\section{ Introduction}

The conjectured duality between
the type IIB superstring theory on the AdS$_5\times$S$^5$ space
(AdS superstring)
and
$D=4,~{\cal N}=4$ Yang-Mills theory
\cite{M,GKP,W}
has been driven not only
studies of variety of background theories
but also studies of basic aspects such as integrability.
The approach of the pp-wave background superstring theory \cite{MTpp}
was explored
by Berenstein, Maldacena and Nastase \cite{BMN} and
developed in, for example
\cite{GKP2,FT}.
For further development Mandal, Suryanarayan and Wadia
pointed out the relevance with the integrability \cite{MSW},
and Bethe anzatz approach was explored
 by Minahan and Zarembo \cite{MZ} and in for example \cite{B,DNW,BFST}.
The integrability
is a powerful aspect expected in the large N QCD \cite{Lipatov}
and shown to exist in
 the IIB superstring theory on the AdS$_5\times$S$^5$ space
by Bena, Polchinski and Roiban \cite{BPR}.
The integrability provides hidden symmetry generated by
an infinite number of
conserved ``non-local" charges \cite{LP,BIZZ}
as well as
an infinite number of  conserved ``local" charges
\cite{pol2} which are related by a spectral parameter
at different points.
Related aspects on the integrability of the AdS superstring were
   discussed in \cite{new18}.

Recently the conformal symmetry of AdS superstrings
was conjectured due to  the $\kappa$ symmetry
\cite{polyakov}.
The classical conformal symmetry of the AdS superstring theory
also  leads to an infinite number of conserved Virasoro operators.
The naive questions are
how the conformal generator  is related to the infinite number of conserved
``local" currents, and how many independent conserved currents exist.
For principal chiral models the stress-energy tensor
is written by trace of the square of the conserved flat current;
for  reviews see refs.
\cite{EHMM,MSW}.
For the AdS superstring theory
the Wess-Zumino term and the $\kappa$ symmetry make a difference.
Recently issues related to
the integrability and the conformal symmetry
of the AdS superstring theory have been discussed
\cite{MP,Mh,AAT}.
In this paper we will obtain the expression of the conformal generator,
which is the stress-energy tensor relating to
 the lowest spin ``local" current,
  and we calculate the higher spin  ``local"  currents
to clarify independent components.

The AdS space contains the Ramond/Ramond flux which causes
difficulty of
the standard Neveu-Schwarz-Ramond
(NSR) formulation of the superstring theory.
The AdS superstring was described in the Green-Schwarz (GS) formalism
by Metsaev and Tseytlin based on the coset
PSU(2,2$\mid$4)/[SO(4,1)$\times$SO(5)]
\cite{MT}.
Later  Roiban and Siegel reformulate it in terms of the unconstrained
GL(4$\mid$4) supermatrix coordinate based on an alternative
coset GL(4$\mid$4)/[Sp(4)$\times$GL(1)]$^2$  \cite{RS}.
In this formalism the local Lorentz is gauged,
and it turns out that this treatment is essential for
separation into  $+/-$ modes (right/left moving modes) easier.
Furthermore the fermionic constraint including the first class and second class
is necessary for
separation of the fermionic modes into $+/-$ modes.
As the first step toward the CFT formulation of the AdS superstring,
the affine Sugawara construction \cite{Halpern},
the Virasoro algebra and the algebra of currents carrying the
space-time indices are also listed.

The organization of this paper is the following;
in the next section the notation is introduced.
In section 3 we analyze the superparticle in the AdS$_5\times$S$^5$ space,
and the relation between
the reparametrization constraint and the conserved right invariant (RI) current
is given.
In section 4 we analyze the superstring in the AdS$_5\times$S$^5$ space,
and the infinite number of conserved currents are presented
both from the conformal point of view and
from the integrability point of view.
We show that the stress-energy tensor
is written by
the ``supertrace" of the square of the RI current
 as the lowest spin ``local" current.
Then we  calculate higher spin ``local" currents
to clarify independent components of the ``local" currents.

\par\vskip 6mm
\section{ GL(4${\mid}$4) covariant coset}

We review the Roiban-Siegel formulation of the  AdS$_5\times$S$^5$ coset
\cite{RS} and follow the notation in \cite{HKAdS}.
The coset GL(4$\mid$4)/[GL(1)$\times$Sp(4)]$^2$ is used instead of
PSU(2,2$\mid$4)/[SO(4,1)$\times$SO(5)] for the linear realization of the
global symmetry after Wick rotations and introducing the auxiliary variables.
A coset element  $Z_M{}^A$
 is an unconstrained matrix defined on a world-volume
 carrying indices $M=(m,\bar{m}),~A=(a,\bar{a})$ with
 $m,\bar{m},a,\bar{a}=1,\cdots,4$.
The left invariant (LI) current, $L^L$, is invariant under the left action
$Z_M{}^A~\to~\Lambda_M{}^NZ_N{}^A
$ with
 a global parameter GL(4$\mid$4)$\ni \Lambda$
\begin{eqnarray}
(J^L)_A{}^B=(Z^{-1}d Z)_A{}^B~~.
\end{eqnarray}
The LI current satisfies the flatness condition by definition
\begin{eqnarray}
dJ^L=-J^LJ^L~~~.
\end{eqnarray}
The right  invariant (RI) current, $J^R$, is invariant under the right action
$Z_M{}^A~\to~Z_M{}^B\lambda_B{}^A
$ with a local parameter
[Sp(4)$\otimes$GL(1)]$^2$ $\ni\lambda$
\begin{eqnarray}
(J^R)_M{}^N=({\cal D}ZZ^{-1})_M{}^N~~,~~({\cal D}Z)_M{}^A\equiv
dZ_M{}^A+Z_M{}^BA_B{}^A
\end{eqnarray}
with
\begin{eqnarray}
A~\to~\lambda A\lambda^{-1}+(d\lambda) \lambda^{-1}~~,
\end{eqnarray}
and
\begin{eqnarray}
dJ^R=J^RJ^R+Z(dA-AA)Z^{-1}~~~.\label{dAAA}
\end{eqnarray}
Originally $A$ is bosonic  [Sp(4)$\otimes$GL(1)]$^2$
$\ni$$A$, but we will show that
the fermionic constraint i.e. $\kappa$ symmetry
gives fermionic components of $A$.

The conjugate momenta are introduced
\begin{eqnarray}
\{
Z_M{}^A,\Pi_B{}^N
\}=(-)^A\delta_B^A\delta_M^N~~~
\end{eqnarray}
as the graded Poisson bracket and
 $\{q,p\}=-(-)^{qp}\{p,q\}$.
There are also two types of differential operators;
 the global symmetry generator (left action generator), $G_M{}^N$,
and
the  supercovariant derivatives (right action generator), $D_A{}^B$,
\begin{eqnarray}
G_M{}^N=Z_M{}^A\Pi_A{}^N~~,~~D_A{}^B=\Pi_A{}^M Z_M{}^B~~~.\label{DGDG}
\end{eqnarray}
In our coset approach $8\times 8=64$ variables for $Z_M{}^A$ are introduced
and auxiliary variables are eliminated by the following constraints
corresponding to the stability group [Sp(4)$\times $GL(1)]$^2$,
 \begin{eqnarray}
({\bf D})_{(ab)}=(\bar{\bf D})_{(\bar{a}\bar{b})}={\rm tr}~{\bf D}=
{\rm tr}~\bar{\bf D}\equiv 0~~~\label{DSp4GL1}~~~,
\end{eqnarray}
where the bosonic components are denoted by  boldfaced characters as
 ${\bf D}_{ab}\equiv D_{ab}$ and $\bar{\bf D}_{\bar{a}\bar{b}}\equiv
 D_{\bar{a}\bar{b}}$ of \bref{DGDG}.
The number of the coset constraints is $10+10+1+1=22$,
so the number of the coset parameters is $64-22=42$
where $10$ bosons and $32$ fermions.
The $[Sp(4)]^2$ invariant metric is anti-symmetric
and a matrix is decomposed into
 trace part, anti-symmetric-traceless part and the symmetric part,
denoted by
\begin{eqnarray}
{\bf M}_{ab}=-\frac{1}{4}\Omega_{ab}{\bf M}^c{}_c+{\bf M}_{\langle ab\rangle}+{\bf M}_{(ab)}\equiv - \frac{1}{4}\Omega~{\rm tr}{\bf M}
+\langle {\bf M}\rangle+({\bf M})~~~,
\end{eqnarray}
with $M_{(ab)}=\frac{1}{2}(M_{ab}+M_{ba})$,
and similar notation for the barred sector.

Both $G_M{}^N$ and $D_A{}^B$  in \bref{DGDG} satisfy GL(4$\mid$4) algebra.
If we focus on the AdS superalgebra part,
the global symmetry generators $G_M{}^N$ satisfies the global AdS superalgebra
\begin{eqnarray}
\left\{Q_{A\alpha},Q_{B,\beta}\right\}&=&-2\left[
\tau_3{}_{AB}P_{\alpha\beta} +\epsilon_{AB}
M_{\alpha\beta}\right]\label{QQPM}\\
Q_{1\alpha}&=&G_{m\bar{m}}+G_{\bar{m}m
\nn\\
Q_{2\alpha}&=&G_{m\bar{m}}-G_{\bar{m}m
\nn\\
P_{\alpha\beta}
&=&{G}_{\langle mn\rangle}\Omega_{\bar{m}\bar{n}}
-G_{\langle\bar{m}\bar{n}\rangle}\Omega_{mn}\cdots {\rm total~ momentum}\nn\\
M_{\alpha\beta}&=&-G_{(mn)}\Omega_{\bar{m}\bar{n}}
+G_{(\bar{m}\bar{n})}\Omega_{mn}\cdots{\rm total~ Lorentz}\nn~~~.
\end{eqnarray}
The right hand side of \bref{QQPM} can not be diagonalized
by the real SO(2) rotation of $Q_A$'s
because of the total Lorentz charge term with $\epsilon_{AB}$.
On the other hand the local AdS supersymmetry algebra is given by
\begin{eqnarray}
\left\{d_{A\alpha},d_{B,\beta}\right\}&=&2\left[
\tau_3{}
_{AB}
\tilde{p}_{\alpha\beta} +
\epsilon_{AB}m_{\alpha\beta}\right]\\
d_{1\alpha}&=&{D}_{a\bar{a}}+{\bar{D}}_{\bar{a}a}
\nn
\\
d_{2\alpha}&=&{D}_{a\bar{a}}-{\bar{D}}_{\bar{a}a}
\nn
\\
\tilde{p}_{\alpha\beta}&=&
{\bf D}
_{\langle ab\rangle}\Omega_{\bar{a}\bar{b}}
-{\bar{\bf D}}
_{\langle \bar{a}\bar{b}\rangle}\Omega_{ab}
\cdots {\rm local~LI~momentum}\nn
\\m_{\alpha\beta}&=&
-{\bf D}_{(ab)}\Omega_{\bar{a}\bar{b}}
+\bar{\bf D}
_{(\bar{a}\bar{b})}\Omega_{a{b}}\cdots{\rm local~Lorentz}\nn
~~~.
\end{eqnarray}
In our coset approach the local Lorentz generator
is a constraint \bref{DSp4GL1},
so  the
local supercovariant derivative $d_{A\alpha}$'s
can be separated into;
\begin{eqnarray}
\left\{d_{1\alpha},d_{2\beta}\right\}=
2 m_{\alpha\beta}\equiv 0~,~~
 \left\{d_{1\alpha},d_{1\beta}\right\}=
2 \tilde{p}_{\alpha\beta}~~,~~
\left\{d_{2\alpha},d_{2\beta}\right\}=
-2\tilde{p}_{\alpha\beta}
  \end{eqnarray}
Although the global superalgebra can not be separated into
irreducible algebras in the AdS background,
the local superalgebra can be separated into
irreducible sets on the GL(4${\mid}$4) covariant coset approach.
This property allows simpler description of the AdS superstring
as the flat case at least in the classical mechanics level.

\par\vskip 6mm
\section{ AdS Superparticle}

We begin with the action for a superparticle in the AdS$_5\times$S$^5$
\begin{eqnarray}
&S=\displaystyle\int  d\tau~\displaystyle\frac{1}{2e}
\left\{-{\bf J}_\tau^{\langle ab\rangle}{\bf J}_{\tau, \langle ab\rangle}
+\bar{\bf J}_\tau^{\langle \bar{a}\bar{b}\rangle}\bar{\bf J}_{\tau, \langle \bar{a}\bar{b}\rangle}
\right\}&~~~.\label{RS}
\end{eqnarray}
Here we omit the upper-subscript $L$ for the LI currents and their components are denoted as
\begin{eqnarray}
(J^L_{~\mu})_A{}^B
=\left(
\begin{array}{cc}
{\bf J}_{\mu,}{}_a{}^b&j_{\mu,}{}_a{}^{\bar{b}}\\\bar{j}_{\mu,}{}_{\bar{a}}{}^b&\bar{\bf J}_{\mu,}{}_{\bar{a}}{}^{\bar{b}}
\end{array}
\right)~~~.
\end{eqnarray}
From the definition of the canonical conjugates,
$
\Pi_A{}^M={\delta S}/{\delta \partial_\tau Z_M{}^A} (-)^A
$,
we have
the following primary constraints \cite{HKAdS}
\begin{eqnarray}
{\cal A}_{\rm P}=\frac{1}{2}{\rm tr}\left[
\langle{\bf D}\rangle^2-
\langle\bar{\bf D}\rangle^2\right]=0~~,~~
D_{a\bar{b}}=\bar{D}_{\bar{a}{b}}=0~~~\label{ApDD}
\end{eqnarray}
with
\begin{eqnarray}
D_A{}^B=\left(
\begin{array}{cc}{\bf D}_a{}^b&D_a{}^{\bar{b}}\\
\bar{D}_{\bar{a}}{}^b&\bar{\bf D}_{\bar{a}}{}^{\bar{b}}
\end{array}
\right)~~~.
\end{eqnarray}

The Hamiltonian is chosen as
\begin{eqnarray}
{\cal H}=-{\cal A}_{\rm P}=- \frac{1}{2}{\rm tr}\left[
\langle{\bf D}\rangle^2-
\langle\bar{\bf D}\rangle^2\right]
\end{eqnarray}
and the $\tau$-derivative is determined by
the Poisson bracket with ${\cal H}$,
 $\partial_\tau{\cal O}=\{{\cal O},{\cal H}\}$.
The fact that a half of the fermionic constraints  is second class
requires the Dirac bracket in general.
Fortunately the Dirac bracket with the Hamiltonian is equal to
its Poisson bracket because the fermionic
constrains are ${\cal H}$ invariant.

The LI current is calculated as
\begin{eqnarray}
J^L_\tau=Z^{-1}\partial_\tau Z=\left(
\begin{array}{cc}
\langle{\bf D}\rangle&0\\0&\langle \bar{\bf D}\rangle
\end{array}
\right)~~,~~\partial_\tau J^L=0~~~.
\end{eqnarray}
The RI current, generating the global GL(4$\mid$4) symmetry,
is given as
\begin{eqnarray}
J^R_\tau\equiv Z\Pi=Z\left(J^L_\tau+A_\tau\right)Z^{-1}~~,~~
A_\tau=
\left(
\begin{array}{cc}
({\bf D})-\frac{1}{4}\Omega {\rm tr}{\bf D}&D
\\\bar{D}&(\bar{\bf D})-\frac{1}{4}\Omega {\rm tr}\bar{\bf D}
\end{array}
\right)~~~.\label{JRSP}
\end{eqnarray}
Although the stability group does not contain fermionic components originally,
the fermionic components of the gauge connection $A$ in \bref{JRSP} is induced.
It is noted that ``$A$" is the gauge connection
distinguishing from
 the reparametrization
constraint ``${\cal A}$".
The RI current is conserved, since the Hamiltonian is written
by LI currents which are manifestly global symmetry invariant
\begin{eqnarray}
\partial_\tau J^R=0~~~.
\end{eqnarray}

The $\kappa$ symmetry generators are half of the fermionic constraints
by projecting out with the null vector as
\begin{eqnarray}
{\cal B}_{\rm P}{}_a{}^{\bar{b}}=\langle{\bf D}\rangle_a{}^b D_b{}^{\bar{b}}
+D_a{}^{\bar{a}}\langle\bar{\bf D}\rangle_{\bar{a}}{}^{\bar{b}}~~,~~
\bar{\cal B}_{\rm P}{}_{\bar{a}}{}^{{b}}=\langle\bar{\bf D}\rangle_{\bar{a}}{}^{\bar{b}}\bar{D}_{\bar{b}}{}^{{b}}
+\bar{D}_{\bar{a}}{}^{{a}}\langle{\bf D}\rangle_{a}{}^{b}~~~.
\end{eqnarray}
If we construct the closed algebra including these $\kappa$ generators
with keeping
 the bilinear of the fermionic constraints,
the $\tau$-reparametrization constraint, ${\cal A}_{\rm P}$, is modified to \cite{HKAdS}
\begin{eqnarray}
\tilde{\cal A}_{\rm P}&=&\frac{1}{2}{\rm tr}\left[
\langle{\bf D}\rangle^2-
\langle\bar{\bf D}\rangle^2
+2D\bar{D}
\right]~~~~~.
\end{eqnarray}
This expression appears in the Poisson bracket of
${\cal B}$ with $\bar{\cal B}$,
when we keep the bilinear of fermionic constrains.
The  RR flux is responsible for the last term ``$D\bar{D}$".
The term which is bilinear of the constraints
does not change the Poisson bracket
since its bracket with an arbitrary variable
gives terms proportional to the constraints
which are zero on the constrained surface.
In another word ${\cal A}_{\rm P}$ has an ambiguity of bilinear of the constraints,
and the $\kappa$ invariance fixes it.
On the original coset constrained surface \bref{DSp4GL1}
it is also rewritten as
\begin{eqnarray}
\tilde{\cal A}_{\rm P}=\frac{1}{2}~{\rm Str}~[D_A{}^B]^2=\frac{1}{2}~{\rm Str}~[J^R_\tau]^2~~~.
\end{eqnarray}
This is zero-mode contribution of the classical Virasoro constraint
for a superstring
in the AdS$_5\times$S$^5$ background.

\par\vskip 6mm
\section{ AdS Superstring}
\subsection{Conserved currents}

We take the action for a superstring in the AdS$_5\times$S$^5$ given by
\begin{eqnarray}
&S=\displaystyle\int  d^2\sigma~\frac{1}{2}\left\{
-\sqrt{-g}g^{\mu\nu}({\bf J}_\mu^{\langle ab\rangle}{\bf J}_{\nu, \langle ab\rangle}
-\bar{\bf J}_\mu^{\langle \bar{a}\bar{b}\rangle}\bar{\bf J}_{\nu, \langle \bar{a}\bar{b}\rangle}
)
+\frac{k}{2}\epsilon^{\mu\nu}(
E^{1/2}j_\mu^{a\bar{b}}j_{\nu, a\bar{b}}
-
E^{-1/2}\bar{j}_\mu^{\bar{a}{b}}\bar{j}_{\nu, \bar{a}{b}}
)\right\}&\nn\\\label{RS}
\end{eqnarray}
where ``$k$" represents the WZ term contribution with $k=1$ and
 $E={\rm sdet} Z_M{}^A$.
The consistent $\tau$ and $\sigma$ reparametrization generators \cite{HKAdS} are
\begin{eqnarray}
{\cal A}_\perp&=&{\cal A}_{0\perp} +k~{\rm tr}\left[-E^{1/4}Fj_\sigma+E^{-1/4}\bar{F}\bar{j}_\sigma\right]\nn\\
{\cal A}_\parallel&=&{\cal A}_{0\parallel} +k~{\rm tr}\left[E^{-1/4}F\bar{j}_\sigma-E^{1/4}\bar{F}{j}_\sigma\right]\label{Aperp}
\end{eqnarray}
with the following primary constraints
\begin{eqnarray}
{\cal A}_{0\perp}&=&\frac{1}{2}{\rm tr}\left[
(\langle{\bf D}\rangle^2+\langle{\bf J}_\sigma\rangle^2)-
(\langle\bar{\bf D}\rangle^2+\langle\bar{\bf J}_\sigma\rangle^2)
\right]=0\nn\\
{\cal A}_{0\parallel}&=&{\rm tr}\left[
\langle{\bf D}\rangle\langle{\bf J}_\sigma\rangle-
\langle\bar{\bf D}\rangle\langle\bar{\bf J}_\sigma\rangle
\right]=0\\
F_{a\bar{b}}&=&E^{1/4}D_{a\bar{b}}
+\frac{k}{2}E^{-1/4}(\bar{j}_\sigma)_{\bar{b}a}=0\nn\\
\bar{F}_{\bar{a}{b}}&=&E^{-1/4}\bar{D}_{\bar{a}{b}}+\frac{k}{2}E^{1/4}({j}_\sigma)_{{b}\bar{a}}=0~~~.\label{FermionicF}
\end{eqnarray}
Their Poisson brackets are
\begin{eqnarray}
\left\{{\cal A}_\perp(\sigma),{\cal A}_\perp(\sigma')\right\}&=&
2{\cal A}_\parallel(\sigma)\partial_\sigma\delta(\sigma-\sigma')+
\partial_\sigma{\cal A}_\parallel(\sigma)\delta(\sigma-\sigma')\nn\\
\left\{{\cal A}_\perp(\sigma),{\cal A}_\parallel(\sigma')\right\}&=&
2{\cal A}_\perp(\sigma)\partial_\sigma\delta(\sigma-\sigma')+
\partial_\sigma{\cal A}_\perp(\sigma)\delta(\sigma-\sigma')\\
\left\{{\cal A}_\parallel(\sigma),{\cal A}_\parallel(\sigma')\right\}&=&
2{\cal A}_\parallel(\sigma)\partial_\sigma\delta(\sigma-\sigma')+
\partial_\sigma{\cal A}_\parallel(\sigma)\delta(\sigma-\sigma')\nn~~~.
\end{eqnarray}

The Hamiltonian is chosen as
\begin{eqnarray}
{\cal H}&=&-\int d\sigma {\cal A}_\perp\label{HamiltonianSUST}\\&=&
-\int d\sigma {\rm tr}\left[
\frac{1}{2}\left\{
\langle{\bf D}\rangle^2+\langle{\bf J}_\sigma\rangle^2-
\langle\bar{\bf D}\rangle^2-\langle\bar{\bf J}_\sigma\rangle^2\right\}
+\left(kE^{-1/2}\bar{D}\bar{j}_\sigma-k E^{1/2}Dj_\sigma
+j_\sigma\bar{j}_\sigma\right)
\right]
~~~.\nn
\end{eqnarray}
From now on $E=1$ gauge is taken using the local GL(1) invariance.
The global GL(1) symmetry is broken by the WZ term.

Using the Hamiltonian in  \bref{HamiltonianSUST}
the $\tau$-derivative of ${\cal A}_{\perp}$ and ${\cal A}_\parallel$ are given as
\begin{eqnarray}
\partial_\tau {\cal A}_\perp=\partial_\sigma {\cal A}_\parallel~~,~~
\partial_\tau {\cal A}_\parallel=\partial_\sigma {\cal A}_\perp~~~.\label{dAdA}
\end{eqnarray}
Although the coset parameter $Z_M{}^A$ does not satisfy
the world-sheet free wave equation, it is essential to introduce
the world-sheet lightcone
coordinate
\begin{eqnarray}
\sigma^\pm=\tau \pm \sigma~~,~~
\partial_\pm=\frac{1}{2}(\partial_\tau \pm \partial_\sigma)~~~.
\end{eqnarray}
The differential equations \bref{dAdA} are rewritten as
\begin{eqnarray}
\partial_- {\cal A}_+=0~~,~~
\partial_+ {\cal A}_-=0~~,~~
{\cal A}_\pm={\cal A}_\perp\pm {\cal A}_\parallel~~~,
\end{eqnarray}
so the infinite number of the conserved
currents are
\begin{eqnarray}
\partial_- \left[f(\sigma^+){\cal A}_+\right]=0~~,~~
\partial_+ \left[f(\sigma^-){\cal A}_-\right]=0~~\label{conformal}
\end{eqnarray}
with an arbitrary function $f$.
Then there exist infinite number of conserved  charges
\begin{eqnarray}
\partial_- \left[\displaystyle\int d\sigma~
f(\sigma^+){\cal A}_+\right]=0~~,~~
\partial_+ \left[\displaystyle\int d\sigma~
f(\sigma^-){\cal A}_-\right]=0~~~.
\end{eqnarray}

On the other hand the integrability of the superstring
will provide the infinite number of ``local" charges as well as the
``non-local" charges written down in \cite{HY}.
The LI currents
is given by
\begin{eqnarray}
\left\{\begin{array}{ccl}
J^L_\tau&=&\left(
\begin{array}{cc}
\langle{\bf D}\rangle&-k\bar{j}_\sigma\\-kj_\sigma&\langle \bar{\bf D}\rangle
\end{array}
\right)
=\left(
\begin{array}{cc}
\langle{\bf D}\rangle&
2D-2F\\2\bar{D}-2\bar{F}&\langle \bar{\bf D}\rangle
\end{array}
\right)
\approx
\left(
\begin{array}{cc}
\langle{\bf D}\rangle&
2D\\2\bar{D}&\langle \bar{\bf D}\rangle
\end{array}
\right)\nn\\
J^L_\sigma&=&
\left(
\begin{array}{cc}
{\bf J}_{\sigma}
&j_{\sigma}\\
\bar{j}_{\sigma}&
\bar{\bf J}_{\sigma}
\end{array}
\right)
\end{array}\right.~~~
\end{eqnarray}
where the $\tau$ component is determined by \bref{HamiltonianSUST}.
The LI currents satisfy the flatness condition but
does not satisfy the conservation law.
The RI currents are obtained in \cite{HY} as
\begin{eqnarray}
\left\{\begin{array}{ccl}
J^R_\tau&=&ZDZ^{-1}=Z(J^L_\tau+A_\tau)Z^{-1}\label{RISUST}\\
J^R_\sigma&=&Z(J^L_\sigma+A_\sigma)Z^{-1}~~,~~
J^L_\sigma+A_\sigma=
\left(
\begin{array}{cc}
\langle{\bf J}_{\sigma}\rangle
&\bar{F}+\frac{1}{2}j_{\sigma}\\
F+\frac{1}{2}\bar{j}_{\sigma}&
\langle\bar{\bf J}_{\sigma}\rangle
\end{array}
\right)
\end{array}\right.
\end{eqnarray}
where the gauge connection $A_\mu$ is
\begin{eqnarray}
\left\{\begin{array}{ccl}
A_\tau&=&\left(
\begin{array}{cc}
({\bf D})-\frac{1}{4}\Omega {\rm tr}{\bf D}&-D\\-\bar{D}&(\bar{\bf D})-\frac{1}{4}\Omega {\rm tr}\bar{\bf D}
\end{array}
\right)\nn\\
A_\sigma&=&
\left(
\begin{array}{cc}
-({\bf J}_\sigma)+\frac{1}{4}\Omega {\rm tr}{\bf J}_\sigma&
\bar{F}-\frac{1}{2}j_\sigma
\\F-\frac{1}{2}\bar{j}_\sigma
&-(\bar{\bf J}_\sigma)+\frac{1}{4}\Omega {\rm tr}\bar{\bf J}_\sigma
\end{array}
\right)
\end{array}\right.~~~.
\end{eqnarray}
The fermionic components of $A_\mu$ appear again.
In this paper
the fermionic constraints, $F$ and $\bar{F}$,
in
the fermionic components of $A_\sigma$  are kept
while they were absent in our previous paper \cite{HY}
depending on the treatment of the constraint bilinear terms.
Then the integrability of the superstring
leads to the current conservation and the flatness condition
for the RI current;
\begin{eqnarray}
\partial_\tau J^R_\tau=\partial_\sigma J^R_\sigma~~,~~
\partial_\tau J^R_\sigma-\partial_\sigma J^R_\tau=
2\left[J^R_\tau, J^R_\sigma\right]~~~\label{dJRdJR}~~~.
\end{eqnarray}
They are rewritten as
\begin{eqnarray}
\partial_-J^R_+=\left[J^R_-,J^R_+\right]~~,~~
\partial_+J^R_-=\left[J^R_+,J^R_-\right]~~,~~
J^R_\pm=J^R_\tau\pm J^R_\sigma~~~.\label{JRpm}
\end{eqnarray}
Taking the supertrace, denoting ``Str", leads to
the infinite number of conserved ``local"
currents because $J^R_\mu$ are supermatrices,
\begin{eqnarray}
\partial_-~{\rm Str}\left[(J^R_+)^n\right]=0~~,~~
\partial_+~{\rm Str}\left[(J^R_-)^n\right]=0~~,n=1,2,\cdots~~~.\label{JRn}
\label{integrablity}
\end{eqnarray}
It gives the infinite number of conserved ``local" charges
\begin{eqnarray}
\partial_\tau \left[\displaystyle\int d\sigma~
 f(\sigma^+){\rm Str}(J^R_+)^n
\right]=0~~,~~
\partial_\tau \left[\displaystyle\int d\sigma~
 f(\sigma^-){\rm Str}(J^R_-)^n
\right]=0~~~.\end{eqnarray}

In this way classical 2-dimensional conformal symmetry and
integrability of AdS superstring lead to two infinite sets of
 conserved currents,
\bref{conformal} and \bref{integrablity}.
In next sections the relation between them is examined.

\subsection{Stress-energy tensor ($n=2$)}

The ``$+/-$" (right/left moving) modes of the RI currents
on the original coset constrained space
\bref{DSp4GL1}
are written as
\begin{eqnarray}
J^R_\pm=
Z\left(
\begin{array}{cc}
\langle{\bf D}_\pm \rangle&D\pm (\bar{F}+\frac{1}{2}j_\sigma)
\\
\bar{D}\pm(F+\frac{1}{2}\bar{j}_\sigma)&\langle\bar{\bf D}_\pm\rangle
\end{array}
\right)Z^{-1}=
Z\left(
\begin{array}{cc}
\langle{\bf D}_\pm\rangle&d_\pm+\frac{1}{2}j_\pm
\\
\pm(d_\pm -\frac{1}{2}j_\pm)
&\langle\bar{\bf D}_\pm\rangle
\end{array}
\right)Z^{-1}\nn\\
\end{eqnarray}
  with
\begin{eqnarray}
{\bf D}_\pm={\bf D}\pm{\bf J}_\sigma~~,~~
\bar{\bf D}_\pm=\bar{\bf D}\pm\bar{\bf J}_\sigma~~,~~d_\pm=F\pm\bar{F}~~,~~
j_\pm=j_\tau\pm j_\sigma=-\bar{j}_\sigma\pm j_\sigma
\label{pmpm}
\end{eqnarray}
carrying the LI currents indices, $AB$.
This is supertraceless, Str$J^R_\pm=0$, so $n=1$ case of
\bref{JRn} gives just trivial equation.

Let us look at the $n=2$ case of \bref{JRn},
 ${\rm Str}\left[(J^R_\pm)^2\right]$.
Then the ``+" sector  is written as
\begin{eqnarray}
\frac{1}{2}{\rm Str}\left[(J^R_+)^2\right]&=&
\frac{1}{2}{\rm Str}\left[
\left(
\begin{array}{cc}
\langle{\bf D}_+\rangle&d_++\frac{1}{2}j_+
\\
d_+-\frac{1}{2}j_+
&\langle\bar{\bf D}_+\rangle
\end{array}
\right)^2
\right]\nn\\
&=&\frac{1}{2}{\rm tr}\left[
\langle{\bf D}_+\rangle^2-\langle\bar{\bf D}_+\rangle^2
+2(d_++\frac{1}{2}j_+
)(d_+-\frac{1}{2}j_+
)
\right]\nn\\
&=&{\rm tr}\left[\frac{1}{2}\left(
\langle{\bf D}_+\rangle^2-\langle\bar{\bf D}_+\rangle^2\right)
+j_+d_+\right]~~~.\label{419}
\end{eqnarray}
The ``$-$" sector is
\begin{eqnarray}
\frac{1}{2}{\rm Str}\left[(J^R_-)^2\right]&=&
{\rm tr}\left[\frac{1}{2}\left(
\langle{\bf D}_-\rangle^2-\langle\bar{\bf D}_-\rangle^2\right)
-j_-d_-
\right]~~~.\label{420}
\end{eqnarray}

On the other hand the conformal symmetry generator ${\cal A}_\pm$ is
rewritten from the relation \bref{Aperp} and \bref{pmpm} as
\begin{eqnarray}
{\cal A}_\pm&=&
{\rm tr}
\left[\frac{1}{2}\left(
\langle{\bf D}_\pm \rangle^2-\langle\bar{\bf D}_\pm \rangle^2\right)
\pm j_\pm d_\pm
\right]~=~
\frac{1}{2}{\rm Str}\left[(J^R_\pm)^2\right]
~~~.\label{421}
\end{eqnarray}
If we take care of the square of the fermionic constraints,
the closure of the first class constraint set
including the $\kappa$ symmetry
generators,
\begin{eqnarray}
{\cal B}_\pm&=&
\langle{\bf D}_\pm\rangle d_\pm
+d_\pm\langle\bar{\bf D}_\pm\rangle\label{kappaSUST}
~~
\end{eqnarray}
determines the ambiguity of  bilinear of the constraints as
\begin{eqnarray}
\tilde{\cal A}_\pm=
{\rm tr}\left[\frac{1}{2}\left(
\langle{\bf D}_\pm \rangle^2-\langle\bar{\bf D}_\pm \rangle^2\right)
\pm(\frac{1}{2}d_\mp +j_\pm)d_\pm
\right]=
{\cal A}_\pm+{\rm tr}F\bar{F}~~
\end{eqnarray}
obtained in \cite{HKAdS} as a generator of the ${\cal ABCD}$
constraint set
known to exist for a superstring in a flat space
 \cite{WSmech,ABCD}.
Then the stress-energy tensor is
\begin{eqnarray}
T_{\pm\pm}\equiv\tilde{\cal A}_\pm
\approx
{\cal A}_\pm
={\rm Str}J^R_\pm J^R_\pm
~~~.\label{423}
\end{eqnarray}
This is $\kappa$ symmetric stress-energy tensor
in a supersymmetric generalization of
 Sugawara form.

\subsection{Supercovariant derivative algebra}

Existence of the conformal invariance should present the
irreducible coset components of supercovariant derivatives
\cite{HKAdS};
\begin{eqnarray}
\langle{\bf D}_\pm\rangle&=&\langle{\bf D}\rangle\pm\langle{\bf J}_\sigma\rangle
~~,~~
\langle\bar{\bf D}_\pm\rangle~=~\langle\bar{\bf D}\rangle\pm\langle\bar{\bf J}_\sigma\rangle\nn\\
d_\pm&=&F\pm\bar{F}=(D\pm\frac{1}{2}j_\sigma)\pm(\bar{D}\pm\frac{1}{2}\bar{j}_\sigma)
\nn~~~.
\end{eqnarray}

On the  constraint surface \bref{DSp4GL1} and \bref{FermionicF}
the $+/-$ sector supercovariant derivatives are separated as
\begin{eqnarray}
\left\{\langle{\bf D}_+\rangle_{ab}(\sigma),
\langle{\bf D}_-\rangle_{cd}(\sigma')\right\}
&=&2\Omega_{\langle c|\langle b}({\bf D})_{a\rangle|d\rangle}
\delta(\sigma-\sigma')
\equiv 0\nn\\
\left\{\langle{\bf D}_+\rangle_{ab}(\sigma),
d_{-,c\bar{d}}(\sigma')\right\}
&=&\Omega_{c\langle b}d_{+,a\rangle \bar{d}}
\delta(\sigma-\sigma')=
\Omega_{c\langle b}(F+\bar{F})_{a\rangle \bar{d}}
\delta(\sigma-\sigma')
\approx 0\nn\\
\left\{d_{+,a\bar{b}}(\sigma),
d_{-,c\bar{d}}(\sigma')\right\}
&=&2\left[
\Omega_{ac}(\bar{\bf D})_{\bar{b}\bar{d}}
+\Omega_{\bar{b}\bar{d}}({\bf D})_{ac}
\right]
\delta(\sigma-\sigma')
\equiv 0\nn
\end{eqnarray}
with
analogous relation for  the barred sector, $\langle\bar{\bf D}_\pm\rangle$.

The ``+" sector supercovariant derivative algebra is
\begin{eqnarray}
\left\{\langle{\bf D}_+\rangle_{ab}(\sigma),
\langle{\bf D}_+\rangle_{cd}(\sigma')\right\}
&=&2\Omega_{\langle c|\langle b}\Omega_{a\rangle|d\rangle}
\delta'(\sigma-\sigma')+4\Omega_{\langle c|\langle b}
({\bf J}_\sigma)_{a\rangle|d\rangle}
\delta(\sigma-\sigma')\nn\\
&\equiv& 2\Omega_{\langle c|\langle b} \nabla_{a\rangle|d\rangle}
\delta(\sigma-\sigma')
\nn\\
\left\{d_{+,a\bar{b}}(\sigma),
d_{+,c\bar{d}}(\sigma')\right\}
&=&2\left[
\Omega_{\bar{b}\bar{d}}\langle{\bf D}_+\rangle_{ac}
-\Omega_{ac}\langle\bar{\bf D}_+\rangle_{\bar{b}\bar{d}}
\right]
\delta(\sigma-\sigma')
\nn\\
\left\{\langle{\bf D}_+\rangle_{ab}(\sigma),
d_{+,c\bar{d}}(\sigma')\right\}
&=&
\Omega_{c\langle b}(d_-+2j_+)_{a\rangle \bar{d}}
\delta(\sigma-\sigma')\approx
2\Omega_{c\langle b}\omega_{+,a\rangle \bar{d}}
\delta(\sigma-\sigma')
\nn\\
\left\{d_{+,a\bar{b}}(\sigma),
\omega_{+,c\bar{d}}(\sigma')\right\}
&=&-2\Omega_{\bar{b}\bar{d}}\Omega_{ac}\delta'(\sigma-\sigma')
2\left[
-\Omega_{\bar{b}\bar{d}}({\bf J}_\sigma)_{ac}
-\Omega_{ac}(\bar{\bf J}_\sigma)_{\bar{b}\bar{d}}\right]\delta(\sigma-\sigma')\nn\\
&\equiv& -2\nabla_{\bar{b}\bar{d};ac}\delta(\sigma-\sigma')\nn\\
\label{sucovder}\\
\left\{\langle{\bf D}_+\rangle_{ab}(\sigma),
\omega_{+,c\bar{d}}(\sigma')\right\}
&=&
\Omega_{c\langle b}\omega_{-,a\rangle \bar{d}}
\delta(\sigma-\sigma')
\nn\\
\left\{\omega_{+,a\bar{b}}(\sigma),
\omega_{+,c\bar{d}}(\sigma')\right\}
&=&0~~~\nn
\end{eqnarray}
where
\begin{eqnarray}
\omega_\pm&=&j_\pm=-\bar{j}_\sigma\pm j_\sigma~~~.
\end{eqnarray}
This is comparable with the flat case where
the non-local term, $\partial_\sigma\delta(\sigma-\sigma')$, is replaced by the
local Lorentz (~[Sp(4)]$^2$~) covariant non-local term,
$\nabla_\sigma\delta(\sigma-\sigma')$.
For the fifth Poisson bracket,
 $\left\{\langle{\bf D}_+\rangle,
\omega\right\}
$,
it is zero for the flat case but it is not for the AdS case.
For a superstring in a flat space
the consistency of the $\kappa$ symmetry constraint
requires
the first class constraint set,
namely ``${\cal ABCD}$" constraint,
which are bilinear of the supercovariant derivatives
 \cite{WSmech,ABCD}.
 For the AdS case the situation is completely the same,
despite of this anomalous term \cite{HKAdS}.

\par\vskip 6mm

\subsection{``Local" currents ($n\geq 3$)}

Next let us look at $n\geq 3 $ cases
of
the infinite number of conserved ``local" current \bref{JRn}.
For simplicity we focus on the ``+" sector and replace
$``+"$ by $``~\hat{~}~"$, as
$J_+ \to \hat{J}$.
The first  three powers of the RI current,
$(J^R)^n$ with $n=1,2,3$, are listed as below:
\begin{eqnarray}
\left[Z^{-1}\hat{J}^R Z\right]_{AB}&=&\left(
\begin{array}{cc}
\langle\hat{\bf D}\rangle_{\langle ab\rangle}&
(\hat{d}+\frac{1}{2}\hat{j})_{a\bar{b}}
\\
\pm(\hat{d} -\frac{1}{2}\hat{j})_{b\bar{a}}
&\langle\hat{\bar{\bf D}}\rangle_{\bar{a}\bar{b}}
\end{array}
\right)
\end{eqnarray}
\begin{eqnarray}
\left[Z^{-1}(\hat{J}^R)^2 Z\right]_{AB}&=&-\frac{1}{4}
\left(
\begin{array}{cc}
\Omega_{ab}~{\rm tr}(
\langle\hat{\bf D}\rangle^2
+\hat{j}\hat{d})
&\\&
\Omega_{\bar{a}\bar{b}}~{\rm tr}(
\langle\hat{\bar{\bf D}}\rangle^2
-\hat{j}\hat{d})
\end{array}
\right)\\
&&+
\left(
\begin{array}{cc}
(\hat{d}^2-\frac{1}{4}\hat{j}^2)_{(ab)}
+\langle \hat{j}\hat{d}\rangle_{\langle ab \rangle}
&\hat{\cal B}_{a\bar{b}}
+\frac{1}{2}(\langle \hat{\bf D}\rangle \hat{j}
+\hat{j}\langle \hat{\bar{\bf D}}\rangle
)_{a\bar{b}}
\\
\hat{\cal B}_{b\bar{a}}
-\frac{1}{2}(\langle \hat{\bf D}\rangle \hat{j}
+\hat{j}\langle \hat{\bar{\bf D}}\rangle
)_{b\bar{a}}
&(\hat{d}^2-\frac{1}{4}\hat{j}^2)_{(\bar{a}\bar{b})}
-\langle\hat{j}\hat{d}\rangle_{\langle \bar{a}\bar{b} \rangle}
\end{array}
\right)\nn
\end{eqnarray}
\begin{eqnarray}
&&\left[Z^{-1}(\hat{J}^R)^3 Z\right]_{AB}~=~
\frac{1}{4}
\left(
\begin{array}{cc}
\Omega_{ab}~{\rm tr}\left[\hat{\cal B}\hat{j}-
(\langle\hat{\bf D}\rangle \hat{d} )\hat{j}
\right]
&\\&
-\Omega_{\bar{a}\bar{b}}~{\rm tr}\left[\hat{\cal B}\hat{j}
-(\hat{d}\langle\hat{\bar{\bf D}}\rangle)\hat{j}
\right]
\end{array}
\right)\\
&&-
\left(
\begin{array}{cc}
\left[\frac{1}{4}{\rm tr}(\langle\hat{\bf D}\rangle^2+\hat{j}\hat{d})~
\langle\hat{\bf D}\rangle
-\langle\hat{\bf D}\rangle (\hat{j} \hat{d})
+\hat{\cal B}\hat{j}
\right]_{\langle ab\rangle}
&
\frac{1}{4}{\rm tr}(\langle\hat{\bf D}\rangle^2
-\langle\hat{\bar{\bf D}}\rangle^2)~(\hat{d}+\frac{1}{2}\hat{j})_{a\bar{b}}
\\
\frac{1}{4}{\rm tr}(\langle\hat{\bf D}\rangle^2
-\langle\hat{\bar{\bf D}}\rangle^2)~(\hat{d}-\frac{1}{2}\hat{j})_{b\bar{a}}
&
\left[\frac{1}{4}{\rm tr}(\langle\hat{\bar{\bf D}}\rangle^2
+\hat{j}\hat{d})~
\langle\hat{\bar{\bf D}}\rangle
-(\hat{j} \hat{d})
\langle\hat{\bar{\bf D}}\rangle -\hat{\cal B}\hat{j}
\right]_{\langle \bar{a}\bar{b}\rangle}
\end{array}
\right)\nn\\
&&+
\left(
\begin{array}{cc}
\left[
2(\hat{d}^2-\frac{1}{4}\hat{j}^2)\langle\hat{\bf D}\rangle
+\hat{d}\langle\hat{\bar{\bf D}}\rangle\hat{d}
-\frac{1}{4}\hat{j}\langle\hat{\bar{\bf D}}\rangle \hat{j}
\right]_{(ab)}&
-\frac{1}{4}{\rm tr}(\hat{j}\hat{d})
(\hat{d}+\frac{1}{2}\hat{j})_{a\bar{b}}
+
\left[\langle\hat{{\bf D}}\rangle
(\hat{d}+\frac{1}{2}\hat{j})\langle\hat{\bar{\bf D}}\rangle
\right]_{a\bar{b}}
\\
\frac{1}{4}{\rm tr}(\hat{j}\hat{d})
(\hat{d}-\frac{1}{2}\hat{j})_{b\bar{a}}
+
\left[\langle
\hat{{\bf D}}\rangle
(\hat{d}-\frac{1}{2}\hat{j})\langle\hat{\bar{\bf D}}\rangle
\right]_{b\bar{a}}
&
\left[
2(\hat{d}^2-\frac{1}{4}\hat{j}^2)\langle\hat{\bar{\bf D}}\rangle
+\hat{d}\langle\hat{{\bf D}}\rangle\hat{d}
-\frac{1}{4}\hat{j}\langle\hat{{\bf D}}\rangle \hat{j}
\right]_{(\bar{a}\bar{b})}
\end{array}
\right)\nn\\
&&+
\left(
\begin{array}{cc}
&
\left[\left\{
(\hat{d}^2-\frac{1}{4}\hat{j}^2)
+\langle\hat{j}\hat{d}
\rangle
\right\}(\hat{d}+\frac{1}{2}\hat{j})
\right]_{a\bar{b}}
\\
\left[\left\{
(\hat{d}^2-\frac{1}{4}\hat{j}^2)
-\langle\hat{j}\hat{d}
\rangle
\right\}(\hat{d}-\frac{1}{2}\hat{j})
\right]_{\bar{a}b}
&
\end{array}
\right)\nn
\end{eqnarray}
In this computation
 5-dimensional $\gamma$-matrix relations are used,
for example
${\bf V}^{\langle ab\rangle}{\bf U}_{\langle bc\rangle}
+{\bf U}^{\langle ab\rangle}{\bf V}_{\langle bc\rangle}
=\frac{1}{2}\delta^a_c~{\rm tr}{\bf V}{\bf U}$ for bosonic
vectors ${\bf V},~{\bf U}$.

The conserved ``local" current with $n=3$ becomes
\begin{eqnarray}
{\rm Str}
(\hat{J}^R)^3 &=&
{\rm tr}\left[2\hat{\cal B}\hat{j}
-(\langle\hat{\bf D}\rangle \hat{d} )\hat{j}
-(\hat{d}\langle\hat{\bar{\bf D}}\rangle)\hat{j}
\right]
=~{\rm tr}
~(\hat{\cal B}\hat{j}) \label{430}
\end{eqnarray}
where $\hat{\cal B}$ is the $\kappa$ generating constraint
 \bref{kappaSUST}.
 The conserved ``local" current with $n=4$ becomes
\begin{eqnarray}
{\rm Str}
(\hat{J}^R)^4 &=&
-\frac{1}{2}
{\rm tr}\left(
\langle\hat{\bf D}\rangle^2+
\langle\hat{\bar{\bf D}}\rangle^2
\right)
\hat{\cal A}+\left(~\cdots~\right){\rm tr}
~(\hat{\cal B}\hat{j})~~.
\end{eqnarray}
The conserved ``local" current with $n=5,6$ are given as;
 Str$(\hat{J}^R)^5=$(
$\hat{\cal B}$ dependent terms),
 Str$(\hat{J}^R)^6=$(
$\hat{\cal A}$ and $\hat{\cal B}$ dependent terms).
In general for even $n=2m$ its bosonic part is given as
\begin{eqnarray}
{\rm Str}
(\hat{J}^R)^{2m} \mid_{\rm bosonic}&=&
\left({\rm tr}\langle\hat{\bf D}\rangle^2\right)^m-
\left({\rm tr}\langle\hat{\bar{\bf D}}\rangle^2\right)^m\nn\\
&=&{\rm tr}\left(\langle\hat{\bf D}\rangle^2-\langle\hat{\bar{\bf D}}\rangle^2
\right)~\left\{\left(
{\rm tr}\langle\hat{\bf D}\rangle^2\right)^{m-1}
+\cdots +
\left({\rm tr}\langle\hat{\bar{\bf D}}\rangle^2\right)^{m-1}
\right\}\nn\\
&\Rightarrow& (\cdots)\hat{\cal A}+\left(~\cdots~\right){\rm tr}
~(\hat{\cal B}\hat{j})
\end{eqnarray}
where the last equality is guaranteed by the $\kappa$ invariance.
It is also pointed out that
the conserved supertraces of multilinears in the currents factorize in traces of lower number of currents and that for an even number of currents one of the factors
is the stress tensor in \cite{BPR}.
For odd $n=2m+1$ its bosonic part is given as
\begin{eqnarray}
{\rm Str}
(\hat{J}^R)^{2m+1} \mid_{\rm bosonic}~=~0
~\Rightarrow~\left(~\cdots~\right){\rm tr}
~(\hat{\cal B}\hat{j})
\end{eqnarray}
where the possible fermionic variable dependence is
a term proportional to $\hat{\cal B}$
 guaranteed by the $\kappa$ invariance.

In this way, after taking supertrace the even
$n$-th power of $J^R$
reduces terms proportional to ${\cal A}$ and ${\cal B}$,
and the odd   $n$-th power of $J^R$
reduces a term proportional to ${\cal B}$ only.
In this paper
${\cal CD}$ constraints in the ${\cal ABCD}$ first class constraint set
are not introduced for simpler argument,
and set to zero because they are bilinears of constraints.

\par\vskip 6mm
\section{ Conclusion and discussions}

We obtained
the expression of
the conserved ``local" currents
derived from the integrability of a superstring
in the AdS$_5\times$S$^5$ background.
The infinite number of the conserved ``local" currents
 are written by the supertrace of the $n$-th power of the RI currents.
 The lowest nontrivial case, $n=2$, is nothing but the stress-energy tensor
  which is also Virasoro constraint,
  Str$(J^R)^2_\pm$ in \bref{419} and \bref{420}.
  For even $n$ the ``local" current reduces to terms proportional to the
  Virasoro constraint and the $\kappa$ symmetry constraint.
 For odd $n$ it reduces to a term   proportional to the
 $\kappa$ symmetry constraint.
In another word  the integrability
reduces to
the ${\cal AB}({\cal CD})$ first class constraint set
where ${\cal A}$ is the Virasoro generator
and ${\cal B}$ is the $\kappa$ symmetry generator.
The ${\cal ABCD}$ first class constraint set
is the local symmetry generator of superstrings both on the flat space
and on the AdS space.
It is natural that the physical degrees of freedom of
a superstring is common locally,
  independently of  flat or AdS backgrounds.

 It seems that the combination of the  ${\cal B}_\pm j_\pm$
 in \bref{430}
plays the role of the world-sheet supersymmetry operator
in a sense of the grading of the conformal generator.
However it is not
straightforward
to construct the worldsheet supersymmetry operator.
As in the flat case where the lightcone gauge makes
the relation between
the GS fermion and the NSR fermion more transparent,
the $\kappa$ gauge fixing will be a clue to
make a connection to the world-sheet supersymmetry.
We leave this problem in addition to the quantization
problem for future investigations.

\par\vskip 6mm

\noindent{\bf Acknowledgments}

The author thanks to K. Kamimura, S. Mizoguchi and K. Yoshida for fruitful discussions.

\par\vskip 6mm