text:
\begin{document}

\newcommand{\wta}{{\rm {wt} }  a }
\newcommand{\R}{\frak R}
\newcommand{\Lat}{ R}
\newcommand{\qtrp}{\overline{V_L}}

\newcommand{\trp}{\mathcal{W}(p)}

\newcommand{\wtb}{{\rm {wt} }  b }
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\be}{\begin {equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\g}{\frak g}
\newcommand{\hg}{\hat {\frak g} }
\newcommand{\hn}{\hat {\frak n} }
\newcommand{\h}{\frak h}
\newcommand{\V}{\Cal V}
\newcommand{\hh}{\hat {\frak h} }
\newcommand{\n}{\frak n}
\newcommand{\Z}{\Bbb Z}
\newcommand{\N}{{\Bbb Z} _{> 0} }
\newcommand{\Zp} {\Z _ {\ge 0} }

\newcommand{\C}{\Bbb C}
\newcommand{\Q}{\Bbb Q}
\newcommand{\1}{\bf 1}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}

\newcommand{\wt}{{\rm {wt} }   }

\newcommand{\cA}{\Cal A}
\newcommand{\non}{\nonumber}
\def \l {\lambda}
\baselineskip=14pt
\newenvironment{demo}[1]
{\vskip-\lastskip\medskip
  \noindent
  {\em #1.}\enspace
  }
{\qed\par\medskip
  }

\title[]{The vertex algebra $M(1) ^+$ and certain affine vertex algebras of level $-1$}

  \subjclass[2000]{
Primary 17B69, Secondary 17B67, 17B68, 81R10}
\author{Dra\v{z}en  Adamovi\' c and Ozren Per\v{s}e }
\address{Department of Mathematics, University of Zagreb, Croatia}
\email{adamovic@math.hr; perse@math.hr}
\maketitle

\begin{abstract}
We give  a coset realization of the vertex operator algebra $M(1)
^+$ with central charge $\ell$. We realize $M(1) ^+$ as a commutant
of certain affine vertex algebras of level $-1$ in the vertex
algebra $L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes
L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)$. We  show that the
simple vertex algebra $L_{C_{\ell} ^{(1)}}(-\Lambda_0)$ can be
(conformally) embedded into $L_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$
and find the corresponding decomposition. We also study certain
coset subalgebras inside $L_{C_{\ell} ^{(1)}}(-\Lambda_0)$.
\end{abstract}

\section{Introduction}

In the last few years various types of $\mathcal{W}$--algebras have been studied in the framework of vertex operator algebras (see \cite{Ar}, \cite{AdM-aim}, \cite{AdM-cmp}, \cite{DLY}, \cite{KWak}).
In this paper we will be focused on ${\mathcal W}$--algebras which admit coset realization.
To any vertex algebra $V$ and its subalgebra $U$, one can associate
a new vertex algebra

\bea {\rm Com}(U,V)= \{ v \in V \ | \ u_n v =0 \mbox{ for all } u
\in U, n \geq 0 \} \nonumber \eea

called the {\it commutant} (or {\it coset}) of $U$ in $V$. This is a
very important construction in the theory of vertex operator
algebras, because it gives a realization of a large family of ${\mathcal W}$--algebras.  Another important construction is the {\it
orbifold} construction, where a new vertex operator algebra is
obtained as invariants in a given vertex operator algebra with
respect to the finite automorphism group. As we shall see in our paper, some vertex algebras admit both realizations, coset and orbifold.

In this paper we consider certain coset vertex algebras for vertex
algebras associated to affine Lie algebras. Let ${\frak g}$ be a
simple Lie algebra of type $X_{n}$, $\hat{\frak g}$ the associated
affine Lie algebra of type $X_{n}^{(1)}$, and $L_{X_{n}^{(1)}}(k
\Lambda _0)$ the simple vertex operator algebra associated to
$\hat{\frak g}$ of level $k \in \C$, $k \neq - h^{\vee}$. For $k,m
\in \N$, $L_{X_{n}^{(1)}}((k+m) \Lambda _0)$ is a subalgebra of
$L_{X_{n}^{(1)}}(k \Lambda _0) \otimes L_{X_{n}^{(1)}}(m \Lambda
_0)$, and one has the associated coset vertex operator algebra \bea
\label{coset-1} && {\rm Com}(L_{X_{n}^{(1)}}((k+m) \Lambda _0) ,
L_{X_{n}^{(1)}}(k \Lambda _0) \otimes L_{X_{n}^{(1)}}(m \Lambda
_0)). \eea Although there are no precise general results (to the
best of our knowledge) about the structure of these cosets, it is
believed that these vertex operator algebras are finitely generated
and rational.

  In \cite{AP-2010},  we present a vertex-algebraic proof of the fact that  in  the case $k=1$ and affine Lie algebras of types $D_n ^{(1)}$ and $B_{n} ^{(1)}$,
  the coset (\ref{coset-1})  is isomorphic to the rational vertex operator algebra $V_L ^{+}$.

The situation is even more complicated for general $k,m \in \C$,
such that $k,m,k+m \neq - h^{\vee}$. Then one has the coset vertex
operator algebra
\bea
\label{coset-2} {\rm Com}({\widetilde L}_{X_{n}^{(1)}}((k+m) \Lambda _0) ,
L_{X_{n}^{(1)}}(k \Lambda _0) \otimes L_{X_{n}^{(1)}}(m \Lambda
_0)),
\eea
where ${\widetilde L}_{X_{n}^{(1)}}((k+m) \Lambda _0)$ is a certain
affine vertex operator algebra associated to $X_{n}^{(1)}$ of level
$k+m$ (not necessarily simple). In this paper we identify some
special cases of such cosets, and it turns out that they are not
rational.

   The construction in \cite{AP-2010} is based on fermionic construction of vertex  algebras and certain conformal embeddings.
   In the present paper we use bosonic construction of vertex  algebras  and construct new conformal embeddings
   of affine vertex algebras at level $-1$. By applying the bosonic realization  of
the affine vertex  algebras $L_{A_1 ^{(1)}}(-\tfrac{1}{2}\Lambda_0)$
and $L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)$ (cf. \cite{FF}) we consider coset
vertex   algebras

\bea && \mbox{Com}( L_{A_1 ^{(1)}}(- \Lambda_0),L_{A_1
^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes L_{A_1
^{(1)}}(-\tfrac{1}{2}\Lambda_0)) \quad \mbox{and} \label{coset-uvod} \\
&& {\rm Com} ( {\widetilde L}_{C_{\ell} ^{(1)}}(-\Lambda_0),
L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes L_{C_{\ell}
^{(1)}}(-\tfrac{1}{2}\Lambda_0) ). \nonumber \eea

It is interesting that these cosets   have central charge $1$. We
show that these cosets are isomorphic to  $M(1) ^+$, where $M(1)$ is
the Heisenberg vertex operator algebra of rank $1$, and $M(1)^+$ is
the $\Z _2$--orbifold vertex algebra studied in \cite{DN1}. The
structure theory of $M(1) ^+$ shows that these cosets are irrational
vertex operator algebras and isomorphic to $W(2,4)$--algebra with
central charge $c=1$.

By combining results from \cite{A1} and the present paper, we classify irreducible ordinary  $\widetilde{L}_{C_{\ell} ^{(1)}}(-\Lambda_0)$--modules. We believe that the (tensor) category of  $\widetilde{L}_{C_{\ell} ^{(1)}}(-\Lambda_0)$--modules is related  to the (tensor) category of $M(1) ^+$--modules. We plan to address this correspondence in our forthcoming publications.

Generalizing (\ref{coset-uvod}), we use a natural realization of the
vertex operator algebra $L_{A_1 ^{(1)}}(- \Lambda_0) ^{\otimes
\ell}$ as a subalgebra of $L_{C_{\ell}
^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes L_{C_{\ell}
^{(1)}}(-\tfrac{1}{2}\Lambda_0)$, and prove that

$$ \mbox{Com} (L_{A_1 ^{(1)}}(- \Lambda_0) ^{\otimes
\ell}, L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes
L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)) $$

is isomorphic to $M(1) ^+$, where $M(1)$ is the Heisenberg vertex
operator algebra of rank $\ell$.

Our construction is based on a new, interesting conformal  embedding of affine vertex operator algebras at level $-1$ which can be of independent interest. We show that $L_{C_{\ell} ^{(1)}}(-\Lambda_0)$ is
conformally embedded into $L_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$
and that
$$ L_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0) \cong  L _{C_{\ell} ^{(1)}}
(-\Lambda_0) \oplus  L _{C_{\ell} ^{(1)}} (-2\Lambda_0 +
\Lambda_2),$$

which implies that $L _{C_{\ell} ^{(1)}} (-\Lambda_0)$ is $\Z
_2$--orbifold of $L_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$.

By using conformal embeddings we study certain categories of
$A_{2\ell-1} ^{(1)}$--modules of level $-1$  from \cite{AP} as a
$C_{\ell} ^{(1)}$--modules. It turns out that  irreducible highest
weight $A_{2\ell-1} ^{(1)}$--modules $L_{ A_{2\ell -1} ^{(1)} } (-
(n+1) \Lambda_0 + n \Lambda_{1} )$ and $L_{ A_{2\ell -1} ^{(1)} } (-
(n+1) \Lambda_0 + n \Lambda_{2 \ell -1} )$ $(n \in {\N} )$ are also
irreducible as $C_{\ell} ^{(1)}$--modules. This result is an affine
analogue of the isomorphism of finite-dimensional $
C_{\ell}$--modules:

$$ V_{A_{2\ell -1} } (n \omega_{2 \ell -1})
\cong V_{A_{2\ell -1} } (n \omega_1) \cong V_{C_{ \ell  } } (n
\omega_1).
$$

Using these conformal embeddings  we also  show that the coset
$${\rm Com} (   L _{A_1 ^{(1)}} (-\Lambda_0) ^{\otimes \ell}  ,L_{C_{\ell}  ^{(1)}} (-\Lambda_0) ) $$
is isomorphic to $M(1) ^+$, where $M(1)$ is the Heisenberg vertex
operator algebra of rank $\ell -1$.

\section{Preliminaries}

Let $V$ be a vertex algebra (\cite{B}, \cite{FHL}, \cite{FLM},
\cite{LL}). For a subalgebra $U$ of $V$ denote by \bea {\rm
Com}(U,V)= \{ v \in V \ | \ u_n v =0 \mbox{ for all } u \in U, n
\geq 0 \} \eea
 the commutant of $U$ in $V$ (cf.
\cite{FZ}, \cite{GKO}, \cite{LL}). Then, ${\rm Com}(U,V)$ is a
subalgebra of $V$ (also called coset vertex algebra).

Let ${\frak h}$ be a finite-dimensional vector space equipped with a
nondegenerate symmetric bilinear form $\la \cdot, \cdot \ra$,
considered as an abelian Lie algebra. Let $\hat{\frak h} = {\frak h}
\otimes {\C}[t,t^{-1}] \oplus {\C} K$ be its affinization with the
center $K$. Then the free bosonic Fock space

$M(1) = S({\frak h} \otimes t^{-1} \C[t^{-1}])$ is a simple vertex
operator algebra of central charge $\ell = \dim {\frak h}$, with
Virasoro vector \bea \label{Vir-Heisenberg}
 \omega =\frac{1}{2} \sum _{i=1}^{\ell} h^{(i)}(-1)^2 {\1},\eea where
$\{ h^{(1)}, \ldots , h^{(\ell)} \} $ is any orthonormal basis of
${\frak h}$ (cf. \cite{FLM}, \cite{LL}). We shall also use the
notation $M_{\frak h}(1)$ to emphasize the associated vector space
${\frak h}$.

Vertex algebra $M(1)$  has an order 2 automorphism which is lifted
from the map $h \mapsto -h$, for $h \in {\frak h}$. Denote by  $M(1)
^+$ (or $M_{\frak h}(1) ^+$) the subalgebra of invariants of that
automorphism. The irreducible modules for $M(1) ^+$ were classified
in \cite{DN1} and \cite{DN2}. For $\ell =1$, it was proved in
\cite{DG} that $M(1) ^+$ is generated by $\omega$ and one primary
vector of conformal weight $4$, so it is isomorphic to a
$W(2,4)$--algebra with central charge $1$.

\vskip 5mm

Let $\frak g$ be the simple Lie algebra of type $X_{n}$, and
$\hat{\frak g}$ the associated affine Lie algebra of type
$X_{n}^{(1)}$. For any weight $\Lambda$ of $\hat{\frak g}$, denote
by $L_{X_{n} ^{(1)}}(\Lambda)$ the irreducible highest weight
$\hat{\frak g}$--module. Denote by $\Lambda _i$, $i=0, \ldots n$ the
fundamental weights of $\hat{\frak g}$ (cf. \cite{K1}). We shall
also use the notation $V_{X_n}(\mu)$ for a highest weight $\frak
g$--module of highest weight $\mu$, and $\omega _i$, $i=1, \ldots n$
for the fundamental weights of $\frak g$.

For any $k \in \C$, denote by $N_{X_{n} ^{(1)}}(k \Lambda _0)$ the
generalized Verma $\hat{\frak g}$--module with highest weight $k
\Lambda _0$. Then, $N_{X_{n} ^{(1)}}(k \Lambda _0)$ is a vertex
operator algebra of central charge $\frac{k \dim \frak
g}{k+h^{\vee}}$, for any $k \neq - h^{\vee}$, with Virasoro vector
obtained by Sugawara construction:
\begin{eqnarray} \label{Vir-Sugawara}
\omega=\frac{1}{2(k+h^{\vee})}\sum_{i=1}^{\dim {\frak g}}
a^{i}(-1)b^{i}(-1){\bf 1},
\end{eqnarray}
where $\{a^{i}\}_{i=1, \dots, \dim {\frak g}}$ is an arbitrary basis
of ${\frak g}$, and $\{b^{i}\}_{i=1, \dots, \dim {\frak g}}$  the
corresponding dual basis of ${\frak g}$ with respect to the
symmetric invariant bilinear form, normalized by the condition that
the length of the highest root is $\sqrt{2}$. (cf. \cite{FZ},
\cite{L}, \cite{FrB}, \cite{K}, \cite{LL}).

It follows that any quotient of $N_{X_{n} ^{(1)}}(k \Lambda _0)$ is
a vertex operator algebra, for $k \neq - h^{\vee}$. Specially,
$L_{X_{n} ^{(1)}}(k \Lambda _0)$ is a simple vertex operator
algebra, for any $k \neq - h^{\vee}$.

\section{Simple Lie algebras of type $C_{\ell}$ and $A_{2\ell -1}$}

Consider two $2 \ell$--dimensional vector spaces $A_1 = \oplus
_{i=1} ^{2 \ell}{\C} a_i ^+$, $ A_2 = \oplus _{i=1}^ {2 \ell} {\C}
a_i^-$ and let $A=A_1 \oplus A_2$. The Weyl algebra $W_{2 \ell}$ is
the complex associative algebra generated by $A$ and non-trivial
relations
\[
[a_i ^+,a_j^-]=\delta_{i,j} , \qquad 1 \le i,j \le 2 \ell.
\]

The normal ordering on $A$ is defined by
\[
:\!xy\!:  \ = \frac 1 2 (xy+yx),\ \qquad x,y \in A .
\]
Define
\[
e_{\epsilon_i-\epsilon_j}^A=\ :\!a_i ^+ a_j^-\!:, \quad
f_{\epsilon_i-\epsilon_j}^A=\ :\!a_j ^+ a_i^-\!:, \quad 1 \le i,j
\le 2 \ell; \ i<j,
\]
and
\[
H_i=-:a_i ^+ a_i^-:,  \qquad 1 \le i \le 2 \ell.
\]
Then the Lie algebra $\frak g _1$ generated by the set
\[
\{ e_{\epsilon_i-\epsilon_j}^A, f_{\epsilon_i-\epsilon_j}^A \ \vert
\ 1 \le i,j \le 2 \ell, \ i < j \}
\]
is the simple Lie algebra of type $A_{2 \ell -1}$ (cf. \cite{Bou}
and \cite{FF}). The Cartan subalgebra $\frak h _1$ is spanned by
\[
\{ H_i - H_{i+1}  \ \vert \ 1 \le i \le 2 \ell -1 \}.
\]

Let $\theta$ be the automorphism of $W_{2 \ell}$ of order two given
by

$$  a_{i} ^+ \mapsto  a_{2\ell+1-i} ^{-}, \quad a_{2\ell+1-i} ^-  \mapsto  a_{i} ^{+}, \quad
a_{i} ^-  \mapsto - a_{2\ell+1-i} ^+, \quad a_{2\ell+1-i} ^+
\mapsto - a_{i} ^-,$$

for $i =1, \dots, \ell$. Clearly, $\frak g _1$ is
$\theta$--invariant and
$$ \theta (e_{\epsilon_i-\epsilon_j}^A)=- e_{\epsilon_{2\ell +1-j}-\epsilon_{2\ell
+1-i}}^A, \quad \theta (e_{\epsilon_i-\epsilon_{2\ell +1-j}}^A)=
e_{\epsilon_{j}-\epsilon_{2\ell +1-i}}^A,
$$

and similarly for root vectors associated to negative roots.

The subalgebra $\frak g$ of $\frak g _1$ generated by

\bea
 && e_{2 \epsilon _{i}} =a_i ^+  a_{2\ell +1-i} ^-, \quad
    f_{2 \epsilon _{i}}=a_i ^-  a_{2\ell +1-i} ^+,
 \nonumber \\
&& e_{\epsilon _{i}+ \epsilon _{j}}=\frac{1}{2}(a_i ^+
 a_{2\ell +1-j} ^- + a_j ^+  a_{2\ell +1-i} ^-), \quad
 f_{\epsilon _{i}+ \epsilon _{j}}=\frac{1}{2}(a_i ^- a_{2\ell +1-j}
^+ + a_j ^-  a_{2\ell +1-i} ^+), \nonumber \\
&& e_{\epsilon _{i}- \epsilon _{j}}=\frac{1}{2}(a_i ^+
 a_{j} ^- - a_{2\ell +1-j} ^+
 a_{2\ell +1-i} ^-), \quad
f_{\epsilon _{i}- \epsilon _{j}}=\frac{1}{2}(a_j ^+
 a_{i} ^- - a_{2\ell +1-i} ^+
 a_{2\ell +1-j} ^-), \nonumber
\eea

for $i,j =1, \dots, \ell$, $i<j$, is the simple Lie algebra of type
$C_{\ell}$. The Cartan subalgebra $\frak h$ is spanned by
\[
\{ H_i - H_{2\ell +1-i}  \ \vert \ 1 \le i \le  \ell \}.
\]
Clearly, $\theta$ acts as $1$ on $\frak g$. Furthermore,
$$ e_{\epsilon _{1}+ \epsilon _{2}} ^*=\frac{1}{2}(a_1 ^+
 a_{2\ell -1} ^- - a_2 ^+  a_{2\ell} ^-) \in \frak g _1
$$
is a highest weight vector for $\frak g$, which generates the
irreducible $\frak g$--module $V_{C_{\ell}}(\omega _2)$. Clearly,
$\theta$ acts as $-1$ on $V_{C_{\ell}}(\omega _2)$. We obtain the
decomposition

\bea \label{decomp-fin1}
 \frak g _1 \cong \frak g \oplus V_{C_{\ell}}(\omega _2). \eea

Since

$$ \dim V_{A_{2\ell -1}} (n \omega_1) = \dim V_{C_{\ell }} (n \omega_1),$$
one easily concludes that the irreducible $\frak g _1$--module
$V_{A_{2\ell -1}} (n \omega_1)$ remains irreducible when restricted
to $\frak g$. Thus,

\bea \label{decomp-fin2} V_{A_{2\ell -1} } (n \omega_1) \cong V_{C_{
\ell  } } (n \omega_1) \quad \mbox{for } n \in \Zp. \eea

Similarly

\bea \label{decomp-fin3} V_{A_{2\ell -1} } (n \omega_{2 \ell -1})
\cong V_{C_{ \ell  } } (n \omega_1) \quad \mbox{for } n \in \Zp.
\eea

We will consider certain affine analogues of relations
(\ref{decomp-fin1})--(\ref{decomp-fin3}).

In what follows we shall need the following
decompositions of $\frak g$--modules:

\bea

&&V_{C_{\ell }} (\omega_2) \otimes V_{C_{\ell }} ( \omega_2) \cong
V_{C_{\ell }} (2 \omega_2)\oplus V_{C_{\ell }} ( \omega_1 +
\omega_3) \oplus V_{C_{\ell }} (\omega_4)
\nonumber \\
&& \qquad \qquad \qquad \qquad \quad
 \oplus V_{C_{\ell }} (2\omega_1) \oplus
V_{C_{\ell}} ( \omega_2) \oplus V_{C_{\ell }} (0)  \quad ( \ell \ge 4),  \nonumber \\
&& V_{C_{3}} (\omega_2) \otimes V_{C_{3}} ( \omega_2) \cong
 V_{C_{3}} (2 \omega_2)\oplus V_{C_{3}} ( \omega_1 +
\omega_3) \oplus V_{C_{3}} (2\omega_1) \oplus
V_{C_{3}} ( \omega_2) \oplus V_{C_{3}} (0), \nonumber \\
&& V_{C_{2}}(\omega _2) \otimes V_{C_{2}}(\omega _2) \cong
V_{C_{2}}(2 \omega _2) \oplus V_{C_{2}}(2 \omega _1) \oplus V_{C_{2}}(0), \nonumber \\
 &&V_{C_{\ell }} (n \omega_1) \otimes V_{C_{\ell }} ( \omega_2)
\cong V_{C_{\ell }} (n \omega_1 + \omega_2)\oplus V_{C_{\ell }} (
(n-1)\omega_1 + \omega_3) \nonumber \\
 && \qquad \qquad \qquad \qquad \quad
  \oplus V_{C_{\ell }} (n \omega_1)
 \oplus V_{C_{\ell }} ((n-2)\omega_1 + \omega_2) \quad (\ell \ge 3, n \ge 2), \nonumber \\
 &&V_{C_{\ell }} (\omega_1) \otimes V_{C_{\ell }} ( \omega_2) \cong  V_{C_{\ell }} ( \omega_1 + \omega_2)
 \oplus V_{C_{\ell }} (\omega_3) \oplus V_{C_{\ell }} ( \omega_1) \quad ( \ell \ge 3). \nonumber \\
&& \label{tens-pr-decomp} \mbox{}
 \eea

\section{Weyl vertex algebras and symplectic affine Lie algebras}

The Weyl algebra $W_{\ell}( \tfrac{1}{2} + {\Z})$ is a complex
associative algebra generated by

$$  a ^{\pm} _{i}(r),  \quad   r  \in \tfrac{1}{2} + {\Z}, \ 1 \le i \le \ell$$ and non-trivial relations

\bea

&& [a_{i} ^{+}(r)  , a_{j} ^{-}(s)  ]  =\delta_{r+s,0} \delta_{i,j}
\nonumber

\eea

where $r, s \in {\tfrac{1}{2}}+ {\Z}$, $ i, j \in \{1, \dots, \ell
\}$.

 Let $M_{\ell}$ be the irreducible $W_{\ell}( \tfrac{1}{2} + {\Z})$--module generated by
 the
cyclic vector ${\1}$ such that

$$ a^{\pm} _i  (r) {\1}     =  0 \quad
\mbox{for} \ \ r > 0 ,  \ 1 \le i \le \ell.$$

Define the following   fields on $M_{\ell}$

$$   a_i ^{\pm }(z) = \sum_{ n \in   {\Z}
 } a_i ^{\pm }(n+ \tfrac{1}{2} )  z ^{-n- 1}.$$

 The fields $a_{i} ^{\pm }(z)$, $i =1, \dots, \ell$ generate on $M_{\ell}$  the
unique structure of a simple vertex algebra (cf. \cite{K},
\cite{FrB}). Let us denote the corresponding vertex operator by $Y$.

We have the following Virasoro vector in $M_{\ell}$:

\bea \label{vir-bos} &&\omega=\frac{1}{2} \sum_{i=1} ^{\ell} ( a_i
^- (- \frac{3}{2}) a_i ^{+} (-\frac{1}{2} ) -a_i ^+ (-  \frac{3}{2})
a_i ^{-} (-\frac{1}{2} ) ) {\1}.\eea

Let $Y(\omega,z) = \sum_{n \in {\Z} } L(n) z ^{-n-2}$.

Then $M_{\ell}$ is a $\tfrac{1}{2}{\Zp}$--graded with respect to
$L(0)$:

$$ M_{\ell} := \bigoplus_{ m \in \tfrac{1}{2} \Zp} M_{\ell} (m), \quad M_{\ell} (m)= \{ v \in M_{\ell} \ \vert \  L(0) v = m v \}.$$

Note that $M_{\ell} (0) = {\C} {\1}$. For $ v \in M_{\ell} (m)$ we
shall write $\wt (v) = m$.

The following result is well-known.

\begin{theorem}[\cite{FF}] \label{ff}

We have

$$  M_{\ell} \cong
L_{C_{\ell} ^{(1)}}( -\tfrac{1}{2}\Lambda _{0}) \bigoplus
L_{C_{\ell} ^{(1)}}( -\tfrac{3}{2}\Lambda _{0} + \Lambda_1). $$

In the case $\ell =1$ we have
$$  M_1 \cong
L_{A_1 ^{(1)}}( -\tfrac{1}{2}\Lambda _{0}) \bigoplus L_{A_1 ^{(1)}}(
-\tfrac{3}{2}\Lambda _{0} + \Lambda_1). $$
\end{theorem}

\begin{remark} The highest weights of modules from Theorem \ref{ff}
are admissible in the sense of \cite{KW}. Representations of vertex
operator algebras associated to affine Lie algebras of type $A_1
^{(1)}$ and $C_{\ell} ^{(1)}$ with admissible highest weights were
studied in \cite{AM} and \cite{A-1994}.
\end{remark}

We shall now consider the vertex algebra $M_{2 \ell}$ and its
subalgebra $L_{C_{\ell} ^{(1)}}( -\tfrac{1}{2}\Lambda _{0}) \otimes
L_{C_{\ell} ^{(1)}}( -\tfrac{1}{2}\Lambda _{0})$. For $\ell =1$,
$L_{A_1 ^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes L_{A_1
^{(1)}}(-\tfrac{1}{2}\Lambda_0)$ is a subalgebra of $M_{2}$.

Let $\theta : M_{2 \ell} \rightarrow M_{2 \ell}$ be the automorphism
of order two of  the vertex algebra $M_{2 \ell}$ which is lifted
from the following automorphism of the Weyl algebra

\bea & a_{i} ^+ (s) \mapsto  a_{2\ell+1-i} ^{-} (s), \quad
&a_{2\ell+1-i} ^- (s) \mapsto  a_{i} ^{+} (s), \nonumber \\
& a_{i} ^-(s) \mapsto - a_{2\ell+1-i} ^+ (s), \quad &a_{2\ell+1-i}
^+(s) \mapsto - a_{i} ^- (s), \nonumber \eea

for $i =1, \dots, \ell$ and  $s \in \tfrac{1}{2} + {\Z}$.

If we have a subalgebra $U \subset M_{2 \ell}$ which is
$\theta$--invariant, we define

$$ U ^0 = \{ u \in U \ \vert \ \theta (u) = u \}, \quad U ^1 = \{ u \in U \ \vert \ \theta (u) = -u \}.$$

Define the following vectors in $M_{2 \ell}$:
\bea

&b_{i} ^+ = \frac{1}{\sqrt{2}}( a_{i} ^+ (-\tfrac{1}{2}) +
a_{2\ell+1-i} ^- (-\tfrac{1}{2}) ) {\1},
&b_{2\ell+1-i} ^+ = \frac{\sqrt{-1}}{\sqrt{2}}( a_{i} ^+ (-\tfrac{1}{2}) - a_{2\ell+1-i} ^- (-\tfrac{1}{2}) ) {\1} \nonumber \\

&b_{i} ^- = \frac{1}{\sqrt{2}}( a_{i} ^- (-\tfrac{1}{2}) -
a_{2\ell+1-i}^+ (-\tfrac{1}{2}) ) {\1}, &b_{2\ell+1-i} ^- =
\frac{-\sqrt{-1}}{\sqrt{2}}( a_{i} ^- (-\tfrac{1}{2}) +
a_{2\ell+1-i} ^+ (-\tfrac{1}{2}) ) {\1}, \nonumber

\eea for $i =1, \dots, \ell$. Then the subalgebra generated by $b_i
^+, b _i ^-$ (resp. $b_{2\ell+1-i} ^+, b _{2\ell+1-i} ^-$), for $i
=1, \dots, \ell$, is isomorphic to $M_{\ell}$.

Since

$$ \theta (b_i ^{\pm}) = b_i ^{\pm}, \quad  \theta (b_{2\ell+1-i} ^{\pm}) = - b_{2\ell+1-i} ^{\pm},$$

for $i =1, \dots, \ell$, we have
$$ M_{2 \ell}^{0} \cong M_{\ell} \otimes M_{\ell} ^0 \cong M_{\ell} \otimes L_{C_{\ell} ^{(1)}}( -\tfrac{1}{2}\Lambda _{0}),$$
and for $\ell =1$:
$$ M_{2}^{0} \cong M_{1} \otimes M_{1} ^0 \cong M_{1} \otimes L_{A_1 ^{(1)}}( -\tfrac{1}{2}\Lambda _{0}).$$

\section{Commutant of $L_{A_1 ^{(1)}}(- \Lambda_0) ^{\otimes \ell}$ in  $L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)
\otimes L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)$}

In this section we use Weyl vertex algebra $M_{2 \ell}$ to study
certain subalgebras of $L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)
\otimes L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)$. We determine
the commutants
$$ {\rm Com} ( L_{A_1 ^{(1)}}(- \Lambda_0) ^{\otimes \ell},L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)
\otimes L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0))$$

and
$$ {\rm Com} ( L_{A_1 ^{(1)}}(-\Lambda_0),L_{A_1 ^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes L_{A_1 ^{(1)}}(-\tfrac{1}{2}\Lambda_0)
).$$

Let $U$ be the vertex subalgebra  of $M_{\ell} ^0 \otimes M_{\ell}
^0 \subset M_{2 \ell} ^0$ generated by

\bea \label{gen-1-higher} && e^{(i)}=a_i ^+ (-\tfrac{1}{2}) a_{
2\ell +1-i} ^-( -\tfrac{1}{2}) {\1}, \quad  f^{(i)}=a_ i ^-
(-\tfrac{1}{2}) a_{2\ell +1-i} ^+(
-\tfrac{1}{2}) {\1} \quad \mbox{and} \nonumber \\
&& h^{(i)}= (-a_ i ^+ (-\tfrac{1}{2}) a_i^-( -\tfrac{1}{2}) +
a_{2\ell +1-i} ^+ (-\tfrac{1}{2}) a_{2\ell +1-i} ^-( -\tfrac{1}{2}))
{\1},
 \eea

 $i =1, \dots, \ell$. It is clear that $U$ is isomorphic to the tensor product
 $$\underbrace{ L_{A_1 ^{(1)}}( -\Lambda _{0}) \otimes \cdots \otimes  L_{A_1 ^{(1)}}( -\Lambda _{0})}_{\ell \ \  \mbox{times} }$$

of $\ell$ copies of the affine vertex algebra $ L_{A_1
^{(1)}}(-\Lambda _{0})$.

 Define also

 $$ H ^{(i)} =   (a_i ^+ (-\tfrac{1}{2}) a_i ^- (-\tfrac{1}{2})+a_{2\ell +1-i} ^+ (-\tfrac{1}{2})
 a_{2\ell +1-i} ^- (-\tfrac{1}{2}) ) {\1}, \quad (i=1, \dots, \ell).$$
 Then
 $$ H ^{(i)} \in \mbox{Com} ( U, M_{2\ell}), \quad (i=1, \dots, \ell).$$

 Let $$ {\frak h}= \bigoplus_{i =1 } ^{ \ell} {\C} H ^{(i)}.$$ Then ${\frak h}$ can be
 considered as an abelian Lie algebra, and the components of the vertex operators

 $$ Y(h,z) = \sum_{\n \in {\Z} } h(n) z ^{-n-1} \quad (h \in {\frak h})$$

 define a representation of the associated Heisenberg algebra $\widehat{\frak h}$.
 Moreover, ${\frak h}$ generates the subalgebra of the $M_{2 \ell}$ which is isomorphic to the
 Heisenberg vertex algebra $M_{\frak h} (1)$ with central charge $\ell$.
 We also note that $\la H ^{(i)}, H ^{(j)} \ra = H ^{(i)}(1)H ^{(j)}= -2 \delta _{ij}$.

Using relations (\ref{gen-1-higher}) one can show that the Virasoro
vector (\ref{vir-bos}) (in $M_{2 \ell}$) can be written in a form:

$$ \omega =  \omega_1   + \omega_2$$

where
$$\omega_1 =\frac{1}{2} \sum_{i = 1 } ^{\ell} (e ^{(i)}(-1)f^{(i)}(-1)+f ^{(i)} (-1)e^{(i)}(-1)+ \frac{1}{2} h ^{(i)} (-1)^2){\1}$$
is the  Virasoro vector  in $U$  and

$$\omega_2 = -\frac{1}{4}\sum_{i = 1 } ^{\ell} H ^{(i) }(-1) ^{2} {\1}$$
is the Virasoro vector   in $M_{\frak h} (1)$.

For $i=1,2$ let

$Y(\omega_i , z) = \sum_{n\in {\Z} } L_i (n) z ^{-n-2}$.

  \begin{proposition} \label{com-bos-higher} We have:

$$ {\rm Com} (U, M_{2\ell}) = M_{\frak h} (1).$$

\end{proposition}
\begin{proof}
It is clear that
 $W= \mbox{Com} ( U, M_{2\ell})$ contains a subalgebra isomorphic to $M_{\frak h}
(1)$.

Assume now that $M_{\frak h} (1) \ne W$. Then there is a vector

$ w \in W, \ {\wt} (w) > 0$ such that

$$ H ^{(i)}(n) w =  \delta_{n,0} \lambda _{(i)} w \quad (n \in {\Zp}, \ \lambda _{(i)} \in {\Z}, \quad i = 1, \dots, \ell). $$

(Note that each  $H^{(i)}(0)$ acts semisimply on $M_{2\ell}$ with eigenvalues in ${\Z}$.)

By definition of $W$ we have that $L_1 (0) w = 0$. Therefore

$$L(0) w = L_2(0) w = - \frac{1}{4} \sum_{i = 1 } ^{\ell} (\lambda _{(i)})  ^2 w. $$
This contradicts the fact that $\wt (w) > 0$. So, $W= M_{\frak h} (1)$.
\end{proof}

Since $\theta (H^{(i)}) = -H^{(i)}$ for $i=1, \dots, \ell$ we have that
$M_{\frak h} (1)^+ =M_{\frak h}(1) ^0$ is a subalgebra of $M_{\ell} ^0 \otimes M_{\ell} ^0$.
 Therefore the  vertex algebra $L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)$ contains a subalgebra isomorphic
to $U \otimes M_{\frak h} (1) ^+$. By using
Proposition \ref{com-bos-higher} we obtain the following theorem.

\begin{theorem} \label{coset-bos-r-higher}We have:

$$ {\rm Com} ( L_{A_1 ^{(1)}}(- \Lambda_0) ^{\otimes \ell},L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)
\otimes L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0) ) \cong M_{\frak h} (1) ^+.$$
\end{theorem}

In the case $\ell =1$, ${\frak h}$ is one-dimensional vector space
${\frak h}= {\C} H$, where
$$ H = (a_1 ^+ (-\tfrac{1}{2}) a_1 ^- (-\tfrac{1}{2})+a_2 ^+ (-\tfrac{1}{2}) a_2 ^- (-\tfrac{1}{2}) ) {\1}$$
and
\begin{theorem} \label{coset-bos-r} We have:

$$ {\rm Com} ( L_{A_1 ^{(1)}}(-\Lambda_0),L_{A_1 ^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes L_{A_1 ^{(1)}}(-\tfrac{1}{2}\Lambda_0) ) \cong M_{\frak h} (1) ^+.$$
\end{theorem}

In the next section we generalize Theorem \ref{coset-bos-r} in
another way.

\section{Commutant of level $-1$ type $C_{\ell} ^{(1)}$  affine vertex algebra in  $L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)
\otimes L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)$ }

In this section we study another subalgebra of $L_{C_{\ell}
^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes L_{C_{\ell}
^{(1)}}(-\tfrac{1}{2}\Lambda_0)$. The subalgebra of $M_{\ell} ^0
\otimes M_{\ell} ^0 \subset M_{2 \ell} ^0$ generated by \bea
\label{gen-1-C}
 && e_{2 \epsilon _{i}} (=e^{(i)})=a_i ^+ (-\tfrac{1}{2}) a_{2\ell +1-i} ^-( -\tfrac{1}{2})
 {\1}, \quad   f_{2 \epsilon _{i}}(=f^{(i)})=a_i ^- (-\tfrac{1}{2}) a_{2\ell +1-i} ^+( -\tfrac{1}{2}) {\1},
 \nonumber \\
&& e_{\epsilon _{i}+ \epsilon _{j}}=\frac{1}{2}(a_i ^+
(-\tfrac{1}{2}) a_{2\ell +1-j} ^-( -\tfrac{1}{2}) + a_j ^+
(-\tfrac{1}{2}) a_{2\ell +1-i} ^-( -\tfrac{1}{2})){\1}, \nonumber \\
&& f_{\epsilon _{i}+ \epsilon _{j}}=\frac{1}{2}(a_i ^-
(-\tfrac{1}{2}) a_{2\ell +1-j} ^+( -\tfrac{1}{2}) + a_j ^-
(-\tfrac{1}{2}) a_{2\ell +1-i} ^+( -\tfrac{1}{2})){\1}, \nonumber \\
&& e_{\epsilon _{i}- \epsilon _{j}}=\frac{1}{2}(a_i ^+
(-\tfrac{1}{2}) a_{j} ^-( -\tfrac{1}{2}) - a_{2\ell +1-j} ^+
(-\tfrac{1}{2}) a_{2\ell +1-i} ^-( -\tfrac{1}{2})){\1}, \nonumber \\
&& f_{\epsilon _{i}- \epsilon _{j}}=\frac{1}{2}(a_j ^+
(-\tfrac{1}{2}) a_{i} ^-( -\tfrac{1}{2}) - a_{2\ell +1-i} ^+
(-\tfrac{1}{2}) a_{2\ell +1-j} ^-( -\tfrac{1}{2})){\1}, \nonumber
\\
&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad  \mbox{for } i,j =1, \dots, \ell, i<j,
  \eea
is level $-1$ affine vertex operator algebra associated to $C_{\ell}
^{(1)}$. We denote it by ${\widetilde L}_{C_{\ell} ^{(1)}}(-
\Lambda_0)$.

Let

$$ H = \sum_{i = 1 } ^{\ell} H ^{(i)} =\sum _{i=1}^{2 \ell} a_i ^+ (-\tfrac{1}{2}) a_i ^- (-\tfrac{1}{2}){\1}. $$

Set ${\h}_1 = {\C} H \subset {\h}$. Let $M_{ {\h}_1} (1)$ be the
Heisenberg vertex algebra generated by $H$. Clearly $\la H, H \ra =
-2 \ell$.

Since  $H \in \mbox{Com} ({\widetilde L}_{C_{\ell} ^{(1)}}(-
\Lambda_0), M_{2 \ell})$, we have that
  $M_{2 \ell}$ contains a
subalgebra isomorphic to $ {\widetilde L}_{C_{\ell} ^{(1)}}(-
\Lambda_0) \otimes M_{{\h}_1 }(1)$.

\begin{proposition} \label{Vir-decomp-C}
The Virasoro vector (\ref{vir-bos}) in $M_{2 \ell}$ can be written
in a form:

$$ \omega = \omega_1 + \omega_2$$

where $ \omega_1$ is the  Virasoro vector (\ref{Vir-Sugawara}) in
${\widetilde L}_{C_{\ell} ^{(1)}}(- \Lambda_0)$ obtained by the
Sugawara construction and

$$\omega_2 = -\frac{1}{4 \ell} H(-1) ^{2} {\1}$$ is the Virasoro vector in $M_{ {\h}_1} (1)$.
\end{proposition}
\begin{proof} Formula (\ref{Vir-Sugawara}) implies that
\bea \label{proof-Vir-1}
&& \omega_1 = \frac{1}{2 \ell} \Big(
\sum_{i=1}^{\ell} (e_{2 \epsilon _{i}}(-1)f_{2 \epsilon
_{i}}(-1)+f_{2 \epsilon
_{i}}(-1)e_{2 \epsilon _{i}}(-1)){\1} \nonumber \\
&& +2 \sum_{i,j=1 \atop i<j}^{\ell} (e_{\epsilon _{i}+ \epsilon
_{j}}(-1)f_{\epsilon _{i}+ \epsilon _{j}}(-1)+ f_{\epsilon _{i}+
\epsilon _{j}}(-1)e_{\epsilon _{i}+ \epsilon _{j}}(-1)){\1} \nonumber \\
&& +2 \sum_{i,j=1 \atop i<j}^{\ell} (e_{\epsilon _{i}- \epsilon
_{j}}(-1)f_{\epsilon _{i}- \epsilon _{j}}(-1)+ f_{\epsilon _{i}-
\epsilon _{j}}(-1)e_{\epsilon _{i}- \epsilon _{j}}(-1)){\1} \nonumber \\
&& +\frac{1}{2}\sum_{i=1}^{\ell} h_{2 \epsilon _{i}}(-1)^2 {\1}
\Big),
 \eea
where

\bea h_{2 \epsilon _{i}} (=h^{(i)})= (-a_i ^+ (-\tfrac{1}{2}) a_i^-(
-\tfrac{1}{2})+ a_{2\ell +1-i} ^+ (-\tfrac{1}{2}) a_{2\ell +1-i}^-(
-\tfrac{1}{2}) ){\1}. \nonumber \eea

It follows from relations (\ref{gen-1-C}) that \bea
\label{proof-Vir-2} && (e_{2 \epsilon _{i}}(-1)f_{2 \epsilon
_{i}}(-1)+f_{2
\epsilon _{i}}(-1)e_{2 \epsilon _{i}}(-1)){\1} \nonumber \\
&& = 2 a_i ^+ (-\tfrac{1}{2})a_i ^- (-\tfrac{1}{2}) a_{2\ell +1-i}
^+ (-\tfrac{1}{2}) a_{2\ell +1-i} ^- (-\tfrac{1}{2}){\1} \nonumber \\
&& + a_i ^- (-\tfrac{3}{2})a_i ^+ (-\tfrac{1}{2}){\1}  + a_{2\ell
+1-i} ^- (-\tfrac{3}{2})a_{2\ell +1-i} ^+ (-\tfrac{1}{2}){\1} \nonumber \\
&& -a_i ^+ (-\tfrac{3}{2})a_i ^- (-\tfrac{1}{2}){\1}  - a_{2\ell
+1-i} ^+ (-\tfrac{3}{2})a_{2\ell +1-i} ^- (-\tfrac{1}{2}){\1}, \eea

\bea \label{proof-Vir-3}
 && 2(e_{\epsilon _{i}+ \epsilon
_{j}}(-1)f_{\epsilon _{i}+ \epsilon _{j}}(-1)+ f_{\epsilon _{i}+
\epsilon _{j}}(-1)e_{\epsilon _{i}+
\epsilon _{j}}(-1)){\1} \nonumber \\
&& = a_i ^+ (-\tfrac{1}{2})a_i ^- (-\tfrac{1}{2}) a_{2\ell +1-j} ^+
(-\tfrac{1}{2}) a_{2\ell +1-j} ^- (-\tfrac{1}{2}){\1} \nonumber \\
&& + a_i ^+ (-\tfrac{1}{2})a_j ^- (-\tfrac{1}{2}) a_{2\ell +1-j} ^-
(-\tfrac{1}{2})
a_{2\ell +1-i} ^+ (-\tfrac{1}{2}){\1} \nonumber \\
&& + a_i ^- (-\tfrac{1}{2})a_j ^+ (-\tfrac{1}{2}) a_{2\ell +1-i} ^-
(-\tfrac{1}{2}) a_{2\ell +1-j} ^+ (-\tfrac{1}{2}){\1} \nonumber \\
&& + a_j ^+ (-\tfrac{1}{2})a_j ^- (-\tfrac{1}{2}) a_{2\ell +1-i} ^+
(-\tfrac{1}{2})
a_{2\ell +1-i} ^- (-\tfrac{1}{2}){\1} \nonumber \\
&& +\frac{1}{2} \Big( a_i ^- (-\tfrac{3}{2}) a_i ^+
(-\tfrac{1}{2}){\1} + a_j ^- (-\tfrac{3}{2}) a_j ^+
(-\tfrac{1}{2}){\1} + a_{2\ell +1-i} ^- (-\tfrac{3}{2}) a_{2\ell
+1-i} ^+
(-\tfrac{1}{2}){\1}  \nonumber \\
&& +a_{2\ell +1-j} ^- (-\tfrac{3}{2}) a_{2\ell +1-j} ^+
(-\tfrac{1}{2}){\1} - a_i ^+ (-\tfrac{3}{2}) a_i ^-
(-\tfrac{1}{2}){\1} - a_j ^+ (-\tfrac{3}{2}) a_j ^-
(-\tfrac{1}{2}){\1} \nonumber \\
 && - a_{2\ell +1-i} ^+ (-\tfrac{3}{2}) a_{2\ell
+1-i}^- (-\tfrac{1}{2}){\1} - a_{2\ell +1-j} ^+ (-\tfrac{3}{2})
a_{2\ell
+1-j} ^- (-\tfrac{1}{2}){\1} \Big), \nonumber \\
&& \mbox{}
  \eea
and \bea \label{proof-Vir-4}
 && 2(e_{\epsilon _{i}- \epsilon
_{j}}(-1)f_{\epsilon _{i}- \epsilon _{j}}(-1)+ f_{\epsilon _{i}-
\epsilon _{j}}(-1)e_{\epsilon _{i}-
\epsilon _{j}}(-1)){\1} \nonumber \\
&& = a_i ^+ (-\tfrac{1}{2})a_i ^- (-\tfrac{1}{2}) a_{j} ^+
(-\tfrac{1}{2}) a_{j} ^- (-\tfrac{1}{2}){\1} \nonumber \\
&& - a_j ^+ (-\tfrac{1}{2})a_i ^- (-\tfrac{1}{2}) a_{2\ell +1-j} ^+
(-\tfrac{1}{2})
a_{2\ell +1-i} ^- (-\tfrac{1}{2}){\1} \nonumber \\
&& - a_i ^+ (-\tfrac{1}{2})a_j ^- (-\tfrac{1}{2}) a_{2\ell +1-i} ^+
(-\tfrac{1}{2}) a_{2\ell +1-j} ^- (-\tfrac{1}{2}){\1}  \nonumber \\
&& +a_{2\ell +1-i} ^+ (-\tfrac{1}{2})a_{2\ell +1-i} ^-
(-\tfrac{1}{2}) a_{2\ell +1-j} ^+ (-\tfrac{1}{2})
a_{2\ell +1-j} ^- (-\tfrac{1}{2}){\1} \nonumber \\
&& +\frac{1}{2} \Big( a_i ^- (-\tfrac{3}{2}) a_i ^+
(-\tfrac{1}{2}){\1} + a_j ^- (-\tfrac{3}{2}) a_j ^+
(-\tfrac{1}{2}){\1} + a_{2\ell +1-i} ^- (-\tfrac{3}{2}) a_{2\ell
+1-i} ^+
(-\tfrac{1}{2}){\1}  \nonumber \\
&& +a_{2\ell +1-j} ^- (-\tfrac{3}{2}) a_{2\ell +1-j} ^+
(-\tfrac{1}{2}){\1} - a_i ^+ (-\tfrac{3}{2}) a_i ^-
(-\tfrac{1}{2}){\1} - a_j ^+ (-\tfrac{3}{2}) a_j ^-
(-\tfrac{1}{2}){\1} \nonumber \\
 && - a_{2\ell +1-i} ^+ (-\tfrac{3}{2}) a_{2\ell
+1-i}^- (-\tfrac{1}{2}){\1} - a_{2\ell +1-j} ^+ (-\tfrac{3}{2})
a_{2\ell
+1-j} ^- (-\tfrac{1}{2}){\1} \Big), \nonumber \\
&& \mbox{}
  \eea
for all $i,j =1, \dots, \ell$, $i<j$. Using relations
(\ref{proof-Vir-1}), (\ref{proof-Vir-2}), (\ref{proof-Vir-3}) and
(\ref{proof-Vir-4}) one can obtain \bea \omega _{1}= \frac{1}{2}
\sum_{i=1} ^{2 \ell} ( a_i ^- (- \frac{3}{2}) a_i ^{+} (-\frac{1}{2}
) -a_i ^+ (- \frac{3}{2}) a_i ^{-} (-\frac{1}{2} ) ) {\1} +
\frac{1}{4 \ell} H(-1) ^{2} {\1}, \nonumber \eea which implies the
claim of Proposition.
\end{proof}

Using Proposition \ref{Vir-decomp-C} and applying similar arguments
as in the proof of Proposition \ref{com-bos-higher}, we obtain:

\begin{proposition} \label{com-bos-C} We have:

$$ {\rm Com} ({\widetilde L}_{C_{\ell} ^{(1)}}(-
\Lambda_0), M_{2 \ell}) = M_ { {\h}_1} (1).$$

\end{proposition}

\vskip 5mm Since $\theta (H) = -H$, we have that $M_ { {\h}_1} (1)^+ =M_ { {\h}_1} (1)
^0$ is a subalgebra of $M_{\ell} ^0 \otimes M_{\ell} ^0$.
 Therefore the  vertex algebra $L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes
 L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)$ contains a subalgebra isomorphic
to ${\widetilde L}_{C_{\ell} ^{(1)}}(- \Lambda_0) \otimes M_ { {\h}_1} (1)
^+$. By using Proposition \ref{com-bos-C} we obtain the following
theorem.

\begin{theorem} \label{coset-bos-r-C}We have:

$$ {\rm Com} ( {\widetilde L}_{C_{\ell} ^{(1)}}(-\Lambda_0),
L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes L_{C_{\ell}
^{(1)}}(-\tfrac{1}{2}\Lambda_0) ) \cong M_ { {\h}_1}  (1) ^+.$$
\end{theorem}

\section{The classification of ordinary modules for  ${\widetilde L}_{C_{\ell} ^{(1)}}(-
\Lambda_0)$} \label{sect.class.C}

In this section we obtain a classification of irreducible
${\widetilde L}_{C_{\ell} ^{(1)}}(- \Lambda_0)$--modules, which we
use in the following sections.

The following vertex algebra was considered in \cite{A1}: Let $\ell
\ge 3$ and
$$V_{C_{\ell} ^{(1)}} (-\Lambda_0)= \frac{N_{C_{\ell} ^{(1)}}
(-\Lambda_0)}{< \Delta_3 (-1) {\1} > },$$

where $< \Delta_3 (-1) {\1} >$ is the ideal generated by the singular vector
$\Delta_3 (-1) {\1}$, such that $\Delta_3(-1)$ is given by the following
determinant:

\bea
 \Delta_3(-1) =  \left| \begin{array}{ccc} e_{ 2\epsilon_1} (-1) & e_{
\epsilon_1 + \epsilon_2} (-1) &  e_{ \epsilon_1 + \epsilon_3} (-1)
\\ e_{ \epsilon_1 + \epsilon_2} (-1) & e_{ 2
\epsilon_2} (-1)&  e_{ \epsilon_2 + \epsilon_3} (-1)
\\e_{ \epsilon_1
+ \epsilon_3} (-1) & e_{ \epsilon_2 + \epsilon_3} (-1) & e_{ 2
\epsilon_3} (-1)
\end{array} \right|.
   \nonumber
 \eea

Using relations (\ref{gen-1-C}), one can easily check that \bea
 \left| \begin{array}{ccc} e_{ 2\epsilon_1} (-1) & e_{
\epsilon_1 + \epsilon_2} (-1) &  e_{ \epsilon_1 + \epsilon_3} (-1)
\\ e_{ \epsilon_1 + \epsilon_2} (-1) & e_{ 2
\epsilon_2} (-1)&  e_{ \epsilon_2 + \epsilon_3} (-1)
\\e_{ \epsilon_1
+ \epsilon_3} (-1) & e_{ \epsilon_2 + \epsilon_3} (-1) & e_{ 2
\epsilon_3} (-1)
\end{array} \right| {\1}=0
   \nonumber
 \eea
in $\widetilde{L}_{C_{\ell} ^{(1)}} (-\Lambda_0)$, so
$\widetilde{L}_{C_{\ell} ^{(1)}} (-\Lambda_0)$ is a certain quotient
of $V_{C_{\ell} ^{(1)}} (-\Lambda_0)$, for $\ell \ge 3$. Thus, any
irreducible module for $\widetilde{L}_{C_{\ell} ^{(1)}}
(-\Lambda_0)$ is an irreducible module for $V_{C_{\ell} ^{(1)}}
(-\Lambda_0)$. The classification of all irreducible modules in the category $\mathcal{O}$  for
$V_{C_{3} ^{(1)}} (-\Lambda_0)$ was obtained in \cite[Example
4.1]{A1}. We will apply this result to obtain a classification of all irreducible ordinary  $\widetilde{L}_{C_{\ell} ^{(1)}} (-\Lambda_0)$--modules.
(Recall that a module is called ordinary  if $L(0)$ acts semisimply with finite-dimensional weight spaces).

\begin{proposition} \label{classif-C-0}Let $\ell \ge 3$.
The set

\bea \label{skup}&& \{ L_{C_{\ell} ^{(1)}}( (-n-1 ) \Lambda_0 + n
\Lambda_1) \ \vert \ n \in {\Zp} \}  \cup \{ L_{C_{\ell} ^{(1)}}(- 2
\Lambda_0 + \Lambda_2) \}

 \eea
 provides a complete list of irreducible ordinary modules for the vertex operator algebras $V_{C_{\ell} ^{(1)}}
(-\Lambda_0)$ and $\widetilde{L}_{C_{\ell} ^{(1)}} (-\Lambda_0)$.
\end{proposition}

\begin{proof}
We use the well-known method   for classification of highest weights
of $V_{C_{\ell} ^{(1)}} (-\Lambda_0)$--modules as solutions of
certain polynomial equations arising from the singular vectors (cf.
\cite{A-1994}, \cite{AM}, \cite{AP}, \cite{MP}, \cite{P1},
\cite{P2}). The highest weights of ordinary modules are of the form
$- \Lambda_0 + \mu$, where $\mu = \sum_{i=1}^{\ell}h_{i} \epsilon
_{i}$. Clearly, $h_{i} \in \Zp$, for $i=1, \ldots , \ell$. Using
polynomials from \cite[Example 4.1]{A1}, and the fact that
$$\frac{1}{2}f_{\epsilon _{3}- \epsilon _{i}}(0)^2 \Delta_3(-1){\1}=
\left| \begin{array}{ccc} e_{ 2\epsilon_1} (-1) & e_{ \epsilon_1 +
\epsilon_2} (-1) &  e_{ \epsilon_1 + \epsilon_i} (-1)
\\ e_{ \epsilon_1 + \epsilon_2} (-1) & e_{ 2
\epsilon_2} (-1)&  e_{ \epsilon_2 + \epsilon_i} (-1)
\\e_{ \epsilon_1
+ \epsilon_i} (-1) & e_{ \epsilon_2 + \epsilon_i} (-1) & e_{ 2
\epsilon_i} (-1)
\end{array} \right| {\1}, $$
for $i=4, \ldots , \ell$, we obtain that the weights $\mu$ are
annihilated by the polynomials \bea

&& p_i( \mu) = (h_1 +1) (h_2 + \frac{1}{2}) h_i,  \nonumber \\

&& q_i( \mu) = (h_1 +1) ( 4h_i + ( h_2 + h_i) ( h_2 + h_i -1) ),
\nonumber
\\

&& r_i( \mu) = 4 h_i ( h_2 +1) + ( h_1 + h_i -1) ( h_2 + h_i + h_2 (
h_1 + h_i)), \nonumber \eea

for $i=3, \ldots , \ell$. Then one easily obtains that $h_i=0$, for
$i=3, \ldots , \ell$, and that either $h_2=0$ and $h_1=n$, for $n
\in {\Zp}$ or $h_2=1$ and $h_1=1$. So we have proved that any irreducible ordinary $V_{C_{\ell} ^{(1)}}
(-\Lambda_0)$--module (resp. $\widetilde{L}_{C_{\ell} ^{(1)}} (-\Lambda_0)$--module) must belong to the set (\ref{skup}).

Since $a_1 ^+ (-1/2) ^{n} {\1} \in M_{2 \ell}$ is a singular vector
of highest weight $-(n+1) \Lambda_0 + n \Lambda_1$, and
$$ e_{\epsilon _{1}+ \epsilon _{2}} ^*=\frac{1}{2}(a_1 ^+
(-\tfrac{1}{2}) a_{2\ell -1} ^-( -\tfrac{1}{2}){\1} - a_2 ^+
(-\tfrac{1}{2}) a_{2\ell} ^-( -\tfrac{1}{2}) {\1}) \in M_{2 \ell}
$$

is a singular vector of highest weight $-2\Lambda_0 + \Lambda_2$,
we have that every module from the set (\ref{skup}) is a module for
these vertex operator algebras. The proof is now complete.
\end{proof}

\begin{corollary} \label{classif-C-1} Let $\ell \ge 3$.
The set of irreducible ordinary  $ L_{C_{\ell} ^{(1)}}
(-\Lambda_0)$--modules is a subset of the set

\bea && \{ L_{C_{\ell} ^{(1)}}( (-n-1 ) \Lambda_0 + n \Lambda_1) \
\vert \ n \in {\Zp} \}  \cup \{ L_{C_{\ell} ^{(1)}}(- 2 \Lambda_0 +
\Lambda_2) \}. \nonumber

 \eea
\end{corollary}

\section{Conformal embedding of $C_{\ell} ^{(1)}$ into $A_{2 \ell -1} ^{(1)}$ at level $-1$}

In this section we show that $L_{C_{\ell} ^{(1)}} (-\Lambda_0)$ is a
$\Z _2$--orbifold of vertex operator algebra $L_{A_{2 \ell -1}
^{(1)}} (-\Lambda_0)$, and determine the corresponding
decomposition.

The subalgebra of $M_{2 \ell}$ generated by

\bea && e_{\epsilon _{i}- \epsilon _{j}}^{A} = a_i ^+
(-\tfrac{1}{2}) a_{j} ^-( -\tfrac{1}{2})
 {\1}, \quad f_{\epsilon _{i}- \epsilon _{j}}^{A}=  a_i ^- (-\tfrac{1}{2}) a_{j} ^+( -\tfrac{1}{2})
 {\1},  \nonumber \\
&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad  \mbox{for }
  i,j =1, \dots, 2 \ell, i<j,\nonumber
\eea

is level $-1$ affine vertex operator algebra associated to the affine Lie algebra ${\hat \g_1}$ of type $A_{2
\ell -1}^{(1)}$. We denote it by $\widetilde{L}_{A_{2 \ell -1}
^{(1)}} (-\Lambda_0)$. Clearly, $\widetilde{L}_{C_{\ell} ^{(1)}}
(-\Lambda_0)$ is a subalgebra of $\widetilde{L}_{A_{2 \ell -1}
^{(1)}} (-\Lambda_0)$.  Let ${\hg}$ be the affine Lie algebra of type $C_{\ell} ^{(1)}$.

As before, denote by $ \omega_1$ the
Virasoro vector in ${\widetilde L}_{C_{\ell} ^{(1)}}(- \Lambda_0)$,
and by $ \omega_1 ^A$ the  Virasoro vector in $\widetilde{L}_{A_{2
\ell -1} ^{(1)}} (-\Lambda_0)$.

\begin{proposition} We have
$$\omega_1 = \omega_1
^A.$$
\end{proposition}
\begin{proof} Similarly as in Proposition \ref{Vir-decomp-C}, one can show that
\bea \omega _{1}^A= \frac{1}{2} \sum_{i=1} ^{2 \ell} ( a_i ^- (-
\frac{3}{2}) a_i ^{+} (-\frac{1}{2} ) -a_i ^+ (- \frac{3}{2}) a_i
^{-} (-\frac{1}{2} ) ) {\1} + \frac{1}{4 \ell} H(-1) ^{2} {\1},
\nonumber \eea which implies the claim of Proposition.
\end{proof}

Furthermore, the vector $ e_{\epsilon _{1}+ \epsilon _{2}} ^*$ from
the proof of Proposition \ref{classif-C-0} is a singular vector for
${\hg}$  in $\widetilde{L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$
which generates $\widetilde{L}_{C_{ \ell} ^{(1)}}
(-\Lambda_0)$--module $\widetilde{L}_{C_{\ell} ^{(1)}} (-2\Lambda_0
+ \Lambda_2)$, whose top component is irreducible $\frak g$--module
$V_{C_{\ell}}(\omega _2)$. Clearly, $\theta$ acts as $1$ on
$\widetilde{L}_{C_{\ell} ^{(1)}} (-\Lambda_0)$ and as $-1$ on
$\widetilde{L}_{C_{\ell} ^{(1)}} (-2\Lambda_0 + \Lambda_2)$.

\begin{lemma} \label{lem-ext}
Let $u,v \in \widetilde{L}_{C_{\ell} ^{(1)}} (-2\Lambda_0 +
\Lambda_2)$. Then $u_n v \in \widetilde{L}_{C_{\ell} ^{(1)}}
(-\Lambda_0)$, for any $n \in \Z$.
\end{lemma}
\begin{proof} It suffices to prove the lemma for $u$ and $v$ from top component $R(0)$  of $\widetilde{L}_{C_{\ell} ^{(1)}} (-2\Lambda_0 +
\Lambda_2)$. Then the statement will follow from the associator formulae.

First we notice that $$u _0 v \in \widetilde{L}_{C_{\ell} ^{(1)}}
(-\Lambda_0),$$

(since  vectors of conformal weight $1$ with bracket $[u,v] =u_0 v$
span   Lie algebra ${\g}_1$ of type $A_{2 \ell -1}$, and fixed point
subalgebra ${\g}$ is a Lie algebra of type $C_{\ell}$).

Assume now that    $$u_{n_0} v \notin \widetilde{L}_{C_{\ell} ^{(1)}}
(-\Lambda_0)$$

for certain $u, v \in R(0)$ and $n_0 \in {\Z}$. Take maximal $n_0$ with this property.

 Then  $u_{n_0}  v$ has nontrivial component  in some highest weight $\widetilde{L}_{C_{\ell} ^{(1)}}
(-\Lambda_0)$--module $W$  of highest weight $-\Lambda_0 + \mu$, and
therefore there is a nontrivial intertwining operator of type

\bea {W \choose \widetilde{L}_{C_{\ell} ^{(1)}} (-2\Lambda_0 +
\Lambda_2) \quad \widetilde{L}_{C_{\ell} ^{(1)}} (-2\Lambda_0 +
\Lambda_2) }. \label{op-1}\eea

One can associate to this intertwining operator, a non-trivial $\g$--homomorphism
$$ f: V_{C_{\ell}}(\omega _2) \otimes V_{C_{\ell}}(\omega _2)  \rightarrow V_{C_{\ell}}(\mu).$$
In particular, $V_{C_{\ell}}(\mu)$ must appear in the decomposition
of tensor product $V_{C_{\ell}}(\omega _2) \otimes
V_{C_{\ell}}(\omega _2)$.

First consider the case $\ell =2$. Using the decomposition of tensor
product $V_{C_{2}}(\omega _2) \otimes V_{C_{2}}(\omega _2)$ from
(\ref{tens-pr-decomp}) and the fact that the lowest conformal
weights of modules of highest weights $-\Lambda_0 + 2 \omega _2$ and
$-\Lambda_0 + 2 \omega _1$ are $\frac{5}{2}$ and $\frac{3}{2}$,
respectively, we conclude that these modules cannot appear inside
$\widetilde{L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$. Thus, $u_{n_0}
v \in \widetilde{L}_{C_{\ell} ^{(1)}} (-\Lambda_0)$.

Now, let $\ell \geq 3$. Relation (\ref{tens-pr-decomp}) implies that
the only cases of weights $-\Lambda_0 + \mu$ (aside from $\mu =0$)
from Proposition \ref{classif-C-0} such that $\mu$ appears in the
decomposition of $V_{C_{\ell}}(\omega _2) \otimes
V_{C_{\ell}}(\omega _2)$ are when $\mu = 2 \omega _1$ or $\mu =
\omega _2$. Since  $u_{n_0} v$ is $\theta$--invariant, it does not
contain component inside $\widetilde{L}_{C_{\ell} ^{(1)}}
(-2\Lambda_0 + \Lambda_2)$. On the other hand, the lowest conformal
weight of module of highest weight $-\Lambda_0 + 2 \omega _1$ is
$\frac{\ell + 1}{\ell}$, which is not an integer. Thus, this module
cannot appear inside $\widetilde{L}_{A_{2 \ell -1} ^{(1)}}
(-\Lambda_0)$. Thus, $u_{n_0} v \in \widetilde{L}_{C_{\ell} ^{(1)}}
(-\Lambda_0)$.
\end{proof}

\begin{theorem} \label{thm-ext} We have:
$$\widetilde{L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0) = \widetilde{L}_{C_{\ell} ^{(1)}}
(-\Lambda_0) \oplus \widetilde{L}_{C_{\ell} ^{(1)}} (-2\Lambda_0 +
\Lambda_2).$$
In particular,
\bea
&& \widetilde{L}_{C_{\ell} ^{(1)}} (-\Lambda_0)=
\widetilde{L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0) ^{0}, \quad
  \widetilde{L}_{C_{\ell} ^{(1)}} (-2\Lambda_0 + \Lambda_2)=
\widetilde{L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0) ^{1}. \nonumber
\eea
\end{theorem}
\begin{proof}
Lemma \ref{lem-ext} shows that $\widetilde{L}_{C_{\ell} ^{(1)}}
(-\Lambda_0) \oplus \widetilde{L}_{C_{\ell} ^{(1)}} (-2\Lambda_0 +
\Lambda_2)$ is a vertex subalgebra of $\widetilde{L}_{A_{2 \ell -1}
^{(1)}} (-\Lambda_0)$. But this subalgebra clearly contains
generators of $\widetilde{L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$,
which implies the claim of Theorem.
\end{proof}

The classification of irreducible ordinary $\widetilde{L}_{A_{2 \ell
-1} ^{(1)}} (-\Lambda_0)$--modules follows from the results from
\cite{AP} and similar arguments as in Section \ref{sect.class.C}:

\begin{proposition}
The set

\bea \{ L_{A_{2\ell -1} ^{(1)}}( (-n-1 ) \Lambda_0 + n \Lambda_1) \
\vert \ n \in {\Zp} \}  \cup \{ L_{A_{2\ell -1} ^{(1)}}( (-n-1 )
\Lambda_0 + n \Lambda_{2 \ell -1}) \ \vert \ n \in {\Zp} \}
\nonumber \eea
 provides a complete list of irreducible ordinary modules for the vertex operator algebra
 $\widetilde{L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$.
\end{proposition}

The following result shows that most irreducible $\widetilde{L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$--modules remain irreducible when we restrict them on $\widetilde{L}_{C_{\ell} ^{(1)}} (-\Lambda_0)$.
\begin{theorem}
Assume that $\ell \ge 3$, $n \in {\N}$. Then we have the following isomorphism of $\widetilde{L} _{C_{\ell} ^ {(1)} } (-\Lambda_0)$--modules:

$${L}_{A_{2 \ell -1} ^{(1)}} (-(n+1) \Lambda_0 + n \Lambda_1) \cong L_{C_{\ell} ^{(1)}} (-(n+1) \Lambda_0 + n \Lambda_1), $$

$${L}_{A_{2 \ell -1} ^{(1)}} (-(n+1) \Lambda_0 + n \Lambda_{2\ell -1} ) \cong L_{C_{\ell} ^{(1)}} (-(n+1) \Lambda_0 + n \Lambda_1). $$
\end{theorem}
\begin{proof}
Set $ W= {L}_{A_{2 \ell -1} ^{(1)}} (-(n+1) \Lambda_0 + n
\Lambda_1)$ and let $w$ denote the highest weight vector. Then the
top component $W(0)$ is isomorphic to the irreducible highest weight
${\g}_1$--module $V_{A_{2\ell -1}} (n \omega_1) \cong U(\g_1) w$.
Relation (\ref{decomp-fin2}) implies that

$$ W(0) \subset W_1 =U(\hg). w. $$

As before, denote by $R(0)$ the top component of
$\widetilde{L}_{C_{\ell} ^{(1)}} (-2\Lambda_0 + \Lambda_2)$, and let
$u \in R(0)$, $v \in W(0)$. Using the decomposition of tensor
product $V_{C_{\ell }} (n \omega_1) \otimes V_{C_{\ell }}
(\omega_2)$ from relation (\ref{tens-pr-decomp}), the classification
of $\widetilde{L}_{C_{\ell} ^{(1)}} (-\Lambda_0)$--modules, and
standard fusion rules argument we see that

$$ u_n v \in W_1 \quad (n \in {\Z}). $$

This easily implies that

\bea && \widetilde{L}_{C_{\ell} ^{(1)}} (-2\Lambda_0 + \Lambda_2) .  W   \subseteq W_1  .  \label{rel-pom-1} \eea

Let now $v \in W$.

The irreducibility of $W$ implies that

$$ W= \widetilde{L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0). v = \mbox{span}_{\C} \{ u_n v \ \vert \ u \in \widetilde{L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0) \}. $$
Using relation (\ref{rel-pom-1}) we get

$$ W= \widetilde{L}_{C_{\ell } ^{(1)}} (-\Lambda_0 ) . v = \mbox{span}_{\C} \{ u_n v \ \vert \ u \in \widetilde{L}_{C_{\ell } ^{(1)}} (-\Lambda_0) \}. $$
Therefore, $W$ is irreducible $\widetilde{L}_{C_{\ell } ^{(1)}}
(-\Lambda_0)$--module. The proof of the second isomorphism is
analogous.
\end{proof}

 We believe that vertex operator algebras  $\widetilde{L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$ and  $\widetilde{L}_{C_{\ell} ^{(1)}}
(-\Lambda_0)$ are simple.  But even without these simplicity results we can conclude that
the analogous of Theorem \ref{thm-ext} also holds for simple vertex
operator algebras ${L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$ and
$L_{C_{\ell} ^{(1)}} (-\Lambda_0)$.

\begin{corollary} \label{coro-orbifold} We have:
 $$ L_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0) =  L _{C_{\ell} ^{(1)}}
(-\Lambda_0) \oplus  L _{C_{\ell} ^{(1)}} (-2\Lambda_0 +
\Lambda_2).$$
\end{corollary}
\begin{proof}
First we notice that the automorphism $\theta$ also naturally acts on a simple vertex operator algebra
${L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$.  Theorem \ref{thm-ext} implies that
$${L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0) = V ^0 \oplus V ^1$$
where $V^0$ (resp. $V^1$) is a quotient of $\widetilde{L}_{C_{\ell}
^{(1)}} (-\Lambda_0)$ (resp. $\widetilde{L}_{C_{\ell} ^{(1)}}
(-2\Lambda_0 + \Lambda_2)$). By using the fact that
${\Z}_2$--orbifold components of simple vertex operator algebra are
simple (cf. \cite{DM}), we get that

$V ^0 =  L _{C_{\ell} ^{(1)}} (-\Lambda_0)$ and $V ^1 =L_{C_{\ell}
^{(1)}} (-2\Lambda_0 + \Lambda_2)$. The proof follows.
 \end{proof}

\section{Commutant of $L _{A_1 ^{(1)}} (-\Lambda_0) ^{ \otimes \ell }$ in
$L_{C_{\ell} ^{(1)}} (-\Lambda_0) $ }

We shall now  study vertex operator algebra $L_{C_{\ell} ^{(1)}}
(-\Lambda_0) $. Corollary \ref{coro-orbifold} implies that
$L_{C_{\ell} ^{(1)}} (-\Lambda_0) $ is ${\Z}_2$--orbifold of vertex
operator algebra $L_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$. Clearly,
$L_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$ is a quotient of
$\widetilde{L}_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$ modulo the
maximal submodule of $\widetilde{L}_{A_{2 \ell -1} ^{(1)}}
(-\Lambda_0)$ (which is possibly zero).

As before, we denote

\bea && H^{(i)}= (a_ i ^+ (-\tfrac{1}{2}) a_ i ^{-} (-\tfrac{1}{2})
+ a_{ 2\ell  +1- i}  ^+ (-\tfrac{1}{2}) a_{2\ell +1- i}  ^ - (-
\tfrac{1}{2}) ){\bf 1} , \quad (i = 1, \dots, \ell) \nonumber
\\
&&  H = H^{(1)} +  \cdots + H ^{ (\ell)}. \nonumber \eea

Also, let

\bea
&& \overline{H}^{(i)} =  H ^{(i)} - H ^{(i+1)} \quad (i =1, \dots, \ell -1) \quad \mbox{and} \nonumber \\
&& \overline{\h}_1 = {\C} \overline{H}^{(1)} + \cdots +
{\C}\overline{H} ^{( \ell-1)}. \nonumber \eea

Clearly $\langle \overline{H} ^{(i)} , \overline{H} ^{(j)} \rangle =-4 \delta_{i,j} $.

\begin{theorem}
We have:

$${\rm Com} (   L _{A_1 ^{(1)}} (-\Lambda_0) ^{\otimes \ell}  ,L_{C_{\ell}  ^{(1)}} (-\Lambda_0) )    \cong M_ { \overline{\h}_1}  (1) ^+.  $$
\end{theorem}
\begin{proof}
By using same arguments as in the proof of Proposition \ref{com-bos-higher} we get that:
$$   {\rm Com}  (   L _{A_1 ^{(1)}} (-\Lambda_0) ^{\otimes \ell}  ,\widetilde{ L}_{A_{2 \ell-1}  ^{(1)}} (-\Lambda_0) ) = M_ { \overline{\h}_1}  (1). $$

 Since generators $\overline{H} ^{(i)}$ don't belong to the maximal submodule of  $\widetilde{ L}_{A_{2 \ell-1}  ^{(1)}} (-\Lambda_0) $
  (which is possibly zero) we conclude that
\bea \label{prva-tv} {\rm Com} (   L _{A_1 ^{(1)}} (-\Lambda_0) ^{\otimes \ell}  , L_{A_{2 \ell -1 }  ^{(1)}} (-\Lambda_0) )
\cong M_ { \overline{\h}_1}  (1)  .  \eea

Let $\theta$ be the automorphism of order two of the vertex operator
algebra $L_{A_{2 \ell -1 }  ^{(1)}} (-\Lambda_0) $ as in Corollary
\ref{coro-orbifold}. Then $$\theta ( \overline{H}  ^{(i)} ) = -
\overline{H} ^{(i)} \quad (i = 1, \dots, \ell -1), $$

and we get that  \bea \label{druga-tv} M_ { \overline{\h}_1}  (1) ^+ \subset L_{A_{2 \ell -1 }  ^{(1)}} (-\Lambda_0)  ^0
= L_{C_{\ell}  ^{(1)}} (-\Lambda_0) . \eea
  Now proof of the theorem follows from relations (\ref{prva-tv})-(\ref{druga-tv}).
\end{proof}


TITLE: Writing direct sum of a normed space as $X=\mathcal{N}(T_{\lambda}^{r})\bigoplus T_{\lambda}^{r} (X)$
QUESTION [0 upvotes]: I'm reading about some spectral properties of compact operators and I'm doing some problems about it. There is one that needs next theorem:
Theorem. Let $T: X ~ X$ be a
compact linear operator on a normed space $X$, and let $\lambda\neq 0.$ Let $n=r$ be the smallest integer (depending of $\lambda$) such that
$$\mathcal{N}(T_{\lambda}^{r})=\mathcal{N}(T_{\lambda}^{r+1})=\mathcal{N}(T_{\lambda}^{r+2})=\ldots$$ and $$T_{\lambda}^{r} (X)=T_{\lambda}^{r+1} (X)=T_{\lambda}^{r+2} (X)=\ldots.$$
Then $X$ can be represented in the form $X=\mathcal{N}(T_{\lambda}^{r})\bigoplus T_{\lambda}^{r} (X).$
The problem says this:
Derive the representation of the above theorem in the case of the linear operator
$T: \mathbb{R}^2\rightarrow\mathbb{R}^2$ represented by the matrix
\begin{pmatrix}
1 & -1 \\
-1 & 1
\end{pmatrix}
I'm stuck solving this. I know that $T$ is a linear bounded operator with finite rank, so $T$ is compact with linear form $T(x,y)=(x-y,-y+x).$ I'm tryng to compute $\mathcal{N}(T_{\lambda}^{r})$ and $T_{\lambda}^{r} (X)$ but I don't get any useful.
Any kind of help is thanked in advanced.

REPLY [0 votes]: First show that $T$ has the eigenvalues $0$ and $2$.
Then $T- \lambda I$ is inverible if $0 \ne \lambda \ne 2$, hence $r=0$ and we have
$\mathbb R^2=\mathcal{N}(T_{\lambda}^{0})\bigoplus T_{\lambda}^{0} (X)$ if $0 \ne \lambda \ne 2$.
Now let $ \lambda=0$. Then $T - \lambda I=T$. Show that $T^2=2T$. Therefore $r=1$ , hence $\mathbb R^2=\mathcal{N}(T)\bigoplus T(X)$.
Now let $ \lambda=2$. Then $T - \lambda I=T-2I$. Show that $(T-2I)^2=-2(T-2I)$. Therefore $r=1$ , hence $\mathbb R^2=\mathcal{N}(T-2I)\bigoplus (T-2I)(X)$.


text:
\begin{abstract}
A long-standing problem in the theory of stochastic gradient descent (\sgd) is to prove that its without-replacement version \rsgd converges faster than the usual with-replacement version. We present the \emph{first} (to our knowledge) non-asymptotic solution to this problem, which shows that after a ``reasonable'' number of epochs \rsgd indeed converges faster than \sgd. Specifically, we prove that under strong convexity and second-order smoothness, the sequence generated by \rsgd converges to the optimal solution at the rate $\mathcal{O}\bigl(\frac{1}{T^2} + \frac{n^3}{T^3}\bigr)$, where $n$ is the number of components in the objective, and $T$ is the total number of iterations. This result shows that after a reasonable number of epochs \rsgd is \emph{strictly better} than \sgd (which converges as $\mathcal{O}\bigl(\frac{1}{T}\bigr)$). The key step toward showing this better dependence on $T$ is the introduction of $n$ into the bound; and as our analysis will show, in general a dependence on $n$ is unavoidable without further changes to the algorithm. We show that for sparse data \rsgd has the rate $\mathcal{O}\left(\frac{1}{T^2}\right)$, again strictly better than \sgd. Furthermore, we discuss extensions to nonconvex gradient dominated functions, as well as non-strongly convex settings.
\end{abstract}

\section{Introduction}
We consider stochastic optimization methods for the finite-sum problem
\begin{align}
  F\left(x\right) := \frac{1}{n} \sum_{i=1}^n f_i\left(x\right),\label{1}
\end{align}
where each function $f_i : \reals^d\rightarrow \reals$ is smooth and convex, and the sum $F$ is strongly convex. A classical approach to solving~\eqref{1} is stochastic gradient descent (\sgd). At each iteration \sgd independently samples an index $i$ uniformly from $\{1,\ldots,n\}$, and uses the (stochastic) gradient $\nabla f_i$ to compute its update. The stochasticity makes each iteration of \sgd cheap, and the uniformly independent sampling of $i$ makes $\nabla f_i$ an unbiased estimator of the full gradient $\nabla F$. These properties are central to \sgd's effectiveness in large scale machine learning, and underlie much of its theoretical analysis (see for instance,  \citep{sra2012optimization,rakhlin2012making,bertsekas2011incremental,bottou2016optimization,shalev2014understanding}).

However, what is actually used in practice is the \emph{without replacement} version of \sgd, henceforth called \rsgd. Specifically, at each epoch \rsgd samples a random permutation of the $n$ functions uniformly independently (some implementations shuffle the data only once at load, rather than at each epoch). Then, it iterates over these functions according to the sampled permutation and updates in a manner similar to \sgd. Avoiding the use of random sampling at each iteration, \rsgd can be computationally more practical~\citep{bottou2012stochastic}; furthermore, as one would expect, empirically \rsgd is known to converge faster than \sgd~\citep{bottou2009curiously}.

This discrepancy between theory and practice has been a long-standing problem in the theory of \sgd. It has drawn renewed attention recently, with the goal of better understanding convergence of \rsgd. The key difficulty is that without-replacement leads to statistically non-independent samples, which greatly complicates analysis.  Two extreme case positive results are however available:  \citet{shamir2016without} shows that \rsgd is not much worse than usual \sgd, provided the number of epochs is not too large; while \citet{gurbuzbalaban2015random} show that \rsgd converges faster than \sgd asymptotically at the rate $\mathcal{O}(\frac{1}{T^2})$.

But it remains unclear what happens in between, after a reasonable finite number of epochs are run. This regime is the most compelling one to study, since in practice one runs neither one nor infinitely many epochs. This motivates the central question of our paper:
\begin{center}
  \it Does \rsgd converge faster than \sgd after a reasonable number of epochs?
\end{center}
We answer this question positively in this paper; our results are more precisely summarized below.

\begin{table}
  \caption{\small Comparison of convergence rates of \sgd and \rsgd. The first three functions considered are strongly convex. We omit all the constants from the rate (for details on  constants, please see Section~\ref{discuss}). Under the sparse setting (sparsity level $\rho$), we are not aware of specialized results corresponding to \sgd. For the LP condition functions, see definition in section~\ref{lpcondition}. Here the criterion of LP condition functions is based on suboptimal of function values, while the other three are based on distance to the unique optimal solution.}
  \label{sample-table}
  \centering
  \begin{tabular}{lllll}
    \toprule
  Algorithm & Quadratic & Lipschitz Hessian & Sparse Data & LP Condition \\
    \midrule
  \sgd & $\Oc(1/T)$& $\Oc(1/T)$ & $\Oc(1/T)$ & $\Oc(1/T)$\\
  \addlinespace[0.2em]

  \rsgd & \textcolor{red}{$\Oc(1/T^2 + n^3/T^3)$} &  \textcolor{red}{$\Oc(1/T^2 + n^3/T^3)$}& \textcolor{red}{$\Oc(1/T^2 + \rho^2 n^3 / T^3)$} & \textcolor{red}{$\Oc(1/T^2 + n^3/T^3)$}\\
    \bottomrule
  \end{tabular}
\end{table}

\subsection{Summary of results}
We follow the common practice of reporting convergence rates depending on $T$, the number of calls to the (stochastic / incremental) gradient oracle.
For instance, \sgd converges at the rate $\mathcal{O}(\frac{1}{T})$ for  solving~\eqref{1}, ignoring logarithmic terms in the bound~\citep{rakhlin2012making}. The underlying argument is to view \sgd as stochastic approximation with noise~\citep{nemirov09}, therefore ignoring the finite-sum structure of~\eqref{1}. Our key observation for \rsgd is that one should reasonably include dependence on $n$ into the bound (see Section~\ref{dpn}). Such a compromise leads to a better dependence on $T$, which further shows how \rsgd beats \sgd after a finite number of epochs. Our main contributions are the following:

\begin{list}{{\small$\blacktriangleright$}}{\leftmargin=2em}
\item Under a mild assumption on second order differentiability, and assuming strong-convexity, we establish a convergence rate of $\mathcal{O}\bigl(\frac{1}{T^2} + \frac{n^3}{T^3}\bigr)$ for \rsgd, where $n$ is the number of components in~\eqref{1}, and $T$ is the total number of iterations (Theorem~\ref{thm41} and \ref{thm43}). From the bounds we can calculate the precise number of epochs after which \rsgd is \emph{strictly better} than \sgd.
\item We prove that a dependence on $n$ is necessary for beating the \sgd rate $\Oc(\nf{1}{T})$. This tradeoff precludes the possibility of proving a convergence rate of the type $\mathcal{O}\bigl(\nf{1}{T^{1+\delta}}\bigr)$ with some $\delta>0$ in the general case, and justifies our choice of introducing $n$ into the rate (Theorem~\ref{thm42}).
\item Assuming a sparse data setting common in machine learning, we further improve the convergence rate of \rsgd to $\mathcal{O}\bigl(\nf{1}{T^2}\bigr)$. This rate is \emph{strictly better} than \sgd, indicating \rsgd's advantage in such cases (Theorem~\ref{thm52}).
\item We extend our results to the non-convex function class with Polyak-\L ojasiewicz condition, establishing a similar $\mathcal{O}\bigl(\frac{1}{T^2} + \frac{n^3}{T^3}\bigr)$ rate for \rsgd (Theorem~\ref{thpl}).
\item We show a class of examples where \rsgd is provably faster than \sgd after arbitrary number (even less than one epoch) of iterations (Theorem~\ref{last}).
\end{list}

We provide a detailed discussion of various aspects of our results in Section~\ref{discuss}, including explicit comparisons to \sgd, the role of condition numbers, as well as some limitations. Finally, we end by noting some extensions and open problems in Section~\ref{sec:conc}. As one of the extensions, for non-strongly convex problems, we prove that \rsgd achieves a comparable convergence rate as \sgd, with possibly smaller constant in the bound under certain parameter paradigms (Theorem~\ref{ext1}).

\subsection{Related work}
\citet{Recht2012BeneathTV} conjecture a tantalizing matrix AM-GM inequality that underlies \rsgd's superiority over \sgd. While limited progress on this conjecture has  been reported~\citep{israel2016arithmetic,zhang2014note}, the correctness of the full conjecture is still wide open. With the technique of transductive Rademacher complexity, \citet{shamir2016without} shows that \sgd is not worse than \rsgd provided the number of iterations is not too large. Asymptotic analysis is provided in~\citep{gurbuzbalaban2015random}, which proves that \rsgd limits to a $\mathcal{O}\bigl(\nf{1}{T^2}\bigr)$ rate for large $T$. \citet{ying2018stochastic} show that for a fixed step size, \rsgd converges to a distribution closer to optimal than \sgd asymptotically.

When the functions are visited in a deterministic order (e.g., cyclic), the method turns into Incremental Gradient Descent (\igd), which has a long history~\citep{bertsekas2011incremental}. \citet{kohonen1974adaptive} shows that \igd converges to a limit cycle under constant step size and quadratic functions.  Convergence to neighborhood of optimality for more general functions is studied in several works, under the assumption that step size is bounded away from zero (see for instance~\citep{solodov1998incremental}). With properly diminishing step size, \citet{nedic2001convergence} show that an $\mathcal{O}\bigl(\nf{1}{\sqrt{T}}\bigr)$ convergence rate in terms of distance to optimal can be achieved under strong convexity of the finite-sum. This rate is further improved in \citep{gurbuzbalaban2015convergence} to $\mathcal{O}(\nf{1}{T})$ under a second order differentiability assumption.

In the real world, \rsgd has been proposed as a standard heuristic~\citep{bottou2012stochastic}. With numerical experiments, \citet{bottou2009curiously} notices an approximately $\mathcal{O}(\nf{1}{T^2})$ convergence rate of \rsgd. Without-replacement sampling also improves data-access efficiency in distributed settings, see for instance~\citep{feng2012towards, lee2015distributed}. The permutation-sampling idea has been further embedded into more complicated algorithms; see~\citep{de2016efficient,defazio2014finito,shamir2016without} for variance-reduced methods, and \citep{shalev2013stochastic} for decomposition methods.

Finally, we note a related body of work on coordinate descent, where a similar problem has been studied: \emph{when does random permutation over coordinates behave well?} \citet{Grbzbalaban2017WhenCC} give two kinds of quadratic problems when cyclic version of coordinate descent beats the with replacement one, which is a stronger result indicating that random permutation also beats the with replacement method. However, such a deterministic version of the algorithm suffers from poor worst case. Indeed, in \citep{sun2016worst} a setting is analyzed where cyclic coordinate descent can be dramatically worse than both with-replacement and random permutation versions of coordinate descent. \citet{lee2016random} further study this setting, and analyze how the random permutation version of coordinate descent avoids the slow convergence of cyclic version. In~\citep{wright2017analyzing}, Wright et el.\ propose a more general class of quadratic functions where random permutation outperforms cyclic coordinate descent.

config: arxiv


TITLE: Decidability of transcendental numbers
QUESTION [13 upvotes]: I have a question, whose answer is probably well known, but I can't seem to find anything meaningful after a bit of searching, so I would appreciate some help.
My question is whether it is known that deciding whether a number is transcendental is undecidable.
Possibly, one assumes as input, say a program that returns the i^th bit of the number.
Thanks in advance for any pointers.

REPLY [11 votes]: The subset of transcedental numbers is not decidable. We assume here that reals are represented in a standard way, so that we can compute limits of sequences of reals which are computably Cauchy.
Recall that a sequence $(a_n)_n$ is computably Cauchy if there is a computable map $f$ such that, given any $k$ we have $|a_{m} - a_{n}| < 2^{-k}$ for all $m, n \geq f(k)$. The standard representations of reals are like that, for example the one where a real is represented by a machine that computes an arbitrarily good rational approximation. (We can also speak in terms of computing digits, but then we have to allow negative digits. This is a well known issue in computability theory of the reals.)

Theorem: Suppose $S \subseteq \mathbb{R}$ is a subset such that there exists a computable sequence $(a_n)_n$ which is computably Cauchy and its limit $x = \lim_n a_n$ is outside $S$. Then the question "is a real number $x$ an element of $S$" is undecidable.

Proof.
Suppose $S$ were decidable. Given any Turing machine $T$, consider the sequence $b_n$ defined as
$$b_n = \begin{cases}
a_n & \text{if $T$ has not halted in the first $n$ steps,}\\\\
a_m & \text{if $T$ has halted in step $m$ and $m \leq n$.}
\end{cases}$$
It is easy to check that $b_n$ is computably Cauchy, therefore we can compute its limit $y = \lim_n b_n$. Now we have $y \in S$ iff $T$ halts, so we can solve the Halting Problem. QED.
There is a dual theorem in which we assume the sequence is outside $S$ but its limit is in $S$.
Examples of sets $S$ satisfying these conditions are: an open interval, a closed interval, the negative numbers, the singleton $\lbrace 0 \rbrace$, rational numbers, irrational numbers, transcedental numbers, algebraic numbers, etc.
A set which does not satisfy the conditions of the theorem is $S = \lbrace q + \alpha \mid q \in \mathbb{Q}\rbrace$ of rational numbers translated by a non-computable number $\alpha$. Exercise: is $S$ decidable?

REPLY [1 votes]: The set of transcendentals is not open in $\mathbf R$ (in particular, it is dense and codense in $\mathbf R$. Hence it is undecidable.