%Paper: hep-th/9403080
%From: "J.W.van de Leur" <vdleur@math.ruu.nl>
%Date: Tue, 15 Mar 1994 14:03:32 +0100
%Date (revised): Tue, 17 May 1994 14:40:01 +0200


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\topmatter
\title   KdV Type Hierarchies, the String Equation and $W_{1+\infty}$
Constraints
\endtitle
\author
Johan van de Leur
\endauthor
\thanks
\;The author was partly supported by a fellowship of the Royal Netherlands
Academy of
Arts and Sciences. E-mail: vdleur\@math.ruu.nl
\endthanks
\address
J.W. van de Leur, Kievitdwarsstraat 22,  3514VE Utrecht, The Netherlands
\endaddress
\abstract
To every partition $n=n_1+n_2+\cdots+n_s$ one can associate a vertex
operator realization of the Lie algebras $a_{\infty}$ and $\hat{gl}_n$.
Using this construction  we make reductions of the $s$--component KP hierarchy,
reductions which are related to these partitions.
In this way we obtain matrix KdV type
equations. Now assuming that (1) $\tau$ is a $\tau$--function of the
$[n_1,n_2,\ldots,n_s]$--th reduced KP hierarchy and
(2) $\tau$  satisfies a `natural' string equation,
we prove that $\tau$ also satisfies the vacuum constraints of the
$W_{1+\infty}$ algebra.
\endabstract
\endtopmatter

\subheading{\S 0. Introduction}
\vskip 10pt

In recent years KdV type hierarchies have been related to 2D gravity. To be
slightly more precise  (see [Dij] for the details and references),
the square root of the partition function of the Hermitian $(n-1)$--matrix
model
in the continuum limit is the $\tau$--function of the $n$--reduced
Kadomtsev Petviashvili (KP) hierarchy. Hence, the $(n-1)$--matrix  model
corresponds to $n$--th Gelfand Dickey hierarchy. For $n=2,3$ these hierarchies
are better known as the KdV-- and Boussinesque hierarchy, respectively.
The partition function is then characterized by the so-called string
equation:
%
$$L_{-1}\tau=\frac{1}{n}\frac{\partial \tau}{\partial x_1},\tag{0.1}$$
%
where $L_{-1}$ is an element of  the $c=n$  Virasoro algebra,
wich is related to the principal realization of the affine lie algebra
$\hat{sl}_n$,
or rather $\hat{gl}_n$. Let $\alpha_k=-kx_{-k},\ 0,\ \frac{\partial}{\partial
x_k}$ for
$k<0,\ k=0,\ k>0$, respectively, then
%
$$L_k=\frac{1}{2n}\sum_{\ell\in {\Bbb Z}}:\alpha_{-\ell}\alpha_{\ell+nk}:
+\delta_{0k}\frac{n^2-1}{24n}.
\tag{0.2}
$$


By making the shift $x_{n+1}\mapsto x_{n+1}+{n\over n+1}$, we modify the
origin of the $\tau$--function and thus obtain the following
form of the string equation:
$$L_{-1}\tau=0.\tag{0.3}$$


Actually, it can be shown ([FKN] and [G]) that the above conditions,
$n$--th reduced KP and equation (0.3) (which from now on we will call
the string equation),
on a $\tau$--function of the KP hierarchy imply
more general constraints, viz. the vacuum constraints of the $W_{1+\infty}$
algebra. This last condition is reduced to the vacuum conditions of the $W_n$
algebra when some redundant variables are eliminated.

The $W_{1+\infty}$ algebra is the central extension of the Lie algebra of
differential
operators on ${\Bbb C}^{\times}$.
This central extension was discovered by Kac and Peterson in 1981 [KP3] (see
also [Ra], [KRa]). It has as basis the operators
$W^{(\ell+1)}_k=-t^{k+\ell}(\frac{\partial}{\partial t})^{\ell}$, $\ell\in{\Bbb
Z}_+$,
$k\in{\Bbb Z}$, together with the central element $c$. There is a well-known
way
how to express these elements in the elements of the Heisenberg algebra,
the $\alpha_k$'s. The $W_{1+\infty}$ constraints then are
%
$$\hat W^{(\ell+1)}_k\tau=\{W^{(\ell+1)}_k +\delta_{k,0}c_{\ell}\}\tau=0
\ \text{for } -k\le \ell\ge 0.
\tag{0.4}
$$
%
For the above $\tau$--function, $\hat W^{(1)}_k=-\alpha_{nk}$ and
$\hat W^{(2)}_k=L_k-\frac{nk+1}{n}\alpha_{nk}$.

It is well--known that the $n$--reduced KP hierarchy is related to the
principal realization (a vertex realization) of the basic module of
$\hat{sl}_n$. However there are many inequivalent vertex realization.
Kac and Peterson [KP1] and independently Lepowsky [L] showed that
for the basic representation of a simply--laced affine Lie algebra
these different realizations are parametrized by the conjugacy classes
of the Weyl group of the corresponding finite dimensional Lie algebra.
Hence, for the case of $\hat{sl}_n$ they are parametrized by the partitions
$n=n_1+n_2+\cdots+n_s$ of $n$. An explicit description of these realizations
was given in [TV] (see also $\S 2$). There the construction was given in such a
way
that it was possible to make reductions of the KP--hierarchy.
In all these constructions a `natural' Virasoro algebra played an important
role.
A natural question now is: If $\tau$ is a $\tau$--function of this
$[n_1,n_2,\ldots,n_s]$--th
reduced KP hierarchy and $\tau$ satisfies the string equation (0.3),
where $L_{-1}$ is an element of this new Virasoro algebra, does $\tau$ also
satisfy some
corresponding $W_{1+\infty}$ constraints? In this paper we give a positive
answer to this question. As will be shown in $\S 6$, there exists a`natural'
$W_{1+\infty}$  algebra for which (0.4) holds.

This paper is organized as follows. Sections 1--3 give results wich
were obtained in [KV] and [TV] (see also [BT]). Its major part is an
exposition of the $s$--component KP hierarchy following [KV].
In $\S1$, we describe the semi--infinite
wedge representation of the group $GL_{\infty}$ and the Lie algebras
$gl_{\infty}$ and $a_{\infty}$. We  define the KP hierarchy
in the so--called fermionic picture. The loop algebra $\hat{gl}_n$ is
introduced in
$\S 2$. We obtain it as a subalgebra of $a_{\infty}$.
Next we construct to every partition  $n=n_1+n_2+\cdots+n_s$ of $n$ a vertex
operator
realization of $a_\infty$ and $\hat{gl}_n$. $\S 3$ is devoted to the
description of $s$--component
KP hierarchy in terms of formal pseudo--differential operators. $\S 4$
describes
reductions of this $s$--component KP hierarchy related to the above
partitions.
In $\S 5$ we introduce the string equation and deduce its consequences in
terms of the pseudo--differential operators. Using the results of $\S 5$ we
deduce
in $\S 6$ the $W_{1+\infty}$ constraints. $\S 7$ is devoted to a geometric
interpretation of the string equation  on the Sato Grassmannian,
which is similar to that of [KS].

Notice that since the Toda lattice hierarchy of [UT] is related to the
2--component
KP hierarchy, some results of this paper also hold for certain reductions
of the Toda lattice hierarchy.

Finally, I would like to thank Victor Kac for valuable discussions, and
the Mathematical Institute of the University of Utrecht
for the computer and e-mail facilities. I want to dedicate this paper to the
memory of my father, Noud van de Leur, who died quite suddenly in the period
that I was writing this article.

\vskip 10pt
\subheading{\S 1.  The semi-infinite wedge
representation of the group $GL_{\infty}$ and the KP hierarchy in
the fermionic picture}

\vskip 10pt
{\bf 1.1.}  Consider the infinite complex matrix group
%
$$GL_{\infty} = \{ A = (a_{ij})_{i,j \in {\Bbb Z}+\frac{1}{2}}|A\
\text{is invertible and all but a finite number of}\ a_{ij} -
\delta_{ij}\ \text{are}\ 0\}$$
%
and its Lie algebra
%
$$gl_{\infty} = \{ a = (a_{ij})_{i,j \in {\Bbb Z}+\frac{1}{2}}|\
\text{all but a finite number of}\ a_{ij}\ \text{are}\ 0\}$$
%
with bracket $[a,b] = ab-ba$.  This Lie algebra has a
basis consisting of matrices $E_{ij},\ i,j \in {\Bbb Z} + \frac{1}{2}$, where
$E_{ij}$ is the matrix with a $1$ on the $(i,j)$-th entry and zeros
elsewhere. Now $gl_{\infty}$ is a subalgebra of the bigger Lie algebra
%
$$\overline{gl_{\infty}} = \{ a = (a_{ij})_{i,j \in {\Bbb Z}+\frac{1}{2}}
|\  a_{ij}=0\  \text{if}\   |i-j|>>0\}.$$
%
This Lie algebra $\overline{gl_{\infty}}$ has a universal central extension
$a_{\infty} := \overline{gl_{\infty}}\bigoplus {\Bbb C}c$
with Lie bracketdefined by
%
$$[a+\alpha c,b+\beta c]=ab-ba+\mu (a,b)c , \tag{1.1.1}$$
%
for $a,b\in \overline{gl_{\infty}}$ and $\alpha ,\beta\in{\Bbb C}$;
here $\mu$ is the following 2--cocycle:
%
$$\mu (E_{ij},E_{kl})=\delta_{il}\delta_{jk}(\theta(i)-\theta(j)),
\tag{1.1.2}$$
%
where the linear mapping $\theta :{\Bbb R}\to {\Bbb C}$ is defined by
%
$$\theta(i):=\cases 0 & \text{if}\ i>0,\\
		    1 & \text{if}\ i\le 0.\endcases \tag{1.1.3}$$
%
Let ${\Bbb C}^{\infty} = \bigoplus_{j \in {\Bbb Z}+\frac{1}{2}} {\Bbb
C} v_{j}$ be an infinite dimensional complex vector space with fixed
basis $\{ v_{j}\}_{j \in {\Bbb Z}+\frac{1}{2}}$.  Both the group
$GL_{\infty}$ and the Lie algebras $gl_{\infty}$ and $a_{\infty}$ act linearly
on
${\Bbb C}^{\infty}$ via the usual formula:
%
$$E_{ij} (v_{k}) = \delta_{jk} v_{i}.$$
%
We introduce, following [KP2], the corresponding semi-infinite wedge space $F =
\Lambda^{\frac{1}{2}\infty} {\Bbb C}^{\infty}$, this is the vector space
with a basis consisting of all semi-infinite monomials of the form
$v_{i_{1}} \wedge v_{i_{2}} \wedge v_{i_{3}} \ldots$, where $i_{1} >
i_{2} > i_{3} > \ldots$ and $i_{\ell +1} = i_{\ell} -1$ for $\ell >>
0$.  We can now define representations $R$ of $GL_{\infty}$ on $F$ by
%
$$
R(A) (v_{i_{1}} \wedge v_{i_{2}} \wedge v_{i_{3}} \wedge \cdots) = A
v_{i_{1}} \wedge Av_{i_{2}} \wedge Av_{i_{3}} \wedge \cdots . \tag{1.1.4}
$$
%
In order to describe representations of the Lie algebras we find it convenient
to define  wedging and contracting
operators $\psi^{-}_{j}$ and $\psi^{+}_{j}\ \ (j \in {\Bbb Z} +
\frac{1}{2})$ on $F$ by
%
$$\align
&\psi^{-}_{j} (v_{i_{1}} \wedge v_{i_{2}} \wedge \cdots ) = \cases 0
& \text{if}\ j = i_{s} \text{for some}\ s \\
(-1)^{s} v_{i_{1}} \wedge v_{i_{2}} \cdots \wedge v_{i_{s}} \wedge
v_{-j} \wedge v_{i_{s+1}} \wedge \cdots &\text{if}\ i_{s} > -j >
i_{s+1}\endcases \\
&\psi^{+}_{j} (v_{i_{1}} \wedge v_{i_{2}} \wedge \cdots ) = \cases 0
&\text{if}\ j \neq i_{s}\ \text{for all}\ s \\
(-1)^{s+1} v_{i_{1}} \wedge v_{i_{2}} \wedge \cdots \wedge
v_{i_{s-1}} \wedge v_{i_{s+1}} \wedge \cdots &\text{if}\ j = i_{s}.
\endcases
\endalign
$$
%
Notice that the definition of $\psi^{\pm}_{j}$ differs from the one in [KV].
The reason for this will become cclear in $\S$ 7 where we describe the
connection with the Sato Grassmannian.
These  wedging and contracting operators satisfy the following relations
$(i,j \in {\Bbb Z}+\frac{1}{2}, \lambda ,\mu = +,-)$:
%
$$\psi^{\lambda}_{i} \psi^{\mu}_{j} + \psi^{\mu}_{j}
\psi^{\lambda}_{i} = \delta_{\lambda ,-\mu} \delta_{i,-j}, \tag{1.1.5}$$
hence they generate a Clifford algebra, which we denote by ${\Cal C}\ell$.

Introduce the following elements of $F$ $(m \in {\Bbb Z})$:
%
$$|m\rangle = v_{m-\frac{1}{2} } \wedge v_{m-\frac{3}{2} } \wedge
v_{m-\frac{5}{2} } \wedge \cdots .$$
It is clear that $F$ is an irreducible ${\Cal C}\ell$-module such that
%
$$\psi^{\pm}_{j} |0\rangle = 0 \ \text{for}\ j > 0 . \tag{1.1.6}$$
%
We are now able to define representations $r$, $\hat r$ of $gl_{\infty}$,
$a_{\infty}$ on $F$ by
%
$$r(E_{ij})=\psi^-_{-i}\psi^+_j,\quad \hat
r(E_{ij})=:\psi^-_{-i}\psi^+_j:,\quad \hat r(c)=I,
$$
%
where $:\ :$ stands for the {\it normal ordered product} defined in
the usual way $(\lambda ,\mu = +$ or $-$):
%
$$:\psi^{\lambda (i)}_{k} \psi^{\mu (j)}_{\ell}: = \cases \psi^{
\lambda (i)}_{k}
\psi^{\mu (j)}_{\ell}\ &\text{if}\ \ell \ge k \\
-\psi^{\mu (j)}_{\ell} \psi^{\lambda (i)}_{k} &\text{if}\ \ell <
k.\endcases \tag{1.1.7}$$
%
\vskip 10pt
{\bf 1.2.} Define the {\it charge decomposition}
%
$$F = \bigoplus_{m \in {\Bbb Z}} F^{(m)} \tag{1.2.1}$$
by letting
%

$$\text{charge}(|0\rangle ) = 0\ \text{and charge} (\psi^{\pm}_{j}) =
\pm 1. \tag{1.2.2}$$
%
It  is easy to see that each $F^{(m)}$ is irreducible with
respect to $g\ell_{\infty}$, $a_{\infty}$ (and $GL_{\infty}$).  Note that
$|m\rangle$ is its highest weight vector, i.e.,
$\hat r(E_{ij})=r(E_{ij})-\delta_{ij}\theta(i)$ and
%
$$\align
&r(E_{ij})|m\rangle = 0 \ \text{for}\ i < j, \\
&r(E_{ii})|m\rangle = 0\  (\text{resp.}\ = |m\rangle ) \ \text{if}\ i > m\
(\text{resp. if}\ i < m).
\endalign
$$
%
Let
${\Cal O}
= R(GL_{\infty})|0\rangle \subset F^{(0)}$
be the $GL_{\infty}\text{-orbit}$
of the vacuum vector $|0\rangle$, then one has

\proclaim{Proposition 1.1 ([KP2])}  A non-zero element $\tau$ of $F^{(0)}$
lies in ${\Cal O}$ if and only if the following equation holds in $F \otimes
F$:
%
$$\sum_{k \in {\Bbb Z}+\frac{1}{2}} \psi^{+}_{k} \tau \otimes \psi^{-}_{-k}
\tau = 0. \tag{1.2.3}$$
\endproclaim

\demo{Proof}For a proof  see [KP2] or [KR].\ \ \ $\square$
\enddemo

Equation (1.2.3) is called the {\it KP hierarchy in the fermionic
picture}.

\vskip 10pt
\subheading{\S 2.  The loop algebra $\hat{gl}_n$, partitions of $n$ and vertex
operator constructions}

\vskip 10pt
{\bf 2.1}  Let $\tilde{gl}_n =gl_n({\Bbb C}[t,t^{-1}])$ be the loop algebra
associated to $gl_n({\Bbb C})$. This algebra has a natural representation
on the vector space $({\Bbb C}[t,t^{-1}])^n$. Let $\{ w_i\}$
 be the standard basis of ${\Bbb C}^n$, by identifying  $({\Bbb
 C}[t,t^{-1}])^n$ over ${\Bbb C}$ with ${\Bbb C}^{\infty}$ via
%
$v_{nk+j-\frac{1}{2}}=t^{-k}w_j$
%
we obtain an embedding $\phi :\tilde{gl}_n \to \overline{gl_{\infty}}$;
%
$$\phi (t^k e_{ij})=\sum_{\ell\in {\Bbb Z}} E_{n(\ell -k)+i-\frac{1}{2},
n\ell+j-\frac{1}{2}},$$
%
here $e_{ij}$ is a basis of $gl_n({\Bbb C})$.

A straightforward calculation shows that the restriction of the cocycle $\mu$
to $\phi (\tilde {gl}_n)$ induces the following 2--cocycle
 on
$\tilde{gl}_n$:
%
$$\mu (x(t),y(t))= \text{Res}_{t=0} dt\ \text{tr} ({dx(t)\over dt}y(t)).$$
Here and further $\text{Res}_{t=0} dt \sum_j f_jt^j$ stands for $f_{-1}$. This
gives a central extension $\hat{gl}_n=\tilde{gl}_n\bigoplus {\Bbb C}K$, where
the bracket is defined by
%
$$[t^\ell x+\alpha K,t^my+\beta K]=t^{\ell+m}(xy-yx)+\ell
\delta_{\ell,-m}\text{tr}(xy)K.$$
%
In this way we have an embedding $\phi :\hat{gl}_n\to a_{\infty}$, where
$\phi(K)=c$.

Since $F$ is a module for $a_{\infty}$, it is clear that with this
embedding we also have a representation of $\hat{gl}_n$
on this semi-infinite wedge space. It is well--known that the level one
representations of  the affine Kac--Moody algebra
$\hat{gl}_n$ have a lot of inequivalent realizations. To be more precise, Kac
and Peterson [KP1] and independently Lepowsky [L] showed
that to every conjugacy class of the Weyl group of $gl_n({\Bbb C})$
or rather $sl_n({\Bbb C})$ there exists an inequivalent vertex operator
realization of the same level one module. Hence to every partition
of $n$, there exists such a construction.

We will now sketch how one can  construct these vertex realizations of
$\hat{gl}_n$,
following [TV]. From now on let $n=n_1+ n_2+\cdots+n_s$ be a partition of $n$
into $s$ parts,  and denote by
$N_a=n_1+n_2+\cdots +n_{a-1}$.
We begin by relabeling the basis vectors $v_j$ and with them the corresponding
fermionic (wedging and contracting) operators:
($1\le a \le s,\ 1\le p\le n_a,\ j\in {\Bbb Z}$)
%
$$\align
v^{(a)}_{n_aj-p+\frac{1}{2}}&=v_{nj-N_a-p+\frac{1}{2}} \\
\psi^{\pm (a)}_{n_a \mp p\pm \frac{1}{2}}&=\psi^{\pm }_{n_aj\mp N_a\mp
p\pm\frac{1}{2}}
\tag{2.1.1}
\endalign$$
%
Notice that with this relabeling we have:
%
$\psi^{\pm (a)}_{k}|0\rangle = 0\ \text{for}\ k > 0.$
%
We also rewrite the $E_{ij}$'s:
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
$$E^{(ab)}_{n_aj-p+\frac{1}{2},n_bk-q+\frac{1}{2}}=E_{nj-N_a-p+\frac{1}{2},nk-N_b-q+\frac{1}{2}}.$$
%
The corresponding Lie bracket on $a_{\infty}$ is given by
%
$$[E^{(ab)}_{jk},E^{(cd)}_{\ell m}]=\delta_{bc}\delta_{kl}E^{(ad)}_{jm}-
\delta_{ad}\delta_{jm}E^{(db)}_{\ell
k}+\delta_{ad}\delta_{bc}\delta_{jm}\delta_{k\ell}(\theta (j)-\theta (k))c,$$
%
and $\hat r(E^{(ab)}_{jk})=:\psi^{-(a)}_{-j}\psi^{+b}_k:$.

Introduce the  fermionic fields $(z \in {\Bbb C}^{\times})$:
%
$$\psi^{\pm (a)}(z) \overset{\text{def}}\to{=} \sum_{k \in {\Bbb
Z}+\frac{1}{2}} \psi^{\pm
(a)}_{k} z^{-k-\frac{1}{2}}.\tag{2.1.2}$$
%
Let $N$ be the least common multiple of $n_1,n_2,\ldots,n_s$.
It was shown in [TV] that the modes of the fields
%
$$:\psi^{+(a)}(\omega_a^p z^{N\over n_a})\psi^{-(b)}(\omega_b^q z^{N\over
n_b}):, \tag{2.1.3}$$
for $1\le a,b\le s$, $1\le p\le n_a$, $1\le q\le n_b$, where $\omega_a=e^{2\pi
i/n_a}$, together with the identity,
generate a representation of $\hat{gl}_n$ with  $K=1$.

Next we introduce special bosonic fields ($1\le a\le s$):
%
$$\alpha^{(a)}(z) \equiv \sum_{k \in {\Bbb Z}} \alpha^{(a)}_{k} z^{-k-1}
\overset{def}\to{=} :\psi^{+(a)}(z) \psi^{-(a)}(z):. \tag{2.1.4}$$
%
The operators $\alpha^{(a)}_{k} $
satsify the canonical commutation relation of the associative
oscillator algebra,  which we
denote by ${\frak a}$:
%
$$[\alpha^{(i)}_{k},\alpha^{(j)}_{\ell}] =
k\delta_{ij}\delta_{k,-\ell},\tag{2.1.5}$$
and one has
%
$$\alpha^{(i)}_{k}|m\rangle = 0 \ \text{for}\ k > 0.\tag{2.1.6}$$
%
It is easy to see that restricted to $\hat{gl}_n$,
$F^{(0)}$ is its basic highest weight representation (see [K, Chapter
12]).

In order to express the fermionic fields $\psi^{\pm (i)}(z)$ in terms of
the bosonic fields $\alpha^{(i)}(z)$, we need some additional operators
$Q_{i},\ i = 1,\ldots ,s$, on $F$.  These operators are uniquely defined by
the following conditions:
%
$$Q_{i}|0\rangle = \psi^{+(i)}_{-\frac{1}{2}} |0\rangle ,\ Q_{i}\psi^{\pm
(j)}_{k} = (-1)^{\delta_{ij}+1} \psi^{\pm
(j)}_{k\mp \delta_{ij}}Q_{i}.\tag{2.1.7}$$
They satisfy the following commutation relations:
%
$$Q_{i}Q_{j} = -Q_{j}Q_{i}\ \text{if}\ i \neq j,\ [\alpha^{(i)}_{k},Q_{j}] =
\delta_{ij} \delta_{k0}Q_{j}.\tag{2.1.8}$$

\proclaim{Theorem 2.1}  ([DJKM1], [JM])
$$\psi^{\pm (i)}(z) = Q^{\pm 1}_{i}z^{\pm \alpha^{(i)}_{0}} \exp
(\mp \sum_{k < 0} \frac{1}{k} \alpha^{(i)}_{k}z^{-k})\exp(\mp
\sum_{k > 0} \frac{1}{k} \alpha^{(i)}_{k} z^{-k}). \tag{2.1.9}$$
\endproclaim

\demo{Proof} See [TV].
\enddemo

The operators on the right-hand side of (2.1.9) are called vertex
operators.  They made their first appearance in string theory (cf.
[FK]).

If one substitutes (2.1.9) into (2.1.3), one obtains the vertex operator
realization of $\hat{gl}_n$ which is related to the partition
$n=n_1+n_2+\cdots+n_s$ (see [TV] for more details).

\vskip 10pt
{\bf 2.2.} The realization  of $\hat{gl}_n$, described in the previous section,
has a natural Virasoro algebra. In [TV], it was shown that the following two
sets of operators have the same action on $F$.
%
$$\align
&L_k=\sum_{i=1}^s\{ \sum_{j\in{\Bbb Z}}{1\over
2n_i}:\alpha_{-j}^{(i)}\alpha_{j+n_ik}^{(i)}:+\delta_{k0}{{n_i^2-1}\over
24n_i}\},\tag{2.2.1} \\
&H_k=\sum_{i=1}^s\{\sum_{j\in{\Bbb Z}+\frac{1}{2}}({j\over n_i}+{k\over
2}):\psi^{+(i)}_{-j}\psi^{-(i)}_{j+n_ik}:+\delta_{k0}{{n_i^2-1}\over
24n_i}\}, \tag{2.2.2} \endalign
$$
%
So $L_k=H_k$,
$$[L_k,\psi_j^{\pm (i)}]=-(\frac{j}{n_i}+\frac{k}{2})\psi_{j+n_ik}^{\pm(i)}
\tag{2.2.3}
$$
 and
%
$$[L_k,L_\ell ]=(k-\ell)L_{k+\ell}+\delta_{k,-\ell}{{k^3-k}\over 12}n. $$

\vskip 10pt

{\bf 2.3.} We will now use the results of \S 2.1 to
describe  the $s$-component boson-fermion
correspondence.  Let ${\Bbb C}[x]$ be the space of polynomials in
indeterminates $x = \{ x^{(i)}_{k}\},\ k = 1,2,\ldots ,\ i =
1,2,\ldots ,s$.  Let $L$ be a lattice with a basis $\delta_{1},\ldots
,\delta_{s}$ over ${\Bbb Z}$ and the symmetric bilinear form
$(\delta_{i}|\delta_{j}) = \delta_{ij}$, where $\delta_{ij}$ is the
Kronecker symbol.  Let
%
$$\varepsilon_{ij} = \cases -1 &\text{if $i > j$} \\
1 &\text{if $i \leq j$.} \endcases \tag{2.3.1}$$
%
Define a bimultiplicative function $\varepsilon :\ L \times L @>>> \{
\pm 1 \}$ by letting
%
$$\varepsilon (\delta_{i}, \delta_{j}) = \varepsilon_{ij}.
\tag{2.3.2}$$
Let $\delta = \delta_{1} + \ldots + \delta_{s},\  Q= \{ \gamma \in
L|\ (\delta | \gamma ) = 0\}$, $\Delta = \{ \alpha_{ij} :=
\delta_{i}-\delta_{j}| i,j = 1,\ldots ,s,\ i \neq j \}$.  Of course
$Q$ is the root lattice of $sl_{s}({\Bbb C})$, the set $\Delta$
being the root system.

Consider the vector space ${\Bbb C}[L]$ with basis $e^{\gamma}$,\
$\gamma \in L$, and the following twisted group algebra product:
%
$$e^{\alpha}e^{\beta} = \varepsilon (\alpha ,\beta)e^{\alpha +
\beta}. \tag{2.3.3}$$
Let $B = {\Bbb C}[x] \otimes_{\Bbb C} {\Bbb C}[L]$ be the tensor
product of algebras.  Then the $s$-component boson-fermion
correspondence is the vector space isomorphism
%
$$\sigma :F @>\sim >> B, \tag{2.3.4}$$
given by
%
$$\sigma (\alpha^{(i_{1})}_{-m_{1}} \ldots
\alpha^{(i_{r})}_{-m_{r}}Q_1^{k_{1}}\ldots Q_s^{k_{s}}|0 \rangle ) = m_{1}
\ldots
m_{s}x^{(i_{1})}_{m_{1}} \ldots x^{(i_{r})}_{m_{r}} \otimes
e^{k_{1}\delta_{1} + \ldots + k_{s}\delta_{s}} . \tag{2.3.5}$$
%
The transported charge then will be as follows:
%
$$
\text{charge}(p(x)\otimes e^{\gamma}) = (\delta |\gamma).
\tag{2.3.6}
$$
We denote the transported charge decomposition by
%
$$B = \bigoplus_{m \in {\Bbb Z}} B^{(m)}.$$


The transported action of the operators $\alpha^{(i)}_{m}$ and $Q_{j}$ looks
as follows:
%
$$\cases
\sigma \alpha^{(j)}_{-m}\sigma^{-1}(p(x) \otimes e^{\gamma}) =
mx^{(j)}_{m}p(x)\otimes e^{\gamma},\ \text{if}\ m > 0, &\  \\
\sigma \alpha^{(j)}_{m} \sigma^{-1}(p(x) \otimes e^{\gamma}) = \frac{\partial
p(x)}{\partial x_{m}} \otimes e^{\gamma},\ \text{if}\ m > 0, &\  \\
\sigma \alpha^{(j)}_{0} \sigma^{-1} (p(x) \otimes e^{\gamma}) =
(\delta_{j}|\gamma ) p(x) \otimes e^{\gamma} , &\ \\
\sigma Q_{j} \sigma^{-1} (p(x) \otimes e^{\gamma}) = \varepsilon
(\delta_{j},\gamma)  p(x) \otimes e^{\gamma + \delta_{j}}
. & \
\endcases \tag{2.3.7}
$$
%
For notational conveniences, we introduce
$\delta_j=\sigma\alpha_0^{(j)}\sigma^{-1}$.
Notice that $e^{\delta_j}=\sigma Q_j\sigma^{-1}$.
\vskip 10pt
{\bf 2.4.}  Using the isomorphism $\sigma$ we can reformulate the KP hierarchy
(1.2.3) in the bosonic picture.

We start by observing that (1.2.3) can be rewritten as follows:
%
$$\text{Res}_{z=0}\ dz ( \sum^{s}_{j=1} \psi^{+(j)}(z)\tau
\otimes \psi^{-(j)}(z)\tau ) = 0,\ \tau \in F^{(0)}.
\tag{2.4.1}$$
%
Notice that for $\tau \in F^{(0)},\ \sigma (\tau) = \sum_{\gamma \in Q}
\tau_{\gamma}(x)e^{\gamma}$.
 Here and further  we write $\tau_{\gamma}(x)e^{\gamma}$ for
$\tau_{\gamma} \otimes
e^{\gamma}$.  Using Theorem 2.1, equation (2.4.1) turns under $\sigma
\otimes \sigma :\ F \otimes F \overset\,\,\sim\to\longrightarrow
{\Bbb C}[x^{\prime},x^{\prime \prime}]
\otimes ({\Bbb C}[L^{\prime}] \otimes {\Bbb C}[L^{\prime \prime}])$ into the
following set of equations;
 for all $\alpha ,\beta \in L$ such that $(\alpha
|\delta ) = -(\beta |\delta ) = 1$ we have:
%
$$\aligned
&\text{Res}_{z=0} ( dz
 \sum^{s}_{j=1} \varepsilon (\delta_{j}, \alpha-\beta)
z^{(\delta_{j}|\alpha - \beta - 2\delta_{j})}  \\
 &\times \exp
(\sum^{\infty}_{k=1} (x^{(j)^{\prime}}_{k} - x^{(j)^{\prime
\prime}}_{k})z^{k})
\exp (-\sum^{\infty}_{k=1} (\frac{\partial}{\partial x^{(j)^{\prime}}_{k}}
 - \frac {\partial}{\partial x^{(j)^{\prime
\prime}}_{k}})\frac{z^{-k}}{k}) \\
& \tau_{\alpha -
\delta_{j}}(x^{\prime})(e^{\alpha})^{\prime}\tau_{\beta + \delta_{j}}(x^{\prime
\prime})(e^{\beta})^{\prime\prime})
= 0  . \endaligned \tag{2.4.2}$$




\vskip 10pt
\subheading{\S 3. The algebra of formal pseudo-differential operators and the
$s$-component KP hierarchy as a dynamical system}

\vskip 10pt
{\bf 3.0.} The KP hierarchy and its $s$-component generalizations admit
several  formulations.  The one we will give here was  introduced by Sato [S],
it is given in the
language of formal pseudo-differential operators.  We will show that this
formulation follows from the $\tau$-function formulation given by
equation (2.4.2).

\vskip 10pt
{\bf 3.1.} We shall work over the algebra ${\Cal A}$ of formal power
series over ${\Bbb C}$ in indeterminates $x = (x^{(j)}_{k})$, where
$k = 1,2,\ldots $ and $j = 1,\ldots ,s$.  The indeterminates
$x^{(1)}_{1},\ldots ,x^{(s)}_{1}$ will be viewed as variables and
$x^{(j)}_{k}$ with $k \geq 2$ as parameters.  Let
%
$$\partial = \frac{\partial}{\partial x^{(1)}_{1}} + \ldots +
\frac{\partial}{\partial x^{(s)}_{1}}.$$
{\it A formal} $s \times s$ {\it matrix pseudo-differential
operator} is an expression of the form
%
$$P(x,\partial ) = \sum_{j \leq N} P_{j}(x)\partial^{j}, \tag{3.1.1}$$
where $P_{j}$ are $s \times s$ matrices over ${\Cal A}$.   Let $\Psi$ denote
the vector
space over ${\Bbb C}$ of all expressions (3.1.1).  We have a linear
isomorphism $S :\ \Psi @>>> \text{Mat}_{s}({\Cal A}((z)))$ given by
$S(P(x,\partial )) = P(x,z)$.  The matrix series
$P(x,z)$ in indeterminates $x$ and $z$ is called the {\it symbol} of
$P(x,\partial )$.

Now we may define a product $\circ$ on $\Psi$ making it an associative
algebra:
%
$$S(P \circ Q) = \sum^{\infty}_{n = 0} \frac{1}{n!}
\frac{\partial^{n}S(P)}{\partial z^{n}} \partial^{n}S(Q).\tag{3.1.2}$$
{}From now on, we shall drop the multiplication sign $\circ$ when no ambiguity
may arise.
One defines the differential part of $P(x,\partial )$ by
$P_{+}(x,\partial ) = \sum^{N}_{j=0} P_{j}(x) \partial^{j},$
and let $P_{-} = P-P_{+}$.  We have the corresponding vector space
decomposition:
%
$$\Psi = \Psi_{-} \oplus \Psi_{+}. \tag{3.1.3}$$
%
One defines a linear map $*: \Psi \rightarrow \Psi$ by the
following formula:
$$(\sum_{j} P_{j}\partial^{j})^{*} = \sum_{j} (-\partial )^{j}
\circ ^{t}\! P_{j}. \tag{3.1.4}$$
Here and further $^{t}P$ stands for the transpose of the matrix $P$.
Note that $*$ is an anti-involution of the algebra
$\Psi$.


\vskip 10pt
{\bf 3.2.}  Introduce the following notation
%
$$z \cdot x^{(j)} = \sum^{\infty}_{k=1} x^{(j)}_{k} z^{k},\ e^{z
\cdot x} = diag (e^{z\cdot x^{(1)}} ,\ldots ,e^{z\cdot x^{(s)}} ).$$
The algebra $\Psi$ acts on the space $U_{+}$ (resp.
$U_{-}$) of formal oscillating matrix functions of the form
%
$$\sum_{j \leq N} P_{j}z^{j}e^{z\cdot x}\ \ \ (\text{resp.}\ \sum_{j
\leq N} P_{j}z^{j} e^{-z\cdot x}), \ \text{where}\ \ P_{j} \in \
\text{Mat}_{s} ( {\Cal A} ),$$
in the obvious way:
%
$$P(x)\partial^{j}e^{\pm z\cdot x} = P(x)(\pm z)^{j} e^{\pm z\cdot x}.$$
One has the following fundamental lemma (see [KV]).

\proclaim{Lemma 3.1} If $P,Q \in \Psi$ are such that
%
$$Res_{z=0} (P(x,\partial) e^{z\cdot x})  \ ^{t}
(Q(x^{\prime},\partial^{\prime}) e^{-z\cdot x^{\prime}})  dz = 0,
\tag{3.2.1}$$
then $(P \circ Q^{*})_{-} = 0$.
\endproclaim

\vskip 10pt
{\bf 3.3.}  We proceed now to rewrite the formulation (2.4.2) of the
$s$-component KP hierarchy in terms of formal pseudo-differential
operators.

For each $\alpha \in \ \text{supp}\ \tau :=\{\alpha\in Q| \tau=\sum_{\alpha\in
Q}\tau_{\alpha}e^{\alpha}, \tau_{\alpha}\ne 0\}$ we define the (matrix
valued) functions
$$V^{\pm} (\alpha ,x,z) = (V^{\pm}_{ij}(\alpha ,x,z))^{s}_{i,j=1}
\tag{3.3.1}$$
as follows:
%
$$\aligned
&V^{\pm}_{ij}(\alpha ,x,z) \overset{\text{def}}\to{=}
\varepsilon (\delta_{j} , \alpha + \delta_{i})
 z^{(\delta_{j}|\pm \alpha + \alpha_{ij})} \\
& \times \exp (\pm \sum^{\infty}_{k=1} x^{(j)}_{k} z^{k})
\exp(\mp \sum^{\infty}_{k=1} \frac{\partial}{\partial
x^{(j)}_{k}} \frac{z^{-k}}{k}) \tau_{\alpha  \pm
\alpha_{ij}}  (x)/\tau_{\alpha}(x) .
\endaligned \tag{3.3.2}
$$
It is easy to see that equation (2.4.2) is equivalent to the
following bilinear identity:
%
$$Res_{z=0}V^{+}(\alpha ,x,z)\ ^{t}V^{-}(\beta ,x^{\prime},z)dz = 0\
\text{for all}\ \alpha ,\beta \in Q. \tag{3.3.3}$$
%
Define $s \times s$ matrices $W^{\pm (m)} (\alpha ,x)$ by the
following generating series (cf. (3.3.2)):
%
$$
\sum^{\infty}_{m=0}
W^{\pm (m)}_{ij} (\alpha ,x)(\pm z)^{-m}
= \varepsilon_{ji}z^{\delta_{ij}-1} (\exp \mp
\sum^{\infty}_{k=1} \frac{\partial}{\partial x^{(j)}_{k}}\frac{z^{-k}}{k})
\tau_{\alpha \pm
\alpha_{ij}} (x))/\tau_{\alpha} (x). \tag{3.3.4}
$$

We see from (3.3.2) that $V^{\pm}(\alpha ,x,z)$ can be written in the
following form:
%
$$V^{\pm}(\alpha ,x,z) = (\sum^{\infty}_{m=0}
W^{\pm (m)}(\alpha ,x)R^{\pm}(\alpha ,\pm z)(\pm
z)^{-m})e^{\pm z \cdot x}, \tag{3.3.5}$$
where
%
$$R^{\pm}(\alpha ,z) = \sum^{s}_{i=1}
\varepsilon (\delta_{i}, \alpha ) E_{ii} (\pm z)^{\pm
(\delta_{i}|\alpha )}. \tag{3.3.6}$$
Here and further $E_{ij}$ stands for the $s \times s$ matrix whose
$(i,j)$ entry is $1$ and all other entries are zero.  Now it is clear
that $V^{\pm}(\alpha ,x,z)$ can be
written in
terms of formal pseudo-differential operators
%
$$P^{\pm}(\alpha ) \equiv P^{\pm} (\alpha ,x,\partial ) =
I_{n} + \sum^{\infty}_{m=1} W^{\pm (m)} (\alpha ,x)\partial^{-m}\
\text{and}\ R^{\pm}(\alpha ) = R^{\pm}(\alpha ,\partial) \tag{3.3.7}$$
as follows:
$$V^{\pm}(\alpha ,x,z) = P^{\pm } (\alpha )
R^{\pm}(\alpha)e^{\pm z \cdot x} . \tag{3.3.8}$$
Since obviously
$R^{-}(\alpha ,\partial)^{-1} = R^{+}(\alpha
,\partial)^{*}$,
using Lemma 3.1 we deduce from the bilinear identity
(3.3.3):
$$\align
&P^{-}(\alpha  ) = (P^{+}(\alpha  )^{*})^{-1} ,
\tag{3.3.9}\\
&(P^{+}(\alpha)R^{+}(\alpha - \beta)P^{+}(\beta)^{-1})_{-} = 0\
\text{for all}\ \alpha ,\beta \in \text{supp}\ \tau . \tag{3.3.10}
\endalign$$

Victor Kac and the author showed in [KV] that
given $\beta \in \text{supp}\ \tau$, all the
pseudo-differential operators $P^+(\alpha )$, $\alpha \in \text{supp}\
\tau$, are
completely determined by $P^+(\beta )$ from equations (3.3.10). They also
showed that $P=P^+(\alpha)$ satisfies the Sato equation:
%
$$
\frac{\partial P}{\partial x^{(j)}_{k}} = -(PE_{jj} \circ
\partial^{k} \circ P^{-1})_{-} \circ P. \tag{3.3.11}$$
%
To be more precise, one has the following

\proclaim{Proposition 3.2}  Consider the formal oscillating functions
$V^{+}(\alpha ,x,z)$ and $V^{-}(\alpha ,x,z),$ \newline $\alpha \in Q$, of the
form (3.3.8), where $R^{\pm}(\alpha ,z)$ are given by (3.3.6) and
$P^{\pm}(\alpha ,x,\partial ) \in I_{s} + \Psi_{-}$.  Then the
bilinear identity (3.3.3) for all $\alpha ,\beta \in \ \text{supp}\
\tau$ is equivalent to the Sato equation (3.3.11) for each $P =
P^{+}(\alpha )$ and the matching conditions (3.3.9-10)  for
all $\alpha ,\beta \in \ \text{supp} \ \tau$.
\endproclaim

\vskip 10pt
{\bf 3.4.}  Fix $\alpha \in Q$, introduce the following formal
pseudo-differential
operators $L(\alpha),\  C^{(j)}(\alpha )$, and
differential operators  $B^{(j)}_{m}(\alpha)$:
$$\aligned
L \equiv L(\alpha )
  & = P^{+}(\alpha) \circ \partial \circ P^{+}(\alpha)^{-1}, \\
C^{(j)} \equiv C^{(j)}(\alpha ) &=
P^{+}(\alpha)E_{jj} P^{+}(\alpha)^{-1}, \\
B^{(j)}_{m} \equiv B^{(j)}_{m}(\alpha) &=
(P^{+}(\alpha)E_{jj} \circ \partial^{m} \circ
P^{+}(\alpha)^{-1})_{+}.
\endaligned \tag{3.4.1}
$$
%
Then
%
$$\aligned
L &= I_{s} \partial + \sum^{\infty}_{j=1} U^{(j)}(x) \partial^{-j} , \\
C^{(i)} &= E_{ii} + \sum^{\infty}_{j=1} C^{(i,j)}(x)
\partial^{-j},\ i = 1,2,\cdots ,s,
\endaligned \tag{3.4.2}$$
subject to the conditions
%
$$
\sum^{s}_{i=1} C^{(i)} = I_{s}, \
C^{(i)}L = LC^{(i)},\ C^{(i)}C^{(j)} = \delta_{ij} C^{(i)}.
 \tag{3.4.3}$$
They satisfy the following set of equations for some $P \in I_{n} + \Psi_{-}$:
%
$$
\cases LP = P\partial &\ \\
C^{(i)}P = PE_{ii} &\ \\
\displaystyle{\frac{\partial P}{\partial x^{(i)}_{k}} = -(L^{(i)k})_{-} P,\
\text{where}\  L^{(i)} = C^{(i)}L.}
\endcases
\tag{3.4.4}$$

\proclaim{Proposition 3.3}  The system of equations (3.4.4) has a
solution $P \in I_{s} + \Psi_{-}$ if and only if we can find a formal
oscillating function of the form
%
$$W(x,z) = (I_{s} + \sum^{\infty}_{j=1} W^{(j)}(x)z^{-j})e^{z\cdot x}
\tag{3.4.5}$$
that satisfies the linear equations
%
$$
LW = zW ,\
C^{(i)}W = WE_{ii} , \
\frac{\partial W}{\partial x^{(i)}_{k}} = B^{(i)}_{k}W .
 \tag{3.4.6}$$
\endproclaim
%
%
And finally, one has the following
\proclaim{Proposition 3.4} If for every $\alpha \in Q$ the
formal pseudo-differential operators $L \equiv L(\alpha)$ and $C^{(j)} \equiv
C^{(j)}(\alpha)$ of the form (3.4.2) satisfy conditions (3.4.3) and if the
equations (3.4.4) have a solution $P \equiv P(\alpha) \in I_{s} +
\Psi_{-}$, then the
differential operators $B^{(j)}_{k} \equiv B^{(j)}_{k}(\alpha)$
satisfy one of the following equivalent
conditions:
%
$$\left\{ \matrix \displaystyle{\frac{\partial L}{\partial x^{(j)}_{k}} =
[B^{(j)}_{k},L],}  & \   \\
\displaystyle{\frac{\partial C^{(i)}}{\partial x^{(j)}_{k}} =
[B^{(j)}_{k},C^{(i)}],} & \
\  \endmatrix \right. \tag{3.4.7}$$
%
$$\frac{\partial L^{(i)}}{\partial x^{(j)}_{k}} =
[B^{(j)}_{k},L^{(i)}], \tag{3.4.8}$$
%
$$\frac{\partial B^{(i)}_{\ell}}{\partial x^{(j)}_{k}} -
\frac{\partial B^{(j)}_{k}}{\partial x^{(i)}_{\ell}} =
[B^{(j)}_{k},B^{(i)}_{\ell}]. \tag{3.4.9}$$
%
Here $L^{(j)}\equiv L^{(j)}(\alpha)=C^{(j)}(\alpha)\circ L(\alpha)$.
\endproclaim
Equations (3.4.7) and (3.4.8) are called {\it Lax type} equations.
Equations (3.4.9) are called the {\it Zakharov-Shabat type}
equations.  The latter
are the compatibility conditions for the linear problem (3.4.6).

\vskip 10pt
\subheading{\S4 . $[n_1,n_2,\ldots ,n_s]$-reductions of the $s$-component KP
hierarchy}

\vskip 10pt
{\bf 4.1.} Using (2.1.9), (2.1.3), (2.3.5) and (2.3.7),
we obtain the to the partition
$n=n_1+n_2+\cdots +n_s$ related vertex operator realization of
$\hat{gl}_n$ in the vector space $B^{(m)}$.
Now, restricted to $\hat{sl}_n$, the representation in $F^{(m)}$ is not
irreducible anymore, since $\hat{sl}_n$ commutes with the operators
%
$$\beta_{kn_s}^{(s)}=\sqrt{n_s\over N}\sum_{i=1}^s \alpha_{kn_i}^{(i)},\quad
k\in {\Bbb Z}.
\tag{4.1.1}$$
%
In order to describe the irreducible part of the representation of $\hat{sl}_n$
in $B^{(0)}$ containing the vacuum vector 1, we choose the complementary
generators of the oscillator algebra $\frak a$
contained in $\hat{sl}_n$ ($k\in {\Bbb Z}$):
%
$$\beta^{(j)}_{k} = \cases \alpha^{(j)}_{k} &\text{if $k \notin
n_j{\Bbb Z}$,} \\
{{N_{j+1}\alpha_{\ell n_{j+1}}^{(j+1)}-n_{j+1}(\alpha_{\ell n_1}^{(1)}
+\alpha_{\ell n_2}^{(2)}+\cdots +\alpha_{\ell n_j}^{(j)})}\over
\sqrt{N_{j+1}(N_{j+1}-n_{j+1})}}&\text{if $k=\ell n_j$ and
$1 \leq j < s$,}
\endcases \tag{4.1.2}$$
%
so that the operators (4.1.1 and 2) also satisfy relations (2.1.5). Hence,
introducing the new indeterminates
%
$$y^{(j)}_{k} = \cases x^{(j)}_{k} &\text{if $k \notin n_j{\Bbb N}$}, \\
{{N_{j+1}x_{\ell n_{j+1}}^{(j+1)}-(n_1x_{\ell n_1}^{(1)}+n_2x_{\ell
n_2}^{(2)}+\cdots +n_jx_{\ell n_j}^{(j)})}\over
\sqrt{N_{j+1}(N_{j+1}-n_{j+1})}}&\text{if $k=\ell n_j$ and $1 \leq j < s$,}\\
{{n_1x_{\ell n_1}^{(1)}+n_2x_{\ell n_2}^{(2)}+\cdots +n_sx_{\ell
n_s}^{(s)}}\over \sqrt{Nn_s}} &\text{if $k=\ell n_s$ and $j=s$},
\endcases \tag{4.1.3} $$
we have: ${\Bbb C}[x] = {\Bbb C}[y]$ and
%
$$\sigma (\beta^{(j)}_{k}) = \frac{\partial}{\partial y^{(j)}_{k}}\
\text{and}\ \sigma (\beta^{(j)}_{-k}) = ky^{(j)}_{k} \ \text{if}\ k
> 0. \tag{4.1.4}$$
Now it is clear that the irreducible with respect to
$\hat{sl}_n$ subspace of $B^{(0)}$ containing the
vacuum $1$ is the vector space
%
$$B^{(0)}_{[n_1,n_2,\ldots ,n_s]} = {\Bbb C}[y^{(j)}_{k}|1 \leq j < s,\ k \in
{\Bbb
N},\ \text{or}\ j = s,\ k \in {\Bbb N}\backslash n_s{\Bbb Z}] \otimes
{\Bbb C}[Q]. \tag{4.1.5}$$
The vertex operator realization of $\hat{sl}_n$ in
the vector space $B^{(0)}_{[n_1,n_2,\ldots ,n_s]}$ is then obtained
by expressing the
fields (2.1.3) in terms of vertex
operators (2.1.9), which are expressed via (4.1.2) in
the operators (4.1.4), the
operators $e^{\delta_i -\delta_j}$ and $\delta_i-\delta_j\
(1 \leq i < j \leq s)$ (see [TV] for details).

The $s$-component KP hierarchy of  equations  (2.4.2) on $\tau
\in B^{(0)} = {\Bbb C}[y] \otimes {\Bbb C}[Q]$ when restricted to
$\tau \in B^{(0)}_{[n_1,n_2,\ldots ,n_s]}$ is called the {\it $[n_1,n_2,\ldots
,n_s]$-th reduced} KP {\it
hierarchy}.  It is obtained from the $s$-component KP hierarchy by
making the change of variables (4.1.3) and putting zero all terms
containing partial derivates by
$y^{(s)}_{n_s},y^{(s)}_{2n_s},y^{(s)}_{3n_s},\ldots$.

The totality of solutions of the $[n_1,n_2,\ldots ,n_s]$-th reduced KP
hierarchy is given
by the following

\proclaim{Proposition 4.1}  Let ${\Cal O}_{[n_1,n_2,\ldots ,n_s]}$ be the orbit
of $1$
under the (projective) representation of the loop group
$SL_{n}({\Bbb C}[t,t^{-1}])$ corresponding to the representation of
$\hat{sl}_n$ in $B^{(0)}_{[n_1,n_2,\ldots ,n_s]}$.  Then
%
$${\Cal O}_{[n_1,n_2,\ldots ,n_s]} = \sigma ({\Cal O}) \cap
B^{(0)}_{[n_1,n_2,\ldots ,n_s]}.$$
In other words, the $\tau$-functions of the $[n_1,n_2,\ldots ,n_s]$-th reduced
KP hierarchy are
precisely the $\tau$-functions of the KP hierarchy in the variables
$y^{(j)}_{k}$, which are independent of the variables
$y^{(s)}_{\ell n_s},\ \ell \in {\Bbb N}$.
\endproclaim

\demo\nofrills{Proof} \ is the same as of a similar statement in [KP2].
\ \ \ $\square$
\enddemo

\vskip 10pt
{\bf 4.2.} It is clear from the definitions and results of \S 4.1 that the
condition on the $s$-component KP hierarchy to be $[n_1,n_2,\ldots ,n_s]$-th
reduced
is equivalent to
%
$$\sum^{s}_{j=1} \frac{\partial \tau}{\partial x^{(j)}_{kn_j}}
= 0,\quad\text{for all $k\in {\Bbb N}$}\tag{4.2.1}$$
Using the Sato equation (3.3.11), this
implies the
following  two equivalent conditions:
%
%
$$\sum^{s}_{j=1} \frac{\partial W(\alpha )}{\partial x^{(j)}_{kn_j}} =
W(\alpha)\sum_{j=1}^sz^{kn_j}E_{jj}, \tag{4.2.2}$$
%
$$(\sum_{j=1}^sL(\alpha)^{kn_j}C^{(j)})_{-}=0.\tag{4.2.3}$$

\vskip 10pt
\subheading{\S 5. The string equation}

\vskip 10pt
{\bf 5.1.} {\it From now on we assume that $\tau$ is any solution of the KP
hierarchy.} In particular, we no longer assume that $\tau_{\alpha}$ is a
polynomial. For instance the soliton and dromion solutions of [KV, \S 5] are
allowed. Of course this means that the corresponding wave functions
$V^{\pm}(\alpha ,z)$ will be of a more general nature than before.

Recall from \S 3 the wave function $V(\alpha,z)\equiv
V^+(\alpha,z)=P(\alpha)R(\alpha)e^{z\cdot x}=$\break
$
P^+(\alpha)R^+(\alpha)e^{z\cdot x}$. It is natural to compute
%
$$\align
{{\partial V(\alpha,z)}\over \partial z}
&={\partial\over\partial z}P(\alpha)R(\alpha)e^{z\cdot x}\\
&=P(\alpha)R(\alpha){\partial\over\partial z}e^{z\cdot x}\\
&=P(\alpha)R(\alpha)\sum_{a=1}^s\sum_{k=1}^{\infty} kx_k^{(a)}
\partial^{k-1}E_{aa}R(\alpha)^{-1}P(\alpha)^{-1}V(\alpha,z).
\endalign$$
Define
%
$$M(\alpha):=P(\alpha)R(\alpha)\sum_{a=1}^s\sum_{k=1}^{\infty} kx_k^{(a)}
\partial^{k-1}E_{aa}R(\alpha)^{-1}P(\alpha)^{-1},\tag{5.1.1}$$
%
then one easily checks that $[L(\alpha),M(\alpha)]=1$ and
%
$$[\sum_{a=1}^s L(\alpha)^{n_a}C^{(a)}(\alpha), M(\alpha)\sum_{a=1}^s {1\over
n_a}L(\alpha)^{1-n_a}C^{(a)}(\alpha)]=1.\tag{5.1.2}$$
%

Next, we calculate the $(i,j)$--th coefficient of $(M(\alpha)\sum_{a} {1\over
n_a}L(\alpha)^{1-n_a}C^{(a)}(\alpha))_{-}P(\alpha)R(\alpha)$.
Let $P=S(P(\alpha))$ and $R=S(R(\alpha))$, then
$$\align
&S((M(\alpha)\sum_{a=1}^s{1 \over n_a}L(\alpha)^{1-n_a}C^{(a)}(\alpha))_-
P(\alpha)R(\alpha))_{ij})= \\
&S(P(\alpha)R(\alpha)\sum_{a=1}^s\sum_{k\in{\Bbb N}}{1\over n_a
}kx_k^{(a)}\partial^{k-n_a}
E_{aa}R(\alpha)^{-1}P(\alpha)^{-1})_-P(\alpha)R(\alpha))_{ij})= \\
&({{\partial PR}\over\partial z}\sum_{a=1}^s{1\over n_a}E_{aa}z^{1-n_a}+
\sum_{a=1}^s\{
 \sum_{k=1}^{n_a}{k\over n_a}x_k^{(a)}PRE_{aa}z^{k-n_a}
  -x^{(a)}_{n_a}E_{aa}PR \\
  &\qquad -\sum_{k=1}^{\infty} {{k+n_a}\over n_a}x^{(a)}_{k+n_a}{{\partial
  PR}\over \partial x_k^{(a)}}
\})_{ij}
\endalign$$
%
Define
$$\align
\tilde\tau_{\alpha+\delta_i-\delta_j}&=\exp(-\sum_{k=1}^{\infty}
{\partial\over\partial x_k^{(j)}}{{z^{-k}}\over
k})\tau_{\alpha+\delta_i-\delta_j}
(x)\\
&=\tau_{\alpha+\delta_i-\delta_j}(\ldots,x_k^{(b)}-{{\delta_{jb}}\over
kz^k},\ldots),
\endalign
$$
%
then
$$(PR)_{ij}=\epsilon(\delta_j |\alpha+\delta_i)z^{\delta_{ij}-1+(\delta_j
|\alpha)}
{\tilde\tau_{\alpha+\delta_i-\delta_j}\over \tau_{\alpha}}
$$
%
and hence
$$
\align
&S((M(\alpha)\sum_{a=1}^s{1\over
n_a}L(\alpha)^{1-n_a}C^{(a)}(\alpha))_-P(\alpha)R(\alpha))_{ij})=
{{\epsilon(\delta_j|\alpha+\delta_i)}\over n_j}\times \\
&\{ {1\over\tau_{\alpha}}(\sum_{k=1}^{\infty}{\partial\over\partial x_k^{(j)}}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
z^{-k-n_j}+(\delta_{ij}-1+(\delta_j|\alpha))z^{-n_j})\tilde\tau_{\alpha+\delta_i-\delta_j}
+\sum_{k=1}^{n_j} kx_k^{(j)}{{\tilde{\tau}_{\alpha+\delta_i-\delta_j}}\over
\tau_{\alpha}}z^{k-n_j} \\
&-n_jx^{(i)}_{n_i}{{\tilde{\tau}_{\alpha+\delta_i-\delta_j}}\over\tau_{\alpha}}
-\sum_{a=1}^s {{n_j}\over n_a}\sum_{k=1}^{\infty} (k+n_a)x_{k+n_a}^{(a)}
{\partial\over \partial
x_k^{(a)}}({{\tilde{\tau}_{\alpha+\delta_i-\delta_j}}\over \tau_{\alpha}})
\} z^{\delta_{ij}-1+(\delta_j|\alpha)}.
\tag{5.1.3}
\endalign
$$
%
\vskip 10pt
{\bf 5.2.} We introduce the natural generalization of the string equation
(0.3).
Let $L_{-1}$ be given by (2.2.1), the string equation is the following
constraint
on $\tau\in F^{(0)}$:
%
$$L_{-1}\tau=0. \tag{5.2.1}$$
%
Using (2.3.7) we rewrite $L_{-1}$ in terms of operators on $B^{(0)}$:
%
$$L_{-1}=\sum_{a=1}^s \{
\delta_a x_{n_a}^{(a)}+
{1\over 2n_a}\sum_{p=1}^{n_a-1} p(n_a-p)x_p^{(a)}x_{n_a-p}^{(a)}
+{1\over n_a}\sum_{k=1}^{\infty}(k+n_a)x_{k+n_a}^{(a)}{\partial\over\partial
x_k^{(a)}}\}.
$$
%
Since $\tau=\sum_{\alpha\in Q}\tau_{\alpha}e^{\alpha}$ and $L_{-1}\tau=0$,
we find that for all $\alpha\in Q$:
%
$$\sum_{a=1}^s \{
(\delta_a|\alpha)x_{n_a}^{(a)}+
{1\over 2n_a}\sum_{p=1}^{n_a-1} p(n_a-p)x_p^{(a)}x_{n_a-p}^{(a)}
+{1\over n_a}\sum_{k=1}^{\infty}(k+n_a)x_{k+n_a}^{(a)}{\partial\over\partial
x_k^{(a)}}\}
\tau_{\alpha}=0.
\tag{5.2.2}
$$
%
Clearly, also $L_{-1}\tilde\tau_{\alpha+\delta_i-\delta_j}=0$, this gives
(see e.g. [D]):
%
$$\align
&\sum_{a=1}^s \{
(\delta_a|\alpha+\delta_i-\delta_j)(x_{n_a}^{(a)}-{\delta_{aj}\over
n_jz^{n_j}})+
{1\over 2n_a}\sum_{p=1}^{n_a-1} p(n_a-p)(x_p^{(a)}-{\delta_{aj}\over
pz^p})\times \\
&(x_{n_a-p}^{(a)}-{\delta_{aj}\over (n_j-p)z^{n_j-p}})+{1\over
n_a}\sum_{k=1}^{\infty}(k+n_a)(x_{k+n_a}^{(a)}-{\delta_{aj}\over
(k+n_j)z^{k+n_j}}){\partial\over\partial x_k^{(a)}}\}
\tilde\tau_{\alpha+\delta_i-\delta_j}=0.
\tag{5.2.3}
\endalign$$
%
So, in a similar way as in [D], one deduces from (5.2.2 and 3) that
$$\tilde\tau_{\alpha+\delta_i-\delta_j}\tau_{\alpha}^{-2}L_{-1}\tau_{\alpha}
-\tau_{\alpha}^{-1}L_{-1}\tilde\tau_{\alpha+\delta_i-\delta_j}=0.
$$
%
Hence, we find that for all $\alpha\in Q$ and $1\le i,j\le s$:
%
$$\align
\ &{1\over n_j}
\{ {1\over\tau_{\alpha}}(\sum_{k=1}^{\infty}{\partial\over\partial x_k^{(j)}}
z^{-k-n_j}+(\delta_{ij}-1+(\delta_j|\alpha)+{1\over 2}-{{n_a}\over
2})z^{-n_j}-n_jx^{(i)}_{n_i}
)\tilde{\tau}_{\alpha+\delta_i-\delta_j}\\
&+\sum_{k=1}^{n_j} kx_k^{(j)}{{\tilde{\tau}_{\alpha+\delta_i-\delta_j}}\over
\tau_{\alpha}}z^{k-n_j}
-\sum_{a=1}^s {{n_j}\over n_a}\sum_{k=1}^{\infty} (k+n_a)x_{k+n_a}^{(a)}
{\partial\over \partial
x_k^{(a)}}({{\tilde{\tau}_{\alpha+\delta_i-\delta_j}}\over \tau_{\alpha}})
\}=0.
\tag{5.2.4}
\endalign
$$
%
Comparing this with (5.1.3), one finds
$$S((\sum_{a=1}^s \{ ({1\over n_a}M(\alpha)L(\alpha)^{1-n_a}C^{(a)}(\alpha))_-
-{{n_a-1}\over 2n_a}L(\alpha)^{-n_a}C^{(a)}(\alpha)\} P(\alpha)R(\alpha))_{ij})
=0
$$
%
We  thus conclude that the string equation induces for all $\alpha\in Q$:
$$
\sum_{a=1}^s \{ ({1\over n_a}M(\alpha)L(\alpha)^{1-n_a}C^{(a)}(\alpha))_-
-{{n_a-1}\over 2n_a}L(\alpha)^{-n_a}C^{(a)}(\alpha)\}
=0.
\tag{5.2.5}
$$
%
So, if (5.2.5) holds
$$N(\alpha):=
\sum_{a=1}^s \{ {1\over n_a}M(\alpha)L(\alpha)^{1-n_a}C^{(a)}(\alpha)
-{{n_a-1}\over 2n_a}L(\alpha)^{-n_a}C^{(a)}(\alpha)\}
$$
%
is a differential operator that satisfies
%
$$[\sum_{a=1}^s L(\alpha)^{n_a}C^{(a)}(\alpha),N(\alpha)]=1.$$

\vskip 10pt
\subheading{\S 6. $W_{1+\infty}$ constraints}

\vskip 10pt
{\bf 6.1.} Let $e_i$, $1\le i\le s$ be a basis of ${\Bbb C}^s$.
In a similar way as in \S 2, we identify $({\Bbb C}[t,t^{-1}])^s$ with ${\Bbb
C}^{\infty}$,
viz., we put
%
$$v^{(a)}_{-k-\frac{1}{2}}=t^ke_a.\tag{6.1.1}$$
%
We can associate to $({\Bbb C}[t,t^{-1}])^s$ $s$--copies of the Lie algebra
of differential operators on ${\Bbb C}^{\times}$,
it has as basis the operators (see [Ra] or [KRa]):
%
$$-t^{k+\ell}({\partial\over\partial t})^{\ell}e_{ii},
\quad\text{for}\ k\in {\Bbb Z},\ \ell\in{\Bbb Z}_+,\ 1\le i\le s.$$
%
We will denote this Lie algebra by $D^s$. Via (6.1.1) we can embed this algebra
into $\overline{gl_{\infty}}$
and also into $a_{\infty}$, one finds
%
$$-t^{k+\ell}({\partial\over\partial t})^{\ell}e_{ii}
\mapsto \sum_{m\in{\Bbb
Z}}-m(m-1)\cdots(m-\ell+1)E^{(ii)}_{-m-k-\frac{1}{2},-m-\frac{1}{2}}.
\tag{6.1.2}
$$
%
It is straightforward, but rather tedious, to calculate the corresponding
2--cocycle, the result is as follows (see also [Ra] or [KRa]).
Let $f(t), g(t)\in {\Bbb C}[t,t^{-1}]$ then
%
$$\mu(f(t)({\partial\over\partial t})^{\ell}e_{aa},g(t)({\partial\over\partial
t})^m e_{bb})
=\delta_{ab}{{\ell
!m!}\over(\ell+m+1)!}\text{Res}_{t=0}dt\ f^{(m+1)}(t)g^{(\ell)}(t).
$$
Hence in this way we get a central extension $\hat D^s=D^s\oplus {\Bbb C}c$ of
$D^s$
with Lie bracket
%
$$\align
&[f(t)({\partial\over\partial t})^{\ell}e_{aa}+\alpha
c,g(t)({\partial\over\partial t})^m e_{bb}+\beta c]=\\
&\delta_{ab}\{
(f(t)({\partial\over\partial t})^{\ell}g(t)({\partial\over\partial t})^m -
g(t)({\partial\over\partial t})^m f(t)({\partial\over\partial t})^{\ell})e_{aa}
+{{\ell !m!}\over(\ell+m+1)!}\text{Res}_{t=0}dt\ f^{(m+1)}(t)g^{(\ell)}(t)c\}.
\tag{6.1.3}\endalign
$$
%
Since we have the representation $\hat r$ of $a_{\infty}$, we find that
%
$$\hat r (-t^{k+\ell}({\partial\over\partial t})^{\ell}e_{aa})=
\sum_{m\in{\Bbb Z}}m(m-1)\cdots (m-\ell+1):\psi^{+(a)}_{-m-\frac{1}{2}}
\psi^{-(a)}_{m+k+\frac{1}{2}}:.
$$
%
In terms of the fermionic fields (2.1.2), we find
%
$$\sum_{k\in{\Bbb Z}}\hat r (-t^{k+\ell}({\partial\over\partial
t})^{\ell}e_{aa})z^{-k-\ell-1}=
:{{\partial^{\ell}\psi^{+(a)}(z)}\over z^{\ell}}
\psi^{-(a)}(z):. \tag{6.1.4}
$$
%

\vskip 10pt
{\bf 6.2.} We will now express $-t^{k+\ell}({\partial\over\partial
t})^{\ell}e_{aa}$
in terms of the oscillators $\alpha^{(a)}_k$. For this purpose, we
first calculate
%
$$\align
:(y-z)\psi^{+(a)}(y)\psi^{-(a)}(z):&=(y-z)\psi^{+(a)}(y)\psi^{-(a)}(z)-1 \\
&=X_a(y,z)-1,\\
\endalign$$
%
where
%
$$X_a(y,z)={(\frac{y}{z})}^{\alpha^{(a)}_0}
\exp(-\sum_{k<0}\frac{\alpha^{(a)}_k}{k}(y^{-k}-z^{-k}))
\exp(-\sum_{k>0}\frac{\alpha^{(a)}_k}{k}(y^{-k}-z^{-k})).
\tag{6.2.1}
$$
%
Then
$$:{{\partial^{\ell}\psi^{+(a)}(z)}\over \partial z^{\ell}}\psi^{-(a)}(z):
=\frac{1}{\ell+1} \frac{\partial^{\ell+1}X_a(y,z)}{\partial
y^{\ell+1}}|_{y=z}.
\tag{6.2.2}
$$
%
Notice that the right--hand--side of this formula is some normal
ordered expression in the $\alpha^{(a)}_k$'s. For some explicit formulas of
(6.2.2), we refer to the appendix of [AV].

\vskip 10pt
{\bf 6.3.} In the rest of this section, we will show that $\hat D^s$ has a
subalgebra
that will provide the extra constraints, the so called $W$--algebra constraints
on $\tau$.

{\it From now on we assume that $\tau$ is a $\tau$--function of the
$[n_1,n_2,\ldots,n_s]$-th
reduced KP hierarchy, which satisfies the string equation.}
So, we assume that (4.2.3) and (5.2.1) holds. Hence,
for all  $\alpha\in \text{supp}\ \tau$ both
%
$$\align
Q(\alpha):=&\sum_{a=1}^s L(\alpha)^{n_a}C^{(a)}(\alpha)\quad\text{and} \\
 N(\alpha)=&
\sum_{a=1}^s \{ {1\over n_a}M(\alpha)L(\alpha)^{1-n_a}C^{(a)}(\alpha)
-{{n_a-1}\over 2n_a}L(\alpha)^{-n_a}C^{(a)}(\alpha)\}
\endalign
$$
are differential operators. Thus, also
$N(\alpha)^pQ(\alpha)^q$ is a differential operator, i.e.,
%
$$((\sum_{a=1}^s \{ {1\over n_a}M(\alpha)L(\alpha)^{1-n_a}
-{{n_a-1}\over 2n_a}L(\alpha)^{-n_a})^pL(\alpha)^{qn_a}C^{(a)}(\alpha))_-=0
\quad\text{for } p,q\in {\Bbb Z}_+.\tag{6.3.1}$$
%
Using (6.3.1),we are able to prove the following
%
\proclaim{Lemma 6.1}
For all $\alpha\in Q$ and $p,q\in {\Bbb Z}_+$
%
$$\text{Res}_{z=0}dz\sum_{a=1}^s z^{qn_a}({1\over n_a}
z^{{1-n_a}\over 2}{\partial\over \partial z}z^{{1-n_a}\over 2})^p
V^+(\alpha,x,z)E_{aa}^{\phantom{aa}t} V^-(\alpha,x',z) =0.
\tag{6.3.2}$$
%
\endproclaim
%
\demo{Proof}
%
Using Taylor's formula we rewrite the right--hand--side of (6.3.2):
%
$$\text{Res}_{z=0}dz\sum_{a=1}^s z^{qn_a}({1\over n_a}
z^{{1-n_a}\over 2}{\partial\over \partial z}z^{{1-n_a}\over 2})^p
V^+(\alpha,x,z)E_{aa}
\exp (\sum_{\ell =1}^s\sum_{k=1}^{\infty}(x_k^{(\ell)\prime}-x_k^{(\ell)})
{\partial\over\partial x_k^{(\ell)}})^tV^-(\alpha,x,z).
\tag{6.3.3}$$
%
Since
%
$${{\partial V^-(\alpha,x,z)}\over\partial x_k^{(\ell)}}=
-(P^-(\alpha,x,z)E_{\ell\ell}\partial^k P^-(\alpha,x,z)^{-1})_+
V^-(\alpha,x,z),
$$
%
it suffices to prove that for all $m\ge 0$
%
$$\text{Res}_{z=0}dz\sum_{a=1}^s z^{qn_a}({1\over n_a}
z^{{1-n_a}\over 2}{\partial\over \partial z}z^{{1-n_a}\over 2})^p
V^+(\alpha,x,z)E_{aa}\partial^{mt}V^-(\alpha,x,z) =0.
\tag{6.3.4}$$
%
Now, let
$\sum_{a=1}^s z^{qn_a}({1\over n_a}
z^{{1-n_a}\over 2}{\partial\over \partial z}z^{{1-n_a}\over 2})^p
V^+(\alpha,x,z)E_{aa}
=\sum_i S_i\partial^{-i}e^{x\cdot z}$
and
\break
$V^-(\alpha,x,z)=\sum_j T_j\partial^{-j}e^{-x\cdot z}$,
then (6.3.4) is equivalent to
$$\align
0&=\text{Res}_{z=0}dz \sum_{i,j} S_i z^{-i}e^{x\cdot z}\partial^m
(e^{-x\cdot zt}T_j(-z)^j)\\
&=\text{Res}_{z=0}dz \sum_{i,j}\sum_{\ell=0}^m (-1)^{m-\ell+j}{m\choose\ell}
S_i\partial^{\ell}(^tT_j)z^{m-i-j-\ell}\\
&=\sum_{\scriptstyle 0\le \ell\le m\atop\scriptstyle i+j+\ell=m+1}
(-1)^{\ell+j}{m\choose \ell}S_i\partial^{\ell}(^tT_j).
\tag{6.3.5}\endalign
$$
%
On the other hand (6.3.1) implies that
%
$$\align
0&=
\sum_i S_i\partial^{-i}\sum_j(-\partial)^{-jt}T_j)_-\\
&=\sum_{\scriptstyle i,j\atop\scriptstyle \ell\ge 0}(-1)^j{-i-j\choose \ell}
S_i\partial^{\ell}(^tT_j)\partial^{-i-j-\ell})_-.
\endalign
$$
%
Now let $i+j+\ell=m+1$, then we obtain that for every $m\ge 0$
$$\align
0&=\sum_{\scriptstyle 0\le \ell\atop\scriptstyle i+j+\ell=m+1}
(-1)^{j}{-i-j\choose \ell}S_i\partial^{\ell}(^tT_j)\\
&=\sum_{\scriptstyle 0\le \ell\le m\atop\scriptstyle i+j+\ell=m+1}
(-1)^{\ell+j}{m\choose \ell}S_i\partial^{\ell}(^tT_j),
\endalign
$$
which proves (6.3.5)\ \ \ $\square$
\enddemo

Taking the $(i,j)$--th coefficient of (6.3.3) one obtains
%%
\proclaim{Corollary 6.2}
For all $\alpha\in Q$, $1\le i,j\le s$  and $p,q\in{\Bbb Z}_+$ one has
%
$$\text{Res}_{z=0}dz\sum_{a=1}^s z^{qn_a}({1\over n_a}
z^{{1-n_a}\over 2}{\partial\over \partial z}z^{{1-n_a}\over 2})^p
\psi^{+(a)}(z)\tau_{\alpha+\delta_i-\delta_a}\otimes
\psi^{-(a)}(z)\tau_{\alpha+\delta_a-\delta_j}=0.
\tag{6.3.6}
$$
%
\endproclaim
Notice that (6.3.6) can be rewritten as infinitely many generating series
of Hirota bilinear equations, for the case that $p=q=0$ see [KV].

\vskip 10pt
{\bf 6.4.} The following lemma gives a generalization of an
identity of Date, Jimbo, Kashiwara and Miwa [DJKM3] (see also
 [G]):
%
\proclaim
{Lemma 6.3}
Let $X_b(y,w)$ be given by (6.2.1), then
%
$$
\align
\text{Res}_{z=0}dz&\sum_{a=1}^s
\psi^{+(a)}(z)
X_b(y,w)\tau_{\alpha+\delta_i-\delta_a}e^{\alpha+\delta_i-\delta_a}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\otimes\psi^{-(a)}(z)\tau_{\alpha+\delta_a-\delta_j}e^{\alpha+\delta_a-\delta_j}=\\
&(w-y)\psi^{+(b)}(y)\tau_{\alpha+\delta_i-\delta_b}e^{\alpha+\delta_i-\delta_b}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\otimes\psi^{-(b)}(w)\tau_{\alpha+\delta_b-\delta_j}e^{\alpha+\delta_b-\delta_j}.
\tag{6.4.1}\endalign
$$
%
\endproclaim
\demo
{Proof}
The left--hand--side of (6.4.1) is equal to
%
$$\align\text{Res}_{z=0}&dz
\sum_{a=1}^s \epsilon(\delta_a, \delta_i+\delta_j)
z^{(\delta_a|\delta_i+\delta_j)-2}
y^{(\delta_b|\alpha+\delta_i-\delta_a)}
w^{-(\delta_b|\alpha+\delta_i-\delta_a)}
({{z-y}\over z-w})^{\delta_{ab}}\times\\
&e^{x^{(b)}\cdot y- x^{(b)}\cdot w+x^{(a)}\cdot z}
\exp (-\sum_{k=1}^{\infty}{1\over k}(y^{-k}-w^{-k})
{\partial\over\partial x^{(b)}_k}+{1\over k}z^{-k}
{\partial\over\partial x^{(a)}_k})
\tau_{\alpha+\delta_i-\delta_a}e^{\alpha+\delta_i}
\otimes \\
&e^{-x^{(a)}\cdot z}
\exp (\sum_{k=1}^{\infty}{1\over k}z^{-k}
{\partial\over\partial x_k^{(a)}})
\tau_{\alpha+\delta_a-\delta_j}e^{\alpha-\delta_j}.
\endalign
$$

Recall the bilinear identity for $\beta=\alpha$:
%
$$\text{Res}_{z=0}dz\psi^{+(a)}(z)
\tau_{\alpha+\delta_i-\delta_a}e^{\alpha+\delta_i-\delta_a}
\otimes
\psi^{-(a)}(z)\tau_{\alpha+\delta_a-\delta_j}e^{\alpha+\delta_a-\delta_j}
=0.
$$
Let $X_b(y,w)\otimes 1$ act on this identity, then
%
$$\align
\text{Res}_{z=0}dz&
\sum_{a=1}^s \epsilon(\delta_a, \delta_i+\delta_j)
z^{(\delta_a|\delta_i+\delta_j)-2}
y^{(\delta_b|\alpha+\delta_i)}
w^{-(\delta_b|\alpha+\delta_i)}
({{y-z}\over w-z})^{\delta_{ab}}
e^{x^{(b)}\cdot y- x^{(b)}\cdot w+x^{(a)}\cdot z}\times \\
&\exp (-\sum_{k=1}^{\infty}{1\over k}(y^{-k}-w^{-k})
{\partial\over\partial x^{(b)}_k}+{1\over k}z^{-k}
{\partial\over\partial x^{(a)}_k})
\tau_{\alpha+\delta_i-\delta_a}e^{\alpha+\delta_i}
\otimes \\
&e^{-x^{(a)}\cdot z}
\exp (\sum_{k=1}^{\infty}{1\over k}z^{-k}
{\partial\over\partial x_k^{(a)}})
\tau_{\alpha+\delta_a-\delta_j}e^{\alpha-\delta_j}
=0.
\endalign$$
%
Now, using this and the fact that
${{z-y}\over z-w}={{1-y/z}\over1-w/z}={{w-y}\over z}\delta(w/z) +
({y\over w}){{1-z/y}\over 1-z/w}$,
where
$\delta(w/z)=\sum_{k\in{\Bbb Z}}(w/z)^k$
(such that $f(w,z)\delta(w/z)=f(w,w)\delta(w/z)$),
we obtain that the left--hand--side  of (6.4.1) is equal to
%
$$\align
(w-y)&
y^{(\delta_b|\alpha+\delta_i-\delta_b)}
e^{x^{(b)}\cdot y}
\exp (-\sum_{k=1}^{\infty}{1\over k}y^{-k}
{\partial\over\partial x^{(b)}_k}
\tau_{\alpha+\delta_i-\delta_b}e^{\alpha+\delta_i}
\otimes
w^{-(\delta_b|\alpha+\delta_b-\delta_j)}
e^{-x^{(b)}\cdot w}\times \\
&\exp (\sum_{k=1}^{\infty}{1\over k}w^{-k}
{\partial\over\partial x_k^{(b)}})
\tau_{\alpha+\delta_b-\delta_j}e^{\alpha-\delta_j},
\endalign
$$
%
which is equal to the right--hand--side of (6.4.1)\ \ \ $\square$
\enddemo

Define $c_b(\ell,p)$ as follows
%
$$(z^{{1-n_b}\over 2}{\partial\over\partial z}z^{{1-n_b}\over 2})^p
=\sum_{\ell=0}^p c_b(\ell,p)z^{-n_bp+\ell}({\partial\over\partial z})^{\ell}.
\tag{6.4.2}
$$
%
One also has
%
$$(M(\alpha)L(\alpha)^{-n_b +1}-{{n_b-1}\over 2}L(\alpha)^{-n_b})^p=
\sum_{\ell=0}^p c_b(\ell,p) M(\alpha)^{\ell}L(\alpha)^{-n_bp+\ell}.
\tag{6.4.3}
$$
%
Then it is straightforward to show that
%
$$\align
c_b(\ell,p)=&\sum_{0\le q_0<q_1\cdots<q_{p-\ell-1}\le p-1}
[(q_0+\frac{1}{2})(1-n_b)]\times \\
&[(q_1+\frac{1}{2})(1-n_b)-1]\cdots
[(q_{p-\ell -1}+\frac{1}{2})(1-n_b)-(p-\ell-1)].
\tag{6.4.4}\endalign
$$
%
Now using (6.4.2) and removing the tensor product symbol in
(6.3.6), where we write $x$, respectively $x'$, for the first,
respectively the second, component of the tensor product, one
gets:
%
$$\text{Res}_{z=0}dz \sum_{a=1}^s (\frac{1}{n_a})^pz^{qn_a}
\sum_{\ell=0}^p c_a(\ell,p) z^{-n_ap+\ell}
\frac{\partial^{\ell}\psi^{+(a)}(z)}{\partial z^{\ell}}
\tau_{\alpha+\delta_i-\delta_a}(x)
\psi^{-(a)}(z)'\tau_{\alpha+\delta_a-\delta_j}(x')
=0
$$
%
Using Lemma 6.3, this is equivalent to
%
$$\align
\text{Res}_{\scriptstyle y=0\atop\scriptstyle z=0}dydz&
\sum_{a=1}^s \psi^{+(a)}(y)
\sum_{b=1}^s
(\frac{1}{n_b})^pz^{qn_b}
\sum_{\ell=0}^p \frac{c_b(\ell,p)}{\ell+1} z^{-n_bp+\ell}
\frac{\partial^{\ell+1}X_b(w,z)}{\partial z^{\ell+1}}|_{w=z}\times \\
&\tau_{\alpha+\delta_i-\delta_a}(x)e^{\alpha+\delta_i-\delta_a}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\psi^{-(a)}(y)'\tau_{\alpha+\delta_a-\delta_j}(x')(e^{\alpha+\delta_a-\delta_j})'
=0.
\endalign
$$
%
Now, recall (6.1.4) and (6.2.2), then
%
$$\align
\ \ \ &\text{Res}_{z=0}dz
\sum_{b=1}^s
(\frac{1}{n_b})^pz^{qn_b}
\sum_{\ell=0}^p \frac{c_b(\ell,p)}{\ell+1} z^{-n_bp+\ell}
\frac{\partial^{\ell+1}X_b(w,z)}{\partial w^{\ell+1}}|_{w=z}
=\\
&\text{Res}_{z=0}dz
\sum_{b=1}^s
(\frac{1}{n_b})^pz^{qn_b}
\sum_{\ell=0}^p c_b(\ell,p) z^{-n_bp+\ell}
:\frac{\partial^{\ell}\psi^{+(b)}(z)}{\partial z^{\ell}}\psi^{-(b)}(z):
=\\
&\sum_{b=1}^s
(\frac{1}{n_b})^p
\sum_{\ell=0}^p {c_b(\ell,p)}
\hat r (-t^{(q-p)n_b+\ell}(\frac{\partial}{\partial t})^{\ell}e_{bb})
=\\
&-\sum_{b=1}^s
(\frac{1}{n_b})^p
\hat r (t^{n_bq}(t^{\frac{1-n_b}{2}}\frac{\partial}{\partial t}
t^{\frac{1-n_b}{2}})^pe_{bb})
=\\
&-\sum_{b=1}^s
\hat r ((t^{\frac{n_b-1}{2}}t^{n_bq}(\frac{\partial}{\partial t^{n_b}})^p
t^{\frac{1-n_b}{2}})e_{bb})
=\\
&\sum_{b=1}^s
\hat r (t^{\frac{n_b-1}{2}}(-\lambda_b^{q}(\frac{\partial}{\partial
\lambda_b})^p)
t^{\frac{1-n_b}{2}})e_{bb})\qquad\text{where}\ \ \ \ \lambda_b=t^{n_b}
\\
\overset{def}\to{=}&W^{(p+1)}_{q-p}.\tag{6.4.5}
\endalign
$$
%
Hence, (6.3.6) is equivalent to
%
$$\text{Res}_{y=0}dy
\sum_{a=1}^s \psi^{+(a)}(y)
W^{(p+1)}_{q-p}
\tau_{\alpha+\delta_i-\delta_a}(x)e^{\alpha+\delta_i-\delta_a}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\psi^{-(a)}(y)'\tau_{\alpha+\delta_a-\delta_j}(x')(e^{\alpha+\delta_a-\delta_j})'
=0.
\tag{6.4.6}
$$
%

If we ignore the cocycle term for a moment, then it is obvious from,
the sixth line of (6.4.5), that the elements $W^{(p+1)}_q$ are the generators
of the W--algebra $W_{1+\infty}$ (the cocycle term, however, will be slightly
different). Upto some  modification of the elements $W^{(p+1)}_0$, one gets
the standard commutation relations of $W_{1+\infty}$, where $c=nI$.

As the next step, we take in (6.4.6) $x_k^{(i)}=x_k^{(i)\prime}$, for all
$k\in{\Bbb N},\ 1\le i\le s$, we then obtain
%
$$\cases
\frac{\partial}{\partial
x_1^{(i)}}(\frac{W^{(p+1)}_{q-p}\tau_{\alpha}}{\tau_{\alpha}})=0&
\text{if } i=j,\\
\tau_{\alpha+\delta_i-\delta_j}W^{(p+1)}_{q-p}\tau_{\alpha}=
\tau_{\alpha}W^{(p+1)}_{q-p}\tau_{\alpha+\delta_i-\delta_j}&
\text{if }i\ne j.
\endcases
\tag{6.4.7}
$$
%
The last equation means that for all $\alpha,\beta\in \text{supp}\ \tau$ one
has
%
$$\frac{W^{(p+1)}_{q-p}\tau_{\alpha}}{\tau_{\alpha}}=
\frac{W^{(p+1)}_{q-p}\tau_{\beta}}{\tau_{\beta}}.
\tag{6.4.8}
$$
%
Next we divide (6.4.6) by $\tau_{\alpha}(x)\tau_{\alpha}(x')$, of course
only for $\alpha\in \text{supp}\ \tau$, and use (6.4.8). Then for all
$\alpha, \beta\in\text{supp}\ \tau$ and $p,q\in {\Bbb Z}_+$ one has
$$\align
\text{Res}_{z=0}dz&
\sum_{a=1}^s \exp(-\sum_{k=1}^{\infty}\frac{z^{-k}}{k}
\frac{\partial}{\partial x_k^{(a)}})
(\frac{W^{(p+1)}_{q-p}\tau_{\beta}(x)}{\tau_{\beta}(x)})
\frac{\psi^{+(a)}(z)\tau_{\alpha+\delta_i-\delta_a}(x)}{\tau_{\alpha}(x)}
e^{\alpha+\delta_i-\delta_a}\times
\\
&\frac{\psi^{-(a)}(z)'\tau_{\alpha+\delta_a-\delta_j}(x')}{\tau_{\alpha}(x')}
(e^{\alpha+\delta_a-\delta_j})'
=0.\endalign
$$
%
Since one also has the bilinear identity (3.3.3) (see also (2.4.1-2), we can
subtract that part and thus
obtain the following
\proclaim
{Lemma 6.4}
For all
$\alpha, \beta\in\text{supp}\ \tau$ and $p,q\in {\Bbb Z}_+$ one has
$$\align
\text{Res}_{z=0}dz&
\sum_{a=1}^s \{\exp(-\sum_{k=1}^{\infty}\frac{z^{-k}}{k}
\frac{\partial}{\partial x_k^{(a)}})-1\}
(\frac{W^{(p+1)}_{q-p}\tau_{\beta}(x)}{\tau_{\beta}(x)})\times
\\
&\frac{\psi^{+(a)}(z)\tau_{\alpha+\delta_i-\delta_a}(x)}{\tau_{\alpha}(x)}
e^{\alpha+\delta_i-\delta_a}
\frac{\psi^{-(a)}(z)'\tau_{\alpha+\delta_a-\delta_j}(x')}{\tau_{\alpha}(x')}
(e^{\alpha+\delta_a-\delta_j})'
=0.
\tag{6.4.9}\endalign
$$
%
\endproclaim
Define
%
$$S(\beta,p,q,x,z):=\sum_{a=1}^s
\{\exp(-\sum_{k=1}^{\infty}\frac{z^{-k}}{k}
\frac{\partial}{\partial x_k^{(a)}})-1\}
(\frac{W^{(p+1)}_{q-p}\tau_{\beta}(x)}{\tau_{\beta}(x)})E_{aa}.
$$
Notice that the first equation of (6.4.7) implies that
$\partial\circ S(\beta,p,q,x,\partial)= S(\beta,p,q,x,\partial)\circ\partial$.
Then viewing (6.4.9) as the $(i,j)$--th entry of a matrix, (6.4.9) is
equivalent to
%
$$\text{Res}_{z=0}dz P^+(\alpha)R^+(\alpha)S(\beta,p,q,x,\partial)e^{x\cdot z}
\ ^t(P^-(\alpha)'R^-(\alpha)'e^{-x'\cdot z})=0.
\tag{6.4.10}
$$
%
Now using Lemma 3.1, one deduces
%
$$(P^+(\alpha)R^+(\alpha)S(\beta,p,q,x,\partial)
R^+(\alpha)^{-1}P^+(\alpha)^{-1})_-=0,
\tag{6.4.11}
$$
hence
$$P^+(\alpha)S(\beta,p,q,x,\partial)
P^+(\alpha)^{-1}=(P^+(\alpha)S(\beta,p,q,x,\partial)
P^+(\alpha)^{-1})_-=0.
$$
So $S(\beta,p,q,x,\partial)=0$
and therefore
$$\{\exp(-\sum_{k=1}^{\infty}\frac{z^{-k}}{k}
\frac{\partial}{\partial x_k^{(a)}})-1\}
(\frac{W^{(p+1)}_{q-p}\tau_{\beta}(x)}{\tau_{\beta}(x)})=0.
$$
{}From which we conclude that
$$W_k^{(p+1)}\tau_{\beta}=\text{constant }\tau_{\beta}\quad\text{for all}\
-k\le p\ge 0.
\tag{6.4.12}
$$
In order to determine the constants on the right--hand--side of (6.4.12)
we calculate the Lie brackets
%
$$[W_{-1}^{(2)},\frac{-1}{k+p+1}W_{k+1}^{(p+1)}]\tau_{\beta}=0
\tag{6.4.13}
$$
%
and thus obtain the main result
%
%
\proclaim
{Theorem 6.5}
The following two conditions for $\tau\in F^{(0)}$ are equivalent:

\noindent (1) $\tau$ is a $\tau$--function of the $[n_1,n_2,\ldots,n_s]$--th
reduced
$s$--component KP hierarchy which satisfies the string equation (5.2.1).

\noindent (2) For all $-k\le p\ge 0$:
$$(W_k^{(p+1)}+\delta_{k0}c_p)\tau=0,
\tag{6.4.14}$$
where
$$\align
c_p=&\frac{1}{2p+2}
\sum_{a=1}^s (\frac{-1}{n_a})^{p+1}\sum_{\ell=0}^p
\ell\cdot\ell !{n_a+\ell\choose\ell+2}
\sum_{0\le q_0<q_1\cdots<q_{p-\ell-1}\le p-1}
[(q_0+\frac{1}{2})(n_a-1)]\times \\
&[(q_1+\frac{1}{2})(n_a-1)+1]\cdots
[(q_{p-\ell -1}+\frac{1}{2})(n_a-1)+p-\ell-1].
\tag{6.4.15}\endalign
$$
\endproclaim
%
%


For $p=0,1$, the constants $c_p$ are equal to 0, respectively
$\sum_{a=1}^s\frac{n_a^2-1}{24n_a}$.

\demo{Proof of Theorem 6.5}
The case (2) $\Rightarrow$ (1) is trivial. For the implication
(1) $\Rightarrow$ (2), we only have to calculate the left--hand--side of
(6.4.13). It is obvious that this is equal to
$(W_k^{(p+1)}+c_{p,k})\tau_{\beta}$, where
$c_{p,k}=\mu(W_{-1}^{(2)},\frac{-1}{k+p+1}W_{k+1}^{(p+1)})$.
It is clear from (6.1.3) that $c_{p,k}=0$ for $k\ne 0$. So from now
on we assume that $k=0$ and $c_p=c_{p,0}$. Then
$$\align
c_p&=\frac{-1}{p+1}\mu(W_{-1}^{(2)},W_1^{(p+1)})
\\
&=\frac{-1}{p+1}\sum_{a=1}^s (\frac{1}{n_a})^{p+1}
\mu(\frac{1-n_a}{2}t^{-n_a}+t^{1-n_a}\frac{\partial}{\partial t},
\sum_{\ell=0}^p c_a(\ell,p)t^{n_a+\ell}(\frac{\partial}{\partial t})^{\ell})
\\
&=\frac{1}{2p+2}
\sum_{a=1}^s (\frac{1}{n_a})^{p+1}
\sum_{\ell=0}^p  (-1)^{\ell+1}
\ell\cdot\ell !{n_a+\ell\choose\ell+2}
c_a(\ell,p),
\endalign
$$
which equals (6.4.15).\ \ \ $\square$
\enddemo

\vskip 10pt

\subheading{\S 7. A geometrical interpretation of the string equation on the
Sato
Grassmannian}

\vskip 10pt
{\bf 7.1.} It is well--known that every $\tau$--function of the
1--component KP hierarchy corresponds to a point of the Sato
 Grassmannian $Gr$ ( see e.g. [S]).
%
Let $H$ be the space of formal Laurent series $\sum a_n t^n$ such that
$a_n=0$ for $n>>0$. The points of $Gr$ are those linear subspaces $V\subset H$
for which the naturel projection $\pi_+$ of $V$ into
$H_+=\{\sum a_nt^n\in H|a_n=0 \text{ for all } n<0\}$ is a Fredholm
operator. The big cell $Gr^0$ of $Gr$ consists of those $V$ for which $\pi_+$
is an isomorphism.

The connection between $Gr$ and the semi--infinite wedge space is made as
follows.
Identify $v_{-k-\frac{1}{2}}=t^k$.
Let $V$ be a point of $Gr$ and $w_0(t),w_{-1}(t),\ldots $ be a basis of $V$,
then we associate to $V$ the following element in the semi--infinite wedge
space
$$w_0(t)\wedge w_{-1}(t)\wedge w_{-2}(t)\wedge \ldots .$$
If $\tau$ is a $\tau$-- function of the $n$--th KdV hierarchy, then $\tau $
corresponds to a point of $Gr$ that satisfies
$t^nV\subset V$ (see e.g. [SW], [KS]).

In the case of the $s$--component KP hierarchy and its
$[n_1,n_2,\ldots,n_s]$--reduction
we find it convenient to represent the Sato Grassmannian slightly different.
Let now $H$ be the space of formal laurent series $\sum a_nt^n$ such that
$a_n \in {\Bbb C}^s$ and $a_n=0$ for $n>>0$. The points $Gr$ are those linear
subspaces
$V\subset H$ for which the projection $\pi_+$ of $V$ into
$H_+=\{\sum a_nt^n\in H|a_n=0 \text{ for all } n<0\}$ is a Fredholm
operator. Again, the big cell $Gr^0$ of $Gr$ consists of those $V$ for which
$\pi_+$
is an isomorphism. The connection with the semi--infinite wedge space is
of course given in a similar way via (2.1.1):
$$v_{nj-N_a-p+\frac{1}{2}}=v^{(a)}_{n_aj-p+\frac{1}{2}}=t^{-n_aj+p-1}e_a,$$
here $e_a$, $1\le a\le s$ is an orthonormal basis of ${\Bbb C}^s$.

It is obvious that $\tau$--functions of the $[n_1,n_2,\ldots,n_s]$--th reduced
$s$--component KP
hierarchy correspond to those subspaces $V$ for which
$$(\sum_{a=1}^s t^{n_a}E_{aa})V\subset V.
\tag{7.1.1}
$$

\vskip 10pt
{\bf 7.2.} The proof that there exists a $\tau$--function of the
$[n_1,n_2,\ldots n_s]$--th
reduced KP hierarchy that satisfies the string equation is in great details
similar
to the proof of Kac and Schwarz [KS] in the principal case, i.e., the
$n$--th KdV case.

Recall the string equation $L_{-1}\tau=H_{-1}\tau=0$. Now modify the origin
by replacing $x_{n_a+1}$ by $x_{n_a+1}+1$ for all $1\le a\le s$.
Then the string equation transforms to
$$(L_{-1}+\sum_{a=1}^s \frac{n_a+1}{n_a} x_1^{(a)})\tau =0,
$$
%
or equivalently
$$(H_{-1}+\sum_{a=1}^s \frac{n_a+1}{n_a} x_1^{(a)})\tau =0.
$$
%
In terms of elements of $\hat D$ this is
$$\hat r(-A)\tau=0\tag{7.2.1}$$
where
$$A=\sum_{a=1}^s\frac{1}{n_a}((n_a+1)t+t^{1-n_a}\frac{\partial}{\partial t}
-\frac{n_a-1}{2}t^{-n_a})E_{aa}.
\tag{7.2.2}
$$
Hence for $V\in Gr$, this corresponds to
$$AV\subset V.\tag{7.2.3}
$$
Now we will prove that there exists a subspace $V$ satisfying
(7.1.1) and (7.2.3).
We will first start by assuming that $m=n_1=n_2=\cdots =n_s$
(this is the case that $L(\alpha)^m$ is a differential operator).
For this case we will show that there exists a unique point in the
big cell $Gr^0$ that satisfies both
(7.1.1) and (7.2.3). So assume that $V\in Gr^0$ and that $V$ satisfies
these two conditions. Since the projection $\pi_+$ on $H_+$ is an isomorphism,
there
exist $\phi_a\in V$, $1\le a\le s$, of the form
$\phi_a=e_a+\sum_{i,a}c_{i,a}t^{-i}$,
with $c_{i,a}=\sum_{b=1}^s c_{i,a}^{(b)} e_b\in {\Bbb C}^s$.
Now $A^p\phi_a=t^pe_a+$lower degree terms, hence these functions for $p\ge 0$
and $1\le a\le s$ form a basis of $V$. Therefore, $t^m\phi_a$ is a linear
combination of
$A^p\phi_b$; it is easy to observe that $A^m\phi_a=\text{constant }t^m\phi_a$.
Using this we find a recurrent relation for the $c_{i,a}^{(b)}$'s:
%
$$(\frac{m+1}{m})^{m-1}ic_{i,a}^{(b)}
=\sum_{\ell=1}^{m-1} d_{m,i,\ell}c_{i-\ell(m+1),a},
\tag{7.2.4}
$$
here the $d_{m,i,\ell}$ are coefficients depending on $m,i,\ell$,
which can be calculated explicitly using (7.2.2).
Since $c^{(b)}_{0,a}=\delta_{ab}$ and $c^{(b)}_{i,a}=0$ for $i<0$
one deduces from (7.2.4) that $c_{i,a}^{(b)}=0$ if $b\ne a$,  and
$c_{i,a}^{(a)}=0$ if $i\ne (m+1)k$ with $k\in{\Bbb Z}$.
So the $\phi_a$ for $1\le a\le s$ can be determined uniquely.
More explicitly, all $\phi_a$ are of the form $\phi_a=\phi^{(m)}e_a$,
with
%
$$\phi^{(m)}=\sum_{i=1}^{\infty} b_i^{(m)}t^{-(m+1)i},
\tag{7.2.5}
$$
%
where the $b_i$ do not depend on $a$ and satisfy
%
$$(\frac{m+1}{m})^{m-1} i(m+1)b_i^{(m)}=\sum_{\ell=1}^{m-1} d_{m,i, \ell}
b^{(m)}_{i-\ell}.
$$
%
Thus the space $V\in Gr^0$ is spanned by $t^{km}A^{\ell}\phi_a$ with
$1\le a\le s$ , $k\in {\Bbb Z}_+$, $0\le \ell<m$.

Notice  that in the case that all $n_a=1$ we find that $V=H_+$,
meaning that the only solution of
(7.1.1) and (7.2.3) in $Gr^0$ is $\tau=\text{constant }e^0$,
corresponding to the vacuum vector $|0\rangle$.

If not all $n_a$ are the same, then it is ovious that there still is a $V\in
Gr^0$
satisfying (7.1.1) and (7.2.3), viz., $V$ spanned by
$t^{kn_a}A^{\ell_a}\phi^{(n_a)}e_a$,
with $1\le a\le s$, $k\in{\Bbb Z}_+$, $0\le \ell_a<n_a$,
where $\phi^{(n_a)}$ is the unique solution determined by (7.2.5).
However, at the present moment we do not know if this $V\in Gr^0$ is still
unique in $Gr^0$.

\break
\Refs
\widestnumber\key{\bf DJKM3}

\ref \key{\bf AV} \by M. Adler and P. van Moerbeke\paper A Matrix Integral
Solution to Two--dDmensional $W_p$--Gravity\jour Comm. Math. Phys.\vol 147\yr
1992 \pages 25--56\endref

\ref\key{\bf BT}\by M. Bergvelt and  A. ten Kroode\paper Partitions, Vertex
Operator
Constructions and Multi--Component KP Equations\jour preprint University of
Illinois,
to appear in Pacific Journal of Mathematics.
\endref

\ref \key{\bf DJKM1} \by E. Date, M. Jimbo, M. Kashiwara and T. Miwa
\paper Operator approach to the Kadomtsev-Petviashvili equation.
Transformation groups for soliton equations. III \jour J. Phys. Soc. Japan
\vol 50 \yr 1981 \pages 3806--3812\endref

\ref \key{\bf DJKM2} \by E. Date, M. Jimbo, M. Kashiwara and T. Miwa
\paper Transformation groups for soliton equations.  Euclidean Lie
algebras and reduction of the KP hierarchy \jour Publ. Res. Inst.
Math. Sci. \vol 18 \yr 1982 \pages 1077--1110\endref

\ref \key{\bf DJKM3} \by E. Date, M. Jimbo, M. Kashiwara and T. Miwa
\paper Transformation groups for soliton equations \jour in:
Nonlinear integrable systems---classical theory and quantum theory
eds M. Jimbo and T. Miwa, World Scientific, 1983\pages 39--120\endref

\ref \key{\bf D} \by  L.A. Dickey \paper Additional symmetries of KP,
Grassmannian, and the string equation II \jour preprint University of Oklahoma
\yr 1992\endref

\ref \key{\bf Dij} \by R. Dijkgraaf \paper Intersection Theory, Integrable
Hierarchies and Topological Field Theory \jour preprint IASSNS--HEP--91, hep-th
9201003 \endref

\ref \key{\bf FK} \by I.B. Frenkel and V.G. Kac \paper Basic
representations of affine Lie algebras and dual resonance models
\jour Invent. Math. \vol 62 \yr 1980 \pages 23--66\endref

\ref\key{\bf FKN}\by M. Fukuma, H. Kawai and R. Nakayama\paper
Infinite Dimensional Grassmannian Structure of Two--Dimensional Quantum
Gravity\jour Comm. Math. Phys.\vol 143\yr 1992\pages 371--403\endref

\ref \key{\bf G} \by J. Goeree\paper $W$--cinstraints in 2d quantum
gravity\jour Nucl. Phys.\vol B358\yr 1991\pages 737--157\endref



\ref \key{\bf JM} \by M. Jimbo and T. Miwa \paper Solitons and infinite
dimensional Lie algebras \jour Publ. Res. Inst. Math. Sci. \vol 19
\yr 1983 \pages 943--1001\endref

\ref \key{\bf K} \by V.G. Kac \paper Infinite dimensional Lie algebras
\jour Progress in Math., vol. 44, Brikh\"{a}user, Boston, 1983; 2nd
ed., Cambridge Univ. Press, 1985; 3d ed., Cambridge Univ. Press,
1990\endref



\ref \key{\bf KP1} \by V.G. Kac and D.H. Peterson \paper 112 constructions
of the basic representation of the loop group of $E_{8}$ \jour in
Proc. of the  Symposium
``Anomalies, Geometry, Topology'', Argonne, eds. W.A. Bardeen, A.R.
White, World Scientific, 1985, pp.
276--298\endref

\ref \key{\bf KP2} \by V.G. Kac and D.H. Peterson \paper Lectures on the
infinite wedge representation and the MKP hierarchy \jour Sem. Math.
Sup., vol. 102, Presses Univ. Montreal, Montreal, 1986, pp.
141--184\endref

\ref \key{\bf KP3} \by V.G. Kac and D.H. Peterson \paper Spin and wedge
representations of infinite dimensional Lie algebras and groups
\jour Proc. Nat. Acad. Sci U.S.A.\yr 1981 \pages 3308--33012
\endref


\ref \key{\bf KR} \by V.G. Kac and A.K. Raina \paper Bombay lectures on
highest weight representations of infinite-dimen-\break sional Lie algebras
\jour Advanced Ser. in Math. Phys., vol. 2, World Scientific,
1987\endref

\ref\key{\bf KRa}\by V. Kac and A. Radul \paper Quasifinite highest weight
modules over the Lie algebra of differential operators on the circle
\jour Comm. Math. Phys.\vol 157 \yr 1993 \pages 429-457
\endref

\ref \key{\bf KS}\by V. Kac and A. Schwarz\paper Geometric Interpretation of
Partition Function of 2D Gravity\jour Phys. Lett.\vol B257
\yr 1991\pages 363\endref



\ref\key{\bf KV}\by V. Kac and J. van de Leur\paper The $n$--Component KP
hierarchy and Representation Theory.\jour in Important Developments in Soliton
Theory, eds. A.s. Fokas and V.E. Zakharov. Springer Series in Nonlinear
Dynamics,\yr 1993\pages 302--343\endref

\ref \key{\bf L} \by J. Lepowsky\paper  Calculus of twisted vertex
operators\jour Proc. Nat. Acad. Sci. U.S.A.
\vol 82 \yr 1985 \pages 8295--8299\endref

\ref\key{\bf Ra} \by A.O. Radul\paper Lie algebras of differential operators,
their central extensions, and W--algebras \jour Funct. Anal. and its Appl.\vol
25
\yr 1991 \pages 33--49\endref

\ref \key{\bf S} \by M. Sato \paper Soliton equations as dynamical systems
on infinite dimensional Grassmann manifolds \jour Res. Inst. Math.
Sci. Kokyuroku \vol 439 \yr 1981 \pages 30--46\endref

\ref \key{\bf SW} \by G. Segal and G. Wilson \paper Loop groups and
equations of $KdV$ type \jour Inst. Hautes Etudes Sci. Publ. Math.
\vol 63 \yr 1985 \pages 1--64\endref

\ref \key{\bf TV} \by F. ten Kroode and J. van de Leur \paper Bosonic and
fermionic realizations of the affine algebra $\hat{gl}_{n}$ \jour
Comm. Math. Phys.\ \vol 137 \yr 1991 \pages 67--107\endref

\ref \key{\bf UT} \by K. Ueno and K. Takasaki \paper Toda lattice hierarchy
\jour Adv. stud. Pure Math., vol. 4, North-Holland, 1984 \pages
1--95\endref


\endRefs





\enddocument

