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\vspace*{6cm}

\begin{center}
{\bf{A NOTE ON CLOSED STRINGS}}
\end{center}

\begin{center}
Shervgi S. Shahverdiyev$^*$
\end{center}

\vspace{0.5cm}



\begin{quote}
\noindent
Special kind of closed strings is considered. It is shown that these closed strings
behave as two (an even number of) open strings at the classical level and almost one open string 
at the quantum level. They contain photons in their spectrum and can lie on D branes. 
Some properties of closed string field effective action are declared.
\end{quote}

\vspace{1cm}
\noindent
Keywords:D--Branes, closed strings, open strings, closed string field effective action 

\noindent
PACS: 11.25.-w,11.25.Sq   
\vfill

$^*$882 N Mentor Ave Pasadena CA 91104

e-mail:shervgi@yahoo.com
%,Phone:(626)4059406

\newpage
\section{Introduction}
In this paper two problems in string theory are considered.
One of them is the presence of open strings in the type II closed string theories.  
The second is the formulation of effective action for closed string fields.

With the appearance of D -- Branes \cite{p} in string theory, the open strings ending on them were added   
to type II theories. Natural appearance of open strings in the theory of closed strings remains unclear, to my best knowledge. 

More progress has been done in understanding the role of open string tachyon field both in the framework of Cubic \cite{1} and Boundary \cite{2}-\cite{4} Open String Field theories (see \cite{6}-\cite{Tsey} and references therein). However, we do not know much about effective action for closed string fields. 

In the present paper, we consider a subset of closed strings and show that they  behave as two (an even number of) open strings at the classical level. After quantization these closed strings contain photons in their spectrum. 
Massive fields in their spectrum have masses twice as much as the masses of fields in the spectrum of an open string. Therefore these closed strings behave almost as one open string at the quantum level. We apply this result to the effective action for closed string fields.


In the next section  we give a new look to closed strings and show that there are closed strings
that behave as two open strings.
In section 3, we  quantize the closed strings in question and 
show that they contain photons in their  spectrum. Also, we investigate 
properties of closed string spectrum and find some properties of closed string field effective action.  
Summary of results is given in section 4. 
In section 5 we make two remarks and give solutions that represent closed strings behaving as an even number of open strings at the classical level and almost one open string at the quantum level.

 


\section{Closed-open string relationship}
The action for string 
is
\begin{equation}\label{a}
S =-\frac{1}{4\pi\alpha}\int d \tau d \sigma 
(-\gamma)^{1/2}\gamma^{ab}\partial_a X^\mu\partial_b X^\nu
\eta_{\mu\nu}
\end{equation}
and 
$$
\delta S =
\frac{1}{2\pi\alpha}\int d \tau d \sigma \partial_a \{ 
(-\gamma)^{1/2}\gamma^{ab}\partial_b X_\mu\}\delta X^\mu - 
\frac{1}{2\pi\alpha}\int d \tau \partial_{\sigma} X_\mu\ \delta X^\mu|_{\sigma=0}^{\sigma=\pi}
$$
so we have the following boundary condition \cite{GSW}-\cite{12} 
$$
\partial_{\sigma} X_\mu\ \delta X^\mu(\tau, 0)-
\partial_{\sigma} X_\mu\ \delta X^\mu(\tau, \pi) =0.
$$
One of the solutions to this equation (for closed string) is
\begin{equation}\label{b}
\partial_{\sigma} X^\mu(\tau, 0)=\partial_{\sigma}X^\mu(\tau, \pi),\quad 
X^{\mu}(\tau, 0)=X^{\mu}(\tau,\pi).
\end{equation}

Solution to the equation of motion obtained from action (\ref{a}) in the conformal gauge with (\ref{b}) 
can be represented in the form
\begin{equation}\label{d}
X^{\mu }=x^{\mu }+2\alpha ^{\prime }p^{\mu }\tau +i(\alpha ^{\prime}/2)^{1/2}
\sum_{n\neq 0}\frac{1}{n}\left( \alpha _{n}^{\mu }e^{2in(\sigma
-\tau )}+{\tilde{\alpha}}_{n}^{\mu }e^{-2in(\sigma +\tau )}\right). 
\end{equation}

We choose the following subset of conditions (\ref{b}). 
\begin{equation}\label{c}
\partial_{\sigma} X^\mu(\tau, 0)=\partial_{\sigma}X^\mu(\tau, \pi) =0,\quad
X^{\mu}(\tau, 0)=X^{\mu}(\tau,\pi).
\end{equation}
The condition
$
\partial_{\sigma} X^\mu(\tau, 0) =0
$
gives us that 
${\tilde{\alpha}}_{n}^{\mu }= \alpha _{n}^{\mu }$
and this leads to the solution. (in order to stress that these solutions represent closed strings we 
denote them by $ X^{\mu}_{closed}$) 
 
\begin {equation}\label{1}
X^{\mu}_{closed}=x^{
\mu}+ 2\alpha^{\prime}p^{\mu}\tau+ i(2\alpha^\prime)^{1/2}\sum_{n \ne 0}
\frac{1}{n}\alpha^{\mu}_n e^{-2in\tau}cos(2n\sigma).
\end {equation}
We see that this solution satisfies the condition
$$
 \partial_\sigma X^{\mu}_{closed}
(\tau,\pi/2)=0.
$$
Note that this condition is satisfied at point $\sigma =\pi/2$ and (\ref{1}) also satisfies
$$
X^{\mu}_{closed}(\tau, 0)=X^{\mu}_{closed}(\tau,\pi)
$$
which means that string is closed and behaves as two open strings because Neumann boundary conditions are satisfied in two different 
points $\sigma=0$ and $\sigma=\pi/2$.











\section{On closed string field effective action}
In this section we quantize  closed strings represented by (\ref{1}) and investigate spectrum of closed strings
represented by (\ref{d}).

To quantize (\ref{1}) we impose 
$$
[X^\mu_{closed}(\tau, \sigma), \Pi^\nu_{closed}(\tau, \sigma^\prime)]=i\eta^{\mu\nu}\delta(\sigma-\sigma^\prime),\quad 
\Pi^\nu_{closed}=\frac{1}{2\pi\alpha^\prime}\dot{X}^\nu_{closed}
$$
which leads to the following commutation relations on the oscillators and the zero modes
$$
[x^\mu, p^\nu]=2i\eta^{\mu\nu},\quad [\alpha_n^\mu, \alpha _m^\nu]=n\delta_{n+m}\eta^{\mu\nu}.
$$
From  equation 
$$
(L_0-1)|\psi_{phys}>=0, \quad
L_0=\frac{\alpha^\prime}{4}p^2+\sum_{n=1}^{\infty}
\alpha_{-n}\alpha_{n}+ \frac{D}{2}\sum_{n=1}^{\infty}n 
$$
we can read the expression for the mass operator
$$
{\bf M}^2=\frac{4}{\alpha^\prime}\left(\sum_{n=1}^{\infty}\alpha_{-n}%
\alpha_{n} -1\right), 
$$
As it is seen closed strings (\ref{1}) contain massless vector fields $ A_\mu\sim \alpha_{-1}^\mu|0,k>$, photons, in their spectrum.  
We see that they behave as two open strings at the classical level and almost one at the quantum level, because the masses of fields are as twice as much as the ones in the spectrum of an open string. 


Now, we assume that closed strings (\ref{d}) can transform to closed strings (\ref{1}).
Let us show that this corresponds to process ${\tilde{\alpha}}_{n}^{\mu }\to \alpha _{n}^{\mu }$. 
The massless spectrum of closed strings (\ref{d}) is
$$
G_{\mu\nu} \sim (\alpha_{-1}^\mu\tilde{\alpha}_{-1}^\nu+\alpha_{-1}^\nu
\tilde{\alpha}_{-1}^\mu)|0,k> , \quad M^2(G_{\mu\nu})=0,
$$
$$
B_{\mu\nu} \sim (\alpha_{-1}^\mu\tilde{\alpha}_{-1}^\nu-\alpha_{-1}^\nu%
\tilde{\alpha}_{-1}^\mu)|0,k> , \quad M^2(B_{\mu\nu})=0,
$$
$$
\Phi \sim \alpha_{-1}^\mu\tilde{\alpha}_{-1}^\mu|0,k> , \quad M^2(\Phi)=0.
$$
After the limit ${\tilde{\alpha}}_{n}^{\mu }\to \alpha _{n}^{\mu }$ the field content changes. Fields $G_{\mu\nu}$
and $\Phi $ became massive and new field $ A_\mu$ appears.
$$
M^2=\frac{2}{\alpha^\prime}\left(\sum_{n=1}^{\infty}\left(
\alpha_{-n}\alpha_{n}+ \tilde{\alpha}_{-n}\tilde{\alpha}_{n}\right)-2\right)
\to {\bf M}^2=\frac{4}{\alpha^\prime}\left(\sum_{n=1}^{\infty}\alpha_{-n}%
\alpha_{n} -1\right), 
$$
$$
G_{\mu\nu} \to {\bf G_{\mu\nu}} \sim \alpha_{-1}^\mu\alpha_{-1}^\nu|0,k> , \quad {\bf M}^2({\bf G_{\mu\nu})}= \frac{4}{\alpha^\prime},
$$
$$
B_{\mu\nu} \to 0, 
$$
$$
\Phi \to {\bf \Phi} \sim \alpha_{-1}^\mu\alpha_{-1}^\mu|0,k>, \quad {\bf M}^2({\bf \Phi)}= \frac{4}{\alpha^\prime},
$$
$$
A_\mu\sim \alpha_{-1}^\mu|0,k>, \quad {\bf M}^2(A_\mu)=0.
$$
We see that as ${\tilde{\alpha}}_{n}^{\mu }\to \alpha _{n}^{\mu }$ spectrum of closed strings (\ref{d}) coincides with the spectrum of 
closed strings (\ref{1}).
According to this result we can state that the 
limit ${\tilde{\alpha}}_{n}^{\mu }\to \alpha _{n}^{\mu }$ corresponds, in the language of fields,  to 
\begin{equation}\label{eq}
S (G_{\mu\nu}, B_{\mu\nu}, \Phi, ...)\to {\bf S} ({\bf G_{\mu\nu}},
A_\mu, {\bf {\Phi}}, ...),
\end{equation}
where $S$ is the closed string field action.
This means that there is mechanism that can happen, and then the massless fields became massive and
new  massless vector field appears. This resembles Higgs mechanism. 
More about all of the above mentioned see \cite{10}.

 



\section{Summary of results}
The main idea in this paper is that the open strings attached to D--Branes (in type II theories) do not satisfy the condition
$
X^{\mu}(\tau, 0)=X^{\mu}(\tau,\pi).
$
This condition should be satisfied because it is assumed from the beginning of the type II theories.
We suggest the way of avoiding this contradiction.

Closed strings  (\ref{1}) 

{{\bf{i)}} behave as two open strings at the classical level  because they satisfy 
open string boundary conditions in two different points $\sigma=0$ and $\sigma =\pi/2$,

{\bf{ii)}} can lie on D-branes, because  after T duality on i-th direction  Neumann boundary condition becomes Dirichlet boundary condition on i-th direction. Therefore, two points of closed strings (\ref{1}),
$X^i_{closed}(\tau, \sigma=0)$ and $X^i_{closed}(\tau, \sigma=\pi/2)$,
 can lie on D--Branes,

{\bf{iii)}}} contain photons in their spectrum and behave almost as one open string at the quantum level.


The next result is  the following: if there is transformation between closed strings (\ref{d})
and closed strings  (\ref{1})(this corresponds to the limit 
${\tilde{\alpha}}_{n}^{\mu }\to \alpha _{n}^{\mu }$ in (\ref{d})) 
then the closed string field effective action has  property (\ref{eq}).




\section{Remarks}
It is known that two or more open strings can join to form a closed string. However, there is no evidence that 
the resulting closed string will behave as two or more open strings. Moreover, even and odd number of open strings 
can join to form a closed string. 
In contrast, there are solutions to the equation of motion with (\ref{c})
\begin {equation}\label{y}
X^{\prime\mu}_{closed}=x^{
\mu}+ 2\alpha^{\prime}p^{\mu}\tau+ i(2\alpha^\prime)^{1/2}\sum_{n \ne 0}
\frac{1}{n}\alpha^{\mu}_n e^{-2kin\tau}cos(2kn\sigma),\quad k\subset N
\end {equation}
which obey Neumann boundary conditions in 2k points 
$$
\sigma=(0, \pi/2k, \pi/k, 3\pi/2k, ..., \pi).
$$
Hence, (\ref{y}) presents closed string behaving as 2k number of open strings. Therefore, there are no closed strings 
behaving as an odd number of open strings. (\ref{1}) is (\ref{y}) with $k=1$.

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\end{thebibliography}
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