%Paper: hep-th/9311139
%From: Hiroshi Ishikawa <ishikawa@hep1.c.u-tokyo.ac.jp>
%Date: Tue, 23 Nov 93 13:25:45 +0900

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%  Note on N=0 string as N=1 string  (H.Ishikawa and M.Kato)             %
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%       1993.11.22                                                       %
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\def\del{\partial}
\def\pbar{\bar p}
\def\xbar{\bar x}
\def\unit{{\bf 1}}
\def\picunit#1{{\bf 1}^{#1}}
\def\dualunit#1{{\bf 1}_{*}^{#1}}
\def\singular{\mbox{\bf S}}
\def\ie{{\em i.e., }}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\beqa{\begin{eqnarray}}
\def\eeqa{\end{eqnarray}}
\def\vac#1{|#1\rangle}
\def\upvac#1{|#1, \uparrow \rangle}
\def\fock#1{{\cal F}_#1}
%------------------------------------------------------------------------%

\begin{titlepage}

    \begin{normalsize}
     \begin{flushright}
                 UT-Komaba/93-23 \\
                 November 1993
     \end{flushright}
    \end{normalsize}
    \begin{LARGE}
       \vspace{1cm}
       \begin{center}
         Note on $N=0$ string as $N=1$ string \\
       \end{center}
    \end{LARGE}

  \vspace{5mm}

\begin{center}
           Hiroshi I{\sc shikawa}
           \footnote{E-mail address:
              ishikawa@hep1.c.u-tokyo.ac.jp}
                \ \  and \ \
           Mitsuhiro K{\sc ato}
           \footnote{E-mail address:
              katom@tkyvax.phys.s.u-tokyo.ac.jp} \\
      \vspace{4mm}
        {\it Institute of Physics, College of Arts and Sciences} \\
        {\it University of Tokyo, Komaba}\\
        {\it Meguro-ku, Tokyo 153, Japan}\\
      \vspace{1cm}

    \begin{large} ABSTRACT \end{large}
        \par
\end{center}
\begin{quote}
 \begin{normalsize}
\ \ \ \
A similarity transformation, which brings a particular class of the
$N=1$ string to the $N=0$ one, is explicitly constructed. It enables
us to give a simple proof for the argument recently proposed by
Berkovits and Vafa.
The $N=1$ BRST operator is turned into the direct sum of
the corresponding $N=0$ BRST operator and that for an additional topological
sector. As a result, the physical
spectrum of these $N=1$ vacua is shown to be isomorphic to
the tensor product of the $N=0$ spectrum and the topological sector
which consists of only the vacuum.
This transformation manifestly keeps the operator algebra.

 \end{normalsize}
\end{quote}


\end{titlepage}
\vfil\eject


% Introduction

The aim of this short note is to give a simple proof to the argument
initiated by Berkovits and Vafa\cite{BV} that the $N=0$ (bosonic)
string can be regarded as a special case of the $N=1$ (fermionic)
string.

The idea of ref.~\cite{BV} is the following. For any given $c=26$ matter
with its energy-momentum tensor $T_m(z)$, we have $c=15$ ($\hat c=10$) system
by coupling spin $(3/2,-1/2)$ fermionic ghosts which we denote as $(b_1, c_1)$.
Then this system can be considered as a matter system of critical
$N=1$ fermionic string because of the existence of $N=1$ super
conformal algebra in the system:
\beqa\label{N=1op}
  T_{N=1} &=& T_m - \frac{3}{2} b_1 \del c_1 - \frac{1}{2} \del b_1 \, c_1
        + \frac{1}{2} \del^2 (c_1 \del c_1) \,,\\
  G_{N=1} &=& b_1 + c_1 (T_m + \del c_1 \, b_1) + \frac{5}{2} \del^2 c_1
  \,,
\eeqa
which satisfy
\beqa
  T_{N=1}(z) T_{N=1}(w) & \sim & \frac{15/2}{(z-w)^4}
        + \frac{2}{(z-w)^2} T_{N=1}(w) + \frac{1}{z-w} \del T_{N=1}(w) \,,\\
  T_{N=1}(z) G_{N=1}(w) & \sim & \frac{3/2}{(z-w)^2} G_{N=1}(w)
                        + \frac{1}{z-w} \del G_{N=1}(w) \,,\\
  G_{N=1}(z) G_{N=1}(w) & \sim & \frac{10}{(z-w)^3} +
       \frac{2}{z-w} T_{N=1}(w)  \,.
\eeqa
Tensoring spin $(2,-1)$ reparametrization ghosts $(b,c)$ and spin
$(3/2,-1/2)$ super ghosts $(\beta , \gamma)$ to the above, we can
construct whole Fock space of $N=1$ fermionic string
\begin{equation}\label{N=1space}
  \fock{{N=1}} = (T_m) \bigotimes (b_1, c_1) \bigotimes (b, c) \bigotimes
  (\beta, \gamma)
\,.
\end{equation}
Now their claim is that this $N=1$ string\footnote[1]{Hereafter we
  mean this special class of $N=1$ string by simply saying $N=1$
  string.} is equivalent to the $N=0$ bosonic string with the matter
$T_m(z)$ after integrating out the $(b_1,c_1)$ and $(\beta , \gamma)$.
The equivalence stands not only for the reduction of the Fock space
(\ref{N=1space}) to the $N=0$ Fock space
\begin{equation}
  \fock{{N=0}} = (T_m) \bigotimes (b, c)
\,,
\end{equation}
but also for the one to one correspondence between the physical
observables of each theory such as scattering amplitude.

The authors of ref.~\cite{BV} gave a plausible argument for how $N=1$
amplitude is reduced to the corresponding one in the $N=0$ for the
case of the physical vertex made from the spin 1 operator of the
matter in $T_m(z)$. But generic physical vertex operators are
expressed as mixed form of matter and ghosts, {\it e.g.} ground ring
operators in the two dimensional string. So more general proof is
desired to establish the equivalence.

Ref.~\cite{F} is an attempt toward this end, but is still insufficient
as the author himself admitted. In particular, it is not clear whether
the operator product algebras of the physical vertex operators in
$N=1$ and $N=0$ are isomorphic each other, which is must for the
coincidence of amplitude.

In the following we will construct a simple similarity transformation
which maps the $N=1$ theory to the tensor product of the $N=0$ theory and a
trivial topological
sector whose physical state is only its vacuum. The $N=1$ BRST charge
is mapped to the direct sum of the one in the $N=0$ theory and the one
in the topological sector, each of which acts on its own sector
exclusively. Therefore the relation between the BRST cohomology of the
$N=1$ and that of the $N=0$ is very clear as will be shown.
Moreover, algebra isomorphism is manifest due to the similarity
transformation. This is exactly the same technique used in
ref.~\cite{IK}
in which present authors have shown that the equivalence of BRST
cohomologies of $SL(2,R)/U(1)$ black hole and $c=1$ string.


% Main part
Let us start with the definition of the BRST charge for the $N=1$
string obtained from $c=26$ matter in the above:
\begin{eqnarray}
  Q_{N=1} &=& \oint\! dz \left(
        c T_{N=1} - \frac{1}{2} \gamma G_{N=1} + b c \del c
                - \frac{1}{4} b \gamma^2
        + \frac{1}{2} \del c \, \beta \gamma - c \beta \del \gamma
                \right) \nonumber \\
          &=&   \oint\! dz
        \left[
         \left(  c - \frac{1}{2} \gamma c_1 \right) T_m
        + c \left( - \frac{3}{2} b_1 \del c_1 - \frac{1}{2} \del b_1 \, c_1
        + \frac{1}{2} \del^2 (c_1 \del c_1) \right) \right. \nonumber \\
           & &\left.
        -\frac{1}{2} \gamma \left(
                b_1 + b_1 c_1 \del c_1 + \frac{5}{2} \del^2 c_1
        \right)
        + b c \del c - \frac{1}{4} b \gamma^2
        + \frac{1}{2} \del c \, \beta \gamma - c \beta \del \gamma\,
        \right]
\,.
\end{eqnarray}
The physical states of the $N=1$ string are defined as $Q_{N=1}$ cohomology.
Due to the coupled terms of the field in $\fock{{N=0}}$ and the fields
in the topological sector
$\fock{{top}} = (b_1, c_1) \bigotimes (\beta, \gamma)$,
it is not so trivial to show that the $Q_{N=1}$
cohomology is reduced to the $N=0$ one by means of this form of
$Q_{N=1}$.

Our main result is the following similarity transformation which maps
$Q_{N=1}$ into very simple form:
\begin{equation}\label{transf}
  e^R Q_{N=1} e^{-R} = Q_{N=0} + Q_{top} \,,
\end{equation}
where
\begin{equation}
    R = \oint\! dz\, c_1 \left(
        \frac{1}{2} \gamma b - 3 \del c \, \beta - 2 c \del \beta
        - \frac{1}{2} \del c_1 c b
        + \frac{1}{4} \beta \gamma \del c_1 \right) \,,
\end{equation}
and
\begin{equation}
  Q_{N=0} = \oint\! dz \left( c T_m + b c \del c \right) \,,
\end{equation}
\begin{equation}
  Q_{top} = \oint\! dz \left( -\frac{1}{2} b_1 \gamma \right) \,.
\end{equation}
$Q_{N=0}$ is the BRST operator in the $N=0$ string which acts
only on $\fock{{N=0}}$ and $Q_{top}$ is the one in the topological sector
which also acts only on its own sector $\fock{{top}}$.
Now, with this form of the BRST operator, it is clear that $Q_{N=1}$
cohomology is direct product of $Q_{N=0}$ cohomology and $Q_{top}$
cohomology.
And $Q_{top}$ cohomology only consists of the vacuum in $\fock{{top}}$,
because $b_1$, $\gamma$ are $Q_{top}$-exact and $c_1$, $\beta$ are not
$Q_{top}$-closed so that all the states in $\fock{{top}}$ but the
vacuum are unphysical or $Q_{top}$-exact. Thus we obtain one to one
correspondence between the $Q_{N=1}$ and $Q_{N=0}$ cohomologies. Since
(\ref{transf}) is similarity transformation, it manifestly keeps the
operator algebra. As a result, any correlation functions in the $N=1$
theory are reduced to the corresponding ones in the $N=0$ theory.
These establish the equivalence of $N=0$ string and a special class of
$N=1$ string constructed from $\fock{{N=1}}$.

It may be helpful to see how $N=1$ physical vertex operators are
mapped to corresponding ones in $N=0$ theory in our transformation.
For example, a vertex operator in the ghost number one picture
treated in \cite{BV}, which is expressed as
$c V - {1 \over 2}\gamma c_1 V$ in our notation, is mapped as
\begin{equation}
  e^R (c V - {1 \over 2}\gamma c_1 V) e^{-R} = c V \,.
\end{equation}
The right hand side is desired form in the $N=0$ string. On the other
hand, the ghost number zero picture operator is invariant under the
transformation, so that n-point amplitude on a sphere satisfies
\begin{eqnarray}
&&\langle(c\ e^{-\phi}c_1V_1)  (c\ e^{-\phi}c_1V_2)
  (c\ V_3 - {1 \over 2}\gamma c_1 V_3)
  \int V_4 ...\int V_n \rangle_{\fock{{N=1}}} \nonumber \\
&&= \langle (c\ V_1)  (c\ V_2) (c\ V_3) \int V_4 ...\int V_n
  \rangle_{\fock{{N=0}}} \times \langle (e^{-\phi}c_1)(e^{-\phi}c_1)
  \rangle_{\fock{{top}}} \,.
\end{eqnarray}
The second factor in the right hand side is just a trivial factor
which is usually taken to be one. These argument, of course, can be
apparently extended to any vertex operators in any pictures using our
transformation.


% Discussion

Some remarks are in order.

(a) As is seen from eq.(\ref{N=1op}), the energy-momentum tensor of the
$N=1$ theory has a non-standard form for the $(b_1, c_1)$-sector; modified by
a total derivative term made of $c_1$. However, this term
disappears after the transformation and the energy-momentum tensor of
$N=0$ and topological sector turn into the standard ones
\beq
  e^R T_{total} e^{-R} = T_m - 2 b \del c - \del b \, c
        - \frac{3}{2} b_1 \del c_1 - \frac{1}{2} \del b_1 \, c_1
        - \frac{3}{2} \beta \del \gamma - \frac{1}{2} \del \beta \, \gamma\, ,
\eeq
where we denote the total energy-momentum tensor of the $N=1$ string as
$T_{total}$.

(b) After the transformation by $R$, the physical states take the
form $|phys \rangle_{N=0} \bigotimes |0 \rangle_{top} $.
Since $R$ has only ($b_1,c_1,\beta,\gamma$)-ghost number even
terms, the transformation does not change even-odd property with
respect to this ghost number (let us denote it by $N_F$).
Therefore, it is obvious from $R$-transformed expression that all the
physical states are $N_F$ even.

(c) GSO projection can be defined with this $N_F$ in the $N=1$
theory.
The physical spectrum consists of $N_F$ even states
only, as is seen from the above. So, the physical contents of the theory would
be empty if we choose the $N_F$ odd sector in the projection.
In the Ramond sector, similarly, the sector with the same $N_F$ parity
as the ground state
has to be singled out.


% Acknowledgements
\vspace{0.5in}
\section*{Acknowledgements}
We have been much benefitted from {\tt ope.math} the package for the
operator product expansion developed by A.~Fujitsu.
The research of M.K. is supported in part by
the Grant-in-Aid for Encouragement of Young Scientists (\# 05740168)
and by that for Scientific Research on Priority Areas ``Infinite Analysis''
(\# 05230011) from the Ministry of Education, Science and Culture.


%\newpage
\begin{thebibliography}{99}

\bibitem{BV} N.~Berkovits and C.~Vafa,
{\em On the Uniqueness of String Theory},
preprint HUTP-93/A031, KCL-TH-93-13, hep-th/9310170 (October 1993).

\bibitem{F} J.~M.~Figueroa-O'Farrill,
{\em On the Universal String Theory},
preprint QMW-PH-93-29, hep-th/9310200 (October 1993).

\bibitem{IK} H.~Ishikawa and M.~Kato, Phys.~Lett. {\bf B302} (1993) 209.

\end{thebibliography}

\end{document}

