
\nopagenumbers
\
\vskip .5cm
\centerline{\bf LEVEL ONE REPRESENTATIONS OF THE AFFINE LIE ALGEBRA 
$B_n^{(1)}$}
\vskip 1cm
\centerline {FONS TEN KROODE\footnote *{Current address: Nieuwe Prinsengracht $77^1$, 1018 VR Amsterdam, The Netherlands.}}
\vskip .1cm
\centerline {\sl Department of Mathematics,}
\centerline {\sl University of Illinois at Urbana Champaign,}
\centerline{\sl 1409 West Green Street, Urbana, IL~61801, USA}
\vskip .5cm
\centerline {JOHAN VAN DE LEUR\footnote {**}{E-mail:
vdleur@math.ruu.nl
\hfil \break
The research of J. van de Leur has been made possible by a fellowship
of the Royal Netherlands Academy of Arts and Sciences.}} 
\vskip .1cm
\centerline {\sl Mathematical Institute,}
\centerline {\sl University of Utrecht,}
\centerline {\sl P.O. Box 80.010, 3508~TA, Utrecht, The Netherlands}
\vskip 2cm
\centerline {\bf Abstract}
\vskip .1cm
\midinsert\narrower\narrower\narrower\narrower
\noindent We give an explicit description of the Kac-Peterson-Lepowsky construction of the
basic representation for the affine Lie algebra $B_n^{(1)}$. 
Using the conjugacy classes of the Weyl group of $B_n$, we describe
all inequivalent maximal Heisenberg subalgebras of the corresponding
affine Lie algebra. We associate to these Heisenberg subalgebras multicomponent charged and
neutral free fermionic fields. The boson-fermion correspondence for
these fields provides us with fermionic vertex operators, whose `normal
ordered products' give the (twisted) vertex operators of the
Kac-Peterson-Lepowsky construction.
\overfullrule=0pt
\vskip 1cm
\noindent{\bf Keywords}:\quad affine algebra, fermionic fields, Heisenberg
algebra, Kac-Moody algebra, Kac-Peterson-Lepowsky construction,
vertex operator, Virasoro algebra, Weyl group
\vskip .5cm
\overfullrule=5pt
\noindent{\bf AMS classification}:\quad 17B65, 17B67, 22E65
\endinsert








\baselineskip=22pt plus.1pt
\count30=0                      %chapterno = 0
\count31=0                      %sectionno = 0
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0

\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count33}

\def\vac{\mid \overline 0 >}
\def\a#1#2{\hat\alpha_{#1}({N\over n_{#1}}#2)}




\beginsection \chapno . {\bf  Introduction and summary}
\par
The Korteweg de Vries (KdV)-equation
$${\partial f\over \partial t}-{3\over 2}f{\partial f \over \partial x}-{1\over 4}{\partial^3 f\over \partial x^3}=0\forno$$
is one of the most celebrated examples of so-called soliton equations. It was introduced in 1895 by the two dutch mathematicians Korteweg en de Vries in order to describe the propagation of shallow water waves in a narrow channel. The dependent variable $f=f(x,t)$ describes the height of waves in the channel as a function of the space coordinate $x$ along the channel  and a time coordinate $t$.

It is not difficult to verify that (1.1) has the solution
$$f(x,t)=8u^2/(e^{u(x+u^2t+\alpha)}+e^{-u(x+u^2t+\alpha)})^2=2u^2{\rm sech}^2u(x+u^2t+\alpha)\forno$$
where $u$ and $\alpha$ are arbitrary real parameters. This solution describes the motion of a wave, which moves in time without loss of shape; for $t=0$ it looks like $2u^2{\rm sech}^2u(x+\alpha)$, which is a single `pulse' peaked at $x=0$, and this profile propagates with a velocity $u^2$ in the negative $x$-direction. It is worthwhile to notice that the velocity is proportional to the amplitude of the wave. Such a wave is called a solitary wave or soliton.

In the 20th century the KdV-equation was forgotten for a long time. It took until the 1960's for the equation to reappear in the numerical work of Kruskal and Zabusky  [KZ] on nonlinear interactions in chrystals and plasmas. Their work triggered a lot of further research in the field of nonlinear models. At present one knows that there are many other nonlinear partial differential equations, which allow soliton-like solutions.

Since the pioneering work of Date, Jimbo, Kashiwara and Miwa [DJKM] in the early 80's, it is well known that infinite dimensional Lie algebras and groups play an important role in the theory of soliton equations. The infinite dimensional group acts as a `symmetry group' on the space of solutions of a soliton equation, i.e., it transfers solutions into each other. This means that, if one can identify the the symmetry group for a particular soliton equation and its corresponding action on the solutions, it should be possible to construct solutions starting from  a trivial one ($f=0$).

Date, Jimbo, Kashiwara and Miwa studied infinite dimensional Lie algebras and groups and were able to obtain from  level one representations of classical affine Kac-Moody algebras systems of partial differential equations. Conversely, it was proved by Kac and Peterson [KP83] that the orbit through the highest weight vector in a level one highest weight representation of an affine Kac-Moody group can be described by a family of integrable partial differential equations.
Kac and Wakimoto elaborated on this result. In [KW89] they associate to the following data
$$\eqalign{(1)&\hbox{ an affine Kac-Moody algebra }\underline g,\cr
(2)&\hbox{ an integrable highest weight representation }V\hbox{ of }\underline g,\cr
(3)&\hbox{ a vertex operator construction }R\hbox{ of }V.\cr}$$
a hierarchy of soliton equations that describe the orbit through the highest weight vector of the corresponding Kac-Moody group.
A crucial point is that these describing equations can look different in different vertex realizations $R$ of $V$, whence one can associate different integrable systems to the same representation $V$. This was illustrated in [K],  [KW89] and [KrB] for the simple case   of the basic representation of $A_1^{(1)}$; this representation has two different vertex realizations, one (the principal realization) yields the KdV hierarchy, the other one (the homogeneous realization) the so-called Toda-AKNS hierarchy (in [KW89] this last one is called the nonlinear Schr\"odinger-hierarchy). In order to study the connection between these integrable systems, it is important to describe all different vertex operator construction $R$ of the representation $V$. This problem was studied by Kac and Peterson [KP85]  and, from a somewhat different point of view by Lepowsky [L]. It turned out that each  vertex realization $R$ of $V$ depends on the choice of a  maximal Heisenberg subalgebra 
$\hat{\underline s}$ of $\hat{\underline g}$, i.e.  a
subalgebra  on a basis  $\{p_i,q_i\}_{i\in {\bf N}}$ and the canonical central
element $c$ of $\hat {\underline g}$, whose elements satisfy the
commutation relations $[p_i,q_j]=\delta_{ij}c$. Such a subalgebra $\hat{\underline s}$ is an
important tool in the construction of irreducible highest weight
representations of $\hat{\underline g}$. We will briefly elaborate on this.


It is very easy to construct an irreducible representation of 
$\hat{\underline s}$. Just take the ring of polynomials in all
variables, ${\bf C}[x_i;i\in{\bf N}]$ and let the elements of $\hat{\underline
s}$ act  by: $p_i\mapsto {\partial \over \partial  x_i}, q_i\mapsto x_i,
c\mapsto 1$. It is well known that any irreducible representation of
$\hat{\underline s}$ in which $c$ act as the identity operator and which
contains a vacuum vector, i.e. a vector which is annihilated by all the
$p_i$'s, is isomorphic to this one.

Now let $V$ be an irreducible highest weight representation of
$\hat{\underline g}$ in which the central element $c$ is represented by the
identity operator. Define the vacuum space $V^+$ as the collection of all
vectors, which are annihilated by the $p_i$'s. If for all $v\in V$ there
exists a number $N$ such that $p_{i_1}p_{i_2}\ldots p_{i_k}.v=0$ whenever
$i_1+i_2+\cdots+i_k>N$, the representation $V$ is completely reducible with
respect to  $\hat{\underline s}$, i.e., it can be identified with the tensor
product $V^+\otimes  {\bf C}[x_i]$. The next step is of course to study the
structure of the vacuum space, $V^+$. Roughly speaking, this is done by
constructing a sufficient supply of operators in ${\cal U}(\hat{\underline
g})$ that commute with the action of  $\hat{\underline s}$. Such operators form
an algebra and the vacuum space is an irreducible representation of this
algebra (see, e.g. [LW84,KP85]).

Lepowsky and Wilson [LW78] gave the first explicit construction of the basic
representation of the simplest affine Lie algebra $A^{(1)}_1$ in this manner.
Their results were soon generalized in [KKLW] to arbitrary simply-laced and
twisted affine Lie algebras. In these papers one uses the so-called principal
Heisenberg subalgebra. By a character theoretical argument one can show that
the basic representation of a simply-laced or twisted affine Lie algebra
remains irreducible when restricted to this Heisenberg subalgebra. This means
that the vacuum space is one dimensional and that the representation space
can be identified with a polynomial ring. The action of the affine algebra on
this polynomial ring is given by a set of formal differential operators,
the so-called vertex operators. A different construction of the same
representation, using the homogeneous Heisenberg subalgebra, was found by
Frenkel-Kac [FK] and independently Segal [S]. In this case the vacuum space
is identified with the group algebra of the coroot lattice of the underlying
finite dimensional Lie algebra.

A uniform description of these different realizations of the basic
representation of simply-laced or twisted affine algebras was given by
Kac and Peterson [KP85] and Lepowsky [L]. Kac and Peterson explain how the inequivalent Heisenberg
subalgebras of an affine Lie algebra can be related to the conjugacy classes
in the Weyl group of the underlying finite dimensional Lie algebra and
present a vertex operator construction of the basic representation for each
such conjugacy class. The homogeneous and principal realizations mentioned
above are extremes of this construction; the former is related to the conjugacy
class of the identity, the latter to the conjugacy class of Coxeter elements.
It turns out that the vacuum space is an irreducible representation of the
centralizer of the Heisenberg subalgebra in the Kac-Moody group associated to
$\hat{\underline g}$. In the case of the principal Heisenberg subalgebra this
centralizer coincides with the center of the Kac-Moody group, whence the vacuum
space must be one dimensional.


For non simply-laced untwisted affine Lie algebras the basic representation
does not remain irreducible when restricted to the principal Heisenberg
subalgebra, which means that the centralizer of the Heisenberg algebra in
the Kac-Moody group is not large enough to construct the vacuum space. This
makes the construction of representations inherently more difficult. In this
paper we consider the orthogonal affine algebra $B^{(1)}_n$ and
present for each equivalence class of Heisenberg subalgebras a  vertex
operator construction of the spinor
representations of this algebra, i.e. $L(\Lambda_0), L(\Lambda_1)$ and
$L(\Lambda_n)$. Some of them, such as the principal and homogeneous
constructions were found before, see [F80,F81,FF,KP81,LM,M,BT-M,GNOS,KW89].

Applying the method of Kac and Wakimoto [KW89], one can use our vertex operator constructions to obtain new hierarchies of soliton equations, describing the Kac-Moody group orbit through the highest weight vector of one of the above level one representation of $B_n^{(1)}$.  The hierarchies in the principal realization of $B_n^{(1)}$, for $n=1$, were calculated in [KW89]. This provides level two hierarchies for $A_1^{(1)}$. The corresponding system of equations turn out to be a hierarchy of super Hirota bilinear equations.


The basic idea of this paper is directly taken from [KP85] and [L]; we  introduce a basis
for the affine algebra, which is adapted to the Heisenberg  subalgebra  under
consideration (see formula (1.6) below).  Next we construct embeddings of the
affine algebra $B^{(1)}_n$ into the infinite rank affine algebras $B_\infty$ 
and $D_\infty$. The well known spinor representations of $B_\infty$ and
$D_\infty$ can then be restricted to the subalgebra $B^{(1)}_n$, which yields
a fermionic construction of the $B^{(1)}_n$- modules $L(\Lambda_0), L(\Lambda_1)$ and
$L(\Lambda_n)$ (see main result A below). The vertex operators, generating the
action of the affine algebra, are represented as normal ordered products of
fermionic fields.

It turns out that these fermionic fields transform in a nice way under the
Heisenberg algebra and under an associated Virasoro algebra. Using these
transformation properties, one is able to express  all but one of the fermionic
fields in elements of the Heisenberg subalgebra and certain fermionic
translation and chirality operators (see main result B below). The fact that
one of the fermionic fields cannot be expressed in elements of the Heisenberg
subalgebra is precisely the reason that the basic representation does not
remain irreducible when restricted to a Heisenberg subalgebra. In fact, the
components of this fermionic field generate an infinite dimensional
Clifford algebra whose elements commute with the Heisenberg algebra.

Below we will give a fairly detailed description of our method, 
serving  both as a summary and  as a guideline for the reader.




\vskip.75cm
\noindent We will start to recall some well known facts about the classical
Lie algebra $B_n$ and its Weyl group. 
Throughout the paper we will think of $B_n$ as  $so_{2n+1}({\bf C})$, 
the collection of skew symmetric $(2n+1)\times (2n+1)$ matrices with complex entries.
As usual we take a set of commuting matrices $\epsilon_i\in so_{2n+1}({\bf C}),
1\le i\le n$, orthonormal with respect to the normalized 
trace form $(A|B):={1\over 2}{\rm trace}(AB), 
A,B\in so_{2n+1}({\bf C})$. The ${\bf C}$-span of these matrices 
is a Cartan subalgebra  ${\underline h}:=\oplus_{i=1}^n{\bf C}\epsilon_i$
of $so_{2n+1}({\bf C})$. 
We will denote the common eigenvectors with respect to the adjoint 
action of ${\underline h}$ on $so_{2n+1}({\bf C})$ by $G_{jk}(\lambda,\mu)$, $G_j(\lambda)$
$(\lambda,\mu\in\{\pm 1\}, 1\le j,k \le n))$. 
With this notation we have the commutation relations
$$\eqalign{[H,G_{jk}(\lambda,\mu)]
  &=(\lambda\epsilon_j+\mu\epsilon_k\mid H)G_{jk}(\lambda,\mu)\cr
[H,G_{j}(\lambda)]
  &=(\lambda\epsilon_j\mid H)G_{j}(\lambda)\cr}
\qquad \forall H\in{\underline h}\forno$$

\noindent
The $G_{jk}(\lambda,\mu)$'s together with the  $G_{j}(\lambda)$'s span the Lie algebra $so_{2n+1}({\bf C})$ 
($\epsilon_i=G_{ii}(+,-)$).

The Weyl group $W(so_{2n+1}({\bf C}))$ consists of all permutations  
of the $\epsilon_i$'s and of all sign changes. 
Carter [C] has proved that the conjugacy classes in  this group
are  in one-one correspondence with so-called `signed' or `colored' partitions of $n$. 
Any such partition is determined by the choice of $r$  `white' numbers $n_1,n_2\cdots , n_r$ and $s-r$ `black'
numbers ${\overline n}_{r+1},\cdots , {\overline n}_s$ adding up to $n$. It will be denoted by
$[n_1n_2\cdots n_r{\overline n}_{r+1}\cdots{\overline n}_s]$

Any partition of the number $n$ in $s$ different parts 
$n_1,\ldots , n_s$ determines a direct sum decomposition 
${\bf C}^{2n+1}\cong {\bf C}^{2n_1}\oplus{\bf C}^{2n_2}
\oplus\ldots{\bf C}^{2n_s}\oplus{\bf C}$, 
and an associated block decomposition of a $(2n+1)\times (2n+1)$ matrix.
In accordance with this we write:
$$\eqalign{G^{ab}_{jk}(\lambda,\mu)&:=
  G_{n_1+n_2+\cdots n_{a-1}+j,n_1+n_2+\cdots n_{b-1}+k}(\lambda,\mu)\cr
 G^a_j(\lambda)&:=G_{n_1+\cdots
+n_{a-1}+j}(\lambda)\cr}
  \qquad\qquad 1\le a,b\le s, 1\le j\le n_a,1\le k\le n_b\forno$$

\noindent Obviously, the $G^{ab}_{jk}(\lambda,\mu)$'s and $G^{a}_{j}(\lambda)$'s still span 
$so_{2n+1}({\bf C})$.

Now choose a representative 
$w\in W(so_{2n+1}({\bf C}))$ for each conjugacy class $[w]$. 
Such a $w$ acts on ${\underline h}\cong{\underline h}^*$. 
It is easy to see though, that its action can be extended to the 
full Lie algebra $so_{2n+1}({\bf C})$; 
there exists a matrix $\hat w\in SO_{2n+1}({\bf C})$ (not unique) such that
${\rm Ad}(\hat w)\lower 3pt\hbox{$|{\underline h}$} = w$.
In other words: the transformation 
${\rm Ad}(\hat w):so_{2n+1}({\bf C})\rightarrow so_{2n+1}({\bf C})$ 
is an automorphism extending the action of $w$ on $\underline h$. 
We will exploit the non-uniqueness of the
matrix $\hat w$ and construct two distinct lifts ${\rm Ad}(\hat
w^\epsilon$) depending on a parameter $\epsilon$, which can take the values $0$
or ${1\over 2}$.

The matrices $\hat w^\epsilon$ can be chosen in such a way that the transformations
${\rm Ad}(\hat w^\epsilon)$ have finite order. 
Like any other inner automorphism of finite order they are conjugate to automorphisms
of the form ${\rm exp}(2\pi i {\rm ad} H^\epsilon_{[w]})$ for some 
$H^\epsilon_{[w]}\in {\underline h}$ (see [H,K]).
We will construct  an orthogonal matrix $S_{[w]}$ (not depending on $\epsilon$)
such that $${\rm Ad}(\hat w^\epsilon)={\rm Ad}\ S_{[w]}{\rm exp}(2\pi i {\rm ad}
H^\epsilon_{[w]}){\rm Ad}\ S_{[w]}^{-1}\forno$$ This matrix is used to introduce
a  new Cartan subalgebra  ${\underline h}_{[w]}:=S_{[w]}^{-1}{\underline
h}S_{[w]}$ and new root vectors
$$\eqalign{A^{ab}_{jk}(\lambda,\mu)&=S_{[w]}^{-1}G^{ab}_{jk}(\lambda,\mu)S_{[w]}\cr
A^{a}_{j}(\lambda)&=S_{[w]}^{-1}G^{a}_{j}(\lambda)S_{[w]}\cr} \forno$$

Let $N$ be the order of  ${\rm Ad}(\hat w^\epsilon)$ and set  $\omega:=e^{2\pi i /N}$.
 The ${\bf Z}_N$ gradation of $so_{2n+1}({\bf C})$ associated 
to $e^{2\pi i {\rm  ad}H^\epsilon_{[w]}}$ is defined by:
$$\eqalign{&so_{2n+1}({\bf C})=
  \oplus_{\overline\jmath\in{\bf Z}_N}so_{2n+1}({\bf C})_{\overline \jmath}\cr
  &so_{2n+1}({\bf C})_{\overline \jmath}:=
\{y\in so_{2n+1}({\bf C})\mid e^{2\pi i {\rm  ad}H^\epsilon_{[w]}}(y)=
\omega^j y\}\cr}\forno$$

\noindent The set
$\{A^{ab}_{jk}(\lambda,\mu)_{\overline l},\ A^{a}_{j}(\lambda)_{\overline l}
 \mid 1\le a,b\le s,  1\le j\le n_a, 
  1\le k\le n_b, \overline l\in{\bf Z}_N\}$ 
is clearly the most natural spanning set for $so_{2n+1}({\bf C})$ 
with respect to the automorphism $e^{2\pi i {\rm ad }H^\epsilon_{[w]}}$. Similarly, the elements
$$\eqalign{\hat A^{ab}_{jk}(\lambda,\mu)(l)&:= 
  e^{-i\theta{\rm ad}H^\epsilon_{[w]}}e^{i{l\over N}\theta}
  A^{ab}_{jk}(\lambda,\mu)_{\overline l}-\delta_{l0}
  (H^\epsilon_{[w]}|A^{ab}_{jk}(\lambda,\mu))c\cr
\hat A^{a}_{j}(\lambda)(l)&:= 
  e^{-i\theta{\rm ad}H^\epsilon_{[w]}}e^{i{l\over N}\theta}
  A^{a}_{j}(\lambda)_{\overline l}-\delta_{l0}
  (H^\epsilon_{[w]}|A^{a}_{j}(\lambda))c\cr}
\forno$$

\noindent together with a central element $c$ span the affine 
algebra $\hat{so}_{2n+1}({\bf C})$ (see section 3 or [KP85]). 

As usual one introduces the following generating series in a 
formal indeterminate $z$:
$$\eqalign{\hat A^{ab}_{jk}(\lambda,\mu)(z)&:=
  \sum_{l\in{\bf Z}}\hat A^{ab}_{jk}(\lambda,\mu)(l)z^{-l}\cr
\hat A^{a}_{j}(\lambda)(z)&:=
  \sum_{l\in{\bf Z}}\hat A^{a}_{j}(\lambda)(l)z^{-l}\cr}
\forno$$

\noindent Constructing a unitary representation for 
$\hat{so}_{2n+1}({\bf C})$ now amounts to finding operators 
on a (pre-)Hilbert space representing the action of these  
$\hat A^{ab}_{jk}(\lambda,\mu)(z)$'s  and $\hat A^{a}_{j}(\lambda)(z)$'s  (`vertex operators').

In order to achieve this, we choose an embedding of 
$\hat{so}_{2n+1}({\bf C})$ into the infinite rank affine algebra 
$B_\infty$ or $D_\infty$ and make use of the well known spin representation 
of the latter algebras. This representation can be described as follows; 
let $\epsilon=0$ for $B_\infty$ and $1\over 2$  for $D_\infty$ and  let $Cl^\epsilon$ be the Clifford algebra on generators 
$\phi(i), i\in {\epsilon+\bf Z}$, satisfying the anticommutation relations
$$\{\phi(i),\phi(j)\}:=\phi(i)\phi(j)+\phi(j)\phi(i)=\delta_{i,-j}\forno$$

\noindent The spin module $V^\epsilon$ (also called fermionic Fock space) is 
the unique irreducible $Cl^\epsilon$-module generated by a vacuum vector 
$|{\overline 0}>$, which is annihilated by all $\phi(i)$'s with $i< 0$:
$$\phi(i)|{\overline 0}>=0\qquad\forall i< 0\forno$$

\noindent $V^\epsilon$ is thus simply the vector space on a basis 
$\{\phi(i_1)\phi(i_2)\ldots \phi(i_k)|{\overline 0}>
\mid i_1>i_2>\cdots >i_k\ge 0\}$. 
It can be made into a (pre-)Hilbert space by introducing an inner 
product such that:
$$\eqalign{&(|{\overline 0}>,|{\overline 0}>)
  \equiv <{\overline 0}\mid {\overline 0}>=1\cr
  &\phi(i)^\dagger=\phi(-i)\cr}\forno$$

It is well known that the so called normal ordered products 
$:\phi(i)\phi(j):,\ i,j\in \epsilon+{\bf Z}$ define a representation of 
$B_\infty$ or $D_\infty$ on $V^\epsilon$. Moreover, this representation breaks into two 
irreducible components:
$$V^\epsilon=V_{\overline 0}^\epsilon\oplus V_{\overline 1}^\epsilon\forno$$

\noindent where $ V_{\overline 0}^\epsilon$ ($ V_{\overline 1}^\epsilon$) is the 
subspace of $V^\epsilon$ generated by the action of an even (odd) number of 
fermions on the vacuum $|{\overline 0}>$.

The embedding $\hat{so}_{2n+1}({\bf C})\hookrightarrow B_\infty\hbox{ or } D_\infty$ and 
the spin representation of $B_\infty$ and $D_\infty$ on $V^0$, $V^{1\over 2}$, respectively  now provides 
a fermionic realization of the fields $\hat A^{ab}_{jk}(\lambda,\mu)(z)$  and $\hat A^{a}_{j}(\lambda)(z)$.
It turns out that these fields can --up to a multiple of the identity-- 
be written as the normal ordered product of two fermionic fields. 
This  fermionic realization of $\hat{so}_{2n+1}({\bf C})$ on $V^\epsilon$ is 
the first main result of this paper;

\vskip.3cm
\noindent{\bf Main result A.}

{\sl \noindent Let $\lambda_n={1\over 2}(\epsilon_1+\cdots +\epsilon_n)$ 
be the $n^{th}$ fundamental weight of $so_{2n+1}({\bf C})$. On $V^\epsilon$ we
have:} $$\eqalign{\hat
A^{ab}_{jk}(\lambda,\mu)(z)+(H^\epsilon_{[w]}-(1-2\epsilon)\lambda_n\vert
A^{ab}_{jk}(\lambda,\mu))&=C^a_j(\lambda)C^b_k(\mu):\phi_a^\lambda(\omega^{-j}z)\phi_b^\mu(\omega^{-k}z):\cr
\hat A^{a}_{j}(\lambda)(z)+(H^\epsilon_{[w]}-(1-2\epsilon)\lambda_n\vert
A^{a}_{j}(\lambda))&=C^a_j(\lambda):\phi_a^\lambda(\omega^{-j}z)\sigma(z):\cr}\forno$$
{\sl\noindent The $C_{j}^a(\lambda)$'s in this formula are constants,  which
will be defined in section 4.4. The fermionic fields  $\phi_a^\lambda(z),
\break 1\le a\le s,\lambda=\pm$  and $\sigma(z)$ have the following formal
expansions $$\eqalign{\phi_a^\lambda(z)&:=\cases{\displaystyle{\sum_{k\in
{1\over 2}+{\bf Z}}\phi_a^\lambda(k)z^{{N\over n_a}k}}&if\ $1\le a \le
r$\cr\cr\displaystyle{ \sum_{k\in{\bf Z}}\phi_a^\lambda(k)z^{{N\over
2n_a}k}}&if\ $r< a \le s$\cr}\cr\cr
\sigma(z)&:=\sum_{k\in(1-(-)^{s-r})/4+\epsilon+{\bf
Z}}\sigma(k)z^{Nk}\cr}\forno$$

\noindent where the fermions $\phi_a^\lambda (l)$  and $\sigma (l)$ can be expressed in 
terms of (linear combinations of) the old fermions $\phi(i)$, 
determined by the signed partition 
$[n_1n_2\ldots n_r{\overline n}_{r+1}\ldots {\overline n}_s]$ 
(see section 4.4). Moreover, the fermion $\phi_a^\lambda(l)$  and $\sigma (l)$ is a creation, 
annihilation and a combination of a creation and annihilation operator for
$l>0$, $l<0$, $l=0$, respectively.}

So our construction associates fermionic fields 
$\phi_a^\lambda(z)$ to each cycle in the conjugacy class $[w]$. 
The distinction between positive and negative cycles will be 
apparent from the different properties of the associated fermions. 
For $1\le a \le r$ we have: $\phi_a^+(l)^\dagger = \phi_a^-(-l)$
 or equivalently:
$\phi_a^+(z)^\dagger=\phi_a^-(z)$. The $\phi_a^+(l)$'s and $\phi_a^-(l)$'s are
 independent. They satisfy the anticommutation relations:
$$\eqalign{&\{ \phi_a^+ (k),\phi_b^+(l)\} =0= \{ \phi_a^- (k),\phi_b^-(l)\}\cr
  &\{ \phi_a^+ (k),\phi_b^-(l)\} =\delta_{ab}\delta_{k,-l}\cr}
  \qquad\qquad 1\le a,b\le r,\ k,l\in {\textstyle{1\over 2}}+{\bf Z}\forno$$

\noindent In physics the fermions $\phi_a^+ (k)$ and $\phi_a^- (k)$ 
are associated to two fermionic particles of opposite charge. 
Hence, one speaks of charged  free fermions in this case.

For $r<a\le s$ 
we have
$\phi_a^+ (l)^\dagger=\phi_a^-(-l)=
  (-)^l\phi_a^+(-l)$ or equivalently:
$\phi_a^+(z)^\dagger=\phi_a^-(z)=
  \phi_a^+(\omega^{n_a}z)$, so there is only one (neutral) particle in this
case.   The anticommutation relations are
$$\{\phi_a^+(k),\phi_a^+(l)\}=
  (-)^l\delta_{ab}\delta_{k,-l}\qquad\qquad r<a,b\le s,\ k,l\in{\bf Z}\forno$$

\noindent These fermions are usually called neutral free fermions.

Finally, the $\sigma (l)$'s are also neutral free fermions. They satisfy
the relations $\sigma  (l)^\dagger=\sigma(-l)$
or equivalently:
$\sigma(z)^\dagger=\sigma (z)$
The anticommutation relations are in this case:
$$\{\sigma(k),\sigma(l)\}=\delta_{k,-l}\forno$$
\vskip.75cm

\noindent The second main result of this paper deals with 
a bosonization procedure for the fermionic fields $\phi_a^\lambda(z)
, 1\le a \le s,\ \lambda=\pm$. In order to describe it, 
we introduce bosonic fields $\hat\alpha_a(z), 1\le a\le s$ by:
$$\hat\alpha_a(z)=\cases{:\phi_a^+(z)\phi_a^-(z):&if $1\le a\le r$\cr
           {1\over 2}:\phi_a^+(z)\phi_a^-(z):
           &if $r<a\le s$ \cr}\forno$$

\noindent With the formal expansions (1.17) it is immediate that 
these $\hat\alpha_a(z)$'s have the following expansions:
$$\hat\alpha_a(z)=\cases{\sum_{k\in{\bf Z}} 
  z^{-{N\over n_a}k}\hat\alpha_a( {N\over n_a}k)
  &if $1\le a\le r$\cr\sum_{k\in {1\over 2}+{\bf Z}}
  z^{-{N\over n_a}k}\hat\alpha_a({N\over n_a}k)&if $r<a\le s$ \cr}\forno$$

\noindent From (1.6) we see that  $\hat\alpha_a(z)$ is 
--up to multiples of the identity and multiplicative factors-- 
equal to the field $\hat A^{aa}_{n_an_a}(+,-)(z)$. 
Therefore, the `modes' $\hat\alpha_a({N\over n_a}k)$ are elements 
of the affine algebra $\hat{so}_{2n+1}({\bf C})$. 
It is easy to verify that they satisfy the commutation relations for 
a set of bosonic oscillators:
$$[\hat\alpha_a({N\over n_a}k) ,
  \hat\alpha_b({N\over n_b}l)]
  =k\delta_{ab}\delta_{k,-l}\forno$$

\noindent The subalgebra 
$\hat{\underline s}_{[w]}:=
 \oplus_{k\ne 0}\oplus_{a=1}^s{\bf C}\hat\alpha_a({N\over n_a}k)
 \oplus{\bf C}c$ 
is a Heisenberg subalgebra of $\hat{so}_{2n+1}({\bf C})$ and in fact, 
the mapping $[w]\rightarrow\hat{\underline s}_{[w]}$ 
establishes the correspondence between conjugacy classes in 
$W(so_{2n+1}({\bf C}))$ and conjugacy classes of Heisenberg subalgebras
mentioned before.  We can now formulate

\vskip.3cm
\noindent{\bf Main result B.}

\noindent{\sl Let $1\le a\le r<b\le s$, $1\le d,e\le s$ and $\lambda =+(1),-(1)$, then
$$\eqalign{\phi_a^\lambda(z)=&Q_a^\lambda z^{{N\over
n_a}\left(\lambda\hat\alpha_a(0)
           +{1\over 2}\right)}
           {\rm exp}\left(-\lambda\sum_{k<0}{1\over k}
           z^{-{N\over n_a}k}\a ak\right)
           {\rm exp}\,\left(-\lambda\sum_{k>0}{1\over k}
           z^{-{N\over n_a}k}\a ak\right)
           \cr
           \phi_b^\lambda(z)=&Q_b
           {\rm exp}\left(-\lambda\sum_{k<0}{1\over k}
           z^{-{N\over n_b}k}\a bk\right)
           {\rm exp}\,\left(-\lambda\sum_{k>0}{1\over k}
           z^{-{N\over n_b}k}\a bk\right)
           \cr}
           \forno$$
where the operators $Q_d :V^\epsilon\rightarrow V^\epsilon$ are defined by:
$$\eqalign{Q_d\phi_e^\lambda(k)&=-\phi_e^\lambda(k)Q_d\quad\hbox{if } 
           e\ne d\cr
           Q_a\phi_a^\lambda(k)&=\phi_a^\lambda(k+\lambda)Q_a\cr
           Q_b\phi_b^\lambda(k)&=\phi_b^\lambda(k)Q_b\cr
	   Q_d\sigma(k)&=-\sigma(k)Q_d\cr
           }
  \forno$$
and
$$Q_d\vac=
           \cases{\phi_d^+({1\over2})\vac
                  &if $1\le d\le r$\cr\cr
                  \phi_d^+(0)\vac
                  &if $r<d\le s$\cr}
  \forno$$
Moreover,
$$\eqalign{Q_a^\dagger &=Q_a^{-1}\cr
           Q_b^\dagger &=Q_b ={1\over 2}Q_b^{-1}\cr}
  \forno$$
and 
$$[\a dk,Q_e]=\delta_{de}\delta_{k0}Q_e\forno$$
The following anticommutation relations are satisfied:
$$\{Q_d,Q_e\}=0
  \quad\hbox{for } d\ne e\forno$$}




The exponential dependence of the fermionic fields on the oscillators 
is of course quite familiar. 
Important is the appearance of the fermionic operators $Q_a$. 
There is again a big difference between $Q$'s associated to positive 
and to negative cycles; formula (1.23) shows that the operators 
$Q_a$ for $1\le a\le r$ have infinite period. For this reason we 
will call them fermionic translation operators. 
For $r< a\le s$ however, (1.25) shows that we have $Q_a^2={1\over 2}$. 
We will call these $Q$'s fermionic chirality operators.


Combining results A and B, we find for each conjugacy class 
$[w]$ a realization of $\hat{so}_{2n+1}({\bf C})$ on the spin module 
$V^\epsilon$, 
in terms of the oscillators $\hat\alpha_a({N\over n_a}k)$, the fermions
 $\sigma (k)$ and the operators $Q_a^\lambda,\ 1\le a\le s,\lambda=\pm 1$.
The action of the $\sigma (k)$'s and the $Q_a$'s generate the vacuum space
$V^{\epsilon,+}$ of $\hat{\underline s}_{[w]}$ in $V^\epsilon$. In section 7 we
will show that $V^{\epsilon,+}$ has the structure of the tensor product of the
group algebra of a lattice and an infinite Grassmann algebra.

All these
realizations are isomorphic as $\hat {so}_{2n+1}({\bf C})$  modules. In fact, by
considering the so called principal realization,  i.e. the one associated to the
partition $[{\overline {n}}]$ we will be able to prove the following
 
\break
{\noindent\bf Corollary C.}

{\noindent\sl The irreducible $B_\infty$-modules 
$V_{\overline 0}^0$ and $V_{\overline 1}^0$ and the  irreducible $D_\infty$-modules 
$V_{\overline 0}^{1\over 2}$ and $V_{\overline 1}^{1\over 2}$  remain irreducible when 
restricted to $\hat{so}_{2n+1}({\bf C})\subset B_\infty$ or $D_\infty$. 
They can be identified with the fundamental 
$\hat{so}_{2n+1}({\bf C})$ modules  $L(\Lambda_n)$, $L(\Lambda_n)$,
$L(\Lambda_0)$ and  $L(\Lambda_1)$, respectively.}





 \baselineskip=22pt plus.1pt



\count30=1                      %chapterno = 1 
\count31=0                      %sectionno = 0 
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0

\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count31
.\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count31 .\number\count33}


\long\def\proof#1{\noindent{\bf{Proof.}}\quad#1\qedd}
\def\qedd{\hfill\hbox{\vrule\vbox{\hrule\kern2truept\hbox{\kern2truept \vbox
to2pt{\hsize=2pt}\kern2truept}\kern2truept\hrule}\vrule}\vskip .5cm}



\def\so{so_{2n+1}({\bf C})} \def\SO{SO_{2n+1}({\bf C})}

\def\ad{{\rm ad}} \def\Ad{{\rm Ad}} \def\trace{{\rm trace}}

\def\E{{\cal E}} \def\F{{\cal F}} \def\tn{\hat \tau_{\underline n}}
\def\tw{\hat \tau_{[w]}} \def\Hn{H_{\underline n}} \def\Hw{H_{[w]}}
\def\Sn{S_{\underline n}} \def\Sw{S_{[w]}} \def\Xw{X_{[w]}} \def\sw{\hat
s_{[w]}} \def\hw{{\underline h}_{[w]}} \def\gl{gl_{2n+1}({\bf C})}

\def\oi{o(\infty )} \def\gli{gl(\infty )} \def\glib{\overline {gl(\infty )}}
\def\oib{\overline {o(\infty )}} \def\Ai{A_{\infty}} \def\Di{D_{\infty}}
\def\glt{\tilde g l_{2n+1}({\bf C})} \def\glh{\hat g l_{2n+1}({\bf C})}
\def\sot{\tilde s o_{2n+1}({\bf C})} \def\soh{\hat s o_{2n+1}({\bf C})}
\def\vac{\mid \overline 0 >}



\beginsection \chapno .  The Lie algebra $so_{2n+1}(\bf C)$\par \beginsection
\secno .  Introduction.\par
 Define $F_{ij}= E_{ij}-E_{ji}$, where $E_{ij}$ is the $(2n+1)\times (2n+1)$ matrix with a 1  on the
$(i,j)^{th}$ entry and zeros elsewhere, then the $F_{ij}$ with  $1\le i <j \le
2n+1$ form a basis of $\so$. The matrices $E_{ij}$ and $F_{ij}$ act on the
space of $(2n+1)$-dimensional column vectors, which we denote by ${\bf
C}^{2n+1}$. As a basis for this vector space ${\bf C}^{2n+1}$ we take the
collection $\{ e_i\}_{1\le i\le 2n+1}$ where $e_i$ is the vector with a one on
the $i^{th}$ entry and zeros elsewhere. In the sequel it will be convenient to
have another basis of ${\bf C}^{2n+1}$; 
  we define for $\lambda =+$ or $-$ and
for $1\le j\le n$ (we sometimes also write $\lambda = +1$ or $-1$ instead of
$+$ or $-$, respectively):
$$f_j(\lambda ) := {1\over \sqrt 2}(e_{2j-1}-i\lambda e_{2j})\forno$$
\noindent Clearly, the elements of $\{ f_j(+), f_j(-)\}_{1\le j\le
n}\cup\{e_{2n+1}\}$ form a new basis of ${\bf C}^{2n+1}$.

Let $\epsilon_j =iF_{2j-1,2j}$ $1\le j \le n$, then the subspace  $\underline
h=\oplus_{i=1}^n {\bf C}\epsilon_i$ is a Cartan subalgebra of $\so$. Notice that
$$(\epsilon_i \vert \epsilon_j)= \delta_{ij} \forno$$
Introduce the following elements of $\so$ ($1\le j,k \le n,\lambda ,\mu \in \{
+,-\}$):
$$\eqalign{G_{jk}(\lambda ,\mu ) &= 
          f_j(\lambda )f_k^t(\mu )-f_k(\mu )f_j^t(\lambda )\cr
G_j(\lambda)&=f_j(\lambda)e_{2n+1}^t-e_{2n+1}f_j(\lambda)^t\cr} \forno$$

\noindent where $v^t$ denotes the transpose of a vector $v\in {\bf C}^{2n+1}$.
The $G_{jk}(\lambda,\mu)$'s and $G_j(\lambda)$'s span the Lie algebra $\so$.
They are, however, not linearly independent; we have $G_{jk}(\lambda ,\mu 
)=-G_{kj}(\mu ,\lambda )$, hence $G_{jj}(\lambda ,\lambda )=0$. Notice that
$\epsilon_j = G_{jj}(+,-)$.  The commutation relations with $H \in \underline
h$ are:
$$\eqalign{[H,G_{jk}(\lambda,\mu)]&= (\lambda\epsilon_j + \mu\epsilon_k\vert
H)G_{jk}(\lambda,\mu) \cr [H,G_{j}(\lambda)]&= (\lambda\epsilon_j\vert
H)G_{j}(\lambda) \cr} \forno$$


\noindent This shows that the {\it root space decomposition} of $\so$ is given
by:
$$\so = \bigoplus_{1\le j\le n} {\bf C}\epsilon_j \bigoplus
  \bigoplus_{{1\le j< k\le n\atop \lambda,\mu=\pm} }{\bf
C}G_{jk}(\lambda,\mu)\bigoplus   
  \bigoplus_{{1\le j\le n\atop \lambda=\pm}}{\bf C}G_{j}(\lambda)
  \forno$$

\noindent and that the {\it set of roots} can be identified with:
$$\Delta =\{ \pm (\epsilon_i \pm \epsilon_j) \vert 1 \le i\not= j \le n\}\cup
\{\pm \epsilon_j\vert 1\le j \le n\}\forno$$











\beginsection\secno .  The Weyl group of $\so$

\noindent To every $\alpha \in \Delta$ we associate a linear transformation  of
$\underline h$ : $$r_\alpha (h) = h-(\alpha \vert h)\alpha 
  \qquad h \in \underline h
  \forno$$

\noindent Then $r_\alpha$ is a reflection in the hyperplane $H_\alpha = \{ x
\in \underline h \vert (\alpha \vert x) = 0\}$. Denote by $W$ the subgroup of
$GL(\underline h )$ generated by all reflections $r_\alpha$, $\alpha \in
\Delta$. This is called the {\it Weyl group} of $\so$. It  will play an
important role in the sequel.

The elements of $W$ act on the orthonormal basis $\{ \epsilon_i \}_{1\le i \le
n}$ of $\underline h$ by permuting the basis vectors and changing the signs of
an arbitrary number of them. So $W$ is isomorphic to the semidirect product of
$({\bf Z}/2{\bf Z})^{n}$ and $S_n$, the symmetric group on n letters.  If we
ignore the sign-changes, then each  element $w \in W$ determines a permutation
in $S_n$. Such a permutation can be expressed as a product of $s$ disjoint
cycles. We will write $(k_1 k_2 \cdots k_l)$ for the cycle $k_1\mapsto
k_2\mapsto\cdots\mapsto k_l\mapsto k_1$.  The action of the corresponding
$W$-element on $\underline h$ is then given by:
$$w(\epsilon_{k_i} )=
  \cases{\pm \epsilon_{k_{i + 1}} &if $1\le i<l$\cr &\cr
         \pm \epsilon_1 &if $i=l$\cr}$$

We now adjoin a sign to the cycle $(k_1 k_2 \cdots k_l)$ and call it {\it
positive}  (resp. {\it negative}) if $w^l(\epsilon_{k_1} )=\epsilon_{k_1}$
(resp. $w^l(\epsilon_{k_1} )= -\epsilon_{k_1}$). Suppose that $w$ is the
product of $r$ disjoint positive cycles of length $n_i$ ($1\le i\le r$) and
$s-r$ disjoint negative cycles of length $n_j$ ($r<j\le s$). Since disjoint
cycles commute, we can assume that $n_1\ge n_2\ge\cdots\ge n_r$ and $n_{r+1}
\ge n_{r+2}\ge\cdots\ge n_s$. Hence, to each  $w\in W$ we can associate a
so-called {\it signed cycle type}
$$[n_1 n_2 \cdots n_r \overline n_{r+1} \overline n_{r+2}
  \cdots \overline n_s] \qquad \sum_{i=1}^sn_i=n,\qquad  n_1\ge n_2\ge\cdots
n_r, n_{r+1} \ge n_{r+2}\ge\cdots \ge n_s\forno$$
 
\noindent  (A bar means that the cycle is negative.) It is clear that conjugate
cycles have the same signed cycle type.

Conversely, we can assign a conjugacy class $[w]$ to such a  signed cycle type
by the following representative $w\in W$:
$$w(\epsilon_{n_1 +\cdots + n_i + p}) =
           \cases{\epsilon_{n_1 +\cdots + n_i + p +1}&if $1\le p<n_{i+1}$
                  \cr &\cr
                  \epsilon_{n_1 +\cdots + n_i + 1}&if $p= n_{i+1} 
                  \hbox{and } 1\le i <r$
                  \cr & \cr
                  -\epsilon_{n_1+\cdots +n_i+1}
                  &if $p=n_{i+1} \hbox{and } r\le i<s$
                  \cr}\forno$$

The following proposition is a special case of the classification of the
conjugacy classes in the Weyl group of an arbitrary simple Lie algebra, due to
Carter [C].



\proclaim Proposition  \prono. The correspondence between the conjugacy classes
in the Weyl group $W$ of $\so$ and the signed cycle types (2.2.2) sketched
above, is a bijection.










\beginsection \secno . Lifting $w\in W$ 

\noindent Let $w$ be an arbitrary element of $W\subset {\rm Aut}({\underline
h})$. Such a transformation of the Cartan subalgebra can be lifted to an
element of the group ${\rm Aut}\left(\so\right)$. In the forthcoming sections
we will construct for any representative $w$ of the form (2.2.3) a $\hat w\in
\SO$ such that
$$\Ad (\hat w)(h):=\hat w h\hat w^{-1} =w(h)\qquad
  \hbox{for all } h\in \underline h$$

\noindent  So  $\Ad (\hat w)$  lifts the action of $w$ on $\underline h$ to the
full algebra $\so$. 

\par Because $\hat w\in\SO$, there exists a matrix
$X_{[w]}\in\so$ such that
$$\hat w = e^{2\pi iX_{[w]}}\forno$$ 

\noindent It is not hard to find such a matrix $X_{[w]}$. For any $\hat
w\in \SO$ there exists an orthonormal basis $\{s_i\}_{1\le i\le 2n+1}$ such
that
$$\eqalign{\hat w(s_{2k-1})&=\phantom{-}\cos(-\theta_k) s_{2k-1}+\sin(-\theta_k)
s_{2k}\cr \hat w(s_{2k})&=-\sin(-\theta_k) s_{2k-1}+\cos(-\theta_k) s_{2k}\cr
\hat w(s_{2n+1})&=s_{2n+1}\cr}\forno$$

\noindent showing that, with respect to this basis, $\hat w$ has the form
${\rm exp}\{2\pi i{\rm ad}H_{[w]}\}$, where
$H_{[w]}=\sum_{k=1}^n\theta_k\epsilon_k\in {\underline h}$. In other words: if
$S_{[w]}$ is the orthogonal matrix with columns $s_i, 1\le i\le 2n+1$, we have
$$\hat w = S_{[w]}e^{2 \pi i H_{[w]}}S_{[w]}^{-1}=e^{2 \pi i S_{[w]}H_{[w]}S_{[w]}^{-1}}\forno$$

\noindent In the sequel we will construct both $S_{[w]}$ and $H_{[w]}$.

Of course $\hat w$  and hence $X_{[w]}$ are not unique; $\hat w$ can be multiplied
from the right by an arbitrary element of the Cartan subgroup $H={\rm
exp}\{\underline h\}$ of $\SO$. Using this freedom, we will construct two
 lifts ${\rm Ad} (\hat w^\epsilon), \epsilon=0,{1\over 2}$ of one and
the same element $w\in W$. (The reason for this will become clear later.) Both 
lifts will be seen to have finite order as
automorphisms of the full algebra $\so$ (although not necessarily the same as
the order of $w$ on $\underline h$).






Now let $w$ be the representative (2.2.3) associated to the signed cycle type
$[n_1n_2\cdots n_r \overline n_{r+1}\overline n_{r+2}\cdots\overline n_s]$. A
quick glance at (2.2.3) shows that the matrix $\hat w$ should have the
following block diagonal form:
$$\pmatrix{\hat w_1\cr&\hat w_2\cr&&\ddots\cr&&&\hat w_s\cr&&&&\gamma\cr}$$

\noindent where $\hat w_i\in O_{2n_i}({\bf C})$ and $\gamma=\pm 1$. We will see
that ${\rm det}\,\hat w_i=1$ if $1\le i\le r$ and ${\rm det}\,\hat w_i=-1$ if
$r+1\le i\le s$. Hence $\gamma=1$ if $s-r$ is even and $\gamma=-1$ if $s-r
$ is odd. Related to this block form is the following partition of
$(2n+1)\times (2n+1)$ skew symmetric matrices:
$$\bordermatrix{&{\scriptstyle 2n_1\atop\scriptstyle\longleftrightarrow}
                &{\scriptstyle 2n_2\atop\scriptstyle\longleftrightarrow}
                &&{\scriptstyle 2n_s\atop\scriptstyle\longleftrightarrow}
                &{\scriptstyle 1\atop\scriptstyle\longleftrightarrow}\cr
                {\scriptstyle 2n_1\updownarrow }&B_{11}&B_{12}&\ldots&B_{1s}
                &b_{1}
                \cr
                {\scriptstyle 2n_2\updownarrow }&B_{21}&B_{22}&\ldots&B_{2s}
                &b_{2}
                \cr
                &\vdots&\vdots&\ddots&\vdots
                \cr
                {\scriptstyle 2n_s\updownarrow }&B_{s1}&B_{s2}&\ldots&B_{ss}
                &b_{s}
                \cr
                {\phantom{2n}\scriptstyle 1\hfill\updownarrow}              
                &-b^t_{1}&-b^t_2&\ldots
                &-b^t_s&0
                \cr
                }\forno$$ 

\noindent where $B_{ij}$ is a block of size $2n_i\times 2n_j$ and $b_i$ a
$2n_i$-column vector. Furthermore, $B_{ij}^t=-B_{ji}$. Clearly, the adjoint
action of $\hat w$ leaves this partition invariant.

With this blockform in mind we relabel the elements  $E_{kl}$, $F_{kl}$,
$G_{kl}(\lambda ,\mu )$, $G_k(\lambda)$ and define:
$$\eqalign{E^{ij}_{pq}&:= E_{2n_1+\cdots 2n_{i-1}+p,
           2n_1+\cdots +2n_{j-1}+q}\qquad 1\le p\le 2n_i, 1\le q\le 2n_j
           \cr
           F^{ij}_{pq}&:=F_{2n_1+\cdots +2n_{i-1}+p,
           2n_1+\cdots +2n_{j-1}+q}\qquad 1\le p\le 2n_i, 1\le q\le 2n_j
           \cr
           G^{ij}_{pq}(\lambda ,\mu )&:=G_{n_1+\cdots +n_{i-1}+p,
           n_1+\cdots +n_{j-1}+q}(\lambda ,\mu )           
           \qquad 1\le p\le n_i, 1\le q\le n_j \cr
            G^i_p(\lambda)&:=G_{n_1+\cdots
+n_{i-1}+p}(\lambda)\qquad\qquad\qquad\qquad 1\le p\le n_i\cr
e_p^i&:=e_{2n_1+\cdots +2n_{i-1}+p}\qquad 1\le
p\le 2n_i\cr
  f_p^i(\lambda )&:=f_{n_1+\cdots +n_{i-1}+p}(\lambda )\qquad 1\le p\le n_i\cr}
  \forno$$

\noindent Then 
$$\eqalign{G^{ij}_{pq}(\lambda ,\mu )&=f_p^i(\lambda )f_q^j(\mu )^t-
                             f_q^j(\mu )f_p^i(\lambda )^t\cr
G^i_p(\lambda)&=f_p^i(\lambda )e_{2n+1}^t-e_{2n+1}f_p^i(\lambda )^t\cr}\forno$$

\noindent Finally, we also redefine the roots:
$$\eqalign{\pm (\epsilon^i_p \pm \epsilon^j_q) &:=
  \pm (\epsilon_{n_1+\cdots +n_{i-1}+p}\pm \epsilon_{n_1+\cdots
  +n_{j-1}+q})\cr \pm \epsilon^i_p&:=\pm\epsilon_{n_1+\cdots
+n_{i-1}+p}\cr}\forno$$










\beginsection\secno .  The case of one positive cycle \par In this section we
will describe the case that $[w]$ corresponds to one positive cycle of length
$n$. Take the representative $w\in W$ defined by (2.2.3);
$$w(\epsilon_i) =
           \cases{\epsilon_{i +1}&if $1\le i<n$
                  \cr & \cr
                  \epsilon_1&if $i=n$
                  \cr}\forno$$ 

\noindent The following lemma was proved in [KrL91b].

\proclaim Lemma  \prono. Let $w$ be given by (2.4.1) and let $\hat w$ be the
matrix $$\eqalign{\hat w=&\cos ({n+1\over n}\pi )(E_{1,2n-1}+ E_{2,2n} 
                   \sum_{k=1}^{n-1}(E_{2k+2,2k} +E_{2k+1,2k-1}))+\cr
                  &+\sin ({n+1\over n}\pi )(E_{2,2n-1}-E_{1,2n} +
                   \sum_{k=1}^{n-1}(E_{2k+2,2k-1}-E_{2k+1,2k}))\cr
                  &+E_{2n+1,2n+1}\cr}\forno$$ \noindent Then ${\rm Ad}(\hat w)$
is a lift of $w$ to $Aut \,(\so)$. Define the orthogonal matrix $S$ by
$$S:={1\over\sqrt n}\sum_{j,k=1}^n \cos ({2jk\over n}\pi ) 
                                   (E_{2j-1,2k-1}+E_{2j,2k})
                                  +\sin ({2jk\over n}\pi) 
                                   (E_{2j,2k-1}-E_{2j-1,2k})+E_{2n+1,2n+1}
                                   \forno$$ Then $\hat w= e^{2\pi i X}$, with
$X=SHS^{-1}$ and $$H={1\over 2n}\sum_{k=1}^n (n-2k+1)\epsilon_k\forno$$


\par It is easy to construct another lift from this one. We just multiply the
matrix $\hat w$ from the right by the matrix

$$e^{2\pi i({1\over
2}\sum_{j=1}^n\epsilon_j)}=-\sum_{j=1}^{2n}E_{jj}+E_{2n+1,2n+1}\forno$$

\noindent This matrix clearly commutes with $S$. With this remark the following
lemma follows immediately from lemma 2.4.1.

\proclaim Lemma  \prono. Let $\epsilon$ be $0$ or ${1\over 2}$. Set $\hat
w^\epsilon:=\hat w \, .\, {\rm exp}\{2\pi i({1\over
2}-\epsilon)\sum_{j=1}^n\epsilon_j\}$, where $\hat w$ is given by (2.4.2). Then
${\rm Ad}(\hat w^\epsilon)$ is a lift of the element $w\in W$ given by (2.4.1).
Moreover $\hat w^\epsilon= e^{2\pi i X^\epsilon}$, with
$X^\epsilon=SH^\epsilon S^{-1}$, S given by (2.4.3) and $$H^\epsilon={1\over
2n}\sum_{k=1}^n (2n(1-\epsilon)-2k+1)\epsilon_k\forno$$


The matrix $S$ is an orthogonal matrix, meaning that the mapping ${\rm
Ad}(S^{-1}):\so\to\so$ is an isomorphism of $\so$. In particular, it maps the
standard Cartan subalgebra $\underline h$ to another Cartan subalgebra, which
we denote by $\underline h_{[n]}$. So we have the following commutative diagram

\def\normalbaselines{\baselineskip20pt
       \lineskip3pt \lineskiplimit3pt }
  \def\mapright#1{\smash{
      \mathop{\longrightarrow}\limits^{#1}}}
  \def\rmapdown#1{\Big\downarrow
      \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
  \def\lmapdown#1{\Big\downarrow
      \llap{$\vcenter{\hbox{$\scriptstyle#1\quad$}}$}}
 $$\matrix{\underline h&\mapright{w=e^{2\pi i{\rm ad} X^\epsilon}}&\underline
h\cr
          \lmapdown{{\rm Ad }(S^{-1})}&&\rmapdown{{\rm Ad} (S^{-1})}\cr
           \underline h_{[n]}&\mapright{e^{2\pi i{\rm ad} H^\epsilon}}
            &\underline h_{[n]}\cr}\forno$$ showing that $e^{2\pi i\ad
H^\epsilon}$ corresponds to $w$ on $\underline h_{[n]}$. We now want to
determine the root vectors with respect to this new Cartan subalgebra. Of
course they are  $$\eqalign{A_{jk}(\lambda ,\mu )&:=S^{-1}G_{jk}(\lambda ,\mu
)S\cr
            A_j(\lambda)&:=S^{-1}G_{j}(\lambda)S\cr} \forno$$ Let $\hat
f_j(\lambda ):=S^{-1}f_j(\lambda ), \hat e_{2n+1} =S^{-1}e_{2n+1}$; because $S$
is orthogonal, we can write: $$\eqalign{A_{jk}(\lambda ,\mu )&=\hat f_j(\lambda
)\hat f_k(\mu )^t-
                        \hat f_k(\mu )\hat f_j(\lambda )^t\cr
A_{j}(\lambda)&=\hat f_j(\lambda )\hat e_{2n+1}^t-
                        \hat e_{2n+1}\hat f_j(\lambda )^t\cr}\forno$$ \proclaim
Lemma  \prono. Let $\omega =e^{2\pi i\over n}$, then $$\hat f_j(\lambda
)={1\over\sqrt n}\sum _{p=1}^n
                     \omega^{-\lambda jp}f_p(\lambda );\qquad\hat
e_{2n+1}=e_{2n+1}\forno$$ Consequently: $$\eqalign{A_{jk}(\lambda ,\mu
)&={1\over n}\sum_{p,q=1}^n
  \omega^{-\lambda jp -\mu kq }G_{pq}(\lambda , \mu )\cr
A_{j}(\lambda)&={1\over \sqrt{n}}\sum_{p=1}^n
  \omega^{-\lambda jp}G_{p}(\lambda)\cr}\forno$$ \par \proof
{A straightforward calculation.}






\beginsection\secno .  The case of two negative cycles \par In this section we
consider the special case of two negative cycles. We assume that $n=n_1+n_2$,
and that $w$ is given by (see (2.2.3)): $$w(\epsilon_p^i
)=\cases{\epsilon_{p+1}^i& if $p<n_i$\cr
                          -\epsilon_1^i   & if $p=n_i$\cr}\forno$$
 
\noindent The following lemma was proved in [KrL91b].

\proclaim Lemma  \prono. Define the matrix $\hat w\in SO_{2n+1}({\bf C})$ by:
$$\hat w= \sum_{p=1}^2 (E_{1,2n_p}^{pp}+E_{2,2n_p-1}^{pp}+
                             \sum_{j=1}^{2n_p-2} E_{j+2,j}^{pp})+E_{2n+1,2n+1}
\forno$$  Then ${\rm Ad}\,(\hat w)$ is a lift of the element $w$ defined in
(2.5.1). Let $S$ be the ortogonal matrix  $$\eqalign{S=&{1\over\sqrt
{n_1}}\sum_{i=1}^{n_1}\Biggl\{
              {1\over\sqrt 2}\Bigl(
              (-1)^iE_{2i-1,1}^{11} +(-1)^{n_1+i}E_{2i,1}^{11} +
              E_{2i-1,2n_2-1}^{12} +E_{2i,2n_2-1}^{12}\Bigr)+\cr
             &\qquad\qquad
              \sum_{j=2}^{n_1}\Bigl(\cos ({i(n_1-j+1)\over {n_1}}\pi )
              (E_{2i-1,2j-1}^{11} +(-1)^{n_1-j+1}E_{2i,2j-1}^{11})+\cr
             &\qquad\qquad\qquad\sin ({i(n_1-j+1)\over {n_1}}\pi )
              (E_{2i-1,2j}^{11} +(-1)^{n_1-j+1}E_{2i,2j}^{11})\Bigr)\Biggr\}+
             \cr
             &{1\over\sqrt {n_2}}\sum_{i=1}^{n_2} \Biggl\{{1\over\sqrt
2}\Bigl(              (-1)^iE_{2i-1,2}^{21} +(-1)^{n_2+i}E_{2i,2}^{21} +
              E_{2i-1,2n_2}^{22} +E_{2i,2n_2}^{22}\Bigr)+\cr
             &\qquad\qquad
              \sum_{j=1}^{n_2-1}\Bigl(\cos ({i(n_2-j)\over {n_2}}\pi )
              (E_{2i-1,2j-1}^{22} +(-1)^{n_2-j}E_{2i,2j-1}^{22})+\cr
             &\qquad\qquad\qquad\sin ({i(n_2-j)\over {n_2}}\pi )
              (E_{2i-1,2j}^{22} +(-1)^{n_2-j}E_{2i,2j}^{22})\Bigr)\Biggr\}
             \cr&+E_{2n+1,2n+1}
              \cr}\forno$$ Then $\hat w= e^{2\pi i X}$, with $X=SHS^{-1}$ and
$$H= \sum_{j=1}^{n_1}
       {n_1-j+1\over 2n_1}\epsilon_j^1+
     \sum_{j=1}^{n_2}
       {n_2-j\over 2n_2}\epsilon_j^2
     \forno$$

As in the previous section we determine  $\hat f_j^a(\lambda ):=
S^{-1}f_j^a(\lambda ), \hat e_{2n+1}:=S^{-1}e_{2n+1}$  and the new root vectors
$A_{jk}^{ab}(\lambda ,\mu ):=S^{-1}G_{jk}^{ab}(\lambda ,\mu )S, A_{j}^{a}
(\lambda):=S^{-1}G_{j}^{a}(\lambda)S$.


\proclaim Lemma  \prono. Let $\omega_a =e^{2\pi i\over {2n_a}}$ and define
$$\eqalign{\alpha_{bj}^{1k}(\lambda ,\mu):=& 
  \cases{(-1)^j{1-i\lambda (-1)^{n_1}\over 2}&if $k=1$ and $b=1$\cr
         \omega_1^{\mu j(n_1-k+1)}{1-i\lambda (-1)^{n_1-k+1}\over\sqrt 2}
         &if $1<k\le n_1$ and $b=1$\cr
         1-i\lambda\over 2&if $k=n_2$ and $b=2$\cr
         0&in all other cases\cr}\cr
  \alpha_{bj}^{2k}(\lambda , \mu):=& 
  \cases{(-1)^j\mu {i+\lambda (-1)^{n_2}\over 2}&if $k=1$ and $b=1$\cr
         \omega_2^{\mu j(n_2-k)}{1-i\lambda (-1)^{n_2-k}\over\sqrt 2}
         &if $1\le k< n_2$ and $b=2$\cr
         \mu {i+\lambda\over 2}&if $k=n_2$ and $b=2$\cr
         0&in all other cases\cr}\cr}\forno$$ Then $$\hat f_j^a(\lambda
)={1\over\sqrt {2n_a}}\sum _{b=1}^2
                 \sum_{k=1}^{n_b}\sum_{\mu = +,-}    
                 \alpha_{bj}^{ak}(\lambda ,\mu )f_k^b(\mu ),\qquad \hat e_{2n+1}
=e_{2n+1}\forno$$ Consequently $$\eqalign{A_{jk}^{ab}(\lambda ,\mu )&={1\over
{2\sqrt {n_an_b}}}
  \sum_{c,d=1}^2\sum_{p=1}^{n_c}\sum_{q=1}^{n_d}\sum_{\nu ,\xi =+,-}
  \alpha_{cj}^{ap}(\lambda ,\nu)\alpha_{dk}^{bq}(\mu ,\xi)
  G_{pq}^{cd}(\nu ,\xi )\cr A_{j}^{a}(\lambda)&={1\over {2\sqrt {n_a}}}
  \sum_{c=1}^2\sum_{p=1}^{n_c}\sum_{\nu=+,-}
  \alpha_{cj}^{ap}(\lambda ,\nu)
  G_{p}^{c}(\nu)\cr}\forno$$ \par \proof {A straightforward calculation.}






\beginsection \secno . The case of one negative cycle\par

\noindent Here  we consider the case where $[w]$ corresponds to a single
negative cycle. As a representative for this conjugacy class we take the
element $w\in W$ defined in (2.2.3):

$$w(\epsilon_i)=\cases{\epsilon_{i+1}&if\ $1\le i \le n$\cr-\epsilon_1&if\
$i=1$\cr}\forno$$


\proclaim Lemma \prono.
 Define the matrix $\hat w\in SO_{2n+1}({\bf C})$ by $$\hat
w:=E_{1,2n}+E_{2,2n-1}+\sum_{j=1}^{2n-2}E_{j+2,j}-E_{2n+1,2n+1}\forno$$ Then
${\rm Ad}\,(\hat w)$ is a lift of the element $w$ defined in (2.6.1). Let $S$ be
the ortogonal matrix $$\eqalign{S= {1\over
\sqrt{n}}\sum_{j=1}^{n}\sum_{k=2}^{n}&\Bigl\{\cos\left({j(n-k+1)\pi\over
n}\right)(E_{2j-1,2k-1}+(-)^{n-k+1} E_{2j,2k-1})\cr
&+\sin\left({j(n-k+1)\pi\over n}\right)(E_{2j-1,2k}+(-)^{n-k+1}
E_{2j,2k})\Bigr\}\cr &+{1\over\sqrt{2n}}\sum_{j=1}^{n}(-)^j(E_{2j-1,1}+(-)^{n} 
E_{2j,1})+{1\over\sqrt{2n}}\sum_{j=1}^{n} (E_{2j-1,2n+1}+
E_{2j,2n+1})\cr&+E_{2n+1,2}\cr }\forno$$ Then $\hat w= e^{2\pi i X}$, with
$X=SHS^{-1}$ and $$H={1\over 2n}\sum_{j=1}^n(n-j+1)\epsilon_j\forno$$ \par\proof
{Completely similar to the proofs of lemmas 2.4.1 and 2.5.1.}

 


Again we compute $\hat f_j(\lambda)=S^{-1}f_j(\lambda), \hat
e_{2n+1}=S^{-1}e_{2n+1}$ and the elements $A_{jk}(\lambda ,
\mu)=S^{-1}G_{jk}(\lambda , \mu)S$ and  $A_{j}(\lambda)=S^{-1}G_{j}(\lambda)S$.


\proclaim Lemma \prono. Let  $\omega:=e^{2\pi i/2n}$ and define
$$\eqalign{\alpha_j^k(\lambda,\mu)&={1-i\lambda (-)^{n-k+1}\over
\sqrt{2}}\omega^{\mu j(n-k+1)}\left({1\over \sqrt{2}}\right)^{\delta_{k1}}\cr
\beta_j(\lambda)&:={1-i\lambda \over
\sqrt{2}}\cr\gamma^k(\mu)&:={i\mu\over\sqrt{2}}{\delta_{k1}}\cr}\forno$$ Then:
$$\eqalign{\hat
f_j(\lambda)&={1\over\sqrt{2n}}\sum_{k=1}^n\sum_{\mu=\pm}\alpha_j^k(\lambda,\mu)f_k(\mu)+{1\over\sqrt{2n}}\beta_j(\lambda)e_{2n+1}\cr
\hat e_{2n+1}&=\sum_{k=1}^n\sum_{\mu=\pm}\gamma^k(\mu)f_k(\mu)\cr}\forno$$
Consequently: $$\eqalign{A_{jk}(\lambda,\mu)&={1\over
2n}\sum_{p,q=1}^n\sum_{\nu,\xi=\pm}\alpha_j^p(\lambda,\nu)\alpha_k^q(\mu,\xi)G_{pq}(\nu,\xi)\cr&+
{1\over 2n}\sum_{p=1}^n\sum_{\nu=\pm}(\alpha_j^p(\lambda,\nu)\beta_k(\mu)-
\alpha_k^p(\mu,\nu)\beta_j(\lambda))G_{p}(\nu)\cr
A_j(\lambda)&={1\over\sqrt{2n}}\sum_{p,q=1}^n\sum_{\nu,\xi=\pm}
\alpha_j^p(\lambda,\nu)\gamma^q(\xi)G_{pq}(\nu,\xi)\cr
&-{1\over\sqrt{2n}}\sum_{q=1}^n\sum_{\xi=\pm}\beta_j(\lambda)\gamma^q(\xi)G_{q}(\xi)\cr}\forno$$
\par\proof {An easy calculation.}





\beginsection \secno .  The general case

\noindent In the previous three sections we have discussed lifts of elements
$w\in W$ whose conjugacy classes correspond to the signed cycle types $[n],
[\overline n_1\overline n_2]$ and $[\overline n]$. Here we will combine the
results obtained in these special cases to treat an arbitrary signed cycle type
$[n_1n_2\cdots n_r\overline n_{r+1}\overline n_{r+2}\cdots \overline n_s]$. We
start by giving some definitions.


Denote by $w_a(1\le a\le r) $, $w_{ab}(r< a\ne b\le s)$ and $w_\eta(\eta=\pm)$
the following elements of the Weyl group: $$\eqalign{w_a(\epsilon_j^b
)=&(1-\delta_{ab})\epsilon_j^b +
           \delta_{ab}(1-\delta_{j,n_a})\epsilon_{j+1}^a+
           \delta_{ab}\delta_{j,n_a}\epsilon_1^a\cr
           w_{ab}(\epsilon_j^c )=&(1-\delta_{ac})(1-\delta_{bc})\epsilon_j^c+
           \delta_{ac}(1-\delta_{j,n_a})\epsilon_{j+1}^a-
           \delta_{ac}\delta_{j,n_a}\epsilon_1^a\cr
           &\qquad+\delta_{bc}(1-\delta_{j,n_b})\epsilon_{j+1}^b-
           \delta_{bc}\delta_{j,n_b}\epsilon_1^b\cr
           w_\eta(\epsilon^a_j)=&\epsilon^a_j+{\eta-1\over
2}\delta_{as}\Bigl(\epsilon^s_j-(1-\delta_{j,n_s})\epsilon_{j+1}^s+\delta_{j,n_s}\epsilon_1^s\Bigr)
\cr}\forno$$ Define: $$\eqalign{\hat w_a^\epsilon :=&(4\epsilon-1)\cos
({n_a+1\over n_a}\pi )
                      (E_{1,2n_a-1}^{aa}+ E_{2,2n_a}^{aa} +
                      \sum_{k=1}^{n_a-1}
                      (E_{2k+2,2k}^{aa} +E_{2k+1,2k-1}^{aa}))+\cr
                     &(4\epsilon-1)\sin ({n_a+1\over n_a}\pi )
                      (E_{2,2n_a-1}^{aa}-E_{1,2n_a}^{aa} +
                      \sum_{k=1}^{n_a-1}
                      (E_{2k+2,2k-1}^{aa}-E_{2k+1,2k}^{aa}))\qquad\qquad
(\epsilon=0,{\textstyle 1\over  \textstyle 2})\cr
               \hat w_{ab}:=&\sum_{p=a,b} (E_{1,2n_p}^{pp}+E_{2,2n_p-1}^{pp}+
                             \sum_{j=1}^{2n_p-2} E_{j+2,j}^{pp})\cr \hat
w_\eta:=&{1-\eta\over
2}(E^{ss}_{1,2n_s}+E^{ss}_{2,2n_s-1}+\sum_{j=1}^{2n_s-2}E^{ss}_{j+2,j})+\eta
E_{2n+1,2n+1}\cr} \forno$$   

$$S_a:={1\over\sqrt n_a}\sum_{j,k=1}^{n_a} \cos ({2jk\over {n_a}}\pi ) 
                                   (E_{2j-1,2k-1}^{aa}+E_{2j,2k}^{aa})
                                  +\sin ({2jk\over {n_a}}\pi) 
                                   (E_{2j,2k-1}^{aa}-E_{2j-1,2k}^{aa})\forno$$

$$\eqalign{S_{ab}:=&{1\over\sqrt {n_a}}\sum_{i=1}^{n_a}\Biggl\{{1\over\sqrt 2}
              \Bigl((-1)^iE_{2i-1,1}^{aa} +(-1)^{n_a+i}E_{2i,1}^{aa} +
              E_{2i-1,2n_b-1}^{ab} +E_{2i,2n_b-1}^{ab}\Bigr)+\cr
             &\qquad\qquad
              \sum_{j=2}^{n_a}\Bigl(\cos ({i(n_a-j+1)\over {n_a}}\pi )
              (E_{2i-1,2j-1}^{aa} +(-1)^{n_a-j+1}E_{2i,2j-1}^{aa})+\cr
             &\qquad\qquad\qquad\sin ({i(n_a-j+1)\over {n_a}}\pi )
              (E_{2i-1,2j}^{aa} +(-1)^{n_a-j+1}E_{2i,2j}^{aa})
             \Bigr)\Biggr\}+\cr
             &{1\over\sqrt {n_b}}\sum_{i=1}^{n_b}\Biggl\{{1\over\sqrt 2}
              \Bigl((-1)^iE_{2i-1,2}^{ba} +(-1)^{n_b+i}E_{2i,2}^{ba} +
              E_{2i-1,2n_b}^{bb} +E_{2i,2n_b}^{bb}\Bigr)+\cr
             &\qquad\qquad
              \sum_{j=1}^{n_b-1}\Bigl(\cos ({i(n_b-j)\over {n_b}}\pi )
              (E_{2i-1,2j-1}^{bb} +(-1)^{n_b-j}E_{2i,2j-1}^{bb})+\cr
             &\qquad\qquad\qquad\sin ({i(n_b-j)\over {n_b}}\pi )
              (E_{2i-1,2j}^{bb} +(-1)^{n_b-j}E_{2i,2j}^{bb})
             \Bigr)\Biggr\}\cr}
              \forno$$

 $$\eqalign{S_\eta:={1-\eta\over 2}\Bigl\{ {1\over
\sqrt{n_s}}\sum_{j=1}^{n_s}\sum_{k=2}^{n_s}&\Bigl\{\cos\left({j(n_s-k+1)\pi\over
n_s}\right)(E^{ss}_{2j-1,2k-1}+(-)^{n_s-k+1} E^{ss}_{2j,2k-1})\cr
&+\sin\left({j(n_s-k+1)\pi\over n_s}\right)(E^{ss}_{2j-1,2k}+(-)^{n_s-k+1}
E^{ss}_{2j,2k})\Bigr\}\cr
+{1\over\sqrt{2n_s}}\sum_{j=1}^{n_s}&(-)^j(E^{ss}_{2j-1,1}+(-)^{n_s} 
E^{ss}_{2j,1})+{1\over\sqrt{2n_s}}\sum_{j=1}^{n_s} (E^{ss}_{2j-1,2n_s+1}+
E^{ss}_{2j,2n_s+1})+E^{ss}_{2n_s+1,2}\Bigr\}\cr +{1+\eta\over
2}E_{2n+1,2n+1}\cr}\forno$$

$$\eqalign{H_a^\epsilon&:={1\over 2n_a}\sum_{k=1}^{n_a}
(2n_a(1-\epsilon)-2k+1)\epsilon_k^a\cr H_{ab}&:= \sum_{k=1}^{n_a}{n_a-k+1\over
2n_a}\epsilon_k^a+
     \sum_{k=1}^{n_b}{n_b-k\over 2n_b}\epsilon_k^b\cr H_\eta&:={1-\eta\over
2}{1\over 2n_s}\sum_{j=1}^{n_s}(n_s-j+1)\epsilon^s_j\cr}     \forno$$

\proclaim Lemma  \prono. Let $[w]=[n_1 n_2 \cdots n_r \overline n_{r+1}
\overline n_{r+2}
  \cdots \overline n_s] $ be a signed cycle type of $n$, then the element
$$w:=w_{(-)^{s-r}}\prod_{a=1}^r w_a \prod_{a=1}^{\left[s-r\over 2\right]}
w_{r+2a-1,r+2a}\forno$$ is a representative of the conjugacy class of the Weyl
group that corresponds to this signed cycle type.  For $\epsilon=0,{1\over 2}$
set $$\hat w^\epsilon:=\sum_{a=1}^r \hat w_a^\epsilon
+\sum_{a=1}^{\left[s-r\over 2\right]}
  \hat w_{r+2a-1,r+2a}+\hat w_{(-)^{s-r}}\forno$$ Then ${\rm Ad}\,(\hat
w^\epsilon)$ is a lift of $w$. Define an orthogonal matrix
$$S_{[w]}:=\sum_{a=1}^r S_a +\sum_{a=1}^{\left[s-r\over 2\right]}
S_{r+2a-1,r+2a}+S_{(-)^{s-r}}\forno$$ Then $\hat w^\epsilon=e^{2\pi
iX_{[w]}^\epsilon}$ with
$X_{[w]}^\epsilon=S_{[w]}H_{[w]}^\epsilon S_{[w]}^{-1}\in\so$  and
$$H_{[w]}^\epsilon:=\sum_{a=1}^r H_a^\epsilon +\sum_{a=1}^{\left[s-r\over
2\right]} H_{r+2a-1,r+2a}+H_{(-)^{s-r}}\forno$$ \noindent Let
$$\omega_a=\cases{e^{2\pi i\over n_a}&for $1\le a\le r$\cr
                  e^{2\pi i\over 2n_a}&for $r< a\le s$\cr}$$ Define the
constants $\alpha_{bj}^{ak}(\lambda ,\mu),\beta^a_j(\lambda),
\gamma^j_a(\lambda),\delta$ as follows:

\noindent for $1\le a\le r$: $$\alpha_{bj}^{ak}(\lambda ,\mu)=
         \cases{\sqrt 2\omega_a^{-\lambda jk}&if $a=b$ and $\lambda =\mu$\cr
                0&in all other cases\cr}\forno$$ for $1\le a\le
\left[{s-r+1\over 2}\right]$: $$\alpha_{bj}^{r+2a-1,k}(\lambda ,\mu)= 
  \cases{(-1)^j{1-i\lambda (-1)^{n_{r+2a-1}}\over 2}&if $k=1$ and $b=r+2a-1$\cr
         \omega_{r+2a-1}^{\mu j(n_{r+2a-1}-k+1)}
         {1-i\lambda(-1)^{n_{r+2a-1}-k+1}\over\sqrt 2}
         &if $1<k\le n_{r+2a-1}$ and $b=r+2a-1$\cr
         {1-i\lambda\over 2}&if $k=n_{r+2a}$ and $        
b=r+2a$\cr
         0&in all other cases\cr}\forno$$ for $1\le a\le \left[{s-r\over
2}\right]$:
 $$ \alpha_{bj}^{r+2a,k}(\lambda ,\mu)= 
  \cases{(-1)^j\mu {i+\lambda (-1)^{n_{r+2a}}\over 2}&if $k=1$ and $b=r+2a-1$\cr
         \omega_{r+2a}^{\mu j(n_{r+2a}-k)}
         {1-i\lambda (-1)^{n_{r+2a}-k}\over\sqrt 2}
         &if $1\le k< n_{r+2a}$ and $b=r+2a$\cr
         \mu {i+\lambda\over 2}&if $k=n_{r+2a}$ and $b=r+2a$\cr
         0&in all other cases\cr}\forno$$
for $1\le a\le s$:
$$\eqalign{\beta^a_j(\lambda)&={1-(-)^{s-r}\over
2}{1-i\lambda\over\sqrt{2}}\delta_{as}\cr\gamma^j_a(\lambda)&={1-(-)^{s-r}\over
2}{i\lambda\over\sqrt{2}}\delta_{as}\delta_{j1}\cr\delta&={1+(-)^{s-r}\over
2}\cr}\forno$$ Then: $$\eqalign{\hat
f^a_j(\lambda)&={1\over\sqrt{2n_a}}\sum_{b=1}^s\sum_{k=1}^{n_b}
\sum_{\mu=\pm}\alpha_{bj}^{ak}(\lambda,\mu)f^b_k(\mu)+{1\over\sqrt{2n_a}}\beta^a_j(\lambda)e_{2n+1}\cr
\hat
e_{2n+1}&=\sum_{a=1}^s\sum_{j=1}^{n_a}\sum_{\mu=\pm}\gamma^j_a(\mu)f_j^a(\mu)+\delta
e_{2n+1}\cr}\forno$$ Consequently: $$\eqalign{A_{jk}^{ab}(\lambda,\mu)&={1\over
2\sqrt{n_an_b}}\sum_{c,d=1}^s\sum_{p=1}^{n_c}
\sum_{q=1}^{n_d}\sum_{\nu,\xi=\pm}\alpha_{cj}^{ap}(\lambda,\nu)\alpha_{dk}^{bq}(\mu,\xi)G^{cd}_{pq}(\nu,\xi)\cr&+
{1\over
2\sqrt{n_an_b}}\sum_{c=1}^s\sum_{p=1}^{n_c}\sum_{\nu=\pm}(\alpha_{cj}^{ap}(\lambda,\nu)\beta_k^b(\mu)-
\alpha_{ck}^{bp}(\mu,\nu)\beta^a_j(\lambda))G_p^c(\nu)\cr
A_j^a(\lambda)&={1\over\sqrt{2n_a}}\sum_{c,d=1}^s\sum_{p=1}^{n_c}\sum_{q=1}^{n_d}\sum_{\nu,\xi=\pm}
\alpha_{cj}^{ap}(\lambda,\nu)\gamma^q_d(\xi)G_{pq}^{cd}(\nu,\xi)\cr
&-{1\over\sqrt{2n_a}}\sum_{d=1}^s\sum_{q=1}^{n_d}\sum_{\xi=\pm}\beta_j^a(\lambda)\gamma^q_d(\xi)G^d_q(\xi)+{1\over\sqrt{2n_a}}\sum_{d=1}^s\sum_{p=1}^{n_c}\sum_{\nu=\pm}\alpha_{cj}^{ap}(\lambda,\nu)\delta
G^c_p(\nu)\cr}\forno$$ \proof {A straightforward generalization of the results
of the previous
        two sections.}


The lemma tells us that the lift ${\rm Ad}\,(\hat w^\epsilon)$ of $w\in W$ can
be diagonalized to $\hat\tau_{[w]}^\epsilon:={\rm exp}(2\pi i {\rm ad}
H_{[w]}^\epsilon)$ by conjugation with the automorphism ${\rm Ad}(S_{[w]}^{-1})$.
In the sequel we will work with this $\hat\tau_{[w]}^\epsilon$ rather then with
${\rm Ad}(\hat w^\epsilon)$. From the commutative diagram

\def\normalbaselines{\baselineskip20pt
       \lineskip3pt \lineskiplimit3pt }
  \def\mapright#1{\smash{
      \mathop{\longrightarrow}\limits^{#1}}}
  \def\rmapdown#1{\Big\downarrow
      \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
  \def\lmapdown#1{\Big\downarrow
      \llap{$\vcenter{\hbox{$\scriptstyle#1\quad$}}$}}
 $$\matrix{\underline h&\mapright{{\rm Ad}(\hat w^\epsilon)}&\underline h\cr
          \lmapdown{{\rm Ad}(S_{[w]}^{-1})}&&\rmapdown{{\rm Ad}(S_{[w]}^{-1})}\cr
           \underline h_{[w]}&\mapright{\hat\tau_{[w]}^\epsilon}
            &\underline h_{[w]}\cr}\forno$$

\noindent it is clear that the restriction of $\hat\tau_{[w]}^\epsilon$ to the
new Cartan subalgebra $\underline h_{[w]}:={\rm Ad}(S_{[w]}^{-1})\left( \underline{h}\right)$
corresponds to the  action of the Weyl element $w$ on $\underline h$. The Cartan subalgebra ${\underline h}$ has a basis $\{\epsilon^a_j\}_{1\le a\le
s,1\le j\le n_a}$. The image of this basis under the isomorphism ${\rm
Ad}(S_{[w]}^{-1})$ is of course $\{A^{aa}_{jj}(+,-)\}_{1\le a\le
s,1\le j\le n_a}$.  The root
vectors with respect to $\underline h_{[w]}$ are the elements
$A^{ab}_{jk}(\lambda,\mu)$ and $A^{a}_{j}(\lambda)$ from the lemma.

Let $N$ be the order of $\hat\tau_{[w]}^\epsilon$. (Actually, $N$ depends on
$\epsilon$, as we will see below, and it would be more precise to write
$N^\epsilon$ instead of $N$. To avoid cumbersome notation in the sequel, we
will suppress this $\epsilon$-dependence.) Like any other finite order
automorphism $\hat\tau_{[w]}^\epsilon$ induces a ${\bf Z}_N\equiv {\bf Z}/N{\bf
Z}$ gradation of $\so$:
$$\so =\bigoplus_{{\overline \jmath}\in {\bf Z}_N} \so_{\overline
\jmath}\forno$$

\noindent where the eigenspace $\so_{\overline \jmath}$ is defined by
$$\so_{\overline \jmath}:=\{ y\in \so \mid \hat\tau_{[w]}^\epsilon  (y)=\omega^j y\}\forno$$

\noindent Here we have introduced the $N^{th}$ root of unity $\omega :=e^{2\pi
i/N}$ and the notation $\overline \jmath := j\hbox{mod } N$. Notice that
$[\so_{\overline \jmath} ,\so_{\overline k}] =\so_{\overline {\jmath +k}}$.
Because $\hat\tau_{[w]}^\epsilon$ is diagonal, its  eigenvectors and eigenvalues are
easily found; one has
$$\eqalign{\hat\tau_{[w]}^\epsilon\left(G^{ab}_{jk}(\lambda,\mu)\right)&=e^{2\pi
i(\lambda H^a_j( \epsilon)+\mu H^b_k(
\epsilon))}G^{ab}_{jk}(\lambda,\mu)\cr\hat\tau_{[w]}^\epsilon\left(G^{a}_{j}(\lambda)\right)&=e^{2\pi
i\lambda H^a_j(\epsilon)}G^{a}_{j}(\lambda)\cr}\forno$$

\noindent where
 $$H^a_j(\epsilon):=(H_{[w]}^\epsilon\vert \epsilon^a_j)\qquad\qquad  1\le a\le
s, 1\le j\le n_a\forno$$ 

\noindent From these relations one easily reads off the order of
$\hat\tau_{[w]}^\epsilon$.

\proclaim Lemma  \prono . Let $[w]=[n_1 n_2 \cdots n_r \overline n_{r+1}
\overline n_{r+2}
  \cdots \overline n_s] $ be a signed cycle type of $n$, $N'$ the order of $w$,
i.e. the least common multiple of the numbers $n_1, n_2,\cdots, n_r, 2n_{r+1},
2n_{r+2}, \cdots, 2n_s$ and $N$ the order of the automorphism $\tw^\epsilon$, then:
$$N=\cases{N'&if $N'{2n_a(1-\epsilon)+1\over n_a}\in 2{\bf Z}$ for all $1\le
a\le r\le s$\cr
           &if $r=0$ \cr
           2N'&in all other cases\cr}\forno$$









\baselineskip=22pt plus.1pt

\count30=2                      %chapterno = 2
\count31=0                      %sectionno = 0
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0

\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count31 .\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count31 .\number\count33}


\long\def\proof#1{\noindent{\bf{Proof.}}\quad#1\qedd}
\def\qedd{\hfill\hbox{\vrule\vbox{\hrule\kern2truept\hbox{\kern2truept
\vbox to2pt{\hsize=2pt}\kern2truept}\kern2truept\hrule}\vrule}\vskip .5cm}



\def\so{so_{2n+1}({\bf C})}
\def\SO{SO_{2n+1}({\bf C})}

\def\ad{{\rm ad}}
\def\Ad{{\rm Ad}}
\def\trace{{\rm trace}}

\def\E{{\cal E}}
\def\F{{\cal F}}
\def\tn{\hat \tau_{\underline n}}
\def\tw{\hat \tau_{[w]}^\epsilon}
\def\Hn{H_{\underline n}}
\def\Hw{H_{[w]}^\epsilon}
\def\Sn{S_{\underline n}}
\def\Sw{S_{[w]}}
\def\Xw{X_{[w]}}
\def\sw{\hat s_{[w]}}
\def\hw{{\underline h}_{[w]}}
\def\gl{gl_{2n+1}({\bf C})}

\def\oi{o(\infty )}
\def\gli{gl(\infty )}
\def\glib{\overline {gl(\infty )}}
\def\oib{\overline {o(\infty )}}
\def\Ai{A_{\infty}}
\def\Di{D_{\infty}}
\def\glt{\tilde {gl}_{2n+1}({\bf C})}
\def\glh{\hat {gl}_{2n+1}({\bf C})}
\def\sot{\tilde {so}_{2n+1}({\bf C})}
\def\soh{\hat {so}_{2n+1}({\bf C})}
\def\vac{\mid \overline 0 >}





\beginsection \chapno .  The affine algebra $\soh$
\par
\beginsection \secno .  Introduction
\par
In this section we will briefly review the realizations of the affine
algebra as algebras of loops twisted by the automorphisms $\tw$ (see [K]).
We will also introduce so-called vertex operators as
generating series in a formal indeterminate $z$, whose coefficients describe
the action of the affine algebra in an integrable highest weight
representation. In section 6 we will see that these operators are
precisely the vertex operators one encounters in physics.

\beginsection \secno .  Twisted realizations of $\sot$ [K, H]\par
Let $\sot:=\bigoplus_{k\in{\bf Z}}e^{ik\theta}\so$ be the loop algebra
associated to $\so$.
Using the automorphism $\tw:\so\to\so$, one
defines the twisted realization $L(\so,\tw )$ of $\sot$ as
the space
$$L(\so,\tw ):=\bigoplus_{k\in {\bf Z}}\lambda^k\so_{\overline k}\forno$$
with commutation relations
$$[\lambda^kx,\lambda^jy]:=\lambda^{k+j}[x,y]\qquad\hbox{for all } 
  x\in\so_{\overline k},y\in\so_{\overline j}\forno$$
It is well known [K, H] that $\sot$ and
$L(\so,\tw )$  are isomorphic; an isomorphism is given by:
$$\Phi_{[w]}^\epsilon:\lambda^ky\mapsto e^{-i\theta \ad\Hw}e^{i{k\over N}\theta}y
  \qquad \hbox{for all } y\in\so_{\overline k}\forno$$
This suggests to introduce the following spanning set of
$\sot$ (see [KP85]):
$$y(k):=e^{-i\theta \ad\Hw} e^{i{k\over N}\theta}y_{\overline
k}\qquad y\in \so, k\in{\bf Z}\forno$$
where $y_{\overline k}$ denotes the component of $y$, which is
homogeneous of degree $\overline k$ in the ${\bf Z}_N$ gradation of $\so$ 
induced by $\tw$ (see (2.7.18-19)).


The affine algebra $\soh:=\sot\bigoplus {\bf C}c$ is defined as the 
one dimensional central extension of $\sot$ with commutation relations
$$[\tilde x+\alpha c,\tilde y+\beta c]:=[\tilde x,\tilde y]
  +\nu(\tilde x,\tilde y)c\forno$$
where the two cocycle $\nu :\sot\times\sot\rightarrow{\bf C}$ is
given by:
$$\nu (\tilde x,\tilde y):={1\over 2\pi i}\int^{2\pi}_0
  d\theta\left({d\tilde x(\theta )\over d\theta}\vert
  \tilde y(\theta )\right)\forno$$
Recall that $(A\vert B)={1\over 2}\trace (AB)$, for all $A,B\in\so$. 

The corresponding twisted realization $\hat L(\so,\tw )$ is
defined as $ L(\so,\tw )\bigoplus {\bf C}c$
with commutation relations
$$[\lambda^ky_{\overline k}+\alpha c,\lambda^lz_{\overline l}+\beta c]
=\lambda^{k+l}[y_
{\overline k},z_{\overline l}]+{k\over N}\delta_{k+l,0}(y_{\overline k}\vert
z_{\overline l})c\forno$$
The isomorphism
$\Phi_{[w]}^\epsilon:L(\so,\tw ) \rightarrow \sot$ can be
extended to an isomorphism $\hat\Phi_{[w]}^\epsilon$ between  
$\hat L(\so,\tw )$ and $\soh$:
$$\hat\Phi_{[w]}^\epsilon(\lambda^ky_{\overline k}+\alpha c):=
  \hat y(k)+\alpha c\forno$$
where the elements $\hat y(k)\in\soh$ are defined by
$$\hat y(k):=y(k)-\delta_{k,0}(H_{[w]}^\epsilon\vert y)c\forno$$



\beginsection \secno.  Vertex operators\par
Recall from section 2.7 the vectors $A^{ab}_{jk}(\lambda ,\mu )$ and $A^{a}_{j}(\lambda)$, which span the finite dimensional Lie  algebra $\so$. It is
clear that the affine algebra $\soh$ is
spanned by the corresponding loops $\hat A^{ab}_{jk}(\lambda ,\mu )(l)$ and $\hat A^{a}_{j}(\lambda)(l)$ and the central element $c$. In the sequel we will be interested in representations
$\rho :\soh\rightarrow {\rm End}\,V$ of the affine algebra on a vector space 
$V$. In the construction of representations one uses the following formal generating
series:
$$\eqalign{\hat A^{ab}_{jk}(\lambda ,\mu)(z)&:=\sum_{l\in {\bf
Z}}z^{-l}\rho\left(\hat A^{ab}_{jk}(\lambda ,\mu )(l)\right)\cr
\hat A^{a}_{j}(\lambda)(z)&:=\sum_{l\in {\bf
Z}}z^{-l}\rho\left(\hat A^{a}_{j}(\lambda)(l)\right)}\forno$$

\noindent The parameter $z$ in these series is just a formal parameter.
 If one can somehow find explicit expressions for the operators
$\hat A^{ab}_{jk}(\lambda ,\mu)(z)$ and $\hat A^{a}_{j}(\lambda)(z)$,
 one can --in principle-- 
extract the coefficients of $z^{-l}$ and find the action of
the affine algebra on the representation space $V$.
In fact we will only be interested in integrable highest weight representations
of the affine algebra, in which the central element c is represented by the
identity operator. So from now on we will take $\rho (c)=I$.

Below we give the explicit form of the vertex operators
in terms of the elements $e^{im\theta }G_{jk}^{ab}(\lambda ,\mu )$ and $e^{im\theta }G_{j}^{a}(\lambda)$.
We have:
$$\eqalign{\hat A_{jk}^{ab}(\lambda ,\mu )(z)&=\sum_{l\in{\bf Z}}z^{-l}\rho
  \left(e^{i{l\over N}\theta }e^{-i\theta\ad\,\Hw}
  \left(A^{ab}_{jk}(\lambda ,\mu )\right)_{\overline l}\right)
  -\left(\Hw\vert A^{ab}_{jk}(\lambda ,\mu )\right)I\cr
\hat A_{j}^{a}(\lambda)(z)&=\sum_{l\in{\bf Z}}z^{-l}\rho
  \left(e^{i{l\over N}\theta }e^{-i\theta\ad\,\Hw}
  \left(A^{a}_{j}(\lambda)\right)_{\overline l}\right)
  -\left(\Hw\vert A^{a}_{j}(\lambda)\right)I\cr}
  \forno$$

\noindent Using (2.7.16) and (2.7.20), the first term in the right-hand-side of (3.3.2) can be worked out as:
$$\eqalign{2&\sqrt{n_an_b}
  \sum_{l\in{\bf Z}}z^{-l}\rho\left(e^{i{l\over N}
  \theta}e^{-i\theta\ad\,\Hw}\left(A^{ab}_{jk}(\lambda ,\mu )
  \right)_{\overline l}\right)=\cr
&\phantom{=+}\sum_{m,l\in{\bf Z}}
  \sum_{c,d=1}^s
  \sum^{n_c}_{p=1}\sum^{n_d}_{q=1}\sum_{\nu ,\xi =+,-}
  z^{-l}\alpha_{cj}^{ap}(\lambda ,\nu )\alpha_{dk}^{bq}(\mu ,\xi )
  \rho\left(e^{i{l\over N}\theta}e^{-i\theta\ad\,\Hw}
  G^{cd}_{pq}(\nu ,\xi )\right)
  \delta_{l,N(m+\nu H^c_p(\epsilon)+\xi H^d_q(\epsilon))}
  \cr
&\phantom{=}+\sum_{m,l\in{\bf Z}}
  \sum_{c=1}^s
  \sum^{n_c}_{p=1}\sum_{\nu=+,-}
  z^{-l}\alpha_{cj}^{ap}(\lambda ,\nu )\beta_{k}^{b}(\mu)
  \rho\left(e^{i{l\over N}\theta}e^{-i\theta\ad\,\Hw}
  G^{c}_{p}(\nu)\right)
  \delta_{l,N(m+\nu H^c_p(\epsilon))}
  \cr
&\phantom{=}-\sum_{m,l\in{\bf Z}}
  \sum_{d=1}^s
  \sum^{n_d}_{q=1}\sum_{\xi =+,-}
  z^{-l}\beta_{j}^{a}(\lambda)\alpha_{dk}^{bq}(\mu ,\xi )
  \rho\left(e^{i{l\over N}\theta}e^{-i\theta\ad\,\Hw}
  G^{d}_{q}(\nu)\right)
  \delta_{l,N(m+\xi H^d_q(\epsilon))}
  \cr
&=\phantom{+}\sum_{m\in{\bf Z}}
  \sum_{c,d=1}^s
  \sum^{n_c}_{p=1}\sum^{n_d}_{q=1}\sum_{\nu ,\xi =+,-}
  z^{-N(m+\nu H^c_p(\epsilon)+\xi H^d_q(\epsilon))}
  \alpha_{cj}^{ap}(\lambda ,\nu )\alpha_{dk}^{bq}(\mu ,\xi ) 
  \rho\left(e^{im\theta}G^{cd}_{pq}(\nu ,\xi )\right)
  \cr
&\phantom{=}+\sum_{m\in{\bf Z}}
  \sum_{c=1}^s
  \sum^{n_c}_{p=1}\sum_{\nu=+,-}
  z^{-N(m+\nu H^c_p(\epsilon))}\alpha_{cj}^{ap}(\lambda ,\nu )\beta_{k}^{b}(\mu)
  \rho\left(e^{im\theta}
  G^{c}_{p}(\nu)\right)
  \cr
&\phantom{=}-\sum_{l\in{\bf Z}}
  \sum_{d=1}^s
  \sum^{n_d}_{q=1}\sum_{\xi =+,-}
  z^{-N(m+\xi H^d_q(\epsilon))}\beta_{j}^{a}(\lambda)\alpha_{dk}^{bq}(\mu ,\xi )
  \rho\left(e^{im\theta}
  G^{d}_{q}(\xi)\right)
  \cr}\forno$$


A similar calculation yields:
$$\eqalign{\sqrt{2n_a}
  \sum_{l\in{\bf Z}}z^{-l}\rho&\left(e^{i{l\over N}
  \theta}e^{-i\theta\ad\,\Hw}\left(A^{a}_{j}(\lambda)
  \right)_{\overline l}\right)=\cr
&\sum_{m\in{\bf Z}}
  \sum_{c,d=1}^s
  \sum^{n_c}_{p=1}\sum^{n_d}_{q=1}\sum_{\nu ,\xi =+,-}
  z^{-N(m+\nu H^c_p(\epsilon)+\xi H^d_q(\epsilon))}
  \alpha_{cj}^{ap}(\lambda ,\nu )\gamma_{d}^{q}(\xi ) 
  \rho\left(e^{im\theta}G^{cd}_{pq}(\nu ,\xi )\right)
  \cr
&+\sum_{m\in{\bf Z}}
  \sum_{c=1}^s
  \sum^{n_c}_{p=1}\sum_{\nu =+,-}
  z^{-N(m+\nu H^c_p(\epsilon))}\alpha_{cj}^{ap}(\lambda,\nu)\delta
  \rho\left(e^{im\theta}
  G^{c}_{p}(\nu)\right)
  \cr
& -\sum_{m\in{\bf Z}}\sum_{d=1}^s
  \sum^{n_d}_{q=1}\sum_{\xi=+,-}
  z^{-N(m+\xi H^d_q(\epsilon))}\beta_{j}^{a}(\lambda)\gamma_{d}^{q}(\xi)
  \rho\left(e^{im\theta}
  G^{d}_{q}(\xi)\right)
  \cr
}\forno$$

Using lemma 2.7.1, it is straightforward, but tedious, to calculate the central terms; we give the result in
the following lemma:
\proclaim Lemma  \prono.
\overfullrule=0pt
$$\eqalign{(\Hw\vert A_{jk}^{ab}(\lambda ,\mu ))&=
  \cases{\lambda\delta_{\lambda ,-\mu}({1\over 2} -\epsilon)
         &if $1\le a=b\le r$ and $j= k$
         \cr & \cr\lambda\delta_{\lambda ,-\mu}
         {1\over {n_a}} 
         {\omega_a^{\lambda (k-j)}\over {1-\omega_a^{\lambda (k-j)}}}
         &if $1\le a=b\le r$ and $j\ne k$
         \cr & \cr
         (-1)^{j-k}i{(1-i\lambda (-1)^{n_a})(1-i\mu (-1)^{n_b})
         \over 8\sqrt {n_an_b}}
         \left({\lambda\mu\sin ({j-k \over {n_a}}\pi )
         \over {1+\lambda\mu\cos ({j-k\over {n_a}}\pi )}}\right)
         &if $r<a=b\le s$ and $j\ne k$
         \cr & \cr
         (-1)^{j-k}i{(1-i\lambda (-1)^{n_a})(1-i\mu (-1)^{n_b})
         \over {8\sqrt {n_an_b}}}(-1)
         &if $r+1<a+1=b=r+2c\le s$
         \cr & \cr
         (-1)^{j-k}i{(1-i\lambda (-1)^{n_a})(1-i\mu (-1)^{n_b})
         \over {8\sqrt {n_an_b}}}(1)
         &if $r+1<b+1=a=r+2c\le s$
         \cr &\cr
         0&in all other cases
         \cr}\cr & \cr
(\Hw\vert A_{j}^{a}(\lambda))&=
  \cases{(-)^ji{(1-i\lambda (-1)^{n_a})
         \over {4\sqrt {n_a}}}(-1)
         \phantom{{(1-i\lambda (-1)^{n_a})
         \over {2\sqrt {n_a}}}\lambda\left({\sin ({j-k \over {n_a}}\pi )
         \over {1+\lambda\mu\cos ({j-k\over {n_a}}\pi )}}\right)}
         &if $a=s$ and $s-r$ is odd\cr & \cr 0&in all other     cases\cr}\cr}\forno$$
\overfullrule=5pt
\par
In the next section we will construct a much simpler expression
for the operators $\hat A^{ab}_{jk}(\lambda ,\mu )(z)$ and $\hat A^{a}_{j}(\lambda)(z)$ in terms of
`fermionic fields'.






\beginsection \secno.  Heisenberg algebras\par
\noindent Here we consider the operators $\hat A^{aa}_{jj}(+,-)(z)$ where 
$1\le a\le s, 1\le j\le n_a$. We will see that the components $\rho\bigl(\hat
A^{aa}_{jj}(+,-)(l)\bigr)$ span a so-called oscillator algebra. Leaving out the
`zero modes', one obtains a Heisenberg algebra.

First of all we remark that it is sufficient to consider $\hat A^{aa}_{n_an_a}
(+,-)(z)$;

\proclaim Lemma \prono.
$$\hat A^{aa}_{jj}(+,-)(z)=\cases{\phantom{-}\hat A^{aa}_{n_an_a}(+,-)(\omega
^{-j}z)&if\  $1\le a\le r$\cr & \cr-\hat A^{aa}_{n_an_a}(+,-)(\omega^{-j}z)&if\  $r<
a\le s$\cr}\forno$$

\proof{For $1\le a\le r$ we have:
$$\eqalign{A^{aa}_{jj}(+,-)&=S_{[w]}^{-1}\epsilon^a_jS_{[w]}\cr
&=S_{[w]}^{-1}\left({\rm Ad}\,(\hat
w^\epsilon)\right)^j\left(\epsilon^a_{n_a}\right)S_{[w]}\cr &=\left(\hat
\tau_{[w]}^\epsilon\right)^j\left(A^{aa}_{n_an_a}(+,-)\right)}\forno$$ From this result 
one easily derives
$A_{jj}^{aa}(+,-)(l)=\omega^{jl}A^{aa}_{n_an_a}(+,-)(l)$. Using (3.2.9) and
lemma 3.3.1, one shows that $\hat
A_{jj}^{aa}(+,-)(l)=A_{jj}^{aa}(+,-)(l)-\delta_{l0}({1\over 2}-\epsilon)$ (for
$1\le a\le r, 1\le j\le n_a$), showing that we also have:
$\hat A_{jj}^{aa}(+,-)(l)= \omega^{jl}\hat A^{aa}_{n_an_a}(+,-)(l)$. This
proves the first relation of (3.4.1). The second relation is proved
similarly; the minus sign arises because ${\rm Ad}\,(
\hat w^\epsilon)\left(\epsilon^a_{n_a}\right)=-\epsilon^a_1$ if $r<a\le s$.}


The following lemma tells us which of the components $\hat A^{aa}_{n_an_a}(+,-)(l)$ are nonzero.

\proclaim Lemma \prono.
For $1\le a\le r$ we have:
$$\left(A^{aa}_{n_an_a}(+,-)\right)_{\overline l}=\cases{0&if\ $l\notin{N\over n_a}{\bf Z}$\cr & \cr
{1\over n_a}\sum_{j=1}^{n_a}\omega^{-lj}A^{aa}_{jj}(+,-)&if\ $l\in{N\over n_a}{\bf Z}$\cr}\forno$$
\noindent For $r< a\le s$ we have:
$$\left(A^{aa}_{n_an_a}(+,-)\right)_{\overline l}=\cases{0&if\ $l\notin{N\over 2n_a}(2{\bf Z}+1)$\cr & \cr
-{1\over n_a}\sum_{j=1}^{n_a}\omega^{-lj}A^{aa}_{jj}(+,-)&if\ $l\in{N\over 2n_a}(2{\bf Z}+1)$\cr}\forno$$


\proof{For any $x\in\so$ the homogeneous component of degree $\overline l$
in the ${\bf Z}/N{\bf Z}$ gradation induced by $\hat
\tau_{[w]}^\epsilon$ can be obtained by:
$$x_{\overline l}={1\over N}\sum_{q=1}^N\omega^{-lq}\left(\hat
\tau_{[w]}^\epsilon\right)^q(x)\forno$$

\noindent Recall that $\hat\tau_{[w]}^\epsilon$ corresponds to $w$ on the Cartan subalgebra
$\underline h_{[w]}$. Since $A^{aa}_{jj}(+,-)={\rm
Ad}S^{-1}_{[w]}\bigl(\epsilon^a_j\bigr)$, we have (cf (2.7.1)):
$$\hat\tau_{[w]}^\epsilon\left(A^{aa}_{jj}(+,-)\right)=\cases{A^{aa}_{j+1,j+1}(+,-)&if\
$j<n_a$\cr & \cr A^{aa}_{11}(+,-)&if\ $1\le a\le r$\ and\  $j=n_a$\cr & \cr
-A^{aa}_{11}(+,-)&if\ $r<a\le s$\ and\  $j=n_a$\cr}\forno$$

\noindent Substituting $x=A^{aa}_{n_an_a}(+,-)$ in (3.4.5), we find for $1\le
a\le r$:
$$\eqalign{\left(A^{aa}_{n_an_a}(+,-)\right)_{\overline l}&={1\over N}\sum_{k=0}^
{N/n_a-1}\sum_{j=1}^{n_a}\omega^{-l(n_ak+j)}\left(\hat\tau_{[w]}^\epsilon\right)
^j\left(A^{aa}_{n_an_a}(+,-)\right)\cr
&={1\over N}\sum_{k=0}^{N/n_a-1}\omega^{-ln_ak}\sum_{j=1}^{n_a}\omega^{-lj}
A^{aa}_{jj}(+,-)\cr}\forno$$

\noindent which proves (3.4.3). Similarly, we find for $r<a\le s$:
$$\eqalign{\left(A^{aa}_{n_an_a}(+,-)\right)_{\overline l}&={1\over N}\sum_{k=0}
^{N/2n_a-1}\sum_{j=1}^{2n_a}\omega^{-l(2n_ak+j)}\left(\hat\tau_{[w]}^\epsilon\right)^j
\left(A^{aa}_{n_an_a}(+,-)\right)\cr
&={1\over
N}\sum_{k=0}^{N/2n_a-1}\omega^{-l2n_ak}\sum_{j=1}^{n_a}(\omega^{-n_al}-1)\omega^{-lj}
A^{aa}_{jj}(+,-)\cr}\forno$$

\noindent from which one easily reads off (3.4.4).}

Notice that the elements $A^{aa}_{n_an_a}(+,-)_{\overline l}$  belong to the Cartan subalgebra
$\underline h_{[w]}=\oplus_{a=1}^s\oplus_{j=1}^{n_a}{\bf C}A^{aa}_{jj}(+,-)$. It follows from (3.2.7-8) that the commutator $[\hat A^{aa}_{n_an_a}(+,-)(k),
\hat A^{aa}_{n_an_a}(+,-)(l)]$ must be central. In fact we have:

\proclaim Lemma \prono.
Set 
$$\hat\alpha_a\left({N\over n_a}k\right):=(-1)^{2k}n_a\rho\left(\hat
A^{aa}_{n_an_a}(+,-)\left({N\over n_a}k\right)\right)\qquad (k\in{\bf Z}\
\hbox{if}\ 1\le a\le r, k\in{1\over 2}+{\bf Z}\ \hbox{if}\ r< a\le s)\forno$$
\noindent then: $$[\hat\alpha_a\left({N\over
n_a}k\right),\hat\alpha_b\left({N\over
n_b}l\right)]=k\delta_{ab}\delta_{k+l,0}\forno$$

\proof{Use (3.2.7-8) and lemma 3.4.2.}

The relations (3.4.10) are the commutation relations for a set of bosonic
oscillators. Hence one speaks of an oscillator algebra. If we leave out the
zero modes $\hat\alpha_a(0), 1\le a\le r$, we obtain a so-called Heisenberg
algebra, i.e. an algebra on a basis $\{p_i,q_i\}_{i\in{\bf N}}$ and commutation
relations
$$[p_i,q_j]=\delta_{ij}I\forno$$

\noindent The mapping $[w]\mapsto \hat{\underline{s}}_{[w]}:=\oplus_{a=1}^s\oplus_{k\ne 0}{\bf C}\hat\alpha_a\left({N\over n_a}k\right)$ establishes the one-one correspondence between conjugacy classes in $W$ and conjugacy classes of Heisenberg subalgebras of $\soh$ mentioned in the introduction. (Notice that, since 
$[H_{[w]}^0,A_{jj}^{aa}(+,-)]=[H_{[w]}^{1\over 2},A_{jj}^{aa}(+,-)]$, the elements of the Heisenberg algebra do not depend on $\epsilon$.)
















\baselineskip=22pt plus.1pt





\count30=3                      %chapterno = 3
\count31=0                      %sectionno = 0
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0
\overfullrule=0pt


\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count31 .\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count31 .\number\count33}


\long\def\proof#1{\noindent{\bf{Proof.}}\quad#1\qedd}
\def\qedd{\hfill\hbox{\vrule\vbox{\hrule\kern2truept\hbox{\kern2truept
\vbox to2pt{\hsize=2pt}\kern2truept}\kern2truept\hrule}\vrule}\vskip .5cm}



\def\so{so_{2n+1}({\bf C})}
\def\SO{SO_{2n+1}({\bf C})}

\def\ad{{\rm ad}}
\def\Ad{{\rm Ad}}
\def\trace{{\rm trace}}

\def\E{{\cal E}}
\def\F{{\cal F}}
\def\tn{\hat \tau_{\underline n}}
\def\tw{\hat \tau_{[w]}}
\def\Hn{H_{\underline n}}
\def\Hw{H_{[w]}}
\def\Sn{S_{\underline n}}
\def\Sw{S_{[w]}}
\def\Xw{X_{[w]}}
\def\sw{\hat s_{[w]}}
\def\hw{{\underline h}_{[w]}}
\def\shw{\hat {\underline s}_{[w]}}
\def\sh{\hat {\underline s}}
\def\gl{gl_{2n+1}({\bf C})}
\def\oi{o_\epsilon(\infty )}
\def\gli{gl(\infty )}
\def\glib{\overline {gl(\infty )}}
\def\oib{\overline {o_\epsilon(\infty )}}
\def\Ai{A_{\infty}}
\def\Di{D_{\infty}}
\def\Bi{B_{\infty}}
\def\glt{\tilde{gl}_{2n+1}({\bf C})}
\def\glh{\hat{gl}_{2n}({\bf C})}
\def\sot{\tilde{so}_{2n+1}({\bf C})}
\def\soh{\hat{so}_{2n+1}({\bf C})}
\def\vac{\mid \overline 0 >}



\beginsection \chapno .  Fermionic fields\par
\beginsection \secno .  Introduction\par
In this section we will briefly review the Lie algebras $\glib$, $\oib$ ($\epsilon =0,{1\over 2}$)
and their central extensions $\Ai$, $\Bi$ ($\epsilon =0$) or $\Di$ ($\epsilon={1\over 2}$), respectively.
We will also discuss the embeddings $\sot  \hookrightarrow\oib$ and
$\soh \hookrightarrow \Bi$ or $\Di$, which, together with the spin
representation of $\Bi$ and $\Di$, will enable us to find expressions for the vertex 
operators 
$\hat A_{jk}^{ab}(\lambda ,\mu )(z)$ and $\hat A_{j}^{a}(\lambda)(z)$
in terms of normal ordered products of fermionic fields. Some parts 
of section 4.2 and section 4.3 
have been taken from [K], [KP86] and [Y].




\beginsection \secno .  Lie algebras of infinite matrices\par
Let $\glib$ be the Lie algebra of complex infinite matrices such that all
non zero entries  are within a finite distance from the main diagonal,
i.e.,
$$\glib := \{ (g_{ij})_{i,j \in {\bf Z}} \mid g_{ij}=0 
  \hbox{ if } \mid i-j\mid >>0\}\forno$$
The elements $\E_{ij}$ (the matrix with the $(i,j)^{th}$ entry $1$ 
and $0$ elsewhere), for $i,j\in {\bf Z}$, form a basis of a subalgebra
$\gli \subset \glib$. The Lie algebra $\glib$ has a universal central extension
$\Ai :=\glib \bigoplus {\bf C}c_{\Ai}$ with the Lie bracket defined by
$$[a + \alpha c_{\Ai},b+\beta c_{\Ai}]=ab-ba+\mu (a,b)c_{\Ai}\forno$$
for $a,b\in \glib$ and $\alpha ,\beta \in {\bf C}$; here $\mu$ is the
following 2-cocycle:
$$\mu (\E_{ij},\E_{kl}) =\delta_{il} \delta_{jk}
      <\theta ,\E_{ii}- \E_{jj}>\forno$$
where the linear mapping $\theta :\bigoplus_{i\in {\bf Z}} {\bf C}\E_{ii}
\to {\bf C}$ is defined by
$$<\theta ,\E_{ii}>:=\cases{0&if $i>0$\cr &\cr 1&if $i\le 0$\cr}\forno$$

Let $\epsilon =0$ or $1\over 2$; the Lie algebra $\oib$  is defined as a subalgebra of $\glib$:
$$\oib :=\{(g_{ij})\in \glib \mid g_{ij}=-g_{-j+2\epsilon ,-i+2\epsilon}\}\forno$$
We denote by $\oi =\oib \cap \gli$; it is generated by the elements
$\F_{ij}:= \E_{i+\epsilon,-j+\epsilon} -\E_{j+\epsilon,-i+\epsilon}$, for $i,j\in \epsilon+{\bf Z}$. Notice
that $\F_{ij} =-\F_{j,i}$.

The 2-cocycle $\mu$ on $\glib$
induces a 2-cocycle on $\oib$, and hence we can define a central 
extension $\Bi:= \overline{o_0(\infty)} \bigoplus {\bf C}c_0$  and $\Di :=\overline{o_{1\over 2}(\infty)} \bigoplus {\bf C}c_{1\over 2}$ of $\oib$, with
Lie bracket
$$[a+\alpha c_{\epsilon},b+\beta c_{\epsilon}]=ab-ba +{1\over 2}\mu (a,b)c_{\epsilon}\forno$$
for $a,b\in \oib$ and $\alpha ,\beta \in {\bf C}$.
 
We also introduce the vector space ${\bf C}^{\infty}$ of infinite column 
vectors $(w_i)_{i\in {\bf Z}}$ such that all but a finite number of   
$w_i$'s are zero. As a basis for this vector space we take the collection 
$\{ v_i\} _{i\in {\bf Z}}$ where $v_i$ is the vector with a one on the 
$i^{th}$ entry and zeros elsewhere. The Lie algebra $\glib$ clearly acts on the 
vector space ${\bf C}^{\infty}$ by:
$$\E_{ij} v_k =\delta_{jk} v_i\forno$$



Now consider the loop algebra 
$\tilde{so}_{2n+1}({\bf C}) \cong \bigoplus_{k\in {\bf Z}} e^{ik\theta}so_{2n+1}({\bf C})$. The `natural'
representation for this Lie algebra is of course the space 
${\bf \tilde C}^{2n+1}$ of all polynomial loops in the vector space 
${\bf C}^{2n+1}$;
$${\bf \tilde C}^{2n+1} := \{ x:[0,2\pi ]\to {\bf C}^{2n+1}\mid
  x(\theta )= \sum_{k=N}^M e^{ik\theta}x_k\qquad x_k\in {\bf C}^{2n+1}
  \qquad N,M\in {\bf Z}\}\forno$$

\noindent The action of $\tilde{so}_{2n+1}({\bf C})$ on ${\bf \tilde C}^{2n+1}$ is given by
$$e^{ik\theta} A (e^{il\theta}x) := e^{i(k+l)\theta}Ax
\qquad\hbox{for } k,l\in {\bf Z},\  A\in so_{2n+1}({\bf C}) ,\  x\in {\bf C}^{2n+1}\forno$$

The vector spaces ${\bf \tilde C}^{2n+1}$ and ${\bf C}^{\infty}$ are
isomorphic; we choose the following isomorphism:
$$\eqalign{e^{-ik\theta}f_j(\lambda )&\mapsto v_{(2n+1)k-\lambda(n+1-j)+2\epsilon (n+1)}\cr
e^{-ik\theta}e_{2n+1}&\mapsto v_{(2n+1)k+2\epsilon (n+1)}\cr} 
  \forno$$

\noindent Because the loop algebra
$\tilde{so}_{2n+1}({\bf C})$ acts on ${\bf \tilde C}^{2n+1} \cong {\bf C}^{\infty}$,
its elements can be seen as endomorphisms of ${\bf C}^{\infty}$.
In this way we obtain an embedding  $\iota_\epsilon :\tilde{so}_{2n+1}({\bf C}) \hookrightarrow \glib$.
Even more is true: the image of $\iota_\epsilon$ lies in $\oib$. 
\proclaim Lemma  \prono .
The mapping $\iota_\epsilon :\tilde{so}_{2n+1}({\bf C}) \to \oib$ defined by
$$\eqalign{\iota_\epsilon : e^{il\theta}G_{jk}(\lambda ,\mu )&\mapsto 
  \sum_{m\in {\epsilon+\bf Z}}
  \F_{(2n+1)m-\lambda (n+1-j),
      -(2n+1)(l+m)-\mu(n+1-k)}\cr
\iota_\epsilon : e^{il\theta}G_{j}(\lambda  )\phantom{,\mu_k}&\mapsto 
  \sum_{m\in {\epsilon+\bf Z}}
  \F_{(2n+1)m-\lambda (n+1-j),
      -(2n+1)(l+m)}\cr}
\forno$$                  
is a Lie algebra homomorphism, mapping $\tilde{so}_{2n+1}({\bf C})$ injectively into
$\oib$.
The 2-cocycle on $\oib$ induces the following 2-cocycle on $\tilde{so}_{2n+1}({\bf C})$ ($k,l\in {\bf Z},A,B\in so_{2n+1}({\bf C})$):
$$\eqalign{
  {1 \over 2}\mu\left(\iota_\epsilon (e^{ik\theta}A),\iota_\epsilon (e^{il\theta}B)\right)
&=\nu\left(e^{ik\theta}A,e^{il\theta}B\right)+(1-2\epsilon)\delta_{k,-l}(\lambda_n|[A,B])\cr
  &= \delta _{k,-l}\{k(A\mid B)+(1-2\epsilon)(\lambda_n|[A,B])\}
  \cr}\forno$$
where $\lambda_n={1\over 2}\sum_{j=1}^n \epsilon_j$, the $n^{th}$ fundamental weight of $\so$.
The homomorphism $\iota _\epsilon $ can be extended to a homomorphism $\hat\iota_\epsilon: \hat{so}_{2n+1}({\bf C})\hookrightarrow \Bi$  or $\Di$ by defining $\hat {\iota}_\epsilon (c)=c_{\epsilon}$ and
$$\hat\iota_\epsilon(e^{ik\theta}A) =\iota_\epsilon(e^{ik\theta}A)+\delta_{k,0}(1-2\epsilon)(\lambda_n |A)c_\epsilon
\forno$$
\par
\proof{A straightforward calculation.} 
We compute the value of $(\lambda_n|A^{ab}_{jk}(\lambda,\mu ))$ and $(\lambda_n|A_j^a(\lambda))$:
\proclaim Lemma  \prono .
\overfullrule=0pt
$$\eqalign{(\lambda_n\vert A_{jk}^{ab}(\lambda ,\mu ))&=
  \cases{{\lambda \over 2}\delta_{\lambda ,-\mu}         &if $1\le a=b\le r$ and $j=k$\cr & \cr        ( (-1)^{j-k+\delta_{\lambda \mu}n_a}-1)i{(1-i\lambda) (1-i\mu )
         \over 8\sqrt {n_an_b}}
         \left({\lambda\mu\sin ({j-k \over {n_a}}\pi )
         \over {1+\lambda\mu\cos ({j-k\over {n_a}}\pi )}}\right)
         &if $r<a=b\le s$ and $j\ne k$
         \cr & \cr
         i{(-1)^{j-k}(1-i\lambda (-1)^{n_a})(1-i\mu (-1)^{n_b})+(1-i\lambda )(1-i\mu )
         \over {8\sqrt {n_an_b}}}(-1)
         &if $r+1<a+1=b=r+2c\le s$
         \cr & \cr
         i{(-1)^{j-k}(1-i\lambda (-1)^{n_a})(1-i\mu (-1)^{n_b})+(1-i\lambda )(1-i\mu )
         \over {8\sqrt {n_an_b}}}(1)
         &if $r+1<b+1=a=r+2c\le s$
         \cr &\cr
         0&in all other cases
         \cr}\cr & \cr
(\lambda_n\vert A_{j}^{a}(\lambda))&=
  \cases{(-)^ji{(1-i\lambda (-1)^{n_a})
         \over {4\sqrt {n_a}}}(-1)
         \phantom{{(\lambda (-1)^{n_a})
         \over {2\sqrt {n_a}}}\lambda\left({\sin ({j-k \over {n_a}}\pi )
         \over {1+\lambda\mu\cos ({j-k\over {n_a}}\pi )}}\right)}
         &if $a=s$ and $s-r$ is odd\cr & \cr 0&in all other     cases\cr}\cr}\forno$$

\overfullrule=5pt
\beginsection \secno .  The spin representation\par
In this subsection we consider highest weight representations for
$\oi$, $\Bi$ and $\Di$. For this purpose we introduce the Clifford algebra 
$Cl^{\epsilon}$ as the associative algebra on generators $\phi (j)$ ($j\in \epsilon+{\bf Z}$) 
with defining relations
$$\{ \phi (i), \phi (j)\} := \phi (i)\phi (j) +\phi (j) \phi (i) 
                           =\delta_{i,-j}\forno$$
Since the $\phi (j)$'s anti-commute, we call them {\it fermions.}
  
We define the {\it spin module} $V^{\epsilon}$ over $Cl^{\epsilon}$ as the irreducible 
module generated by a vector $\mid \overline 0 >$ satisfying
$$\phi (j)\mid \overline 0 >=0\qquad\hbox{for all }j< 0\forno$$
The vector $\mid \overline 0 >$ is called the {\it vacuum vector}.
The elements
$$\phi (j_p) \phi (j_{p-1})\cdots \phi (j_2) \phi (j_1)\mid \overline 0 >
\qquad \hbox{with}\qquad j_p>j_{p-1}> \cdots>j_2>j_1\ge 0\forno$$
form a basis of $V^{\epsilon}$. 

We use these fermions to define a representation $\tau^{\epsilon}$ of $\oi$ on $V^{\epsilon}$:
$$\eqalign{\tau^{\epsilon} (\F_{ij})&={\textstyle {1\over 2}}
  (\phi (i)\phi (j)-\phi (j)\phi (i))
  \cr
  &=\phi (i)\phi(j)-{\textstyle {1\over 2}}\delta_{i,-j}I\cr}\forno$$
Then restricted to $\oi$, the spin module breaks into a direct sum of 
two irreducible modules. To describe this decomposition, we define a
${\bf Z}_2$-gradation on $V^{\epsilon}$ by introducing a {\it chirality operator} $\chi$
satisfying:
$$\chi \vac =\vac\forno$$
$$\{\chi ,\phi (j)\}=0\qquad\hbox{for all  }j\in \epsilon+{\bf Z}\forno$$
Then,
$$V^{\epsilon}=\bigoplus_{\alpha \in {\bf Z}_2} V_{\alpha}^{\epsilon}\forno$$
where 
$$V_{\alpha}^{\epsilon}=\{ v\in V^{\epsilon}\mid \chi v=(-1)^{\alpha} v\} \forno$$
Each module is an irreducible highest weight module, with highest
weight vectors $\mid \overline 0 >$ and $\mid \overline 1> :=\break
\phi (\epsilon)\mid \overline 0 >$ for $V_{\overline 0}^{\epsilon}$, $V_{\overline 1}^{\epsilon}$,
respectively, in the sense that
$$\tau^{\epsilon}(\F_{i,-j})\mid \alpha >=0 \qquad \hbox{for}\qquad i<j,
  \ \alpha\in{\bf Z}_2\forno$$
$$\tau^{0}(\F_{i,-i})|\alpha > =\cases{-{1\over 2}|\alpha>&if $i>0$\cr &\cr
0&if $i=0$\cr &\cr
{1\over 2}|\alpha >&if $i<0$\cr}\forno$$
$$\tau^{{1\over 2}} (\F_{i,-i}) \mid \overline 0 >=\cases{-{1\over 2}\mid \overline 0 >&
  if $i>0$\cr  &\cr {1\over 2}\mid \overline 0>& if $i< 0$\cr}\qquad
  \tau^{{1\over 2}} (\F_{i,-i}) \mid \overline 1 >=\cases{-{1\over 2}\mid \overline 1 >&
  if $i>{1\over 2}$ or $i=-{1\over 2}$\cr &\cr  {1\over 2}\mid \overline 1 >& if $i< -{1\over 2}$ or $i={1\over 2}$\cr}\forno$$
We denote these representations by $\tau_{\alpha}^{\epsilon}$.

If we try to extend the representation $\tau_{\alpha}^{\epsilon}$ to $\oib$, 
we notice that it is not well defined, e.g. ,
$$\tau^{\epsilon} (\sum_{i>0} \F_{i,-i})\mid \overline 0 >=
 (\sum_{i>0} -{1\over 2})\mid \overline 0 >$$ 
We encounter what physicists call an anomaly.
This problem  can be solved as follows. Define
$$\eqalign{\pi_{\alpha}^{\epsilon}(\F_{ij})&:= \tau_{\alpha}^{\epsilon} (F_{ij})
  \phantom{+ {1\over 2}I}\,\qquad
  \hbox{for } i\not= -j\hbox{ or }i=j=0\cr
  \pi_{\alpha}^{\epsilon} (\F_{i,-i})&:=\tau_{\alpha}^{\epsilon}(\F_{i,-i})+{\textstyle {1\over 2}}I
  \qquad
  \hbox{for }i>0\cr 
  \pi_{\alpha}^{\epsilon} (\F_{i,-i})&:=\tau_{\alpha}^{\epsilon}(\F_{i,-i})-{\textstyle {1\over 2}}I
  \qquad
  \hbox{for }i< 0\cr}\forno$$
Extending $\pi_{\alpha}^{\epsilon}$ by linearity, we get a projective representation
of $\oib$ on $V_{\alpha}^{\epsilon}$. By defining $\pi_{\alpha}^{\epsilon}(c_{\epsilon}):=I$,
we obtain a level $1$  representation of the Lie algebras $\Bi$, $\Di$
on $V_{\alpha}^{\epsilon}$ for $\epsilon=0$, $1\over 2$, respectively.
To be more precise $V^0_\alpha \cong L(\Lambda_0)$ as $\Bi$-module and $V^{1\over 2}_{\overline 0}\cong L(\Lambda_0)$, $V^{1\over 2}_{\overline 1}\cong L(\Lambda_1)$ as $\Di$-module.
Using (4.2.11,13) and (4.3.10-12) one can  also compute the weight of the highest weight vector of $V^\epsilon_\alpha$ as $\soh$-module; the result is that $|\alpha >$ has weight  $\Lambda_n$, $\Lambda_n$, $\Lambda_0$, $\Lambda_1$ for $(\epsilon ,\alpha )= (0,\overline 0)$,  $(0,\overline 1)$,  $({1\over 2},\overline 0)$,  $({1\over 2},\overline 1)$, respectively.

In physics the procedure above, i.e., removing the anomaly, is usually formulated in terms of
a normal ordering prescription on the fermions; one writes
$$\pi^\epsilon (\F_{ij})= :\phi (i)\phi (j):\forno$$
where the {\it normal ordered product}  $:\phi (k)\phi (l):$ is defined by:
$$:\phi (k)\phi (l):\ \buildrel {\rm def} \over =\cases{\phi (k) \phi (l) &if $l< 0$\cr &\cr
0&if $k=l=0$\cr &\cr
                         -\phi (l) \phi (k)&in all other cases\cr}\forno$$


\beginsection \secno .  Vertex operators as normal ordered products of fermionic fields\par
\noindent Here we return to the expressions (3.3.1) for the vertex operators
$\hat A_{jk}^{ab}(\lambda ,\mu )(z)$ and  $\hat A_{j}^{a}(\lambda)(z)$. For the representation $\rho$ occuring in this formula we take $\rho_\epsilon:=\pi^{\epsilon} \circ \hat {\iota}_\epsilon$.
So we first embed the algebra $\soh$ in $\Bi$ or $\Di$ by the homomorphism 
$\hat {\iota}_\epsilon$ ($\epsilon = 0$, $1\over 2$, respectively) and then we represent it on the spin module $V^{\epsilon}$ by the representation $\pi^{\epsilon}$.

Let us start with the field $\hat A_{jk}^{ab}(\lambda ,\mu )(z)$. Recall from (3.3.3) that we have:
$$\eqalign{2\sqrt{n_an_b}\Bigl\{\hat A_{jk}^{ab}(\lambda ,\mu )(z)&+(H_{[w]}^\epsilon\vert
A_{jk}^{ab}(\lambda ,\mu ))\Bigr\}=\cr
&\sum z^{-N(l+\nu H^c_p(\epsilon)+\xi H^d_q(\epsilon))}\alpha^{ap}_{cj}(\lambda,\nu)\alpha^{bq}_{dk}(\mu,\xi)\rho_\epsilon\Bigl( e^{il\theta}G^{cd}_{pq}(\nu,\xi)\Bigr)\cr
+&\sum z^{-N(l+\nu H^c_p(\epsilon))}\alpha^{ap}_{cj}(\lambda,\nu)\beta^{b}_{k}(\mu)\rho_\epsilon\Bigl(e^{il\theta}G^{c}_{p}(\nu)\Bigr)\cr
-&\sum z^{-N(l+\xi H^d_q(\epsilon))}\beta^{a}_{j}(\lambda)\alpha^{bq}_{dk}(\mu,\xi)\rho_\epsilon\Bigl(e^{il\theta}G^{d}_{q}(\xi)\Bigr)\cr}\forno$$

\noindent where the various summations  run over $l\in {\bf Z}, 1\le c,d\le s, 
1\le p\le n_c, 1\le q\le n_d, \nu,\xi=\pm$. Next we use the embedding (4.2.11-13) $\hat \iota_\epsilon :\soh\hookrightarrow B_\infty$ or $\Di$:
$$\eqalign{e^{il\theta}G^{cd}_{pq}(\nu,\xi)\mapsto\sum_{m\in\epsilon +{\bf Z}}&\F_{(2n+1)m-\nu(n+1-(n_1+n_2+\cdots +n_{c-1}+p)),-(2n+1)(m+l)-\xi(n+1-(n_1+n_2+\cdots +n_{d-1}+q))}\cr
&+\delta_{l,0}(1-2\epsilon)(\lambda_n|G_{pq}^{cd}(\nu,\xi))\cr
e^{il\theta}G^{c}_{p}(\nu)\mapsto\sum_{m\in\epsilon +{\bf Z}}&\F_{(2n+1)m-\nu(n+1-(n_1+n_2+\cdots +n_{c-1}+p)),-(2n+1)(m+l)}\cr}\forno$$





\noindent With the spin representation $\pi^\epsilon :\F_{ij}\mapsto :\phi(i)\phi(j):$, formula (4.4.1) can be rewritten as:
$$\eqalign{2\sqrt{n_an_b}\Bigl\{&\hat A_{jk}^{ab}(\lambda ,\mu )(z)+(H_{[w]}^\epsilon-(1-2\epsilon)\lambda_n\vert
A_{jk}^{ab}(\lambda ,\mu ))\Bigr\}=\cr
:\sum &z^{N(m-\nu H^c_p(\epsilon))}
\alpha^{ap}_{cj}(\lambda,\nu)\phi((2n+1)m+\nu(n_1+n_2+\cdots+n_{c-1}+p-n-1))\cr
&z^{-N(l+m+\xi H^d_q(\epsilon))}
\alpha^{bq}_{dk}(\mu,\xi)\phi(-(2n+1)(m+l)+\xi(n_1+n_2+\cdots+n_{d-1}+q-n-1)):\cr+
:\sum &z^{N(m-\nu H^c_p(\epsilon))}
\alpha^{ap}_{cj}(\lambda,\nu)\phi((2n+1)m+\nu(n_1+n_2+\cdots+n_{c-1}+p-n-1))\cr
&z^{-N(l+m)}
\beta^{b}_{k}(\mu)\phi(-(2n+1)(m+l)):\cr-
:\sum &z^{N(m-\xi H^d_q(\epsilon))}
\alpha^{bq}_{dk}(\mu,\xi)\phi((2n+1)m+\xi(n_1+n_2+\cdots+n_{d-1}+q-n-1))\cr
&z^{-N(l+m)}
\beta^{a}_{j}(\lambda)\phi(-(2n+1)(m+l)):\cr}\forno$$



 \noindent (various summations over $l\in {\bf Z},m\in\epsilon+{\bf Z}, 1\le c,d\le s, 
1\le p\le n_c, 1\le q\le n_d, \nu,\xi=\pm$.)

The expression for the field $\hat A_{j}^{a}(\lambda )(z)$ in terms of normal ordered products of fermions is obtained in a completely similar way. The result is
$$\eqalign{\sqrt{2n_a}\Bigl\{&\hat A_{j}^{a}(\lambda)(z)+(H_{[w]}^\epsilon-(1-2\epsilon)\lambda_n\vert
A_{j}^{a}(\lambda))\Bigr\}=\cr
:\sum &z^{N(m-\nu H^c_p(\epsilon))}
\alpha^{ap}_{cj}(\lambda,\nu)\phi((2n+1)m+\nu(n_1+n_2+\cdots+n_{c-1}+p-n-1))\cr
&z^{-N(l+m+\xi H^d_q(\epsilon))}
\gamma^{q}_{d}(\xi)\phi(-(2n+1)(m+l)+\xi(n_1+n_2+\cdots+n_{d-1}+q-n-1)):\cr+
:\sum &z^{N(m-\nu H^c_p(\epsilon))}
\alpha^{ap}_{cj}(\lambda,\nu)\phi((2n+1)m+\nu(n_1+n_2+\cdots+n_{c-1}+p-n-1))\cr
&z^{-N(l+m)}
\delta\phi(-(2n+1)(m+l)):\cr-
:\sum &z^{N(m-\xi H^d_q(\epsilon))}
\gamma^{q}_{d}(\xi)\phi((2n+1)m+\xi(n_1+n_2+\cdots+n_{d-1}+q-n-1))\cr
&z^{-N(l+m)}
\beta^{a}_{j}(\lambda)\phi(-(2n+1)(m+l)):\cr}\forno$$



\noindent (various summations over $l\in {\bf Z}, m\in\epsilon+{\bf Z},  1\le c,d\le s, 
1\le p\le n_c, 1\le q\le n_d, \nu,\xi=\pm$.)

These expressions motivate the introduction of the  auxiliary fields $\phi_{ja\lambda}(z)$ and $\sigma(z)$ by:
$$\eqalign{\phi_{ja\lambda}(z):=&\sum z^{N(l-\nu H^c_p(\epsilon))}
\alpha^{ap}_{cj}(\lambda,\nu)\phi((2n+1)l+\nu(n_1+n_2+\cdots+n_{c-1}+p-n-1))+\cr&
\sum z^{Nl}
\beta^{a}_{j}(\lambda)\phi((2n+1)l)\cr\cr
\sigma(z):=&\sum z^{N(l-\xi H^d_q(\epsilon))}
\gamma^{q}_{d}(\xi)\phi((2n+1)l+\xi(n_1+n_2+\cdots+n_{d-1}+q-n-1))+\cr&
\sum z^{Nl}\delta\phi((2n+1)l)\cr}\forno$$

\noindent (various summations over $l\in \epsilon+{\bf Z}, 1\le c,d\le s, 
1\le p\le n_c, 1\le q\le n_d, \nu,\xi=\pm$.) With these fields (4.4.3) and (4.4.4) become
$$\eqalign{2\sqrt{n_an_b}\Bigl\{\hat A_{jk}^{ab}(\lambda ,\mu )(z)+(H_{[w]}^\epsilon-(1-2\epsilon)\lambda_n\vert A_{jk}^{ab}(\lambda ,\mu ))\Bigr\}&=
:\phi_{ja\lambda}(z)\phi_{kb\mu}(z):\cr
\sqrt{2n_a}\Bigl\{\hat A_{j}^{a}(\lambda)(z)+(H_{[w]}^\epsilon-(1-2\epsilon)\lambda_n\vert
A_{j}^{a}(\lambda))\Bigr\}&=
:\phi_{ja\lambda}(z)\sigma(z):\cr}\forno$$


The next step is to analyze the components of the fields $\phi_{ja\lambda}(z)$ and $\sigma(z)$. Using the definitions (2.7.11-14) of the constants $\alpha^{ap}_{cj}(\lambda,\mu), \beta^a_j(\lambda), \gamma^q_d(\xi)$ and $\delta$, it is straightforward (but tedious) to derive the following
\proclaim Lemma  \prono .
Define
$$\eqalign{\nu_(p):&={1-i(-)^p\over \sqrt{2}}\cr
\sigma_a^\epsilon(p):&={\nu_(p)\over \nu(2\epsilon n_a)}=\cases{i(-)^{2\epsilon n_a}&if\ $p+2\epsilon n_a\in 2{\bf Z}+1$\cr &\cr 1&if\ $p+2\epsilon n_a\in 2{\bf Z}$\cr}\cr}\forno$$
Then
$$\eqalign{\phi_{ja\lambda}(z)&=\cases{\displaystyle{\sqrt{2}\omega^{-{\lambda Nj\over 2n_a}}
\sum_{k\in{\bf Z}+1/2}(\omega^{-j}z)^{{N\over n_a}k}\phi_a^\lambda(k)}&if\ $1\le a\le r$\cr\cr\displaystyle{(-)^{2\epsilon j}\nu(2\epsilon n_a)^\lambda\sum_{k\in{\bf Z}}(\omega^{-j}z)^{{N\over 2n_a}k}\phi_a^\lambda(k)}&if\ $r<a\le s$\cr}\cr\cr
\sigma(z)&=\sum_{k\in(1-(-)^{s-r})/4+\epsilon+{\bf Z}}z^{Nk}\sigma(k)\cr}\forno$$
 where the fermions $\phi_a^\lambda(k)$ and  $\sigma(k)$ are defined as follows:\hfil\break
\noindent for $1\le a\le r$, $l\in {\bf Z}$:
$$\eqalign{\phi_a^+(n_al-p+{1\over 2})&:=\phi((2n+1
)(l-\epsilon)-(n_{a+1}+n_{a+2}+\cdots+ n_s
+p))\cr
\phi_a^-(n_al+p-{1\over 2})&:=\phi((2n+1)(l+\epsilon)+n_{a+1}+n_{a+2}+\cdots n_s+p)\cr}\qquad 1\le p\le n_a\forno$$
for $1\le a\le \Bigl[{s-r+1\over 2}\Bigr]$ $m\in \epsilon+{\bf Z}$:
$$\eqalign{&\phi_{r+2a-1}^\lambda(2n_{r+2a-1}m+p):=\cr&\lambda^{p+2\epsilon n_{r+2a-1}}\sigma_{r+2a-1}^\epsilon(p)
\cases{
\eqalign{{1\over \sqrt{2}}\Bigl\{\phi((2n&+1)m+n-(n_{1}+\cdots n_{r+2a-2}))\cr+\phi&((2n+1)(m+1)-n+n_{1}+\cdots n_{r+2a-2})\Bigr\}\cr}&if\ $p=n_{r+2a-1}$\cr\cr
\eqalign{&\eqalign{{1-\delta_{r+2a-1,s}\over \sqrt{2}}&\Bigl\{\phi((2n+1)m-n-1+n_{1}+\cdots n_{r+2a})\cr+\phi&((2n+1)m+n+1 -(n_{1}+\cdots n_{r+2a}))\Bigr\}\cr}\cr
&+\delta_{r+2a-1,s}\phi((2n+1)m)\cr}
&if\ $p=0$\cr\cr
\phi((2n+1)m+n-(n_{1}+\cdots +n_{r+2a-1})+p)&if\ $1\le p
<n_{r+2a-1}$\cr\cr
\phi((2n+1)m-n +n_{1}+\cdots n_{r+2a-1}+p)
&if\
$-n_{r+2a-1}< p <0$\cr}\cr}\forno$$
for $1\le a\le \left[{s-r\over 2}\right]$, $m\in\epsilon+{\bf Z}$:
$$\eqalign{&\phi_{r+2a}^\lambda(2n_{r+2a}m+p):=\cr&\lambda^{p+2\epsilon n_{r+2a}}\sigma_{r+2a}^\epsilon(p)
\cases{\eqalign{{i\over \sqrt{2}}
\Bigl\{\phi((2n&+1)(m+1)-n+n_1+\cdots n_{r+2a-2})\cr&-\phi((2n+1)m+n-(n_{1}+\cdots n_{r+2a-2}))\Bigr\}}&if\ $p=n_{r+2a}$\cr\cr
\eqalign{{i\over \sqrt{2}}\Bigl\{\phi((2n&+1)m-n-1+n_{1}+\cdots n_{r+2a})\cr&-\phi((2n+1)m+n+1-(n_{1}+\cdots n_{r+2a}))\Bigr\}}&if\ $p=0$\cr\cr
\phi((2n+1)m+n+1-(n_{1}+\cdots n_{r+2a})+p)&if\ $1\le p
<n_{r+2a}$\cr\cr\phi((2n+1)m-n-1+n_{1}+\cdots +n_{r+2a}+p)&if\
$-n_{r+2a}< p <0$\cr}\cr}\forno$$
for $m\in \epsilon+{\bf Z}$:
$$\eqalign{
\sigma(m+{\textstyle{ 1-(-)^{s-r}\over4}}):=&{\textstyle{1+(-)^{r-s}\over 2}}\phi((2n+1)m)+\cr&\quad{\textstyle{1-(-)^{r-s}\over 2}}{i\over\sqrt{2}}
\Bigl\{\phi((2n+1)(m+1)-n_s)-\phi((2n+1)m+n_s)\Bigr\}
\cr}\forno$$

We can now formulate the following theorem, which provides us with a fermionic realization of $\soh$ on the spin module $V^\epsilon$.
\proclaim Theorem  \prono .
Let $\phi_a^\lambda(k)$ and $\sigma(k)$ be defined by (4.4.9-12). Set
$$\eqalign{\phi_a^\lambda(z)&:=\cases{\displaystyle{\sum_{k\in{\bf Z}+1/2}\phi_a^\lambda(k)z^{{N\over n_a}k}}&if\ $1\le a \le r$\cr\cr\displaystyle{
\sum_{k\in{\bf Z}}\phi_a^\lambda(k)z^{{N\over 2n_a}k}}&if\ $r< a \le s$\cr}\cr\cr
\sigma(z)&:=\sum_{k\in(1-(-)^{s-r})/4+\epsilon+{\bf Z}}\sigma(k)z^{Nk}\cr}\forno$$
Then:
$$\eqalign{\hat A^{ab}_{jk}(\lambda,\mu)(z)+(H_{[w]}^\epsilon-(1-2\epsilon)\lambda_n\vert A^{ab}_{jk}(\lambda,\mu))&=C^a_j(\lambda)C^b_k(\mu):\phi_a^\lambda(\omega^{-j}z)\phi_b^\mu(\omega^{-k}z):\cr
\hat A^{a}_{j}(\lambda)(z)+(H_{[w]}^\epsilon-(1-2\epsilon)\lambda_n\vert A^{a}_{j}(\lambda))&=C^a_j(\lambda):\phi_a^\lambda(\omega^{-j}z)\sigma(z):\cr}\forno$$
where
$$C^a_j(\lambda):=\cases{{1\over\sqrt{n_a}}\omega^{-{\lambda Nj\over 2n_a}}&if\ $1\le a\le r$\cr &\cr (-)^{2\epsilon j}{\nu(2\epsilon n_a)^\lambda\over\sqrt{2n_a}}&if\ $r<a\le s$\cr}\forno$$

Let us now study the new fermions $\phi_a^\lambda(k)$ and $\sigma(k)$ in somewhat
more detail. First the $\phi_a^\lambda(k)$ for $1\le a\le r$; it is clear from the definition (4.4.9) that $\phi_a^+(k)$ and $\phi_a^-(k)$ are independent. The anticommutation relations are also easily read off from (4.4.9). One has:
$$\{\phi_a^\lambda(k),\phi_b^\mu(l)\}=\delta_{ab}\delta_{\lambda,-\mu}\delta_{k,-l}\qquad 1\le a,b\le r, k,l\in{\textstyle {1\over 2}}+{\bf Z}, \lambda,\mu=\pm\forno$$

\noindent This type of fermions is usually called {\it charged free} fermions in physics. The charge conjugation operation will be discussed in the next section.

For $r< a\le s$ the situation is quite different. It is immediate from (4.4.10-11) that
$$\phi_a^{-\lambda}(k)=(-)^k\phi_a^\lambda(k)\forno$$
hence there is in fact only one {\it neutral free} fermion. The relevant anticommutation relations are:
$$\{\phi_a^+(k),\phi_b^+(l)\}=(-)^k\delta_{ab}\delta_{k,-l}\qquad r< a,b\le s, k,l\in{\bf Z}\forno$$

Next, we consider the fermions $\sigma(k)$. Notice that $k$ runs over ${1\over 4}(1-(-)^{s-r})+\epsilon+{\bf Z}$. In particular, if ${1\over 4}(1-(-)^{s-r})=\epsilon$ there is a zero mode $\sigma(0)$. As ususal we call this Ramond fermions. If ${1\over 4}(1-(-)^{s-r})\ne \epsilon$,  $k$ runs over ${1\over 2}+{\bf Z}$  and we obtain so-called Neveu-Schwarz fermions. One easily verifies that
$$\{\sigma(k),\sigma(l)\}=\delta_{k,-l}\forno$$

\noindent All other anticommutators, in particular $\{\phi_a^\lambda(k),\phi_b^\mu(l)\}$ for $a\ne b$ and $\{\phi_a^\lambda(k),\sigma(l)\}$ are zero.

Recall that, in terms of the fermions $\phi(k)$, $k\in \epsilon+{\bf Z}$ the module $V^\epsilon$ is characterized as the unique irreducible $Cl^\epsilon$-module generated by $\vac$ satisfying $\phi(k)\vac =0$ for $k<0$. In terms of the new fermions $\phi_a^\lambda (k)$, $\sigma(k)$ this condition becomes:
$$\eqalign{\phi_a^\lambda(k)&\vac=0\qquad\hbox{for }k<0,\ 1\le a\le s\cr
\sigma (k)&\vac=0\qquad\hbox{for }k<0\cr
\{\phi_{r+2a-1}^+ (0)+(4\epsilon -1)i\phi_{r+2a}^+ (0)\}&\vac=0\qquad\hbox{for }1\le a\le [{s-r\over 2}]\cr
\{\phi_s^+(0)+i\sigma (0)\}&\vac=0\qquad\hbox{for }\epsilon={1\over 2},\ s-r\in 1+2{\bf Z}\cr}\forno$$

\beginsection \secno .  Hermitian structure and normal ordering
\par
It is well known that the spin module $V^\epsilon$ can be equipped
with an inner product, giving it the structure of a pre-Hilbert space;

\proclaim Theorem  \prono.
There exists a unique positive definite Hermitian form $(\ ,\
):V^\epsilon\times V^\epsilon\rightarrow{\bf C}$ such that:
$$\eqalign{&a)\quad\phi(j)^\dagger=\phi(-j)\cr&b)\quad(\vac,\vac)=1\cr}
\forno$$
\par
Instead of $(\ ,\ )$ we will also use Dirac's `bra-ket' notation;
$$(v,w)\equiv<v\mid w>\forno$$
We will leave out the symbols $\pi$ and $\pi\circ\hat {\iota}$ when
a representation is understood.

With the inner product one formulates the following lemma:
\proclaim Lemma  \prono.
$$\eqalign{
  &a)\quad \phi_a^+(j)^\dagger=\phi_a^-(-j), \sigma(j)^\dagger=\sigma(-j)\cr
  &b)\quad \F_{ij}^\dagger=\F_{-j,-i}\cr
  &c)\quad\left(e^{il\theta}G_{jk}(\lambda ,\mu)\right)^\dagger
 =e^{-il\theta}G_{kj}(-\mu ,-\lambda)=-e^{-il\theta}G_{jk}(-\lambda ,-\mu)\cr
&d)\quad \left(e^{il\theta}G_{j}(\lambda)\right)^\dagger=
-e^{-il\theta}G_{j}(-\lambda)\cr}\forno$$
\par
\proof{For a) we use the definitions (4.4.9-12) and the fact that $\sigma_a^\epsilon (p)^*=(-)^{p+2\epsilon n_a}\sigma_a^\epsilon (p)$. Relation b) is immediate from (4.3.13). c) and d) follow from b) and (4.2.11,13).}



Notice that relation (4.5.3a) and the definitions (4.4.13) imply
$$\eqalign{\phi_a^+(z)^\dagger&=\phi_a^-(z)\cr\sigma(z)^\dagger&=\sigma(z)}\forno$$


We use the Hermitian structure to define normal ordering for arbitrary
operators on the spin module $V$. For this we note that the
definition (4.3.14) can be rewritten as:
$$:\phi(i)\phi(j): = \phi(i)\phi(j)-<\overline 0\mid \phi(i)\phi(j)\vac
  \forno$$
The notation
$<\overline 0\mid A\vac :=(\vac ,A\vac)$ in this formula is called
the vacuum expectation value of the operator $A$. For future use we compute the vacuum expectation value of two fermionic fields. 

\proclaim Lemma  \prono. Let $1\le c\le [{s-r\over 2}]$, then the `contractions' $<\overline 0\mid\phi_a^\lambda(u)\phi_b^\mu(v)\vac$ 
and $<\overline 0\mid\phi_a^\lambda(u)\sigma(v)\vac$ 
are
equal to
$$\eqalign{<\overline 0\mid\phi_a^\lambda(u)\phi_b^\mu(v)\vac &=
  \cases{\delta_{\lambda ,-\mu}
         \sum_{k\ge 0}\left(v/u\right)^{{N\over n_a}(k+{1\over 2})}
         =\delta_{\lambda ,-\mu} 
         {\left(v/u\right)^{{N\over 2n_a}}\over 1-\left(v/u\right)^{N\over
         n_a}}
         &if $1\le a=b\le r$
         \cr &\cr
         {1\over 2}+\sum_{k>0}\left(-\lambda\mu\left(v/u\right)
         ^{{N\over 2n_a}}\right)^k
         ={1\over 2}{1-\lambda \mu\left(v/u\right)^{N\over 2n_a}\over
         1+\lambda \mu\left(v/u\right)^{N\over 2n_a}}  
         &if $r<a=b\le s$
         \cr & \cr
         {4\epsilon-1\over 2}i
         &if $a+1=b=r+2c$ 
         \cr & \cr
         -{4\epsilon-1\over 2}i
         &if $a=b+1=r+2c$ 
         \cr &\cr
         0
         &in all other cases
         \cr}
         \cr &\cr
<\overline 0\mid\phi_a^\lambda(u)\sigma(v)\vac &=
\delta_{as}\epsilon {\scriptstyle {1-(-)^{s-r}\over 2}}i\cr}\forno$$


Next we extend the normal
ordering prescription to arbitrary operators on the spin module;
\proclaim Definition  \prono.
Let $A$ and $B$ be operators on the spin module $V$, then
the normal ordered product of these operators is defined by:
$$:AB:{\buildrel {\rm def}\over =}AB-<\overline 0\mid AB\vac\forno$$
\par
\noindent Of course this definition is different from the usual one, because we
subtract only a $c$-number from the ordinary product 
and no operator structures.
Our motivation for this definition is that it produces the right ordering
prescription for a pair of bosonic oscillators as we will show later on.




\beginsection \secno . Bosonic oscillators on the spin module\par
Introduce the bosonic field $\hat\alpha_a(z):=n_a\hat A^{aa}_{n_an_a}(+,-)(\omega^{n_a}z)$ (see (3.4.9)); from lemma 3.4.2 we know that this field has the mode expansion
$$\hat\alpha_a(z)=\cases{\sum_{k\in{\bf Z}}\hat\alpha_a\left({N\over n_a}k\right)z^{-{N\over n_a}k}&if $1 \le a\le r$\cr &\cr
\sum_{k\in{1\over 2}+{\bf Z}}\hat\alpha_a\left({N\over n_a}k\right)z^{-{N\over n_a}k}&if $r < a\le s$\cr}\forno$$

\noindent and that the modes satisfy the commutation relations (3.4.10):
$$[\hat\alpha_a\left({N\over n_a}k\right),\hat\alpha_b\left({N\over n_b}l\right)]=k\delta_{ab}\delta_{k+l,0}\forno$$

We will now express these bosonic oscillators in terms of normal ordered products of fermions;
using the lemmas 4.2.2  and  3.3.1, we obtain  $$(H_{[w]}^\epsilon-(1-2\epsilon)\lambda_n \vert A^{aa}_{n_an_a}(+,-))=0\forno$$
Hence,  theorem 4.4.2 yields
$$\hat\alpha_a(z)=\cases{:\phi_a^+(z)\phi_a^-(z):&if $1\le a\le r$\cr &\cr
{1\over 2}:\phi_a^+(z)\phi_a^-(z):&if $r< a\le s$\cr}\forno$$

\noindent In terms of the modes this reads:
$$\hat\alpha_a\left({N\over n_a}k\right)=\cases{\sum_{j\in{1\over 2}+{\bf Z}}:\phi_a^+(j)\phi_a^-(-j-k):&if $1 \le a\le r$\cr &\cr
{1\over 2}
\sum_{j\in{\bf Z}}:\phi_a^+(j)\phi_a^-(-j-2k)):&if $r < a\le s$\cr}\forno$$

\noindent Notice (formula (4.6.5)) that $\hat\alpha_a(0)$ is the normal ordered product of fermions, this is essential; in order to obtain this, we had to choose two lifts of $w$ (see section 2.3).

We conclude this section with a lemma, which  is fundamental.
\def\a#1#2{\hat\alpha_{#1}({N\over n_{#1}}#2)}
\proclaim Lemma \prono.
$$\eqalign{
  [\a ak ,\phi_b^+(z)]=&\delta_{ab}z^{{N\over n_a}k}\phi_b^+(z)\cr
  [\a ak ,\phi_b^-(z)]=&-\delta_{ab}z^{{N\over n_a}k}\phi_b^-(z)\cr
  }\forno$$
\par

\proof{We use the expressions (4.6.5) for $\a ak$ in terms of fermions. For $1\le b\le r$ we may assume that $1\le a\le r$:
$$\eqalign{[\a ak,\phi_b^+(l)]
  &=\sum_{j\in {1\over2}+{\bf Z}}[:\phi_a^+(j)\phi_a^-(-j-k):,\phi_b^+(l)]\cr
  &=\sum_{j>-k}[\phi_a^+(j)\phi_a^-(-j-k),\phi_b^+(l)]-
\sum_{j<-k}[\phi_a^-(-j-k)\phi_a^+(j),\phi_b^+(l)]\cr
  &=\sum_{j\in{1\over 2}+{\bf Z}}\delta_{ab}\delta_{j+k,l}\phi_b^+(j)\cr
  &=\delta_{ab}\phi_b^+(l-k)\cr}\forno$$


\noindent For $r< b\le s$ we assume that  $r< a\le s$. We can then leave out the normal ordering sign in the second relation of (4.6.5) and compute:
$$\eqalign{[\a ak,\phi_b^+(l)]
  &={1\over 2}\sum_{j\in {\bf Z}}[\phi_a^+(j)\phi_a^-(-j-2k),\phi_b^+(l)]\cr
  &={1\over 2}\sum_{j\in {\bf Z}}
     [(-1)^{-j-2k}\phi_a^+(j)\phi_a^+(-j-2k),\phi_b^+(l)]\cr
  &={1\over 2}\sum_{j\in{\bf Z}}[\delta_{ab}\delta_{j+2k,l}\phi_b^+(j)
     -\delta_{ab}\delta_{l,-j}(-1)^{-2k}\phi_b^+(-j-2k)]\cr
  &=\delta_{ab}\phi_b^+(l-2k)\cr}\forno$$
Notice that in (4.6.8) $r<b\le s$, hence, if $a=b$, then $k\in {1\over 2}+{\bf Z}$
and in particular $(-1)^{-2k}=-1$.

Using (4.4.13),
the first relation
of (4.6.6) follows. The second relation is proved by Hermitian conjugacy of the
first.}





\baselineskip=22pt plus.1pt



  

\count30=4                      %chapterno = 4
\count31=0                      %sectionno = 0
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0

\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count31 .\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count31 .\number\count33}


\long\def\proof#1{\noindent{\bf{Proof.}}\quad#1\qedd}
\def\qedd{\hfill\hbox{\vrule\vbox{\hrule\kern2truept\hbox{\kern2truept
\vbox to2pt{\hsize=2pt}\kern2truept}\kern2truept\hrule}\vrule}\vskip .5cm}



\def\so{so_{2n+1}({\bf C})}
\def\SO{SO_{2n+1}({\bf C})}

\def\ad{{\rm ad}}
\def\Ad{{\rm Ad}}
\def\trace{{\rm trace}}

\def\E{{\cal E}}
\def\F{{\cal F}}
\def\tn{\hat \tau_{\underline n}}
\def\tw{\hat \tau_{[w]}}
\def\Hn{H_{\underline n}}
\def\Hw{H_{[w]}^\epsilon}
\def\Sn{S_{\underline n}}
\def\Sw{S_{[w]}}
\def\Xw{X_{[w]}}
\def\sw{\hat s_{[w]}}
\def\hw{{\underline h}_{[w]}}

\def\gl{gl_{2n+1}({\bf C})}
\def\oi{o(\infty )}
\def\gli{gl(\infty )}
\def\glib{\overline {gl(\infty )}}
\def\oib{\overline {o(\infty )}}
\def\Ai{A_{\infty}}
\def\Di{D_{\infty}}
\def\glt{\tilde{gl}_{2n+1}({\bf C})}
\def\glh{\hat{gl}_{2n+1}({\bf C})}
\def\sot{\tilde{so}_{2n+1}({\bf C})}
\def\soh{\hat{so}_{2n+1}({\bf C})}
\def\vac{\mid \overline 0 >}
\def\sh{\hat {\underline s}}
\def\shw{\hat {\underline s}_{[w]}}
\def\a#1#2{\hat\alpha_{#1}({N\over n_{#1}}#2)}
\def\L{{\cal F}}
\def\D{{\cal D}}

\beginsection \chapno .  Conformal symmetry
\par
\beginsection \secno .  Introduction
\par
In this section we will review some well known constructions of the
Virasoro algebra. In sections 5.2 and 5.3 we will describe representations of this algebra in terms of bosonic and fermionic oscillators. In section 5.4 we will consider the so-called Sugawara construction of the Virasoro algebra.      

These three different constructions of the Virasoro algebra are in fact closely related. We will discuss this relation, which will play a crucial role in the sequel, in section 5.5.





\beginsection \secno . Bosonic oscillator representations of the Virasoro algebra
\par
Recall that the Virasoro algebra is the universal one
dimensional central extension of the conformal algebra. It has a basis
consisting of elements $d_k, k\in{\bf Z}$ and a central element $c_{vir}$,
which satisfy the commutation relations
$$[d_k,d_l]=(k-l)d_{k+l}+{{ 1\over
  12}}\delta_{k+l,0}\left(k^3-k\right)c_{vir}\forno$$
The following construction of the Virasoro algebra is standard in
physics.
\proclaim Lemma  \prono .
Let $\{a_i\}_{i\in{\cal Z}}$ (${\cal Z}={\bf Z}$ or ${1\over 2}+{\bf Z}$) 
be a collection of operators on a vector
space V, such that $[a_i,a_j]=i\delta_{i+j,0}I$ and $a_i(v)=0$, for all 
$v\in V$ and $ i\gg 0$. Define normal ordering by 
$:a_ia_j:=a_ia_j$ if $i\le j$ and
$:a_ia_j:=a_ja_i$ if $i>j$, then the assignment $d_k\mapsto {\cal B}_k$,
where the operators ${\cal B}_k$ are defined by
$${\cal B}_k:={ 1\over 2}\sum_{j\in{\cal Z}}:a_{-j}a_{j+k}:+\delta_{k,0}\beta\forno$$
and 
$$\beta:=\cases{0&if ${\cal Z}={\bf Z}$\cr &\cr
               {1\over 16}&if ${\cal Z}={1\over 2}+{\bf Z}$\cr}\forno$$
is a representation of the Virasoro algebra with $c_{vir}\mapsto I$.
Moreover, we have:
$$[{\cal B}_k,a_i]=-ia_{i+k}\forno$$
\par
\proof{see, e.g., [FLM], [KR].}
Translated to our situation, we get a $c_{vir}=1$ representation of the
Virasoro algebra on the spin module $V^\epsilon$ by:
$$\tilde{\cal B}_k^{(a)}:={ 1\over 2}\sum_{j\in{\cal Z}_a}
  :\hat\alpha_a(-{N\over n_a}j)\a a{(j+k)}:+\delta_{k,0}\beta_a
  \qquad 1\le a\le s\forno$$
where
$${\cal Z}_a:=\cases{{\bf Z}&if $1\le a\le r$\cr &\cr
                      {1\over 2}+{\bf Z}&if $r<a\le s$\cr}
  \qquad\hbox{and  }\beta_a:=\cases{0&if $1\le a\le r$\cr &\cr
                                    {1\over 16}&if $r<a\le s$\cr}
  \forno$$
Starting from these operators, we construct $c_{vir}=n_a$
representations of the Virasoro algebra in the following manner; set
$\hat{\cal B}_k^{(a)}:=1/n_a\tilde{\cal B}_{n_ak}^{(a)}$, then these $\hat{\cal B}_k^{(a)}$'s satisfy:
$$[\hat{\cal B}_k^{(a)},\hat{\cal B}_l^{(a)}]=(k-l)\hat{\cal B}_{k+l}^{(a)}+{ 1\over 12}n_a\delta_{k+l,0}
  \left(k^3-k\right)I+{ 1\over 12}\delta_{k+l,0}k{n_a^2-1\over
  n_a}I\forno$$
The extra term in the right hand side of (5.2.7) is a two coboundary
and hence we define operators ${{\cal B}}_k^{(a)}:=\hat{\cal B}_k^{(a)}+{1\over 24}\delta_{k,0}
{n_a^2-1\over n_a}I$ to obtain the commutation relations in their standard
form:
$$[{{\cal B}}^{(a)}_k,{{\cal B}}^{(a)}_l]=(k-l){{\cal B}}^{(a)}_{k+l}+{ 1\over
  12}\delta_{k+l,0}n_a\left(k^3-k \right)I\forno$$


Since the operators ${{\cal B}}_k^{(a)}$ and ${{\cal B}}_l^{(b)}$ commute for $a\ne b$,
we can construct a $c_{vir}=n$ representation by summation over $a$.






\proclaim Proposition  \prono .
Let the operators $B_k$ be defined by
$$B_k:=\sum_{a=1}^s{{\cal B}}^{(a)}_k=\sum_{a=1}^s{1\over n_a}
\Bigl\{{ 1\over 2}\sum_{j\in {\cal Z}_a}:\hat\alpha_a(-{N\over n_a}j)\a a{(j+n_ak)}:+\delta_{k,0}\Bigl( {n_a^2-1\over 24}+\beta_a\Bigr)I\Bigr\}
       \forno$$
Then the assignment $d_k\mapsto B_k$ is a  (bosonic oscillator) representation of the
Virasoro algebra with $c_{vir}\mapsto nI$. Moreover, we have:
$$[B_k,\a aj]=-{j\over n_a}\a a{(j+n_ak)}\forno$$

\beginsection \secno . Fermionic  oscillator constructions of the Virasoro algebra


\noindent We start this section with some fermionic oscillator representations for the Virasoro algebra, which are standard in physics.

\proclaim Lemma \prono .
Let $\{\psi_k^+,\psi_k^-\}_{k\in{1\over 2}+{\bf Z}}$ be a collection of operators on a vector space $V$ satisfying $\{\psi_k^+,\psi_j^+\}=0=\{\psi_k^-,\psi_j^-\}, \{\psi_k^+,\psi_j^-\}=\delta_{k+j,0}$. Suppose $\psi_k^\pm(v)=0$ for all $v \in V$ and $k>>0$. Define normal ordering by $:\psi_k^+\psi^-_j:=\psi_k^+\psi^-_j$ if $k<j$ and $-\psi_j^-\psi^+_k$ if $k>j$. Then the assignment $d_k\mapsto {\cal F}_k$, where the operators ${\cal F}_k$ are defined by
$${\cal F}_k=\sum_{j\in{1\over 2}+{\bf Z}}(j+{k\over 2}):\psi_{-j}^+\psi_{j+k}^-:, \qquad k\in {\bf Z}\forno$$
\noindent defines a representation of the Virasoro algebra with $c_{vir}\mapsto 1$. Moreover, we have:
$$[{\cal F}_k,\psi_j^\pm]=-\left(j+{k\over 2}\right)\psi_{j+k}^\pm\forno$$

\proof{This result can essentially be found in [KR].}

\proclaim Lemma \prono .
Let $\{\psi_i\}_{i\in {\cal Z}}$  (${\cal Z}={\bf Z}$ or ${1\over 2}+{\bf Z}$) be a collection of operators on a vector
space $V$, such that $\{\psi_i,\psi_j\} =\delta_{i+j}I$
and $\psi_i(v)=0$ for all $v\in V$ and $i>>0$. Define
normal ordering by $:\psi_i\psi_j:= \psi_i\psi_j$ if $i\le j$
not $i=j=0$, $:\psi_0\psi_0:=0$ and $:\psi_i\psi_j:=-\psi_j\psi_i$
if $i>j$, then the assignment $d_k\mapsto \tilde{\cal F}_k$, where
the operators $\tilde{\cal F}_k$ are defined by
$$\tilde{\cal F}_k={1\over 2}\sum_{j\in{\cal Z}}j:\psi_{-j}\psi_{j+k}:
+\delta_{k0}\gamma I\forno$$
where
$$\gamma:=\cases{{1\over 16}&if ${\cal Z}={\bf Z}$\cr &\cr0&if ${\cal Z}={1\over 2}+{\bf Z}$\cr}\forno$$
is a representation of the Virasoro algebra with $c_{vir}\mapsto {1\over
2}I$.
Moreover, we have:
$$[\tilde{\cal F}_k,\psi_i ]=-\left(i+{k\over 2}\right)\psi_{i+k}\forno$$
\par
\proof{see, e.g. [KR].}


The first lemma can be used to construct  fermionic oscillator representations of the Virasoro algebra in terms of the fermions $\phi^+_a(k),\phi^-_a(k), 1\le a\le r$. For a fixed $a$ ($1\le a\le r$) we identify $\phi^+_a(k)=\psi^+_{-k},\phi^-_a(k)=\psi^-_{-k}$. In this way we get a $c_{vir}=1$ representation of the Virasoro algebra. To obtain a $c_{vir}=n_a$ representation, we set
$${\cal F}^{(a)}_k:={1\over n_a}\sum_{j\in{1\over 2}+{\bf Z}}(j+{n_ak\over 2}):\phi_{a}^+(j)\phi_{a}^-(-j-n_ak):+{1\over 24}\delta_{k0}{n_a^2-1\over n_a}I\qquad 1\le a\le r\forno$$

The second lemma can be employed in a similar fashion;  for a fixed $a$ ($r<a\le s$) we identify $\phi_a^+(2k)=\psi_{-2k}$ and $\phi_a^+(2k+1)=i\psi_{-2k-1}$, which yields a $c_{vir}={1\over 2}$ Virasoro algebra. The operators 
$${\cal F}_k^{(a)}:={1\over 4n_a}\sum_{j\in {\bf Z}}j:\phi_a^+(j)\phi_a^-(-j-2n_ak):+\delta_{k0}{1\over 16n_a}I+{1\over 24}\delta_{k0}{n_a^2-1\over n_a}I\qquad r<a\le s\forno$$

\noindent then provide a $c_{vir}=n_a$ construction of the Virasoro algebra.


 

 All this yields the following proposition.

\proclaim Proposition  \prono . The operators
$$\eqalign{F_k:=&\sum_{a=1}^r\sum_{j\in {1\over 2}+{\bf Z}}
  {1\over n_a}\left(j+{n_ak\over 2}\right)
  :\phi_a^+(j)\phi_a^-(-j-n_ak):+{1\over 24}\delta_{k0}{n_a^2-1\over n_a}I\cr
  &+{1\over 2}\sum_{a=r+1}^s\sum_{j\in {\bf Z}}{j\over 2n_a}
  :\phi_a^+(j)\phi_a^-(-j-2n_ak):+\delta_{k0}{1\over 16n_a}I+{1\over 24}\delta_{k0}{n_a^2-1\over n_a}I\cr}\forno$$
define  a $c_{vir}=n$ (fermionic oscillator) representation of the Virasoro algebra.
Set $m_a:=n_a$ if $1\le a\le r$ and $m_a:=2n_a$ if $r< a\le s$, then
$$\eqalign{[F_k,\phi_a^+(l)]
  &={1\over m_a}\left(l-{m_ak\over 2}\right)\phi_a^+(l-m_ak)\cr
  [F_k,\phi_a^-(l)]
  &={1\over m_a}\left(l-{m_ak\over 2}\right)\phi_a^-(l-m_ak)\cr
  }\forno$$
Moreover
$$[F_k,\hat\alpha_a({N\over n_a}l)]
  =-{l\over n_a}\hat\alpha_a({N\over
  n_a}(l+n_ak))\forno$$

\proof{The relations (5.3.9) are a direct consequence of (5.3.2) and (5.3.5). To prove (5.3.10), we use the expression (4.6.5) of the bosonic oscillators in terms of the fermions.}


Notice that we didn't use the fermions $\sigma(k), k\in{1-(-)^{s-r}\over 4}+\epsilon+{\bf Z}$ up to now. Using lemma 5.3.2 once again, we get an additional $c_{vir}={1\over 2}$ Virasoro algebra by
$$F_k^\prime:={1\over 2}\sum_{j\in{1-(-)^{s-r}\over 4}+\epsilon+{\bf Z}}j:\sigma(j)\sigma(-j-k)+\delta_{k0}\delta_{{1-(-)^{s-r}\over 4},\epsilon}{\textstyle 1\over 16}I\forno$$

\noindent From (5.3.5) it is also clear that
$$[F_k^\prime,\sigma(j)]=\left(j-{k\over 2}\right)\sigma(j-k)\forno$$

\noindent Clearly, $[F_k,F_l^\prime]=0$ and hence the operators $F_k+F_k^\prime$   define a $c_{vir}=n+{1\over 2}$ representation of the Virasoro algebra. We can now formulate the following lemma, which states the conformal transformation properties of the fermionic fields $\phi_a^\lambda(z), \sigma(z)$ and the vertex operators $\hat A^{ab}_{jk}(\lambda,\mu)(z), \hat A^{a}_{j}(\lambda)(z)$.

\proclaim Lemma \prono .
$$\eqalign{[F_l+F_l^\prime,\phi_a^\lambda(z)]&={1\over N}z^{Nl}\left(z{d\over dz}+N{l\over 2}\right)\phi_a^\lambda(z)\cr
[F_l+F_l^\prime,\sigma(z)]&={1\over N}z^{Nl}\left(z{d\over dz}+N{l\over 2}\right)\sigma(z)\cr
[F_l+F_l^\prime,\hat A^{ab}_{jk}(\lambda,\mu)(z)]&={1\over N}z^{Nl}\left(z{d\over dz}+{Nl}\right)\hat A^{ab}_{jk}(\lambda,\mu)(z)\cr
[F_l+F_l^\prime,\hat A^{a}_{j}(\lambda)(z)]&={1\over N}z^{Nl}\left(z{d\over dz}+{Nl}\right)\hat A^{a}_{j}(\lambda)(z)\cr}\forno$$

\proof{The first two relations are easily derived by means of (5.3.9) and (5.3.12). The third relation is proved as follows. If $a=b$, $j=k$ and $\lambda=-\mu$, this relation can be verified by means of $\hat A_{jj}^{aa}(\lambda,-\lambda)(z)=\pm {\lambda\over n_a}\hat\alpha_a(\omega^{-j}z)$ and (5.3.10). In all other cases we multiply the commutator $[F_l, \hat A_{jk}^{ab}(\lambda,\mu)(z)]$ by the factor $(1+\lambda\mu\delta_{ab}\omega_a^{j-k})^2$ and compute:
$$\eqalign{(1+\lambda\mu\delta_{ab}\omega_a^{j-k})^2[F_l, \hat A_{jk}^{ab}(\lambda,\mu)(z)]&=
(1+\lambda\mu\delta_{ab}\omega_a^{j-k})^2[F_l, C_j^a(\lambda)C_k^b(\mu):\phi_a^\lambda(\omega^{-j}z)\phi_b^\mu(\omega^{-k}z):]\cr
&=\lim_{u\to \omega^{-j}z \atop v \to \omega^{-k}z}(1+\lambda\mu\delta_{ab}(v/u)^{N\over m_a})^2[F_l, C_j^a(\lambda)C_k^b(\mu):\phi_a^\lambda(u)\phi_b^\mu(v):]\cr}\forno$$
We can now leave out the normal ordering because the contraction $<\overline 0|\phi_a^\lambda (u)\phi_b^\mu(v)\vac$ is a multiple of the identity. This allows us to compute the commutator by means of the first relation of the lemma. After reinsertion of the normal ordering we find:
$$\eqalign{\lim_{u\to \omega^{-j}z \atop v \to \omega^{-k}z}&(1+\lambda\mu\delta_{ab}(v/u)^{N\over m_a})^2\Bigl\{
{1\over N}(u^{Nl+1}{d\over du}+v^{Nl+1}{d\over dv})+{l\over 2}(u^{Nl}+v^{Nl})\Bigr\}C_j^a(\lambda)C_k^b(\mu)<\overline 0|\phi_a^\lambda (u)\phi_b^\mu(v)\vac \cr
&+(1+\lambda\mu\delta_{ab}\omega_a^{j-k})^2{1\over N}z^{Nl}(z{d\over dz}+Nl)C_j^a(\lambda)C_k^b(\mu):\phi_a^\lambda(\omega^{-j}z)\phi_b^\mu(\omega^{-k}z):\cr}\forno$$
Now using lemmas 3.3.1, 4.2.2 and  4.5.3, and formula (4.4.15), we deduce that
$$\lim_{u\to \omega^{-j}z \atop v \to \omega^{-k}z}(1+\lambda\mu\delta_{ab}(v/u)^{N\over m_a})^2
{1\over N}(u^{Nl+1}{d\over du}+v^{Nl+1}{d\over dv})<\overline 0|\phi_a^\lambda (u)\phi_b^\mu(v)\vac =0$$ and that
$$\eqalign{\lim_{u\to \omega^{-j}z \atop v \to \omega^{-k}z}&(1+\lambda\mu\delta_{ab}(v/u)^{N\over m_a})^2{l\over 2}(u^{Nl}+v^{Nl})C_j^a(\lambda)C_k^b(\mu)<\overline 0|\phi_a^\lambda (u)\phi_b^\mu(v)\vac \cr
&=-l(1+\lambda\mu\delta_{ab}\omega_a^{j-k})^2(H^\epsilon_{[w]}-(1-2\epsilon)\lambda_n|A_{jk}^{ab}(\lambda, \mu))\cr}$$
Using (3.2.9) and (4.2.13), we finally find the desired result.

The fourth relation can be proved in an analogous way.}



\beginsection \secno .  The Sugawara construction of the Virasoro algebra
\par
Let $\{u_m\}_{1\le m\le 2n^2+n}$ be a basis of $\so$ and let $\{u^m\}_
{1\le m\le 2n^2+2n}$ be the dual basis with respect to the normalized
trace form $(\ \mid \ )$ on $\so$.
The following lemma is known as the Sugawara construction of the Virasoro
algebra.

\proclaim Lemma  \prono .
Define the operators ${\cal S}_k:V^\epsilon\rightarrow V^\epsilon$ by:
$${\cal S}_k:=\Bigl\{{ 1\over 4n}\sum_{j\in{\bf Z}}
  \sum_{m=1}^{2n^2+n}:\left(e^{-ij\theta}u_m\right) \left(e^{i(j+k)\theta}{u^m}\right):\Bigr\}+(1-2\epsilon)\delta_{k,0}{2n+1\over 16}I\forno$$
The assignment $d_k\mapsto {\cal S}_k$ defines a representation of the
Virasoro algebra with $c_{vir}\mapsto (n+{\textstyle {1\over 2}})I$. Moreover, we have:
$$[{\cal S}_k,e^{il\theta}x]=-le^{i(k+l)\theta}x
  \qquad\hbox{for all}\quad k,l\in {\bf Z}, x\in\so\forno$$
\par
\proof{We have $\sum_{m=1}^{2n^2+n}[u_m,u^m]=0$, whence:
$$\sum_{m=1}^{2n^2+n}[e^{ik\theta}u_m,e^{ij\theta}u^m]
  =k\delta_{k+j,0}\sum_{m=1}^{2n^2+n}(u_m\mid u^m)\forno$$
It is also clear that the operators $e^{ik\theta}u^m, k> 0\ (k\ge 0)$ for $\epsilon=0$ (resp. $\epsilon={1\over 2}$)
annihilate the vacuum $\vac$. 
If $k=j=0$, then $\sum_{m=1}^{2n^2+2}u_mu^m$ is the Casimir operator for $\so$. Since $\vac$ has $\soh$-weight $\Lambda_n$, $\Lambda_0$ for $\epsilon=0,\ {1\over 2}$, respectively, its weight as $\so$-module  is $\lambda_n$, $0$, respectively. Hence: 
$$\sum_{m=1}^{2n^2+2}u_mu^m\vac =(1-2\epsilon)(\lambda_n +2\rho |\lambda_n)\vac \forno$$
where $\rho$ is the sum of the fundamental weights of $\so$. Moreover
$$(\lambda_n+2\rho |\lambda_n)={2n^2+n\over 4}\forno$$
With these remarks and the normal ordering
definition 4.5.4 one easily finds:
$$\sum_{m=1}^{2n^2+n}:\left(e^{ik\theta}u_m\right)
  \left(e^{ij\theta}u^m\right):=
  \cases{\sum_{m=1}^{2n^2+2}u_mu^m-(1-2\epsilon){2n^2+n\over 4}&if $j=k=0$\cr &\cr
  \sum_{m=1}^{2n^2+2}
  \left(e^{ik\theta}u_m\right)\left(e^{ij\theta}u^m\right)
  &if\ $k\le j$ and not $j=k=0$\cr &\cr
  \sum_{m=1}^{2n^2+n}\left(e^{ij\theta}u^m\right)\left(e^{ik\theta}u_m\right)
  &if\ $k>j$\cr}\forno$$
The rest of the proof is standard; see, e.g., [KR].}

The lemma tells us that the Virasoro operators ${\cal S}_k$ act as
$ie^{ik\theta}{d\over d\theta}$ on the affine algebra.
Now let us take an element
$\hat x(l)$ of the affine algebra (see (3.2.9)). It is clear then that:
$$[{\cal S}_k,\hat x(l)]=e^{ik\theta}\ad\,\Hw\left(x(l)\right)
  -{l\over N}x(l+Nk)\forno$$
In view of the right hand side of this formula it is natural to
consider the operator ${\cal S}_k-e^{ik\theta}\Hw$.
Since
$e^{ik\theta}\Hw=\Hw(Nk)$, we have:
$$[{\cal S}_k-e^{ik\theta}\Hw,\hat x(l)]=-{l\over N}x(l+Nk)-\nu
  (\Hw(Nk),x(l))I\forno $$
The value of the two cocycle $\nu$ in the right hand side of this
formula is easily computed with the help of (3.2.6); using the ad-invariance of
the normalized trace form, we get:
$$\nu (\Hw(Nk),x(l))
  =-{l\over N}\delta_{l+Nk,0}\left(\Hw\mid x_{\overline l}\right)\forno$$
Recalling the definition (3.2.9) of $\hat x(l+Nk)$, we find:
$$[{\cal S}_k-e^{ik\theta}\Hw,\hat x(l)]=-{l\over N}\hat x(l+Nk)\forno$$

Again we summarize the results in a proposition.
\proclaim Proposition  \prono .
Define the operators $ S_k:V^\epsilon\rightarrow V^\epsilon$ by:
$$ S_k:=
\Bigl\{{ 1\over 4n}\sum_{j\in{\bf Z}}
  \sum_{m=1}^{2n^2+n}:\left(e^{-ij\theta}u_m\right)
  \left(e^{i(j+k)\theta}{u^m}\right):\Bigr\}-e^{ik\theta}\Hw 
+\delta_{k0}\Bigl\{(1-2\epsilon){2n+1\over 16}+{1\over 2}\mid \Hw \mid^2\Bigr\}I\forno$$
then the assignment $d_k\mapsto S_k$ is a representation of the
Virasoro algebra with $c_{vir}\mapsto (n+{\textstyle {1\over 2}})I$. Moreover, we have:
$$[S_k,\hat x(l)]=-{l\over N}\hat x(l+Nk)\forno$$
and hence:
$$[S_k,\hat \alpha_a\left({N\over n_a}l\right)]=-{l\over n_a}\hat \alpha_a\left({N\over n_a}(l+n_ak)\right)\forno$$
\proof{An easy exercise.}


\beginsection \secno . Relation between $B_k, F_k$ and $S_k$
\par
\proclaim Theorem \prono .
$$\eqalign{B_k&=F_k\cr S_k&=F_k+F_k^\prime\cr}\forno$$

\proof{To prove the first relation, we consider the difference $\Delta_k:=F_k-B_k$. Using (5.2.10) and (5.3.10), we find $[\Delta_k,\hat \alpha({N\over n_a}k]=0$ and hence $[\Delta_k,B_k]=0$. Consequently, the $\Delta_k$'s span a $c_{vir}=0$ representation of the Virasoro algebra (i.e. a conformal algebra):
$$\eqalign{[\Delta_k,\Delta_l]&=[\Delta_k,F_l]\cr
&=[F_k,F_l]-[B_k,F_l]\cr&=[F_k,F_l]-[B_k,F_l-B_l]-[B_k,B_l]\cr&=(k-l)\Delta_{k+l}\cr}\forno$$

\noindent where we have used the fact that the central charges of the bosonic and fermionic constructions of the Virasoro algebra coincide. It is well known (see [KW88], [GO]) that the only unitary representation of the Virasoro algebra with these properties is the trivial one, meaning that $\Delta_k\equiv 0$.

Similarly, we set $\delta_k:=F_k+F_k^\prime-S_k$.  Using (5.3.13) and (5.4.12), we see that these $\delta_k$'s commute with the affine algebra and hence $[\delta_k,S_l]=0$. A calculation identical to (5.5.2) shows:
$$[\delta_k,\delta_l]=(k-l)\delta_{k+l}\forno$$
Hence, $\delta_k\equiv 0$}













\baselineskip=22pt plus.1pt

\count30=5                      %chapterno = 5

\count31=0                      %sectionno = 0
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0

\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count31 .\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count31 .\number\count33}


\long\def\proof#1{\noindent{\bf{Proof.}}\quad#1\qedd}
\def\qedd{\hfill\hbox{\vrule\vbox{\hrule\kern2truept\hbox{\kern2truept
\vbox to2pt{\hsize=2pt}\kern2truept}\kern2truept\hrule}\vrule}\vskip .5cm}



\def\so{so_{2n+1}({\bf C})}
\def\SO{SO_{2n+1}({\bf C})}

\def\ad{{\rm ad}}
\def\Ad{{\rm Ad}}
\def\trace{{\rm trace}}

\def\E{{\cal E}}
\def\F{{\cal F}}
\def\tn{\hat \tau_{\underline n}}
\def\tw{\hat \tau_{[w]}}
\def\Hn{H_{\underline n}}
\def\Hw{H_{[w]}}
\def\Sn{S_{\underline n}}
\def\Sw{S_{[w]}}
\def\Xw{X_{[w]}}
\def\sw{\hat {\underline s}_{[w]}}
\def\hw{{\underline h}_{[w]}}

\def\gl{gl_{2n}({\bf C})}
\def\oi{o_\epsilon(\infty )}
\def\gli{gl(\infty )}
\def\glib{\overline {gl(\infty )}}
\def\oib{\overline {o_\epsilon(\infty )}}
\def\Ai{A_{\infty}}
\def\Bi{B_{\infty}}
\def\Di{D_{\infty}}
\def\glt{\tilde{gl}_{2n}({\bf C})}
\def\glh{\hat{gl}_{2n}({\bf C})}
\def\sot{\tilde{so}_{2n+1}({\bf C})}
\def\soh{\hat{so}_{2n+1}({\bf C})}
\def\vac{\mid \overline 0 >}
\def\sh{\hat {\underline s}}
\def\shw{\hat {\underline s}_{[w]}} 
\def\a#1#2{\hat\alpha_{#1}({N\over n_{#1}}#2)}
\def\L{{\cal L}}
\def\D{{\cal D}}

\beginsection \chapno .  Vertex operators
\par
\beginsection \secno .  Introduction
\par
In this section we will show that the fermionic fields $\phi_a^+(z)$
and $\phi_a^-(z)$ can be expressed in terms of so called 
fermionic translation operators $Q_a:V^\epsilon_\alpha\to V^\epsilon_{\alpha +\overline 1}$ for 
$1\le a\le r$, 
fermionic chirality operators $Q_a:V^\epsilon_\alpha\to V^\epsilon_{\alpha+\overline 1}$
for $r<a \le s$
and the bosonic oscillators $\a ak$. This leads to bosonic expressions for the
vertex operators $\hat A^{ab}_{jk}(\lambda,\mu)(z)$ and $\hat
A^{a}_{j}(\lambda)(z)$.
\beginsection \secno . 
Expressions for $\phi_a^+(z)$ and $\phi_a^-(z)$ \par 
Recall the commutation relations (4.6.6):
$$[\a ak,\phi_b^\lambda(z)]
  =\lambda\delta_{ab}z^{{N\over n_a}k}\phi_b^\lambda(z)\forno$$
These relations can be seen as formal eigenvalue equations for the
adjoint action of the HSA $\sw$. They determine the formal eigenvectors
$\phi_a^+(z)$ and $\phi_a^-(z)$ up to operators 
that commute with the action
of the HSA. These operators are in their turn determined by the conformal
transformation properties of the fermionic fields  (5.3.9):
$$[F_k,\phi_a^\lambda(z)]={1\over N}z^{Nk}\left(z{d\over
           dz}+{1\over 2}Nk\right)\phi_a^\lambda(z)\forno$$
and Theorem 5.5.1;

\proclaim Theorem  \prono .
Let $1\le a\le r<b\le s$ and $1\le d,e\le s$, then
$$\eqalign{\phi_a^\lambda(z)=&Q_a^\lambda z^{{N\over
n_a}\left(\lambda\hat\alpha_a(0)
           +{1\over 2}\right)}
           {\rm exp}\left(-\lambda\sum_{k<0}{1\over k}
           z^{-{N\over n_a}k}\a ak\right)
           {\rm exp}\,\left(-\lambda\sum_{k>0}{1\over k}
           z^{-{N\over n_a}k}\a ak\right)
           \cr
           \phi_b^\lambda(z)=&Q_b
           {\rm exp}\left(-\lambda\sum_{k<0}{1\over k}
           z^{-{N\over n_b}k}\a bk\right)
           {\rm exp}\,\left(-\lambda\sum_{k>0}{1\over k}
           z^{-{N\over n_b}k}\a bk\right)
           \cr}
           \forno$$
where the operators $Q_d :V^\epsilon\rightarrow V^\epsilon$ are defined by:
$$\eqalign{Q_d\phi_e^\lambda(k)&=-\phi_e^\lambda(k)Q_d\quad\hbox{if } 
           e\ne d\cr
           Q_a\phi_a^\lambda(k)&=\phi_a^\lambda(k+\lambda)Q_a\cr
           Q_b\phi_b^\lambda(k)&=\phi_b^\lambda(k)Q_b\cr
						Q_d\sigma(k)&=-\sigma(k)Q_d\cr
           }
  \forno$$
and
$$Q_d\vac=
           \cases{\phi_d^+({1\over2})\vac
                  &if $1\le d\le r$\cr\cr
                  \phi_d^+(0)\vac
                  &if $r<d\le s$\cr}
  \forno$$
Moreover,
$$\eqalign{Q_a^\dagger &=Q_a^{-1}\cr
           Q_b^\dagger &=Q_b ={1\over 2}Q_b^{-1}\cr}
  \forno$$
and 
$$[\a dk,Q_e]=\delta_{de}\delta_{k0}Q_e\forno$$
The following anticommutation relations are satisfied:
$$\{Q_d,Q_e\}=0
  \quad\hbox{for } d\ne e\forno$$
\par

\proof{The proof of this theorem can be found in [KrL91a-b].} 
        

Notice that the field $\sigma(z)$ cannot be expressed in terms of bosonic
oscillators. The reason for this is clearly that the $\a ak$'s commute with the
$\sigma(j)$'s. If we want to bosonize the fields $\sigma(z)$, we have to extend
the oscillator algebra. For example, if ${1-(-)^{s-r}\over 4}=\epsilon$, we
introduce $\beta(z)= \sum_{k\in{1\over 2}+{\bf Z}}\beta_k
z^{-Nk}=:\sigma(z)\sigma (\omega^{1\over 2}z):$. The  $\beta_k$'s span an
oscillator algebra, commuting with $\hat{\underline s}_{[w]}$, and we can derive
an expression like (6.2.3) for the field $\sigma(z)$. The problem is of course
that the $\beta_k$'s belong to the infinite rank algebras $B_\infty$ ($\epsilon
=0)$ or $D_\infty$ ($\epsilon={1\over 2})$, but not to the subalgebra
$\hat{so}_{2n+1}$. (We do not know of a similar procedure for the case 
${1-(-)^{s-r}\over 4}\ne\epsilon$.)

From (6.2.6) and (6.2.7) it is clear that the operators $Q_a,1\le a\le s$ are
invertible operators, centralizing the HSA $\hat{\underline s}_{[w]}$. Let 
$\tilde{\underline s}_{[w]}$ be the image of $\hat{\underline s}_{[w]}$ under
the canonical projection of $\hat{so}_{2n+1}({\bf C})$ on the loop algebra
$\tilde{so}_{2n+1}({\bf C})$. The centralizer of $\tilde{\underline s}_{[w]}$
in $\tilde{SO}_{2n+1}({\bf C})$ is easily determined; it is generated by the
loops

$$\tilde Q_a:=\cases{{\rm exp}(-i\theta {\rm ad}\ H_{[w]}^\epsilon){\rm exp}(i{\theta\over
n_a}\sum_{k=1}^{n_a}\epsilon^a_k-2\pi i
\sum_{k=1}^{n_a}kA^{aa}_{kk}(+,-))&if $1\le a\le r$\cr\cr
{\rm exp}(-i\theta {\rm ad}\ H_{[w]}^\epsilon){\rm exp}(\pi i
\sum_{k=1}^{n_a}A^{aa}_{kk}(+,-))&if $\ r<a\le s$\cr}\forno$$

\noindent and the group $H_{\overline
0}:=\{{\rm exp}(h),h\in{\underline h}_{\overline 0}\}$.
Now denote by $G$ the connected and simply connected covering group of
$SO_{2n+1}({\bf C})$. The elements $\tilde Q_a$ cannot be lifted to closed loops
in $G$. Therefore, we take products $\tilde Q_a^\lambda\tilde Q_b^\mu$, which
can be lifted to elements of $\tilde G$. Together with a lift of $H_{\overline
0}$ and the center of $G$, they generate the centralizer of $\tilde{\underline
s}_{[w]}$ in $\tilde G$. Similarly, the operators $Q_a^\lambda Q_b^\mu$, the
group $\{{\rm exp}\pi^\epsilon((h)),h\in{\underline h}_{\overline 0}\}$ and
${\bf C}^\times$ generate the centralizer $\hat{C}_{[w]}$ of $\hat{\underline
s}_{[w]}$ in the Kac-Moody group associated to the representation $\pi^\epsilon$.
It is well known (see [KP85]), that the level one representations of a simply
laced or twisted affine Lie algebra remain irreducible when restricted to
the Heisenberg system $(\hat{\underline s}_{[w]},\hat{C}_{[w]})$. For the non
simply laced algebra $\hat{so}_{2n+1}({\bf C})$ this is not true, since there
are many other operators in ${\cal U}(\hat{\underline g})$  centralizing the HSA,
such as $Q_a\sigma(k)$, $\sigma(k)\sigma(l)$ etc. 


\beginsection \secno .  Vertex operators for $\soh$

\noindent Consider the normal ordered operator products $:\phi_a^\lambda(u)
\phi_b^\mu(v):$ and $:\phi_a^\lambda(u)\sigma(v):$ in two formal variables $u$
and $v$. With the normal ordering definition 4.5.4 these products can be
rewritten as

$$\eqalign{:\phi_a^\lambda(u)\phi_b^\mu(v):&=\phi_a^\lambda(u)\phi_b^\mu(v)
  -<\overline 0\mid\phi_a^\lambda (u)\phi_b^\mu (v)\vac\cr
:\phi_a^\lambda(u)\sigma(v):&=\phi_a^\lambda(u)\sigma(v)
  -<\overline 0\mid\phi_a^\lambda (u)\sigma(v)\vac\cr}\forno$$

\noindent The contractions $<\overline 0\mid\phi_a^\lambda (u)\phi_b^\mu
(v)\vac$ and $<\overline 0\mid\phi_a^\lambda (u)\sigma(v)\vac$ are formal
series in $u$ and $v$ (see lemma 4.5.3). The ordinary products
$\phi_a^\lambda(u)\phi_b^\mu(v)$ and $\phi_a^\lambda(u)\sigma(v)$ can be
rewritten by means of theorem 6.2.1.

\proclaim Proposition  \prono .
Define
$$\eqalign{E_a^{(\pm)}(z)&:={\rm exp}\,\left(-\sum_{\pm k>0}{1\over k}
  z^{-{N\over n_a}k}\a ak\right)\cr\cr
F_{a,\lambda}(z)&:=\cases{Q_a^\lambda 
  z^{{N\over n_a}\left(\lambda\hat\alpha_a(0)+{1\over 2}\right)}&if $1\le a\le
r$\cr\cr Q_a&if $r< a\le s$\cr}\cr\cr
C_{ab}(z)&:=\cases{1-\delta_{ab}z^{{N\over n_a}}&if $1\le a\le r$\cr\cr
{1-\delta_{ab}z^{N\over 2n_a}\over 1+\delta_{ab}z^
{N\over 2n_a}}&if $r< a\le s$\cr}\cr
 }\forno$$
Then
$$\phi_a^\lambda(u)\phi_b^\mu(v)=C_{ab}(v/u)^{\lambda\mu}F_{a,\lambda}(u)
F_{b,\mu}(v)E_a^{(-)}(u)^\lambda E_b^{(-)}(v)^\mu E_a^{(+)}(u)^\lambda
E_b^{(+)}(v)^\mu
  \forno$$
and 
$$\phi_a^\lambda (u)\sigma(v)= 
  F_{a,\lambda}(u)E_a^{(-)}(u)^\lambda E_a^{(+)}(u)^\lambda \sigma (v)\forno$$
\proof{An easy formal calculation shows that
$E_a^{(+)}(u)E_b^{(-)}(v)=C_{ab}(v/u)E_b^{(-)}(v)E_a^{(+)}(u)$. 
The theorem follows immediately from this result and theorem 6.2.1.}

In order to find a bosonic expression for the vertex operator $\hat
A_{jk}^{ab}(\lambda,\mu)(z)$, we would like to replace the formal variables $u$
and $v$ in (6.3.1) by $\omega^{-j}z$ and $\omega^{-k}z$, respectively. This has
to be done with some care, because  the formal
series $(1+\lambda \mu\delta_{ab}(v/u)^{{N\over m_a}})^{-1}
=1+\delta_{ab}\sum_{k>0}(-\lambda\mu (v/u)^{{N\over m_a}})^k$ can occur in
(4.5.6) and (6.3.3). Taking a limit $u\rightarrow\omega^{-j}z$ and 
$v\rightarrow\omega^{-k}z$ clearly is an ill-defined procedure for such formal
series. To avoid these problems, we write for all possible choices of
$a,b,\lambda,\mu,j,k$ except the cases $a=b\wedge\lambda=-\mu\wedge j=k$:


$$\eqalign{\hat
A_{jk}^{ab}(\lambda,\mu)(z)=&{1\over1+\delta_{ab}\lambda\mu\omega_a^{j-k}}
\lim_{u\to \omega^{-j}z \atop v \to \omega^{-k}z}(1+\lambda\mu\delta_{ab}(v/u)^
{N\over m_a})C_j^a(\lambda)C_k^b (\mu):\phi_a^\lambda (u)\phi_b^\mu(v):\cr
&+(H^\epsilon_{[w]}-(1-2\epsilon)\lambda_n|A_{jk}^{ab}(\lambda,
\mu))\cr}\forno$$ 

\noindent Notice, that there is no problem to take the limit because of the
normal ordering sign. Using (6.3.1) and the fact that
$$\lim_{u\to \omega^{-j}z \atop v \to
\omega^{-k}z}(1+\lambda\mu\delta_{ab}(v/u)^ {N\over m_a})C_j^a(\lambda)C_k^b
(\mu)<\overline 0|\phi_a^\lambda (u)\phi_b^\mu(v)\vac 
=-(1+\lambda\mu\delta_{ab}\omega_a^{j-k})(H^\epsilon_{[w]}-(1-2\epsilon)\lambda_n
|A_{jk}^{ab}(\lambda, \mu))\forno$$
we find
$$\eqalign{\hat
A&_{jk}^{ab}(\lambda,\mu)(z)={1\over1+\delta_{ab}\lambda\mu\omega_a^{j-k}}
\lim_{u\to \omega^{-j}z \atop v \to \omega^{-k}z}(1+\lambda\mu\delta_{ab}(v/u)^
{N\over m_a})C_j^a(\lambda)C_k^b (\mu)\phi_a^\lambda (u)\phi_b^\mu(v)\cr
&=C_{ab}(\omega^{j-k})^{\lambda\mu}C_j^a(\lambda)C_k^b (\mu)F_{a,\lambda}
(\omega^{-j}z) F_{b,\mu}(\omega^{-k}z)E_a^{(-)}(\omega^{-j}z)^\lambda
E_b^{(-)}(\omega^{-k}z) ^\mu E_a^{(+)} (\omega^{-j}z)^\lambda
E_b^{(+)}(\omega^{-k}z)^\mu\cr}\forno$$ 
In this formula, $C_{ab}(z)$ is of course to be interpreted as a
genuine meromorphic function in $z$. The vertex operator $\hat
A_{j}^{a}(\lambda)(z)$ is more straightforward; we find:
$$\hat
A_{j}^{a}(\lambda)(z)=C_j^a(\lambda)F_{a,\lambda}(\omega^{-j}z)E_a^{(-)}
(\omega^{-j}z)^\lambda E_a^{(+)}(\omega^{-j}z)^\lambda \sigma
(\omega^{-k}z)\forno$$



We obtain the following important consequence of these results.
\proclaim Theorem  \prono .
The spin module $V_\alpha^\epsilon$ is an irreducible $\soh$-module, with highest
weights $\Lambda_0$, $\Lambda_1$, $\Lambda_n$, $\Lambda_n$ for $(\epsilon ,\alpha)=({1\over 2},\overline 0)$, $({1\over 2} ,\overline 1)$, $(0,\overline 0)$, $(0,\overline 1)$, respectively.
\par
\proof{We only have to show that $V_\alpha^\epsilon$ remains irreducible for one
realization; we choose the principal realization, i.e.,
$[w]=[\overline n]$. In that case $r=0$, $s=1$ and $n_s=n$. 
 The action of $B_\infty$ or $D_\infty$ on the spin module $V^\epsilon$ is
generated by the homogeneous components of the normal ordered products
$:\phi^+_1(u)\phi^+_1(v):$, $:\phi^+_1(u)\sigma(v):$ and
$:\sigma(u)\sigma(v):$. These products can clearly be expressed in elements
of the Heisenberg algebra $\sw$ and $ \tau(k):=i\sqrt 2
Q_1\sigma (k)$, $k\in {1\over 2}+\epsilon+{\bf Z}$. Notice that $\{\tau (k),\tau
(l)\}=\delta_{k,-l}$ and $\{\tau (k),\phi^+_1(l)\}=0$ whence
$[\tau(k),\hat\alpha_1(k)]=0$. Because $\tau (k)|\alpha >=0$ for $k<0$ and $\tau
(0)|\alpha >={1\over \sqrt{2}}(-1)^{\alpha+1} |\alpha >$, we only have to show
that the vectors $$\tau (k_p)\tau (k_{p-1})\cdots \tau (k_1)|\alpha >\forno$$
with $k_p>k_{p-1}>\cdots >k_1 > 0$ can be obtained from the highest weight vector
$|\alpha >$ by the action of the affine Lie algebra $\soh$. For this purpose we
define $$\eqalign{ \tau(z):&=i\sqrt{2}Q_1\sigma(z)\cr X(z)&:=E_1^{(-)}(z)\tau
(z)E_1^{(+)}(z)\cr
\phantom{:}&=i\sqrt{2}C_1^n(-)^{-1}\hat A_n^1(z)\cr
\phantom{:}&=:\sum_{k\in {\bf Z}} X(k)z^{-k}\cr
\sum_{k\ge 0}S_k z^k:&=E_1^{(-)}(z)^{-1}\cr}\forno$$ 
From (6.3.10) we derive 
$$\tau(z)=E_1^{(-)}(z)^{-1}X(z)E_1^{(+)}(z)^{-1}\forno$$ Now suppose that we
have already obtained an element of the form (6.3.9) by the action of $\soh$.
Then it is straightforward to check that for $l>k_p$:  $$\tau (l)\tau (k_p)\tau
(k_{p-1})\cdots \tau (k_1)|\alpha >=\Bigl(\sum_{k=0}^{2nl} S_kX(k-2nl)\Bigr)\tau
(k_p)\tau (k_{p-1})\cdots \tau (k_1)|\alpha >\forno$$
 Thus
the irreducible $\Bi$- or $D_\infty$-module $V_\alpha^\epsilon$ 
remains irreducible when restricted to $\soh$.


The calculation of the corresponding highest weight is straightforward,
and will be left to the reader (see also section 4).}

The proof of the theorem yields the principal realization for
the irreducible $\hat{so}_{2n+1}$-modules $V_\alpha^\epsilon$;

\proclaim Corollary  \prono .
The assignments
$$\eqalign{\hat \alpha_1(l)&\mapsto{\partial\over\partial x_l}\qquad\qquad
l\in 2{\bf N}-1\cr \hat \alpha_1(-l)&\mapsto{lx_l}\qquad\qquad
l\in 2{\bf N}-1\cr
i\sqrt{2}Q_1\sigma(-k)&\mapsto {\partial\over\partial \xi_k}\qquad\qquad
0<k\in{1\over 2}+\epsilon+{\bf Z}\cr 
i\sqrt{2}Q_1\sigma(k)&\mapsto \xi_k\qquad\qquad
0<k\in{1\over 2}+\epsilon+{\bf Z}\cr
i\sqrt{2}Q_1\sigma(0)&\mapsto
{1\over\sqrt{2}}(-)^{\alpha+1}\qquad\qquad {\rm if}\
\epsilon={1\over 2}\cr}\forno$$
realize the $\hat{so}_{2n+1}$-modules $V_\alpha^\epsilon$ on the tensor product
of the polynomial ring ${\bf C}[x_{2k-1};k\in{\bf N}]$ and the exterior
algebra ${\bf C}[\xi_k;0<k\in{1\over 2}+\epsilon+{\bf Z}]$

\noindent We will discuss more general realizations in the next section.

If $n=1$, then $\soh$ becomes $\hat{sl}_2({\bf C})$ and its level 1 modules become level 2 $\hat{sl}_2({\bf C})$-modules. To be more precise (see e.g. [LW84], [KW89]):
\proclaim Corollary  \prono .
The spin module  $V_\alpha^\epsilon$ is an irreducible $\hat{sl}_2({\bf C})$-module, with highest
weights $2\Lambda_0$, $2\Lambda_1$, $\Lambda_0+\Lambda_1$, $\Lambda_0+\Lambda_1$ for $(\epsilon ,\alpha)=({1\over 2},\overline 0)$, $({1\over 2} ,\overline 1)$, $(0,\overline 0)$, $(0,\overline 1)$, respectively.
\par







\baselineskip=22pt plus.1pt

\count30=6                      %chapterno = 6
\count31=0                      %sectionno = 0
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0

\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count31 .\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count31 .\number\count33}


\long\def\proof#1{\noindent{\bf{Proof.}}\quad#1\qedd}
\def\qedd{\hfill\hbox{\vrule\vbox{\hrule\kern2truept\hbox{\kern2truept
\vbox to2pt{\hsize=2pt}\kern2truept}\kern2truept\hrule}\vrule}\vskip .5cm}



\def\so{so_{2n+1}({\bf C})}
\def\SO{SO_{2n+1}({\bf C})}

\def\ad{{\rm ad}}
\def\Ad{{\rm Ad}}
\def\trace{{\rm trace}}

\def\E{{\cal E}}
\def\F{{\cal F}}
\def\tn{\hat \tau_{\underline n}}
\def\tw{\hat \tau_{[w]}}
\def\Hn{H_{\underline n}}
\def\Hw{H_{[w]}}
\def\Sn{S_{\underline n}}
\def\Sw{S_{[w]}}
\def\Xw{X_{[w]}}
\def\sw{\hat {\underline s}_{[w]}}
\def\hw{{\underline h}_{[w]}}

\def\gl{gl_{2n}({\bf C})}
\def\oi{o_\epsilon(\infty )}
\def\gli{gl(\infty )}
\def\glib{\overline {gl(\infty )}}
\def\oib{\overline {o_\epsilon(\infty )}}
\def\Bi{B_{\infty}}
\def\Di{D_{\infty}}
\def\glt{\tilde{gl}_{2n}({\bf C})}
\def\glh{\hat{gl}_{2n}({\bf C})}
\def\sot{\tilde{so}_{2n+1}({\bf C})}
\def\soh{\hat{so}_{2n+1}({\bf C})}
\def\vac{\mid \overline 0 >}
\def\sh{\hat {\underline s}}
\def\shw{\hat {\underline s}{[w]}}
\def\a#1#2{\hat\alpha_{#1}({N\over n_{#1}}#2)}
\def\L{{\cal L}}
\def\D{{\cal D}}
\def\Cxrs{{\bf C}[x_k^{(a)};\ 1\le a\le s,\ k\in {\bf N} \hbox{ if }
          1\le a\le r,\ k\in {\bf N}-{1\over 2} \hbox{ if } r<a\le s]}
\def\Cx{{\bf C}[x]}
\def\CL{{\bf C}(L^*)}
\def\Ct{{\bf C}_\epsilon [\theta ,\xi ]}
\def\ht{{\rm ht}}

\beginsection \chapno .  Bosonic realization of the spin module,
and $q$-dimension formulas
\par
\beginsection \secno .  Introduction
\par
In this section we realize the spin module $V^\epsilon$ as the tensor
product of a polynomial algebra in infinitely many variables,
an infinite dimensional Grassmann algebra and the group algebra of a lattice.
In section  7.3 we will give the $q$-dimension formulas of the 
spin module
that correspond to different realizations.


\beginsection \secno .
  The spin module  as Fock space
\par
Recall from section 3.4  that the elements
$\hat\alpha_a({N\over n_a}k)$ with $k\in {\bf Z}$, $k\ne 0$
if $1\le a\le r$
and $k\in {1\over 2}+{\bf Z}$ if $r<a\le s$ together with $c$  form 
a basis of the Heisenberg subalgebra $\sw$. They satisfy the commutation relations
$$[\hat\alpha_a({N\over n_a}k),\hat\alpha_b({N\over n_b}l)]=k\delta_{ab}\delta_{k,-l}c$$
This Heisenberg subalgebra has a canonical representation $\sigma$ in the space of
polynomials in infinitely many variables $\Cx:=\Cxrs$. It is given by:
$$\rho \left(\hat\alpha_a({N\over n_a}k)\right)=
  \cases{{\partial\over\partial x_k^{(a)}}&if $k>0$\cr
         -kx_k^{(a)}&if $k<0$\cr}
  \forno$$

If we consider $V^\epsilon$ as an $\sw$--module, this module contains a collection of 
vectors which are annihilated by all $\hat\alpha_a({N\over n_a}k)$
($k>0$).
This {\it vacuum space} $V^{\epsilon,+}$ for $\sw$ is the linear span of the vectors
$$Q_1^{k_1}Q_2^{k_2}\cdots Q_r^{k_r}Q_{r+1}^{l_{r+1}}\cdots Q_s^{l_s}\sigma (m_1)\sigma (m_2)\cdots\sigma (m_t)\vac
  \forno$$
where $k_a\in {\bf Z}$, $l_a=0$ or $1$ and $m_j\in {1-(-)^{s-r}\over 4}+\epsilon+{\bf Z}$, $m_1>m_2>\cdots >m_t\ge 0$.
Unfortunately, these vectors are not linearly independent: we have  (see
(4.4.20) and (6.2.5)):
$$\eqalign{\{Q_{r+2a-1}+(4\epsilon -1)iQ_{r+2a}\}\vac&=0\cr
 \{Q_s+i\sigma (0)\}\vac&=0\qquad\hbox{if }\epsilon={1\over 2}\hbox{ and }s-r\hbox{ odd}\cr}\forno$$
Motivated by these relations we introduce operators $q_a$; $a\in {\bf Z}$, $\epsilon\le a\le [{s-r\over 2}+\epsilon]$ as follows:
$$q_a:=\cases{{1\over\sqrt 2}\{Q_{r+2a-1}-(4\epsilon -1)iQ_{r+2a}\}&if $1\le a\le [{s-r\over 2}]$\cr\cr
Q_s&if $a=\epsilon=0$ and $s-r$  odd\cr\cr \sigma (0)&if $a=\epsilon=0$ and  $s-r$  even\cr\cr
{1\over\sqrt 2}\{Q_s-i\sigma (0)\}&if $a=[{s-r\over 2}]+1$, $\epsilon ={1\over 2}$ and $s-r$  odd\cr}\forno$$
Similarly, we set $q_{-a}:=q_a^\dagger$. The following anticommutation relations are immediate from the definition above:
$$\eqalign{\{q_a,q_b\}=&\delta_{a,-b}\cr
\{q_a,\sigma(k)\}=&0\qquad\qquad\hbox{for }k\ne 0\cr}\forno$$
 We are now able to give a basis for the vacuum space $V^{\epsilon,+}$:
\proclaim Theorem \prono .
The vectors
$$v_{k_1,\cdots,k_r,l_1,\cdots,l_p,m_1,\cdots,m_t}:=
Q_1^{k_1}\cdots Q_r^{k_r}q_{l_1}\cdots q_{l_p}
 \sigma (m_1)\cdots\sigma (m_t)\vac \forno$$
with $k_i\in {\bf Z}$, $\epsilon\le l_1<l_2<\cdots\le [{s-r\over 2}+\epsilon]$
and $m_1>m_2>\cdots>m_t>0$ form a basis of the vacuum space $V^{\epsilon,+}$.

\proof{From  the definition of the operators $q_a$ and (4.4.9-12) one derives ($\epsilon\le a\le [{s-r\over 2}+\epsilon]$):
$$q_a\vac=\cases{0&if $a<0$\cr\cr
\phi(0)\vac=|\overline 1>&if $a=\epsilon=0$\cr\cr
\phi(n+1-(n_1+\cdots+n_{r+2a-1}))\vac&if $\epsilon=0$ and $a>0$\cr\cr
\phi (n_1+\cdots +n_{r+2a-2}+{1\over 2})\vac&if $\epsilon={1\over 2}$ and $a>0$\cr}\forno$$
From this and (4.4.12) we see that the vectors
$q_{l_1}\cdots q_{l_p}
 \sigma (m_1)\cdots\sigma (m_t)\vac $
with 
 $\epsilon\le l_1<l_2<\cdots\le [{s-r\over 2}+\epsilon]$
and $m_1>m_2>\cdots>m_t>0$
are linearly independent. The relation
$$\hat \alpha_a(0)v_{k_1,\cdots,k_r,l_1,\cdots,l_p,m_1,\cdots,m_t}=k_av_{k_1,\cdots,k_r,l_1,\cdots,l_p,m_1,\cdots,m_t}\forno$$
concludes the proof.}

Now consider the tensor product ${\bf C}[x]\otimes V^{\epsilon,+}$. It can be
identified in a natural way with a subspace of the spin module $V^\epsilon$. 
From Theorem 6.2.1, it is clear that the components $\phi_a^+(k)$,
$\phi_a^-(k)$, $\sigma (k)$ of the fields $\phi_a^+(z)$, $\phi_a^-(z)$ and $\sigma (z)$ leave this
subspace invariant. Hence ${\bf C}[x]\otimes V^{\epsilon,+}$ must be isomorphic to $V^\epsilon$.



Next we will show that the vacuum space $V^{\epsilon,+}$ is isomorphic to the tensor
product of a group algebra of a lattice
and a Grassmann algebra. 
Consider the lattice $L$ in the vector space $\underline h$ 
generated by the $\hat\alpha_i:=\hat\alpha_i(0)$, i.e.,
$$L=\sum_{i=1}^r {\bf Z}\hat\alpha_i\forno$$
The normalized trace form on $\so$ induces an integral positive definite  bilinear form on $L$:
$$\eqalign{(\hat\alpha_i|\hat\alpha_j)&=
  (n_i\hat A^{ii}_{n_in_i}(0)|n_j\hat A_{n_jn_j}^{jj}(0))\cr
  &= (\sum_{p=1}^{n_i}G_{pp}^{ii}(+,-)|\sum_{q=1}^{n_j}G_{qq}^{jj}(+,-))\cr
  &=\sum_{p=1}^{n_i}\sum_{q=1}^{n_j}(\epsilon_p^i | \epsilon_q^j )\cr
  &=n_i\delta_{ij}\cr}\forno$$
Let $L^*$ be the dual lattice of $L$; if we identify 
$\underline h^*$ with $\underline h$, we obtain as basis for $L^*$
the elements $\alpha_i:=\hat\alpha_i^*={1\over n_i}\hat\alpha_i$.
So
$$L^*=\sum_{i=1}^r {\bf Z}\alpha_i\forno$$
and
$$(\alpha_i|\alpha_j)={\delta_{ij}\over n_i}\forno$$
showing that $L^*$ is again positive definite, but in general no longer integral.
To the lattice $L^*$ we associate its group algebra $\CL$,
i.e., a ${\bf C}$--algebra 
with basis $e^\gamma$ for $\gamma\in L^*$ and multiplication 
$e^\beta e^\gamma =e^{\beta +\gamma}$. We also define some `height functions':
$$\ht_i(k_1\alpha_1 +k_2\alpha_2 +\cdots +k_r\alpha_r ):=
  k_1+k_2+\cdots +k_{i-1}\forno$$
and 
$$\ht(k_1\alpha_1 +k_2\alpha_2 +\cdots +k_r\alpha_r ):=
  k_1+k_2+\cdots +k_r\forno$$   

Let $\Ct={\bf C}[\theta_i ,\xi_j ;i\in {\bf Z},\ \epsilon\le i\le [{s-r\over 2}+\epsilon ],\ 0<j\in  {1-(-)^{s-r}\over 4}+\epsilon+{\bf Z}]$ be the infinite Grassmann
algebra, i.e., the $\theta_i$, $\xi_i$ satisfy the
following anticommutation relations
$$\theta_i\theta_j +\theta_j\theta_i= \theta_i\xi_j +\xi_j\theta_i=\xi_i\xi_j +\xi_j\xi_i=0\forno$$
One can also introduce the derivations ${\partial\over\partial\theta_i}$ and ${\partial\over\partial\xi_i}$
on this algebra;
$$\eqalign{{\partial\over\partial\theta_i}(\theta_j)=\delta_{ij}-
  \theta_j{\partial\over\partial\theta_i},&\qquad {\partial\over\partial\theta_i}(\xi_j)=-
  \xi_j{\partial\over\partial\theta_i}\cr
{\partial\over\partial\xi_i}(\xi_j)=\delta_{ij}-
  \xi_j{\partial\over\partial\xi_i},&\qquad{\partial\over\partial\xi_i}(\theta_j)=-
  \theta_j{\partial\over\partial\xi_i}\cr}\forno$$
and hence
$${\partial\over\partial\theta_i}{\partial\over\partial\theta_j}
  +{\partial\over\partial\theta_j}{\partial\over\partial\theta_i}=
{\partial\over\partial\theta_i}{\partial\over\partial\xi_j}
  +{\partial\over\partial\xi_j}{\partial\over\partial\theta_i}=  
{\partial\over\partial\xi_i}{\partial\over\partial\xi_j}
  +{\partial\over\partial\xi_j}{\partial\over\partial\xi_i}=0\forno$$






We are now ready to define the isomorphism $\rho$ between the
spin module $V^\epsilon$ and the {\it Fock space} $F^\epsilon$:
$$\rho :V^\epsilon\to F^\epsilon :=\Cx\otimes\CL\otimes\Ct\forno$$
It is given by:
$$\eqalign{\rho :\hat\alpha_{i_1}(-{N\over n_{i_1}}j_1)&\cdots
          \hat\alpha_{i_q}(-{N\over n_{i_q}}j_q)
         v_{k_1,\cdots,k_r,l_1,\cdots,l_p,m_1,\cdots,m_t}
  \mapsto\cr & \Bigl({1\over \sqrt 2}\Bigr)^{\delta_{l_1,0}} j_1\cdots j_q x_{j_1}^{\left( i_1\right)}\cdots
          x_{j_q}^{\left( i_q\right)}
          \otimes
          e^{k_1\alpha_1 +\cdots +k_r\alpha_r}
          \otimes
          \theta_{l_1}\cdots \theta_{l_p}\xi_{m_1}\xi_{m_2}\cdots\xi_{m_t}\cr}
  \forno$$
and in particular
$$\rho :\vac\mapsto 1\otimes 1\otimes 1\forno$$
This is an isomorphism between vector spaces. Let $p\in\Cx$,
$\gamma\in L^*$ and $q\in\Ct$, then we have the following transported
action
$$\eqalign{\rho Q_i \rho^{-1}(p\otimes e^\gamma\otimes q)=&
           (-1)^{\ht_i(\gamma)}p\otimes e^{\gamma +\alpha_i}\otimes q
           \cr
           \rho Q_i^{-1} \rho^{-1}(p\otimes e^\gamma\otimes q)=&
           (-1)^{\ht_i(\gamma)}p\otimes e^{\gamma -\alpha_i}\otimes q
           \cr
           \rho q_0\rho^{-1}(p\otimes e^\gamma\otimes q)=&
           (-1)^{\ht(\gamma)}p\otimes e^\gamma \otimes {1\over \sqrt 2}(\theta_{0}+{\partial\over\partial\theta_0})(q)\cr
           \rho q_j\rho^{-1}(p\otimes e^\gamma\otimes q)=&
           (-)^{\ht(\gamma)}p\otimes e^\gamma \otimes \theta_jq
           \cr
           \rho q_{-j}\rho^{-1}(p\otimes e^\gamma\otimes q)=&
           (-1)^{\ht(\gamma)}p\otimes e^\gamma \otimes 
           {\partial q\over\partial\theta_j}
           \cr
           \rho \hat\alpha_k(-{N\over n_k}l) \rho^{-1}
           (p\otimes e^\gamma\otimes q)=&
           l x_l^{(k)}p\otimes e^\gamma \otimes q
           \cr
           \rho \hat\alpha_k({N\over n_k}l) \rho^{-1}
           (p\otimes e^\gamma\otimes q)=&
           {\partial p\over \partial x_l^{(k)}}\otimes e^\gamma \otimes q
           \cr
           \rho \hat\alpha_i(0) \rho^{-1}(p\otimes e^\gamma\otimes q)=&
           (\hat\alpha_i|\gamma )p\otimes e^\gamma \otimes q
           \cr
           \rho \sigma (-l) \rho^{-1}(p\otimes e^\gamma\otimes q)=&
           (-1)^{\ht(\gamma)}p\otimes e^\gamma \otimes \xi_lq
           \cr
           \rho \sigma (l) \rho^{-1}(p\otimes e^\gamma\otimes q)=&
           (-1)^{\ht(\gamma)}p\otimes e^\gamma \otimes 
           {\partial q\over\partial\xi_l}
           \cr}
  \forno$$
here $1\le i\le r$, $1\le j\le [{s-r\over 2}+\epsilon ]$, $1\le k\le s$ and $l>0$.
\beginsection \secno .  q-dimension formulas \par
Recall the operator $ S_0=F_0+F'_0=B_0+F_0'$ (see theorem 5.5.1). Using (5.2.9),  (5.3.11-12) and (7.2.8), one finds:
$$\eqalign{&S_0v_{k_1,\cdots k_r,l_1\cdots l_{p},m_1,\cdots ,m_t}=\cr&
  \left\{\sum_{a=1}^s \Bigl( {n_a^2-1\over 24n_a}+{\beta_a\over n_a}\Bigr)+{\delta_{{1-(-)^{s-r}\over 4},\epsilon}\over 16}+{1\over 2}\left(
  {k_1^2\over n_1}+\cdots {k_r^2\over n_r}\right)+m_1+\cdots +m_t\right\}
 v_{k_1,\cdots k_r,l_1\cdots l_{p},m_1,\cdots ,m_t}\cr}
  \forno$$
Combining this with (5.2.10), we find the following
form of the `q-dimension' formula for $V^\epsilon$:

$$\eqalign{&{\rm trace}\lower 8pt\hbox{$V^\epsilon$}q^{S_0}=\cr
  &2^{[{s-r\over 2}+\epsilon ]+1-2\epsilon}
  {\displaystyle q^{ \left\{\sum_{a=1}^s \Bigl( {n_a^2-1\over 24n_a}+{\beta_a\over n_a}\Bigr)+{\delta_{{1-(-)^{s-r}\over 4},\epsilon}\over 16}\right\}}
\prod_{0\ne j\in {1-(-)^{s-r}\over 4}+\epsilon +{\bf Z}_+} (1+q^j)
  \sum_{k_1,k_2,\cdots ,k_r\in {\bf Z}}q^{{1\over 2}\left(
  {k_1^2\over n_1}+\cdots {k_r^2\over n_r}\right)}\over \displaystyle
  \prod_{i=1}^r\prod_{j\in {\bf N}} (1-q^{{j\over n_i}})
  \prod_{i=r+1}^s\prod_{j\in {1\over 2}+{\bf Z_+}} (1-q^{{j\over n_i}})
  }\cr}\forno$$
We can rewrite this formula in terms of some classical functions;
let
$$\eta (q)=q^{1\over 24}\prod_{j\in {\bf N}}(1-q^j)\forno$$
be the well--known {\it Dedekind $\eta$--function}, $$\theta_{L^*}(q)
  =\sum_{\gamma\in L^*}q^{{1\over 2}(\gamma |\gamma )}\forno$$ the {\it $\theta$--function of the lattice} $L^*$
 and
$$\theta (q)=\sum_{k\in {\bf Z}} q^{k^2\over 2}\forno$$
the classical  {\it $\theta$--function}.
Then
$$\theta_{L^*}(q)
 =\prod_{i=1}^r \theta (q^{1\over n_i})
  \forno$$
Now using 
the  identities
$$\eqalign{q^{1\over 16}\prod_{j\in {\bf N}} (1+q^j)&=q^{1\over 48}{\eta(q^2)\over \eta (q)}\cr
\prod_{j\in {\bf N}} (1+q^{j-{1\over 2}})&=q^{1\over 48}{(\eta(q))^2\over \eta (q^{1\over 2})\eta (q^2)}\cr}\forno$$        
we can rewrite (7.3.2) and obtain
$${\rm trace}\lower 8pt\hbox{$V^\epsilon$}q^{S_0}=\cases{{\displaystyle
  2^{[{s-r\over 2}+\epsilon ]+1-2\epsilon}
q^{n+{1\over 2}\over 24}{\eta(q^2)\over \eta (q)} 
  \prod_{i=1}^r {\theta (q^{1\over n_i})\over  \eta (q^{1\over n_i})}
  \prod_{i=r+1}^s 
  {\eta (q^{1\over n_i})\over \eta (q^{1\over 2n_i})}}&if ${1-(-)^{s-r}\over 4}=\epsilon$
  \cr
&\cr &\cr
{\displaystyle
  2^{[{s-r\over 2}+\epsilon ]+1-2\epsilon}
q^{n+{1\over 2}\over 24}{(\eta(q))^2 \over  \eta (q^{1\over 2})\eta (q^2)}
 \prod_{i=1}^r {\theta (q^{1\over n_i})\over  \eta (q^{1\over n_i})}
  \prod_{i=r+1}^s  {\eta (q^{1\over n_i})\over  \eta (q^{1\over 2n_i})}
}&if ${1-(-)^{s-r}\over 4}\not =\epsilon$
  \cr
}
  \forno$$
Notice that $n+{1\over 2}$ is the value of the central charge $c_{vir}$. 

Using (7.3.1) and the fact that the chirality operator $\chi$ commutes with
all oscillators, one easily recovers the
$q$-dimension formulas of $V^\epsilon_\alpha$.
\proclaim Proposition  \prono .
If $s-r =0$ and $\epsilon={1\over 2}$, then
$${\rm trace}\lower 8pt\hbox{$V^{1\over 2}_\alpha$}q^{S_0}=
{q^{n+{1\over 2}\over 24}}\sum_{j=0}^1{{\displaystyle \sum_{k_1+\cdots +k_r\in j+2{\bf Z}}q^{{1\over 2}\left(
  {k_1^2\over n_1}+\cdots {k_r^2\over n_r}\right)}{1\over 2}\left\{ {(\eta(q))^2 \over  \eta (q^{1\over 2})\eta (q^2)}+(-)^{j+\alpha} {\eta(q^{1\over 2})\over \eta (q)}\right\}}\over {\displaystyle \prod_{i=1}^r  \eta (q^{1\over n_i})}}\forno$$
And
$${\rm trace}\lower 8pt\hbox{$V^\epsilon_\alpha$}q^{S_0}=\cases{{\displaystyle
  2^{[{s-r\over 2}+\epsilon ]-2\epsilon}
q^{n+{1\over 2}\over 24}{\eta(q^2)\over \eta (q)} 
  \prod_{i=1}^r {\theta (q^{1\over n_i})\over  \eta (q^{1\over n_i})}
  \prod_{i=r+1}^s 
  {\eta (q^{1\over n_i})\over \eta (q^{1\over 2n_i})}}&if ${1-(-)^{s-r}\over 4}=\epsilon$
  \cr
&\cr &\cr
{\displaystyle
  2^{[{s-r\over 2}+\epsilon ]-2\epsilon}
q^{n+{1\over 2}\over 24}{(\eta(q))^2 \over  \eta (q^{1\over 2})\eta (q^2)}
 \prod_{i=1}^r {\theta (q^{1\over n_i})\over  \eta (q^{1\over n_i})}
  \prod_{i=r+1}^s  {\eta (q^{1\over n_i})\over  \eta (q^{1\over 2n_i})}
}&if ${1-(-)^{s-r}\over 4}\not =\epsilon$
  \cr
}
  \forno$$
in all other cases.


\vskip 1cm
\noindent{\bf Acknowledgements}: Johan van de Leur would like to thank Prof. Victor Kac for valuable conversations (in  1986-1988), which were crucial for the start of this work.






\vskip 1cm

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\bye





































