namely $\Gamma = \mbox{$[k]$}$.
The dimensions of the maximal cycle
$C_{({\scriptsize \mbox{$[k]$}},
\emptyset,\cdots,\emptyset)}$
can be read from
the dimensions of the critical submanifold
${\cal F}_{({\scriptsize \mbox{$[k]$}},
\emptyset,\cdots,\emptyset)}$
and the corresponding Morse index
\begin{eqnarray}
dim~ C_{({\scriptsize \mbox{$[k]$}},
\emptyset,\cdots,\emptyset)} &=&
2(n-1)+2n(k-1)
\nonumber \\
&=&
2nk-2 .
\label{dim of C1}
\end{eqnarray}
Taking complex symplectic description
(\ref{moduli2}) of the moduli space,
an explicit form of the vacuum which belongs
to this maximal cycle will be given by
\footnote{
The configuration given in (\ref{cycle1})
is a representative of the vacuum.
Any configuration which can be obtained from
(\ref{cycle1}) by $GL(k:{\bf C})$-action
(\ref{GL(k:C)action}) describes the same vacuum.}
\begin{eqnarray}
&&
B_1 =
\left(
\begin{array}{ccccc}
0       & 1      & 0       & \ldots & 0        \\
0       & 0      & 1       & \ddots & \vdots   \\
\vdots  & \vdots & \ddots  & \ddots & 0        \\
\vdots  & \vdots &         & \ddots & 1        \\
0       &  0     & \ldots  & \ldots & 0
\end{array}
\right)
~~,~~
B_2 =
\left(
\begin{array}{ccccc}
0       & a_1    & a_2     & \ldots  & a_{k\!-\!1}  \\
\vdots  & 0      & a_1     & \ddots  & \vdots   \\
\vdots  & \vdots & \ddots  & \ddots  & a_2      \\
\vdots  & \vdots &         & \ddots  & a_1        \\
0       & 0      & \ldots  & \ldots  & 0
\end{array}
\right)
\nonumber
\\
&&
H_{\dot{1}} =
\left(
\begin{array}{cccc}
0      & h_{11}    & \ldots & h_{1 n\!-\!1} \\
\vdots & \vdots    &        & \vdots \\
\vdots & \vdots    &        & \vdots \\
0      & h_{k\!-\!1 1} & \ldots & h_{k\!-\!1 n\!-1} \\
1      & h_{k 1}   & \ldots & h_{k n\!-\!1}
\end{array}
\right)
~~, ~~
H_{\dot{2}}=0~~.
\label{cycle1}
\end{eqnarray}
$a_i$ and $h_{ij}$ in (\ref{cycle1})
are the $nk-1$ complex parameters
which constitute the maximal cycle
$C_{({\scriptsize \mbox{$[k]$}},
\emptyset,\cdots,\emptyset)}$.
Explicit form (\ref{cycle1})
may be obtained as follows.
Let $(B_a,H_{\dot{A}})$ be
a fixed point of $T^2$-action
(\ref{torus action BH}).
There exist $n$ Young tableaux
$(\Gamma_1,\Gamma_2,\cdots,\Gamma_n)$
by which $(B_a,H_{\dot{A}})$ is labelled.
Consider an infinitesimal deformation
$(B_a+\delta B_a,H_{\dot{A}}+\delta H_{\dot{A}})$
from this fixed point.
$(\delta B_a,\delta H_{\dot{A}})$ is
a tangent vector of the moduli space
at $(B_a,H_{\dot{A}})$.
The $U(1)\times U(1)_R$-symmetry rotates
tangent vectors at the fixed point.
The $T^2$-action has the form
\begin{eqnarray}
\delta B_1 &&
\mapsto
~~e^{i \phi} \gamma(\phi,\theta)^{-1} \delta B_1
            \gamma(\phi,\theta) ,
\nonumber \\
\delta B_2 &&
\mapsto
~~e^{i \theta} \gamma(\phi,\theta)^{-1}
            \delta B_2 \gamma(\phi,\theta) ,
\nonumber \\
\delta H_{\dot{1}} &&
\mapsto
~~e^{i \phi} \gamma(\phi,\theta)^{-1}
            \delta H_{\dot{1}} ,
\nonumber \\
\delta H_{\dot{2}} &&
\mapsto
~~e^{-i \theta} \gamma(\phi,\theta)^{-1}
             \delta H_{\dot{2}} ,
\label{torus action tangent}
\end{eqnarray}
where $\gamma$ is
the homomorphism from $U(1) \times U(1)_R$ to $U(k)$
which is associated with the fixed point
(cf.(\ref{fixed point})).
It is possible
to diagonalize the tangent space with respect to
$T^2$-action (\ref{torus action tangent}),
which also provides the diagonalization
of the hessian of Morse function
(\ref{Morse function}) at the critical point.
The eigenspaces with the non-positive
eigenvalues will generate the maximal cycle
$C_{(\Gamma_1,\Gamma_2,\cdots,\Gamma_n)}$.
In the case of
$(\Gamma_1,\Gamma_2,\cdots,\Gamma_n)$
$=(\mbox{$[k]$},\emptyset,\cdots,\emptyset)$
we obtain (\ref{cycle1}).
Each one of $a_i$ and $h_{ij}$ in (\ref{cycle1})
parametrizes the eigenspace with the non-positive
eigenvalue in the (holomorphic) tangent space
at the corresponding fixed point.
Notice that configuration (\ref{cycle1})
itself satisfies $D$-flat condition
(\ref{ADHM1}) besides stability condition
(\ref{stability condition}) for any values
of $a_i$ and $h_{ij}$.

            Taking the viewpoint of D5-branes
the eigenvalues of $X^i$ (or $B_a$) will describe
their positions in the four-dimensions.
Therefore,
($B_a,H_{\dot{A}}$) in (\ref{cycle1})
describes the classical vacuum of $k$ D5-branes
degenerate at $(z_1,z_2)=(0,0)$.
It admits the extra
$nk-1$ complex parameters,
$a_i$ and $h_{ij}$,
which constitute the maximal cycle
$C_{({\scriptsize \mbox{$[k]$}},
\emptyset,\cdots,\emptyset)}$.
This means that
{\it any point of the cycle
$C_{({\scriptsize \mbox{$[k]$}},
\emptyset,\cdots,\emptyset)}$ describes
the vacuum of $k$ D5-branes degenerate
at the origin.}
Notice that,
though we are describing the vacuum configuration
$(B_a,H_{\dot{A}})$ using complex symplectic quotient
(\ref{moduli2}),
we can always find an element $g$ of $GL(k:{\bf C})$
so that
$(gB_ag^{-1},gH_{\dot{1}},
H_{\dot{2}}^{\dagger}g^{-1})$
satisfies $D$-flat condition (\ref{ADHM2}).
Without loss of generality we can choose
$g$ such that its lower triangular part are zero
and therefore $gB_ag^{-1}$ can be regarded as
upper triangular matrices.
So, the vacuum
$(gB_ag^{-1},gH_{\dot{1}},
H_{\dot{2}}^{\dagger}g^{-1})$,
according to our previous argument,
goes to zero as $\eta \rightarrow 0$,
which shows that
{\it this topological
cycle disappears at $\eta=0$.}

~

             To study the vacua which belong
to the maximal cycles it is also useful to describe
their characteristics in a convenient form.
For instance,
$B_a$ in (\ref{cycle1}) satisfy the relations
\begin{eqnarray}
B_1B_2 &=& B_2B_1,~~~
\nonumber \\
B_1^{k}&=&0,~~
\nonumber \\
B_2 &=&
\sum_{i=1}^{k-1}a_i B_{1}^i~~~.
\label{ideal1}
\end{eqnarray}
Similar characterizations
are also possible for $H_{\dot{A}}$ (\ref{cycle1}).
In order to describe them
we first remark that moduli space (\ref{moduli})
can be considered as the moduli space of the torsion free
sheaves of rank $n$ on ${\bf C}P_2$
\cite{Okonek},\cite{Nakajima}.
(The complex coordinates $(z_1,z_2)$ of ${\bf C}^2$
are identified with the inhomogeneous coordinates
of ${\bf C}P_2$.)
The idea is the monad construction of torsion free
sheaf.
Consider the following monad complex
\footnote{${\cal O}_{{\bf C}^2}$ is the structure sheaf of
${\bf C}^2$.}:
\begin{eqnarray}
V \otimes {\cal O}_{{\bf C}^2}
\stackrel{a}{\longrightarrow}
\begin{array}{c}
V \otimes {\cal O}_{{\bf C}^2} \\
\oplus \\
V \otimes {\cal O}_{{\bf C}^2} \\
\oplus \\
W \otimes {\cal O}_{{\bf C}^2}
\end{array}
\stackrel{b}{\longrightarrow}
V \otimes {\cal O}_{{\bf C}^2}
\label{monad}
\end{eqnarray}
where
\begin{eqnarray}
a \equiv
\left(
\begin{array}{c}
B_1-z_1 \\
B_2-z_2 \\
\sqrt{2}H_{\dot{2}}^{\dagger}
\end{array}
\right)
,~~~~
b \equiv
\left(
\begin{array}{ccc}
-B_2+z_2, & B_1-z_1,
& \sqrt{2}H_{\dot{1}}
\end{array}
\right).
\end{eqnarray}
Notice that it holds $b \cdot a = \mu_{{\bf C}}=0$
for any vacuum configuration.
The corresponding torsion free sheaf
${\cal E}$ is given by
\begin{eqnarray}
{\cal E}=Ker~b/ Im~a~~~.
\label{torsion free sheaf}
\end{eqnarray}
$GL(k:{\bf C})$-action (\ref{GL(k:C)action})
on $(B_a,H_{\dot{A}})$ is noting but
the linear transform of the bases of $V$.
The torsion free sheaf ${\cal E}$ given by
(\ref{torsion free sheaf}) is invariant under this
transform. Therefore, to each vacuum of $k$ D5-branes
one can attach a torsion free sheaf
of rank $n$ uniquely.

                      Let $B_a$ and $H_{\dot{A}}$
in complex (\ref{monad}) be those given by equations
(\ref{cycle1}).
In such a situation
an element
$\left(\begin{array}{c}v_1(z) \\
 v_2(z) \\ w(z) \end{array} \right)$
of $Ker~b/ Im~a$ satisfies the equation,
\begin{eqnarray}
\sqrt{2}\sum_{j=1}^n w_j(z)H_{\dot{1}}(e_j)
=(B_2-z_2)v_1(z)-(B_1-z_1)v_2(z),
\label{ker b}
\end{eqnarray}
where $w(z)=\sum_{j=1}^nw_j(z)e_j$.
$e_j (1 \leq j \leq n)$ are the bases of
$W (= {\bf C}^n)$ :
\begin{eqnarray}
e_1=
\left( \begin{array}{c}
1 \\ 0 \\ \vdots \\ 0
\end{array}\right),~~
\cdots,~~
e_n=
\left( \begin{array}{c}
0 \\ \vdots \\ 0 \\ 1
\end{array}\right) .
\end{eqnarray}
Since it holds that
\begin{eqnarray}
H_{\dot{1}}(e_j)=
\sum_{i=1}^k
h_{i j\!-\!1}B_1^{i-1}H_{\dot{1}}(e_1) ,
\end{eqnarray}
for $j \geq 2$,
we can find out
$g_j(z) \in {\cal O}_{{\bf C}^2}$ such that
$H_{\dot{1}}(e_j)=g_j(B)H_{\dot{1}}(e_1)$.
($~g_1(z) \equiv 1~$).
Therefore equation (\ref{ker b}) reduces to
\begin{eqnarray}
\sqrt{2}
\sum_{j=1}^nw_j(z)g_j(B)~H_{\dot{1}}(e_1)
=(B_2-z_2)v_1(z)-(B_1-z_1)v_2(z),
\end{eqnarray}
which, by replacing $z_a$ with $B_a$,
implies
\begin{eqnarray}
\sum_{j=1}^nw_j(B)g_j(B)H_{\dot{1}}(e_1)=0.
\end{eqnarray}
Because $H_{\dot{1}}(e_1)$ is cyclic
with respect to $B_a$ and
$B_a$ themselves are commutative,
we can conclude that
\begin{eqnarray}
\sum_{j=1}^nw_j(B)g_j(B)=0 .
\label{relation1}
\end{eqnarray}
Equation (\ref{relation1})
may be simplified by introducing the analogue
of gauge transformation
\begin{eqnarray}
w=
\left( \begin{array}{c}
w_1 \\
\vdots \\
w_n
\end{array} \right)
\mapsto
\tilde{w}=
\left( \begin{array}{c}
\tilde{w}_1 \\
\vdots \\
\tilde{w}_n
\end{array} \right)
=G w~~~,
\end{eqnarray}
where the ``gauge transform" $G(z)$ has the form
\begin{eqnarray}
G(z) =
\left(
\begin{array}{ccccc}
1      & g_2(z) & g_3(z) & \ldots & g_n(z) \\
0      & 1      & 0      & \ldots & 0      \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
\vdots &        & \ddots & \ddots & 0      \\
0      & \ldots & \ldots & 0      & 1
\end{array}\right)~~.
\label{gauge}
\end{eqnarray}
Notice that each $g_j(z) \in {\cal O}_{{\bf C}^2}$
is given by the relation,
$H_{\dot{1}}(e_j)=g_j(B)H_{\dot{1}}(e_1)$.
The data of $H_{\dot{A}}$ are now encoded
into the ``gauge transform" $G$.
By this transformation,
equation (\ref{relation1}) becomes
equivalent to the following constraint
on the first component of $\tilde{w}$,
\begin{eqnarray}
\tilde{w}_1(B)=0.
\end{eqnarray}
Due to relations (\ref{ideal1})
any polynomial which satisfies this constraint
can be given by the combinations
\begin{eqnarray}
r(z)z_1^{k}+s(z)(z_2-\sum_{i=1}^{k-1}a_iz_1^{i}),
\end{eqnarray}
where $r(z),s(z) \in {\cal O}_{{\bf C}^2}$.
Therefore it seems very plausible that
one can distinguish
$B_a$ and $H_{\dot{A}}$ (\ref{cycle1})
by relations (\ref{ideal1}) and
``gauge transform" (\ref{gauge}).

~

          $(B_a,H_{\dot{A}})$ given in
(\ref{cycle1}) describes the vacuum of $k$ D5-branes
degenerate at the origin $(0,0)$.
These vacua form the maximal cycle
$C_{({\scriptsize \mbox{$[k]$}},
\emptyset,\cdots,\emptyset)}$.
By changing the eigenvalues of $B_a$ from $0$ to
$z_a$ in (\ref{cycle1})
it provides the vacuum configuration that
$k$ D5-branes are overlapping at $P=(z_1,z_2)$.
The additional parameters in (\ref{cycle1}) also
constitute a cycle of the moduli space but
it can be topologically identified with
$C_{({\scriptsize \mbox{$[k]$}},\emptyset,
\cdots,\emptyset)}$.
Let us consider the vacua of
$k=k_1+\cdots+k_l$ D5-branes
($k_1 \geq k_2 \geq \cdots \geq k_l$)
in which each $k_i$ pieces are degenerate
at $P_i=(z_1^{(i)},z_2^{(i)})$.
Among them we shall concentrate on the configuration
obtained by superposing the vacua of
$k_i$ D5-branes ($1 \leq i \leq l$)
having forms (\ref{cycle1})
with their eigenvalues shifted.
Explicitly such a configuration may be written as
\begin{eqnarray}
B_a=
\left( \begin{array}{ccc}
B_a^{(1)} &        &           \\
          & \ddots &           \\
          &        & B_a^{(l)}
\end{array} \right)
~~~,~~~
H_{\dot{A}}=
\left( \begin{array}{c}
H_{\dot{A}}^{(1)} \\
\vdots \\
H_{\dot{A}}^{(l)}
\end{array} \right)~~.
\label{cycle2}
\end{eqnarray}
$B_a^{(i)}$ and $H_{\dot{A}}^{(i)}$ are respectively
$k_i \times k_i$ and $k_i \times n$ complex matrices
such that $(B_a^{(i)},H_{\dot{A}}^{(i)})$
describes the vacuum of $k_i$ D5-branes
overlapping at $P_i=(z_1^{(i)},z_2^{(i)})$
which is given (cf. (\ref{cycle1})) by
\begin{eqnarray}
&&
B_1^{(i)} =
\left(
\begin{array}{ccccc}
z_1^{(i)} & 1         & 0       & \ldots & 0        \\
0         & z_1^{(i)} & 1       & \ddots & \vdots   \\
\vdots    &  0        & \ddots  & \ddots & 0        \\
\vdots    & \vdots    & \ddots  & \ddots & 1        \\
0         &  0        & \ldots  &  0     & z_1^{(i)}
\end{array}
\right)
~~,~~
B_2^{(i)} =
\left(
\begin{array}{ccccc}
z_2^{(i)}&a_1^{(i)}& a_2^{(i)}&\ldots& a_{k_i\!-\!1}^{(i)}\\
0       &z_2^{(i)}&a_1^{(i)}&\ddots& \vdots \\
\vdots  &  0     & \ddots  & \ddots  & a_2^{(i)}  \\
\vdots  & \vdots & \ddots  & \ddots  & a_1^{(i)}  \\
0       &  0     & \ldots  &  0      & z_2^{(i)}
\end{array}
\right)
\nonumber
\\
&&
H_{\dot{1}}^{(i)} =
\left(
\begin{array}{cccc}
0      & h_{11}^{(i)}& \ldots & h_{1 n\!-\!1}^{(i)} \\
\vdots & \vdots    &        & \vdots \\
\vdots & \vdots    &        & \vdots \\
0 &h_{k_i\!-\!1 1}^{(i)}&\ldots &h_{k_i\!-\!1 n\!-1}^{(i)} \\
1      & h_{k_i 1}^{(i)} & \ldots & h_{k_i n\!-\!1}^{(i)}
\end{array}
\right)
~~, ~~
H_{\dot{2}}^{(i)}=0~~.
\label{element of cycle2}
\end{eqnarray}
For each $i$, vacuum configuration (\ref{element of cycle2})
admits the additional $nk_i-1$ complex parameters,
$a^{(i)}$ and $h^{(i)}$.
They form the maximal cycle
$C_{({\scriptsize \mbox{$[k_i]$}},\emptyset,\cdots,\emptyset)}$
in the moduli space ${\cal M}(k_i)$.
One can expect that these parameters,
considering them as the additional parameters for the
vacuum of $k=k_1+\cdots+k_i$ D5-branes,
also constitute an appropriate cycle in the moduli space
${\cal M}(k)$.

              It is important to note that explicit form
(\ref{cycle2}) does not always describe a vacuum
when $P_i$ coincides with $P_j$ for some $i$ and $j$.
This is because
the first column vector of $H_{\dot{1}}$ in
(\ref{cycle2}) is no longer a cyclic vector
when $(z_1^{(i)},z_2^{(i)})=(z_1^{(j)},z_2^{(j)})$.
Notice that, as far as all the $l$-points
$P_1,\cdots,P_l$ are different from one another,
the first column vector of $H_{\dot{1}}$
is cyclic and therefore configuration
(\ref{cycle2}) describes the vacuum
of $k$ D5-branes for any values of
the additional complex parameters.
In the case when some of the $l$-points coincide,
to guarantee stability (\ref{stability condition}) of
configuration (\ref{cycle2}),
it may become necessary to introduce a subspace of
$Im~H_{\dot{1}}$ spanned by not only the first column
vector but also the other column vectors of $H_{\dot{1}}$
and then to consider the possibility whether
the vectors of this subspace besides the vectors obtained
from them by the successive actions of $B_a$ can span $V$.
Though this prescription  for restoring the stability
of the configuration seems to work, the introduction
of the column vectors of $H_{\dot{1}}$ other than
the first one turns out to freeze some
of the additional complex parameters in (\ref{cycle2}).
(In fact some of these parameters are absorbed into the
$GL(k:{\bf C})$-symmetry.)

            There is another prescription to
restore the stability condition.
Take an element $q$ of $GL(k:{\bf C})$
which depends on the $l$-points
$P_1,\cdots,P_l$ and becomes singular
when some of these points coincide.
Transform (\ref{GL(k:C)action})
of vacuum (\ref{cycle2}) by $q$ is gauge-equivalent to
the original configuration
as far as these $l$-points are different
from one another.
It might be possible to choose $q$ of
$GL(k:{\bf C})$ so that the transform by $q$
is still well-defined without missing any complex
parameters in (\ref{cycle2}) even when some of
the $l$-points coincide.
The complex parameters will be rescaled such that
the singularities of $q$ are absorbed into them.
One may say that such an element $q$ of
$GL(k:{\bf C})$ gives the change of coordinates
of the cycle.

            These modified forms (which are obtained
by appropriate singular gauge transformations)
will appear naturally in our approach.
It is because our description of
the topological cycles of the moduli space
is based on the consideration of
the fixed points of the $T^2$-action.
Since the $U(1)\times U(1)_R$-symmetry rotates
$P_i=(z_1^{(i)},z_2^{(i)})$ in the way given by
(\ref{torus action}),
the cycle formed by the complex parameters in (\ref{cycle2})
will be captured in our treatment
by examining the limit that the $l$-points in
(\ref{cycle2}) approach to one another
and go to the origin $(0,0)$.
In fact, we will show that
{\it the vacuum which belongs to the maximal
cycle $C_{(\Gamma,\emptyset,\cdots,\emptyset)}$
with $\Gamma= \mbox{$[k_1,\cdots,k_l]$}$
can be identified with
(\ref{cycle2}) under the limit
that all the $l$-different points
$P_1,\cdots, P_l$ in (\ref{cycle2})
go to the origin $(0,0)$. }
For this identification
it is necessary to know
how the vacuum configuration in which
each $k_i$ D5-branes are degenerate at $P_i$
behaves when all these $l$-points approach to
the origin and
then to compare it with the configuration
which belongs to the maximal cycle
mentioned above.

~

              It may be enough to consider the vacuum
in which $m$ D5-branes are overlapping at a point $P$
in the four-dimensions while the other D5-branes
are degenerate at another point, say $Q$,
and then to compare this configuration,
when $P$ goes to $Q$,
with the vacuum
which belongs to the corresponding maximal cycle.
For definiteness
let us suppose that $m$ and $m+m'$ D5-branes
are degenerate respectively at
$P=(0,\lambda)$ and $Q=(0,0)$.
$\lambda \neq 0$.
Their vacuum is denoted
by $(\hat{B}_a,\hat{H}_{\dot{A}})$.
$\hat{B}_a$ have the block-diagonal form
$\left( \begin{array}{cc}
\hat{B}_a^{(1)} & 0 \\
0  & \hat{B}_a^{(2)}
\end{array}\right)$
where $\hat{B}_a^{(1)}$ and $\hat{B}_a^{(2)}$
are respectively $m \times m$ and $(m+m')\times(m+m')$
complex matrices.
$\hat{H}_{\dot{A}}$ also have the form
$\left(\begin{array}{c}
\hat{H}_{\dot{A}}^{(1)} \\
\hat{H}_{\dot{A}}^{(2)}
\end{array}\right)$
where
$\hat{H}_{\dot{A}}^{(1)}$
and
$\hat{H}_{\dot{A}}^{(2)}$
are respectively
$m \times n$ and $(m+m')\times n$
complex matrices.
The constituent $(\hat{B}_a^{(1)},\hat{H}_{\dot{A}}^{(1)})$
($~(\hat{B}_a^{(2)},\hat{H}_{\dot{A}}^{(2)})~$)
describes the contribution of $m$ ($~m+m'~$)
D5-branes degenerate at $P$ ($~Q~$).
The explicit form is given in Table 1.
$(\hat{B}_a,\hat{H}_{\dot{A}})$ in Table 1
admits $n(2m+m')-2$ complex parameters
$\hat{a}_j,\hat{b}_j,\hat{d}_j$ and $\hat{h}_{ij}$.
Among them,
$\hat{b}_j$ and $\hat{h}_{ij}$ $(1 \leq i \leq m)$
are the additional parameters  associated with
the configuration of the first $m$ D5-branes
while $\hat{a}_j$, $\hat{d}_j$ and
$\hat{h}_{ij}$ $(m+1 \leq i \leq 2m+m')$
are those associated with the $m+m'$
D5-branes at $Q$.

            For this vacuum one can provide a
characterization similar to (\ref{ideal1}).
From the form given in Table 1 the following
relations among $\hat{B}_a$ can be found out :
\begin{eqnarray}
\hat{B}_1^{m+m'}&=&0,
\nonumber
\\
\hat{B}_1^m
\left(
\hat{B}_2-\sum_{j=1}^{m'-1}\hat{a}_j\hat{B}_1^j
\right)&=&0,~~
\nonumber
\\
\left(
\hat{B}_2-\sum_{j=1}^{m-1}\hat{b}_j\hat{B}_1^j-\lambda
\right)
\left(
\hat{B}_2-\sum_{j=1}^{m'-1}\hat{a}_j\hat{B}_1^j \right)
&=&
\sum_{j=1}^m\hat{c}_j\hat{B}_1^{m'+j-1},
\nonumber \\
\hat{B}_1\hat{B}_2
&=&
\hat{B}_2\hat{B}_1
\label{ideal2}
\end{eqnarray}
where $\hat{c}_j$ ($1 \leq j \leq m$) are
introduced by
\begin{eqnarray}
\hat{c}_j \equiv
-\lambda \hat{d}_j+\sum_{r+s=j}
(\hat{a}_r-\hat{b}_r)\hat{d}_s.
\label{parameter hatc}
\end{eqnarray}
Since $\lambda$ is not equal to zero,
these $\hat{c}_j$ are equivalent to the parameters
$\hat{d}_j$.

            Let us examine the limit that
$P=(0,\lambda)$ goes to $Q=(0,0)$.
The configuration obtained from
$(\hat{B}_a,\hat{H}_{\dot{A}})$
in Table 1 by setting $\lambda$ equal to zero
does not always describe a vacuum of D5-branes.
This is because, when $\lambda =0$,
the first column vector of $\hat{H}_{\dot{1}}$
in Table 1 is not cyclic with respect to $\hat{B}_a$
and it becomes necessary to compensate it
with the other column vectors of $\hat{H}_{\dot{1}}$
in order to satisfy stability condition
(\ref{stability condition}).
With this compensation some of $\hat{h}_{ij}$
in $\hat{H}_{\dot{1}}$ are frozen or absorbed into the
$GL(2m+m':{\bf C})$-symmetry.
The number of the additional complex parameters
will decrease from $(2m+m')n-2$.
To avoid such a decrease of parameters
we need to change our perspective.
We begin by describing the maximal cycle
$C_{({\scriptsize \mbox{$[m+m',m]$}},
\emptyset,\cdots,\emptyset)}$.
The dimensions of this cycle
can be read as follows
\begin{eqnarray}
dim~ C_{({\scriptsize \mbox{$[m+m',m]$}},
\emptyset,\cdots,\emptyset)}
&=&
2(n-1)+
2 \left\{
n(2m+m'-2)+(n-1)
\right\}
\nonumber \\
&=&
2n(2m+m')-4.
\end{eqnarray}
The vacuum configuration
$(B_a,H_{\dot{A}})$ in
$C_{({\scriptsize \mbox{$[m+m',m]$}},
\emptyset,\cdots,\emptyset)}$
is given in Table 2.
It has $n(2m+m')-2$ complex parameters
$a_j$, $b_j$, $c_j$ and $h_{ij}$.
These parameters constitute the maximal cycle
$C_{({\scriptsize \mbox{$[m+m',m]$}},
\emptyset,\cdots,\emptyset)}$.
After some manipulation we can find out
$B_a$ in Table 2 satisfy the relations
\begin{eqnarray}
B_1^{m+m'}&=&0,
\nonumber
\\
B_1^m
\left(
B_2-\sum_{j=1}^{m'-1}a_jB_1^j
\right)&=&0,~~
\nonumber
\\
\left(
B_2-\sum_{j=1}^{m-1}b_jB_1^j
\right)
\left(
B_2-\sum_{j=1}^{m'-1}a_jB_1^j \right)
&=&
\sum_{j=1}^mc_jB_1^{m'+j-1} ,
\nonumber \\
B_1B_2
&=&
B_2B_1 .
\label{ideal3}
\end{eqnarray}

               Now we can investigate
the behavior of the vacuum
$(\hat{B}_a,\hat{H}_{\dot{A}})$
when $P$ goes to $Q$.
As we have mentioned,
we should not set $\lambda=0$
for the explicit form in Table 1.
But relations (\ref{ideal2}) themselves
might be handled under this limit.
We first rescale $\hat{c}_j$ (\ref{parameter hatc})
such that they do not depend on $\lambda$ and
then introduce the new parameters in stead of
$\hat{a}_j,\hat{b}_j$ and $\hat{d}_j$ :
\begin{eqnarray}
a_j &=& \hat{a}_j,
\nonumber \\
b_j &=& \hat{b}_j,
\nonumber \\
c_j &=&
\hat{c}_j(\hat{a},\hat{b},\hat{d}:\lambda).
\label{newparameter}
\end{eqnarray}
With fixing these new parameters,
relations (\ref{ideal2}) become (\ref{ideal3})
as $\lambda$ goes to zero !
Notice that equations (\ref{newparameter})
can be regarded as the change of coordinates
as far as $\lambda \neq 0$.
These new parameters themselves will survive
even at $\lambda =0$.
The payoff for this transmutation of the parameters
is the change of realization of the vacuum
from those given in Table 1 to those given in Table 2.
To complete the discussion one should also introduce
similar rescalings of the parameters $\hat{h}_{ij}$.
Though the investigation of
``gauge transform" (cf. (\ref{gauge})) provides
a nice description of the rescalings
for these parameters,
it becomes complicated and hard to give a general theory.
So we convince the reader by giving some examples.

\subsubsection*{Example 1 : $(m,m')=(1,0)$}

                    Let us consider the case of
two D5-branes located at two different points,
say $P$ and $Q$. For definiteness we set
$P=(0,\lambda)$ and $Q=(0,0)$.
$\lambda \neq 0$.
The corresponding vacuum configuration,
which is denoted by
$(\hat{B}_a,\hat{H}_{\dot{A}})$,
has the form
\begin{eqnarray}
&&
\hat{B}_1 =
\left(
\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}
\right)
~~,~~
\hat{B}_2=
\left(
\begin{array}{cc}
\lambda & 0 \\
0 & 0
\end{array}
\right) ,
\nonumber \\
&&\hat{H}_{\dot{1}} =
\left(
\begin{array}{cccc}
1 & \hat{h}_{11} & \cdots & \hat{h}_{1 n\!-\!1}
\\
1 & \hat{h}_{21} & \cdots & \hat{h}_{2 n\!-\!1}
\end{array}
\right)~~,~~
\hat{H}_{\dot{2}}=0.
\label{example1-1}
\end{eqnarray}
$\hat{h}_{1j}$ and $\hat{h}_{2j}$ are
respectively the additional parameters
associated with the D5-brane located at
$P$ and $Q$.
Notice that $\hat{B}_2$ in
(\ref{example1-1}) satisfies the relation
\begin{eqnarray}
\hat{B}_2(\hat{B}_2-\lambda)=0 .
\label{eq of B example1-1}
\end{eqnarray}
As far as $\lambda \neq 0$,
$\hat{H}_{\dot{1}}(e_1)
=\left( \begin{array}{c}
1 \\ 1
\end{array}
\right)$ is cyclic with respect to $\hat{B}_2$.
In particular
$\hat{H}_{\dot{1}}(e_j)$ $(j \geq 2)$
are obtainable from
$\hat{H}_{\dot{1}}(e_1)$
by appropriate actions of
$\hat{B}_2$ :
\begin{eqnarray}
\hat{H}_{\dot{1}}(e_j)=
\left\{
\frac{\hat{h}_{1j\!-\!1}-\hat{h}_{2 j\!-\!1}}{\lambda}
\hat{B}_2+\hat{h}_{2j\!-\!1}
\right\}
\hat{H}_{\dot{1}}(e_1).
\end{eqnarray}
With this expression of $\hat{H}_{\dot{1}}(e_j)$
one can see that the corresponding
``gauge transform" (cf. (\ref{gauge}) ),
which we write as $\hat{G}$,
is given by
\begin{eqnarray}
\hat{G}(z) =
\left(
\begin{array}{ccccc}
1      & \hat{g}_2(z) & \hat{g}_3(z) & \ldots &
\hat{g}_n(z) \\
0      & 1      & 0      & \ldots & 0      \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
\vdots &        & \ddots & \ddots & 0      \\
0      & \ldots & \ldots & 0      & 1
\end{array}\right),
\label{gauge example1-1}
\end{eqnarray}
where $\hat{g}_j(z)$
$(j \geq 2)$ are
\begin{eqnarray}
\hat{g}_j(z)=
\frac{\hat{h}_{1j\!-\!1}-\hat{h}_{2 j\!-\!1}}{\lambda}
z_2+\hat{h}_{2j\!-\!1}.
\label{hatg example1-1}
\end{eqnarray}

                 At this stage let us remark
on the configuration obtained from (\ref{example1-1})
by setting $\lambda =0$.
The first column vector of $\hat{H}_{\dot{1}}$,
$\left(\begin{array}{c}
1 \\
1
\end{array}\right)$, is no longer cyclic.
In order that this configuration describes a vacuum
one must impose an appropriate constraint
on the parameters $\hat{h}_{ij}$
so that it satisfies stability condition
(\ref{stability condition}).
It is sufficient if
$\left(\begin{array}{c}
\hat{h}_{11} \\
\hat{h}_{21}
\end{array}\right)$
besides
$\left(\begin{array}{c}
1 \\
1 \\
\end{array}\right)$
span ${\bf C}^2 (\equiv V)$.
In such a situation one can take these two vectors as
the bases of $V$. In terms of these new bases,
vacuum (\ref{example1-1}), setting $\lambda=0$,
acquires the form, $\hat{B}_a=\hat{H}_{\dot{2}}=0$ and
$\hat{H}_{\dot{1}}=
\left(\begin{array}{ccc}
0 & 1 & * \\
1 & 0 & *
\end{array}\right)$,
which means the parameters
$\hat{h}_{11,21}$ are absorbed into the
$GL(2:{\bf C})$-symmetry.

                   Nextly we describe the vacuum
which belongs to the maximal cycle
$C_{({\scriptsize \mbox{$[1,1]$}},
\emptyset,\cdots,\emptyset)}$.
We shall write it $(B_a,H_{\dot{A}})$.
It can be realized by
\begin{eqnarray}
&&
B_1=0~~,~~
B_2=
\left(
\begin{array}{cc}
0 & 1 \\
0 & 0
\end{array}\right),
\nonumber \\
&&
H_{\dot{1}} =
\left( \begin{array}{cccc}
0 & h_{11} & \cdots & h_{1 n\!-\!1} \\
1 & h_{21} & \cdots & h_{2 n\!-\!1}
\end{array} \right) ~~,~~
H_{\dot{2}}=0,
\label{example1-2}
\end{eqnarray}
where $2(n-1)$ complex parameters
$h_{ij}$ constitue the maximal cycle.
Notice that $B_2$ satisfy the relation,
\begin{eqnarray}
B_2^2=0~~.
\label{eq of B example1-2}
\end{eqnarray}
As for ``gauge transform" $G$ in
(\ref{gauge}),
the elements of $G$ turn out to be given by
\begin{eqnarray}
g_j(z)=
h_{1j\!-\!1}z_2+h_{2j\!-\!1} .
\label{g example1-2}
\end{eqnarray}

               Let us study the behavior of
configuration (\ref{example1-1})
when $P$ goes to $Q$
by considering rescalings of
the complex parameters in (\ref{example1-1}).
Namely we rescale $\hat{h}_{ij}$ such that
$\hat{h}_{2j}$ as well as
$(\hat{h}_{1j}-\hat{h}_{2j})/\lambda$
do not depend on $\lambda$.
We may introduce the new parameters
in stead of $\hat{h}_{ij}$ :
\begin{eqnarray}
h_{1j}&=&
\frac{\hat{h}_{1j}-\hat{h}_{2j}}{\lambda}~~,
\nonumber \\
h_{2j}&=&
\hat{h}_{2j}.
\label{rescale example1}
\end{eqnarray}
Equations (\ref{rescale example1})
can be also
regarded as the change of coordinates
as far as $\lambda \neq 0$.
Taking the element $\hat{q}(\lambda)$ of $GL(2:{\bf C})$,
\begin{eqnarray}
\hat{q}(\lambda)=
\left( \begin{array}{cc}
1/\lambda & -1/\lambda \\
0 & 1
\end{array}\right)~~
\end{eqnarray}
the transform of (\ref{example1-1})
by $\hat{q}(\lambda)$
becomes
\begin{eqnarray}
\hat{q}(\lambda)\hat{B}_1\hat{q}(\lambda)^{-1}
&=&
0~,~~
\hat{q}(\lambda)\hat{B}_2\hat{q}(\lambda)^{-1}~=~
\left(\begin{array}{cc}
\lambda & 1 \\
0       & 0
\end{array}\right)~~,
\nonumber \\
\hat{q}(\lambda)H_{\dot{1}}
&=&
\left(\begin{array}{cccc}
0 &
\frac{\hat{h}_{11}\!-\!\hat{h}_{21}}{\lambda} &
\cdots &
\frac{\hat{h}_{1 n\!-\!1}\!-\!\hat{h}_{2 n\!-\!1}}
{\lambda} \\
1 &
\hat{h}_{21} &
\cdots & \hat{h}_{2 n\!-\!1}
\end{array} \right).
\label{example1-3}
\end{eqnarray}
Rewritting  configuration (\ref{example1-3})
by rescaled parameters (\ref{rescale example1}),
we can find that it
coincides with (\ref{example1-2})
as $\lambda$ goes to zero.
This shows that the rescaled parameters
survive even at $\lambda=0$ and that,
due to the singularity at $\lambda=0$ in
$\hat{q}(\lambda)$,
one should change the realization from
(\ref{example1-1}) to (\ref{example1-2}),
at least, at $P=Q$.

\subsubsection*{Example 2 : $(m,m')=(1,1)$}

        As the second example we shall consider
the case of three D5-branes one of which is
located at $P=(0,\lambda)$ and
the other two are overlapping at $Q=(0,0)$.
$\lambda \neq 0$.
The corresponding vacuum
$(\hat{B}_a,\hat{H}_{\dot{A}})$ has the form
\begin{eqnarray}
&&
\hat{B}_1=
\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{array}
\right) ~~~,~~~
\hat{B}_2
=
\left(
\begin{array}{ccc}
\lambda & 0 & 0 \\
0 & 0 & \hat{a} \\
0 & 0 & 0
\end{array}
\right)~~~,
\nonumber \\
&&
\hat{H}_{\dot{1}} =
\left(
\begin{array}{cccc}
1 & \hat{h}_{11} & \cdots & \hat{h}_{1 n\!-\!1}
\\
0 & \hat{h}_{21} & \cdots & \hat{h}_{2 n\!-\!1}
\\
1 & \hat{h}_{31} & \cdots & \hat{h}_{3 n\!-\!1}
\end{array}
\right)~~,~~~
\hat{H}_{\dot{2}}=0~~~,
\label{example2-1}
\end{eqnarray}
where $\hat{a}$ and $\hat{h}_{ij}$ are
$3n-2$ complex parameters.
Taking the diagonal elements of $\hat{B}_a$
into account one can say $\hat{h}_{1j}$
are the additional parameters
associated with the first D5-brane
located at $P$ while
$\hat{a}$ besides $\hat{h}_{2j}$ and $\hat{h}_{3j}$
are those for the two D5-branes overlapping at $Q$.
It is easy to check that
$\hat{B}_a$ in (\ref{example2-1}) satisfy the
relations
\begin{eqnarray}
&&
\hat{B}_1^2=0 ~~,
\nonumber \\
&&
(\hat{B}_2-\lambda)(\hat{B}_2-\hat{a}\hat{B}_1)=0~~,
\nonumber \\
&&
\hat{B}_1\hat{B}_2=\hat{B}_2\hat{B}_1=0~~.
\label{eq of B example2-1}
\end{eqnarray}
Notice that
$\hat{H}_{\dot{1}}(e_1)$ is cyclic with respect to
$\hat{B}_a$.
In particular one can write down
$\hat{H}_{\dot{1}}(e_j)$ $(j \geq 2)$ in the form
\begin{eqnarray}
&&
\hat{H}_{\dot{1}}(e_j)=
\left\{
\frac{\hat{h}_{1j\!-\!1}-\hat{h}_{3j\!-\!1}}{\lambda}
\hat{B}_2+
\left(
\hat{h}_{2j\!-\!1}-
\frac{\hat{a}(\hat{h}_{1j\!-\!1}-\hat{h}_{3j\!-\!1})}
{\lambda}
\right)
\hat{B}_1+\hat{h}_{3j\!-\!1}
\right\}
\hat{H}_{\dot{1}}(e_1)~~.
\nonumber \\
&&~~
\label{Hj example2-1}
\end{eqnarray}
The ``gauge transform" $\hat{G}$,
which has a form similar to (\ref{gauge example1-1}),
is now given by the polynomials
$\hat{g}_j(z)$ $(j \geq 2)$
determined from (\ref{Hj example2-1}):
\begin{eqnarray}
\hat{g}_j(z)=
\frac{\hat{h}_{1j\!-\!1}-\hat{h}_{3j\!-\!1}}{\lambda}
z_2+
\left(
\hat{h}_{2j\!-\!1}-
\frac{\hat{a}(\hat{h}_{1j\!-\!1}-\hat{h}_{3j\!-\!1})}
{\lambda}
\right)
z_1+\hat{h}_{3j\!-\!1} ~~~.
\label{hatg example2-1}
\end{eqnarray}

      Let us remark on the configuration
obtained from (\ref{example2-1})
by setting $\lambda=0$.
In order that this configuration describes a vacuum
it is sufficient that the second column vector of
$\hat{H}_{\dot{1}}$ besides
$\left(\begin{array}{c}
1 \\
0 \\
1
\end{array}\right)$ and
$\hat{B}_1
\left(\begin{array}{c}
1 \\
0 \\
1
\end{array}\right)$
span ${\bf C}^3 (\equiv V)$.
This condition gives the constraint,
$\hat{h}_{11}\neq \hat{h}_{31}$.
If it is satisfied,
one can take a new bases of $V$ so that
vacuum (\ref{example2-1}) (with setting $\lambda=0$)
acquires the form
\begin{eqnarray}
&&
\hat{B}_1=
\left(
\begin{array}{ccc}
0 & 1 & \hat{h}_{31} \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}
\right) ~~~,~~~
\hat{B}_2
=
\left(
\begin{array}{ccc}
0& \hat{a} & \hat{a}\hat{h}_{31} \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}
\right)~~~,
\nonumber \\
&&
\hat{H}_{\dot{1}} =
\left(
\begin{array}{cccc}
0 & 0 & * & *
\\
0 & 1 & * & *
\\
1 & 0 & * & *
\end{array}
\right)~~,~~~
\hat{H}_{\dot{2}}=0~~~,
\end{eqnarray}
which means that the parameters
$\hat{h}_{11,21}$ are absorbed into
the $GL(3:{\bf C})$-symmetry.

             Nextly we describe the vacuum
$(B_a,H_{\dot{A}})$ which belongs
to the maximal cycle
$C_{({\scriptsize \mbox{$[$}2,1\mbox{]}},
\emptyset,\cdots,\emptyset)}$.
It can be realized by the form
\begin{eqnarray}
&&
B_1  =
\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{array}
\right) ~~~,~~~
B_2
=
\left(
\begin{array}{ccc}
0 & 0 & 1 \\
a & 0 & 0 \\
0 & 0 & 0
\end{array}
\right)~~~,
\nonumber \\
&&
H_{\dot{1}} =
\left(
\begin{array}{cccc}
0 & h_{11} & \cdots & h_{1 n\!-\!1}
\\
0 & h_{21} & \cdots & h_{2 n\!-\!1}
\\
1 & h_{31} & \cdots & h_{3 n\!-\!1}
\end{array}
\right)~~,~~~
H_{\dot{2}}=0~~~,
\label{example2-2}
\end{eqnarray}
where $3n-2$ complex parameters
$a$ and $h_{ij}$ constitue the maximal cycle
$C_{({\scriptsize \mbox{$[2,1]$}},
\emptyset,\cdots,\emptyset)}$.
One can check $B_a$ satisfy the relations
\begin{eqnarray}
&&
B_1^2=0~~,
\nonumber \\
&&
B_2^2-aB_1=0~~,
\nonumber \\
&&
B_1B_2=B_2B_1=0~~.
\label{eq of B example2-2}
\end{eqnarray}
As for ``gauge transform" $G$ (\ref{gauge}),
its elements turn out to be given by
\begin{eqnarray}
g_j(z)=h_{1j\!-\!1}z_2+h_{2j\!-\!1}z_1+
h_{3j\!-\!1}.
\label{g example2-2}
\end{eqnarray}

              The behavior of configuration
(\ref{example2-1}) when $P$ goes to $Q$ will be studied
taking account of the rescaling of parameters.
We first rescale $\hat{a}$ such that
$\lambda \hat{a}$ does not depend on $\lambda$.
We may introduce the new parameter by
\begin{eqnarray}
a = -\lambda \hat{a}.
\label{rescale1 example2}
\end{eqnarray}
Notice that relations (\ref{eq of B example2-1}),
with fixing $a$,
become (\ref{eq of B example2-2})
as $\lambda$ goes to $0$.
One can also rescale $\hat{h}_{ij}$ such that
\begin{eqnarray}
&&
h_{1j} =
\frac{\hat{h}_{1j}-\hat{h}_{3j}}{\lambda}~~,
\nonumber \\
&&
h_{2j} =
\hat{h}_{2j}+
\frac{a(\hat{h}_{1j}-\hat{h}_{3j})}{\lambda^2}~~,
\nonumber \\
&&
h_{3j} =
\hat{h}_{3j}~~
\label{rescale2 example2}
\end{eqnarray}
do not depend on $a$ and $\lambda$.
Elements (\ref{hatg example2-1})
in the gauge transform $\hat{G}$,
expressing them in terms of these rescaled $h_{ij}$,
coincide with (\ref{g example2-2}).
Equations
(\ref{rescale1 example2})
and (\ref{rescale2 example2})
can be also regarded as the change of coordinates
as far as $\lambda \neq 0$.
Introduce the element $\hat{q}(\lambda)$ of
$GL(3:{\bf C})$ by
\begin{eqnarray}
\hat{q}(\lambda)=
\left(\begin{array}{ccc}
1/\lambda & 0 & -1/\lambda \\
-\hat{a}/\lambda & 1 & \hat{a}/\lambda \\
0 & 0 & 1
\end{array}\right)~~.
\end{eqnarray}
The transform of (\ref{example2-1}) by
$\hat{q}(\lambda)$ becomes
\begin{eqnarray}
\hat{q}(\lambda) \hat{B}_1 \hat{q}(\lambda)^{-1}
&=&
\left( \begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{array}
\right) ~,~
\hat{q}(\lambda) \hat{B}_2 \hat{q}(\lambda)^{-1}
~=~
\left(\begin{array}{ccc}
\lambda & 0 & 1 \\
-\lambda \hat{a} & 0 & 0 \\
0 & 0 & 0
\end{array}
\right)~~~,
\nonumber \\
\hat{q}(\lambda) \hat{H}_{\dot{1}}
&=&
\left(
\begin{array}{cccc}
0 &
\frac{\hat{h}_{11}\!-\! \hat{h}_{31}}{\lambda} &
\cdots &
\frac{\hat{h}_{1 n\!-\!1}
\!-\! \hat{h}_{3 n\!-\!1}}{\lambda}
\\
0 &
\hat{h}_{21}\!-\!
\frac{\hat{a}(\hat{h}_{11}
\!-\! \hat{h}_{31})}{\lambda} &
\cdots &
\hat{h}_{2 n\!-\!1}\!-\!
\frac{\hat{a}
(\hat{h}_{1n\!-\!1}\!-\!
\hat{h}_{3n\!-\!1})}{\lambda}
\\
1 & \hat{h}_{31} & \cdots & \hat{h}_{3 n\!-\!1}
\end{array}
\right)~~.
\label{example2-3}
\end{eqnarray}
Rewritting (\ref{example2-3}) in terms of rescaled
parameters (\ref{rescale1 example2}) and
(\ref{rescale2 example2}) it exactly
coincides with (\ref{example2-2}) at
$P=Q$.

\subsubsection*{Example 3 : $(m,m')=(2,0)$}

            As the last example we shall
investigate the case
of four D5-branes two of which overlap
at $P=(0,\lambda)$
and the other two are degenerate at $Q=(0,0)$.
$\lambda \neq 0$.
The corresponding vacuum
$(\hat{B}_a,\hat{H}_{\dot{A}})$
has the form
\begin{eqnarray}
&&
\hat{B}_1=
\left( \begin{array}{cccc}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array}\right)~~,~~
\hat{B}_2=
\left( \begin{array}{cccc}
\lambda & \hat{a} & 0 & 0 \\
0     & \lambda & 0 & 0 \\
0 & 0 & 0 & \hat{b} \\
0 & 0 & 0 & 0
\end{array}\right)~~,
\nonumber \\
&&
\hat{H}_{\dot{1}}=
\left( \begin{array}{cccc}
0 & \hat{h}_{11} & \cdots & \hat{h}_{1n\!-\!1} \\
1 & \hat{h}_{21} & \cdots & \hat{h}_{2n\!-\!1} \\
0 & \hat{h}_{31} & \cdots & \hat{h}_{3n\!-\!1} \\
1 & \hat{h}_{41} & \cdots & \hat{h}_{4n\!-\!1}
\end{array}\right)~~,~~
\hat{H}_{\dot{2}}=0~~,
\label{example3-1}
\end{eqnarray}
where $\hat{a},\hat{b}$ and $\hat{h}_{ij}$ are
$4n-2$ complex parameters in which,
taking the diagonal parts of $\hat{B}_a$ into account,
$\hat{a}$,$\hat{h}_{1j}$ and $\hat{h}_{2j}$
are the additional parameters associated with
the two D5-branes degenerate at $P$
while $\hat{b}$, $\hat{h}_{3j}$ and $\hat{h}_{4j}$ are
those for the two D5-branes at $Q$.
It is easy to see that
$\hat{B}_{a}$ in (\ref{example3-1}) satisfy the relations,
\begin{eqnarray}
&&
\hat{B}_1^2=0~~,
\nonumber \\
&&
\left(\hat{B}_2-\hat{a}\hat{B}_1-\lambda \right)
\left(\hat{B}_2-\hat{b}\hat{B}_1\right)=0~~,
\nonumber \\
&&
\hat{B}_1\hat{B}_2=\hat{B}_2\hat{B}_1~~.
\label{eq of B example3-1}
\end{eqnarray}
Since $\hat{H}_{\dot{1}}(e_1)$ is cyclic with respect to
$\hat{B}_a$ one can write down $\hat{H}_{\dot{1}}(e_j)$
$(j \geq 2)$ in the form,
\begin{eqnarray}
\hat{H}_{\dot{1}}(e_j) &=&
\left\{
\left(
\frac{\hat{h}_{1j}-\hat{h}_{3j}}{\lambda}
       -\frac{(\hat{a}-\hat{b})
           (\hat{h}_{2j}-\hat{h}_{4j})}{\lambda^2}
\right)
\hat{B}_1\hat{B}_2+
\left(
\hat{h}_{3j}-
\frac{\hat{b}(\hat{h}_{2j}-\hat{h}_{4j})}{\lambda}
\right)
\hat{B}_1
\right.
\nonumber \\
&& ~~~~~~~~~~~~~~~~~~~~~
\left.+
\frac{\hat{h}_{2j}-\hat{h}_{4j}}{\lambda}\hat{B}_2+
\hat{h}_{4j}
\right\}
\hat{H}_{\dot{1}}(e_1)~~.
\end{eqnarray}
The corresponding ``gauge transform" $\hat{G}$,
which has a similar form as
(\ref{gauge example1-1}),
can be described by the following polynomials
$\hat{g}_j(z)$ $(j \geq 2)$:
\begin{eqnarray}
\hat{g}_{j}(z) &=&
\left(
\frac{\hat{h}_{1j}-\hat{h}_{3j}}{\lambda}
       -\frac{(\hat{a}-\hat{b})
                (\hat{h}_{2j}-\hat{h}_{4j})}
                    {\lambda^2}
\right)
z_1z_2+
\left(
\hat{h}_{3j}-
\frac{\hat{b}(\hat{h}_{2j}-\hat{h}_{4j})}{\lambda}
\right)
z_1
\nonumber \\
&& ~~~~~~~~~~~~~~~~~~~~~~~
+
\frac{\hat{h}_{2j}-\hat{h}_{4j}}{\lambda}z_2+
\hat{h}_{4j}~~.
\label{hatg example3}
\end{eqnarray}

           As regards the configuration obtained from
(\ref{example3-1}) by setting $\lambda =0$,
the first column
vector of $\hat{H}_{\dot{1}}$ is not cyclic.
For this configuration to be a vacuum of D5-branes
it may be enough, say, if,
in addition to the first column
vector $\hat{H}_{\dot{1}}(e_1)$ and its descendant
$\hat{B}_1\hat{H}_{\dot{1}}(e_1)$,
the second column vector $\hat{H}_{\dot{1}}(e_2)$ and
its descendant $\hat{B}_1\hat{H}_{\dot{1}}(e_2)$ span
${\bf C}^4$ ($\equiv V$). This condition leads the
constraint, $\hat{h}_{21} \neq \hat{h}_{41}$.
If it is satisfied,
one can introduce the new bases of $V$ with which
vacuum (\ref{example3-1}) (with setting $\lambda =0$)
acquires the form
\begin{eqnarray}
&&
\hat{B}_1=
\left( \begin{array}{cccc}
0 & 0 & 1 & \hat{h}_{21} \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right)~~,~~
\hat{B}_2=
\left( \begin{array}{cccc}
0 & 0 & \hat{b} & \hat{b}\hat{h}_{21}\!-\!
      \frac{\hat{a}\!-\! \hat{b}}
             {\hat{h}_{21}\!-\! \hat{h}_{41}} \\
0     & 0 & 0 & \hat{a} \\
0 & 0 & 0 & \hat{b} \\
0 & 0 & 0 & 0
\end{array}\right)~~,
\nonumber \\
&&
\hat{H}_{\dot{1}}=
\left( \begin{array}{cccc}
0 & 0 & * & * \\
0 & 0 & * & * \\
0 & 1 & * & * \\
1 & 0 & * & *
\end{array}\right)~~,~~
\hat{H}_{\dot{2}}=0~~.
\end{eqnarray}
This means that parameters $\hat{h}_{11,31}$
are absorbed into the $GL(4:{\bf C})$-symmetry.

          Nextly let us describe the vacuum
$(B_a,H_{\dot{A}})$ which belongs to the maximal cycle
$C_{({\scriptsize \mbox{[2,2]}},
\emptyset,\cdots,\emptyset)}$.
It has the realization :
\begin{eqnarray}
&&
B_1=
\left( \begin{array}{cccc}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array}\right)~~,~~
B_2=
\left( \begin{array}{cccc}
0 & a & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & b & 0 & a \\
0 & 0 & 0 & 0
\end{array}\right)~~,
\nonumber \\
&&
H_{\dot{1}}=
\left( \begin{array}{cccc}
0 & h_{11} & \cdots & h_{1n\!-\!1} \\
0 & h_{21} & \cdots & h_{2n\!-\!1} \\
0 & h_{31} & \cdots & h_{3n\!-\!1} \\
1 & h_{41} & \cdots & h_{4n\!-\!1}
\end{array}\right)~~,~~
H_{\dot{2}}=0~~,
\label{example3-2}
\end{eqnarray}
where complex parameters $a,b$ and $h_{ij}$
constitute the maximal cycle.
$B_a$ in (\ref{example3-2}) satisfy the relations,
\begin{eqnarray}
&&
B_1^2=0~~,
\nonumber \\
&&
B_2^2-2aB_1B_2-bB_1=0~~,
\nonumber \\
&&
B_1B_2=B_2B_1~~.
\label{eq of B example3-2}
\end{eqnarray}
The elements of the ``gauge transform" $G$
in (\ref{gauge}) is now given by
\begin{eqnarray}
g_j(z)=
h_{1j}z_1z_2+(h_{3j}-ah_{2j})z_1+
h_{2j}z_2+h_{4j}~~.
\label{g example3-2}
\end{eqnarray}

           The behavior of
configuration (\ref{example3-1})
when $P$ goes to $Q$ can be studied
by appropriate rescalings of the additional parameters.
Let us first rescale $\hat{a}$ and $\hat{b}$ in
(\ref{example3-1}) such that
$\lambda(\hat{a}-\hat{b})$ and $\hat{a}+\hat{b}$ do not
depend on $\lambda$. Introduce the new parameters by
\begin{eqnarray}
a&=&
\frac{\hat{a}+\hat{b}}{2}~~,
\nonumber \\
b&=&
\frac{\lambda(\hat{a}-\hat{b})}{2}~~.
\label{rescale1 example3}
\end{eqnarray}
With fixing them one can see
relations (\ref{eq of B example3-1}) become
(\ref{eq of B example3-2}) as $\lambda$ goes to zero.
As regards the other additional parameters
in (\ref{example3-1})
we shall rescale them such that
\begin{eqnarray}
&&
h_{1j}=
\frac{\hat{h}_{1j}-\hat{h}_{3j}}{\lambda}-
\frac{(\hat{a}-\hat{b})(\hat{h}_{2j}-\hat{h}_{4j})}
          {\lambda^2}~~,
\nonumber \\
&&
h_{2j}=
\frac{\hat{h}_{2j}-\hat{h}_{4j}}{\lambda}~~,
\nonumber \\
&&
h_{3j}=
\hat{h}_{3j}+
\frac{(\hat{a}-\hat{b})(\hat{h}_{2j}-\hat{h}_{4j})}
            {2\lambda}~~,
\nonumber \\
&&
h_{4j}= \hat{h}_{4j} ~~
\label{rescale2 example3}
\end{eqnarray}
become independent of $\hat{a},\hat{b}$ and $\lambda$.
The elements $\hat{g}_j(z)$ of the gauge transform
$\hat{G}$,
expressing them in terms of these rescaled parameters,
coincide with (\ref{g example3-2}).
Notice that equations (\ref{rescale1 example3}) and
(\ref{rescale2 example3}) can be also regarded as
the change of coordinates as far as $\lambda \neq 0$.
Introduce the element
$\hat{q}(\lambda)$ of $GL(4:{\bf C})$
\begin{eqnarray}
\hat{q}(\lambda)=
\left( \begin{array}{cccc}
1/\lambda & -(\hat{a}-\hat{b})/\lambda^2 &
-1/\lambda & (\hat{a}-\hat{b})\lambda^2 \\
0 & 1/\lambda & 0 & -1/\lambda \\
0 & (\hat{a}-\hat{b})/2\lambda & 1 &
-(\hat{a}-\hat{b})/2\lambda \\
0 & 0 & 0 & 1
\end{array}\right)~~~.
\end{eqnarray}
The transform of (\ref{example3-1})
by $\hat{q}(\lambda)$
becomes
\begin{eqnarray}
\hat{q}(\lambda)\hat{B}_1\hat{q}(\lambda)^{-1}
&=&
\left( \begin{array}{cccc}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array} \right) ~~,~~
\nonumber \\
\hat{q}(\lambda)\hat{B}_2\hat{q}(\lambda)^{-1}
&=&
\left( \begin{array}{cccc}
\lambda & (\hat{a}+\hat{b})/2 & 1 & 0 \\
0 & \lambda & 0 & 1 \\
0 & \lambda(\hat{a}-\hat{b})/2 & 0 &
(\hat{a}+\hat{b})/2 \\
0 & 0 & 0 & 0
\end{array} \right) ~~,~~
\nonumber \\
\hat{q}(\lambda)\hat{H}_{\dot{1}}
&=&
\left(\begin{array}{cccc}
0 &
\frac{\hat{h}_{11}-\hat{h}_{31}}{\lambda}-
\frac{(\hat{a}-\hat{b})(\hat{h}_{21}-\hat{h}_{41})}
{\lambda^2} &
\cdots &
\frac{\hat{h}_{1n\!-\!1}-\hat{h}_{3n\!-\!1}}
        {\lambda}-
\frac{(\hat{a}-\hat{b})
       (\hat{h}_{2n\!-\!1}-\hat{h}_{4n\!-\!1})}
        {\lambda^2}
\\
0 &
\frac{\hat{h}_{21}-\hat{h}_{41}}{\lambda} &
\cdots &
\frac{\hat{h}_{2n\!-\!1}-\hat{h}_{4n\!-\!1}}
        {\lambda}
\\
0 &
\hat{h}_{31}+
\frac{(\hat{a}-\hat{b})(\hat{h}_{21}-\hat{h}_{41})}
        {2\lambda} &
\cdots &
\hat{h}_{3n\!-\!1}+
\frac{(\hat{a}-\hat{b})
  (\hat{h}_{2n\!-\!1}-\hat{h}_{4n\!-\!1})}{2\lambda}
\\
1 &
\hat{h}_{41} &
\cdots &
\hat{h}_{4n\!-\!1}
\end{array}\right)~~.
\nonumber \\
&&~~~~~
\label{example3-3}
\end{eqnarray}
Rewritting (\ref{example3-3})
in terms of the rescaled parameters
we can see that it becomes (\ref{example3-2})
as $P$ goes to $Q$.

\sect{Toward Second-Quantization of D5-Brane}
\subsection*{Superposition of D5-Brane Vacua}

               In the previous section
it is shown that the additional complex parameters
of vacuum (\ref{cycle2}) of $k=k_1+k_2+\cdots+k_l$
D5-branes form a cycle in the moduli space
${\cal M}(k)$ which can be
identified with the maxiaml cycle
$C_{(\Gamma,\emptyset,\cdots,\emptyset)}$
with $\Gamma=\mbox{$[k_1,k_2,\cdots,k_l]$}$.
Vacuum (\ref{cycle2}) is described
by the superposition of vacua of
$k_i$ pieces ($1 \leq i \leq l$).
Each constituent, that is,
vacuum (\ref{element of cycle2})
of $k_i$ D5-branes degenerate at $P_i$,
admits the additional degrees of freedom,
which contributes to the parameters
of (\ref{cycle2}).
Even when some of the positions of these
overlapping D5-branes coincide, say,
$P_i=P_j$,
this superposed vacuum of $k$ D5-branes
is shown to be still well-defined
with an approriate change of realization.
In order to make the positions of
these overlapping D5-branes explicit
let us write the above topological cycle by
$C_{{\scriptsize \mbox{$[k_1,k_2,\cdots,k_l]$}}}
(P_1,P_2,\cdots,P_l)$
where
$P_i=(z_1^{(i)},z_2^{(i)})$ is the position of
$k_i$ coincident D5-branes.
$C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,\cdots,P_l)
\simeq C_{({\scriptsize \mbox{$[k_1,\cdots,k_l]$}},
\emptyset,\cdots,\emptyset)}$.

         It is physically reasonable to consider
such a vacuum of $k+\tilde{k}$ D5-branes
in which the configuration of $k$ pieces
can be regarded as the vacuum given by
(\ref{cycle2})
while the configuration of the other
$\tilde{k}$ pieces is an arbitrary vacuum
of $\tilde{k}$ D5-branes.
Taking it in the reverse order one might say
it should be possible,
at least from the viewpoint of
five-branes, to construct a new vacuum
of $k+\tilde{k}$ D5-branes
from two vacua of $k$ and $\tilde{k}$ pieces.
In spite of this physical speculation
we should note that the superposition
of two vacua,
though it works directly when both vacua
of $k$ and $\tilde{k}$ D5-branes
admit to have forms similar to (\ref{cycle2}),
does not work directly
if either vacuum of $k$ or $\tilde{k}$
pieces is generic
\footnote{We can easily superpose two vacua
in the case of $n=1$. This is because, when $n=1$,
one can set \cite{Nakajima} $H_{\dot{2}}=0$
for all the vacua $(B_a,H_{\dot{A}})$.}.
So, it is necessary to explain
how one can ``superpose" the vacuum of $k$ D5-branes
which belongs to the cycle
$C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,\cdots,P_l)$
with a vacuum of $\tilde{k}$ D5-branes
without restricting the latter.

               Let $(B_a^{(1)},H_{\dot{A}}^{(1)})$
be the vacuum of $k$ D5-branes which has
form (\ref{cycle2}). It belongs to the cycle
$C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,\cdots,P_l)$.
Take a generic vacuum of $\tilde{k}$ D5-branes,
which is denoted by $(B_a^{(2)},H_{\dot{A}}^{(2)})$.
It may be convenient to handle these vacua by using
hyperk\"ahler description (\ref{moduli})
($or$ (\ref{ADHM1}) and (\ref{ADHM2})) of the moduli
spaces. Notice that there exist
$g(\eta)^{(1)} \in GL(k:{\bf C})$
and
$g(\eta)^{(2)} \in GL(\tilde{k}:{\bf C})$
such that the transforms of
$(B_a^{(1)},H_{\dot{A}}^{(1)})$
by $g^{(1)}(\eta)$
and
$(B_a^{(2)},H_{\dot{A}}^{(2)})$
by $g^{(2)}(\eta)$
both satisfy $D$-flat conditions
(\ref{ADHM1}) and (\ref{ADHM2}).
These transforms will be denoted by
$(B_a^{(1,2)}(\eta),H_{\dot{A}}^{(1,2)}(\eta))$
in order to make their dependence on
$\eta$ explicit.
For the same reason
the corresponding moduli spaces,
regarded as the hyperk\"ahler quotients,
will be denoted by
${\cal M}(k)_{\eta}$ and
${\cal M}(\tilde{k})_{\eta}$.
Notice that
$(B_a^{(1)}(\eta),H_{\dot{A}}^{(1)}(\eta))
\in {\cal M}(k)_{\eta}$
and
$(B_a^{(2)}(\eta),H_{\dot{A}}^{(2)}(\eta))
\in {\cal M}(\tilde{k})_{\eta} $.
We first consider their behaviors
as $\eta$ goes to zero.
For the case of $\eta$ being positive
the vacuum
$(B_a^{(1)}(\eta),H_{\dot{A}}^{(1)}(\eta))$
admits additional parameters
as it has in the complex symplectic description.
But,
as $\eta$ goes to zero,
these additional degrees of freedom vanish.
It will become
\begin{eqnarray}
B_a^{(1)}(0)
&\equiv&
\lim_{\eta \rightarrow +0}B_a^{(1)}(\eta)
\nonumber \\
&=&
\left( \begin{array}{cccc}
z_a^{(1)}{\bf 1}_{k_1} & 0 & \cdots & 0 \\
0 & \ddots & \ddots & \vdots \\
\vdots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & z_a^{(l)}{\bf 1}_{k_l}
\end{array} \right)~~,
\nonumber \\
H_{\dot{A}}^{(1)}(0)
&\equiv&
\lim_{\eta \rightarrow +0}H_{\dot{A}}^{(1)}(\eta)
\nonumber \\
&=&
0~~.
\label{small instanton}
\end{eqnarray}
Notice that
$P_i=(z_1^{(i)},z_2^{(i)})$ $(1 \leq i \leq l)$
are the positions of overlapping $k_i$ D5-branes.
Configuration (\ref{small instanton})
can be regarded as a vacuum
at $\eta =0$.
In the terminology of
four-dimensional gauge theory,
it describes the small size limit of
$k$ $SU(n)$-instantons.
$k_i$ of these ideal instantons
are located at $P_i$.
As regards the vacuum
$(B_a^{(2)}(0),H_{\dot{A}}^{(2)}(0))$
$\equiv \lim_{\eta \rightarrow +0}
(B_a^{(2)}(\eta),H_{\dot{A}}^{(2)}(\eta))$,
since it is a generic point of the moduli space
of $\tilde{k}$ D5-branes, it describes
non-singular $\tilde{k}$ $SU(n)$-instantons.
At $\eta =0$ one can easily
superpose these two vacua.
Namely let us introduce
\begin{eqnarray}
B_a(0) \equiv
\left( \begin{array}{cc}
B_a^{(1)}(0) & 0 \\
0 & B_a^{(2)}(0)
\end{array} \right)~~,~~
H_{\dot{A}}(0) \equiv
\left( \begin{array}{c}
H_{\dot{A}}^{(1)}(0) \\
H_{\dot{A}}^{(2)}(0)
\end{array}\right)~~.
\label{superpose at 0}
\end{eqnarray}
Then configuration (\ref{superpose at 0})
can be regarded as a point
of the moduli space
${\cal M}(k+\tilde{k})_{\eta =0}$.
The contribution of the first $k$ D5-branes
makes it a singular point.

           It is convenient to
remark on the relation between the moduli spaces
${\cal M}(k+\tilde{k})_{\eta >0}$ and
${\cal M}(k+\tilde{k})_{\eta =0}$.
As was explained in Section.2,
the moduli space ${\cal M}(k+\tilde{k})_{\eta >0}$
is smooth. It has no singularity.
Under the limit that $\eta$ goes to zero
singularities do appear.
These singularities correspond
to the small size limits of $SU(n)$-insatntons.
This phenomenon shows that
{\it ${\cal M}(k+\tilde{k})_{\eta >0}$
can be regarded as a resolution of
${\cal M}(k+\tilde{k})_{\eta =0}$.}
The projection $\pi$
from ${\cal M}(k+\tilde{k})_{\eta >0}$
to ${\cal M}(k+\tilde{k})_{\eta =0}$
is defined by taking the limit of $\eta$ being zero
in the former moduli space.
As regards configuration (\ref{superpose at 0}),
which is a singular point of
${\cal M}(k+\tilde{k})_{\eta =0}$,
the inverse image of (\ref{superpose at 0}),
that is,
$\pi^{-1}((B_a(0),H_{\dot{A}}(0)))$
gives us the vacua
which admit additional complex parameters
by which the singularity is resolved.
Among these vacua there exists a configuration
$(B_a(\eta),H_{\dot{A}}(\eta))$
which corresponds to a superposition
of the two vacua
$(B_a^{(1)}(\eta),H_{\dot{A}}^{(1)}(\eta))$
and
$(B_a^{(2)}(\eta),H_{\dot{A}}^{(2)}(\eta))$.
It realizes the ``superposition" of the vacuum of
$k$ D5-branes which belongs to the cycle
$C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,\cdots,P_l)$
and a generic vacuum of $\tilde{k}$ D5-branes.

         Let us comment briefly on the case of
$(B_a^{(2)},H_{\dot{A}}^{(2)})$ being a vacuum
which causes a singularity at $\eta=0$.
Suppose
$(B_a^{(2)},H_{\dot{A}}^{(2)})$
$\in C_{\tilde{\Gamma}}
(\tilde{P}_1,\cdots,\tilde{P}_{\tilde{l}})$.
($|\tilde{\Gamma}|=\tilde{k}$ and
$l(\tilde{\Gamma})=\tilde{l}$.)
When all the $\tilde{l}$-positions
$\tilde{P}_1,\cdots,\tilde{P}_{\tilde{l}}$
are different from the $l$-positions
$P_1,\cdots,P_l$ of the first $k$ D5-branes,
their superposition will be given
\footnote{
Here we use the complex symplectic description.}
\begin{eqnarray}
B_a \equiv
\left( \begin{array}{cc}
B_a^{(1)} & 0 \\
0 & B_a^{(2)}
\end{array} \right)~~,~~
H_{\dot{A}} \equiv
\left( \begin{array}{c}
H_{\dot{A}}^{(1)} \\
H_{\dot{A}}^{(2)}
\end{array} \right)~~.
\label{superpose}
\end{eqnarray}
Clearly configuration (\ref{superpose})
belongs to the cycle
$C_{\Gamma \cup \tilde{\Gamma}}
(Q_1,\cdots,Q_{l+\tilde{l}})$
of the moduli space ${\cal M}(k+\tilde{k})$
where
$\Gamma \cup \tilde{\Gamma}$ is the Young tableau
of $k+\tilde{k}$ boxes obtained from
$\Gamma=\mbox{$[k_1,\cdots,k_l]$}$
and
$\tilde{\Gamma}
=\mbox{$[\tilde{k}_1,\cdots,
\tilde{k}_{\tilde{l}}]$}$
by reordering $k_i$ and $\tilde{k}_j$.
$Q_1,\cdots,Q_{l+\tilde{l}}$
are the corresponding rearrangement of
$P_1,\cdots,P_l$
and
$\tilde{P}_1,\cdots,\tilde{P}_{\tilde{l}}$.
In the case that some of the positions of
$\tilde{k}$ D5-branes
coincide with those of the first
$k$ D5-branes,
owing to the discussion given in
the previous section,
it is still possible to give
their superposition in the cycle
$C_{\Gamma \cup \tilde{\Gamma}}
(Q_1,\cdots,Q_{l+\tilde{l}})$.

           The ``superpostion" defines
an inclusion $\iota$ of
$C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,\cdots,P_l)
\times {\cal M}(\tilde{k})$
to ${\cal M}(k+\tilde{k})$ :
\begin{eqnarray}
\iota~~~~~
~~C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
         (P_1,\cdots,P_l)
\times {\cal M}(\tilde{k})
~\hookrightarrow~
{\cal M}(k+\tilde{k})~~~.
\label{inclusion map}
\end{eqnarray}
The image of $\iota$, that is,
$\iota $
$\left(C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,\cdots,P_l)
\times {\cal M}(\tilde{k}) \right)$,
is a noncompact submanifold of the moduli space
${\cal M}(k+\tilde{k})$, which will be denoted by
${\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,$$\cdots,P_l)$.
The dimensions of
${\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,$$\cdots,P_l)$ is
equal to the sum of the dimensions of
the moduli space ${\cal M}(\tilde{k})$ and
the cycle
$C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,$$\cdots,P_l)$,
\begin{eqnarray}
dim~
{\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,\cdots,P_l)
=4n \tilde{k}+2(nk-l).
\label{dim of submfd C(P)}
\end{eqnarray}
Notice that $k_1+\cdots+k_l=k$.
Physically speaking,
any point of the submanifold
${\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,$$\cdots,P_l)$
describes the vacuum of $k+\tilde{k}$ D5-branes
in which the configuration of $k$ pieces,
considering it as a vacuum of $k$ D5-branes,
belongs to the cycle
$C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,$$\cdots,P_l)$
of the moduli space ${\cal M}(k)$.
Each point $P_i$ denotes the position
of overlapping $k_i$ D5-branes.
By letting these $l$-positions
of $k=k_1+k_2+\cdots+k_l$
D5-branes free in the four-dimensions,
we may introduce the noncompact
submanifold
${\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}$
of
${\cal M}(k+\tilde{k})$
by
\begin{eqnarray}
{\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
\equiv \left. \left\{
~{\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
(P_1,\cdots,P_l)
~\right|
~P_1,\cdots,P_l \in {\bf C}^2
~\right\}~~.
\label{submfd C}
\end{eqnarray}
The dimensions of this submanifold is
\begin{eqnarray}
dim~
{\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
=4n\tilde{k}+2(nk+l).
\label{dim of submfd C}
\end{eqnarray}

            Now let us exchange the roles of $k$ and
$\tilde{k}$ D5-branes in the construction of
inclusion map (\ref{inclusion map}).
Consider the topological cycle
$C_{\tilde{\Gamma}}
(\tilde{P}_1,$$\cdots,\tilde{P}_{\tilde{l}})$
of the moduli space ${\cal M}(\tilde{k})$.
$\tilde{\Gamma}
=\mbox{$[\tilde{k}_1,\cdots,\tilde{k}_{\tilde{l}}]$}$
with $|\tilde{\Gamma}|=\tilde{k}$.
Each point $\tilde{P}_i$ denotes the position of
overlapping $\tilde{k}_i$ D5-branes.
By the superposition of vacua we will obtain
the inclusion map,
\begin{eqnarray}
\tilde{\iota}~~~~~
~~{\cal M}(k) \times
C_{{\scriptsize
   \mbox{$[\tilde{k}_1,\cdots,\tilde{k}_{\tilde{l}}]$}}}
(\tilde{P}_1,\cdots,\tilde{P}_{\tilde{l}})
~\hookrightarrow~
{\cal M}(k+\tilde{k})~~~,
\label{inclusion map2}
\end{eqnarray}
which provides, as its image,
the noncompact submanifold
${\cal C}_{{\scriptsize
\mbox{$[\tilde{k}_1,\cdots,\tilde{k}_{\tilde{l}}]$}}}
(\tilde{P}_1,\cdots,\tilde{P}_{\tilde{l}})$
of the moduli space ${\cal M}(k+\tilde{k})$.
Letting the $\tilde{l}$-points
$\tilde{P}_1,\cdots,\tilde{P}_{\tilde{l}}$
free we also obtain the noncompact submanifold
${\cal C}_{{\scriptsize
  \mbox{$[\tilde{k}_1,\cdots,\tilde{k}_{\tilde{l}}]$}}}$.
The dimensions of this submanifold is
equal to $4nk+2(n\tilde{k}+\tilde{l})$.
Notice that any point of
the submanifold ${\cal C}_{{\scriptsize
  \mbox{$[\tilde{k}_1,\cdots,\tilde{k}_{\tilde{l}}]$}}}$
describes the vacuum of $k+\tilde{k}$ D5-branes
in which the configuration of $\tilde{k}$ pieces,
considering it as a vacuum of $\tilde{k}$ D5-branes,
belongs to the cycle
$C_{{\scriptsize
  \mbox{$[\tilde{k}_1,\cdots,\tilde{k}_{\tilde{l}}]$}}}
(\tilde{P}_1,$$\cdots,\tilde{P}_{\tilde{l}})$
of the moduli space ${\cal M}(\tilde{k})$.
By examining two vacua of $k+\tilde{k}$
D5-branes which respectively belong to
${\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}$
and
${\cal C}_{{\scriptsize
 \mbox{$[\tilde{k}_1,\cdots,\tilde{k}_{\tilde{l}}]$}}}$
one can find that
their intersection
in the moduli space ${\cal M}(k+\tilde{k})$
has the form,
\begin{eqnarray}
{\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}}
~\cap~
{\cal C}_{{\scriptsize
   \mbox{$[\tilde{k}_1,\cdots,\tilde{k}_{\tilde{l}}]$}}}~=~
{\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}} \cup
{\scriptsize \mbox{$[\tilde{k}_1,
           \cdots,\tilde{k}_{\tilde{l}}]$}}}~~.
\label{intersection 1}
\end{eqnarray}
The noncompact submanifold
${\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}} \cup
{\scriptsize
\mbox{$[\tilde{k}_1,\cdots,\tilde{k}_{\tilde{l}}]$}}}$
is simply realized as the set of the cycles
$C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}} \cup
{\scriptsize \mbox{$[\tilde{k}_1,
\cdots,\tilde{k}_{\tilde{l}}]$}}}
(Q_1,\cdots,Q_{l+\tilde{l}})$
of the moduli space ${\cal M}(k+\tilde{k})$ :
\begin{eqnarray}
{\cal C}_{{\scriptsize
\mbox{$[k_1,\cdots,k_l]$}} \cup
{\scriptsize \mbox{$[\tilde{k}_1,
\cdots,\tilde{k}_{\tilde{l}}]$}}}
&=&
\left.\left\{
C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}} \cup
{\scriptsize \mbox{$[\tilde{k}_1,
\cdots,\tilde{k}_{\tilde{l}}]$}}}
(Q_1,\cdots,Q_{l+\tilde{l}})~\right|~
Q_1,\cdots,Q_{l+\tilde{l}}
\in {\bf C}^2 \right\}~~~.
\nonumber \\
&&~~~~~~~
\end{eqnarray}
Since
$C_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}} \cup
{\scriptsize \mbox{$[\tilde{k}_1,
\cdots,\tilde{k}_{\tilde{l}}]$}}}
(Q_1,\cdots,Q_{l+\tilde{l}})$
is isomorphic to
$C_{({\scriptsize \mbox{$[k_1,\cdots,k_l]$}} \cup
{\scriptsize \mbox{$[\tilde{k}_1,
\cdots,\tilde{k}_{\tilde{l}}]$}},
\emptyset,\cdots,\emptyset)}$,
we can identify the submanifold
${\cal C}_{{\scriptsize
\mbox{$[k_1,\cdots,k_l]$}} \cup
{\scriptsize \mbox{$[\tilde{k}_1,
\cdots,\tilde{k}_{\tilde{l}}]$}}}$
with
$\left( {\bf C}^2 \right)^{\oplus (l+\tilde{l})}
\times
C_{({\scriptsize \mbox{$[k_1,\cdots,k_l]$}} \cup
{\scriptsize
\mbox{$[\tilde{k}_1,\cdots,\tilde{k}_{\tilde{l}}]$}},
\emptyset,\cdots,\emptyset)}$.
Here
``$\left( {\bf C}^2 \right)^{\oplus (l+\tilde{l})}$"
parametrize the $(l+\tilde{l})$-positions
where $k+\tilde{k}$ D5-branes are overlapping.

            One can also generalize inclusion map
(\ref{inclusion map}) to the following direction.
For each $i$, consider the inclusion map
\begin{eqnarray}
C_{{\scriptsize \mbox{$[k_i]$}}}(P_i)
\times
{\cal M}(\tilde{k}+\sum_{j \neq i}^lk_j)~~
\hookrightarrow~~
{\cal M}(k+\tilde{k})~~,
\label{i-th inclusion}
\end{eqnarray}
which now gives the submanifold
${\cal C}_{{\scriptsize \mbox{$[k_i]$}}}$
of the moduli space ${\cal M}(k+\tilde{k})$.
The dimensions of this submanifold is
\begin{eqnarray}
dim~{\cal C}_{{\scriptsize \mbox{$[k_i]$}}}=
4n(\tilde{k}+\sum_{j \neq i}^lk_i)
+2(nk_i+1).
\label{dim of Ci}
\end{eqnarray}
The intersection of these submanifolds clearly turns out
to be (\ref{submfd C}) :
\begin{eqnarray}
{\cal C}_{{\scriptsize \mbox{$[k_1]$}}} \cap
{\cal C}_{{\scriptsize \mbox{$[k_2]$}}} \cap \cdots \cap
{\cal C}_{{\scriptsize \mbox{$[k_l]$}}}
={\cal C}_{{\scriptsize \mbox{$[k_1,\cdots,k_l]$}}=
         \cup_{i=1}^{l}{\scriptsize \mbox{$[k_i]$}}}~~.
\label{intersection 2}
\end{eqnarray}

\subsection*{Physical Observables of Worldvolume Topological
Field Theory}

         Let us concentrate on the moduli space
${\cal M}(k)$, that is, the classical vacua of
$k$ D5-branes having open strings with the
$U(n)$ Chan-Paton factors.
In \cite{FKN}
worldvolume topological $U(k)$ gauge theory has been
constructed such that its physical content can be
identified with cohomology theory of the moduli
space ${\cal M}(k)$.
In particular,
the physical Hilbert space is realized by the
cohomology group $H^*({\cal M}(k))$.
In this subsection, taking this topological field
theoretical viewpoint,
we introduce a subclass of $H^*({\cal M}(k))$, which
admits,
as a class of the physical observables
of topological field theory,
to have a structure analogous to the Fock space.

                     We begin by considering
the noncompact submanifold ${\cal C}_{\Gamma}$
with $|\Gamma|=k$ and $l(\Gamma)=l$.
It is the set of the cycles
$C_{\Gamma}(P_1,\cdots,P_l)$ with
$P_i$ being arbitrary in the four-dimensions.
Since each cycle $C_{\Gamma}(P_1,\cdots,P_l)$
can be topologically identified with
the maximal cycle
$C_{(\Gamma,\emptyset,\cdots,\emptyset)}$,
it holds that
${\cal C}_{\Gamma} \simeq
\left( {\bf C}^2 \right)^{\oplus l}
\times
C_{(\Gamma, \emptyset,\cdots,\emptyset)}$.
This isomorphism implies that
the noncompact directions of the submanifold
${\cal C}_{\Gamma}$ can be identified with
``$\left( {\bf C}^2 \right)^{\oplus l}$",
which simply parametrize the $l$-positions
$P_1,\cdots,P_l$ where $k$ D5-branes overlap.
Let us take the Poincar\'e dual of
${\cal C}_{\Gamma}$ in
the moduli space ${\cal M}(k)$.
It will be denoted by
${\cal O}_{\Gamma}$.
The support of ${\cal O}_{\Gamma}$ is
on the tubular neighborhood of
${\cal C}_{\Gamma}$ and
therefore it is noncompact.
Taking account of the isomorphism
${\cal C}_{\Gamma} \simeq$
$\left( {\bf C}^2 \right)^{\oplus l}
\times C_{(\Gamma,\emptyset,\cdots,\emptyset)}$
one may modify ${\cal O}_{\Gamma}$
by adding or subtracting
an appropriate exact form
so that it does not depend
on the noncompact directions.
This means that
one can make ${\cal O}_{\Gamma}$
independent of the coordinates of
the $l$-positions
$P_1,\cdots,P_l$ on $C_{\Gamma}(P_1,\cdots,P_l)$.
(It will require
a special care when some of these points
coincide.)
As regards the degrees of
${\cal O}_{\Gamma}$
it is given by
\begin{eqnarray}
deg~ {\cal O}_{\Gamma} &=&
dim~ {\cal M}(k)-~dim~{\cal C}_{\Gamma}
\nonumber \\
&=&
2(nk-l)~~,
\label{dim of O}
\end{eqnarray}
which is equal to
the dimensions of the maximal cycle
$C_{(\Gamma,\emptyset,\cdots,\emptyset)}$.

                       Suppose that
$\Gamma=\mbox{$[k_1,\cdots,k_l]$}$.
Consider the noncompact submanifold
${\cal C}_{{\scriptsize \mbox{$[k_i]$}}}$
of the moduli space ${\cal M}(k)$.
${\cal C}_{{\scriptsize \mbox{$[k_i]$}}}$
is the set of the vacua of $k$ D5-branes
in which the configurations of $k_i$ pieces,
considering it as vacua of
$k_i$ five-branes, belong to the cycle
$C_{{\scriptsize \mbox{$[k_i]$}}}(P)$
of the moduli space ${\cal M}(k_i)$.
$P$ is arbitrary in the four-dimensions.
Let us also take the Poincar\'e dual of
this submanifold in the moduli space
${\cal M}(k)$,
which will be denoted by
${\cal O}_{{\scriptsize \mbox{$[k_i]$}}}$.
The support of
${\cal O}_{{\scriptsize \mbox{$[k_i]$}}}$
is on the tubular neighbourhood of
${\cal C}_{{\scriptsize \mbox{$[k_i]$}}}$.
Through the inclusion map,
$C_{{\scriptsize \mbox{$[k_i]$}}}(P)
\times {\cal M}(k-k_i)$
$\hookrightarrow {\cal M}(k)$,
besides the identification of
$C_{{\scriptsize \mbox{$[k_i]$}}}(P)$
with
$C_{({\scriptsize \mbox{$[k_i]$}},
\emptyset,\cdots,\emptyset)}$
in the moduli space ${\cal M}(k_i)$,
the submanifold
${\cal C}_{{\scriptsize \mbox{$[k_i]$}}}$
can be regarded as
$\left( {\bf C}^2 \right) \times {\cal M}(k-k_i)
\times
C_{({\scriptsize \mbox{$[k_i]$}},
\emptyset,\cdots,\emptyset)}$,
where
``${\bf C}^2$"
parametrize the position of
the overlapping $k_i$ D5-branes.
Under this identification
the noncompact directions of
${\cal C}_{{\scriptsize \mbox{$[k_i]$}}}$
are
$\left( {\bf C}^2 \right) \times {\cal M}(k-k_i)$.
By adding or subtracting an appropriate exact form
one may adjust
${\cal O}_{{\scriptsize \mbox{$[k_i]$}}}$
such that
it does not depend on these noncompact directions.
Recalling
$C_{({\scriptsize \mbox{$[k_i]$}},
\emptyset,\cdots,\emptyset)}$
is the cycle which is added in order to resolve
the singularity of (overlapping)
$k_i$ ideal $SU(n)$-instantons,
this means that
${\cal O}_{{\scriptsize \mbox{$[k_i]$}}}$
can be taken such that
it only depends on the local
data of the resolution.
The degrees of
${\cal O}_{{\scriptsize \mbox{$[k_i]$}}}$
turns out to be
\begin{eqnarray}
deg~{\cal O}_{{\scriptsize \mbox{$[k_i]$}}} &=&
dim~{\cal M}(k)-
~dim~{\cal C}_{{\scriptsize \mbox{$[k_i]$}}}
\nonumber \\
&=& 2(nk_i-1)~~,
\label{deg of Oi}
\end{eqnarray}
which is independent of $k$ and equals to
the dimensions of the cycle
$C_{({\scriptsize \mbox{$[k_i]$}},
\emptyset,\cdots,\emptyset)}$
of the moduli space ${\cal M}(k_i)$.

             Now let us apply formula
(\ref{intersection 2}).
Taking the Poincar\'e dual of
(\ref{intersection 2}) one can
obtain the relation
\begin{eqnarray}
{\cal O}_{\Gamma=
      \cup_{i=1}^l{\scriptsize \mbox{$[k_i]$}}}
=
{\cal O}_{{\scriptsize \mbox{$[k_1]$}}}
\wedge \cdots \wedge
{\cal O}_{{\scriptsize \mbox{$[k_l]$}}}~~.
\label{co-intersection 2}
\end{eqnarray}
One of the implications of
(\ref{co-intersection 2})
is as follows :
As was explained in Section.2,
several nontrivial cycles appear
in the moduli space
${\cal M}(k)$,
which one may classify by using the set of
$n$ Young tableaux
$(\Gamma_1,\Gamma_2,\cdots,\Gamma_n)$
satisfying conditions (\ref{Young tableau}).
Among them
we have investigated the maximal cycles
$C_{(\Gamma,\emptyset,\cdots,\emptyset)}$
with $|\Gamma|=k$.
${\cal O}_{\Gamma}$
are introduced as the Poincar\'e duals
of the submanifolds
${\cal C}_{\Gamma}$
which can be identified with
$\left( {\bf C}^2 \right)^{\otimes l(\Gamma)}
\times C_{(\Gamma,\emptyset,\cdots,\emptyset)}$.
The noncompact degrees of freedom simply
measure the positions
where $k$ D5-branes are overlapping.
One can regard the vector space spanned
by these ${\cal O}_{\Gamma}$
as the cohomological version
of the subspace of the homology group
$H_* \left( {\cal M}(k) \right)$
spanned by the cycles
$C_{(\Gamma,\emptyset,\cdots,\emptyset)}$.
Formula (\ref{co-intersection 2})
implies that the subspace
$\bigoplus_{|\Gamma|=k}{\bf C}{\cal O}_{\Gamma}$
of $H^*({\cal M}(k))$
admits a structure
analogous to the Fock space
\begin{eqnarray}
\bigoplus_{|\Gamma|=k}
{\bf C}{\cal O}_{\Gamma}
=
\bigoplus_{m_1+\cdots+m_l=k}
{\bf C}{\cal O}_{m_1}\wedge \cdots \wedge
{\cal O}_{m_l}~~,
\label{restricted Fock space}
\end{eqnarray}
where we simplify the notation as follows :
\begin{eqnarray}
{\cal O}_{m}\equiv
{\cal O}_{{\scriptsize \mbox{$[m]$}}}~~.
\end{eqnarray}

\subsection*{Second-Quantization of D5-Brane}

            To recover the Fock space structure in
(\ref{restricted Fock space}) it is enough to
consider the direct sum  of these moduli spaces
in stead of the specific moduli space.
This extension is quite reasonable
from the viewpoint of five-branes
because the number of five-branes does not suffer
apriori any restriction.
(Notice that we do not compactify the four-dimensions.)
So, summing up $k$ in (\ref{restricted Fock space})
we obtain
\begin{eqnarray}
{\cal H}_{total}\equiv
\bigoplus_{l}
\bigoplus_{m_1,\cdots,m_l}
{\bf C}
{\cal O}_{m_1}\wedge \cdots \wedge
{\cal O}_{m_l}~~,
\label{Fock space}
\end{eqnarray}
which is a subspace of
$\bigoplus_k H^*({\cal M}(k))$.
One might suspect that our definition of
${\cal O}_m$ given in the previous subsection
seems to depend on each moduli space
${\cal M}(k)$
since it is introduced as the Poincar\'e dual
of the submanifold
${\cal C}_{{\scriptsize \mbox{$[m]$}}}$.
But, by the closer look on
the submanifold
${\cal C}_{{\scriptsize \mbox{$[m]$}}}$,
we can expect that ${\cal O}_m$ will be taken
to depend only on the local data of the resolution
of the singularity at $\eta=0$
caused by overlapping $m$ D5-branes
and therefore it will acquire the form
independent of $k$ \footnote{
This property is discussed
in \cite{FKN} as the universality
of the physical observables.}.

         We would like to propose that
{\it ${\cal H}_{total}$ is
the Fock space of the second-quantized
D5-brane which allows
the $U(n)$ Chan-Paton factors.}
${\cal O}_m$ will be identified with
a marginally stable bound state of
$m$ D5-branes.
This identification seems to be reasonable since
${\cal O}_m$ will be regarded
to be constructed from the local data of the resolution
of the singularity caused by overlapping
$m$ D5-branes.
We may introduce
the creation and annihilation operators
of these bound states.
The creation and annihilation opeartors
of ${\cal O}_m$ are denoted
respectively by $\alpha_{-m}$ and $\alpha_{m}$.
These operators can be
normalized to satisfy the relations
\begin{eqnarray}
\mbox{$[~\alpha_{m_1}~,~\alpha_{m_2}~]$}
=\delta_{m_1+m_2,0}~~.
\label{CCR1}
\end{eqnarray}
Notice that they are bosonic operators
since the degrees of ${\cal O}_m$,
which count the fermionic contributions
to ${\cal O}_m$
\footnote{From the topological field theoretical
viewpoint the degrees of ${\cal O}_m$
can be interpreted as the ghost numbers
.},
are even.

\newpage
\subsection*{Acknowledgements}

        We would like to thank H.Nakajima for sending us
his beautiful lecture note \cite{Nakajima-Lec}.
We benefited from discussions with H.Kunitomo and K.Furuuchi.
We also thank J.A.Harvey for letting us know the paper
\cite{Harvey-Moore} in which Nakajima's results are nicely
reviewed from the viewpoint of string dualities.

redpajama_set_name: RedPajamaArXiv