%Paper: hep-th/9508100
%From: lechtenf@itp.uni-hannover.de
%Date: Mon, 21 Aug 1995 18:12:49 +0200

%
% 	CONTRIBUTION TO PROCEEDINGS OF ``STRINGS '95''
%
%	``On the BRST Cohomology of N=2 Strings''
%			by
%	Olaf Lechtenfeld, Hannover University, Germany
%
%	10 min talk given on Wednesday, 03/15/95, 4:30 p.m.
%

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\begin{document}

\centerline{\normalsize{\bf
ON THE BRST COHOMOLOGY OF $N{=}2$ STRINGS}~\footnote{
Supported in part by the `Deutsche Forschungsgemeinschaft'} }
\vspace*{0.4cm}
\centerline{\footnotesize OLAF LECHTENFELD}
\baselineskip=13pt
\centerline{\footnotesize\it
Institut f\"ur Theoretische Physik, Universit\"at Hannover }
\baselineskip=12pt
\centerline{\footnotesize\it
D--30167 Hannover, Germany }
\centerline{\footnotesize E-mail: lechtenf@itp.uni-hannover.de}
\vspace*{0.6cm}
\abstracts{
We analyze the BRST cohomology of the critical $N{=}2$ NSR string
using chiral bosonization. Picture--changing and spectral flow is
made explicit in a holomorphic field basis. The integration of
fermionic and $U(1)$ moduli is performed and yields picture-- and
$U(1)$ ghost number--changing insertions into the string measure for
$n$-point amplitudes at arbitrary genus and $U(1)$ instanton number.}

\vspace*{0.5cm}
\normalsize\baselineskip=15pt
\setcounter{footnote}{0}
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Strings with two world-sheet supersymmetries have been around for
almost 20 years,~\cite{marcus}
with recently renewed interest.~\cite{new}
The gauge-invariant $N{=}2$ string world-sheet action is given by coupling
$N{=}2$ supergravity to two complex $N{=}2$ scalar matter multiplets
$(X^\m,\j^\m)$, $\m{=}0,1$.
Superconformal gauge fixing produces conformal ghosts ($b,c$),
complex $N{=}2$ superconformal ghosts ($\b,\g$), and real $U(1)$ ghosts
($\tilde b,\tilde c$).
The $N{=}2$ superconformal algebra is generated by the currents
$T_{\rm tot}$, $G^\pm_{\rm tot}$, and $J_{\rm tot}$.

In contrast to the $N{=}1$ string, chiral bosonization of $\j^\m$ and $\b,\g$
depends on the field basis. In the `real basis', one bosonizes real and
imaginary parts of the complex NSR fermions and their ghosts, whereas
holomorphic and antiholomorphic combinations are taken in the `holomorphic
basis'. If not indicated otherwise, we shall work in the holomorphic basis
which has the advantage of diagonalizing the $U(1)$ symmetry. The $U(1)$
charges appear as $\pm$ superscripts.  In this way,
$\j^{\pm\m}$ and $\b^\pm,\g^\pm$ are replaced by two pairs of bosons,
$\f^\pm$ and $\vf^\pm$, plus two auxiliary fermion systems ($\h^\pm,\x^\pm$),
spanning an extended Fock space containing ${\bf Z}{\times}{\bf Z}$
copies of the original fermionic one.
It is easy to see that BRST non-trivial operators must have vanishing conformal
dimension and $U(1)$ charge. Further grading of the cohomology is effected
by the mass level, the total ghost number~$u\in{\bf Z}$,
and two picture numbers~$\p^\pm\in\ha{\bf Z}$, with $\p^+{+}\p^-\in{\bf Z}$.
Integral and half-integral picture numbers correspond to NS and R states,
respectively.
For generic momenta, we find\cite{bkl} that the BRST cohomology on the massless
level consists of {\it four\/} classes of states for each pair $(\p^+,\p^-)$,
labelled by $v{\equiv}u{-}\p^+{-}\p^-\in\{1,2,2,3\}$ and created by
vertex operators of type $c$, $\tilde c$, $c\pa c$ and $\tilde c c\pa c$.

Physical states correspond to {\it classes\/} of BRST cohomology classes,
formed under the following four equivalence relations.
First, $c$-- and $c\pa c$--type vertices are to be identified just as in the
bosonic string.
Second, $\tilde c$--type vertex operators get converted to others by applying
the $U(1)$ ghost number--changing operator~$Z^0$.
Third, two picture--changing operators~$Z^\pm$ raise the picture numbers
of vertex operators by unit amounts.
Fourth, NS and R states are connected by the action of the spectral-flow
operators~$S\!F\!O^\pm$ which move $(\p^+,\p^-)\to(\p^+{\pm}\ha,\p^-{\mp}\ha)$.
These maps are given by
$$
Z^0 = \oint\!\tilde b\ \d\Bigl(\oint\! J_{\rm tot}\Bigr) \quad,\quad
Z^\pm(z) = \d(\b^\pm)\ G^\pm_{\rm tot} \quad,\quad
S\!F\!O^\pm(z) = \exp\{\pm\ha\int^z\! J_{\rm tot}\} \quad,
\eqno(1) $$
and commute with $Q_{\rm BRST}$ but are non-trivial.
In this fashion, each physical state has a representative $V^{\rm can}$ at
$v{=}2$ ($\tilde cc$--type) in the canonical picture $(\p^+,\p^-)=({-}1,{-}1)$.
On the massless level, only a single scalar excitation survives.
For the computation of string amplitudes, however, vertex operators in various
other ghost and picture sectors are useful and have been constructed.\cite{bkl}

Any $n$--point amplitude involves a sum over genera $h\in{\bf Z}^+$ and
$U(1)$ instanton number~$c\in{\bf Z}$.
To compute the contribution for fixed $h$ and~$c$,
one must integrate out $2h{-}2{\pm}c{+}n$ complex fermionic moduli of
$U(1)$ charge ${\pm}1$, respectively, and $h{-}1{+}n$ complex $U(1)$ moduli,
to obtain an integration measure for the remaining $3h{-}3{+}n$ complex
metric moduli.\footnote{
As always, the cases of the sphere and the torus require some modifications.}~
The result vanishes for $|c|{>}2h{-}2{+}n$
and symbolically reads\cite{kl}
$$
\VEV{ \abs{
(\oint\!b)^{3h-3+n}\ (Z^+)^{2h-2+c+n}\ (Z^-)^{2h-2-c+n}\ (Z^0)^{n-1} }^2
\prod_{i=1}^{h}\! \Bigl[ Z^0(a_i)Z^0(b_i) \Bigr]\
V_1^{\rm can}\!\ldots V_n^{\rm can} }
\eqno(2) $$
where $a_i$ and $b_i$ denote the homology cycles.
The picture--changers $Z^\pm$ and $Z^0$ may be used partially to
convert vertex operators to other pictures and/or ghost numbers.

Invariance of correlation functions under spectral flow follows from the
fact that a change in monodromies for the world-sheet fermions is equivalent
to a shift in the integration over $U(1)$ moduli, which are nothing but
the flat $U(1)$ connections on the $n$--punctured genus--$h$ Riemann surface.
It is realized on the vertex operators by\cite{kl}
$$
V(z)\ \longrightarrow\ V^\q(z)\ =\
\exp\Bigl\{ \q\int^z \! J_{\rm tot}(z')\,dz' \Bigr\}\ V(z) \quad,
\eqno(3) $$
with $\q{=}{\pm}\ha$ leading to $S\!F\!O^\pm$ mapping from NS to R$^\pm$
sectors.\footnote{
The two sectors R$^+$ and R$^-$ differ
by a unit change in instanton number~$c$.}~
Stated differently,
$$
\Bigl\langle V_1\,V_2\ldots V_n \Bigr\rangle\ =\
\VEV{V_1^{\q_1}\,V_2^{\q_2}\ldots V_n^{\q_n}} \qquad{\rm for}\qquad
\sum_\ell \q_\ell =0 \quad,
\eqno(4) $$
equating all $n$--point amplitudes with the same values for $h$ and~$c$.

This work was done in collaboration with S.~Ketov.
We acknowledge fruitful discussions with N.~Berkovits, H.~L\"u, H.~Ooguri,
and C.~Pope.

\begin{thebibliography}{9}
\bibitem{marcus} see, e.g., N. Markus, hep-th/9211059, and references therein.
\bibitem{new} N. Berkovits and C. Vafa, \np{433}{95}{123};
H. L\"u and C.N.~Pope, hep-th/9411101;
N. Berkovits, hep-th/9412179 and hep-th/9503099.
\bibitem{bkl} J. Bischoff, S. Ketov and O. Lechtenfeld, \np{438}{95}{373};\\
see also: A.Giveon and M. Ro\v cek, \np{400}{93}{145}.
\bibitem{kl} S. Ketov and O. Lechtenfeld, hep-th/9503232.
\end{thebibliography}

\end{document}


