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\topmatter
\title   The $[n_1,n_2,\ldots,n_s]$--th reduced KP hierarchy and $W_{1+\infty}$ Constraints
\endtitle
\author
Johan van de Leur
\endauthor
\thanks
E-mail: vdleur\@math.ruu.nl
\endthanks
\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 obtain reductions of the $s$--component KP hierarchy,
reductions which are related to these partitions. 
In this way we obtain matrix KdV type
equations. We show that the following two constraints on a KP $\tau$--function
are equivalent (1) $\tau$ is a $\tau$--function of the 
$[n_1,n_2,\ldots,n_s]$--th reduced KP hierarchy which satisfies string equation, $L_{-1}\tau=0$,
(2) $\tau$  satisfies the vacuum constraints of the
$W_{1+\infty}$ algebra.
\endabstract
\endtopmatter
\vskip 10pt

\noindent The results of this paper were presented at the V International
Conference on Mathematical Physics, String Theory and
Quantum Gravity at Alushta, June 10-20 1994 and at the Oberwolfach conference on Integrable Systems from a Quantum Point of View, June 19-25 1994. A longer edition of this paper [V], containing more details and proofs, will appear 
elsewhere.

\vskip 10pt
\subheading{\S 1.  $a_{\infty}$ and the KP hierarchy in
the fermionic picture [KV]}

\vskip 10pt
\noindent
Consider the infinite dimensional complex  Lie algebra
$a_{\infty} := \overline{gl_{\infty}}\bigoplus {\Bbb C}c$, where
%
$$\overline{gl_{\infty}} = \{ a = (a_{ij})_{i,j \in {\Bbb Z}+\frac{1}{2}}
|\  a_{ij}=0\  \text{if}\   |i-j|>>0\},$$
%
with Lie bracket defined by
%
$$[a+\alpha c,b+\beta c]=ab-ba+\mu (a,b)c , \tag{1.1}$$
%
for $a,b\in \overline{gl_{\infty}}$ and $\alpha ,\beta\in{\Bbb C}$.
Here $\mu$ is the following 2--cocycle:
%
$$\mu (E_{ij},E_{kl})=\delta_{il}\delta_{jk}(\theta(i)-\theta(j)), 
\tag{1.2}$$
%
where $E_{ij}$ is the matrix with a $1$ on the $(i,j)$-th entry and zeros
elsewhere and $\theta :{\Bbb R}\to {\Bbb C}$ is the step--function  defined by
%
$$\theta(i):=\cases 0 & \text{if}\ i>0,\\
                    1 & \text{if}\ i\le 0.\endcases \tag{1.3}$$
%
Let ${\Bbb C}^{\infty} = \bigoplus_{j \in {\Bbb Z}+\frac{1}{2}} {\Bbb
C} v_{j}$ be the infinite dimensional complex vector space with fixed
basis $\{ v_{j}\}_{j \in {\Bbb Z}+\frac{1}{2}}$. The Lie algebra $a_{\infty}$ acts linearly on
${\Bbb C}^{\infty}$ via the usual formula:
%
$$E_{ij} (v_{k}) = \delta_{jk} v_{i}.$$

Let ${\Cal C}\ell$ be the Clifford algebra with generators $\psi^{\lambda}_{i}$,
$i \in {\Bbb Z}+\frac{1}{2}, \lambda ,\mu = +,-$, satisfying the following relations:
%
$$\psi^{\lambda}_{i} \psi^{\mu}_{j} + \psi^{\mu}_{j}
\psi^{\lambda}_{i} = \delta_{\lambda ,-\mu} \delta_{i,-j}. \tag{1.4}$$
%
We define an irreducible  ${\Cal C}\ell$--module $F$ by introducing a vacuum
vector $|0\rangle$ such that
%
$$\psi^{\pm}_{j} |0\rangle = 0 \ \text{for}\ j > 0 . \tag{1.5}$$
%
Define  a representation $\hat r$ of $a_{\infty}$ on $F$ by
%
$$ \hat r(E_{ij})=:\psi^-_{-i}\psi^+_j:,\quad \hat r(c)=I,
$$
%
where $:\ :$ stands for the {\it normal ordered product} defined in
the usual way $(\lambda ,\mu = +$ or $-$):
%
$$:\psi^{\lambda (i)}_{k} \psi^{\mu (j)}_{\ell}: = \cases \psi^{
\lambda (i)}_{k}
\psi^{\mu (j)}_{\ell}\ &\text{if}\ \ell \ge k \\
-\psi^{\mu (j)}_{\ell} \psi^{\lambda (i)}_{k} &\text{if}\ \ell <
k.\endcases \tag{1.6}$$

Define the {\it charge decomposition}
%
$$F = \bigoplus_{m \in {\Bbb Z}} F^{(m)} \tag{1.7}$$
by letting
%

$$\text{charge}(|0\rangle ) = 0\ \text{and charge} (\psi^{\pm}_{j}) =
\pm 1. \tag{1.8}$$
%
It  is easy to see that each $F^{(m)}$ is irreducible with
respect to  $a_{\infty}$.

We are now able to define the {\it KP hierarchy in the fermionic
picture}, it is the equation 
$$\sum_{k \in {\Bbb Z}+\frac{1}{2}} \psi^{+}_{k} \tau \otimes \psi^{-}_{-k}
\tau = 0, \tag{1.9}$$
for
$\tau\in F^{(0)}$. One can prove (see e. g. [KP2] or [KR]) that  this equation characterizes  the group orbit of the vacuum vector  $|0\rangle$  for the  the group $GL_{\infty}$. Since the group does not play an important role in this paper, we will not introduce it here.
\vskip 10pt
\subheading{\S 2.  Vertex operator constructions}

\vskip 10pt 
\noindent We will now sketch how one can  construct vertex realizations of 
the affine Lie algebra $\hat{gl}_n$,
following [TV] (see also [KP1] and [L]). 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  operators:
($1\le a \le s,\ 1\le p\le n_a,\ j\in {\Bbb Z}$)
%
$$\align
v^{(a)}_{n_aj-p+\frac{1}{2}}&=v_{nj-N_a-p+\frac{1}{2}} \\
\psi^{\pm (a)}_{n_a j \mp p\pm \frac{1}{2}}&=\psi^{\pm }_{n_aj\mp N_a\mp p\pm\frac{1}{2}}
\tag{2.1}
\endalign$$
%
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:
$$E^{(ab)}_{n_aj-p+\frac{1}{2},n_bk-q+\frac{1}{2}}=E_{nj-N_a-p+\frac{1}{2},nk-N_b-q+\frac{1}{2}},\tag{2.2}$$
%
then  $\hat r(E^{(ab)}_{jk})=:\psi^{-(a)}_{-j}\psi^{+b}_k:$.

Introduce the  fermionic fields $(z \in {\Bbb C}^{\times})$:
%
$$\psi^{\pm (a)}(z) \overset{\text{def}}\to{=} \sum_{k \in {\Bbb
Z}+\frac{1}{2}} \psi^{\pm
(a)}_{k} z^{-k-\frac{1}{2}}.\tag{2.3}$$
%
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
%
$$:\psi^{+(a)}(\omega_a^p z^{N\over n_a})\psi^{-(b)}(\omega_b^q z^{N\over n_b}):, \tag{2.4}$$
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 center $K=1$.
It is easy to see that restricted to $\hat{gl}_n$,
$F^{(0)}$ is its basic highest weight representation (see [K, Chapter
12]).
From (2.4) it  is obvious, that if we obtain vertex operators for the fermionic fields
$\psi^{\pm (a)}(z)$, we also have a vertex operator realization of  $\hat{gl}_n$.


Next we introduce special bosonic fields ($1\le a\le s$):
%
$$\alpha^{(a)}(z) \equiv \sum_{k \in {\Bbb Z}} \alpha^{(a)}_{k} z^{-k-1}
\overset{def}\to{=} :\psi^{+(a)}(z) \psi^{-(a)}(z):. \tag{2.5}$$
%
The operators $\alpha^{(a)}_{k} $
satsify the canonical commutation relation of the associative
oscillator algebra,  which we
denote by ${\frak a}$:
%
$$[\alpha^{(i)}_{k},\alpha^{(j)}_{\ell}] =
k\delta_{ij}\delta_{k,-\ell},\tag{2.6}$$
and one has
%
$$\alpha^{(i)}_{k}|m\rangle = 0 \ \text{for}\ k > 0.$$
%



This realization  of $\hat{gl}_n$,  has a natural Virasoro algebra. In [TV], it was shown that the following two sets of operators have the same action on $F$.
%
$$\align 
&L_k=\sum_{i=1}^s\{ \sum_{j\in{\Bbb Z}}{1\over 2n_i}:\alpha_{-j}^{(i)}\alpha_{j+n_ik}^{(i)}:+\delta_{k0}{{n_i^2-1}\over
24n_i}\},\tag{2.7a} \\
&H_k=\sum_{i=1}^s\{\sum_{j\in{\Bbb Z}+\frac{1}{2}}({j\over n_i}+{k\over 2}):\psi^{+(i)}_{-j}\psi^{-(i)}_{j+n_ik}:+\delta_{k0}{{n_i^2-1}\over
24n_i}\}. \tag{2.7b} \endalign
$$
%
So $L_k=H_k$,
$$[L_k,\psi_j^{\pm (i)}]=-(\frac{j}{n_i}+\frac{k}{2})\psi_{j+n_ik}^{\pm(i)}
\tag{2.8}
$$
 and 
%
$$[L_k,L_\ell ]=(k-\ell)L_{k+\ell}+\delta_{k,-\ell}{{k^3-k}\over 12}n. $$

We will now 
describe the $s$-component boson-fermion
correspondence (see [KV]).  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
%
$$\varepsilon_{ij} = \cases -1 &\text{if $i > j$} \\
1 &\text{if $i \leq j$.} \endcases \tag{2.9}$$
%
Define a bimultiplicative function $\varepsilon :\ L \times L @>>> \{
\pm 1 \}$ by letting
%
$$\varepsilon (\delta_{i}, \delta_{j}) = \varepsilon_{ij}.
\tag{2.10}$$
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:
%
$$e^{\alpha}e^{\beta} = \varepsilon (\alpha ,\beta)e^{\alpha +
\beta}. \tag{2.11}$$
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
%
$$\sigma :F @>\sim >> B, \tag{2.12}$$
given by $\sigma (|0\rangle )=1$ and 
$$\sigma\psi^{\pm (a)}(z) \sigma^{-1}=e^{\pm \delta_a}z^{\pm \delta_a}
\exp(\pm\sum_{k=1}^{\infty}x_k^{(a)}z^k)\exp(\mp\sum_{k=1}^{\infty}
{\partial\over \partial x_k^{(a)}}{z^{-k}\over k}),\tag {2.13}$$
%
where
$$\delta_a (p(x) \otimes e^{\gamma}) =
(\delta_{a}|\gamma ) p(x) \otimes e^{\gamma}.\tag{2.14}$$ 
%
The transported charge then is as follows:
%
$$
\text{charge}(p(x)\otimes e^{\gamma}) = (\delta |\gamma).
$$
We denote the transported charge decomposition by
%
$$B = \bigoplus_{m \in {\Bbb Z}} B^{(m)}.$$
The transported action of the operators $\alpha^{(i)}_{m}$ is given by
%
$$\cases
\sigma \alpha^{(j)}_{-m}\sigma^{-1}(p(x) \otimes e^{\gamma}) =
mx^{(j)}_{m}p(x)\otimes e^{\gamma},\ \text{if}\ m > 0, &\  \\
\sigma \alpha^{(j)}_{m} \sigma^{-1}(p(x) \otimes e^{\gamma}) = \frac{\partial
p(x)}{\partial x_{m}} \otimes e^{\gamma},\ \text{if}\ m > 0, &\  \\
\sigma \alpha^{(j)}_{0} \sigma^{-1} (p(x) \otimes e^{\gamma}) =
(\delta_{j}|\gamma ) p(x) \otimes e^{\gamma} 
. & \
\endcases \tag{2.15}
$$
%

If one substitutes (2.13) into (2.4), 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).
  
Using the isomorphism $\sigma$ we can reformulate the KP hierarchy
(1.9) in the bosonic picture.

We start by observing that (1.9) can be rewritten as follows:
%
$$\text{Res}_{z=0}\ dz ( \sum^{s}_{j=1} \psi^{+(j)}(z)\tau
\otimes \psi^{-(j)}(z)\tau ) = 0,\ \tau \in F^{(0)}.
\tag{2.16}$$
%
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  (2.13), equation (2.16) 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:
%
$$\aligned
&\text{Res}_{z=0} ( dz
 \sum^{s}_{j=1} \varepsilon (\delta_{j}, \alpha-\beta)
z^{(\delta_{j}|\alpha - \beta - 2\delta_{j})}  \\
 &\times \exp
(\sum^{\infty}_{k=1} (x^{(j)^{\prime}}_{k} - x^{(j)^{\prime
\prime}}_{k})z^{k})
\exp (-\sum^{\infty}_{k=1} (\frac{\partial}{\partial x^{(j)^{\prime}}_{k}}
 - \frac {\partial}{\partial x^{(j)^{\prime
\prime}}_{k}})\frac{z^{-k}}{k}) \\
& \tau_{\alpha -
\delta_{j}}(x^{\prime})(e^{\alpha})^{\prime}\tau_{\beta + \delta_{j}}(x^{\prime \prime})(e^{\beta})^{\prime\prime})
= 0  . \endaligned \tag{2.17}$$


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

\vskip 10pt
\noindent We proceed now to rewrite the formulation (2.17) of the
$s$-component KP hierarchy in terms of formal pseudo-differential
operators, generalizing the results of [DJKM] and [JM].
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
$$V^{\pm} (\alpha ,x,z) = (V^{\pm}_{ij}(\alpha ,x,z))^{s}_{i,j=1}
\tag{3.1}$$
as follows:
%
$$\aligned
&V^{\pm}_{ij}(\alpha ,x,z) \overset{\text{def}}\to{=}
\varepsilon (\delta_{j} , \alpha + \delta_{i})
 z^{(\delta_{j}|\pm \alpha + \delta_{i}-\delta_{j})} \\
& \times \exp (\pm \sum^{\infty}_{k=1} x^{(j)}_{k} z^{k})
\exp(\mp \sum^{\infty}_{k=1} \frac{\partial}{\partial
x^{(j)}_{k}} \frac{z^{-k}}{k}) \tau_{\alpha  \pm
(\delta_{i}-\delta_{j})}  (x)/\tau_{\alpha}(x) .
\endaligned \tag{3.2}
$$
It is easy to see that equation (2.17) is equivalent to the
following bilinear identity:
%
$$Res_{z=0}V^{+}(\alpha ,x,z)\ ^{t}V^{-}(\beta ,x^{\prime},z)dz = 0\
\text{for all}\ \alpha ,\beta \in Q. \tag{3.3}$$
%
Define $s \times s$ matrices $W^{\pm (m)} (\alpha ,x)$ by the
following generating series (cf. (3.2)):
%
$$
\sum^{\infty}_{m=0}
W^{\pm (m)}_{ij} (\alpha ,x)(\pm z)^{-m}
= \varepsilon_{ji}z^{\delta_{ij}-1} (\exp \mp
\sum^{\infty}_{k=1} \frac{\partial}{\partial x^{(j)}_{k}}\frac{z^{-k}}{k})
\tau_{\alpha \pm
(\delta_i-\delta_j)} (x))/\tau_{\alpha} (x). \tag{3.4}
$$
We see from (3.2) that $V^{\pm}(\alpha ,x,z)$ can be written in the
following form:
%
$$V^{\pm}(\alpha ,x,z) = (\sum^{\infty}_{m=0}
W^{\pm (m)}(\alpha ,x)R^{\pm}(\alpha ,\pm z)(\pm
z)^{-m})e^{\pm z \cdot x}, \tag{3.5}$$
where
$$z \cdot x^{(j)} = \sum^{\infty}_{k=1} x^{(j)}_{k} z^{k},\ e^{z
\cdot x} = diag (e^{z\cdot x^{(1)}} ,\ldots ,e^{z\cdot x^{(s)}} )$$
and
$$R^{\pm}(\alpha ,z) = \sum^{s}_{i=1}
\varepsilon (\delta_{i}, \alpha ) E_{ii} (\pm z)^{\pm
(\delta_{i}|\alpha )}. \tag{3.6}$$
Here and further $E_{ij}$ stands for the $s \times s$ matrix whose
$(i,j)$ entry is $1$ and all other entries are zero. Let
$$\partial = \frac{\partial}{\partial x^{(1)}_{1}} + \ldots +
\frac{\partial}{\partial x^{(s)}_{1}},$$ we can now rewrite
$V^{\pm}(\alpha ,x,z)$ in
terms of formal pseudo-differential operators
%
$$P^{\pm}(\alpha ) \equiv P^{\pm} (\alpha ,x,\partial ) =
I_{s} + \sum^{\infty}_{m=1} W^{\pm (m)} (\alpha ,x)\partial^{-m}\
\text{and}\ R^{\pm}(\alpha ) = R^{\pm}(\alpha ,\partial) \tag{3.7}$$
as follows:
$$V^{\pm}(\alpha ,x,z) = P^{\pm } (\alpha )
R^{\pm}(\alpha)e^{\pm z \cdot x}. \tag{3.8}$$

As usual one denotes the differential part of $P(x,\partial )$ by
$P_{+}(x,\partial ) = \sum^{N}_{j=0} P_{j}(x) \partial^{j},$
and writes $P_{-} = P-P_{+}$. The  linear  anti-involution   $*$ is defined by the
following formula:
$$(\sum_{j} P_{j}\partial^{j})^{*} = \sum_{j} (-\partial )^{j}
\circ ^{t}\! P_{j}. \tag{3.9}$$
Here and further $^{t}P$ stands for the transpose of the matrix $P$.
Then one has the following fundamental lemma  (see [KV]):

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

Using this Lemma, Victor Kac and the author showed in [KV] that 
given $\beta \in \text{supp}\ \tau$, all the
pseudo-differential operators $P^{\pm} (\alpha )$, $\alpha \in \text{supp}\
\tau$, are
completely determined by $P^+(\beta )$ from the following equations
$$\align
&R^{-}(\alpha ,\partial)^{-1} = R^{+}(\alpha
,\partial)^{*} ,
\tag{3.10}\\
&P^{-}(\alpha  ) = (P^{+}(\alpha  )^{*})^{-1} ,
\tag{3.11}\\
&(P^{+}(\alpha)R^{+}(\alpha - \beta)P^{+}(\beta)^{-1})_{-} = 0\
\text{for all}\ \alpha ,\beta \in \text{supp}\ \tau . \tag{3.12}
\endalign$$
They also showed
 the following

\proclaim{Proposition 3.2}  Consider $V^{+}(\alpha ,x,z)$ and $V^{-}(\alpha ,x,z),$ \newline $\alpha \in Q$, of the
form (3.8), where $R^{\pm}(\alpha ,z)$ are given by (3.6).  Then the
bilinear identity (3.3) for all $\alpha ,\beta \in \ \text{supp}\
\tau$ is equivalent to the Sato equation:
$$
\frac{\partial P}{\partial x^{(j)}_{k}} = -(PE_{jj} \circ
\partial^{k} \circ P^{-1})_{-} \circ P. \tag{3.13}$$
for each $P =
P^{+}(\alpha )$ and the matching conditions (3.10-12)  for
all $\alpha ,\beta \in \ \text{supp} \ \tau$.
\endproclaim

  Fix $\alpha \in Q$, introduce the following formal pseudo-differential
operators $L(\alpha),\  C^{(j)}(\alpha )$, and
differential operators  $B^{(j)}_{m}(\alpha)$:
$$\aligned
L \equiv L(\alpha )
  & = P^{+}(\alpha) \circ \partial \circ P^{+}(\alpha)^{-1}, \\
C^{(j)} \equiv C^{(j)}(\alpha ) &=
P^{+}(\alpha)E_{jj} P^{+}(\alpha)^{-1}, \\
B^{(j)}_{m} \equiv B^{(j)}_{m}(\alpha) &=
(P^{+}(\alpha)E_{jj} \circ \partial^{m} \circ
P^{+}(\alpha)^{-1})_{+}.
\endaligned \tag{3.14}
$$
%
Then
%
%
$$
\sum^{s}_{i=1} C^{(i)} = I_{s}, \
C^{(i)}L = LC^{(i)},\ C^{(i)}C^{(j)} = \delta_{ij} C^{(i)}.
 \tag{3.15}$$
\proclaim{Proposition 3.3} 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 (3.14) satisfy conditions (3.15) and if the
equations 
$$
\cases LP = P\partial &\ \\
C^{(i)}P = PE_{ii} &\ \\
\displaystyle{\frac{\partial P}{\partial x^{(i)}_{k}} = -(L^{(i)k})_{-} P,\
\text{where}\  L^{(i)} = C^{(i)}L.}
\endcases
\tag{3.16}$$
 have a solution $P \equiv P^+(\alpha)$ of the form (3.7), then the
differential operators $B^{(j)}_{k} \equiv B^{(j)}_{k}(\alpha)$ 
satisfies the following
conditions:
%
$$\left\{ \matrix \displaystyle{\frac{\partial L}{\partial x^{(j)}_{k}} =
[B^{(j)}_{k},L],}  & \   \\
\displaystyle{\frac{\partial C^{(i)}}{\partial x^{(j)}_{k}} =
[B^{(j)}_{k},C^{(i)}],} & \
\  \endmatrix \right. \tag{3.17}$$
\endproclaim
Finally, we introduce one more pseudo--differential operator
$$M(\alpha):=P(\alpha)R(\alpha)\sum_{a=1}^s\sum_{k=1}^{\infty} kx_k^{(a)}
\partial^{k-1}E_{aa}R(\alpha)^{-1}P(\alpha)^{-1}.\tag{3.18}$$
Then
$$M(\alpha) V^+(\alpha,x,z)={{\partial V^+(\alpha,x,z)}\over \partial z}\tag{3.19}$$
and  one easily checks that $[L(\alpha),M(\alpha)]=1$

\vskip 10pt
\subheading{\S4 . $[n_1,n_2,\ldots ,n_s]$-reductions of the $s$-component KP hierarchy}
\vskip 10pt
\noindent If one restricts the, to the partition $n=n_1+n_2+\cdots +n_s$ related, vertex
operator construction of $\hat{gl}_n$ in the vector space $B^{(0)}$ to $\hat{sl}_n$, the representation  is not irreducible anymore. In order to obtain
irreducible representations of $\hat{sl}_n$, one has to `remove' all elements
%
$$\sum_{i=1}^s \alpha_{kn_i}^{(i)},\quad k\in {\Bbb Z}.
\tag{4.1}$$
%
Hence, a KP $\tau$--function is an $\hat{sl}_n$ $\tau$--function if
$$\sum^{s}_{j=1} \frac{\partial \tau}{\partial x^{(j)}_{kn_j}}
= 0,\quad\text{for all $k\in {\Bbb N}$}.\tag{4.2}$$
We will call this 
the {\it $[n_1,n_2,\ldots ,n_s]$-th reduced} KP {\it
hierarchy}. 
Using the Sato equation (3.13), this
implies the
following  two equivalent conditions:
%
%
$$\sum^{s}_{j=1} \frac{\partial V^+(\alpha ,x,z)}{\partial x^{(j)}_{kn_j}} =
V^+(\alpha,x,z)\sum_{j=1}^sz^{kn_j}E_{jj}, $$
%
$$Q(\alpha)_-=0,\tag{4.3}$$
where $$Q(\alpha)=\sum_{j=1}^sL(\alpha)^{kn_j}C^{(j)}.\tag{4.4}$$

\vskip 10pt
\subheading{\S 5. The string equation and $W_{1+\infty}$ constraints}

\vskip 10pt
\noindent {\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. Of course this means that the corresponding wave functions 
$V^{\pm}(\alpha ,x,z)$ will be of a more general nature than before.

 
Let $L_{-1}$ be given by (2.7a), the string equation is the following constraint
on $\tau\in F^{(0)}$:
%
$$L_{-1}\tau=0. \tag{5.1}$$
%
Using (2.15) 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 $\tau=\sum_{\alpha\in Q}\tau_{\alpha}e^{\alpha}$,
we find that $L_{-1}\tau_{\alpha}=0$ for all $\alpha\in Q$. 
Using a calculation similar to the one in [D], one deduces from (5.1)
that
$$N(\alpha)_-=0,\tag{5.2}$$
where

$$N(\alpha):=
\sum_{a=1}^s \{ {1\over n_a}M(\alpha)L(\alpha)^{1-n_a}C^{(a)}(\alpha)
-{{n_a-1}\over 2n_a}L(\alpha)^{-n_a}C^{(a)}(\alpha)\}.\tag{5.3}
$$
%
Hence $N(\alpha)$ is a differential operator that satisfies
%
$$[Q(\alpha),N(\alpha)]=1.$$


{\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) and (5.1) holds. Hence,
for all  $\alpha\in \text{supp}\ \tau$ both
%
$Q(\alpha)$ and 
 $N(\alpha)$ are differential operators. Thus, also
$N(\alpha)^pQ(\alpha)^q$ is a differential operator, i.e.,
%
$$((\sum_{a=1}^s \{ {1\over n_a}M(\alpha)L(\alpha)^{1-n_a}
-{{n_a-1}\over 2n_a}L(\alpha)^{-n_a})^pL(\alpha)^{qn_a}C^{(a)}(\alpha))_-=0
\quad\text{for } p,q\in {\Bbb Z}_+.\tag{5.4}$$
%
This leads to
%
\proclaim{Lemma 5.1} 
For all $p,q\in {\Bbb Z}_+$ one has
%
$$\text{Res}_{z=0}dz\sum_{a=1}^s z^{qn_a}({1\over n_a}
z^{{1-n_a}\over 2}{\partial\over \partial z}z^{{1-n_a}\over 2})^p
V^+(\alpha,x,z)E_{aa}^{\phantom{aa}t} V^-(\alpha,x',z) =0.
\tag{5.5}$$
%
\endproclaim
%
Taking the $(i,j)$--th coefficient of (5.5) one obtains
%%
\proclaim{Corollary 5.2}
For all $1\le i,j\le s$  and $p,q\in{\Bbb Z}_+$ one has
%
$$\text{Res}_{z=0}dz\sum_{a=1}^s z^{qn_a}({1\over n_a}
z^{{1-n_a}\over 2}{\partial\over \partial z}z^{{1-n_a}\over 2})^p
\psi^{+(a)}(z)\tau_{\alpha+\delta_i-\delta_a}\otimes
\psi^{-(a)}(z)\tau_{\alpha+\delta_a-\delta_j}=0.
\tag{5.6}
$$
%
\endproclaim

Let
$$\align
W^{(p+1)}_{q-p}&=\text{Res}_{z=0}dz\sum_{a=1}^s z^{qn_a}({1\over n_a}
z^{{1-n_a}\over 2}{\partial\over \partial z}z^{{1-n_a}\over 2})^p
:\psi^{+(a)}(z)\psi^{-(a)}(z):\\
&=-\sum_{a=1}^s
\hat r (\frac{1}{n_a}t^{n_aq}(t^{\frac{1-n_a}{2}}\frac{\partial}{\partial t}
t^{\frac{1-n_a}{2}})^pe_{aa})
\\
&=\sum_{a=1}^s
\hat r (t^{\frac{n_a-1}{2}}(-\lambda_a^{q}(\frac{\partial}{\partial \lambda_a})^p)
t^{\frac{1-n_a}{2}})e_{aa})\qquad\text{where}\ \ \ \ \lambda_a=t^{n_a},
\tag{5.7}
\endalign
$$
and
$$-t^{k+\ell}({\partial\over\partial t})^{\ell}e_{ii}
= \sum_{m\in{\Bbb Z}}-m(m-1)\cdots(m-\ell+1)E^{(ii)}_{-m-k-\frac{1}{2},-m-\frac{1}{2}}.
\tag{5.8}
$$
Using a generalization of an 
identity of Date, Jimbo, Kashiwara and Miwa [DJKM3] (see also 
 [G], [V]), (5.6) is equivalent to
$$\text{Res}_{y=0}dy 
\sum_{a=1}^s \psi^{+(a)}(y)
W^{(p+1)}_{q-p}
\tau_{\alpha+\delta_i-\delta_a}(x)e^{\alpha+\delta_i-\delta_a}
\psi^{-(a)}(y)'\tau_{\alpha+\delta_a-\delta_j}(x')(e^{\alpha+\delta_a-\delta_j})'
=0.
\tag{5.9}
$$
%

If we ignore the cocycle term for a moment, then it is obvious from (5.7), that the elements $W^{(p+1)}_k$ are the generators
of the W--algebra $W_{1+\infty}$ [Ra], [KRa] (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 (5.9) $x_k^{(i)}=x_k^{(i)\prime}$, for all
$k\in{\Bbb N},\ 1\le i\le s$, we then obtain
%
$$\cases
\frac{\partial}{\partial x_1^{(i)}}(\frac{W^{(p+1)}_{q-p}\tau_{\alpha}}{\tau_{\alpha}})=0&
\text{if } i=j,\\
\tau_{\alpha+\delta_i-\delta_j}W^{(p+1)}_{q-p}\tau_{\alpha}=
\tau_{\alpha}W^{(p+1)}_{q-p}\tau_{\alpha+\delta_i-\delta_j}&
\text{if }i\ne j.
\endcases
\tag{5.10}
$$
%
The last equation means that for all $\alpha,\beta\in \text{supp}\ \tau$ one has
%
$$\frac{W^{(p+1)}_{q-p}\tau_{\alpha}}{\tau_{\alpha}}=
\frac{W^{(p+1)}_{q-p}\tau_{\beta}}{\tau_{\beta}}.
\tag{5.11}
$$
%
Next we divide (5.9) by $\tau_{\alpha}(x)\tau_{\alpha}(x')$, of course
only for $\alpha\in \text{supp}\ \tau$, and use (5.11). 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), we can subtract that part and thus
obtain the following 
\proclaim
{Lemma 5.3}
For all
$\alpha, \beta\in\text{supp}\ \tau$ and $p,q\in {\Bbb Z}_+$ one has
$$\align
\text{Res}_{z=0}dz&
\sum_{a=1}^s \{\exp(-\sum_{k=1}^{\infty}\frac{z^{-k}}{k}
\frac{\partial}{\partial x_k^{(a)}})-1\}
(\frac{W^{(p+1)}_{q-p}\tau_{\beta}(x)}{\tau_{\beta}(x)})\times
\\
&\frac{\psi^{+(a)}(z)\tau_{\alpha+\delta_i-\delta_a}(x)}{\tau_{\alpha}(x)}
e^{\alpha+\delta_i-\delta_a}
\frac{\psi^{-(a)}(z)'\tau_{\alpha+\delta_a-\delta_j}(x')}{\tau_{\alpha}(x')}
(e^{\alpha+\delta_a-\delta_j})'
=0.
\tag{5.12}\endalign
$$
%
\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 (5.10) implies that 
$\partial\circ S(\beta,p,q,x,\partial)= S(\beta,p,q,x,\partial)\circ\partial$.
Then viewing (5.12) as the $(i,j)$--th entry of a matrix, (5.12) is
equivalent to
%
$$\text{Res}_{z=0}dz P^+(\alpha)R^+(\alpha)S(\beta,p,q,x,\partial)e^{x\cdot z}
\ ^t(P^-(\alpha)'R^-(\alpha)'e^{-x'\cdot z})=0.
\tag{5.13}
$$
%
Now using Lemma 3.1, one deduces
%
$$(P^+(\alpha)R^+(\alpha)S(\beta,p,q,x,\partial)
R^+(\alpha)^{-1}P^+(\alpha)^{-1})_-=0,
\tag{5.14}
$$
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
$$W_{q-p}^{(p+1)}\tau_{\beta}=\text{constant }\tau_{\beta}\quad\text{for all}\
p,q\ge 0.
\tag{5.15}
$$
In order to determine the constants on the right--hand--side of (5.15)
we calculate the Lie brackets 
%
$$[W_{-1}^{(2)},\frac{-1}{q+1}W_{q-p+1}^{(p+1)}]\tau_{\beta}=0.
\tag{5.16}
$$
%
Notice that the right--hand--side of (5.16) is equal to
$$(W_{p-q}^{(p+1)}+\mu(W_{-1}^{(2)},\frac{-1}{q+1}W_{q-p+1}^{(p+1)}))\tau_{\beta}.$$
Now using (1.2-3), (2.2)  and (5.8) we thus obtain
 the main result, generalizing some of the results of [FKN], see also [AV]:
%
%
\proclaim
{Theorem 5.4}
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 (5.1).

\noindent (2) For all $ p,q\ge 0$:
$$(W_{q-p}^{(p+1)}+\delta_{pq}c_{p+1})\tau=0,
\tag{5.17}$$ 
where
$$c_{p+1}={1\over p+1}\sum_{b=1}^s \sum_{j=1}^{n_b }
({n_b-2j+1\over 2n_b})_{p+1},\tag{5.18}$$
and $(k)_{\ell}=k(k-1)(k-2)\cdots (k-\ell+1)$
\endproclaim
%
%\vskip 20pt
\noindent {\bf Aknowledgements.}  I would like to thank Frits Beukers and
Victor Kac for valuable discussions, and 
the Mathematical Institute of the University of Utrecht 
for the computer and e-mail facilities.

\vskip 1cm
\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 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 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 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 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 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 V}\by J. van de Leur \paper KdV--type hierarchies, the string equation and $W_{1+\infty}$ constraint \jour Utrecht University preprint and 
hep-th 9403080\endref

\endRefs
\vskip 20pt
\noindent J.W. van de Leur 

\noindent Kievitdwarsstraat 22
 
\noindent  3514VE Utrecht, The Netherlands






\enddocument
