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\title{Superspace Representations of SU(2,2/N) Superalgebras and Multiplet
Shortening}

\author{S. Ferrara%\thanks{A footnote may follow the name of the speaker
		%to indicate other collaborators. This is 
		%obtained with the {\tt\ttbs thanks} command.}
		\\
	Theoretical Physics Division, CERN, Geneva 23, Switzerland\\
	E-mail: \email{Sergio.Ferrara@cern.ch}}

\conference{Quantum Aspects of Gauge Theories, Supersymmetry and Unification}

\abstract{We derive massless and massive representations of all SU(2,2/N)
superalgebras by using superfields defined in ``harmonic superspace".  This
method allows one to easily construct ``short superfields" which are relevant
in the analysis of the AdS/CFT correspondence.}

%\dedicated{Dedicated to\ldots\\if you want.}

\begin{document} 

\maketitle %%%%%%%%%% THIS IS IGNORED %%%%%%%%%%%
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\section{Introduction}

The study of superconformal algebras has recently attracted 
renewed interest for their dual role in the ${\rm AdS}_{d+1}/{\rm 
CFT}_d$ correspondence \cite{mal,gkp,wit}, connected to the 
near-horizon geometry of $d-1$-branes. 

A special role is played by $3$-branes since they are related to 
superconformal invariant quantum Yang-Mills theories. These 
theories are the only ones exhibiting conformal symmetry both at 
weak and strong coupling and, in any case, admitting, unlike other 
types of branes, Yang-Mills fields in the conformal regime. 

The bulk and boundary operators in this correspondence are 
classified by highest weight UIR's of $SU(2,2/N)$ algebras 
\cite{fefrza,AF} where $N=1,2$ and 4 in the known examples, since 
supergravity or superstring theory can admit at most 32 ($8N$) 
supersymmetries. Nevertheless, in the study of superconformal 
algebras and their representations different values of $N$ are of 
interest because they help one to exhibit some general features of 
short representations, corresponding to conformal operators with 
protected dimension, but more importantly, because these algebras 
may be relevant for some generalizations of the known schemes in 
which more than 32 supersymmetries may be required \cite{bgg1}. 

Recently \cite{AFSZ} it has been shown that a known 
generalization of ordinary superspace, called ``harmonic 
superspace" \cite{GIK1}-\cite{GIK3}, is particularly suitable to 
build up, in a rather simple and general manner, all possible 
composite operators of superconformal invariant gauge theories 
with $N>1$ extended supersymmetry. 

Other approaches, like ordinary superspace \cite{Siegel,HST} or 
the oscillator construction \cite{bgg}-\cite{gmz1} of highest 
weight representations, although in principle possible, are much 
more complicated to deal with and the complete analysis of all 
possible shortenings would be unnecessarily difficult. 

In fact, the structure of harmonic superspace is powerful enough 
to allow us to extend the analysis of Ref. \cite{AFSZ} to all 
$SU(2,2/N)$ superalgebras  with arbitrary $N$,\footnote{For a 
thorough treatment of the action of superconformal groups on 
harmonic superspaces see Refs. \cite{hh}.} although no dynamical 
theory is known for $N>4$. This report contains results obtained in 
Ref.~\cite{sf}.

{}From a mathematical point of view harmonic superspace is an 
enlarged space where superfields are defined on ``flag manifolds" 
\cite{hh,knapp} 
\begin{equation}\label{flag}
  {\mathcal M} = \frac{SU(N)}{S\left(U(n_1)\times\ldots\times
U(n_p)\right)}\;, 
 \left(\sum_{k=1}^p n_k =N\right). 
\end{equation}
To study the general case of multiplets it is important 
\cite{AFSZ} to use the choice $n_k=1$ ($k=1\ldots N$), i.e. where 
we quotient the group $SU(N)$ by its maximal torus. Then the above 
manifold is the largest flag manifold with complex dimension \\ \hfill 
$N(N-~1)/2$. 

The ultrashort UIR's of $SU(2,2/N)$ superalgebras described by 
analytic harmonic superfields depend only on half of the odd 
coordinates (Grassmann or G-analyticity):
\begin{equation}\label{-5}
   W^{12\ldots k} =  W^{12\ldots k}(\theta_{k+1},\theta_{k+2},\ldots,\theta_{N},
\bar\theta^1,\bar\theta^2,\ldots,\bar\theta^k)\;. 
\end{equation}
In addition, they are annihilated by all the ``step-up" generators 
$E_a$ in the Cartan decomposition of the Lie algebra of $SU(N)$. 
In other words, these superfields correspond to highest weight 
states of $SU(N)$: 
\begin{equation}\label{HW}
  E_a|{\rm HW}\rangle = 0\;.
\end{equation}
In harmonic superspace this irreducibility condition corresponds 
to harmonic (or H-) analyticity.  The crucial point is that the 
$SU(2,2/N)$ algebra acting on such states defines a 
``quasi-primary" superconformal field denoted by 
\begin{equation}\label{calD}
{\cal D} (\ell,J_1,J_2;r;a_1,\ldots, a_{N-1}) 
\end{equation}
where $\ell,J_1,J_2$ are the conformal dimension and spin of the 
state, $r$ is the $U(1)$ $R$ charge and $a_1,\ldots, a_{N-1}$ are 
the $SU(N)$ Dynkin labels. We assign the $R$ charge 
$r_\theta={1\over 2}(1-{4\over N})$ to the Grassmann coordinates 
in order to be consistent with the convention that chiral 
superfields $\Phi(\theta)$ have $l=-r$ for any $N$. This is also 
the charge which naturally appears in the definition of the   \\ 
$SU(2,2/N)$ superalgebra \cite{FKTN}. \footnote{Note that for 
$N=4$, $r_\theta =0$ and the $r$ quantum number becomes a 
``central charge" \cite{dp,bin}. In this case the analysis of 
section 2 refers to the $PSU(2,2/4)$ algebra for $r=0$ and to the 
$PU(2,2/4)$ algebra for $r\neq 0$.} 

The G- and H-analytic superfields (\ref{-5}) have their lowest 
(scalar) component belonging to the rank $k$ antisymmetric 
representation of $SU(N)$ ($k=1\ldots[{N\over 2}]$), have $R$ 
charge $r_k={2k\over N}-1$ and will be shown to describe 
``ultrashort" representations of the $SU(2,2/N)$ superalgebra. If 
the algebra is interpreted as acting on ${\rm AdS}_5$, these 
are the ``supersingleton" representations \cite{ff2}. For $k=0$ the 
superfield is actually ``chiral" and in this case the highest 
weight state may carry a spin label $(J_L,0)$ with $\ell =1+J_L$. 
The chiral superfield is the supersingleton representation when 
the top spin is $J_L={N\over 2}$. For all other analytic 
superfields $(k>1)$ the supersingleton will have top spin 
$J_L={N\over 2}-{k\over 2}$. 

It should be pointed out that the same massless multiplets can be 
described in terms of ordinary but {\it constrained} superfields 
\cite{Siegel, HST}. The reason why we prefer the harmonic 
superspace version is the fact that the superfields  (\ref{-5}) 
are {\it unconstrained} analytic objects. Analyticity is a 
property which is preserved by multiplication. This will allow us 
to tensor the above massless UIR's in a very simple way and thus 
obtain series of short multiplets of $SU(2,2/N)$. We observe that 
from the ${\rm AdS}_5$ point of view, tensoring more than two 
supersingleton reps produces ``massive bulk"  reps, while 
tensoring only two of them produces ``massless bulk"  reps 
\cite{ff2,gmz1}. The latter are the ``supercurrent" multiplets 
discussed in Ref. \cite{HST}. 


\section{Unitarity bounds and shortening of UIR's of $SU(2,2/N)$} 

The unitarity bounds of highest weight UIR's of $SU(2,2/N)$ have 
been derived in Refs. \cite{ff,dp,bin,mss}. They correspond to some bounds on 
the highest weight state (\ref{calD}). Let us define the 
quantities
\begin{equation}\label{mm}
  m_1=\sum_{k=1}^{N-1}a_k\;, \quad m=\sum_{k=1}^{N-1}(N-k)a_k
\end{equation}
and
\begin{eqnarray}\label{XY}
  X(J,r,{2m\over N}) &=& 2+2J-r+{2m\over N}\;, \nonumber \\ 
  Y(r,{2m\over N}) &=& -r+{2m\over N}\;.
\end{eqnarray}
Then we have $(J_1=J_L, \ J_2=J_R)$:
\begin{equation}\label{A}
  {\rm A)}\qquad \ell \geq X(J_2,r,{2m\over N}) \geq X(J_1,-r, 2m_1-{2m\over 
N})
\end{equation}
(or $J_1\rightarrow J_2$, $r\rightarrow -r$, ${2m\over 
N}\rightarrow 2m_1-{2m\over N}$);
\begin{equation}\label{B}
  {\rm B)}\qquad \ell = Y(r,{2m\over N}) \geq X(J_1,-r, 2m_1-{2m\over 
N}) 
\end{equation}
(or $J_1\rightarrow J_2$, $r\rightarrow -r$, ${2m\over 
N}\rightarrow 2m_1-{2m\over N}$); 
\begin{equation}\label{C}
  {\rm C)}\qquad \ell = m_1\;, \quad r={2m\over N}-m_1\;, \quad J_1=J_2=0\;. 
\end{equation}

The massless UIR's correspond to B) for $a_k=0$, $\ell=-r=1+J_L$ 
and to C) for $\ell=m_1=1$, $r_k= {2k\over N}-1$, $1\leq k \leq 
[{N\over 2}]$. Note that the two series overlap for $J_L=0$ in B) 
and $k=0$ in C).

The short multiplets that we shall build in section 4 by tensoring 
massless multiplets from the C) series in the case of $N=2n$ for  
$k=n$ ($r=0$)  will belong to the shortenings in B) and C) 
obtained for $J_1=0$ and $r=0$: 
\begin{eqnarray}
  &{\rm B)}&\ \ell= {2m\over N}\;, \quad  {2m\over N}-m_1 \geq 1\;\nonumber\\
  &{\rm C)}&\ \ell= m_1\;, \quad  {2m\over N}=m_1 \label{BC}
\end{eqnarray}
 
\section{Massless superconformal multiplets}


\subsection{Grassmann analytic superfields}

We consider superfields $$W^{i_1\ldots i_k}(x_{\alpha\dot\alpha}, 
\theta^\alpha_i,\bar\theta^{\dot\alpha i})$$ with $k=1,\ldots, n$ 
(where $n=[{N\over 2}]$) totally antisymmetrized indices in the 
fundamental representation of $SU(N)$. These superfields satisfy 
the following constraints:
\begin{eqnarray}
  &&D^{(j}_\alpha W^{i_1)i_2\ldots i_k}=0\;, \label{1-1}\\
  &&\bar D_{\dot\alpha \{j}W^{i_1\}i_2\ldots i_k}=0  \label{1-2} 
\end{eqnarray}
where $()$ means symmetrization and $\{\}$ means the traceless 
part. The spinor derivatives algebra is 
\begin{equation}\label{spder}
  \{D^{i}_\alpha,\bar D_{\dot\alpha j}\} = i\delta^i_j 
\partial_{\alpha\dot\alpha}
\end{equation}
with $\partial_{\alpha\dot\alpha} = 
\sigma^\mu_{\alpha\dot\alpha}\partial_\mu$. In the cases $N=2,3,4$  
these constraints define the on-shell $N=2$ matter 
(hyper)multiplet \cite{Sohnius} and the $N=3,4$ on-shell 
super-Yang-Mills multiplets \cite{So}. Their generalization to 
arbitrary $N$ has been given in Refs. \cite{Siegel,HST} where it 
has also been shown that they describe on-shell massless 
multiplets.  

Our aim in this section is to rewrite the constraints (\ref{1-1}), 
(\ref{1-2}) in harmonic superspace where they will take the simple 
form of analyticity conditions. Using this fact we will then be 
able to construct tensor products of the corresponding multiplets 
in a very straightforward and easy way (section 4). 

The main purpose of introducing harmonics is to be able to 
covariantly project all the $SU(N)$ indices in (\ref{1-1}), 
(\ref{1-2}) onto a set of $U(1)$ charges. To this end we choose 
the harmonic coset $SU(N)/(U(1))^{N-1}$ described in terms of 
harmonic variables $u^I_i$ and their conjugates $u^i_I = (u^I_i)^* 
$. \footnote{The harmonic notation used here differs from the 
original one of Refs. \cite{GIK1,GIK2}. It is similar to the one 
introduced in Ref. \cite{GIK3} for the case $N=3$ and in Refs. 
\cite{hh} for general $N$.} They form an $SU(N)$ matrix where $i$ 
is an index in the fundamental representation of $SU(N)$ and 
$I=1,\ldots,N$ is a collection of the $N-1$ $U(1)$ charges 
corresponding to the projections of the second index (the harmonic 
$u^i_I$ carries charges opposite to those of $u^I_i$). They 
satisfy the following $SU(N)$ defining conditions: 
\begin{eqnarray}
 &&u^I_i u^i_J=\delta^I_J~, \label{a2}\\ u\in 
SU(N):\quad &&u^I_i u^j_I =\delta^j_i~,\label{a2'}\\ && 
\varepsilon^{i_1\ldots i_N}u^1_{i_1}\ldots u^N_{i_N}=1~. 
\label{a2''} 
\end{eqnarray}

Now, let us use these harmonic variables to split all the $SU(N)$ 
indices in the constraints (\ref{1-1}), (\ref{1-2}) into 
independent $(U(1))^{N-1}$ projections. For example, the 
projection 
\begin{equation}\label{2}
  W^{12\ldots k} = W^{i_1i_2\ldots i_k} u^1_{i_1}u^2_{i_2}\ldots u^k_{i_k}
\end{equation}
satisfies the constraints
\begin{eqnarray}
&&D^{1}_\alpha  W^{12\ldots k} = 
D^{2}_\alpha  W^{12\ldots k} =\ldots =  \nonumber \\
&&D^{k}_\alpha  W^{12\ldots k} = 0\;, 
 \nonumber \\
\label{4a}
\end{eqnarray}
\begin{eqnarray}
&&  \bar D_{\dot\alpha\; k+1} W^{12\ldots k} = 
 \bar D_{\dot\alpha\; k+2} W^{12\ldots k} =\ldots =  \nonumber \\
&&\bar D_{\dot\alpha\; N} W^{12\ldots k} = 0 
\label{4b}
\end{eqnarray}
where $D^I_\alpha = D^i_\alpha u_i^I$ and $\bar D_{\dot\alpha\; 
I}= \bar D_{\dot\alpha\; i}u^i_I$. The first of them, eq. 
(\ref{4a}), is a corollary of the commuting nature of the 
harmonics variables, and the second one, eq. (\ref{4}), of the 
unitarity condition (\ref{a2}). The main achievement in rewriting 
the constraints (\ref{1-1}), (\ref{1-2}) in this new form is that 
they can be explicitly solved by going to an appropriate 
G-analytic basis in superspace: 
\begin{eqnarray}
  &&x^{\alpha\dot\alpha}_A =  x^{\alpha\dot\alpha} 
+ i(\theta^\alpha_1 \bar\theta^{1\dot\alpha} + \ldots + 
\theta^\alpha_k \bar\theta^{k\dot\alpha} \nonumber \\
&& - \theta^\alpha_{k+1} 
\bar\theta^{k+1\dot\alpha}-\ldots - \theta^\alpha_N 
\bar\theta^{N\dot\alpha} )\;,\nonumber\\ 
  &&\theta^\alpha_I = \theta^\alpha_i u^i_I\;, \quad 
\bar\theta^{\dot\alpha I} = \bar\theta^{\dot\alpha i} 
u_i^I\;.\label{5'} 
\end{eqnarray}
In this basis $W^{12\ldots k}$ becomes an unconstrained function 
of $k$ $\bar\theta$'s and $N-k$ $\theta$'s: 
\begin{equation}\label{5}
   W^{12\ldots k} =  W^{12\ldots k}(x_A,\theta_{k+1},\ldots,\theta_{N},
\bar\theta^1,\ldots,\bar\theta^k, u)\;.
\end{equation}
Altogether it depends on half the number of the odd variables of 
$N$-extended superspace and for this reason we call it Grassmann 
(or G-) analytic. We recall that the notion of Grassmann 
analyticity was first introduced in Ref. \cite{GIO}, still in the 
context of ordinary superspace. In $N=2$ harmonic superspace 
\cite{GIK1} this notion became $SU(2)$ covariant. The 
generalization to $N=3$ was given in Ref. \cite{GIK2} and later on 
to general $N$ in Refs. \cite{hh} (under the name of ``$(N,p,q)$ 
superspace"). 

The massless conformal multiplets describe the ordinary massless 
UIR's of the super Poincar\'e group obtained earlier by the Wigner 
method of induced representations (see, for instance, Ref. 
\cite{str}). The self-conjugate $N=8$ multiplet was obtained by 
the oscillator method in Ref. \cite{gm2}. 

\subsection{Harmonic analyticity as $SU(N)$ \\
irreducibility}

It is important to realize that a G-analytic superfield is an 
$SU(N)$ covariant object only because it depends on the harmonic 
variables. In order to recover the original harmonic-independent 
but constrained superfield $ W^{i_1i_2\ldots 
i_k}(x,\theta,\bar\theta)$ (\ref{1-1}), (\ref{1-2}) we need to 
impose differential conditions involving the harmonic variables. 
The harmonic derivatives are made out of the operators 
\begin{equation}\label{a5}
  \partial^{\,I}_J = u^I_i{\partial\over\partial u^J_i} - 
u^i_J{\partial\over\partial u^i_I} 
\end{equation}
which respect the defining relations (\ref{a2}), (\ref{a2'}). 
These derivatives act on the harmonics as follows: 
\begin{equation}
\partial^I_J u^K_i=\delta^K_J u^I_i~,\qquad
\partial^I_J u^i_K=-\delta^I_K u^i_J~.\label{F9}
\end{equation} 
The diagonal ones $\partial^{\,I}_I$ count the $U(1)$ charges, 
\begin{equation}\label{chc}
  \partial^{\,I}_I u^I_i =  u^I_i\;, \qquad \partial^{\,I}_I u_I^i =  
-u_I^i\;.
\end{equation}
The relation (\ref{a2''}) implies that the charge operators 
$\partial^{\,I}_I$ are not independent, 
\begin{equation}\label{a6}
  \sum_{I=1}^N \partial^{\,I}_I = 0
\end{equation}
(this reflects the fact that we are considering $SU(N)$ and not 
$U(N)$). 

A basic assumption in our approach to the harmonic coset 
$SU(N)/U(1)^{N-1}$ is that any harmonic function is homogeneous 
under the action of $U(1)^{N-1}$, i.e., it is an eigenfunction of 
the charge operators $\partial^{\,I}_I$, 
\begin{eqnarray}\label{eigch}
   &&\partial^{\,I}_I f^{K_1\ldots K_q}_{L_1\ldots L_r}(u) = 
(\delta^{K_1}_I+\ldots + \delta^{K_q}_I - \delta_{L_1}^I -\ldots 
\nonumber \\
&&- \delta_{L_r}^I) f^{K_1\ldots K_q}_{L_1\ldots L_r}(u) 
\end{eqnarray}
(note that the charges  $K_1\ldots K_q;L_1\ldots L_r$ are not 
necessarily all different). Thus it effectively depends on the 
$(N^2-1)-(N-1)=N(N-1)$ real coordinates of the coset  
$SU(N)/U(1)^{N-1}$. Then the actual harmonic derivatives on the 
coset are the $N(N-1)/2$ complex derivatives $\partial^{\,I}_J $, 
$I<J$ (or their conjugates $\partial^{\,I}_J $, $I>J$). 

The set of $N^2-1$ derivatives  $\partial^{\,I}_J $ (taking into 
account the linear dependence (\ref{a6})) form the algebra of 
$SU(N)$: 
\begin{equation}\label{a8}
 [\partial^{\,I}_J,\partial^{\,K}_L ]=\delta^K_J \partial^{\,I}_L -\delta^I_L 
\partial^{\,K}_J~.
\end{equation}
The Cartan decomposition of this algebra $L^+ + L^0 + L^-$ is 
given by the sets 
\begin{eqnarray}\label{a9}
  L^+ &= &\{\partial^{\,I}_J \;, \ I<J\}\;, 
  \quad
L^0 = \{\partial^{\,I}_I\;, \ \sum_{I=1}^N \partial^{\,I}_I = 0 
\}\;,\nonumber \\
  L^- &=& \{\partial^{\,I}_J \;, \ I>J\}\;. 
\end{eqnarray}
It becomes clear that imposing the harmonic conditions 
\begin{equation}\label{a10}
  \partial^{\,I}_J f^{K_1\ldots K_q}_{L_1\ldots L_r}(u) = 0\;, \quad I<J
\end{equation}
on a harmonic function with a given set of charges $K_1\ldots 
K_q;L_1\ldots L_r$ defines the highest weight of an $SU(N)$ irrep. 
In other words, the harmonic expansion of such a function contains 
only one irrep which is determined by the combination of charges 
$K_1\ldots K_q;L_1\ldots L_r$. In fact, not all of the derivatives 
$\partial^{\,I}_J \;, \ I<J$ are independent, as follows from the 
algebra (\ref{a8}). The independent set consists of the $N-1$  derivatives 
\begin{equation}\label{rankder}
  \partial^{\,1}_2\;,\ \partial^{\,2}_3\;,\ \ldots,\ \partial^{\,N-1}_N 
\end{equation}
corresponding to the simple roots of $SU(N)$.
Then the $SU(N)$ defining constraint (\ref{a10}) is equivalent to
\begin{equation}\label{a100}
(\partial^{\,1}_2\;,\partial^{\,2}_3\;,\ldots,\ 
\partial^{\,N-1}_N) f^{K_1\ldots K_q}_{L_1\ldots L_r}(u) = 0\;. 
\end{equation}

The coset $SU(N)/U(1)^{N-1}$ can be parametrized by $N(N-1)/2$ 
complex coordinates. In this case the constraints (\ref{a10}) take 
the form of covariant (in the sense of Cartan) Cauchy-Riemann 
analyticity conditions. For this reason we call the set of 
constraints (\ref{a10}) (or the equivalent set (\ref{a100})) 
harmonic (H-)analyticity conditions. The above argument shows that 
H-analyticity is equivalent to defining a highest weight of 
$SU(N)$, i.e. it is the $SU(N)$ irreducibility condition on the 
harmonic functions. 

As an example, take $N=2$ and the function $f^1(u)$ subject to the 
constraint 
\begin{equation}\label{N2ex}
  \partial^{\,1}_2f^1(u) = 0 \ \Rightarrow \ f^1(u) = f^i u^1_i\;.
\end{equation}
So, the harmonic function is reduced to a doublet of $SU(2)$. 
Similarly, for $N=4$ the function $f^{12}(u)$ is reduced to the 
$\underline 6$ of $SU(4)$. Indeed, the constraints 
$\partial^{\,2}_3 f^{12}(u) = \partial^{\,3}_4 f^{12}(u) = 0$ 
ensure that $f^{12}(u)$ depends on $u^1,u^2$ only, $f^{12}(u)= 
f^{ij}u^1_iu^2_j$. Then the constraint $\partial^{\,1}_2 f^{12}(u) 
=f^{ij}u^1_iu^1_j$ = 0 implies $f^{ij} = -f^{ji}$. 

In the G-analytic basis (\ref{5'}) the harmonic derivatives become 
covariant $D^{\,I}_J$. In particular, the derivatives 
\begin{eqnarray}\label{sptn}
  D^{\,I}_{J}&=& \partial^{\,I}_{J} - i\theta^\alpha_{J} 
\bar\theta^{I\;\dot\alpha}\partial_{\alpha\dot\alpha} - 
\theta_J\partial^I + \bar\theta^I\bar\partial_J\;, \nonumber \\ 
I&=&1,\ldots,k, \ J=k+1,\ldots,N 
\end{eqnarray}
acquire space-time derivative terms. The $SU(N)$ commutation 
relations among the $D^{\,I}_{J}$ are not affected by the change 
of basis. The same is true for the commutation relations of the 
$D^{\,I}_{J}$  with the spinor derivatives: 
\begin{equation}\label{3}
  [D^{\,I}_J,D^K_\alpha]=\delta^K_J D^I_\alpha\;, 
\qquad [D^{\,I}_J,\bar D_{\dot\alpha\; K}]=-\delta^I_K \bar 
D_{\dot\alpha\; J}\;. 
\end{equation}
Using these relations one can see that the H-analyticity 
conditions 
\begin{equation}
  \label{a10'}
  D^{\,I}_J W^{12\ldots k} = 0\;, \quad I<J
\end{equation}
or the equivalent set
\begin{equation}\label{a111}
  (D^{\,1}_2\;,D^{\,2}_3\;,\ldots, D^{\,N-1}_N) W^{12\ldots k} = 0 
\end{equation}
are compatible with the G-analyticity ones (\ref{4}). 

\subsection{Analyticity and massless multiplets: \\ ``Singletons"} 

The constraints of H-analyticity (\ref{a10'}) combined with those 
of G-analyticity  (\ref{4}) have important implications for the 
components of the superfield. First of all, they make each 
component an irrep of $SU(N)$. Take, for example, the first 
component 
\begin{equation}\label{6}
   \phi^{12\ldots k}(x,u) =  W^{12\ldots k}\vert_0
\end{equation}
where $\vert_0$ means $\theta=\bar\theta=0$. The constraints 
$\partial^{\,I}_{I+1}\phi^{12\ldots k}(x,u)=0$, $I=k,\ldots,N$ 
imply that $\phi^{12\ldots k}(x,u)$ takes the form
$$
\phi^{12\ldots k}(x,u) = \phi^{i_1i_2\ldots i_k}(x) u^1_{i_1} 
u^2_{i_2}\ldots u^k_{i_k}\;. 
$$
This is a rank $k$ tensor without any symmetry, i.e. a reducible 
representation of $SU(N)$. Further, the constraint, e.g.,  
$$
\partial^1_2 \phi^{123\ldots k}(x,u) = \phi^{113\ldots k}(x,u) = 
$$
$$
\phi^{i_1i_2i_3\ldots i_k}u^1_{i_1} u^1_{i_2}u^3_{i_3}\ldots 
u^n_{i_k} =0 
$$
removes the symmetric part in the first two indices. Similarly, 
the remaining constraints  (\ref{a10'}) remove all the 
symmetrizations and we find the totally antisymmetric rank $k$ 
irrep of $SU(N)$. 

Another example are the spinor components
\begin{eqnarray}\label{spinco}
&&\chi_{\alpha}^{12\ldots k\; k+1}(x,u) = 
D^{k+1}_{\alpha}W^{12\ldots k}\vert_0 \;, \nonumber \\
&&\bar\psi_{\dot\alpha}^{23\ldots k}(x,u) = \bar 
D_{1\dot\alpha}W^{12\ldots k}\vert_0\;. 
\end{eqnarray}
The same harmonic argument shows that these are harmonic 
projections of the totally antisymmetric components 
$\chi_{\alpha}^{[i_1i_2\ldots i_{k+1}]}(x)$ and 
$\bar\psi_{\dot\alpha}^{[i_2i_3\ldots i_k]}(x)$. 

Further important constraints occur at the level of 2 or more 
$\theta$'s:
\begin{eqnarray}
  D^{I\alpha}D^{J}_\alpha W^{12\ldots k}&=& 0,  
I,J=k+1,\ldots,N,\nonumber \\
 \label{lin2}\\ 
  \bar D_{I\dot\alpha}\bar D_{J}^{\dot\alpha} W^{12\ldots k}&=& 0, 
I,J=1,\ldots,k. \label{lin3} 
\end{eqnarray}
The easiest way to see this is to hit the defining constraint 
(\ref{1-1}) with $D^{k\alpha}$ and then project with harmonics. 

The constraints (\ref{lin2}), (\ref{lin3}) imply that the 
components of the type 
\begin{eqnarray}
  &&\chi^{1\ldots k+p}_{(\alpha_1\ldots \alpha_p)} = D^{k+1}_{\alpha_1}\ldots
D^{k+p}_{\alpha_p} W^{12\ldots k}\vert_0\;, 
 p\leq N-k \nonumber \\
\label{top1}\\ 
  &&\bar\psi^{p+1\ldots k}_{(\dot\alpha_1\ldots\dot\alpha_p)} = 
\bar D_{1\dot\alpha_1} \ldots \bar D_{p\dot\alpha_p} W^{12\ldots 
k}\vert_0\;, \quad p\leq k 
\nonumber \\
\label{top2} 
\end{eqnarray}
are totally symmetric in their spinor indices, i.e. they carry
spin $(p/2,0)$ or $(0,p/2)$, correspondingly. Among them one finds 
the 
\begin{eqnarray}\label{top}
&&\mbox{top spin $({N\over 2}-{k\over 2},0)$:}\quad 
\chi_{(\alpha_1\ldots \alpha_{N-k})} = 
\nonumber \\
&&D^{k+1}_{\alpha_1}\ldots 
D^{N}_{\alpha_{N-k}} W^{12\ldots k}\vert_0 
\end{eqnarray}
which is also an $SU(N)$ singlet. Note that in the case $N=2n$,  
$k=n$ the top spin occurs both as $(n/2,0)$ and $(0,n/2)$ (we call 
this a ``self-conjugate" multiplet). Moreover, if $N=4n$ and 
$k=2n$ one can impose a reality condition on the superfield 
$W^{12\ldots 2n}$ which implies, in particular, that 
\begin{equation}\label{real}
  \chi_{(\alpha_1\ldots \alpha_{2n})} = (\psi_{(\dot\alpha_1\ldots \dot\alpha_{2n})})^* 
\;.
\end{equation}

Next, one can show that all the components of the type 
(\ref{top1}), (\ref{top2}) satisfy massless field equations. 
Indeed, from the constraint (\ref{lin2}) and from G-analyticity it 
follows that 
\begin{eqnarray}
  0&=& \bar D_{k+1\;\dot\beta}D^{k+1\;\alpha_1} D^{k+1}_{\alpha_1}\ldots
D^{k+p}_{\alpha_p} W^{12\ldots k}\nonumber\\ 
  &=&2i\partial_{\dot\beta}^{\alpha_1}D^{k+1}_{\alpha_1}\ldots
D^{k+p}_{\alpha_p} W^{12\ldots k}\nonumber\\ &\Rightarrow& 
\partial_{\dot\beta}^{\alpha_1}\chi^{1\ldots k+p}_{(\alpha_1\ldots \alpha_p)} = 0
 \label{nooo} 
\end{eqnarray}
and similarly for $\bar\psi^{p+1\ldots 
k}_{(\dot\alpha_1\ldots\dot\alpha_p)}$. The leading scalar 
component (\ref{6}) satisfies the d'Alembert equation: 
\begin{eqnarray}\label{dale}
 0 &=& (D^1)^2(\bar D_1)^2 W^{12\ldots k} = 4\Box W^{12\ldots k} \ \nonumber \\
 &\Rightarrow& \ 
\Box\phi^{12\ldots k} = 0.
\end{eqnarray}

 
Finally, all the components of mixed type, 
\begin{eqnarray}\label{mity}
&&f^{p+1\ldots k+q}_{\dot\alpha_1\ldots\dot\alpha_p 
\alpha_1\ldots\alpha_q}  \nonumber \\
&& = \bar D_{1\dot\alpha_1}\ldots \bar 
D_{p\dot\alpha_p} D^{k+1}_{\alpha_1}\ldots 
D^{k+q}_{\alpha_q}W^{12\ldots k}\vert_0\;, \nonumber \\
&& p\leq k\;, \ q\leq N-k 
\end{eqnarray}
are expressed in terms of the space-time derivatives of lower 
components. Indeed,
\begin{eqnarray}
&& D^1_{k+q} f^{p+1\ldots k+q}_{\dot\alpha_1\ldots\dot\alpha_p 
\alpha_1\ldots\alpha_q}  \nonumber \\
&&= - \bar D_{k+q\dot\alpha_1}\bar 
D_{2\dot\alpha_2}\ldots D^{k+q}_{\alpha_q}W^{12\ldots 
k}\vert_0\nonumber\\ 
  &&= (-1)^{p+q-1}i\partial_{\dot\alpha_1\alpha_q} \bar D_{2\dot\alpha_2}
\ldots D^{k+q-1}_{\alpha_{q-1}}W^{12\ldots k}\vert_0 
 \nonumber \\
&&\Rightarrow \ f^{p+1\ldots k+q-1\; 1
}_{\dot\alpha_1\ldots\dot\alpha_p \alpha_1\ldots\alpha_q} \nonumber \\
&&=(-1)^{p+q-1}i\partial_{\dot\alpha_1\alpha_q}  g^{1\; p+1\ldots 
k+q-1}_{\dot\alpha_2\ldots\dot\alpha_p 
\alpha_1\ldots\alpha_{q-1}} 
\label{nolabel} 
\end{eqnarray}
 
To summarize, the superfield $W^{12\ldots k}$ subject to the 
constraints of G- and H-analyticity has the following component 
content (the derivative terms are not shown):
\begin{eqnarray}
 && W^{12\ldots k} =\phi^{12\ldots k} \nonumber\\
 &&+\bar\theta^1_{\dot\alpha}\bar\psi^{\dot\alpha\; 23\ldots k} + 
\ldots + \bar\theta^k_{\dot\alpha}\bar\psi^{\dot\alpha\; 12\ldots 
k-1} \nonumber\\
 &&+ \theta^\alpha_{k+1}\chi_{\alpha}^{1\ldots k\; k+1}  + \ldots + 
\theta^\alpha_{N}\chi_{\alpha}^{1\ldots k\; N} \nonumber\\ 
 &&+\bar\theta^1_{\dot\alpha} \bar\theta^2_{\dot\beta} 
\bar\psi^{(\dot\alpha\dot\beta)\; 3\ldots k} + \ldots + 
\bar\theta^{k-1}_{\dot\alpha} \bar\theta^k_{\dot\beta} 
\bar\psi^{(\dot\alpha\dot\beta)\; 1\ldots k-2}\nonumber\\ 
 && + \theta^\alpha_{k+1} \theta^\beta_{ k+2} 
\chi_{(\alpha\beta)}^{1\ldots k\; k+1\; k+2} 
 + \nonumber \\
 && \ldots + \theta^\alpha_{N-1} \theta^\beta_{N} 
\chi_{(\alpha\beta)}^{1\ldots k\; N-1\; N}
\ldots\nonumber \\ 
 && + \bar\theta^1_{\dot\alpha_1}\ldots \bar\theta^k_{\dot\alpha_k} 
\bar\psi^{(\dot\alpha_1\ldots\dot\alpha_k)} \nonumber \\
&& +\theta^{\alpha_1}_{k+1} \ldots \theta^{\alpha_{N-k}}_{N} 
\chi_{(\alpha_1\ldots\alpha_{N-k})}  \label{expan} 
\end{eqnarray}
where all the fields belong to totally antisymmetric irreps of 
$SU(N)$ and satisfy the massless field equations
\begin{eqnarray}
  &&\Box \phi^{[i_1\ldots i_k]}=0\;, \nonumber\\
  &&\partial^{\beta\dot\alpha_1} 
\bar\psi_{(\dot\alpha_1\ldots \dot\alpha_p)}^{[i_1\ldots i_{k-p}]} 
= 0\;,\quad 1\leq p \leq k \label{onshell}\\ 
  &&\partial^{\alpha_1\dot\beta}\chi_{(\alpha_1\ldots \alpha_p)}^{[i_1\ldots i_{p}]}
= 0\;, \quad 1\leq p \leq N-k   \nonumber 
\end{eqnarray}
This is the content of an $N$-extended superconformal multiplet of 
the C) series of section 2. It is characterized by the $SU(N)$ 
irrep of the first component (described by the Young tableau 
$m_1=\ldots=m_k=1,\ m_{k+1}=\ldots=m_{N-1}=0$), by its $R$ charge 
\begin{equation}\label{rch}
  r_k={2k\over N}-1
\end{equation}
and conformal dimension $\ell=1$ and by the top spin 
$J_{\scriptsize\rm top} =({N\over 2}-{k\over 2},0)$. 
 
\subsection{Chiral superfields}

The G-analytic superfields considered above contain at least one 
$\bar\theta$. The case of ``extreme" G-analyticity will be the 
absence of any $\bar\theta$'s. These are the well-known chiral 
superfields \cite{fwz} satisfying the constraint
\begin{equation}\label{chir}
  \bar D_{i\;\dot\alpha}W=0 \quad\Rightarrow \quad W=W(x^{\alpha\dot\alpha}_L,
\theta^\alpha_i)
\end{equation}
where
\begin{equation}\label{chb}
  x^{\alpha\dot\alpha}_L = x^{\alpha\dot\alpha} 
- i\theta^\alpha_i\bar\theta^{i\;\dot\alpha}\;. 
\end{equation}
Note that in this case we do not need harmonic variables, since 
G-analyticity involves a subset of odd coordinates forming an 
entire irrep of $SU(N)$, and not a set of $U(1)$ projections. 
Consequently, in order to put such a superfield on shell, we 
cannot use H-analyticity but need to impose a new type of 
constraint: 
\begin{equation}\label{chshell}
  D^{\alpha\; i}D^{j}_\alpha W = 0\;.
\end{equation}
The resulting components are multispinors of the same chirality 
(cf. eq. (\ref{expan})):
\begin{eqnarray}\label{chexpan}
  W &=&\phi + \theta^\alpha_{i}\chi_{\alpha}^{i}  \nonumber \\
   & + & \ldots + 
\theta^{\alpha_1}_{i_1} \ldots \theta^{\alpha_{n}}_{i_n} 
\chi_{(\alpha_1\ldots\alpha_{n})}^{[i_1\ldots i_n]} + \ldots + 
(\theta)^{2N} \chi \nonumber \\
\end{eqnarray}
satisfying  massless field equations. The tops spin is $({N\over 
2} ,0)$. 

The chiral superfields above are scalar, but there exist 
conformally covariant chiral superfields with an arbitrary 
$(J_L,0)$ index of the highest weight: $W_{\alpha_1\ldots 
\alpha_{2J_L}}$. In this case the masslessness condition is 
\cite{HST} $D^{\alpha_1i}W_{\alpha_1\ldots \alpha_{2J_L}}=0$. 


 

\section{Short superconformal multiplets:  bulk `` massless" and ``massive"
states}

In this section we shall concentrate on the case $N=2n$ for 
reasons of simplicity. The analytic superfield $W^{12\ldots 
n}(\theta_{n+1},\ldots,\theta_{2n}, 
\bar\theta^1,\ldots,\bar\theta^n)$ des\-cribes a superconformal 
multiplet characterized by the Young tableau 
$m_1=\ldots=m_n=1,$ $ m_{n+1}=\ldots=m_{2n-1}=0$ of its first component (a Lorentz 
scalar), by its dimension $\ell=1$ and $R$ charge $r=0$ (see 
(\ref{rch})). Now we shall use this multiplet as a building block 
for constructing other ``short" superconformal multiplets.  

The building block $W^{12\ldots n}$ can be equivalently rewritten 
by choosing different harmonic projections of its $SU(N)$ indices 
and, consequently, different sets of G-analyticity constraints. 
This amounts to superfields of the type  
\begin{equation}\label{15}
  W^{I_1I_2\ldots I_n}
(\theta_{J_1},\ldots, \theta_{J_n}, 
\bar\theta^{I_1},\ldots,\bar\theta^{I_n}) 
\end{equation}
where $I_1,\dots,I_n$ and $J_1,\dots,J_n$ are two complementary 
sets of $n$ indices. Each of these superfields depends on $2N=4n$ 
Grassmann variables, i.e. half of the total number of $4N=8n$. 
This is the minimal size of a G-analytic superspace, so we can say 
that the $W$'s are the ``shortest" superfields (superconformal 
multiplets). Another characteristic of these $W$'s is the absence 
of $R$ charges. 

The idea now is to start multiplying different species of the 
$W$'s of the type (\ref{15}) in order to obtain composite objects 
depending on various numbers of odd variables. The sets 
$I_1,\dots,I_n$ can be chosen in $(2n)!/(n!)^2$ different ways. 
However, we do not need consider all of them. The following choice 
of $W$'s and of the order of multiplication covers all possible 
intermediate types of G-analyticity: 
$$
A(p_1,p_2,\ldots,p_{2n-1})= 
$$ 
\begin{eqnarray}
 &&[W^{1\ldots n}(\theta_{n+1 \ldots  2n} 
\bar\theta^{1  \ldots  n})]^{p_1 + \ldots +p_{2n-1}}\nonumber\\ 
  &&\times [W^{1\ldots n-1\; 
n+1}(\theta_{\underline{n}\;{n+2} \ldots  2n} \nonumber \\
&&~~~\times \bar\theta^{1\ldots 
n-1\; \underline{n+1}}) ]^{p_2 + \ldots +p_{2n-1}}\nonumber\\ 
  &&\times [W^{1\ldots n-1\; 
n+2}(\theta_{n\;  {n+1}\; {n+3} \ldots 2n} \nonumber \\
&&~~~\times \bar\theta^{1\ldots 
n-1\; \underline{n+2}} ) ]^{p_3 + \ldots +p_{2n-1}}\nonumber\\ 
  && \cdots     \nonumber\\ 
  &&\times [W^{1\ldots n-1\; 
2n-1}(\theta_{n \ldots {2n-2} \; 2n}\nonumber \\
&&~~~\times \bar\theta^{1\ldots n-1\; 
\underline{2n-1}})]^{p_n + \ldots +p_{2n-1}}\nonumber\\ 
  &&\times [W^{1\ldots n-2\; n\; n+1}(\theta_{\underline{n-1}\; {n+2} 
\ldots  2n}\nonumber \\
&&~~~\times \bar\theta^{1\ldots n-2\; n\; n+1})]^{p_{n+1} + \ldots 
+p_{2n-1}}\nonumber\\ 
  &&\times [W^{1\ldots n-3\; n-1\; n\; n+1}(\theta_{\underline{n-2}\; n+2
 \ldots 2n} \nonumber \\
&&~~~ \times \bar\theta^{1\ldots n-3\; n-1\; n\; n+1}) ]^{p_{n+2} + \ldots 
+p_{2n-1}}\nonumber\\ 
  && \cdots     \nonumber\\ 
  &&\times [W^{13\ldots n+1}(\theta_{\underline{2}\; n+2 
 \ldots 2n} 
\bar\theta^{13\ldots n+1}) ]^{p_{2n-2} +p_{2n-1}}\nonumber\\  
  &&\times [W^{23\ldots n+1}(\theta_{\underline{1}\; {n+2} \ldots  2n} 
\bar\theta^{23\ldots n+1}) ]^{p_{2n-1}} \;.         
\label{verylong} 
\end{eqnarray}

The power $\sum^{2n-1}_{r=k} p_r$ of the $k$-th $W$ is chosen in 
such a way that each new $p_r$ corresponds to bringing in a new 
type of $W$. As a result, at each step a new $\theta$ or 
$\bar\theta$ appears (they are underlined in (\ref{verylong})), 
thus adding new odd dimensions to the G-analytic superspace. The 
only exception of this rule is the second step at which both a new 
$\theta$ and a new $\bar\theta$ appear. So, the series 
(\ref{verylong}) covers all possible subspaces with $4n, 
4n+4,4n+6,\ldots, 8n-2$ odd coordinates (notice once again the 
missing subspace with $4n+2$ odd coordinates). In this sense we 
can say that the G-analytic superfield 
$A(p_1,p_2,\ldots,p_{2n-1})$ realizes a ``short" superconformal 
multiplet.  

The superfield $A(p_1,p_2,\ldots,p_{2n-1})$ should be submitted to 
the same H-analyticity constraints as one would impose on 
$W^{1\ldots n}$ alone,
\begin{eqnarray}\label{alone}
 && D^{\, I}_{I+1}A(p_1,p_2,\ldots,p_{2n-1})=0\;, \nonumber \\
&&I=1,2,\ldots, 2n-1\;.
\end{eqnarray}
This is clearly compatible with G-analyticity since the conditions 
on a generic $A(p_1,p_2,\ldots,p_{2n-1})$ form a subset of these 
on $W^{1\ldots n}$. As before, H-analyticity makes 
$A(p_1,p_2,\ldots,p_{2n-1})$ irreducible under $SU(N)$. Here is 
the structure of Young tableau which corresponds to the first 
(scalar) component of this superfield (and characterizes the 
supermultiplet as a whole): 
\begin{figure}[h]
\setlength{\unitlength}{2mm}
 \begin{center}
\label{doublerep}     
 %\caption{$(m,\ell -m,m)$ representation:}
\begin{picture}(23,8)
%\caption{$(m,\ell -m,m)$}
\put(0,0){\framebox (4,4){\tiny $1$}}
\put(4,0){\framebox (20,4){\tiny $\cdots$}}
\put(24,0){\framebox (4,4){{\tiny $1$}}}
\put(29,1.5){$m_1$}
\put(0,-4){\framebox (4,4){\tiny $2$}}
\put(4,-4){\framebox (16,4){\tiny $\cdots$}}
\put(20,-4){\framebox (4,4){\tiny $2$}}
\put(25,-2.5){$m_2$}
\put(10,-5.5){$\cdots$}
\put(0,-10){\framebox (4,4){\tiny $k$}}
\put(4,-10){\framebox (12,4){\tiny $\cdots$}}
\put(16,-10){\framebox (4,4){\tiny $k$}}
\put(21,-8.5){$m_k$}
\put(8,-11.5){$\cdots$}
\put(0,-16){\framebox (4,4){\tiny 2n-1}}
\put(4,-16){\framebox (8,4){\tiny $\cdots$}}
\put(12,-16){\framebox (4,4){\tiny 2n-1}}
\put(17,-14.5){$m_{2n-1}$}
\end{picture}
  \end{center}
\end{figure}
\vskip 3cm \noindent The top row is filled with indices projected 
with $u^1_i$ (hence the symmetrization among them), the second row 
- with $u^2_i$, etc. The harmonic conditions (\ref{alone}) remove 
all the symmetrizations among indices belonging to different 
projections (rows). By counting the number of occurrences of the 
projection $1$ in (\ref{verylong}), we easily find the relation 
\begin{equation}\label{m-l}
  m_1 = \ell - p_{2n-1}
\end{equation}
where $\ell$ is the total number of $W$'s (equal to the dimension 
of the superfield $A$, since $\ell_W=1$). Another simple counting 
shows the relation 
\begin{equation}\label{sum-l}
  \sum^{2n-1}_{k=1}m_k = n\ell = {N\over 2}\ell\;.
\end{equation}
If the last $W$ in (\ref{alone}) is not present there is an 
additional relation among the Young tableau labels: 
\begin{equation}\label{simre}
  p_{2n-1}=0 \quad \Rightarrow \quad  m_1  = {2\over 
N}\sum^{2n-1}_{k=1}m_k\;.
\end{equation}
\vfill\eject

Finally, introducing the Dynkin labels $[a_1,\ldots,$ $a_{2n-1}]$ 
where $a_{1}=m_{2n-1}$ and $a_k = m_{2n-k+1}-m_{2n-k}$ for $k\geq 
2$, we find 
\begin{eqnarray}
  && a_{1}= \sum_{k=n}^{2n-1} p_k\;,  \nonumber\\
  && a_{2}= p_{n-1}\;, \quad  \ldots\;, \quad  a_{n-2}= p_{3}\;,  \nonumber\\
  && a_{n-1}= p_2 + \sum_{k=n+1}^{2n-1} (k-n)p_{k}\;, \nonumber\\ 
  && a_{n}= p_{1} \;,  \label{DL}  \\
  &&  a_{n+1}= (n-2)\sum_{k=n+1}^{2n-1} p_k + 
  \sum_{k=2}^{n} (k-1)p_{k}\;,   \nonumber\\
  && a_{n+2}= p_{n+1} \;, \quad  \ldots\;, \quad 
  a_{2n-1}= p_{2n-2}\;.   \nonumber
\end{eqnarray}

   

\setcounter{equation}0 
\section{Conclusion}

In this paper we studied representations of four-dimensional superconformal
algebras with an arbitrary number of supersymmetries.

This analysis also provides the classification of short multiplets of
superalgebras on AdS$_5$ and in particular ``massless" and ``massive" fields in
anti-de Sitter geometries, in terms of boundary ``composite" operatprs

\section*{Acknowledgements}

This  work  has been supported in part by the 
European Commission TMR programme ERBFMRX-CT96-0045 (Laboratori 
Nazionali di Frascati, INFN) and by DOE grant DE-FG03-91ER40662, 
Task C. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



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B. Morel, A. Sciarrino and P. Sorba, {\it Phys. Lett.} {\bf B166} (1986) 
69; erratum B167 (1986) 486.  

\bibitem{Sohnius} M.F. Sohnius, {\it Nucl. Phys.}  {\bf B138} (1978) 109.

\bibitem{So} M.F. Sohnius, {\it Nucl. Phys.}  {\bf B136} (1978) 461.

\bibitem{GIO} A. Galperin, E. Ivanov and V.I. Ogievetsky {\it JETP Lett.} 
{\bf 33} (1981) 168. 

\bibitem{str} J. Strathdee, {\it Int. J. Math. Phys.} {\bf 2} (1987) 273 and refs
therein. 

\bibitem{gm2}
M. Gunaydin and N. Marcus, {\it Class. Quantum Grav.} {\bf 2} (1985) 
L19.

\bibitem{fwz}
S. Ferrara, J. Wess and B. Zumino, {\it Phys. Lett.} {\bf B51} (1974) 
239 





\end{thebibliography}



\end{document}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Here we start...


{\init These pages} provide an example of the format employed for
publishing conference proceedings in the \href{http://jhep.sissa.it}
{JHEP Electronic Journal}.  

Proceedings contributions must be written using \LaTeXe{} and the
class {\sf JHEP}, (provided by the file {\tt JHEP.cls}) using the
option {\tt proceedings}.  This option is implemented starting from the
current version 2.0 of the {\sf JHEP} class.

As with previous versions of {\sf JHEP}, \LaTeX\ files written in {\sf
article} style are typeset without pain simply by changing the class
name (and adding some informations to build the title page).

While respecting the common latex syntax, {\sf JHEP} incorporates
features like automatic in-text figures and hyper-references 
for formul\ae, figures and bibliography. 

Hyper-functionality has been added to the normal \LaTeX{}
instructions so that there is no need to learn new commands, 
aside from new operations like ad-hoc hyper-linking inside or outside the
document, and sending e-mail. 

All these hyper-features are intended to work with viewers which
support HTML specials such as: {\sf xhdvi}, {\sf
\href{http://norma.nikhef.nl/~t16/ndvi_doc.html}{ndvi}} or {\sf
\href{http://www.geom.umn.edu/java/idvi/}{idvi}} for the {\tt dvi}
output; hyper-ghostview for the {\tt ps} format; {\sf pdf} viewers
after distillation.  To view articles on the Web, the use of {\tt dvi}
files is recommended because of their smaller size, and the better
quality of screen fonts.


{\tt JHEP.cls} has been developed to run on a version of \LaTeXe{} newer
than or equal to {\bf `June 96'}, for code compatibility.


\section{Syntax}

The following table shows the structure of a document, listing the
commands which appear in the preamble.

% DO NOT LOOK AT THE FOLLOWING, IT'S SPECIAL, DUE TO \tt.
\TABLE{
\fbox{\parbox{7.1cm}{\baselineskip .8\baselineskip\tt\noindent
					\let\lbr\{\def\{{\char'173}%
						  \def\}{\char'175}%
	\smallskip\ttbs documentclass[proceedings,opts]\{JHEP\}\\
	\smallskip\ttbs conference\{...\}\\
	\ttbs title\{...\}\\
	\ttbs author\{name \ttbs\ttbs\ addr..\ \ttbs\ttbs\ ..addr\}\\
	\ttbs abstract\{...\}\\
	\ttbs dedicated\{...\}\\
	\vdots\\
	\ttbs begin\{document\}\\
	\vdots\\
	\ttbs acknowledgments\\
	\vdots\\
	\ttbs end\{document\}
	}}\caption{Structure of a document}
}

The first command declares that you are using the {\sf JHEP} class
with the {\tt proceedings} option, and provides futher
\textref{options}{options} if needed.

Then the commands \verb!\conference!, \verb!\title!, \verb!\author!,
\verb!\abstract! are required tp build the \textref{titlepage}{title page}.

%A command \verb!\preprint! is active to insert the 
%preprint numbers at the top right-hand corner, unless 
%the \textref{published}{{\tt published} option} is provided.
%
%In this case the paper is assumed to be accepted for publication in JHEP and 
%its history must be supplied with the three commands \verb!\received!,
%\verb!\accepted!, \verb!\JHEP!.
%

This is automatically generated, before the text becomes twocolumn, at
\verb!\begin{document}!.  There is thus no need for a
\verb!\maketitle! command, or {\tt titlepage} option.

\TABULAR{|l|l|}{
\hline 
\tt proceedings	& Necessary to compose the proceedings conribution.\\
\hline
\tt (no)hyper,
		 & To choose between hyper and paper versions.\\
\tt nohyper $\equiv$ paper
		 & Hyper version is hyper-active and oneside.\\
		 & Paper version is not hyper-active.\\
\hline
\tt draft	& Enables the draft mode: labels and citations appear on\\
		 & the margin, processing info at the top of every page,\\
		 & and {\tt epsfig} figures appear as a box with the filename. \\
\hline
\tt a4paper(*),	& Standard paper size options. \\
\tt letterpaper	& Options {\tt a5paper, b5paper, landscape}\\
\tt legalpaper	& are not defined.\\
\tt executivepaper	&\\
\hline 
\tt 10pt(*)	& For proceedings, the fontsize is fixed to {\tt10pt}.\\ 
\hline
\tt oneside(*), twoside
	 & These set the page margins for one or two-side mode \\
	 & and the side of figures, tables and margin notes.\\
\hline
}{Options\label{options}}

An \verb!\acknowledgments! command can be used to insert the relative
section, at a proper place, e.g.\ at the end of the introduction or
after the conclusions.


\section{Options}

Table~\ref{options} shows the options which can be used 
at the \verb!\documentclass! command for {\sf JHEP}:

The option {\tt proceedings} is strictly necessary to produce a
proceedings paper, and usually it is the only one really needed.

Options with an asterisk ({\tt hyper}, {\tt a4paper}, {\tt oneside},
{\tt 10pt}) are the default behavior. Note that {\tt a5paper}, {\tt
b5paper}, {\tt landscape}, {\tt titlepage}, are not provided.

Also {\tt twocoumn} is not available, being  automatic for JHEP
proceedings.


\section{Typographic style}

The following defaults are fixed as the selected format and,
accordingly, some \LaTeX{} commands and options are disabled.


\subsection{Dimensions}

The width and the height of the text depend on the declared paper
type. They amount roughly to 74\% of the paper width and 77\% of the
paper height.

Following the ISO standard specification, the default paper size is
{\tt a4paper}, so the text dimensions are as shown on these pages:
width 15.5~cm, height 40 textlines.

US users which still use {\tt letter} paper, are able to print the
documents without loosing any information, even though they will loose
centering. They should generate their own postscript for letter paper,
starting from the {\tt dvi}.

\subsection{Pagestyle}

Headers and page numbering are as in the present example and
\verb!\pagestyle!, \verb!\thispagestyle! are ignored.

\subsection{Fonts} 

The selected text font for this two-column style is 10 points CM
Roman, and this is the only allowed choice.

The value of \verb!\baselinestretch! is set to 1.1 line spacing, please do not change it.



\subsection{Equations}\label{sec:eq}

Equations are formatted as in plain \LaTeX, like in the example at the
end of these pages. if you are using an hyper viewer, you can follow
automatic hyperlinks (see equation~\ref{eq:pi}).

The spacing around the {\tt\&}'s in the {\tt eqnarray} environment 
is slightly reduced, to match that of the {\tt equation} environment.

The long equations must be splitted across lines, because `twocolumn'
equations are not used.

Please insert the correct punctuation at the end of each group of 
equations, spaced by a ``\verb!\,!''.

%\subsection{Table of contents} \label{sec:toc}
%
%A table of contents is automatically inserted at the top of the second
%page, before the document starts. In case a document had no sections (e.g. 
%letters or short communicaions) the table of contents is omitted.
%
%In preprint mode, the \textref{options}{{\tt notoc}} option can be
%given to suppress this behavior.

 
\section{Title page}\label{titlepage}

The frontmatter informations are automatically inserted in the first
page, in `flushed left' style.  After this the text becomes twocolumn.

Note that the command \verb!\maketitle! and the option
\verb!titlepage!  longer exist.

The top line contains the JHEP logo, plus the conference
identification.  All the information on the title page is supplied
with the following commands which must appear in the preamble.


\subsection{Mandatory commands}

\fverb!\conference{...}!\fverbdo : This provides the conference name.
Please insert the short conference name provided by the organization,
in order for it to appear in the heading through the paper.

\fverb!\title{...}!\fverbdo : The title of the talk, of course.

\fverb!\author{...\\...}!\fverbdo : This command provides the name of
the speaker and his address.  The lines after `\verb!\\!'  are typeset as the
address, where line breaks can be obtained with further `\verb!\\!'
commands.

A single \verb!\author! should be used, assuming that each talk in a
conference is given by a single person. This is to be contrasted with 
the case of JHEP papers, where more \verb!\author! can be used.
Use \verb!\thanks! to list other collaborators.

The \textref{email}{\tt\ttbs email} command may be used (usually
at the end of the address, like in this example) to provide an e-mail link.

\fverb!\abstract{...}!\fverbdo : This provides the abstract, and
substitutes the old \LaTeX{} environment. By choice it has been
decided that only `horizontal' material may be put in it, to prevent
line breaks and display-math formul\ae. In-text formul\ae{} are, of
course, allowed.

The abstract is part of the title contents and, thus, must be declared
{\it before} \verb!\begin{document}!.


\subsection{Optional commands}

\fverb!\thanks{...}!\fverbdo : This command generates a footnote on
the title page and should be used inside \verb!\author! to provide
additional information for the speaker, like research grants etc., 
or  to list the other collaborators for the present work. 

%\fverb!\preprint{...}!\fverbdo : This command is used to provide the
%preprint reference numbers. It inserts this information at the top
%right-hand corner of the title page.

%\fverb!\keywords{...}!\fverbdo : This command inserts a line
%containing the keywords for the actual paper. 
%A \href{http://jhep/JOURNAL/keywords.html}{list of keywords} is 
%provided by the JHEP Journal. They must appear on the same line
%separated by commas.

\fverb!\dedicated{...}!\fverbdo : This command inserts a dedication
below the abstract, it is flushed right in small italic style.

%\fverb!\received{...}, \accepted{...}, \JHEP{...}!\fverbdo: These
%three commands should not be used by authors. They are automatically
%inserted by the Journal to show, in the head of the first page, the
%dates of arrival and acceptance, and to insert a `bop-hook' to show
%the vertical JHEP numbering in the published version.
%


\section{Inside the document}

Please use standard \LaTeX{}. This facilitates the adoption of a
uniform style and the correction of the manuscript. Special attention
is required only in the following cases.

\subsection{Figures and tables}

For document portability, JHEP requires figures to be produced either
in pure \LaTeX{} ({\tt picture} environment) or in encapsulated
postscript ({\tt eps}).

To insert figures and tables please use the commands described in the
following section, such that the style is uniform.

Tables should be light, i.e.\ should avoid triple lines, and even
double ones, if possible.

\subsection{Bibliography}

For published items, the style for JHEP is: the name of authors in
Roman font; the journal info as produced with the
\textref{sec:journ}{shortcuts} described in the appendix; the XXX
number of a preprint version reported at the end and enclosed in
square brackets, i.e.\ [\hepph{9912999}].  Also for these please use
the shortcuts.

For preprints, please indicate also the title of the paper, before the
XXX number (not enclosed in brackets) or any preprint specification.

Separate papers by the same authors with a semicolon `\verb!;!', and
add also a newline `\verb!\\!' if a new group of authors starts.
Terminate each entry with a full stop.

Every \verb!\cite! carries an automatic hyperlink to the relative 
entry. For this reason, do not use the {\sf cite} package.

The bibliography style of JHEP is also available as a {\sf Bib\TeX}
style \href{http://jhep.sissa.it/JOURNAL/JHEP.bst}{\tt JHEP.bst}.


\section{Extra commands}


\subsection{Figures and tables}

The {\sf JHEP} class can insert figures and tables via the two basic
commands {\tt\ttbs FIGURE}, {\tt\ttbs TABLE}. Both of them calculate
the width of their objects, and behave in different ways. In case the
object is larger than a column, it is placed on top of a page,
spanning the two columns (like table~\ref{options}). If the object is
smaller than a column width, it is inserted inside the column, as, for
example, table~\ref{mytable}. In case it is smaller than 60\% of the
column width, it is inserted {\it inside} the text, which flows around,
as for figure~\ref{myfigure} (this is however not recommended in
two-column mode).

Note that as with the \LaTeX{} environments {\tt figure} and {\tt
table}, these commands are just containers of the actual figure or
table.

\marginpar{\small\sffamily FIGURE \ref{myfigure} . . called . . here.}

\fverb!\FIGURE[pos]{body}!\fverbdo
\smallskip
\FIGURE{\epsfig{file=graph.eps,width=2.25cm}% 
        \caption{Made with \tt\ttbs FIGURE.}%
	\label{myfigure}}

\noindent This command inserts a figure like the example on the
\csname if@twoside\endcsname\ifodd\count0left\else right\fi\else
right\fi. The {\tt body} must contain `horizontal' material, {\it
i.e.\ no vskips}, and its width is automatically calculated. To select
a given width one should enclose the content in a \verb!\parbox!.

The \verb!\caption{...}! command can be used inside and works as
in \LaTeX.  A typical form of {\tt body} %, which uses {\sf epsfig}, 
is `\verb!\epsfig{...,width=...}\caption{...}!'.

\medskip

\fverb!\TABLE[pos]{body}!\fverbdo\smallskip\\ 
This command is similar to \verb!\FIGURE!. As with the \verb!table!
environment of \LaTeX{}, authors have to include a {\tt tabular}
environment in the {\tt body} to generate the actual table.

\medskip

With both commands the optional argument {\tt pos} can be used to
change the {\it vertical} placement, as in \LaTeX. By default it 
is {\tt ht}. Only in the case of objects smaller than one column, the
argument can be used to specify {\tt l} (left) or {\tt r} (right). 

%Both commands generate floating objects which are inserted in the
%text as soon as there is enough space on the page. 
%If the object happens to be larger than 60\%{} of the textwidth, 
%it is centered in the page like usual \LaTeX{} floats.
%\marginpar{\small\sffamily Table \ref{mytable} . . called . . here.}

%The optional argument {\tt pos} can be used to change the
%{\it horizontal} positioning, which by default is {\tt p}.
%This corresponds to: right hand side in one-side mode or 
%outer part of the pages in two-side mode. Note that due to 
%the title page, the side of pages is inverted with respect 
%to usual numbering.


\TABULAR{|c|l|}{
	\hline	\multicolumn{2}{|c|}{Vertical case}\\
	\hline	\tt ~ht (*)& {\it here} or {\it top}\\
	\hline	\tt b& {\it bottom}\\
	\hline	\multicolumn{2}{|c|}{Horizontal case}\\
	\hline	\tt ~r (*) & {\it right hand side}~~~~~\\
	\hline	\tt l	   & {\it left hand side}\\
	\hline}{\label{mytable}Placement of figures and tables.}


%In case the figure or table is larger than 60\%, the argument {\tt pos} 
%selects the {\it vertical} positioning as in \LaTeX{} \cite{LC}, with the
%default setting {\tt ht}. As a consequence, by default, the big floats appear
%either at the point where they are called or at the top of subsequent pages. 


%\medskip


Two further commands are provided as a shorthand for usual
forms of \verb!\FIGURE! and \verb!\TABLE!:


\fverb!\EPSFIGURE[pos]{filename}{capt}!\fverbdo\smallskip\\
this command produces a figure by calling the given {\it EPS
file} with \verb!\epsfig!. Thus, to use this command, 
{\sf epsfig} has to be invoked in the preamble with 
\verb!\usepackage{epsfig}!. 

This also gives the opportunity to
scale the figure via the keywords {\tt width} or {\tt height}, e.g.:
\verb!\EPSFIGURE{filename,width=...}{capt}!.
%
%
%It is equivalent to:
%\verb!\FIGURE[pos]{\epsfig{file=filename}\caption{capt}}!. 



\fverb!\TABULAR[pos]{align}{body}{capt}!\fverbdo\smallskip\\
This command can be used to produce a complete table: 
{\tt align} represents the alignment directives as in the
{\tt tabular} environment, {\tt body} contains the table entries, and
{\tt capt} the table caption. 

%This is equivalent to:\\
%\verb!\TABLE[pos]{\begin{tabular}{align}body\end{tabular}\caption{capt}}!

%\medskip

%{\small
%The code to implement these features is a modification of the
%\LaTeXe{} package \verb!floatfig! \cite{LC}, still being developed
%(presumably?) under the new name \verb!floatflt! \cite{fltf}.
%
%The package has been modified to comply with JHEP requests and to
%correct the following problems: incompatibility with a \verb!\marginpar!;
%in two-side mode floats to the following page retained the
%previous (wrong) position; floats were not typeset at top of page.
%
%}% END SMALL.

\subsection{Draft}

The {\tt draft} mode provides the following features, in addition to
the usual \csname if@draft\endcsname~~~~~~~~~~~~~\fi\verb!\overfullrule!:

\paragraph{processing info:} the heading shows various informations: 
class options, job name, time and date of processing.

\paragraph{citations:} for each \verb!\cite! command the corresponding
citation labels are reported in a margin note if possible, or locally.

\paragraph{labels:} for each \verb!\label! command the corresponding
label is reported in the margin if possible, or locally.  (See
eq.~\ref{eq:zero} or table~\ref{mytable}).


\subsection{Acknowledgments}

\fverb!\acknowledgments!\fverbdo : This command starts a new section
where the acknowledgments must be placed. It usually resides at the
end of the paper, before any appendix. E.g.:

\acknowledgments

For the present version {\sf JHEP 2.0.0}, I shall thank Don E. Knuth,
for \TeX{}, but not the \LaTeX{} team. The \LaTeX{} code is really
awkward.

\appendix 

\section{Appendix}

\paragraph{Equations:} 
An {\tt eqnarray} to test draft labels and hyper-references.
\begin{eqnarray}
\sum_{n=-\infty}^\infty x^n &=& \sum_{n=0}^\infty x^{-n} -1
				+\sum_{n=0}^\infty x^n\\
&= & {1\over1-1/x} -1 +{1\over1-x}=0\,.~~\label{eq:zero}
\end{eqnarray}
Of course the radius in eq.~(\ref{eq:zero})\ldots

\medskip
This is the example for sec.~\ref{sec:eq}, Euler's (ubiquitous) proo of the
existence of God, which contains all basic operations and objects of
complex algebra:
\begin{equation} {\rm e }^{i\pi} +1 =0\,. \label{eq:pi}\end{equation}



\paragraph{Shortcuts:}\label{sec:journ} 
The following shortcuts for high energy physics journals are provided, and
authors are encouraged to use them. They all require three arguments which
are, in order, the volume number, the year {\it(four digits)} and the page. 
For JHEP, the three arguments are: the month (two digits), the year (four
digits) and the journal number (three digits).

\begin{verbatim}
\adp   -> Adv. Phys. 
\ap    -> Ann. Phys. (NY)  
\arnps -> Ann. Rev. Nucl. Part. Sci. 
\atmp  -> Adv. Theor. Math. Phys. 
\cmp   -> Comm. Math. Phys. 
\cpc   -> Comput. Phys. Commun. 
\cqg   -> Class. and Quant. Grav. 
\hpa   -> Helv. Phys. Acta  
\ijmpa -> Int. J. Mod. Phys. A 
\ijmpb -> Int. J. Mod. Phys. B 
\jetp  -> Sov. Phys. JETP  
\jetpl -> JETP Lett. 
\jgp   -> J. Geom. Phys. 
\jhep  -> J. High Energy Phys.  
\jmp   -> J. Math. Phys. 
\joth  -> J. Operator Theory  
\mpla  -> Mod. Phys. Lett. A 
\mplb  -> Mod. Phys. Lett. B 
\nc    -> Nuovo Cim. 
\npa   -> Nucl. Phys. A 
\npb   -> Nucl. Phys. B 
\npps  -> Nucl. Phys. ... (Proc. Suppl.)
\pla   -> Phys. Lett. A 
\plb   -> Phys. Lett. B 
\ppnp  -> Prog. Part. Nucl. Phys. 
\pr    -> Phys. Rev. 
\pra   -> Phys. Rev. A 
\prb   -> Phys. Rev. B 
\prc   -> Phys. Rev. C 
\prd   -> Phys. Rev. D 
\pre   -> Phys. Rev. E 
\prep  -> Phys. Rep. 
\prl   -> Phys. Rev. Lett. 
\ptp   -> Prog. Theor. Phys. 
\rmp   -> Rev. Mod. Phys. 
\sjnp  -> Sov. J. Nucl. Phys. 
\yf    -> Yad. Fiz. 
\zetf  -> Zh. Eksp. Teor. Fiz. 
\zpc   -> Z. Physik C 
\ibid  -> ibid. 
\end{verbatim}

In addition, further shortcuts are provided for the {\tt xxx} archives. 
They require one argument, that is the combination of year (last two
digits), month (two digits) and archive number (three digits); 

\begin{verbatim}
\hepth -> {\tt hep-th/#1},  and for 
\hepex, \heplat, \hepph, \accphys, 
\grqc, \quantph, \nlinsys, 
\qalg, \alggeom, \solvint, \suprcon, 
\astroph, \chaodyn, \condmat.
\end{verbatim}


Notice that the shortcuts above also provide hyperlinks to the online data
of the corresponding references. For example:
\medskip

\noindent\verb!\cmp{108}{1987}{535}! \ $\rightarrow\ $\\ 
\null\hfill \ignorespaces\cmp{108}{1987}{535};

\noindent\verb!\jhep{07}{1997}{001}! \ $\rightarrow\ $\\
\null\hfill  \ignorespaces\jhep{07}{1997}{001};

\noindent\verb!\hepth{9604001}     ! \ $\rightarrow\ $ \\
\null\hfill \hepth{9604001}.

\subsection{Hyper-features} 

Unless the {\tt nohyper} option is issued (same as {\tt paper}) the
following hyper-features are activated:

\paragraph{email:} \label{email} A new command: 

\fverb!\email{!\it user@address\verb!}!\fverbdo\\
It inserts a {\tt mailto:user@address} link in the document to send
directly e-mail to the author. 
The e-mail address is always typeset in \verb!tt! font. 

\paragraph{href's:} There are three new commands: 

\fverb!\href{!\it link\verb!}{!text\verb!}!, \verb!\name{!tag\verb!}!,
\verb!\base{!URL\verb!}!\fverbdo\smallskip\\
The first two commands, the building blocks of hyperlinks, implement
the two HTML insertions \verb!<A HREF="link">text</A>! 
and \verb!<A NAME="tag">! \verb!</A>!. The command \verb!\base! provides the
base URL, used by some hyper-viewers to allow the use of relative
URL's. 
Lastly a control sequence \verb!\hash! which gives a \verb!#!, to be
used inside \verb!\href! for linking manually inside documents.

{\small(notice the absence of text inside the HTML tag, to make
target anchors invisible and avoid hyper-nesting problems).}

\paragraph{table of contents:} The table of contents which appears on 
the second page contains hyperlinks to sections and pages. 
{\small(Because the TOC is not customizable, the code has to redefine 
internal \LaTeX{} commands, potentially breaking future compatibility).}

\paragraph{bibliography and citations:} Each \verb!\cite! carries a
hyperlink to the relative item in the bibliography which appears at
the end of the document. See for example \cite{TeXbook,LC}. 
{\small (As for the table of contents, compatibility is broken).} All
references which use the provided \textref{sec:journ}{shortcuts}, are
linked to the corresponding online data on JHEP, {\tt xxx} archives or SPIRES.

\paragraph{pages:} Every page carries a target \verb!NAME=pag!{\it n} at
the top left corner. 

\paragraph{refs:} In addition to the usual behavior, the three commands
\verb!\label!, \verb!\ref!, \verb!\pageref! are hyper-active:
\verb!\label{!{\it mylabel}\verb!}! inserts an HTML tag \verb!<A NAME=ref-mylabel>!
into the text, again with zero width. 
The \verb!\ref{!{\it mylabel}\verb!}! command then carries a link to it. 
Also the \verb!\pageref{!{\it mylabel}\verb!}! command generates a
link to the correct page 
(see the link to fig.~\ref{myfigure} on page~\pageref{myfigure} or to
eqn.~\ref{eq:pi} on page~\pageref{eq:pi}). There is also a new command:

\fverb!\textref{!\it mylabel\verb!}{!text\verb!}!\fverbdo\smallskip\\
The argument {\it text} becomes a link to the given label. This can be used
to produce HTML-like text links, but should be used sparingly, in view of the
paper version. As an example of an internal link see the same
\textref{myfigure}{picture}. 

If you have set your WWWBROWSER environment variable, or are running nDVI,
you can follow the links to other documents. At present an external
document can be called as a 
\href{http://babbage.sissa.it/dvi/hep-th/9702201}{whole}, 
but in the future references to parts shall also be considered: for example
a section like {\it``\href{http://babbage.sissa.it/dvi/hep-th/9702201\hash
section.1}{Lightning review of string theory}''}.

{\small We have chosen to refer to the label explicitly in the link
to avoid problems with potentially dangerous redefinitions of
equation numbers in \verb!\@currentlabel! (for example spaces). 
Note that in this way hyper-referencing across documents will require the 
knowledge of the original label name. }





%


\listoftables		% ONLY DRAFT
\listoffigures		% ONLY DRAFT

\begin{thebibliography}{999}
\bibitem{TeXbook} D.E. Knuth, {\it The \TeX book}, Addison-Wesley 1986.
\bibitem{LC} M. Goossens, F. Mittelbach and A. Samarin, {\it The \LaTeX{}
                Companion}, Addison-Wesley 1994.
\bibitem{fltf} M. Dahlgren, {\it Package {\tt floatflt}},
		1994--1996, distributed with \LaTeXe{} 96/06/01.
\end{thebibliography}

\end{document}

\bye

















hep-th0002141.extracted_bib
\begin{thebibliography}{99}

\bibitem{mal}
J.M. Maldacena, {\it  Adv. Theor. Math. Phys.} {\bf 2} (1998) 231, 
{\tt hep-th/9705104}. 

\bibitem{gkp}
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, {\it Phys. Lett.} 
{\bf B428} (1998) 105, {\tt hep-th/9802109}. 

\bibitem{wit}
E. Witten, {\it Adv. Theor. Math. Phys.} {\bf 2} (1998) 253, 
{\tt hep-th/9802150}. 

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\end{thebibliography}