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\title{String bits in small radius AdS and weakly\\ coupled
${\cal N}=4$ Super Yang-Mills Theory: I}
\author{
Avinash Dhar, Gautam Mandal  and Spenta R. Wadia\\
{\it Department of Theoretical Physics,\\
Tata Institute of Fundamental Research,\\ 
Homi Bhabha Road, Mumbai 400 005, India.}
\\
E-mail: \email{adhar@theory.tifr.res.in,
mandal@theory.tifr.res.in, wadia@theory.tifr.res.in}}
%\received{\today} 		%%
%\accepted{\today}		%% These are for published papers.
%\JHEP{12(2001)999}		%% 

\preprint{\hepth{0304062}\\
TIFR/TH/03-09}
	% OR: \preprint{Aaaa/Mm/Yy\\Aaa-aa/Nnnnnn}
			  	% Use \hepth etc. also in bibliography.  

\abstract{
We study light-cone gauge quantization of IIB strings in AdS$_5
\times$ S$^5$ for small radius in Poincare coordinates. A picture of
strings made up of noninteracting bits emerges in the zero radius
limit. In this limit, each bit behaves like a superparticle moving in
the AdS$_5 \times$ S$^5$ background, carrying appropriate
representations of the super conformal group PSU$(2,2|4)$. The
standard Hamiltonian operator which causes evolution in the light-cone
time has continuous eigenvalues and provides a basis of states which
is not suitable for comparing with the dual super Yang-Mills
theory. However, there exist operators in the light-cone gauge which
have discrete spectra and can be used to label the states.  We
construct states of the free string in this basis.  We include all
possible states arising from discretizing the string into varying
number of bits. A non-zero value of the radius introduces interactions
between the bits and the spectrum of strings gets modified. We discuss
the leading perturbative corrections at small radius for a few simple
cases.  Potential divergences in the perturbative corrections, arising
from strings near the boundary, cancel. We suggest that the divergences
cancel in the computation of anomalous dimensions of \underbar{all}
physical string states because of the large amount of supersymmetry
present in the AdS$_5 \times $ S$^5$ background. Our work provides a
framework for a rigorous and detailed testing of the AdS/CFT
conjecture.}
\keywords{String theory, Gauge theory}
\dedicated{Dedicated to the memory of Bunji Sakita.}

\begin{document}
\section{Introduction} 

The connection between $4$-dim. gauge theories and string theory is an
idea with a long history that goes back to attempts to discover
strings and string dynamics in gauge theories. A part of this history,
in connection with the 1/N expansion, can be found in
\cite{Brezin:eb}. See also \cite{Polyakov:ca,Polyakov:1998ju}.
 The hope underlying such attempts was
to guarantee consistent string dynamics in the context of a theory in
4 spacetime dimensions rather than in 10 or 26 spacetime
dimensions. The main motivation for finding a description of gauge
theories in terms of a string theory was to solve the problem of quark
confinement. This problem remains unsolved even today.
For a semi-popular
account of this problem see \cite{Current}.


It took more than two decades to arrive at a precise formulation of
this idea in the form of the AdS/CFT correspondence proposed by
Maldacena \cite{Mal97}. The classic case of this correspondence is the
duality between $N=4$ supersymmetric Yang-Mills theory in 4-dimensions
and type IIB string theory on AdS$_5 \times$ S$^5$.  The clue came
from the seemingly unrelated problem of understanding thermal
properties of a class of black holes in terms of D-branes\footnote{see
\cite{DMW} for a recent review of this subject.}.

The AdS/CFT correspondence has been precisely formulated in the limit
when the 't Hooft coupling $\lambda=g_{YM}^2 N$ is large
\cite{GKP,W}. On the AdS side this means that $R^2=\alpha'\sqrt\lambda$
is large and the dual type IIB string theory is approximated by
supergravity. A correspondence between local gauge-invariant operators
of the gauge theory and local supergravity fields has been extensively
discussed in the literature \cite{MAGOO}. Besides the local operators,
the correspondence has also been studied for gauge-invariant Wilson
loop operators
\cite{MalWilson,PolRyc,GroDru,Berenstein:1998ij,Semenoff}. Here one is aided
by the fact that $\lambda=g_{YM}^2 N$ is precisely the semi-classical
expansion parameter for the loop equations.

In this paper we study AdS/CFT correspondence in the opposite limit
viz. $\lambda \rightarrow 0$. In this limit the gauge theory can be
analyzed perturbatively but the string theory is more difficult and
the standard semi-classical expansion of strings when
$R^2=\alpha'\sqrt\lambda$ is large breaks down. The problem is
conceptually similar to the treatment of gauge theories on the lattice
\cite{Wilson:1974sk,Polyakov:rs}
where the gluon picture breaks down at strong coupling. 
The zeroth order solution of the lattice gauge theory is obtained by
exactly diagonalizing the kinetic (square of the electric field) term
of the Hamiltonian. Gauge invariance (Gauss' law) then connects the
individual string bits on the links of the lattice into long strings
which can connect charged sources.  The potential energy (magnetic
term) is treated perturbatively and it corrects the energy and wave
function of the string state of the zeroth order Hamiltonian
\cite{KogSus}.

We adopt a similar method, for the strongly coupled world sheet string
theory, when $\lambda \rightarrow 0$. We assume that $N$ is large
enough so that string loops are suppressed and the first quantized
picture is an adequate starting point. We work in the light cone gauge
which leaves only global reparametrizations of the string as a
residual symmetry. We then consider the parameter space of the string
as a discrete lattice consisting of $M$ string bits
\footnote{Bit strings have been discussed  earlier \cite{Giles:1977mp}
and more recently in the pp-wave background \cite{BMN,G7,HV}. The idea of
string bits also appeared in 't Hooft's paper \cite{tHooft} on planar
diagrams.}.  At $\lambda = 0$, the string bits are non-interacting. On
each bit we have a Hilbert space which consists of the states of a
super-particle in AdS$_5 \times S^5$. 
The total wave function of the string is a cyclically symmetric direct
product of states carried by the bits. We explicitly present these
states.

The organization of this paper is as follows. In the next section we
will review the quantization of Green-Schwarz world-sheet type IIB
string theory in the light-cone gauge. This gauge-fixing uses Poincare
co-ordinates. In section 3 we will discuss a limit of this theory in
which the AdS radius vanishes. In this limit the theory reduces to a
system of non-interacting bits. We make this more explicit by
introducing discrete bits of string. Each bit behaves like a
superparticle moving in the AdS$_5 \times$ S$^5$ background, carrying
``appropriate representations'' of the superconformal group
\footnote{\label{fn:1a}In the light cone gauge, some of the generators of 
the superconformal algebra, which involve $\tilde x^-$, the non-zero
mode of $x^-$, do not have well-defined action on a single bit and one
needs to consider at least two bits to define their action.  A single
bit carries a representation of the superconformal algebra only in the
naive sense that $\tilde x^-$, which is defined in terms of gradients
of the independent coordinates or momenta, is taken to be zero by
fiat. We will discuss the action of these generators on multi-bit
states in Appendix D.}. In the light-cone gauge, the operator that
gives evolution in light-cone time has continuous eigenvalues. This
basis is not convenient for comparison with the gauge theory at the
boundary. For this purpose, it is desirable to work with the basis
frequently used in describing the UIR's of the superconformal group
PSU$(2,2|4)$. We identify the operators in the light-cone gauge which
have this basis as eigenstates\footnote{\label{fn:1b} One can
encounter continuous or discrete eigenvalues in a given physical
system depending on which operator is being diagonalized, e.g. in the
case of the two-dimensional rotor we have continuous eigenvalues of
linear momentum but discrete eigenvalues of angular momentum.}. In
section 4 we discuss the states of a single bit in such a basis and in
section 5 we discuss the construction of free string states out of
these bits. We discuss states made from a varying number of bits since
strings with different number of bits will be naturally produced when
string-string interactions are switched on. In section 6 we discuss
corrections due to a small but non-vanishing value of the AdS
radius. A non-vanishing value of the radius introduces bit-bit
interactions which modifies the string spectrum.  We compute the
leading order correction to the simplest, but potentially the most
singular, multi-bit states. We show that potential divergences from
the boundary $z=0$ cancel due to supersymmetry, leaving only finite
corrections (details in Appendix C) to the anomalous
dimensions. Section 7 contains some preliminary remarks on comparison
of our string states with gauge theory operators, a more complete
discussion being deferred to \cite{II}. We end with some discussion
and concluding remarks in section 8. Appendix A contains the continuum
expressions for superconformal generators as realized in our string
theory, and Appendix B the discrete version. Appendix C contains
details of perturbation calculation and the cancellation of
divergences. Appendix D contains a discussion on stringy features of
the superconformal generators and their action on multi-bit states.

In this paper, we will work in the $N\to \infty$ limit and
ignore string-string interactions.

\section{Type IIB strings in AdS background in the light-cone gauge} 

In this section we will review type IIB Green-Schwarz superstring in
the AdS$_5 \times$ S$^5$ background geometry in the light-cone gauge
\cite{MT, MTT}. This will also serve to set our notations and
conventions. We will be using Poincare co-ordinates since the
discussion of light-cone gauge fixing is simplest in these
co-ordinates.  Also, in these co-ordinates comparison with the
boundary gauge theory is the most direct. We will not address global
issues arising out of the use of a Poincare patch in the present work.

The AdS$_5 \times$ S$^5$ metric in Poincare co-ordinates is
\bea
ds^2={{\rm R}^2 \over z^2}(\eta_{\mu\nu} dx^\mu dx^\nu+dz^adz^a),
\label{twoone}
\eea 
where $\eta_{\mu\nu}=(-, +, +, +)$ is the Minkowski metric and the six
co-ordinates $z^a, a=1,2, \cdots, 6$ parametrize the S$^5$ and the AdS
``radial'' direction.  The transformation between the ``Cartesian''
$z^a$ and the more customary $z, \Omega_5$ is given by
\[
z^a= z \n^a, \quad \n^a\n^a=1, 
\]
where the six-dimensional unit vector $\n^a$ represents $S^5$ angles
$\Omega_5$. The $z\to 0$ limit is called the boundary whereas $z\to
\infty$ is called the horizon. 

The AdS radius $R$ is related to the 't Hooft coupling
$\lambda$ and the string scale $\alpha'$ by the standard
expression 
\be 
{\rm R}^2=\alpha'\sqrt\lambda, \quad \lambda=g_{\rm YM}^2 N
\label{coupling}
\ee
Note that in the world-sheet action the AdS radius $R$ naturally
appears in the combination ${\rm R}^2/\alpha'=\sqrt\lambda$.
This is the  effective tension of the string moving in
AdS$_5 \times$ S$^5$. For later convenience we introduce the
parameter
\be
{\rm T}= \frac{\sqrt \lambda}{2\pi}
\label{def-t}
\ee

In addition to the bosonic co-ordinates, appearing in (\ref{twoone}),
the world-sheet superstring action has two left Majorana-Weyl
spinors. In the light-cone gauge, the $\kappa$-symmetry of the
Green-Schwarz action can be used to gauge away half of the fermionic
degrees of freedom of each of these two spinors. The remaining $16$
physical fermionic degrees of freedom can be rearranged into two
fundamental representations of the R-symmetry group SU$(4)$,
$\theta^i, \eta^i, i=1, \cdots, 4$ and their conjugates, $\theta_i,
\eta_i$ . The $\theta^i$ are related to the $8$ linear Poincare
supersymmetries while the $\eta^i$ are related to the $8$ non-linear
``conformal'' supersymmetries of the $psu(2,2|4)$ superconformal
algebra. In fact, $\theta$ appears quadratically in the generators of
this algebra, while they also have terms quartic in $\eta$. These
later terms reflect the presence of curvature and the five form flux.

On the bosonic side, it turns out \cite{MTT} that the usual
light-cone gauge choice
\[
x^+ \equiv {x^3+x^0 \over \sqrt{2}}=p^+ \tau,
\] 
where $\tau$ is the world-sheet time co-ordinate, cannot be combined
with a conformal gauge on the world-sheet metric since that is not
consistent with equations of motion of the system. A modified gauge
choice for the world-sheet metric, $g_{\alpha\beta}$, which is
consistent with the gauge choice on $x^+$, is \cite{MTT}
\be
\sqrt{-g} g^{\tau\tau}={-1 \over \sqrt{-g} g^{\sigma\sigma}}=-{2\pi
z^2 \over \sqrt{\lambda}},
\label{gtt-gauge}
\ee
where $\sigma$ is the world-sheet co-ordinate running along the
string\footnote{This gauge choice is potentially singular near the
boundary, $z \rightarrow 0$. However, as we shall see later in this
paper, physical quantities seem to be regular, so presumably this is
just a gauge artifact. Note that $\lambda\to 0$ is not
a problem since the Hamiltonian \eq{twotwo} is
well-defined in this limit.}.

Before gauge fixing, the Green-Schwarz type IIB superstring moving in
AdS$_5 \times$ S$^5$ background manifestly has the global isometries
of the supergroup PSU$(2,2|4)$. The choice of a light-cone gauge
destroys manifest symmetry, but one can still derive expressions for
the generators of the symmetry algebra in terms of the light-cone
bosonic and fermionic degrees of freedom of the superstring \cite{MTT}. 
For example, the ``Hamiltonian'' which generates evolution in the
light-cone time variable $x^+$, and is one of the generators of the
superalgebra $psu(2,2|4)$, is given by $-\int d\sigma \ {\cal
P}^-(\sigma)$, where
\bea
{\cal P}^-\!\! &=&
\!\!  -{1 \over 2 p^+}\biggl(2 p^x p^{\bar x}- \del_z^2+
{1 \over z^2} \big[ l^i_jl^j_i +
4\eta_i {l^i}_j \eta^j+(\eta_i\eta^i -2)^2-\frac14\big]\biggr)-
{{\rm T}^2 \over 2p^+ z^4}(2x'\x'+(z_a')^2) \nn \\
&& +{{\rm T} \over z^2 p^+} \eta^i\rho^a_{ij}\n^a(\theta'^j-
i{\sqrt{2} \over z} \eta^j x')
+{{\rm T}\over z^2 p^+} \eta_i{\rho^a}^{ij}\n^a
(\theta'_j+i{\sqrt{2} \over z} \eta_j {\bar x}'), 
\label{twotwo}
\eea
Here, T is given by \eq{def-t}. The notation is as in Appendix A, in
particular ${l^i}_j$ is as in \eq{A19}.  In this appendix, for the
convenience of the reader, we have also given the light-cone gauge
expressions for the basic generators of $psu(2,2|4)$ algebra, together
with their (anti)commutation relations.  This algebra, and the
light-cone gauge realization of it, will play a central role in what
follows.

\section{String bits and the limit of zero radius}

We begin by noting that due to the gauge choice \eq{gtt-gauge} the AdS
radius R enters the expression in (\ref{twotwo}) through T only in
the terms that have $\sigma$-derivatives. This is, in fact, the case
with all the generators\footnote{except for generators mentioned in
footnote \eq{fn:1a} on page \pageref{fn:1a} which we discuss in
Appendix D.}, as can be seen from the expressions given in 
Appendix A.
As a consequence of this, in the limit R $\rightarrow 0$ all the
$\sigma$-derivative terms disappear from these generators and they
become pointwise local in $\sigma$. That is, the string ``breaks up''
into independent bits. Since the finite T terms are apparently
singular at the boundary $z \rightarrow 0$, one might worry whether it
makes any sense to think of a non-zero radius in (\ref{twotwo}) as a
perturbation. Later in this paper we will see by explicit calculations
that perturbation expansion in T is finite. With this post facto
justification, we will just go ahead with the naive zero radius limit.

To make the bit picture manifest, let us discretize $\sigma$ into a
lattice of $M$ points with a lattice spacing $\epsilon = l/M$, where
$l$ is the total length $l$ of the string. As usual, we need to
rescale continuum variables by powers of $\epsilon$ to get the lattice
variables. Thus, the fact that the total light cone momentum
$P^+=\int_0^l d\sigma \ p^+_{\rm cont.}  =lp^+_{\rm cont.}$ must be
equated to $M p^+_{\rm latt.}$ implies that the momentum of each bit
must be $p^+_{\rm latt.} = \epsilon p^+_{cont.}$. We will permit
ourselves the abuse of notation $p^+_{latt.}  =p^+$ when we are
discussing bit variables.  Similarly, the momentum density (at any
point $\sigma = n \epsilon$ along the string) in the directions $(x^i,
z^a), i=1,2; a=1,...,6,$  are related to the discrete momenta of the
$n$th bit by relations such as $p^i_n = \epsilon p^i(\sigma)$.  This
ensures that (a) the total momenta reduce to simple sums, e.g.
$P^i=\sum_1^M p^i_n$, and (b) since we do not rescale the $x$'s, the
continuum canonical commutation relations $[x^i(\sigma),
p_j(\sigma')]=i\delta^i_j \delta(\sigma-\sigma')$, translate to
$[x^i_n, p_{m,j}]=i\delta^i_j \delta_{mn}$, which is free of
$\epsilon$.  Here we have used $\delta(\sigma-\sigma')= 1/\epsilon
\delta_{mn}$.  A similar rescaling of the superpartners,
$\sqrt{\epsilon}\theta \rightarrow \theta, \sqrt{\epsilon}\eta
\rightarrow \eta$ removes the lattice spacing from their 
anti-commutation relations. The above rescalings are summarized in
\eq{rescalings}.
 
In terms of these discrete bit variables, the T-independent part of
the Hamiltonian, is given by
\be
P_0^- \equiv \sum^M_{n=1} p^-_{n}=\sum^M_{n=1}
 -{1 \over 2 p^+} \biggl(2p^x_np_n^{\bar x}-
\del_{z_n}^2 +{1 \over z_n^2}\biggl[(l^i_{n,j} l^j_{n,i})+4\eta_{n,i} 
{l^{i}}_{n,j} \eta^j_n+ (\eta_{n,i}\eta^i_n -2)^2-\frac14
\biggr]\biggr)
\label{threeone}
\ee
The subscript on $P_0^-$ denotes that T has been set to zero. 

We see that the lattice spacing has completely disappeared from the
expression on the right hand side of (\ref{threeone}). In fact, this
continues to be so even when finite T parts are included. This
property of the Hamiltonian is actually shared by all the generators
of the superconformal algebra. In the limit of vanishing AdS radius
then, we see that it is in this precise sense that the string ``falls
apart'' into non-interacting discrete bits. The states of the free
string are given by taking a tensor product over all the bits of the
states of a free string bit, appropriately cyclically symmetrized with
respect to the bit index, to satisfy the discrete version of the
residual global reparametrization invariance in the light-cone
gauge. The problem of solving for the spectrum of states of the free
(i.e. $\lambda=0$) superstring then reduces to the problem of solving
for the spectrum of a single string bit\footnote{Provided that
generators which involve $\tilde x^-$, such as $K^x$, which a priori
do not have any well-defined action on a single bit Hilbert space, act
appropriately on these tensor product states. See footnote
\eq{fn:1a}.}. It is to this problem that we will now turn our
attention. In Appendix B we have listed discretized expressions for
all the generators, including the finite T parts. We have also listed
there the canonical commutation relations satisfied by the bits. We
will need these expressions in the discussions that follow.

We end this section with a remark about our attitude towards the
discrete $M$-bit model. Rather than regarding discretization as merely
an approximation to a continuum (in which we are interested only in
the limit $M\to \infty, \epsilon\to 0$), here we wish to consider any
finite value of $M$ as well, including $M=1,2,...$. Thus, we will
consider below string states consisting of 1 bit, 2 bits etc.

\section{States of a single bit}

In thinking of the spectrum of states of a single bit, the first
difficulty that we face is that the spectrum of the single-bit
Hamiltonian $-p^-_{n}$ is continuous. Clearly the eigenstates of
$-p^-_{n}$ are not a good starting point for building free superstring
states since these states cannot be directly compared with the
gauge-invariant operators of the boundary theory. This is because the
latter have definite integer dimensions in the limit of vanishing 't
Hooft coupling $\lambda$. It is, however, easy to resolve this
problem. The unitary irreducible representations \cite{GM,KRN}of the
global symmetry group SO$(4, 2)$ of AdS$_5$ are well-known to be
labeled by three (half)integers corresponding to the representations
of the maximal compact subgroup SO(2) $\times$ SO(4)= SO(2) $\times$
SU(2) $\times$ SU(2). Here SO(2) denotes rotation in the $(-1,0)$
directions (generated by $S_{-1,0}$ of \eq{A1}) while SO(4) denotes
rotation in the remaining four directions (see \eq{A1},\eq{hyperb} for
notation). These quantum numbers are denoted as $(E_0, J_1, J_2)$
(\cite{GM}, see also \cite{ZFF}), where $E_0$ physically means the
energy, conjugate to time translations in global co-ordinates, and
$J_1$ and $J_2$ denote the angular momentum representations associated
with the two SU(2) factors in the SO(4). (see equations around
\eq{hyperb} in Appendix A for more detail). Equations
\eq{e0} and \eq{so4} give the translation of these generators into
the generators of the conformal algebra in $(3+1)$-dimensional
Minkowski space. It turns out from these considerations (see Sec 2.2
of \cite{SM}) the Euclidean dimension operator of the conformal
algebra (equivalently the dimension in the boundary gauge theory)
corresponds to $E_0$.

Clearly, the spectrum should be arranged in terms of simultaneous
eigenfunctions of $(E_0, J_1, J_2)$, or equivalently, $(H_+, H_-,
J_1-J_2)$, where $H_\pm \equiv E_0 \mp (J_1 +J_2)$ (on a suitable
highest weight state, see \eq{j1-j2}). The latter basis is
particularly convenient because these operators have 
simple expressions in the light-cone frame, viz.
\bea
H_+=-P^-+K^+, \quad H_-=-K^-+P^+, \quad J_1-J_2=J^{x {\bar x}}.
\label{threetwo}
\eea
Thus, we should find simultaneous eigenstates of these operators
(and {\em not} of $-P^-$) in order to determine which UIR's of SO$(4,
2)$ appear
\footnote{Similar operators have also been considered
in \cite{Metsaev:2002vr}.}. Representations of the full superalgebra, which
includes additional bosonic generators (of the $SU(4)$ R-symmetry), 
are constructed by acting on the abovementioned  eigenstates 
with supersymmetry ``raising operators'' \eq{raising} (cf.
\cite{GM}). 

\subsection{The spectrum of $h_+$}

In this section we will determine the spectrum of  the single-bit
operator $h_{+,n}$ where 
\be
H_+= \sum_{n=1}^M h_{+,n}
\label{h-def}
\ee
in the limit $\lambda=0$. In a later section we will determine the
perturbative correction to the spectrum at small $\lambda$. Since $H_+
= E_0 - (J_1+J_2)$ (on a highest weight state as mentioned above) and
since $J_1-J_2$, being an integer or a (half)integer, does not receive
perturbative corrections, the perturbative correction to the spectrum
of $E_0$ is the same as that of $H_+$, which, therefore, is the same
as the spectrum of anomalous dimensions in the gauge
theory. Henceforth, we will sometimes use the word ``Hamiltonian'' for
the operator $H_+$.

We will see that it is easy to determine $J^{x \bar x}$ on the above
wavefunctions. In order to determine the triple $(E_0, J_1, J_2)$, we
also need to evaluate $H_-$; this is not easy because it involves
$\tilde x^-$ and is hence non-local even at $\lambda=0$. However, we
will be able to measure $H_-$ indirectly on states of our interest
\footnote{\label{anom} It is important to note that $J_1, J_2$ are both
(half)integers and are immune from perturbative corrections.
Since $H_- - H_+ = 2 (J_1 + J_2)$, it is enough to consider perturbative
correction to $H_+$ which, as argued earlier, represents also
the perturbative correction to $E_0$.}  

The expression for $h_+$, as defined in \eq{h-def},
can be obtained from the equations (\ref{B2}) 
and (\ref{B7}) given in Appendix B, as follows
\be
h_+ =-\frac{\del_x \del_{\bar x}}{2p^+} + p^+ x \bar x
-{\del_z^2 \over 2p^+}+{p^+ \over 2}z^2 
+{1 \over 2p^+z^2} \big[({l^i}_j)^2+4\eta_i {l^i}_j \eta^j+
(\eta_i\eta^i-2)^2-{1 \over 4} \big]
\label{threefive}
\ee
Here and in the rest of this section we will drop the bit index $n$
to simplify the notation. Note that the last term in the above
containing fermions comes entirely from the $-p^-$ part of
$h_+$.

\subsubsection{\label{ho}States with no fermions=
eight bosonic oscillators}

To find the purely bosonic states, it is more instructive first to
transform from the ``polar coordinates'' $(z, \n_a)$ used in
\eq{threefive}, to ``Cartesian coordinates $z_a= z \n_a, a=1,...,6$.
The Hamiltonian appropriate to the ``Cartesian coordinates'', denoted
by $\tilde h_+$, is given by
\bea
\tilde h_+ &&=z^{-5/2} h_+  z^{5/2}
\nn\\
&&=
{p^x p^{\bar x} \over p^+}+ p^+ x {\bar x}+
{(p^a)^2 \over 2p^+}+{p^+ \over 2}(z^a)^2 +{1 \over 2p^+z^2}
(4\eta_i {l^i}_j \eta^j+(\eta_i\eta^i)^2-4\eta_i\eta^i).
\label{threethree}
\eea
Here $h_+$ refers to \eq{threefive}. The eigenfunctions
of \eq{threethree}, $\tilde \psi$, are related to eigenfunctions
$\psi$, by  $\tilde \psi = z^{-5/2} \psi$. Here $p^x
= - i\del_{\bar x}, p^{\bar x}= -i \del_x,
p^a = -i \del/\del z_a$ (see Appendix A and B).

On states that do not contain any fermions, the coefficient of $1/z^2$
in (\ref{threethree}) vanishes. The rest of the terms corresponds to
{\em eight harmonic oscillators} corresponding to the eight transverse
bosonic degrees of freedom of the light-cone gauge-fixed string.  The
spectrum of $\tilde h_+$ (hence of $h_+$) is obviously discrete
(in fact, integer\footnote{\label{integer}
Although each of the triple $(E_0, J_1, J_2)$ can be an
integer or half integer, it turns out that $H_+
\equiv E_0 - (J_1+ J_2)$ is always an integer.}), and is given by
\be
E= \sum_{\iota=1}^8 (n_\iota  + 1/2), \quad n_\iota=0,1,...
\label{ho-spectrum}
\ee
The scale in the harmonic oscillations is set by $p^+$, e.g. $\langle
x\bar x \rangle = \langle (z^a)^2 \rangle= 1/p^+$, although the
spectrum is free of $p^+$. The creation and annihilation operators are
defined in  Appendix B (see \eq{B1a}).

Note that in the supersymmetric theory eight bosonic oscillators are
accompanied by eight fermionic oscillators $\eta_i,
\theta_i, i=1,...,4$, as we expect from supersymmetry.

\subsubsection{General wavefunctions}

Let us now consider the more general case when fermionic degrees of
freedom are excited. We will first write down the result.

The spectrum  of $h_+$ (\eq{threefive}) is given by
(see footnote \eq{integer})
\be
e_+ = \alpha + 2 r + s + \bar s + 2,\quad  \alpha, r, s, \bar s \in 
{\mathbf Z}
\label{gen-spectrum}
\ee
where the corresponding eigenfunction is given by 
\be
\ket \Psi = 
\ket{\eta}\ket{\theta}\times 
\Upsilon^\alpha_{l;s,\bar s,r}(p^+,x,\bar x, z)
\times \ket{l}
\label{gen-wave}
\ee
Here $\ket{\eta}\ket{\theta}$ denotes the  part of the wavefunction
involving fermionic coordinates. $\ket{\eta}$ is  any 
polynomial in the four variables $\eta_i, 
i=1,\ldots, 4$, understood as acting on the fermionic ground state
$\ket{0}$. There are 16 such linearly independent 
polynomials: $1, 
\ \eta_i,\ \eta_i \eta_j, \ldots, 
\eta_1\eta_2\eta_3\eta_4 $. Similarly $\ket{\theta}$ denotes 
any polynomial in $\theta_i, i=1,\ldots, 4$. Recall that 
$\{\theta_i, \theta^j\}=
\{\eta_i, \eta^j\} = \delta^j_i, \; \eta^i\ket{0}=0
=\theta^i\ket{0}$. 

The various parts of the wavefunction depending on the bosonic
coordinates are as follows.
\be
\Upsilon^\alpha_{l;s,\bar s,r}(p^+,x,\bar x, z) := f(p^+)\   
h_{s,\bar s}
(x,\bar x)\ \psi^\alpha_{p^+,r}(z)  
\ee
Here $h_{s,\bar s}(x,\bar x)$ is the standard harmonic oscillator
wavefunction of a pair of complex coordinates (see
\eq{threethree} and Appendix B), at level $s, \bar s$ resp., and
\be
\psi^\alpha_{r}(z) = e^{-p^+ z^2/2}
z^{\alpha+{1\over2}} 
L_r^\alpha(p^+ z^2) \sqrt{\frac{2.r!}{(r+\alpha)!}}
\label{psi-alpha}
\ee 
$L_r^\alpha$ are generalized Laguerre functions, and
$\alpha$ is an integer defined below. 
$\int_{-\infty}^\infty dz\psi^\alpha_r(z)^2=1 $
for $p^+=1$.

The wavefunction $\ket{l}$ depends on the $S^5$ angles,
equivalently on the $\n_a, a=1,..6$, and is given by
the spherical harmonic
\[
\ket{l}= C_{a_1...a_l} \n^{a_1}...\n^{a_l}
\]
where $C_{a_1...a_l}$ is a symmetric, traceless, rank $l$ tensor.
$\ket{l}$ satisfies
$ l^{ab}l^{ba} \ket{l} = l(l+4) \ket{l} $ ($l^{ab}$ is 
defined in \eq{A19}).

The form of the wavefunction $f(p^+)$ is not important for our
discussions in this paper and we will postpone its description to
\cite{II}. 

\noindent\underbar{Definition of $\alpha$}

The integer $\alpha$ appearing in \eq{gen-spectrum} and
\eq{gen-wave} and onwards, depends on the $S^5$ angular
momentum $l$ and the fermion wavefunction $\ket{\eta}$,
as follows \cite{Met}.

\gap2

\noindent 
Case $l=0$: $\alpha= | 2 - F_\eta |$ where $F_\eta$ is the number of
$\eta$'s in the wavefunction $\ket{\eta}$.

\gap2

\noindent
Case $l\not= 0$:  
\begin{enumerate}
\item $F_\eta=0$ or $F_\eta=4$: $\alpha= l+2$.
\item $F_\eta=1$: the canonical choices for
the wavefunction $\ket{\eta}$, viz. $\eta_i, i=1,...,4$
do not diagonalize $h_+$. The operator $h_+$ splits the
4-dimensional space ${\V}_1= {\rm Span}\{\eta_i \} \equiv$
complex linear combinations of $\{\eta_i\}$, into two
2-dimensional spaces given by $P_1 {\V}_1$ and $ Q_1 {\V}_1$,
where $P_1, Q_1$ are orthogonal projection operators,
\be
(P_1)^i_j= \frac{l+4}{2l + 4}
\biggl(\delta^i_j - \frac2{l+4} l^i_j\biggr),
\ (Q_1)^i_j= \frac{l}{2l + 4}
\biggl( \delta^i_j + \frac2{l} l^i_j\biggr)
\label{p1}
\ee
If $\ket{\eta}$ belongs to $P_1 {\V}_1$, then $\alpha=l+1$,
if $\ket{\eta}$ belongs to $Q_1 {\V}_1$, then  $\alpha=l+3$.
\item
$F_\eta=3$: Once again the four canonical 
basis vectors $\eta_i \eta_j \eta_k$
do not diagonalize $h_+$. The operator $h_+$ splits the
4-dimensional space 
${\V}_3= {\rm Span}\{\eta_i \eta_j \eta_k\}$ into two
2-dimensional spaces given by 
$P_3 {\V}_3$ and $ Q_3 {\V}_3$, where $P_3, Q_3$
are orthogonal projection operators,
\bea
&& (P_3)^{ijk}_{mnp}= \frac{l+4}{2l + 4}\biggl(
\delta^i_{[m}\delta^j_n\delta^k_{p]} - \frac6{l+4} 
\delta^i_{[m}\delta^j_n l^k_{p]}
\biggr),
\nn\\ 
&& (Q_3)^{ijk}_{mnp}= \frac{l}{2l + 4}\biggl(
\delta^i_{[m}\delta^j_n\delta^k_{p]}+ \frac6{l} 
\delta^i_{[m}\delta^j_n l^k_{p]}\biggr)
\label{p3}
\eea
Here $[\ ]$ on the subscripts denote antisymmetrization
with weight unity.
If $\ket{\eta}$ belongs to $P_3 {\V}_3$, then $\alpha=l+1$,
if $\ket{\eta}$ belongs to $Q_3 {\V}_3$, then  $\alpha=l+3$.  
\item
$F_\eta=2$: Here the operator $h_+$ splits the
6-dimensional space ${\V}_2= {\rm Span}\{\eta_i \eta_j\}$ into three
2-dimensional spaces given by $P_2 {\V}_2, Q_2 {\V}_2 $ and
$R_2 {\V}_2$, where $P_2, Q_2, R_2$
are orthogonal projection operators,
\bea
(P_2)^{ij}_{mn} &&=\frac{l+4}{4(l+1)} \biggl(
\delta^i_{[m}\delta^j_{n]} -  
\frac{4(l+3)}{(l+2)(l+4)}\delta^i_{[m} l^j_{n]} +
\frac{4}{(l+2)(l+4)}l^i_{[m} l^j_{n]} \biggr)
\nn\\
(Q_2)^{ij}_{mn} &&=  \frac{l(l+4)}{2 (l+1)(l+3)}\biggl(
\delta^i_{[m}\delta^j_{n]}
+ \frac{8}{l(l+4)}\delta^i_{[m} l^j_{n]}
- \frac{4}{l(l+4)}l^i_{[m} l^j_{n]} \biggr)
\nn\\
(R_2)^{ij}_{mn} &&= \frac{l}{4(l+3)}\biggl(
\delta^i_{[m}\delta^j_{n]}
 +\frac{4(l+1)}{l(l+2)}\delta^i_{[m} l^j_{n]} +
\frac{4}{l(l+2)}l^i_{[m} l^j_{n]} \biggr)
\label{p2}
\eea
If $\ket{\eta}$ belongs to $P_2 {\V}_2$, then $\alpha=l$,
if $\ket{\eta}$ belongs to $Q_2 {\V}_2$, then  $\alpha=l+2$,
if $\ket{\eta}$ belongs to $R_2 {\V}_2$, then  $\alpha=l+4$.
\end{enumerate}

\noindent\underbar{The value of $J^{x \bar x}$}

\eq{gen-wave} is also an eigenfunction of $J^{x \bar x}$
(see \eq{A11}), with eigenvalue
\[
J^{x \bar x}= s - \bar s + \frac12 (F_\eta - F_\theta)
\]
 
\noindent\underbar{A note about normalizability}

Note that in \eq{gen-wave} we have considered only {\em normalizable}
solutions 
\footnote{In the light-cone gauge, the time evolution of the states of
a bit is governed by a Schrodinger type evolution equation in flat
space with a potential. The appropriate norm in this case is the usual
Schrodinger norm.}. This is appropriate for constructing the space of
states of the superstring. Alternatively, this can be seen from
requiring that the solution be regular at $z
\rightarrow \infty$. This condition uniquely picks the normalizable
solution because of the harmonic oscillator potential piece in
(\ref{threefive}).

\subsubsection{\label{1bit-comparison}Comparison with supergravity}

The spectrum of states for a bit that we have obtained here is
identical to that of the type IIB supergravity multiplet.
For instance, the 20-plet of states
\be
\ket{A}= A^{ij,kl}\eta_i \eta_j \theta_k \theta_l \ket{0}\times 
\Upsilon^0_{0;0,0,0}(p^+,x,\bar x, z)\ \ket{l=0},
\label{basic-state}
\ee
where the tensor $A^{ij,kl}$ belongs to the ${\bf 20'}$, i.e.
the Dynkin label $(0,2,0)$ of SU(4),
corresponds to the part of the supergravity multiplet
characterized by $(2,0,0| {\mathbf{20'}})$. The fact that
the $so(4,2)$ quantum numbers $(E_0,J_1,J_2)$ are
given by $(2,0,0)$ follows from the fact that
$h_+=2, j^{x\bar x}=0$ and, as we discuss in Appendix D,
$h_-=0$ (considering the local terms only).

Note that since the states of a single bit just constitute the
supergravity multiplet these states are all part of a short
multiplet. This is consistent with the non-renormalization theorems,
at least at the planar level, since a single bit cannot receive
corrections due to bit-bit interactions.


\section{\label{string-spectrum}String spectrum}

Let us now discuss the construction of the states of the superstring
from the states of non-interacting bits obtained in the previous
section. There are three points to note here:

\begin{enumerate}

\item The phase space of the string is given by the canonically
conjugate pairs in \eq{funda-canon}. This shows that $p^+$
does not depend on $\sigma$; consequently all bit wavefunctions
must share the same value of $p^+$.

\item We must include in our discussion multi-string states, the $r$th
string being made up of $m_r$ bits such that $\sum_r m_r=M$. This is
because when string-string interactions are taken into account, the original
string composed of $M$ bits can split into strings with smaller number
of bits. In effect, what this means is that in the single string
sector we should consider all states with any number of bits $m \leq
M$.

\item Physical string states must satisfy the constraint coming from the
residual global reparametrization symmetry in the light-cone gauge. In
the discrete version what this means is that a physical state of the
string with $m$ bits can be obtained by cyclically symmetrizing any
tensor product state of $m$ bits. Thus, a physical state of a 
string with $m$ bits is
\be
\ket{p^+; \Psi_1, \Psi_2, \cdots}
= \sum_{\sigma\in P}
\ket{p^+; \Psi_{\sigma(1)}} \otimes 
\ket{p^+; \Psi_{\sigma(2)}} \otimes \cdots \otimes 
\ket{p^+; \Psi_{\sigma(m)}}.
\label{fourone}
\ee
Here $P$ is the group of cyclic permutations of $m$ variables.  We
have inserted the value of $p^+$ as an explicit reminder of point (1)
above. Henceforth, in future references to \eq{fourone} the $p^+$
factors will not be explicitly written.

\end{enumerate}

\subsection{String spectrum at $\lambda=0$}

For the state \eq{fourone}, the value of $H_+$ at $\lambda=0$ is
clearly given by
\be
E_{+,0} = \sum_{n=1}^m e_{+,n}
\label{multi-spectrum-0}
\ee
where $ e_{+,n}$ denotes the value \eq{gen-spectrum} for
the state  $\ket{p^+; \Psi_n}$.

As an example, we can construct a state \eq{fourone}, where we take
each wavefunction to be of the type given by \eq{basic-state}.
Explicitly, the state will look like
\be
\ket{\Psi_0}= {\cal A}_{i_1, j_1,...,k_m, l_m}\prod_{n=1}^m
\eta_{i_n}\eta_{j_n} \theta_{k_n} \theta_{l_n} \ket{0}
\Upsilon^0_{0;0,0,0}(p^+, x_n, \bar x_n, z_n)\ket{l=0}
\label{basic-multi-bit}
\ee
Here we have allowed for a general polarization ${\cal A}$ rather than
a simple cyclically symmetrized product of individual polarizations
$A^{(n)}_{i_n, j_n; k_n, l_n}$.

If we take the polarization ${\cal A}$ to correspond to a traceless
symmetric product of the individual polarizations $A^{(n)}$'s, then
the state becomes 
\be
\ket{\Psi_{\rm sym}}= {\cal A}^{\rm sym}_{i_1, j_1,...,k_m, l_m} 
\prod_{n=1}^m
\eta_{i_n}\eta_{j_n} \theta_{k_n} \theta_{l_n} \ket{0}
\Upsilon^0_{0;0,0,0}(p^+, x_n, \bar x_n, z_n)\ket{l=0}
\label{sym-wave}
\ee
Under $psu(2,2|4)$, this state transforms as 
$ (2m,0,0| \mathbf{0,2m,0})$ which is a BPS state made out of $m$ bits
(this can be seen by noting that in the notation of Gunaydin et al,
this corresponds to the multiplet based on $\ket{0}$ with $p=2m$
oscillators).

\section{Finite radius correction to string spectrum}

\begin{itemize}

\item Single-bit string states:
As we remarked in section \ref{1bit-comparison}, the single
bit string states are all BPS as they cannot receive
corrections form bit-bit interactions.

\item Multi-bit string states: 

In this section we will compute corrections to the spectrum of string
states due to bit-bit interactions introduced by turning on a small
T$= \sqrt \lambda/(2\pi)$ (see \eq{def-t}).

Let us write
\[
H_+ = H_+^0 + {\rm T} V_1 + {\rm T}^2 V_2
\]
where $H_+^0, V_1, V_2$ are defined in
\eq{h0-v1-v2}. 
The leading correction to the spectrum \eq{multi-spectrum-0} is given
by T$^2\Delta E_+$, where
\be
\Delta E_+ =
\langle \Psi_0 | V_2 | \Psi_0 \rangle -
 {\sum_{int}}' 
\frac{ \langle |\Psi_0 | V_1 | \Psi_{int} 
\rangle|^2}{E_{+,int}-E_{+,0}}
\label{delta-e}
\ee
In Appendix C we present a calculation of $\Delta E_+$
for the state $\ket{\Psi_0}$ given in \eq{basic-multi-bit}.
Here we present the main points. 

\end{itemize}

\subsection{Potential divergence from the region $z\to 0$}

Consider the $V_2$ term in \eq{delta-e}. Let us look at the term
involving $(x_{n+1} - x_n)^2/ z_n^2$ in $V_2$ (see
\eq{h0-v1-v2}). The numerator and the denominator separate out in the
expectation value: the first gives a simple factor of $1/p^+$ and the
second an integral of the kind
\[
\int_0^\infty \frac{dz_n}{z_n^4} |\psi^{\alpha=0}_{r=0}(z_n)|^2
\] 
Using $\psi^0_0 (z) \sim \sqrt z$ near $z=0$ (see \eq{psi-alpha}) the
above integral, summed over $M$ bits, gives a leading divergence
\be
M \int_{z_{min}}^\infty \frac{dz}{z^3} \sim M\ \frac{1}{z_{min}^2}
\label{v2-div}
\ee
Here we have put an infra-red cut-off $z_{min}$ near the boundary 
$z=0$. 

As for the $V_1$ contribution in \eq{delta-e} (see
Appendix C for details), the  matrix element 
$ \langle \Psi_0 | V_1 | \Psi_{int} \rangle$ has
finite $z$-integrals of the type
\be
\sum_n
\int_0^\infty 
\frac{dz_n}{z_n^3} \psi^{\alpha=0}_{r=0}(z_n) \psi^{\alpha=2}_{r}
(z_n) = M C_1(r)
\label{c1-r}
\ee
where $C_1(r)$ also includes the finite angular and fermion matrix
elements. The $z$-integral is finite (esp. at $z\to 0$), being $\sim
\int dz z^{1/2 + 7/2}/z^3$, using the fact that for $z\to 0$,
$\psi^\alpha_r (z) \sim z^{\alpha+1/2}$ (see
\eq{psi-alpha}).  The initial state has $\alpha=0, r=0$.  The reason
for the appearance of $\alpha=2$ in the intermediate state is that
$V_1$ contains (i) a $\n^a$ which converts the initial $l=0$
state to a $l=1$ state, and  (ii) an $\eta_i\eta_j$ term  (or its
conjugate), which converts the initial $F_\eta=2$ state to a 
$F_\eta=4$ (or 0, respectively). For such fermion states,
(see {\it definition of} $\alpha$  following \eq{psi-alpha}),
we have  $\alpha=l+2$ which $=3$ in this case.

However, although we do not have a divergent matrix element, we 
have a sum over an infinite number of intermediate states,
given by
\be
- M \sum_{r=0}^\infty \frac{C^2_1(r)}{E_1(r)} = 
- M\sum_{r}^{r_{max}}
\big(A_1 + \frac{B_1}{r} + O(\frac1{r^2})\big) \sim 
- M \big(A_1 r_{max} + B_1 \ln r_{max}
+ {\rm finite}\big)
\label{v1-div}
\ee
since $E_1(r) \equiv E_{int}(r) -E_0
\sim r, C_1(r) \sim \sqrt r$ 
for large $r$.

\subsection{Cancellation between $V_1$ and $V_2$ contributions}

The two divergences \eq{v2-div} and \eq{v1-div} are
hard to compare since the cut-off's are different. There is
a way, however, to recast the $V_2$ calculation using the
$r_{max}$ cut-off. By using (see \eq{A6}) $\{Q^{-}_i,
Q^{-j}\}= -P^-\delta^j_{i}$ we can write the $V_2$ term 
in \eq{delta-e} as 
\[
\frac14 {\sum_{int}}' 
| \langle \Psi_0 | \sum_{n} q^{-i}_{n+1,n} | \Psi_{int} 
\rangle|^2
\]
In a manner similar to \eq{v1-div}, the matrix element is now finite,
say $C_2(r)$ (which includes a finite $z$-integral !)
\be
M \sum_{r=0}^\infty {C_2^2(r)} = 
M\sum_{r}^{r_{max}}
\big(A_2 + \frac{B_2}{r} + O(\frac1{r^2})\big) \sim 
M \big(A_2 r_{max} + B_2 \ln r_{max}
+ {\rm finite}\big)
\label{v2-div-new}
\ee
As we will find in Appendix C, $A_1=A_2, B_1=B_2$. 

This implies that the divergent contributions from $V_2$ and
$V_1$ in \eq{delta-e} cancel, leaving a finite result
\footnote{It might seem surprising at first that there 
is a cancellation of divergences between two rather different looking
terms. This is, however, standard in supersymmetric theories. E.g. in
a theory with one scalar superfield with $\Phi^3$ interaction the mass
corrections cancel between a second order perturbation coming from
$V_1=g \bar\psi \psi \phi$ and a first order perturbation coming from
$V_2=g^2 \phi^4$.}. Since for the $V_2$ contribution, we have
calculated the same quantity using the $z_{min}$ cut-off as well as
the $r_{max}$ cut-off, these two cut-offs are clearly related
\[
\frac1{z_{min}^2} \sim r_{max}
\]
reflecting a UV-IR relation. This in particular implies that the
divergence from both $V_1$ and $V_2$ can be interpreted as coming from
strings near $z=0$. 

\subsection{Other states}

In the above we have considered as initial state tensor product of
specific single-bit states. Our choice was prompted by the maximally
divergent $z_{min}$ behaviour in the $V_2$ contribution. For instance,
instead of an $F_\eta=2$ state, if we take $F_\eta=1$ or $F_\eta=3$
states, we will have a $\ln z_{min}$ divergence. In fact these two
cases are the only ones with a potential divergence. None of the
infinite number of other states, with additional $s, \bar s$ or $r$
excitations, or $F_\eta=0 $ or $F_\eta=4 $, have any divergent
$z$-integration.  The methods we have employed work pretty much the
same way for the other divergent initial states. 
Calculations to check the cancellation of divergences in
these remaining cases are in progress and will be
reported in \cite{II}. 

\subsection{Summary of this section} 

The fact that the $z\to 0$ divergences cancel implies that our
postulate of treating the string world sheet theory perturbatively
around $\lambda=0$ is the correct one. It is clear that in a purely
bosonic model without fermionic terms such a cancellation would not
have occurred. We believe that the cancellation will persist to all
orders of perturbation theory. It is clearly important to 
explicitly check this.

\section{\label{ads-cft} AdS/CFT}

In this section we make some preliminary remarks on comparison with
the boundary gauge theory. More detailed discussions will be presented
in \cite{II}.

%\begin{enumerate}

\myitem 1. \underbar{Single bit, $m=1$}:
We have remarked in the section \ref{1bit-comparison} that the states
of a single bit are simply the supergravity multiplet in AdS$_5\times
S^5$ background. The latter have been extensively studied \cite{MAGOO}
and shown to be in one-to-one correspondence with the local,
gauge-invariant single-trace chiral operators of the boundary gauge
theory. It appears, therefore, that we have a complete equivalence of
these Yang-Mills  operators with single-bit string states in
our theory, which corresponds to $m=1$ in the notation of point (2)
in section \ref{string-spectrum}. 

\myitem 2. \underbar{$m>1$}:
If we consider $m=2$ in \eq{sym-wave} we get the state
$(E_0, J_1, J_2 |{\mathtt{SU(4)\ repr.}})=(4,0,0|{\mathbf{105}})$, 
which, again, is  BPS. This state
matches the quantum numbers (under $psu(2,2|4)$) of the single trace
operator $O_4=A_{(abcd)|}{\rm Tr} (\phi^a \phi^b\phi^c
\phi^d )$. However as we argued in the
previous paragraph, such a gauge theory operator should already
correspond to a single-bit state.  We conclude, therefore, that the
$m=1$ and $m=2$ states in our construction must correspond to two
independent linear combinations of $O_1$  
and a double-trace operator $O_{2,2}=A_{(abcd)|}{\rm Tr} (\phi^a
\phi^b){\rm Tr}(\phi^c \phi^d )$. In the absence of 
string-string interactions
it is not possible to distinguish between the two choices but 
such interactions are likely to make a distinction \cite{G4}.

\myitem 3. \underbar{Counting of states}:

In the simple discussion above, the number of candidate gauge theory
operators and the number of string states agree. We can generalize
this observation. Consider the gauge theory operator
$O_3=A_{(abc)|}{\rm Tr} (\phi^a \phi^b\phi^c)$.  With these quantum
numbers, viz. (3,0,0$|${\bf 50}), this is the only one, since in a
SU(N) YM theory, we cannot have $O_{1,2}=A_{(abc)|}{\rm Tr}
(\phi^a){\rm Tr}(
\phi^b\phi^c)$. Interestingly the story is identical in our string
construction. As we remarked earlier, corresponding to $O_3= $
(3,0,0$|${\bf 50}), there {\em is} a single-bit string state.
It turns out that  there
cannot be a multi-bit state. This is because even for $m=2$ bits, the
minimum value of $E_0$ will be 4, one higher than that of $O_3$. The
fact that Tr$(\phi)$ is not allowed in the SU(N) theory corresponds to
the string theory statement that the minimum value of $E_0$ in a
single bit is 2 (corresponding to
\eq{basic-state}), not 1. Generalization to operators of higher
dimensions is obvious: at $E_0=5$ we have $O_5= {\rm Tr} \phi^5,
O_{2,3}= {\rm Tr} \phi^2 \ {\rm Tr} \phi^3 $, and there are two string
states, a single bit state and a 2-bit state (for any given rank-5
tensor $A_{(abcde)|}$).

In general, it appears that for a given dimension $\Delta_0$ at
$\lambda=0$, there are the same number of the above type of operators
Tr$(\phi^{\Delta_0})$ in the gauge theory as in the string theory (for
any given PSU(2,2$|$4) quantum numbers), both numbers being equal to
the `restricted partition' of the integer $\Delta_0$. Here `restricted
partition' of a number means partitions which do not use unity. Note
that these statements are made for $E_0$ much less than $N$, the rank
of SU(N). Note also that this counting is different from the number of
PSU(2,2$|$4) representations for $O_{\Delta_0}$.

The above discussion suggests that the BPS states of the string theory
constructed in this paper (in the absence of string-string interactions) are
in one-to-one correspondence with the chiral operators of the
$N=\infty$ super Yang-Mills theory. At finite $N$, Tr$\phi^l, l\ge N$
are not to be counted as independent operators any more. It is an
interesting check on our string theory framework to see how such
finite $N$ effects can be understood.

Generically we expect a multi-bit state to be a non-BPS state
corresponding to a massive string excitation. For the special
multi-bit states discussed above to be BPS it is therefore necessary
to show that their anomalous dimensions indeed vanish.  Verifying this
constitutes an important check on the correspondence mentioned above.

The counting of all gauge theory single trace operators with various
quantum numbers at a given dimension $\Delta_0$ has been performed in
\cite{Polya,Sundborg} using Polya's combinatorial
theorem\footnote{A more refined 
calculation with implications for the thermodynamics
of gauge theories at large N was recently discussed in
\cite{Shiraz}.}. A similar technique applied to our
single bit states shows that the number of string states at high
energies grows exponentially. It is also of interest  to redo the
counting in \cite{Polya} to find out the expected number $N$ of
letters in a gauge invariant ``word'' for a given conformal dimension
$\Delta_0$. This can be done by a simple modification of the free energy
given in \cite{Polya}. The result is shown in a plot here.


\hspace{2ex}
\begin{figure}[ht]%or [htb]
   \vspace{0.5cm}
%\hspace{-1in}
\centerline{
   {\epsfxsize=7cm
   \epsfysize=5cm
   \epsffile{n-vs-delta.eps}}
}
\caption{\sl Expected word-size $\langle L \rangle$ 
(y-axis) at a  given conformal dimension $\Delta_0$
(x-axis) grows linearly.
The fluctuation $\langle L^2 \rangle - \langle L \rangle^2$
becomes smaller for larger values of $\Delta_0$.}
\label{linear}
\end{figure}

\myitem 4. \underbar{Anomalous dimensions}:
As we have shown in the previous section, our construction of
strings at small radius AdS allows us to explicitly compute 
correction to $E_0$ in perturbation theory. This
allows us to compare \cite{II} with anomalous dimensions
of gauge invariant operators in the Yang-Mills theory in
perturbation theory around $g_{YM}=0$. Along with the counting,
this constitutes the most important check of the AdS/CFT 
correspondence.

%\end{enumerate}


\section{Discussion and concluding remarks} 

In this paper we have constructed string theory in AdS at small
radius. The construction uses light cone gauge.  We have computed the
free spectrum as well as demonstrated that to order $O(\lambda)$ the
corrections to the conformal dimension are finite. Potential
divergences which arise from strings near the boundary $z=0$ cancel
because of supersymmetry. 

The main point of our paper is the establishment of a working
framework of string theory in AdS at $\lambda\to 0$. Some of the
possible uses and future problems are listed:
\begin{enumerate}

\item  
One obvious  use is to test the strong ($\lambda\to 0$) version of
Maldacena conjecture, which we have remarked on in section
\ref{ads-cft}. We postpone
a more detailed discussion to \cite{II}. The strong version of the
conjecture allows definition of non\-perturbative string theory (in
AdS) in terms of gauge theory.

\item 
We have here a calculable framework of strings coupled to a noncompact
space whose gravity description approaches a singular limit. This
example is more non-trivial than that of a string theory in
backgrounds where the compact part of the spacetime includes an
orbifold or a conifold singularity. The singularity here is not only
in spacetime but the entire spacetime collapses, in a manner similar
to the big bang singularity. Calculation of string theory correlation
functions in such a background should be interesting.

\item
The success of the light-cone gauge, at least perturbatively,
encourages its use in other backgrounds which have
physical gauge theory duals, especially those which describe a
confining gauge theory like \cite{KleStr}.

\item
It is clearly important to extend the present framework
to take into account  string-string interactions,
which we have ignored in this paper.

\end{enumerate}

\gap5

\noindent{\bf Acknowledgments}

\gap5

\noindent 
We are happy to acknowledge many useful discussions with Shiraz
Minwalla.  Part of this work was done during the String Theory
workshop (December, 2002) at the Harish Chandra Research Institute,
Allahabad. We would like to thank the organizers of the meeting for
providing a stimulating atmosphere and for their excellent
hospitality. One of us (SRW) would like to thank the ASICTP for
hospitality during the final stages of this work.

\newpage

\noindent{\bf Dedication:}

\gap5

This work is dedicated to the memory of Professor Bunji Sakita, who
passed away on 31st August 2002. Prof. Sakita made many pioneering
contributions to high energy physics and in particular to string
theory. His `Reminiscences' \cite{Sakita} are a striking summary of an
exciting period in high energy physics. He was a great mentor and a
deeply humble and generous man. He also contributed significantly to
the high energy physics effort at the Tata Institute of Fundamental
Research in Mumbai. His obituary can be found at
http://www.cerncourier.com/main/article/42/9/20/3.

\newpage

\appendix

\section{Generators of the superconformal algebra in the 
light-cone gauge}

The bosonic part of the $psu(2,2|4)$ superalgebra is the so$(4,2)$
algebra. In terms of the standard hermitian rotation generators,
$S_{AB}$, this algebra is
\bea
[S_{AB},S_{CD}]=-i(\eta_{BC}S_{AB}-\eta_{BD}S_{AC}+\eta_{AD}S_{BC}
-\eta_{AC}S_{BD}),
\label{A1}
\eea where $A,B, \cdots =-1,0,1, \cdots, 4$ and the metric 
$\eta_{AB}$ is diag $[-1,-1,1,1,1,1]$. The $S_{AB}$
acts on AdS$_5$, viewed as the hyperbola
\be
\eta_{AB} Y^A Y^B = -R^2
\label{hyperb}
\ee
Eqn. \eq{A1} can be recast into the standard conformal
algebra in $(3+1)$-dimensions by the redefinitions (see, for example,
\cite{SM})
\bea 
{\tilde D}=S_{-14}, \quad S_{\mu -1}={1 \over
2}({\tilde P}_\mu+ {\tilde K}_\mu), \quad S_{\mu 4} ={1 \over
2}({\tilde P}_\mu-{\tilde K}_\mu), \quad S_{\mu\nu}={\tilde
J}_{\mu\nu},
\label{A2}
\eea
where $\mu, \nu =0,1,2,3$. We will actually be working with a slightly
different set of generators for the conformal algebra. We will use
anti-hermitian generators for the dilations and rotations, while
retaining hermitian generators for translations and special conformal
transformations. That is, we will set 
\bea
{\tilde D}=iD, \quad {\tilde J}_{\mu\nu}=-iJ_{\mu\nu}. 
\label{A2a}
\eea
It will also be convenient to set 
\bea
{\tilde P}_\mu=-\sqrt{2} P_\mu, \quad {\tilde K}_\mu= -\sqrt{2} 
{\tilde K}_\mu.
\label{A2b}
\eea
In terms of these generators the conformal algebra is
\bea
&& [D, P_\mu]=-P_\mu, \quad [D, K_\mu]=K_\mu, \quad [P_\mu, K_\nu]=
\eta_{\mu\nu}D-J_{\mu\nu}, \nn \\
&& [J_{\mu\nu}, P_\gamma]=\eta_{\nu\gamma}P_\mu-\eta_{\mu\gamma}P_\nu,
\quad [J_{\mu\nu}, K_\gamma]=\eta_{\nu\gamma}K_\mu-\eta_{\mu\gamma}K_\nu, 
\nn \\
&& [J_{\mu\nu},J_{\gamma\tau}]=\eta_{\nu\gamma}J_{\mu\tau}
-\eta_{\nu\tau}J_{\mu\gamma}+\eta_{\mu\tau}J_{\nu\gamma}
-\eta_{\mu\gamma}J_{\nu\tau}.
\label{A3}
\eea 
To fix conventions for our light-cone frame, we define the light-cone
variables as follows.
\[
x^\pm \equiv {(x^3 \pm x^0) \over \sqrt{2}}, \quad x={(x^1+ix^2)
\over \sqrt{2}}, \quad \bar x= {(x^1-ix^2)
\over \sqrt{2}}.
\] 
Similarly 
\[
p_\pm \equiv {(p_3
\pm p_0) \over \sqrt{2}}, \quad p^x={(p_1+ip_2) \over \sqrt{2}}, \quad
p^{\bar x}={(p_1-ip_2) \over \sqrt{2}}.
\] 
The corresponding operator statements are in \eq{A7}.
The metric in the light-cone frame is
\[
\eta_{+-}=\eta_{-+}= \eta_{x\bar x}=\eta_{{\bar x} x}=1.
\] 
The algebra in (\ref{A3}) needs to be supplemented by the supersymmetry
generators, $Q^\pm_i$ and $S^\pm_i$ in the light-cone gauge, to get
the full superconformal algebra. The additional commutators are
\bea
&& [D, Q^\pm_i]=-{1 \over 2}Q^\pm_i, \quad [D, S^\pm_i]=
{1 \over 2}S^\pm_i, \nn \\
&& [J^{+-}, Q^\pm_i]=\pm {1 \over 2}Q^\pm_i, \quad 
[J^{+-}, S^\pm_i]=\pm {1 \over 2}S^\pm_i, \quad
[J^{x\bar x}, Q^\pm_i]=\mp {1 \over 2}Q^\pm_i, \quad 
[J^{x\bar x}, S^\pm_i]=\mp {1 \over 2}S^\pm_i, \nn \\
&& [Q^\pm_i, {J^j}_k]=\delta^j_i Q^\pm_k-
{1 \over 4} \delta^j_k Q^\pm_i, \quad
[S^\pm_i, {J^j}_k]=\delta^j_i S^\pm_k-
{1 \over 4} \delta^j_k S^\pm_i, \nn \\
&& [J^{+x}, Q^{-i}]=Q^{+i}, \quad [J^{-\bar x}, Q^{+i}]=-Q^{-i}, \quad
[J^{-x}, S^{+i}]=-S^{-i}, \quad [J^{+\bar x}, S^{-i}]=S^{+i}, \nn \\
&& [S^\mp_i, P^\pm]=Q^\pm_i, \quad  [S^-_i, P^x]=Q^-_i, \quad
[S^+_i, P^{\bar x}]=-Q^+_i, \nn \\
&&  [Q^\mp_i, K^\pm]=-S^\pm_i, \quad  [Q^-_i, K^{\bar x}]=-S^-_i, \quad
[Q^+_i, K^x]=S^+_i.
\label{A4}
\eea
Here ${J^j}_k$ are the generators of su$(4)$ algebra in hermitian basis,
$({J^i}_j)^\dagger={J^j}_i$,
\bea
[{J^i}_j, {J^k}_l]=\delta^k_j {J^i}_l-\delta^i_l {J^k}_j.
\label{A5}
\eea
The anticommutators are
\bea
&& \{Q^{\pm i} ,Q^\pm_j\}=\pm P^\pm \delta^i_j, \quad   
\{Q^{+i},Q^-_j\}= P^x \delta^i_j, \nn \\
&& \{S^{\pm i} ,S^\pm_j\}=\pm K^\pm \delta^i_j, \quad   
\{S^{+i},S^-_j\}= K^{\bar x} \delta^i_j, \nn \\
&& \{Q^{+i} ,S^+_j\}=-J^{+x} \delta^i_j, \quad   
\{Q^{-i},S^-_j\}=-J^{-\bar x} \delta^i_j, \nn \\
&& \{Q^{\pm i},S^{\mp}_j\}={1 \over 2}(J^{+-}+J^{x\bar x} \mp D)\delta^i_j
\mp {J^i}_j.
\label{A6}
\eea
The rest of the (anti)commutation relations are obtained by using the
hermiticity conditions
\bea
&& P^{\pm \dagger}=P^\pm, \quad P^{x\dagger}=P^{\bar x}, \quad 
K^{\pm \dagger}=K^\pm, \quad K^{x\dagger}=K^{\bar x}, \quad 
(Q^{\pm i})^\dagger=Q^\pm_i, \quad (S^{\pm i})^\dagger=S^\pm_i, \nn \\
&& (J^{\pm x})^\dagger=-J^{\pm \bar x}, \quad (J^{+-})^\dagger=-J^{+-},
\quad (J^{x \bar x})^\dagger=J^{x \bar x}, \quad D^\dagger=-D.
\label{A7}
\eea

\subsection{String realization of the superconformal generators}

Expressions for many of the generators have been obtained in \cite{MTT}
\footnote{Expressions for the rest can be obtained from these by using
the (anti)commutation relations given above.}. All the generators may
be written as integrals over the closed string of the corresponding
density, 
\[
G=\int_0^l d\sigma \ {\cal G}(\sigma),
\] 
where $l$ is the
length of the string. Below we reproduce the light-cone gauge
expressions obtained in \cite{MTT} for these densities.  These are
written in terms of the bosonic transverse co-ordinates, $x(\sigma),
{\bar x}(\sigma), z^a(\sigma)$, their momentum conjugates
$p^x(\sigma), p^{\bar x}(\sigma), p^a(\sigma)$, the superpartners
$\theta^i(\sigma), \eta^i(\sigma)$ and their conjugates.
\bea
{\cal P}^- &=& -{1 \over 2 p^+}\biggl(2 p^x p^{\bar x}- \del_z^2+
{1 \over z^2} \big[ l^i_jl^j_i +
4\eta_i {l^i}_j \eta^j+(\eta_i\eta^i -2)^2-\frac14\big]\biggr)-
{{\rm T}^2 \over 2p^+ z^4}(2x'\x'+(z_a')^2) \nn \\
&& +{{\rm T} \over z^2 p^+} \eta^i\rho^a_{ij}\n^a(\theta'^j-
i{\sqrt{2} \over z} \eta^j x')
+{{\rm T}\over z^2 p^+} \eta_i{\rho^a}^{ij}\n^a
(\theta'_j+i{\sqrt{2} \over z} \eta_j {\bar x}'), \label{A8} \\
{\cal J}^{+x} &=& -ixp^+, \label{A9} \\
{\cal J}^{+-} &=& -ix^-p^+-{1 \over 2}(\theta^i\theta_i+\eta^i\eta_i)+2, 
\label{A10} \\
{\cal J}^{x \bar x} &=& i(x p^{\bar x}-{\bar x}p^x)+
{1 \over 2}(\theta^i\theta_i-\eta^i\eta_i), \label{A11} \\
{\cal D} &=& i(x^-p^++x p^{\bar x}+{\bar x}p^x)+
z\del_z+{1 \over 2}(\theta^i\theta_i+\eta^i\eta_i)-{1 \over 2}, \label{A12} \\
{\cal K}^+ &=& {1 \over 2}(z^2+2x{\bar x})p^+, \label{A13} \\
{\cal K}^x &=& {1 \over 2}z^2 p^x-x(x^-p^++x p^{\bar x}-iz\del_z
+\frac{i}2)
+\frac{\theta^i}{\sqrt{p^+}}{\cal S}^+_i, 
\label{A14} \\
{{\cal J}^i}_j &=& {l^i}_j+\theta^i\theta_j+\eta^i\eta_j-
{1 \over 4}\delta^i_j(\theta^k\theta_k+\eta^k\eta_k), \label{A15} \\
{\cal Q}^+_i &=& \sqrt{p^+}\theta_i, \quad {\cal S}^+_i=\sqrt{p^+ \over 2}
z\eta_i+i\sqrt{p^+}x\theta_i, \label{A16} \\
{\cal Q}^-_i &=& {1 \over \sqrt{2p^+}}(\sqrt{2}p^x \theta_i-
\eta_i\del_z- {1 \over z}(-3\eta_i/2 + \eta^j\eta_j\eta_i-2(\eta l)_i))
\nn\\
&&~~~~~~~~~~~~~~~+
{{\rm T} \over \sqrt{2p^+} z^2}\rho^a_{ij}z^a(\theta'^j-
i{\sqrt{2} \over z} \eta^j x').
\label{A17}
\eea
Notation used above: $\rho^a_{ij}$ and $\rho^{aij}$ are SU$(4)$
Clebsh-Gordon coefficients (SO$(6)$ Dirac matrices in the chiral
representation). Some useful identities involving them are
\bea
&& \rho^a_{ij}\rho^{bjk}+\rho^b_{ij}\rho^{ajk}=2\delta^{ab}{\delta^i}_j, 
\nn \\
&& \rho^a_{ij}=-\rho^a_{ji}, \quad \rho^a_{ij}=(\rho^{aji})^*,
\quad \rho^{aij}\rho^a_{km}=2({\delta^i}_m{\delta^j}_k-
{\delta^i}_k{\delta^j}_m)
\nn\\
&& \rho^a_{ij}=\frac12 \epsilon_{ijkl}\ \rho^{a,kl}
\label{A18}
\eea
Also, ${l^i}_j$ is given by
\be
{l^i}_j ={i \over 8}{[\rho^a,\rho^b]^i}_j l^{ab},\quad 
l^{ab}= z^a p^b - z^b p^a,
\label{A19}
\ee
and satisfies the identities
\bea
{l^i}_m{l^m}_j={1\over4}{l^p}_q{l^q}_p{\delta^i}_j+2{l^i}_j, \quad
[{l^i}_j, {l^k}_m]=\delta^k_j {l^i}_m-\delta^i_m {l^k}_j.
\label{A20}
\eea

\gap3

\noindent\underbar{Canonical (anti)commutation relations}:

\gap1

In order to evaluate the above generators
\eq{A10}-\eq{A17} 
on wavefunctions we need the (anti)commutation relations between the
fundamental variables, which are as follows:
\bea
[x^-_0, p^+] &&= i,
\nn\\
~[x(\sigma), p^{\bar x}(\sigma')] &&= 
	[{\bar x}(\sigma), p^{x}(\sigma')]= i \delta(\sigma - \sigma'),
[z^a(\sigma), p^b(\sigma')]= i \delta^a_b \delta(\sigma - \sigma'),
\nn\\
\{ \eta^i(\sigma), \eta_j(\sigma')\} &&= 
\{ \theta^i(\sigma), \theta_j(\sigma')\} =
\delta^i_j \delta(\sigma - \sigma')
\label{funda-canon}
\eea

\subsection{Useful linear combinations}

In addition to the above generators, some special linear combinations
turn out to be useful (see \eq{threetwo}.  We have already used
$H_\pm,E_0$ which are given by:
\be
H_+ = - P^- + K^+, \, H_- = P^+ - K^- , E_0 = \frac12 (H_+ + H_-)
\label{e0}
\ee
We list below some more. Consider the $so(4)= su(2) \times su(2)$
subalgebra of $so(4,2)$; let us call its generators $\I_i, \K_i$. We
write these in terms of the above generators as
\bea
\I_3 &&= \half J^{x\bar x} + \frac14 (H_+ - H_-),\;
\K_3= \half J^{x\bar x} - \frac14 (H_+ - H_-)
\nn\\
\I_+ &&= \frac12 (J^{+x} + J^{-x} - P^x + K^x),\;
\K_+ = \half(J^{+x} + J^{-x} + P^x - K^x),
\nn\\
\I_- &&=\half( -J^{+\bar x} - J^{-\bar x} - P^{\bar x} + K^{\bar x}),\;
\K_- =\half( -J^{+\bar x} - J^{-\bar x} + P^{\bar x} - K^{\bar x})
\label{so4}
\eea
Using the above we get
\be
J^{x\bar x}= \K_3+ \I_3,\, H_\pm = E_0 \mp (\K_3 - \I_3)
\ee 
On a state annihilated by $\I_-, \K_+$, we have $\K_3=J_1,
\I_3= -J_2$, therefore
\be
H_\pm = E_0 \mp (J_1 + J_2), \, J^{x\bar x}= J_1 - J_2
\label{j1-j2}
\ee
We also list here some special linear combinations of
the odd generators which are particularly useful:
\bea
&& \widetilde Q^\pm_i= Q^\pm_i \mp S_i^\mp, \;
\widetilde Q^{i\pm}= Q^{i\pm} \pm S^{i \mp}, 
\label{raising}\\
&& \widetilde S^\pm_i= Q^\pm_i \pm S_i^\mp, \;
\widetilde S^{i\pm}= Q^{i\pm} \mp S^{i \mp}
\label{lowering}
\eea
The first line raises the value of $E_0$, while the second lowers the
value of $E_0$, by $1/2$. The bottom of a supermultiplet is defined by
annihilation of the $\widetilde S$ operators ({\em cf.} these are
similar to the $Q',S'$ operators discussed in \cite{SM}.)

\section{Discretized expressions for the generators}

In this appendix we list the discretized versions of the expressions
for the generators given above. As discussed in the main text, section
3, the lattice spacing $\epsilon$ completely disappears from these
expressions after appropriately rescaling the momenta and the
fermionic variables to have finite canonical commutation relations
among the bit variables. These rescalings are 
\bea
(p_n^+, p_n^x,
p_n^{\bar x}, p_n^a)
&& = \epsilon  \biggl( p^+(\sigma), p^x(\sigma),
p^{\bar x}(\sigma), p^a(\sigma) \biggr), \quad \sigma= n \epsilon
\nn\\
(x^-_0, x_n,
{\bar x}_n, z_n^a)
&& =   \biggl( x^-(\sigma), x(\sigma),
{\bar x}(\sigma), z^a(\sigma) \biggr), 
\nn\\
(\eta^i_n, \eta_{j,n}, \theta^i_n, \theta_{j,n})
&& =
{\epsilon}^{1/2}\biggl(
\eta^i(\sigma), \eta_{j}(\sigma), 
\theta^i(\sigma), \theta_{j}(\sigma)\biggr)
\label{rescalings}
\eea
The discrete version of the
canonical (anti)commutation relations \eq{funda-canon} is
\bea
&& [x_n, \ p_m^{\bar x}]=i\delta_{nm}, \quad [{\bar x_n},
p_m^x]=i\delta_{nm}, \quad [z_n^a, p_m^b]=i\delta_{nm}\delta^{ab}, \nn \\
&& \{\eta_n^i, \eta_{jm}\}=\delta_{nm}\delta^i_j, \quad 
\{\theta_n^i, \theta_{jm}\}=\delta_{nm}\delta^i_j.
\label{B1}
\eea
The oscillators for the two transverse directions $x, \bar x$, parallel
to the boundary, are defined as follows:
\bea 
&& a={p^x-ip^+ x \over
\sqrt{2p^+}}, \quad a^\dagger={p^{\bar x}+ip^+ {\bar x} \over
\sqrt{2p^+}}, \quad [a, a^\dagger]=1, \nn \\ && {\bar a}={p^{\bar
x}-ip^+ {\bar x} \over \sqrt{2p^+}}, \quad {\bar a}^\dagger={p^x+ip^+
x \over \sqrt{2p^+}}, \quad [{\bar a}, {\bar a}^\dagger]=1.
\label{B1a}
\eea 
The oscillators for the $z_a$ are more obvious.

The general form of the discrete version of the generators, in the
limit T $\rightarrow 0$ indicated by the subscript $'0'$ on the
generators, is $G_0=\sum_{n=1}^M g_{0n}$, where $g_{0n}$ is the
corresponding generator for the $n$th bit. We list the latter below,
but omit the bit index for ease of notation.
\bea
 p^-_0 &=& -{1 \over 2 p^+} \biggl(2p^xp^{\bar x}-
\del_z^2 +{1 \over z^2}((l^i_j l^j_i)+4\eta_{i} {l^i}_j \eta^j+
(\eta_i\eta^i-2)^2-\frac14)\biggr), \label{B2} \\
j_0^{+x} &=& -ixp^+, \label{B3} \\
j_0^{+-} &=& -ix^-p^+-{1 \over 2}(\theta^i\theta_i+\eta^i\eta_i)+2, 
\label{B4} \\
j_0^{x \bar x} &=& i(x p^{\bar x}-{\bar x}p^x)+
{1 \over 2}(\eta_i\eta^i-\theta_i\theta^i), \label{B5} \\
d_0 &=& i(x^-p^++x p^{\bar x}+{\bar x}p^x)+
z\del_z+{1 \over 2}(\theta^i\theta_i+\eta^i\eta_i)-{1 \over 2}, \label{B6} \\
k_0^+ &=& {1 \over 2}(z^2+2x{\bar x})p^+, \label{B7} \\
k_0^x &=& {1 \over 2}z^2 p^x-x(x^-p^++x p^{\bar x}-iz\del_z +\frac{i}2)
+\theta^i s^+_{0i}, 
\label{B8} \\
{j_0^i}_j &=& {l^i}_j+\theta^i\theta_j+\eta^i\eta_j-
{1 \over 4}\delta^i_j(\theta^k\theta_k+\eta^k\eta_k), \label{B9} \\
q^+_{0i} &=& \sqrt{p^+}\theta_i, \quad s^+_i=\sqrt{p^+ \over 2}
z\eta_i+i\sqrt{p^+}x\theta_i, \label{B10} \\
q^-_{0i} &=& {1 \over \sqrt{2p^+}}(\sqrt{2}p^x \theta_i -
\eta_i\del_z- {1 \over z}(-3\eta_i/2 +
\eta^j\eta_j\eta_i-2(\eta l)_i)). \label{B11}
\eea  
Others can be obtained using the (anti)commutation relations.

When T is non-zero some of the generators \footnote{These are $P^-,
K^-, J^{-x}, J^{-\x}, Q^{-i}$ and $S^{-i}$, the so-called
dynamical generators. Note that they also receive corrections from
string-string interactions, which we are ignoring here.} get modified by
terms that involve bit-bit interactions. We write these generators as
$G=G_0+G_1$, where $G_1=\sum_n g_{1(n+1,n)}$ vanishes for T$=0$. We
list $g_{1(n+1,n)}$ terms for some of the generators that get
modified.  
\bea
-p^-_{1(n+1,n)} &=& {{\rm T}^2 \over 2p^+}\biggl(2
{|x_{n+1}-x_n|^2 \over z_n^4}+{(z^a_{n+1}-z^a_n)^2 
\over z_{n+1}^2z_n^2}\biggr) 
- {{\rm T} \over z_n^2 p^+}\eta^i_n\rho^a_{ij}\n_n^a 
\biggl((\theta_{n+1}^j-\theta_n^j)-\biggr. \nn\\
&& i{\sqrt{2} \over z_n} \eta_n^j (x_{n+1}-x_n) \biggl.\biggr) 
+{{\rm T}\over z_n^2 p^+} \eta_{ni}{\rho^a}^{ij}\n_n^a
\biggl((\theta_{n+1j}-\theta_{nj})+i{\sqrt{2} \over z_n} 
\eta_{nj} (\x_{n+1}-\x_n) \biggr), 
\nn\\
\label{B12} \\
q^-_{1i(n+1,n)} &=& {{\rm T} \over \sqrt{2p^+}}\rho^a_{ij} 
\biggl({\n^a_n \over z_n} (\theta^j_{n+1}-\theta^j_n)
-i\sqrt{2}{\n^a_n \over z^2_n}\eta^j_n(x_{n+1}-x_n)\biggr).
\label{B13}
\eea
These expressions are obtained by a straightforward discretization of
the corresponding continuum expressions in (\ref{A8}) and
(\ref{A17}). There is of course no a$'$ priori unique discretized
expression. We make a natural choice for a minimal set and then fix
the rest by demanding that the algebra be satisfied. Thus, e.g.,
\eq{B12} can be independently obtained from \eq{B13}
using \eq{A6}: $\{ Q^{-i}, Q^-_j \} = -P^- \delta^i_j$.

\section{Perturbation of $H_+$ spectrum and
cancellation of  $z\to 0$ divergence}

We assume the string to consist of $M$ bits. 
We define  $H_+ = H^0_+ + T V_1 + T^2 V_2$:
\bea
H^0_+ &&
= \sum_{n=1}^M
\biggl[
-\frac{\del_{x_n} \del_{\bar x_n}}{2p^+} 
-{\del_{z_n}^2 \over 2p^+}+{p^+ \over 2}z^2_n 
+{1 \over 2p^+z^2_n} 
\big(
({l^i}_{n,j})^2+4\eta_{n,i} 
{l^i}_{n,j} \eta^j_n+
(\eta_{n,i}\eta^i_n-2)^2-{1 \over 4} 
\big) 
\biggr]
\nn\\
V_2 &&
= \sum_{n=1}^M
\biggl[
{1 \over 2p^+}
\biggl(
2{|x_{n+1}-x_n|^2 \over z_n^4}+{(z^a_{n+1}-z^a_n)^2 
\over z_{n+1}^2z_n^2}
\biggr)  
\biggr]
\nn\\
V_1 &&=\sum_{n=1}^M v_{1(n+1,n)}
\nn\\
&& \equiv \sum_{n=1}^M
\biggl[\biggr. - 
{{1} \over z_n^2 p^+}\eta^i_n\rho^a_{ij}\n_n^a 
\biggl(\biggr. 
(\theta_{n+1}^j-\theta_n^j)-
\nn\\
&& i{\sqrt{2} \over z_n} \eta_n^j (x_{n+1}-x_n) 
\biggl.\biggr) 
+{{1}\over z_n^2 p^+} \eta_{ni}{\rho^a}^{ij}\n_n^a
\biggl(
(\theta_{n+1j}-\theta_{nj})+i{\sqrt{2} \over z_n} 
\eta_{nj} (\x_{n+1}-\x_n) 
\biggr)
\biggl.\biggr]
\label{h0-v1-v2}
\eea 
\subsection{Calculation of $\langle \Psi_0 | V_2 | \Psi_0 
\rangle$}

We will calculate this quantity as
\bea
\langle \Psi_0 | V_2 | \Psi_0 \rangle 
&&=
\frac14  \langle \Psi_0 |\{ Q_{1,i}^-, Q^{-i}_1 \} | \Psi_0 \rangle 
\nn\\
= \frac14 \sum_{n,m}\sum_{int} && \biggl[\biggr.
\langle \Psi_0 | q^-_{(n+1,n)i} | \Psi_{int} \rangle 
\langle \Psi_{int} | q^{-i}_{(n+1,n)} | \Psi_0 \rangle
\nn\\ 
&&  +\langle \Psi_0 | q^{-i}_{(n+1,n)} | \Psi_{int} \rangle 
\langle \Psi_{int} | q^-_{(n+1,n)i} | \Psi_0 \rangle \biggl.\biggr]
\label{v2-vev}
\eea
The first equality can be derived by considering the $T^2$ term in the
equality $-P^- = \frac14 \{ Q_{i}^-, Q^{-i} \}$.
Here, 
\be      
q^-_{(n+1,n)i}= \frac1{\sqrt 2 z_n} (\rho.\n_n)_{ij}
\left[ (\theta^j_{n+1} - \theta^j_n) + 
\frac1{z_n} \eta^j_n \left(  (a_{n+1} - {\bar a}^\dagger_{n+1}) -
(a_{n} - {\bar a}^\dagger_{n}) \right) \right]
\label{q-}
\ee
This expression is as in \eq{B13} except that we have
rescaled $\sqrt{p^+} z_n \to z_n, \sqrt{p^+} x_n \to x_n$
and have used $-i \sqrt 2 x_n = a_n - {\bar a}^\dagger_{n}$.

Since the intermediate states $ | \Psi_{int} \rangle $ involved for
various terms in \eq{q-} are different, we can separately treat the
contribution of each term in \eq{q-} to \eq{v2-vev}. Let us
consider, e.g., the term
\be \frac1{\sqrt 2 z_n} (\rho.\n_n)_{ij} \theta^j_{n+1}
\label{sample}
\ee
The intermediate state that will click here is represented in the
figure below. 
\noindent
%\hspace{-100ex}
%\vbox{
\begin{picture}(500,70)(1,1) %lower left=(1,1), upper right=(501,71)
\put(1,50){ \line(1,0){400}}
\put(15,50){\circle{7}}
\put(100,50){\circle{7}}
\put(197,55){n}
\put(200,50){\circle*{7}}
\put(160,35){\small $\ket{l=1,2\eta,\alpha,r}$}
%\put(185,20){\small $\ket{2\eta,2\theta}$} 
\put(292,55){n+1}
\put(300,50){\circle*{7}}
\put(290,35){\small $\ket{3\theta}$}
%\put(285,20){\small $\ket{2\eta,3\theta}$} 
\put(400,50){\circle{7}}
\end{picture}
%}
\noindent
The wavefunction in the picture differs from the initial state only in
the $n$-th and the $n+1$-th bits (the solid circles) and only
in the quantum numbers exhibited \footnote{The full
$\ket{\Psi_{int}}$ is a cyclic permutation of such wavefunctions.}
($\ket{2\eta}$ denotes $\ket{F_\eta=2}$,
etc). Explanation for the $n$-th bit: $l=1$ appears because $\n^a_n$
carries $l=1$, the values of $\alpha$ can  be $l, l+2$ or $l+4$,
i.e. 1,3 or 5, depending on whether $\ket{2\eta}$ belongs to $P_2
\V_2, Q_2\V_2$ or $R_2 \V_2$ (see \eq{p2} and below).
Only the quantum number $r$ remains
unspecified, any $r=0,1,2...$ clicks.  The corresponding matrix
element of the $z$-wavefunctions is (cf. $C_2(r)$ in \eq{v2-div-new})
\be
C_2^{2\eta,\alpha}(r)=
\int_0^\alpha dz \psi^0_0(z)  \frac1{\sqrt 2 z} \psi^\alpha_r(z)
= \frac1{\sqrt 2} \frac{(r + {\alpha-1 \over 2})!}{r!}
\sqrt{\frac{r!}{(r+\alpha)!}}
\label{c-alpha-r}
\ee
Having done the $z$-(and the trivial $x, \bar x$-)integration let us
now leave the S$^5$ and the fermion matrix elements unevaluated. This
amounts to finding an {\em effective interaction} in the finite
dimensional S$^5$ and fermionic sector, after integrating out the
bosonic degrees of freedom $z, x, \x$.  After carrying out the sum
$\sum_r$ involved in the Eqn. \eq{v2-vev} for each $\alpha=1,3$ or 5,
we get the following contribution of the interaction \eq{sample} to
\eq{v2-vev}:
\be
\frac14 \sum_{n,m}\sum_{\alpha=1,3,5} C_2^{2\eta,\alpha}
\langle \tilde \Psi_0 |  (\rho.\n_n)_{ij} \theta^j_{n+1}|
\tilde \Psi_{int,(2\eta,\alpha)} \rangle
\langle \tilde \Psi_{int,(2\eta,\alpha)} 
|  (\rho.\n_m)^{ki} \theta_{m+1,k}|
\tilde \Psi_{0} \rangle
\label{effective}
\ee
where
\be
C_2^{2\eta,\alpha} = \sum_{r=0}^\infty \left(C_2^{2\eta,\alpha}(r)
\right)^2
\label{c-2eta-alpha}
\ee
In the above, the tildes on the wavefunctions denote that they
depend now only on the $S^5$ and the fermionic coordinates.  The
$(2\eta,\alpha)$ tag on the $\tilde \Psi_{int}$ reminds us that 
we must choose the
appropriate projection ($P_2, Q_2$ or $R_2$) on the wavefunction
$\ket{l=1,2\eta}$ on whichever bit it appears.

What remains now is to compute the equivalent of \eq{effective}
for all the other interaction terms. We will write here only one
other term in \eq{q-} in some detail, viz.
\be
\frac1{\sqrt 2 z_n} (\rho.\n_n)_{ij} \eta^j_{n} a_{n+1}
\label{sample-2}
\ee
The pictorial representation of the intermediate state now is

\noindent
%\hspace{10ex}
%\vbox{
\begin{picture}(500,70)(1,1) %lower left=(1,1), upper right=(501,71)
\put(1,50){ \line(1,0){400}}
\put(15,50){\circle{7}}
\put(100,50){\circle{7}}
\put(197,55){n}
\put(200,50){\circle*{7}}
\put(155,35){\small $\ket{l=1,3\eta,\alpha,r}$}
%\put(185,20){\small $\ket{2\eta,2\theta}$} 
\put(292,55){n+1}
\put(300,50){\circle*{7}}
\put(285,35){\small $\ket{s=1}$}
%\put(285,20){\small $\ket{2\eta,3\theta}$} 
\put(400,50){\circle{7}}
\end{picture}
%}

\noindent
Once again only the changed quantum numbers are shown. $s=1$ means a
harmonic oscillator excitation of the '$a$' type (see \eq{psi-alpha}).
The $\alpha$ value of $\ket{l=1,3\eta}$ is determined ($\alpha=l+1=2$
or $\alpha=l+3=4$), depending on whether the $F_\eta=3$ 
state belongs to $P_3
\V_3$ or $Q_3 \V_3$.
The $z,x,\bar x$ integration gives
\[
C_2^{3\eta,\alpha}(r)=
\int_0^\alpha dz \psi^0_0(z)  \frac1{\sqrt 2 z} \psi^\alpha_r(z)
= \frac{\sqrt 2}{\alpha} 
\frac{(r + {\alpha \over 2})!}{\sqrt{r!(r+\alpha)!}}
%\label{c-3eta-alpha-r}
\]
The contribution to \eq{v2-vev} in terms of $S^5$ and fermions
is, in this case,
\be
\frac14 \sum_{n,m}\sum_{\alpha=2,4}C_2^{3\eta,\alpha}
\langle \tilde \Psi_0 |  (\rho.\n_n)_{ij} \eta^j_{n} a_{n+1}|
\tilde \Psi_{int,(3\eta,\alpha)} \rangle
\langle \tilde \Psi_{int,(3\eta,\alpha)} |  
(\rho.\n_m)^{ki} \eta_{m+1,k}|
\tilde \Psi_{0} \rangle
\label{effective-2}
\ee
where
\be
C_2^{3\eta,\alpha}= \sum_{r=0}^\infty \left(C_2^{3\eta,\alpha}
\right)^2
\label{c-3eta-alpha}
\ee
From these examples, it is clear how to compute 
equations like \eq{effective-2} and \eq{c-3eta-alpha} for all
the other terms in \eq{v2-vev}.

\myitem \underbar{Divergence}:

If we explicitly evaluate $C_2^{2\eta,\alpha},C_2^{3\eta,\alpha}$ using
their definitions, we find that
\bea
C_2^{2\eta,\alpha}
&& = \half \sum_{r=1}^\infty \frac1{r} + F_2(\alpha)
\nn\\
C_2^{3\eta,\alpha} 
&& = A'_2(\alpha) \sum_{r=1}^\infty 1
+ B'_2(\alpha) \sum_{r=1}^\infty \frac1{r} + F'_2(\alpha)
\eea
Thus the two types of divergences that appear here are
$\sum_{r=1}^{r_{max}} 1$ and $\sum_{r=1}^{r_{max}} \frac1{r}$, where
we have replaced the upper limit $r=\infty$ by a cut-off $r_{max}$.
The coefficients of such divergences in \eq{v2-vev} are calculated by
combining equations like \eq{effective} and \eq{effective-2}.

\subsection{The $V_1$ contribution}

This term has an expression
\be
- \frac14 \sum_{n,m}\sum_{int}\frac1{E_{int}-E_0} \biggl[
\langle \Psi_0 | v_{1(n+1,n)} | \Psi_{int} \rangle 
\langle \Psi_{int} | v_{1(n+1,n)} | \Psi_0 \rangle\biggr]
\label{v1-term}
\ee
which is similar to \eq{v2-vev}, except that it has
an additional factor due to the
energy denominator.

Like in the case of the $V_2$ term in the previous subsection,
we will  consider each term of $v_{1(n+1,n)}$ in turn
(see \eq{h0-v1-v2}), consider the intermediate
states that click in each case, and ``integrate out''
the $z, x, \x$ variables, leaving matrix elements depending
on $S^5$ and fermions.

As an example we consider
\[
{1 \over z_n^2 p^+}\eta^i_n(\rho.\n)_{ij} \theta_{n+1}^j
\]
The intermediate state is represented by the diagram

\noindent
%\hspace{-100ex}
%\vbox{
\begin{picture}(500,70)(1,1) %lower left=(1,1), upper right=(501,71)
\put(1,50){ \line(1,0){400}}
\put(15,50){\circle{7}}
\put(100,50){\circle{7}}
\put(197,55){n}
\put(200,50){\circle*{7}}
\put(160,35){\small $\ket{l=1,3\eta,\alpha,r}$}
%\put(185,20){\small $\ket{2\eta,2\theta}$} 
\put(292,55){n+1}
\put(300,50){\circle*{7}}
\put(290,35){\small $\ket{3\theta}$}
%\put(285,20){\small $\ket{2\eta,3\theta}$} 
\put(400,50){\circle{7}}
\end{picture}
%}

\noindent
The values of $\alpha$ here are $l+1=2$ or $l+3=4$.
The $z$-integral involved in the matrix element
is $C_1^{3\eta,\alpha}(r)$, which 
is identical to $C_2^{3\eta,\alpha}(r)$. After performing the
$r$-sum we get the following contribution
\be
\frac14 \sum_{n,m}\sum_{\alpha=2,4}C_1^{3\eta,\alpha}
\langle \tilde \Psi_0 | \eta^i_n (\rho.\n_n)_{ij} \theta^j_{n+1}|
\tilde \Psi_{int,(3\eta,\alpha)} \rangle
\langle \tilde \Psi_{int,(3\eta,\alpha)} |  
(\rho.\n_m)^{kl} \theta_{m+1,k}\eta_{n,l}|
\tilde \Psi_{0} \rangle
\label{effective-3}
\ee
where
\be
C_1^{3\eta,\alpha}= \sum_r \frac{\left[ C_2^{3\eta,\alpha}(r)\right]^2}{
(2r + \alpha + 2)-2}
\label{c-3eta-alpha-1}
\ee
The divergence of this term is of the type $\sum_r^{r_{max}} 1/r$.

\subsection{Cancellation of Divergence}

We now collect the coefficient of $\sum_r^{r_{max}} 1$ and
$\sum_r^{r_{max}} (1/r)$ of all the terms in \eq{v2-vev}
and \eq{v1-term}. We will explicitly write here only the
coefficient of $\sum_r 1$. This is given by
\bea
&& \frac14\sum_{n,m}
\biggl[\biggr.
\langle \tilde \Psi_0 |  (\rho.\n_n)_{ij} \eta^j_{n} |
\tilde \Psi_{int,(3\eta,\alpha=2)} \rangle
\langle \tilde \Psi_{int,(3\eta,\alpha=2)} |  
(\rho.\n_m)^{ki} \eta_{m,k}|
\tilde \Psi_{0} \rangle
\nn\\
&&~~~~~~~~~~+  \frac14 
\langle \tilde \Psi_0 |  (\rho.\n_n)_{ij} \eta^j_{n} |
\tilde \Psi_{int,(3\eta,\alpha=4)} \rangle
\langle \tilde \Psi_{int,(3\eta,\alpha=4)} |  
(\rho.\n_m)^{ki} \eta_{m,k}|
\tilde \Psi_{0} \rangle \biggl.\biggr]
\nn\\
&&\frac14\sum_{n,m}
\biggl[\biggr.
\langle \tilde \Psi_0 |  (\rho.\n_n)^{ij} \eta_{n,i} |
\tilde \Psi_{int,(1\eta,\alpha=2)} \rangle
\langle \tilde \Psi_{int,(1\eta,\alpha=2)} |  
(\rho.\n_m)_{jk} \eta_m^{k}|
\tilde \Psi_{0} \rangle
\nn\\
&&~~~~~~~~~~+ \frac14 
\langle \tilde \Psi_0 |  (\rho.\n_n)^{ij} \eta_{n,i} |
\tilde \Psi_{int,(1\eta,\alpha=4)} \rangle
\langle \tilde \Psi_{int,(1\eta,\alpha=4)} |  
(\rho.\n_m)_{jk} \eta_m^{k}|
\tilde \Psi_{0} \rangle \biggl.\biggr]
\nn\\
 && \!\!\! - \frac14\sum_{n,m}
\langle \tilde \Psi_0 | \eta_n^i (\rho.\n_n)_{ij} \eta_{n}^i |
\tilde \Psi_{int,(4\eta,\alpha=3)} \rangle
\langle \tilde \Psi_{int,(4\eta,\alpha=3)} |  
\eta_{m,k}(\rho.\n_m)^{kl} \eta_{m,l}|
\tilde \Psi_{0} \rangle 
\nn\\
&&  \!\!\!  - \frac14\sum_{n,m}
\langle \tilde \Psi_0 | \eta_{n,i} (\rho.\n_n)^{ij} \eta_{n,i} |
\tilde \Psi_{int,(0\eta,\alpha=3)} \rangle
\langle \tilde \Psi_{int,(0\eta,\alpha=3)} |  
\eta_{m,k}(\rho.\n_m)_{kl} \eta_{m,l}|
\tilde \Psi_{0} \rangle
\label{coeff-rmax}
\nn\\
\eea
By repeatedly using (a) the fundamental anticommutation relations of
the fermions \eq{B1}, (b) the explicit form of the projection operators
$P_1, Q_1, P_3$ and $Q_3$ (see \eq{p1}, \eq{p3}), and (c)
$\n^a$-space vev's such as
\[
\langle \n^a \n^b \rangle = \frac16 \delta^{ab},
\quad \langle \n^a l^i_j \n^b \rangle=\frac1{24}[ \rho^a, \rho^b]^i_j
\]
we find that  the above term \eq{coeff-rmax} indeed
vanishes. Note that the first two terms (with
positive sign) arise from $V_2$ the last two (negative
sign) from $V_1$.

The coefficient of $\sum_r^{r_{max}} 1/r$ vanishes by a similar, but
somewhat lengthier computation.

\section{Nonlocal features of stringy representation
in the light cone gauge}

Note that generators like $K^x$ involve $\tilde x^-$, the nonzero mode
of $x^-$. This implies that even at $T=0$, a representation of such
generators may be achieved only when one takes a linear combination
(over and above the cyclic sum in \eq{fourone}) of naive tensor
product of single-bit states.  In the following we will consider the
simplest possible example of such a state \eq{sym-wave} where a linear
combination ({\it a la} Clebsch-Gordon) is already in place because we
have taken a symmetric and traceless tensor product of the $SU(4)$
polarizations.

In order to see if such states carry a representation of the
superconformal algebra, including the non-local generators, let us
consider the example of the $so(4)$ subalgebra of the AdS$_5$ group
$so(4,2)$.  As shown in \eq{so4}, the generators of this algebra
involve $K^x$ which contains $\tilde x^-$ and the action of $so(4)$
therefore constitutes a non-trivial example. 

The state \eq{sym-wave} is expected to be a singlet of $so(4)$.
Therefore it must be annihilated by $\K^\pm, \I^\pm$. It is a tedious
but straightforward exercise to show that the part of each of these
generators, which involves only local terms (including $x^-_0$,
interpreted as $i \del_{p^+}$), indeed annihilate this state for an
appropriate choice of $f(p^+)$ in \eq{psi-alpha}. What remains is to
show that this is true even after including terms involving $\tilde
x^-$.

Recall that the  Gauss law constraint of
$\sigma$-reparametrization, stated in the light cone gauge, 
is, classically \cite{MTT}
\bea
\tilde x^-(\sigma) 
\equiv x^-(\sigma) - x^-(0) &&= 
-\int_0^{\sigma} {d\tilde \sigma \over2 p^+}  
\biggl[\biggr. x'(\tilde \sigma) p^{\bar x}
(\tilde \sigma) - xp^{\bar x \prime}
+
\bar x' p^x - \x' p^x + 
\nn\\
&& {z^a}' p^a - z^a {p^a}'+ i p^+\big( \eta^i\eta_i' - {\eta^i}'\eta_i +
\theta^i\theta_i' - {\theta^i}'\theta_i  \big)\biggl.\biggr]
\label{x-}
\eea
Here we have interpreted $x^-(0)$ as the constant mode of $x^-$.  To
discretize a combination of the form $x^i(\tilde\sigma) p_i'(\tilde
\sigma)$, we replace the local term $x^i(\tilde \sigma)$ by
$(x^i_{k+1} + x^i_k)/2$, the $\tilde \sigma$-derivative by a discrete
difference, i.e., $p'_i(\tilde \sigma) \to
\epsilon^{-1}(p_{i,k+1}- p_{i,k})$ and perform the rescalings
\eq{rescalings}.  This leads to
\bea
\tilde x^-_n = &&
-\frac1{2p^+}\sum_{k=1}^n \biggl[\biggr. (x_{k+1} p^{\bar x}_{k} 
- x_k \p_{k+1}) +  (\bar
x_{k+1} p^x_{k} - \bar
x_{k} p^x_{k+1} + ({z^a}_{k+1}p^a_{k}- {z^a}_{k}p^a_{k+1}) +
\nn\\
&& i p^+\biggl\{( \eta^i_{k+1} \eta_{k,i} - 
\eta^i_{k} \eta_{k+1,i})
+ ( \theta^i_{k+1} \theta_{k,i} - 
\theta^i_{k} \theta_{k+1,i}) \biggr\} \biggl.\biggr]
\label{discrete-x-}
\eea
The part involving the bosonic coordinates $x, \bar x$
and their conjugate momenta 
clearly annihilates the state \eq{sym-wave}, because, e.g.
$x_{k+1}\del/\del x_k - x_k \del/\del x_{k+1}$
becomes proportional to $x_{k+1} x_k - x_k x_{k+1}=0$.

For $z^{a\prime} p^a- z^a p^{\prime a}$, the argument is
similar to the above if the wavefunction in the Cartesian
coordinates were entirely Gaussian (cf. the ground state of a
six-dimensional harmonic oscillator).  In the present case the $z,
\Omega_5$ are more appropriate coordinates. The S$^5$ part of $\tilde
x^-$ annihilates \eq{sym-wave} since it is a $l=0$ state. For the part
involving $z$ and $p_z$, this happens with a specific choice of
operator ordering: $z^{1/2} (z'p_z - zp_z')z^{-1/2}$.  

The fermion part is the most interesting. Here $\tilde x^-$
annihilates the wavefunction only when one takes
the traceless symmetric linear combination of the fermion
polarizations. For illustration, consider a state which
has only the $\eta_{k,1}\eta_{k,2}$ at each bit $k$.
Consider the $\eta$-part of the r.h.s. of \eq{discrete-x-}.
The term $\eta^1_{k+1} \eta_{k,1} - 
\eta^1_{k} \eta_{k+1,1}$ clearly annihilates the
state by Pauli exclusion principle.

It is clear from the discussion above that the prescription for
discretization as well as for the operator ordering is important. We
do not have a complete understanding of this issue yet and hope to
come back to it.

\myitem\underbar{Summary:}

With the above action of $\tilde x^-$ on $\ket{\Psi_{\rm sym}}$, it is
easy to see that both $H_-$ and the lowering operators $\widetilde S$
(see \eq{lowering}) annihilate $\ket{\Psi_{\rm sym}}$.


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\end{thebibliography}

\end{document}

