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%%          The Holographic Supercurrent Anomaly                  %%
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%%                   M. Chaichian, W.F. Chen                      %%
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%%                         April, 2003                            %%
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\title{The Holographic Supercurrent Anomaly}

\author{M. Chaichian\renewcommand{\thefootnote}{\dagger}\thanks{
E-mail: Masud.Chaichian@helsinki.fi} and W.F.
Chen\renewcommand{\thefootnote}{\ddagger}\thanks{E-mail:
Wen-feng.Chen@helsinki.fi} }


\address{ High Energy Physics Division, Department of Physical Sciences,
University of Helsinki\\
and\\
 Helsinki Institute of Physics, FIN-00014,  Helsinki,
Finland}

\maketitle



\begin{abstract}
The $\gamma$-trace anomaly of supersymmetry
current in a supersymmetric gauge theory  shares a
superconformal anomaly multiplet with the  chiral $R$-symmetry anomaly
and the Weyl anomaly, and its holographic reproduction is a
valuable test to the AdS/CFT correspondence conjecture.
We investigate how the $\gamma$-trace anomaly of the supersymmetry
current of ${\cal N}=1$ four-dimensional supersymmetric gauge theory
in an ${\cal N}=1$ conformal supergravity background
can be extracted out from the ${\cal N}=2$ gauged supergravity
in five dimensions.
It is shown that the reproduction of this super-Weyl anomaly
 originates from the following two facts: First the ${\cal N}=2$
bulk supersymmetry transformation converts into ${\cal N}=1$
superconformal transformation on the boundary, which consists of
${\cal N}=1$ supersymmetry transformation and
special conformal supersymmetry (or super-Weyl) transformation;
second the supersymmetry variation of the bulk action
 of five-dimensional gauged supergravity is a total derivative.
  The non-compatibility
 of supersymmetry and  the super-Weyl transformation invariance yields
 the holographic supersymmetry current anomaly. Furthermore, we speculate
on that the contribution from the external gauge field
to the holographic superconformal anomaly is dominated by the leading
terms in the asymptotic expansion of the bulk fields in terms of
the radial coordinate, while external gravitational background
contribution is determined by those terms with logarithmic function
dependence on the radial coordinate.


\vspace{3mm}

\noindent {\it PACS}: 04.65.+e, 11.30.Pb, 11.15.-q, 11.40.-q
\end{abstract}



\vspace{3ex}




\section{Introduction}


  The AdS/CFT
correspondence conjecture \cite{mald}
  states that  the type IIB string theory compactified on $AdS_5\times S^5$
  theory with $N$ units of $R-R$ flux on $S^5$ describes the same physics
  as ${\cal N}=4$ $SU(N)$ supersymmetric Yang-Mills theory.
  Further, the explicit definition was  given
  in Refs.\,\cite{gkp} and \cite{witt1} as follows. In general, Given
 the type IIB superstring theory in the background
$AdS_{d+1}\times X^{9-d}$,
 with $X^{9-d}$ being a compact Einstein manifold, the boundary effect of
the type IIB superstring  theory must be considered since
$AdS_{d+1}$ has a boundary at spatial infinity, which is actually
a copy of Minkowski space $M_d$ (or its compactfied version $S_d$).  The
partition function of such a string theory with certain boundary
value of a bulk field takes the following form,
\begin{eqnarray}
\left.Z_{\rm
String}\left[\phi\right]\right|_{\phi\rightarrow\phi_{0}}
=\int_{\phi (x,0)=\phi_{0}(x)}{\cal D}\phi (x,r)
\exp\left(-S[\phi (x,r)]\right), \label{spf}
\end{eqnarray}
where $\phi^{(0)}(x)$ is the boundary value of the $AdS_{d+1}$ bulk quantity
$\phi (x,r)$ such as the graviton, gravitino, $NS-NS$ and $R-R$
antisymmetric tensor fields etc..
Since the isometry group $SO(2,d)$ of $AdS_{d+1}$ acts as the
conformal group  on $M_d$, and hence a quantum field theory
defined on the boundary should be conformal invariant. The
$AdS/CFT$ correspondence conjecture means that the type IIB
superstring partition function (\ref{spf}) should be identical to
the generating functional for the correlation functions of the
composite operators of certain conformal field theory
\begin{eqnarray}
Z_{\rm CFT}\left[\phi_{0}\right]&=& \left\langle \exp\int_{M^d}
d^dx
{\cal O} (x)  \phi_{0}(x)\right\rangle \nonumber\\
&=&\sum_n\frac{1}{n!}\int \prod_{i=1}^n d^d x_i \left\langle {\cal
O}_1 (x_1)\cdots {\cal O}_n (x_n) \right\rangle \phi_{0} (x_1)
\cdots \phi_{0} (x_n)\nonumber\\
&{\equiv}& \exp\left(-\Gamma_{\rm CFT} [\phi_{0}]\right),
\label{acc1}
\end{eqnarray}
$\Gamma [\phi_{0}]$ being the quantum effective action describing
the composite operators  interacting with
$\phi_{0}$ background field. That is,
 \begin{eqnarray}
   \left.Z_{\rm
String}\left[\phi\right]\right|_{\phi\rightarrow\phi_{0}}
   =Z_{\rm CFT}\left[\phi_{0}\right] \,.
   \label{acc2}
 \end{eqnarray}
In the large-$N$ case,  the type IIB string correction to
supergravity
is proportional to  $1/\sqrt{g_sN}$, $g_s$ being
the string coupling, thus one can neglect the
string effect and just consider its low-energy effective theory,
the type IIB supergravity. In this case,
the  partition function of the type IIB superstring can be
evaluated as the exponential of the supergravity action in a field
configuration $\phi^{\rm cl}[\phi^{0}]$ which satisfies the
classical equation of motion of the supergravity with the boundary
condition given by $\phi_{0}$, i.e.,
\begin{eqnarray}
\left.Z_{\rm
String}\left[\phi\right]\right|_{\phi\rightarrow\phi_{0}}
 =\exp\left(-S_{\rm SUGRA}[\phi^{\rm cl}[\phi_{0}]]\right) \,.
\label{acc3}
\end{eqnarray}
Comparing \,(\ref{acc3}) with (\ref{acc1}) and (\ref{acc2}), we immediately
conclude that the background effective action of the  large-$N$ limit
of the $d$-dimensional conformal field theory can be approximately
equal to the on-shell classical action of  $AdS_{d+1}$ supergravity with
non-empty boundary,
\begin{eqnarray}
\Gamma_{\rm CFT}[\phi_{0}]=S_{\rm SUGRA}[\phi^{\rm
cl}[\phi_{0}]]=\int d^d x \sum_{n}\phi_0^{(n)} (x)
\left\langle {\cal O}^{(n)}\right\rangle. \label{acc4}
\end{eqnarray}
In the above equation, ${\cal O}^{(r)}$ are the various composite operators
in the superconformal field theory such as the energy-momentum tensor
and chiral $R$-symmetry current etc. and $\phi_0^{(r)}$ are the
corresponding background fields such as the gravitational
and gauge  fields etc., which are boundary values of the corresponding
bulk fields.


Let us emphasize the role of the five-dimensional
gauged supergravities \cite{gst,awada,guna2} in AdS/CFT
correspondence \cite{ferr}.  The $AdS_5\times S^5$
background arises from
the near horizon limit of $D3$-brane solution of type IIB supergravity
\cite{agmo}.
In the $AdS_5\times S^5$ background, the spontaneous
compactification on $S^5$
of the type IIB  supergravity actually occurs \cite{freu,duff}.
With the assumption that there exists
a consistent nonlinear
truncation of the massless modes from
the whole Kluza-Klein spectrum of the
type IIB supergravity  compactified on $S^5$ \cite{duff,marcus,kim},
the resultant theory should be the $SO(6)(\cong SU(4))$ gauged ${\cal N}=8$
$AdS_5$ supergravity since the isometry group $SO(6)$ of the internal
space $S^5$ becomes the gauge group of the compactified theory and
the $AdS_5\times S^5$ background preserves all of the supersymmetries of
type IIB supergravity \cite{guna2}.
Furthermore, if the background for the type IIB supergravity
is  $AdS_5 \times X^5$ with $X^5$ being an
Einstein manifold rather than $S^5$ such as $T^{11}=(SU(2)\times
SU(2))/U(1)$ or certain orbifold, then due to the singularities in
the internal manifold, the number of preserved supersymmetries in
the compactified $AdS_5$ supergravity is reduced and the
isometry group of the theory also changes \cite{roman2,kw2,kach,law}.  One can
thus obtain
the gauged ${\cal N}=2,4$ $AdS_5$ supergravity in five dimensions,
and their dual field theories  are believed to be $N=1,2$ supersymmetric
gauge theories \cite{ferr,kw2,kach,law}.
In a strict sense, a supersymmetric gauge theory with lower supersymmetries
is not a conformal invariant theory since its beta function does not vanish.
However, it was shown that renormalization group flow of this type of
supersymmetric gauge theory has the fixed point, at which the conformal
invariance can arise \cite{kach,law,shif,seib,lei2,early1,early2}.
The $AdS/CFT$ correspondence between the ${\cal N}=2,4$
gauged supergravities in five dimensions and  ${\cal N}=1,2$
supersymmertric
gauge theories can thus be built \cite{ferr,kw2,kach,law}.

With the truncation  from the Kaluza-Klein tower, Eq.\,(\ref{acc4}) can be
 considered as a quantum effective action
describing a superconformal gauge theory in an external supergravity
background \cite{witt1,liu}, only where the external fields are
provided by the boundary values of those in a one-dimension-higher
bulk space-time. It is well known that the superconformal  anomaly
will arise from a classical superconformal gauge
theory  in an external supergravity background.
The reproduction of the  superconformal anomaly from the bulk
gauged supergravity will provide an important support to the above $AdS/CFT$
correspondence conjecture  at low-energy level.

In the next section, we shall briefly introduce the superconformal
anomaly for the ${\cal N}=1$ supersymmetric gauge theory in
an external ${\cal N}=1$ conformal  supergravity background.
Section 3 is devoted to a review of the ${\cal N}=2$
gauged supergravity in five dimensions and its
$AdS_5$ classical solution.
 We also emphasize  the behaviour of the fields
of gauged supergravity near the $AdS_5$ boundary
and the reduction of bulk supersymmetry transformation
into an ${\cal N}=1$ superconformal transformation
on the $AdS_5$ boundary. It is well known that the ${\cal N}=1$
superconformal transformation consists of
the ${\cal N}=1$ four-dimensional supersymmetry transformation
and the special conformal supersymmetry transformation
(or super-Weyl in curved space-time background).
In section 4,
we calculate the supersymmetry variation of the five-dimensional ${\cal N}=2$
gauged supergravity and extract out the surface term. Furthermore,
considering the fact that the four-dimensional supersymmetry and
the super-Weyl symmetry cannot be preserved simultaneously, we
find the  external gauge field part of the
holographic super-Weyl anomaly of ${\cal N}=1$
supersymmetric gauge theory  from the boundary term the five-dimensional
gauged supergravity.
In Section 5 we  summarize the main results, and give some further discussions.
Our results suggest that the contributions to the superconformal anomaly
from the external vector field and gravitational  field have different
holographic origin: the external vector contribution lies in the
  leading term in the near-boundary expansion of the bulk fields, while
the gravitational background contribution comes from the terms
with logarithmic dependence on the radial coordinate in the expansion.


\section{Superconformal Anomaly Multiplet in External
Conformal Supergravity Background}

The four-dimensional ${\cal N}=1$ supersymmetric $SU(N)$ gauge theory
consists of  the ${\cal N}=1$ supersymmetric Yang-Mills theory coupled
with ${\cal N}=1$ massless matter fields in various possible representations
of the gauge group. The classical Lagrangian density is
\begin{eqnarray}
{\cal L}_{\rm }=
\mbox{Tr}\left(-\frac{1}{4}W_{\mu\nu} W^{\mu\nu}
+\frac{1}{2}i\overline{\lambda}\gamma^\mu \nabla_\mu \lambda\right)
+{\cal L}_{\rm matter},
\end{eqnarray}
where $W_{\mu\nu}$ is the field strength for the $SU(N)$ gauge
field and the $\lambda$ is a Majorana spinor
in the adjoint representation of $SU(N)$.
Due to the supersymmetry, its energy-momentum
tensor $\theta^{\mu\nu}$, the supersymmetry current $s^{\mu}$ and
the axial vector (or equivalently chiral) R-current $j^{(5)\mu}$
 lie in a supermultiplet \cite{ilio,anm}.
These currents at classical
level are not only conserved,
\begin{eqnarray}
\partial_\mu \theta^{\mu\nu}=\partial_\mu s^\mu=\partial_\mu
j^{(5)\mu}=0,
\end{eqnarray}
but also satisfy further algebraic constraints
\begin{eqnarray}
\theta^{\mu}_{~\mu}=\gamma_\mu s^\mu=0.
\label{superconf}
\end{eqnarray}
This will promote  the Poincar\'{e} supersymmetry to a superconformal
symmetry since one can construct three more conserved currents,
\begin{eqnarray}
d^\mu {\equiv} x_\nu \theta^{\nu\mu}, ~ k_{\mu\nu}{\equiv} 2
x_\nu x^\rho\theta_{\rho\mu}-x^2\theta_{\mu\nu}, ~ l_\mu{\equiv}
ix^\nu \gamma_\nu s_\mu.
\end{eqnarray}
These three new conserved currents give the generators for
dilatation, conformal boost and special supersymmetry transformation.
 However,  the superconformal symmetry may become
 anomalous at quantum level. In the case that all of them, the trace of
 energy-momentum tensor, $\theta^\mu_{~\mu}$, the $\gamma$-trace
 of supersymmetry current, $\gamma^\mu s_\mu$ and the divergence
 of the chiral $R$-current, $\partial_\mu j^{(5)\mu}$,  get
 contribution from quantum effects,
\begin{eqnarray}
\left(\partial_\mu j^{(5)\mu},\gamma^\mu s_\mu,
\theta^{\mu}_{~\mu}\right) \label{cas}
\end{eqnarray}
 will form a (on-shell) chiral supermultiplet with the $\partial_\mu
j^{(5)\mu}$ playing the role of the lowest component of the
corresponding composite chiral superfield \cite{anm,sibold,grisaru}.


In general, there are two possible sources for above
 chiral supermultiplet anomaly \cite{grisaru}. One is due to the
non-vanishing beta function of ${\cal N}=1$ supersymmetric Yang-Mills
theory; The other one comes from
  the coupling of  above supercurrent multiplet with
the external supergravity fields. In this paper, we shall concentrate
on the superconformal anomaly arising from the latter one. In this
case, the $\gamma$-trace anomaly consists only of the super-Weyl
anomaly since the corresponding special conformal supersymmetry
transformation is just the supersymmetric analog of the Weyl
(or local dilational) transformation.
  Note that in
a supersymmetric gauge theory,
the Poincar\'{e}
 symmetry corresponding to the energy-momentum tensor $\theta_{\mu\nu}$,
 the supersymmetry corresponding to the supersymmetry current
 $s_\mu$, and the chiral $R$-symmetry to the axial vector
 current $j_\mu^{(5)}$ are all
 global symmetries and there no
 gauge fields within the supersymmetric gauge theory itself to
 couple with them. If there are
 some external supergravity fields $g_{\mu\nu}$, axial vector fields $A_\mu$
 and vector-spinor (Rarita-Schwinger) fields $\psi_{\mu}$ couple
 to $\theta^{\mu\nu}$,  $j_\mu^{(5)}$ and $s_{\mu}$,
 respectively,
 \begin{eqnarray}
{\cal L}_{\rm ext}=\int d^4x \sqrt{-g}\left( g_{\mu\nu}\theta^{\mu\nu}
+ A_\mu j^{(5)\mu}+\overline{\psi}_\mu s^{\mu}\right),
\label{clag}
\end{eqnarray}
 there will arise  external superconformal
anomaly chiral supermultiplet.
The action (\ref{clag}) describing the coupling of the external
supergravity fields with the currents of supersymmetric Yang-Mills theory
shows  that the covariant conservations  of
the currents, $\nabla_\mu \theta^{\mu\nu}=\nabla_\mu s^\mu=0$, are
equivalent to the local gauge transformation invariance of the external
supergravity system,
\begin{eqnarray}
\delta g_{\mu\nu} (x) &=& \nabla_\mu \xi_\nu +\nabla_\nu
\xi_\mu ,\nonumber\\
\delta\psi_\mu (x)&=& \nabla_\mu \chi (x).
 \label{egt1}
\end{eqnarray}
Furthermore,
the covariant conservation of the axial vector
current $j_\mu^{(5)}$ and
 the vanishing of both the $\gamma$-trace of supersymmetry current
and the trace of energy-momentum  tensor at classical level,
\begin{eqnarray}
\nabla_\mu j^{(5)\mu}=\gamma^\mu s_\mu =\theta^{\mu}_{~\mu}=0,
\label{supercon}
\end{eqnarray}
mean the Weyl transformation invariance of $g_{\mu\nu}$, the super-Weyl
symmetry and the $U(1)$
chiral gauge symmetry of the corresponding external supergravity system,
\begin{eqnarray}
\delta g_{\mu\nu}&=&g_{\mu\nu} \sigma (x),\nonumber\\
\delta \psi_\mu &=& \gamma_\mu \eta (x),\nonumber\\
\delta A_\mu (x) &=& \partial_\mu \Lambda (x).
\label{busy}
\end{eqnarray}
The transformations (\ref{egt1}) and (\ref{busy}) imply that the external
fields
\begin{eqnarray}
(g_{\mu\nu}, \psi_\mu, A_\mu)
\end{eqnarray}
 constitute
an off-shell ${\cal N}=1$ conformal supergravity multiplet \cite{kaku,fra}.
Therefore, in the context of the $AdS/CFT$ (or
more generally gravity/gauge) correspondence the
 superconformal anomaly   in ${\cal N}=1$
supersymmetric gauge theory due to the supergravity external sources
will be reflected
in the explicit violations of the bulk symmetries
of ${\cal N}=2$ gauged $AdS_5$ supergravity on the boundary
\cite{witt1,hesk,bian1}.



With no consideration on the quantum correction from the dynamics of
the supersymmetric gauge theory, the external superconformal anomaly
is exhausted  at one-loop level. The external superconformal anomaly
multiplet for  ${\cal N}=1$ supersymmetric Yang-Mills theory is listed
as the following \cite{grisaru}:
\begin{eqnarray}
\partial_\mu j^{(5)\mu}
&=&\frac{N^2-1}{128\pi^2}
\left({R}_{\mu\nu\lambda\rho}\widetilde{R}^{\mu\nu\lambda\rho}
-\frac{1}{6} F_{\mu\nu}\widetilde{F}^{\mu\nu}\right),
\nonumber\\
 \gamma_\mu s^\mu &=&\frac{N^2-1}{ 128\pi^2}\left[\frac{1}{2}
 R^{\mu\nu\lambda\rho}\gamma_{\lambda\rho}\left(\nabla_\mu \psi_\nu-
 \nabla_\nu \psi_\mu\right)
 -\epsilon^{\mu\nu\lambda\rho}\gamma_5\psi_\rho A_\mu F_{\nu\lambda}\right],
\nonumber\\
\theta^\mu_{~\mu} &=&
\frac{N^2-1}{128\pi^2}\left(C_{\mu\nu\lambda\rho}
C^{\mu\nu\lambda\rho}-
\widetilde{R}_{\mu\nu\lambda\rho}\widetilde{R}^{\mu\nu\lambda\rho}
-\frac{1}{4}F_{\mu\nu} F^{\mu\nu}\right).
%\nonumber\\
%P &=& -\beta (g)\overline{\lambda}^a\lambda^a, ~~Q=-i\beta (g)
%\overline{\lambda}^a\gamma_5\lambda^a
\label{exone}
\end{eqnarray}
%where $G$ is the square of the four dimensional Weyl tensor and
%$G$ is the Euler number density,
%\begin{eqnarray}
%F_{\mu\nu} &=&\partial_\mu A_\nu-\partial_\nu A_\mu, \nonumber\\
 %H&=& C_{\mu\nu\lambda\rho}C^{\mu\nu\lambda\rho}=
%R_{\mu\nu\lambda\rho}R^{\mu\nu\lambda\rho} -2 R_{\mu\nu}
%R^{\mu\nu}+\frac{1}{3}R^2, \nonumber\\
%G &=&
%\widetilde{R}_{\mu\nu\lambda\rho}\widetilde{R}_{\mu\nu\lambda\rho}
%=R_{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}-4R_{\mu\nu}
%R^{\mu\nu}+R^2
%\end{eqnarray}
In above equations, $F_{\mu\nu}=\partial_\mu A_\nu- \partial_\nu A_\mu$  is
the field strength
corresponding to the external $U_R(1)$ vector field $A_\mu$;
$R_{\mu\nu\lambda\rho}$
and $C_{\mu\nu\lambda\rho}$  are the Riemannian and Weyl
tensors corresponding to the
gravitational field $g_{\mu\nu}$. The factor $N^2-1$
comes from the fact that the gauginoes are in the adjoint representation
 of $SU(N)$ gauge group and hence there are $N^2-1$ copies.

In the context of the AdS/CFT correspondence, the holographical arising
of the  gauge field part of the chiral $U_R(1)$ anomaly
was pointed out by Witten that it should come directly from
the Chern-Simons
term in the five-dimensional gauged supergravity \cite{witt1}.
Further, the holographic Weyl anomaly
was shown in Ref.\,\cite{hesk} through a procedure called
holographic renormalization and in Ref.\,\cite{imbi}
by the holomorphic dimensional regularization and
a special bulk diffeomorphsm preserving the Fefferman-Graham
metric \cite{feff} of an arbitrary $d+1$-dimensional manifold
with boundary topologically isomorphic to $S^d$, respectively.
Actually, there have arisen quite a number of papers to discuss
various aspects of the holographic Weyl anomaly including
the modified models \cite{noji}, the asymptotically $AdS$
space-time \cite{kraus},  the $1/N$ correction \cite{ahar}
 and new calculation
framework such as the Hamilton-Jacobi equation \cite{fuku} etc.
The aim of this paper is to tackle the holographic origin of
the supersymmetry current anomaly $\gamma_\mu s^\mu$.





\section{ ${\cal N}=2$ Gauged Supergravity   in Five Dimensions
and Its $AdS_5$ Boundary Reduction}



To show clearly how the holographic supersymmetry current anomaly arises,
we shall review some typical features of the ${\cal N}=2$
gauged supergravity in five dimensions.


The ungauged ${\cal N}=2$ supergravity in five dimensions has the
same structure as ${\cal N}=1$ eleven-dimensional supergravity \cite{crem1}.
It has a global
$USp(2)\cong SU(2)$ R-symmetry, and it contains a graviton $E_M^{~A}$, two
gravitini $\Psi_M^i$ and a vector field ${\cal A}_M$ \cite{gst,crem2,guna},
$M, A=0,\cdots,5$ are the Riemannian and local Lorentz indices, respectively,
and $i=1,2$ the $SU(2)$ doublet indices.
The gravitini are the
$USp(2)\cong SU(2)$ doublets and the symplectic Majorana spinors. The classical
Lagrangian density takes a simple form \cite{crem2},

\begin{eqnarray}
{E}^{-1} \widetilde{\cal L} &=& -\frac{1}{2}{\cal R}[\Omega(E)]
-\frac{1}{2}\overline{\Psi}_M^i
{\Gamma}^{MNP}{\nabla}_N {\Psi}_{P i}
-\frac{1}{4} a_{00}{\cal F}_{MN}{\cal F}^{MN}
\nonumber\\
&& -\frac{3}{8}\sqrt{\frac{1}{6}}ih_0
\left( \overline{\Psi}_M^i\Gamma^{MNPQ}
\Psi_{N i}{\cal F}_{PQ}+2\overline{\Psi}^{M i}\Psi^N_i
F_{MN}\right)
\nonumber\\
&&
+\frac{C}{6\sqrt{6}}{E}^{-1}\epsilon^{MNPQR}
{\cal F}_{MN}{\cal F}_{PQ}
{\cal A}_R
+\mbox{four-fermi terms},
 \end{eqnarray}
where the covariant derivative on the spinor field is defined with
the spinor connection $\Omega_{M}^{~AB}$,
\begin{eqnarray}
\nabla_M \Psi_N^i=\left(\partial_M +\frac{1}{4}\Omega_{M}^{~AB}\Gamma_{AB}
\right)\Psi_N^i.
 \end{eqnarray}

 The gauging of above supergravity is just turning the $U(1)$ subgroup of
 the global
 $SU(2)$ R-symmetry group into a local gauge group and straightforwardly
 considering the vector
 field as the $U(1)$  gauge field \cite{gst}. The space-time
 covariant derivative  on the gravitini
 will be enlarged to include the $U(1)$ gauge covariant derivative,
\begin{eqnarray}
D_M\Psi_N^i=\nabla_M \Psi_N^i+g {\cal A}_{M}\delta^{ij}\Psi_{N j}.
\end{eqnarray}
The gauged ${\cal N}=2$ supergravity action is
\begin{eqnarray}
E^{-1} {\cal L} ={E}^{-1} \widetilde{\cal L}
+g^2 P_0^2-\frac{i\sqrt{6}}{8}g
\overline{\Psi}_M^i\Gamma^{MN}\Psi^{j}_N
\delta_{ij}P_0.
\label{gaugedf}
\end{eqnarray}
The gauged ${\cal N}=2$ supergravity  has  $AdS_5$ classical
solution that
preserves ${\cal N}=2$ supersymmetry with the cosmological
constant proportional to $P_0$ \cite{gst}.
To make the $AdS_5$ classical solution take the standard form,
\begin{eqnarray}
   ds^2 &=& \frac{l^2}{r^2}\left[g_{\mu\nu}(x,r)dx^\mu dx^\nu
   -\left(dr\right)^2\right],
   \nonumber\\
   {\cal A}_M &=&\Psi_M=0,
   \label{metrican}
   \end{eqnarray}
one must choose the parameters in the Lagrangian (\ref{gaugedf}) as the
following ones \cite{gst,bala}:
 \begin{eqnarray}
g=\frac{3}{4}, ~~h_0=\frac{l}{2}\sqrt{\frac{3}{2}},~~h^0=\frac{1}{h_0},
~~V_0=1, ~~P_0=2 h^0 V_0=\frac{4}{l}\sqrt{\frac{2}{3}}, ~~
a_{00}=\left(h_0\right)^2=\frac{3l^2}{8}.
\end{eqnarray}
Consequently,
the  Lagrangian density (\ref{gaugedf})
  up to the quadratic terms in spinor fields becomes
 \begin{eqnarray}
E^{-1} {\cal L} &=& -\frac{1}{2}{\cal R}-\frac{1}{2}\overline{\Psi}_M^i
\Gamma^{MNP}D_N \Psi_{P i}-\frac{3l^2}{32} {\cal F}_{MN}{\cal F}^{MN}
+\frac{C}{6\sqrt{6}}E^{-1}\epsilon^{MNPQR}
{\cal F}_{MN}{\cal F}_{PQ}{\cal A}_R
\nonumber\\
&&-\frac{3i}{4l}\overline{\Psi}_M^i\Gamma^{MN}\Psi^{N j}
\delta_{ij}
 -\frac{3il}{32}\left( \overline{\Psi}_M^i\Gamma^{MNPQ}
\Psi_{N i}{\cal F}_{PQ}+2\overline{\Psi}^{M i}\Psi^N_i
{\cal F}_{MN}\right)-\frac{6}{l^2}.
\label{gaugedfm}
\end{eqnarray}
The supersymmetry transformations at the leading order in spinor
fields read
\begin{eqnarray}
\delta E_M^{~A}&=&\frac{1}{2}\overline{\cal E}^i\Gamma^A\Psi_{M i},
\nonumber\\
\delta \Psi_{M }^i &=& D_{M}{\cal E}^i+
\frac{il}{16} \left(\Gamma_{M}^{~NP}-4\delta_{M}^{~N}
\Gamma^P \right)
{\cal F}_{NP}{\cal E}^i+\frac{i}{2l}\Gamma_M \delta^{ij}{\cal E}_j,
\nonumber\\
\delta {\cal A}_M &=& \frac{i}{l}\overline{\Psi}_M^i{\cal E}_i.
\label{twostm}
 \end{eqnarray}


To calculate the holographic superconformal anomaly, we need
to expand the fields around the $AdS_5$ vacuum solution
(\ref{metrican}). Geometrically,
this  is actually a process of revealing the asymptotic
behaviour of the bulk fields
near the boundary of $AdS_5$ space-time. Correspondingly, the
various bulk symmetries will be reduced to those on the boundary.
For examples, the bulk diffeormorphism invariance of the bulk
decomposes into  the diffeomorphism  symmetry on the boundary
and the Weyl symmetry \cite{imbi}, and the the bulk
supersymmetry converts into
a superconformal symmetry for an off-shell
 conformal supergravity on the boundary \cite{nish,bala}.

 The procedure of reducing the bulk gauged supergravity to
  the off-shell conformal supergravity on the boundary is
  displayed in a series of works on $AdS_3/CFT_2$,
   $AdS_6/CFT_5$ and $AdS_7/CFT_6$ by Nishimura et al \cite{nish}.
     The key point is  using the equations of motion of the
     bulk fields to
   find their radial coordinate dependence near the boundary
   of $AdS_5$ space. For the spinor field such as
   the gravitino, one should also show how a
  sympletic  Majorana spinor in  five dimensions reduces to
  the chiral spinor on the four-dimensional boundary.
   The reduction from the on-shell
   five-dimensional ${\cal N}=2$ gauged supergravity
   to ${\cal N}=1$ off-shell conformal supergravity
   in four dimensions
   was performed
by Balasubramanian et al. \cite{bala}, so we briefly review their
result and then use it to derive the holographic
super-Weyl anomaly of the supersymmetry current
in ${\cal N}=1$ supersymmetric gauge theory.


 As the first step, one should partially fix the local symmetries of bulk
   supergravity
   in the radial direction. According to the $AdS_5$ solution
   (\ref{metrican}), one can choose \cite{nish,bala}
   \begin{eqnarray}
   E_\mu^{~a} (x,r)&=&\frac{l}{r} {e}_\mu^{~a}
    (x)+{\cal O}(r ), ~~
   E_r^{~a}=E_{\mu}^{~\overline{r}}=0,
   ~~E_r^{~\overline{r}}=\frac{l}{r},
   \label{rgf1}
   \end{eqnarray}
   and
   \begin{eqnarray}
   \Psi_r^i(x,r)=0, ~~~ {\cal A}_r (x,r)=0,
   \label{rgf2}
   \end{eqnarray}
to fix the Lorentz symmetry, the supersymmetry and
the gauge symmetry in the $r$-direction, respectively.
The gauge-fixing choice and the torsion-free condition
$dE^a+\Omega^a_{~b}\wedge E^b=0$, further determine the $r$-dependence
of the spin connections,
\begin{eqnarray}
\Omega^{a}_{~\overline{r}}(x,r)&=& -E^{a}(x,r)
=-\frac{l}{r}{e}^{a}(x),\nonumber\\
{\Omega}^{a}_{~b}(x,r)
&=&{\omega}^{a}_{~b}(x).
\label{redsc}
\end{eqnarray}
In above equations, $\mu,a=0,\cdots,3$
are the Riemannian and local Lorentz indices on the boundary, respectively,
and $\overline{r}$ is the Lorentz index in the radial direction.
We use the lower case quantities
to  denote the boundary values of bulk fields,
i.e., they are independent of the radial
coordinate $r$.


The linearized equation of motion for the gauge
field ${\cal A}_\mu$ near the boundary,
\begin{eqnarray}
E^{-1}\partial_M \left[G^{MN}\partial_N {\cal A}_\mu (x,r) \right]=0,
\end{eqnarray}
implies that at the leading order in $r$
one can choose ${\cal A}_\mu (x,r)$  to be independent of $r$,
\begin{eqnarray}
{\cal A}_\mu (x,r) ={A}_\mu (x).
\label{redg}
\end{eqnarray}
Furthermore, the linearized equation of motion for the gravitino is
\begin{eqnarray}
\Gamma^{MNP}D_N \Psi_{P i}+\frac{3i}{2}\Gamma^{MN}
\Psi_{N}^j\delta_{ij}=0.
\end{eqnarray}
The reduced spin connection (\ref{redsc}) and gauge field (\ref{redg})
as well as the gauge choices (\ref{rgf1}) and (\ref{rgf2})
lead to the boundary reduction of the bulk covariant derivative,
 \begin{eqnarray}
 D_r&=&\partial_r, ~~~
 D_\mu
 =\widetilde{D}_\mu (x)-\frac{1}{2r}{\gamma}_\mu \gamma_{5},
 \nonumber\\
 \widetilde{D}_\mu (x) &{\equiv}& \nabla_\mu
 +\frac{1}{4}{\omega}_\mu^{~ab}
 {\gamma}_{~ab}
 +\frac{3}{4}{A}_\mu,
 \label{rederi}
 \end{eqnarray}
 where the four-dimensional convention is defined as the following,
 \begin{eqnarray}
 {\gamma}_{a}=\Gamma_{a},~~ \Gamma_\mu=
 E_\mu^{~a}\Gamma_{a}=\frac{l}{r}
 {\gamma}_{\mu},
 ~~\Gamma^\mu = \overline{E}^\mu_{~a}\Gamma^{a}
 =\frac{r}{l} {\gamma}^{\mu},
 ~~ \gamma_5=\Gamma^{\overline{r}}
 =\Gamma_{\overline{r}},~~\gamma_5^2=1.
 \label{rega}
 \end{eqnarray}
The linearized gravitino equation reduces to
 \begin{eqnarray}
 {\gamma}^{\mu\nu}\left(\partial_r\delta_{ij}-\frac{1}{r}\delta_{ij}
 -\frac{3}{2r}\gamma_5 \epsilon_{ij}\right)\psi_{\nu}^j(x,r)
 -{\gamma}^{\mu\nu\rho}\widetilde{D}_\nu (x)\gamma_5\psi_{\rho i}
 (x,r)=0,
 \end{eqnarray}
Diagonalizing the above equation by combining the
two components of the symplectic Majorana spinor,
$\Psi_\mu\equiv \Psi_{\mu 1}+i\Psi_{\mu 2}$,
and making the chiral decomposition
$\Psi^R_\mu\equiv \frac{1}{2} (1-\gamma_5)\Psi_\mu$,
 $\Psi^L_\mu \equiv \frac{1}{2} (1+\gamma_5)\Psi_\mu$,
one can  see that near the boundary $r\to 0$, the equation for the
right-handed component reads \cite{bala}
\begin{eqnarray}
\left(\partial_r+\frac{1}{2r}\right)\Psi_\mu^R=0
\end{eqnarray}
and hence the radial dependence of $\Psi_\mu^R$ is
\begin{eqnarray}
\Psi_\mu^R=\left(\frac{2l}{r}\right)^{1/2}{\psi}_\mu^R(x).
\label{redsl}
\end{eqnarray}
The  left-handed  component is not independent, and
its radial coordinate dependent behaviour turns out to be \cite{bala}
\begin{eqnarray}
\Psi_\mu ^L &= & \left(2l\tau\right)^{1/2}{\chi}_\mu^L(x), \nonumber\\
\chi_\mu^L &=& \frac{1}{3}\gamma^\nu\left(\widetilde{D}_\mu\psi_\nu^R
-\widetilde{D}_\nu\psi_\mu^R\right)-\frac{i}{12}
\epsilon_{\mu\nu}^{~~\lambda\rho}
\gamma_5\gamma^\nu \left(\widetilde{D}_\lambda\psi_\rho^R
-\widetilde{D}_\rho\psi_\lambda^L\right).
\label{redsr}
\end{eqnarray}
Therefore, the ${\cal N}=2$  bulk supergravity multiplet
($E_M^{~A},\Psi_M^i, {\cal A}_M$) reduces to the
boundary field (${e}_\mu^{~a}$,
${\psi}_\mu^{R}$, ${A}_\mu$). It is actually the
 8+8 off-shell muiltiplet of ${\cal N}=1$ conformal supergravity
 since the bulk supersymmetry transformation
(\ref{twostm}) reduces to the one for  ${\cal N}=1$ conformal supergravity
 \cite{bala}.


The boundary reduction of the bulk supersymmetric transformation to
that for  the conformal supergravity is the following. First, as done
for the bulk gravitino, one
must redefine the bulk supersymmetric transformation parameter,
${\cal E} (x,r)={\cal E}_1(x,r)+i{\cal E}_2(x,r)$ and
decompose it as the chiral
components. The radial coordinate dependence of ${\cal E}^{L,R}$
 should be the same as
as the bulk gravitino,
\begin{eqnarray}
{\cal E}^R(x,r)=\left(\frac{2l}{r}\right)^{1/2}{\epsilon}^R(x),~~~
{\cal E}^L(x,r)= \left(2lr\right)^{1/2}
{\eta}^L(x).
\label{strp}
\end{eqnarray}
At the leading order in $r$, the bulk supersymmetric
transformations reduces to \cite{bala}
\begin{eqnarray}
\delta {e}_\mu^{~a} &=&-\frac{1}{2}
\overline{\psi}_\mu\gamma^{a}{\epsilon},
\nonumber\\
 \delta {\psi}_\mu
&=&\widetilde{D}_\mu {\epsilon} -\gamma_\mu {\eta}
={\nabla}_\mu {\epsilon}
-\frac{3i}{4}A_\mu \gamma_5 {\epsilon}-\gamma_\mu {\eta},
\nonumber\\
\delta {A}_\mu  &=& i\left(\overline{\psi}_\mu\gamma_5{\eta}
-\overline{\chi}_\mu \gamma_5\epsilon \right),
\label{desut}
\end{eqnarray}
where all the spinorial quantities, $\psi_\mu (x)$,
${\chi}_\mu (x)$
${\epsilon} (x)$ and ${\eta}(x)$
 are Majorana spinors constructed from
their chiral components $\psi^R_\mu (x)$, $\chi^L_\mu (x)$,
${\epsilon}^R (x)$ and ${\eta}^L(x)$. Eq.\,(\ref{desut})
shows that the reduced bulk supersymmetric transformation is
indeed the supersymmetric transformation for ${\cal N}=1$
conformal supergravity with $\epsilon$ and $\eta$ playing
the roles of parameters for supersymmetry and special supersymmetry
transformations, respectively \cite{kaku,fra}.


\section{Holographic Super-Weyl anomaly of supersymmetry current}



The arising of the super-Weyl anomaly in ${\cal N}=1$
 supersymmetric Yang-Mills
theory from the ${\cal N}=2$ gauged $AdS_5$ supergravity
 lies in two aspects. On one hand,
as a supersymmetric field theory, the supersymmetric variation
of the Lagrangian (\ref{gaugedfm}) of the gauged
supergravity is a total derivative.
These total derivative terms cannot be naively ignored due to the existence
of the boundary $AdS_5$. On the other hand, near the $AdS_5$ boundary,
the bulk supersymmetric transformation  decomposes into the
supersymmetry transformation and the super-Weyl transformation
on the boundary.
If we require  supersymmetry on the boundary to be preserved,
the total derivative
terms should yield the super-Weyl anomaly of the supersymmetry
current via the AdS/CFT correspondence given by Eq.\,(\ref{acc4}).

 In the following, we shall calculate the supersymmetric
 variation of the gauged ${\cal N}=2$ five-dimensional supergravity
 (\ref{gaugedfm}) and extract out the total derivative terms. Then we shall
 reduce it to the $AdS_5$ boundary and give the holographic supersymmetry
 current anomaly.  However, it should be emphasized that this surface term
 only yields the gauge field background part of the super-Weyl anomaly.
 One must employ a holographic renormalization procedure to reveal
 the gravitational part, just as how the holographic Weyl anomaly
 was found in Ref.\,\cite{hesk}.

 First, the supersymmetric variation of the pure gravitational term
 and the cosmological term gives
 \begin{eqnarray}
 \delta  S_{\rm GR}&=&\delta \int d^5x
  E\left(-\frac{1}{2}{\cal R}-\frac{6}{l^2}\right)
  =\delta \int d^5x
  E\left(-\frac{1}{2}\overline{E}_A^{~M}\overline{E}_B^{~N}
  {\cal R}_{MN}^{~~~AB}
   -\frac{6}{l^2}\right)
  \nonumber\\
  &=&\int d^5x E\left[ -\frac{1}{2}\overline{\cal E}^i\gamma_A\Psi^{M}_i
  \left( {\cal R}_M^{~A}-\frac{1}{2}E_M^{~A} {\cal R}
  -\frac{6}{l^2}E_M^{~A}\right)
  -\nabla_M \left( \overline{E}_A^{~M}\overline{E}_B^{~N}
  \delta\Omega_N^{~AB}\right)\right],
  \label{superv1}
 \end{eqnarray}
where $\overline{E}_A^{~M}$ is the inverse of $E_M^{~A}$.

 The supersymmetry variation of the pure gauge field terms
 (including Chern-Simons term) yields
 \begin{eqnarray}
\delta S_{\rm GA}&=& \int d^5x \delta
\left[E \left(-\frac{3 l^2}{32}\right){\cal F}_{MN} {\cal F}^{MN}
 +\frac{C}{6\sqrt{6}} \epsilon^{MNPQR} F_{MN}
 {\cal F}_{PQ} {\cal A}_R\right]\nonumber\\
&=& \int d^5 x \left[-\frac{3l^2}{32}\left(\delta E {\cal F}_{MN}
{\cal F}^{MN}
+4{\cal F}^{MN}\nabla_M\delta {\cal A}_N \right)\right.\nonumber\\
&&\left.+\frac{C}{6\sqrt{6}} \epsilon^{MNPQR}
\left(4 \nabla_M \delta {\cal A}_N
 {\cal F}_{PQ} {\cal A}_R+ {\cal F}_{MN}
 {\cal F}_{PQ}\delta {\cal A}_R\right)\right]\nonumber\\
 &=& \int d^5x \left\{E\left[-\frac{3l^2}{64}\overline{\cal E}^i
 \Gamma^M\Psi_{M i} {\cal F}_{NP} {\cal F}^{NP}\right.\right.\nonumber\\
 &&\left.+\frac{3il}{8}\nabla_M
 \left(F^{MN}\overline{\cal E}^i\Psi_{N i}\right)
 -\frac{3il}{8}\left(\nabla_M {\cal F}^{MN}\right)
 \overline{\cal E}^i\Psi_{N i}
 \right]\nonumber\\
 &&\left. -\frac{iC}{\sqrt{6}l}\epsilon^{MNPQR}
 \left[\frac{2}{3}\nabla_M \left({\cal A}_R
 {\cal F}_{PQ}\overline{\cal E}^i\Psi_{N i}\right)
 +\frac{1}{2}{\cal F}_{MN}{\cal F}_{PQ}\overline{\cal E}^i\Psi_{R i}
   \right]\right\}.
   \label{superv2}
 \end{eqnarray}

As for the supersymmtric variations
 of the terms concerning the gravitino, we start from the
 kinetic terms of the gravitino,
 \begin{eqnarray}
 \delta S_{\rm KT} &=& \delta \int d^5 x \left[-\frac{1}{2} E
 \overline{\Psi}^i_M \overline{E}_A^{~M} \overline{E}_B^{~N}
 \overline{E}_C^{~P}
 \Gamma^{ABC} \left(\nabla_N \Psi_{P i}
 -\frac{3}{4} {\cal A}_N \delta_{ij}\Psi_P^j\right)\right]\nonumber\\
 &=& \int d^5x \left(-\frac{1}{2}\right)\left[ \left(\delta E \right)
 \overline{\Psi}^i_M \Gamma^{MNP} D_N \Psi_{P i}
 +E \left(\delta \overline{\Psi}^i_M\right)
 \Gamma^{MNP} D_N \Psi_{P i}
  \right.\nonumber\\
 && +3 E\left(\delta \overline{E}_A^{~M}\right) \overline{E}_B^{~N}
 \overline{E}_C^{~P}
 \overline{\Psi}_{M}^i\Gamma^{ABC} D_N \Psi_{P i}\nonumber\\
 && \left.+\overline{\Psi}^i_M \Gamma^{MNP}\left(
 \nabla_N \delta \Psi_{P i}-\frac{3}{4} \delta_{ij}
 \left(\delta {\cal A}_N
 \Psi_P^j+ {\cal A}_N\delta \Psi_P^j\right)\right) \right].
 \end{eqnarray}
 Considering only the terms at most quadratic in the
 spinor quantities, we have
 \begin{eqnarray}
\delta S_{\rm KT} &=& \int d^5 x \left(-\frac{1}{2}\right) E
\left\{ \left[\nabla_M \overline{\cal E}^i
+\frac{3}{4}{\cal A}_\mu\delta^{ij}\overline{\cal E}_j-\frac{il}{16}
\overline{\cal E}^i
\left(\Gamma_M^{~RS}+4\delta_M^{~R}\Gamma^S
 \right)F_{RS}-
\frac{i}{2l}\delta^{ij}\overline{\cal E}_j\Gamma_M\right]\right.\nonumber\\
&&\times \Gamma^{MNP}\left(\nabla_N\Psi_{P i}
-\frac{3}{4}{\cal A}_N\delta_{ik}\Psi_P^k\right)\nonumber\\
&& + \overline{\Psi}^i_M \Gamma^{MNP}
 \nabla_N \left[\nabla_P {\cal E}_i-\frac{3}{4}{\cal A}_P
 \delta_{ij}{\cal E}^j +\frac{il}{16}
 \left(\Gamma_P^{~RS}
 -4\delta_P^{~R}\Gamma^S \right){\cal E}_i {\cal F}_{RS}
 +\frac{i}{2l}\Gamma_P\delta_{ik}{\cal E}^k\right]\nonumber\\
&&\left.-\frac{3}{4}\delta_{ij}\overline{\Psi}^i_M \Gamma^{MNP}{\cal A}_N
\left[\nabla_P {\cal E}^j+\frac{3}{4}{\cal A}_P
 \delta^{jk}{\cal E}_k +\frac{il}{16}
 \left(\Gamma_P^{~RS}
 -4\delta_P^{~R}\Gamma^S \right){\cal E}^j {\cal F}_{RS}
 -\frac{i}{2l}\Gamma_P\delta^{jk}{\cal E}_k\right] \right\}
 \nonumber\\
 &=& \int d^5x \left(-\frac{1}{2}\right) E
 \left\{ \left[(\nabla_M \overline{\cal E}^i)\Gamma^{MNP}\nabla_N
 \Psi_{P i}+\overline{\Psi}^i_M\Gamma^{MNP}\nabla_N \nabla_P
 {\cal E}_i\right]\right. ~~~~\bigcirc\hspace{-4mm}1 \nonumber\\
 &&-\frac{3}{4}\left[\delta_{ij}{\cal A}_N \left(\nabla_M
 \overline{\cal E}^i
 \Gamma^{MNP} \Psi_P^j+
 \overline{\Psi}_M^i
 \Gamma^{MNP}\nabla_P{\cal E}^j\right)
 \right.\nonumber\\
 &&\left.-{\cal A}_M\delta^{ij}\overline{\cal E}_j
 \Gamma^{MNP}\nabla_M \Psi_{P i}
 +\delta_{ij}\overline{\Psi}^i_M \Gamma^{MNP}\nabla_N
   ({\cal A}_{P}{\cal E}^j)
  \right] \bigcirc\hspace{-4mm}2\nonumber\\
  && +\frac{il}{16}\left[-{\cal F}_{RS}\overline{\cal E}^i
  \Gamma_M^{~RS}\Gamma^{MNP}\nabla_{N}\Psi_{P i}
  +\overline{\Psi}^i_M \Gamma^{MNP}\nabla_N
   (\Gamma_{P}^{~RS}{\cal F}_{RS}{\cal E}_i)
    \right]~~~~ \bigcirc\hspace{-4mm}3 \nonumber\\
  &&+\frac{il}{16}\frac{3}{4}{\cal A}_N {\cal F}_{RS}\delta_{ij}
  \left(\overline{\cal E}^i\Gamma_{M}^{~RS}\Gamma^{MNP}
  \Psi_{P}^j
  -\overline{\Psi}^i_M\Gamma^{MNP}\Gamma_{P}^{~RS}
  {\cal E}^j  \right)~~~~ \bigcirc\hspace{-4mm}4 \nonumber\\
  &&-\frac{il}{4}\left[{\cal F}_{MR}
  \overline{\cal E}^i\Gamma^{R}\Gamma^{MNP}\nabla_N
  \Psi_{P i}+\overline{\Psi}^i_M\Gamma^{MNP}\nabla_N\left(
  \Gamma^{R} F_{PR}{\cal E}_i\right)\right]~~~~
  \bigcirc\hspace{-4mm}5 \nonumber\\
  &&+\frac{il}{4} \frac{3}{4}\delta_{ij} {\cal A}_N
  \left( {\cal F}_{MR}
  \overline{\cal E}^i\Gamma^{R}\Gamma^{MNP}
  \Psi_{P}^j+{\cal F}_{PR}\overline{\Psi}^i_M\Gamma^{MNP}
  \Gamma^{R}{\cal E}^j\right)~~~~ \bigcirc\hspace{-4mm}6 \nonumber\\
  &-&\frac{i}{2l}\left[\delta^{ij}
  \overline{\cal E}_j\Gamma_{\mu}\Gamma^{MNP}\nabla_N
  \Psi_{P i}-\delta_{ij}\overline{\Psi}^i_M\Gamma^{MNP}
  \nabla_N\left(
  \Gamma_P{\cal E}^j\right)\right]
  ~~~~ \bigcirc\hspace{-4mm}7 \nonumber\\
  &&\left.+\frac{3}{4}\frac{i}{2l}{\cal A}_N \delta^i_{~j}\left(
  \overline{\cal E}_i\Gamma_{M}\Gamma^{MNP}
  \Psi_{P}^j+\overline{\Psi}^j_M\Gamma^{MNP}
  \Gamma_P{\cal E}_i\right)
  \right\} ~~~~ \bigcirc\hspace{-4mm}8.
  \label{superv3}
 \end{eqnarray}
 We list the calculation results in the following:
 \begin{eqnarray}
 \bigcirc\hspace{-3mm}1\,&=&\nabla_M
 \left(\overline{\cal E}^i\Gamma^{MNP}\nabla_N
 \Psi_{P i}\right)-\overline{\cal E}^i
 [\nabla_M, \Gamma^{MNP}]\nabla_N
 \Psi_{P i}\nonumber\\
 && -\frac{1}{8}R_{MN AB}
 \overline{\cal E}^i\left(\Gamma^{MNP}\Gamma^{AB}
 +\Gamma^{AB}\Gamma^{MNP}\right)\Psi_{P i}\nonumber\\
 &=& \nabla_M
 \left(\overline{\cal E}^i\Gamma^{MNP}\nabla_N
 \Psi_{P i}\right)
 +\left( R_{M}^{~N}-\frac{1}{2}R\delta_{M}^{~N}\right)
 \overline{\cal E}^i \Gamma^{M}\Psi_{N i}; \nonumber \\
 \bigcirc\hspace{-3mm}2\,
 &=& -\frac{3}{4}\nabla_M \left(\delta^{ij} {\cal A}_N
 \overline{\cal E}_i\Gamma^{MNP}\Psi_{P j}\right)
 +\frac{3}{4}\overline{\cal E}^i\Gamma^{MNP}\Psi_M^j \delta_{ij}
 {\cal F}_{NP}; \nonumber \\
 \bigcirc\hspace{-3mm}3\,&=&\frac{il}{16}
 \nabla_M\left({\cal F}_{RS}\overline{\cal E}^i\Gamma_P^{~RS}
 \Gamma^{MNP}\Psi_{N i}\right)
 -\frac{il}{8}{\cal F}_{RS}\overline{\cal E}^i\Gamma^{NPRS}
 \left(\nabla_N \Psi_{P}\right)_i+
 \frac{3il}{4}{\cal F}^{MN}\overline{\cal E}^i \left(\nabla_M
 \Psi_{N}\right)_i;
\nonumber \\
    \bigcirc\hspace{-3mm}4\,&=&\frac{3il}{32}
    {\cal F}_{RS}{\cal A}_M \delta_{ij}\overline{\cal E}^i
    \Gamma^{MNRS}\Psi_N^j
    -\frac{9il}{16}\delta_{ij}{\cal F}^{MN}
    {\cal A}_M \overline{\cal E}^i\Psi_N^j
    ; \nonumber \\
 \bigcirc\hspace{-3mm}5\, &=&  \frac{il}{4}
 \nabla_M \left( \overline{\cal E}^i\Gamma^S\Gamma^{MNP}
 \Psi_N^i {\cal F}_{PS}\right)
 +\frac{il}{2} {\cal F}_{MR}
 \overline{\cal E}^i\Gamma^{MNPR}
 \left(\nabla_N\Psi_P\right)_i;
 \nonumber \\
    \bigcirc\hspace{-3mm}6\,&=&
   -\frac{3il}{8}
{\cal F}_{MR}{\cal A}_N\delta_{ij}
\overline{\cal E}^i\Gamma^{MNPR}
\Psi_P^j; \nonumber \\
     \bigcirc\hspace{-3mm}7\,&=&
     \frac{i}{2l}\delta_{ij}\left[-3\overline{\cal E}^i
     \Gamma^{MN}\nabla_M\Psi_{N}^j+\overline{\Psi}_M^i
    \left[\Gamma^{MNP},\nabla_N\right]\left(\Gamma_P{\cal E}^j\right)
    +3 \overline{\Psi}_M^i\nabla_N\left(\Gamma^{MN} {\cal E}^j\right)
    \right]
\nonumber\\
&=&\frac{3i}{2l}\nabla_M \left(
\overline{\cal E}^i \Gamma^{MN}\Psi_{N}^j\right)
 -\frac{3i}{l}\delta^{ij}\overline{\cal E}^i\Gamma^{MN}
 \nabla_M\Psi_{N}^j ;\nonumber\\
  \bigcirc\hspace{-3mm}8\,&=&-\frac{9i}{4l}{\cal A}_M \overline{\cal E}^i
  \Gamma^{MN}\Psi_{N i}.
  \label{superv4}
 \end{eqnarray}
 The supersymmetric variation of the gravitino mass term is
 \begin{eqnarray}
 \delta S_{\rm GM} &=& \delta \left[\frac{3i}{4l}
 \int E \overline{\Psi}_M^i \Gamma^{MN}
 \Psi_N^j \delta_{ij} \right]
 =\frac{3i}{4l}\int d^5x E\left[
 \left(\delta \overline{\Psi}_M^i\right) \Gamma^{MN}\Psi_N^j
 +\overline{\Psi}_M^i \Gamma^{MN}\delta \Psi_N^j \right]\delta_{ij}
  \nonumber\\
 &=&\frac{3i}{4l}\int d^5x e\left\{\left[\nabla_M \overline{\cal E}^a
+\frac{3}{4}{\cal A}_M\delta^{ik}\overline{\cal E}_k-\frac{il}{16}
\overline{\cal E}^i
\left(\Gamma_M^{~PQ}+4\delta_M^{~P}\Gamma^Q
 \right)F_{PQ}\right.\right.\nonumber\\
 &&\left.-
\frac{i}{2l}\delta^{ac}\overline{\cal E}_k\Gamma_M\right]
\Gamma^{MN} \Psi_N^b\nonumber\\
&& \left.+\overline{\Psi}_M^i \Gamma^{MN}
\left[\nabla_N {\cal E}^j+\frac{3}{4}{\cal A}_N
 \delta^{jk}{\cal E}_k +\frac{il}{16}
 \left(\Gamma_N^{~}
 -4\delta_N^{~P}\Gamma^Q \right){\cal E}^j {\cal F}_{PQ}
 -\frac{i}{2l}\Gamma_N\delta^{jk}{\cal E}_k\right]\right\}\delta_{ij}
 \nonumber\\
 &=&\frac{3i}{4l}\int d^5x E\left\{\delta_{ij}\left[
 (\nabla_M \overline{\cal E}^i)\Gamma^{MN}\Psi_N^j
  + \overline{\Psi}^i_M\Gamma^{MN}\nabla_N{\cal E}^j\right]
  \right.
 \nonumber\\
 && +\frac{3}{4}\delta^i_{~j}\left({\cal A}_M \overline{\cal E}_i
 \Gamma^{MN}\Psi_N^j+{\cal A}_N \overline{\Psi}_M^j\Gamma^{MN}
 {\cal E}_i \right) \nonumber\\
 && -\frac{il}{16}\delta_{ij} {\cal F}_{RP}\left(
 \overline{\cal E}^i\Gamma_M^{~RP}
 \Gamma^{MN}\Psi_N^j
  -\overline{\Psi}^i_M\Gamma^{MN}\Gamma_N^{~RP}
 {\cal E}^j \right)\nonumber\\
 &&-\frac{il}{4}\delta_{ij}\left(
 \overline{\cal E}^i\Gamma^{P}
 \Gamma^{MN}\Psi_N^j {\cal F}_{MP}
  +\overline{\Psi}^i_M\Gamma^{MN}\Gamma^{P}{\cal F}_{NP}
 {\cal E}^j \right)\nonumber\\
 &&-\frac{2i}{l}\delta^i_{~j}
 \left(\overline{\cal E}_i\Gamma^{M}\Psi_{M}^j
 +\overline{\Psi}^j_M\Gamma^M{\cal E}_i\right)\nonumber\\
&=&\int d^5x E \left[\frac{3i}{2l}
 \left(\nabla_M \overline{\cal E}^i\right)
 \Gamma^{MN}\Psi_N^j \delta_{ij}
 -\frac{9i}{8}{\cal A}_M \overline{\cal E}^i
 \Gamma^{MN}\Psi_{N i}
 \right.\nonumber\\
 &&\left.
 +\frac{3}{16}\delta_{ij}{\cal F}^{MN}\overline{\cal E}^i\Gamma_M\Psi_N^j
 -\frac{3}{16}\delta_{ij}{\cal F}_{MN}
 \overline{\cal E}^i\Gamma^{MNP} \Psi_P^j
 -\frac{3}{l^2}\overline{\cal E}^i\Gamma^M\Psi_{M i}
 \right].
 \label{superv5}
 \end{eqnarray}
 Finally, the supersymmetric variation of the interaction terms between
 the gravitino $\Psi_M$ and the graviphoton $A_M$ gives
 \begin{eqnarray}
 &&\delta \int d^5 x E \left(-\frac{3il}{32}\right)\left(
 \overline{\Psi}_M^i \Gamma^{MNPQ}\Psi_{N i} {\cal F}_{PQ}
 +2 \overline{\Psi}_M^i \Psi_i^N {\cal F}_{MN}\right)\nonumber\\
 &=& -\frac{3il}{32}\int d^5x E  \left\{ {\cal F}_{PQ} \left(
 \left[\nabla_M \overline{\cal E}^i
+\frac{3}{4}{\cal A}_M\delta^{ik}\overline{\cal E}_k-\frac{il}{16}
\overline{\cal E}^i
\left(\Gamma_M^{~RS}+4\delta_M^{~R}\Gamma^S
 \right){\cal F}_{RS}\right.\right.\right.\nonumber\\
&&\left.-
\frac{i}{2l}\delta^{ik}\overline{\cal E}_k\Gamma_M\right]
\Gamma^{MNPQ}\Psi_{N i}\nonumber\\
&&\left.+\overline{\Psi}_M^i \Gamma^{MNPQ}
\left[\nabla_Q {\cal E}_i-\frac{3}{4}{\cal A}_Q
 \delta_{ij}{\cal E}^j +\frac{il}{16}
 \left(\Gamma_Q^{~RS}
 -4\delta_Q^{~R}\Gamma^S \right){\cal E}_i {\cal F}_{RS}
 +\frac{i}{2l}\Gamma_Q\delta_{ik}{\cal E}^k   \right]\right)\nonumber\\
&& +2 {\cal F}^{MN}\left(\left[\nabla_M \overline{\cal E}^i
+\frac{3}{4}{\cal A}_M\delta^{ik}\overline{\cal E}_k-\frac{il}{16}
\overline{\cal E}^i
\left(\Gamma_M^{~RS}+4\delta_M^{~R}\Gamma^S
 \right)F_{RS}-
\frac{i}{2l}\delta^{ik}\overline{\cal E}_k\Gamma_M \right]
\Psi_{N i}\right.\nonumber\\
&& \left.+\overline{\Psi}_{M}^i \left[\nabla_N {\cal E}_i-\frac{3}{4}
{\cal A}_N
 \delta_{ij}{\cal E}^j +\frac{il}{16}
 \left(\Gamma_N^{~RS}
 -4\delta_N^{~R}\Gamma^S \right){\cal E}_i {\cal F}_{RS}
 +\frac{i}{2l}\Gamma_N\delta_{ik}{\cal E}^k   \right]\right)\nonumber\\
&&\left. +\cdots \right\}\nonumber\\
&=& -\frac{3il}{32}\int d^5x E\left\{
2\left[ F^{MN} \left( \nabla_M\overline{\cal E}^i \Psi_{N i}
+\overline{\Psi}_M^i\nabla_N{\cal E}_i\right)
+\frac{3}{4}\delta^{ij}{\cal A}_M {\cal F}^{MN}\left(
\overline{\cal E}_i \Psi_{N j}
+\overline{\Psi}_{N i} {\cal E}_j \right)\right.\right.\nonumber\\
&&+\frac{il}{16}{\cal F}^{MN}{\cal F}^{PQ}\left(
-\overline{\cal E}^i \Gamma_{MPQ}\Psi_{N i}
+\overline{\Psi}^i_M \Gamma_{NPQ}{\cal E}_{i}\right)
-\frac{il}{4}{\cal F}^{MN}\left(
\overline{\cal E}^i \Gamma^{P}\Psi_{N i}{\cal F}_{MP}
+\overline{\Psi}^i_M \Gamma^{P}{\cal E}_{i}{\cal F}_{NP}\right)
\nonumber\\
&&\left.-\frac{i}{2l}{\cal F}^{MN}
\left( \delta^{ij}\overline{\cal E}_i \Gamma_{M}\Psi_{N j}
-\delta_{ij}\overline{\Psi}^i_M \Gamma_{N}{\cal E}^{j}\right) \right]
\nonumber\\
&&+{\cal F}_{PQ}\left[
\left(\nabla_M\overline{\cal E}^i \Gamma^{MNPQ}\Psi_{Ni}
+\overline{\Psi}_M^i \Gamma^{MNPQ}\nabla_N{\cal E}_{i}\right)\right.
\nonumber\\
&& +\frac{3}{4}\left({\cal A}_M\delta^{ij}\overline{\cal E}_i
\Gamma^{MNPQ}{\Psi}_{N j}
-{\cal A}_N\delta_{ij}\overline{\Psi}^i_M
\Gamma^{MNPQ}{\cal E}^{j}\right)\nonumber\\
&&-\frac{il}{16}{\cal F}_{RS}
\left(\overline{\cal E}^i\Gamma_M^{~RS}
\Gamma^{MNPQ}\Psi_{N i}
-\overline{\Psi}^i_M \Gamma^{MNPQ}
\Gamma_N^{~RS}{\cal E}_{i}\right)\nonumber\\
&& -\frac{il}{4}\left( \overline{\cal E}^i\Gamma^{R}
\Gamma^{MNPQ}\Psi_{N i}{\cal F}_{MR}
+\overline{\Psi}^i_M \Gamma^{MNPQ}
\Gamma^{R}{\cal E}_{i} {\cal F}_{NR} \right)\nonumber\\
&&\left.\left.-\frac{i}{2l}\left(\delta^{ij}\overline{\cal E}_i\Gamma_M
\Gamma^{MNPQ}\overline{\Psi}_{N j}
-\delta_{ij}\overline{\Psi}^i_M
\Gamma^{MNPQ}\Gamma_N {\cal E}^{j}\right)\right]
 \right\}\nonumber\\
&=& \int d^5x E\left[-\frac{3il}{8}
\left(\nabla_M \overline{\cal E}\right)^i\Psi_{N i}{\cal F}^{MN}
-\frac{9il}{32}\delta^{ij}{\cal  A}_M {\cal F}^{MN}
\overline{\cal E}_i \Psi_{N j}
\right.\nonumber\\
&&
-\frac{3il}{16} \left(\nabla_M
\overline{\cal E}\right)^i\Gamma^{MNPQ}
\Psi_{N i}{\cal  F}_{PQ}-\frac{9il}{64}{\cal  A}_M {\cal F}_{PQ}\delta^{ij}
\overline{\cal E}_i\Gamma^{MNPQ}\Psi_{Nj}\nonumber\\
&&+\frac{3l^2}{64}{\cal F}^{MN}{\cal F}_{MN}\overline{\cal E}^i\Gamma^P
\Psi_{P i}+\frac{3l^2}{64}E^{-1} \epsilon^{MNPQR}
{\cal F}_{MN}{\cal F}_{PQ}
\overline{\cal E}^i
\Psi_{R i}\nonumber\\
&& +\frac{3l^2}{32}\overline{\cal E}^i\Gamma^{Q}
 \Psi_{N a}{\cal F}^{NP}{\cal F}_{PQ}
 -\frac{3l^2}{32}\overline{\cal E}^i\Gamma_{P}
 \Psi_{N i}{\cal F}^{MP}{\cal F}_{MN}\nonumber\\
&& \left.-\frac{3}{16}\delta^{ij}\overline{\cal E}_i\Gamma_M \Psi_{Nj}
{\cal F}^{MN}+\frac{3}{16}\delta^{ij}
\overline{\cal E}_i\Gamma^{MNP} \Psi_{Mj}
{\cal F}_{NP}\right].
\label{superv6}
 \end{eqnarray}
 Putting the above supersymmetric variations (\ref{superv1})
--- (\ref{superv6}) together, we get
\begin{eqnarray}
\delta S &=& \int d^5 x E \nabla_M \left(
%-e^\mu_{~r}e^\nu_{~s}
%\delta \omega_{\nu}^{~rs}
-\frac{9il}{16}
 \overline{\cal E}^i\Psi_{N i}{\cal F}^{MN}
 -\frac{1}{2}\overline{\cal E}^i\Gamma^{MNP}\nabla_N
 \Psi_{P i}+ \frac{3}{8}
  \overline{\cal E}^i\Gamma^{MNP}
 \Psi_{P}^j\delta_{ij}{\cal A}_N \right.\nonumber\\
 &&\left. -\frac{3il}{32}
 \overline{\cal E}^i\Gamma^{MNPQ}
 \Psi_{N i}{\cal F}_{PQ}
 +\frac{9}{4}\overline{\cal E}^i\Gamma^{MN}
 \Psi_{N}^j\delta_{ij}
 +\frac{2iC}{3\sqrt{6}l}E^{-1}
 \epsilon^{MNPQR}\overline{\cal E}^i \Psi_{R}{\cal A}_N
 {\cal F}_{PQ}\right)
 \nonumber\\
  &&
  +\int d^5x \left(-\frac{iC}{2\sqrt{6}l}+\frac{3l^2}{64}\right)
  \epsilon^{MNPQR}\overline{\cal E}^i
  \Psi_{R}{\cal F}_{MN}{\cal F}_{PQ}.
  \label{var1}
\end{eqnarray}

 In above calculation, we have made use of the following
 relations,
 \begin{eqnarray}
 \Psi^i &=& C^{-1}\Omega^{ij}\overline{\Psi}_j^T=C^{-1}\overline{\Psi}^{iT},
 ~~~\overline{\Psi}^i=-\Psi^{iT}C,\nonumber\\
&&   \overline{\Psi}^i\Gamma_{M_1\cdots M_n}\Phi_i =
 -\Psi^{iT}C\Gamma_{M_1\cdots M_n}C^{-1}\overline{\Phi}_i^T\nonumber\\
&=&\left\{\begin{array}{l}
 -\Psi^{iT}\Gamma_{M_1\cdots M_n}\overline{\Phi}^T_i
=\overline{\Phi}_i\Gamma_{M_1\cdots M_n} \Psi^i
=-\overline{\Phi}^i\Gamma_{M_1\cdots M_n} \Psi_i,~~ n=0,1,4,5,\\
\Psi^{iT}\Gamma_{M_1\cdots M_n}\overline{\Phi}^T_i
=-\overline{\Phi}_i\Gamma_{M_1\cdots M_n} \Psi^a
=\overline{\Phi}^i\Gamma_{M_1\cdots M_n} \Psi_a,~~ n=2,3, \end{array}
\right. , \nonumber\\
\Gamma_{MN}&=&\frac{1}{2}[\Gamma_M,\Gamma_N], ~~
\Gamma^{MNP}=-\frac{1}{2!}E^{-1}\,
\epsilon^{MNPQR}\Gamma_{QR},\nonumber\\
\Gamma^{MNPQ}&=& E^{-1}\,
\epsilon^{MNPQR}\Gamma_{R},
~~\Gamma_{MNPQR}=E\, \epsilon_{MNPQR}.
\nonumber \\
 \Gamma_{MN}\Gamma_{PQ}
&=& E\,\epsilon_{MNPQR}\Gamma^R-
\left(G_{MP} G_{NQ}-G_{MQ}G_{NP}\right),
\nonumber\\
\Gamma_{M}\Gamma_{NP} &=& \Gamma_{MNP}
+G_{MN}\Gamma_P-G_{MP}\Gamma_N, \nonumber\\
\Gamma^{MNP}\nabla_N \nabla_P \Psi_i
&=& \frac{1}{2}\Gamma^{MNP}[\nabla_N, \nabla_P] \Psi_i
= \frac{1}{8}\Gamma^{MNP}{\cal R}_{NP AB}\Gamma^{AB}\Psi_i.
 \end{eqnarray}
 Due to the nocommutativity between $\nabla_M$ and
 $\Gamma_{M_1\cdots M_n}$, we reiteratively use the following operations,
 \begin{eqnarray}
 \Gamma_{M_1\cdots M_n} \nabla _M (\cdots)
 &=&\left[\Gamma_{M_1\cdots M_n},\nabla _M\right] (\cdots)
 +\nabla_M \left[\Gamma_{M_1\cdots M_n} (\cdots)\right].
 \end{eqnarray}
 It is convenient to choose the inertial coordinate system,
 i.e. the Christoffel symbol $\Gamma^{M}_{~NP}=0$. Consequently,
 the metricity condition leads to
 \begin{eqnarray}
 \partial_M E_N^{~A}=0,
 \end{eqnarray}
 and hence the modified spin connection,
 \begin{eqnarray}
 \Omega_{M AB}&=&\frac{1}{2}\overline{E}_A^{~N}\left(\partial_M E_{N B}
 -\partial_N E_{M B}\right)
 -\frac{1}{2}\overline{E}_B^{~N}\left(\partial_M E_{N A}
 -\partial_N E_{M A}\right)\nonumber\\
 && -\frac{1}{2}\overline{E}_A^{~P}
 \overline{E}_B^{~Q}\left(\partial_P E_{Q c}
 -\partial_Q E_{PC}\right)E_{M}^{~C}
 +\frac{1}{4}\left(\overline{\Psi}_M\Gamma_A\Psi_B+
 \overline{\Psi}_A\Gamma_M\Psi_B-\overline{\Psi}_M\Gamma_B\Psi_A
  \right),
 \end{eqnarray}
  keeps only the quadratic fermionic terms.
 We have also considered the Ricci and Bianchi identities for
 the Riemannian curvature tensor and the Abelian gauge field
 \begin{eqnarray}
 \epsilon^{MNPQR} {\cal R}_{SPQR}=0, ~~~
 \epsilon^{MNPQR}\nabla_N {\cal R}_{STPQ}=0,~~~
 \epsilon^{MNPQR}\nabla_N {\cal F}_{QR}=0.
 \end{eqnarray}

If we choose the indefinite Chern-Simons coefficient in Eq.\,(\ref{var1})
as
\begin{eqnarray}
C=-\frac{3i\sqrt{6}l^3}{32},
\label{var2}
\end{eqnarray}
the above supersymmetric variation of ${\cal N}=2$
gauged supergravity action in five dimensions is a total derivative,
\begin{eqnarray}
\delta S &=& \int d^4x \int_{r=\epsilon} dr  \partial_M \left[E\left(
%-e^\mu_{~r}e^\nu_{~s}\delta \omega_{\nu}^{~rs}
-\frac{9il}{16}
 \overline{\cal E}^i\Psi_{N i}{\cal F}^{MN}
 -\frac{1}{2}\overline{\cal E}^i\Gamma^{MNP}\nabla_N
 \Psi_{P i}+ \frac{3}{8}
  \overline{\cal E}^i\Gamma^{MNP}
 \Psi_{P}^j\delta_{ij}{\cal A}_N \right.\right.\nonumber\\
 &&\left. \left.-\frac{3il}{32}
 \overline{\cal E}^i\Gamma^{MNPQ}
 \Psi_{N i}{\cal F}_{PQ}
 +\frac{9}{4}\overline{\cal E}^i\Gamma^{MN}
 \Psi_{N}^i\delta_{ij}
 +\frac{l^2}{32}E^{-1}
 \epsilon^{MNPQR}
 \overline{\cal E}^i \Psi_{R}{\cal A}_N {\cal F}_{PQ}\right)\right],
\label{var3}
\end{eqnarray}
which is the typical feature of a supersymmetric field theory.


 Now we can extract out the holographic super-Weyl anomaly from above
 total derivative
 terms as Witten \cite{witt1} did in finding out the chiral
 $R$-symmetry anomaly. Considering the $r$-dependence of bulk fields
 and of the supersymmetry transformation parameter ${\cal E}^i$ given in
 (\ref{rgf1}), (\ref{rgf2}), (\ref{redsc}), (\ref{redg}), (\ref{rega}),
 (\ref{redsl}), (\ref{redsr}) and (\ref{strp}), and taking
 the boundary limit $\epsilon\to 0$ after we integrate over
 the radial coordinate to the near boundary cut-off $r=\epsilon$ , we
 obtain
 \begin{eqnarray}
 \delta S=\frac{l^3}{32}\int d^4 x \epsilon^{\mu\nu\lambda\rho}
 A_\mu F_{\nu\lambda}\overline{\eta}\gamma_5\psi_\rho.
 \label{fvar}
  \end{eqnarray}
 In deriving Eq.\,(\ref{fvar})  we have used  the fact  that
 the metric on the boundary should be the induced metric
 \begin{eqnarray}
 \widetilde{g}_{\mu\nu}(x,r)=\frac{l^2}{r^2}{g}_{\mu\nu}(x,r)
\end{eqnarray}
rather than ${g}_{\mu\nu}(x,r)$ \cite{bian1}.
If we switch on the overall
gravitational constant factor
$-1/(8\pi G^{(5)})$ on the classical Lagrangian (\ref{gaugedfm}),
and consider
the following relations among the $AdS_5$ radius $l$, string
coupling $g_s$, the number $N$ of the $D3$-branes, the
five- and ten-dimensional
gravitational constants connected  by the compactification
on $S^5$ of radius $l$,
\begin{eqnarray}
G^{(5)}=\frac{G^{(10)}}{\mbox{Volume}\, (S^5)}=\frac{G^{(10)}}{l^5\pi^3}, ~~~
G^{(10)}=8\pi^6g^2_s, ~~~l=\left(4\pi g_s\right)^{1/4},
\end{eqnarray}
Eq.\,(\ref{fvar}) yields the gauge field part of the holographic
super-Weyl anomaly (up to a numerical factor),
\begin{eqnarray}
\gamma_\mu s^\mu = -\frac{1}{8\pi G^{(5)}}\frac{l^3}{32}
\epsilon^{\mu\nu\lambda\rho}A_\mu F_{\nu\lambda}\gamma_5\psi_\rho
=-\frac{N^2}{128\pi^2}\epsilon^{\mu\nu\lambda\rho}
A_\mu F_{\nu\lambda}\gamma_5\psi_\rho.
\label{gtra}
\end{eqnarray}
This is the super-Weyl anomaly of supersymmetry
current contributed from the external gauge field background
at the leading order
of large-$N$ expansion of ${\cal N}=1$ $SU(N)$ supersymmetric gauge
theory.



\section{Summary and Discussion}

We have investigated the super-Weyl anomaly
of ${\cal N}=1$ supersymmetric Yang-Mills theory in the
external ${\cal N}=1$ conformal supergravity background via the AdS/CFT
correspondence. With the speculation that
at low-energy the type IIB supergravity in $AdS_5\times X^5$ background
should reduce to the gauged supergravity in five dimensions since
such a background provides a spontaneous compactification on $X^5$,
there should exists a holographic correspondence between
${\cal N}=2$ conformal supergravity in five dimensions   and
${\cal N}=1$ supersymmetric Yang-Mills theory at the fixed point
of its renormalization group flow. The five-dimensional
${\cal N}=2$ gauged  supergravity  has an $AdS_5$ classical
solution  which preserves the full supersymmetry.
Around this $AdS_5$ vacuum configuration, the five-dimensional
 ${\cal N}=2$ on-shell gauged supergravity multiplet reduces to the
${\cal N}=1$ off-shell conformal supergravity mutiplet on the boundary
of $AdS_5$ space. Correspondingly, the bulk ${\cal N}=2$
supersymmetry transformation converts into the ${\cal N}=1$
superconformal transformation in four dimensions, which consists
of the  supersymmetry transformation for ${\cal N}=1$
Poincar\'{e} supergravity and super-Weyl transformation. With
these facts in mind, we calculate the supersymmetry variation of the ${\cal N}=2$
gauged supergravity in five dimensions and obtain the total derivative
terms. Further, we reduce the total derivative terms to the boundary
of $AdS_5$ space using the boundary reduction of the bulk
fields. Considering the incompatibility of
the ${\cal N}=1$ Poincar\'{e} supersymmetry and the super-Weyl symmetry,
we  extract out the super-Weyl anomaly of ${\cal N}=1$ supersymmetric gauge
theory.

   However, as shown in Eq.\,(\ref{gtra}), we only reveal the contribution
 from the external gauged field when ${\cal N}=1$
supersymmetric gauge theory couples to ${\cal N}=1$ conformal
supergravity background. As shown long-time ago in Ref.\cite{abb},
 there should also arise
a contribution from the external gravitational  background,
\begin{eqnarray}
\gamma_\mu s^{\mu} \sim \frac{1}{128\pi^2}
R^{\mu\nu\lambda\rho}\gamma_{\mu\nu}
\nabla_\lambda\psi_\rho.
\end{eqnarray}
The reason for not having revealed this gravitational background
contribution is
that we only consider the leading order of $r$-dependence
of the bulk fields.  If we make an complete near-boundary
analysis and consider the asymptotic expansion until the logarithmic
term emerges \cite{bian1}, i.e.,
\begin{eqnarray}
{\cal F}(x,r)=r^m\left[ f_{(0)}(x)+ f_{(2)}(x) r+\cdots
r^n \left( f_{(2n)}(x)+ \widetilde{f}_{(2n)} (x)\,\ln r
+\cdots\right)+\cdots
\right],
\end{eqnarray}
the on-shell gauged five-dimensional supergravity action
has an indfrared divergence  near the $AdS_5$ boundary due to its
infinite boundary. One must perform a holographic renormalization
procedure to make it well defined\cite{bian1}.
In this way, we believe that the contribution from the gravitational
background field will arise exactly as the holographic Weyl anomaly does
\cite{hesk}. Actually, in the pioneering work by Witten \cite{witt1},
 it was pointed out that the chiral $R$-symmetry anomaly
 comes from the Chern-Simons term of five-dimensional conformal
 supergravity. This can also give only the contribution from the external
 gauge field. To achieve the contribution to the chiral R-symmetry
 anomaly
 from the gravitational
 background, one must consider the asymptotic expansion of the bulk fields
 until the logarithmic function  of the radial coordinate
 emerges. The superficial  reason  for this  lies in the
 existence of the Chern-Simons term
 consisting only of the vector field.
 The Chern-Simons term is a topological term and becomes
 a total derivative under gauge transformation. Its supersymmetric
 variation is not a total derivative term, but to achieve supersymmetry
 for the bulk gauged supergravity, as shown in
 Eqs.\,(\ref{var1})--(\ref{var3}), one must choose the indefinite
 coefficient of the Chern-Simons term to make its non-derivative
 supersymmetric variation  cancel with the other one. In particular,
 unlike the induced metric $l^2 g_{\mu\nu}(x,r)/r^2$ and
 the gravitino $\Psi_\mu (x,r)$,
 the leading term in the near-boundary asymptotic expansion
 of the gauge field ${\cal A}_\mu (x,r)$ is
 independent of the radial coordinate.
 In Eq.\,(\ref{var3}), it is the
 total derivative term relevant to the supersymmetric variation
 of the Chern-Simons term that leads to the external gauge field
 contribution to the super-Weyl anomaly of supersymmetry current.
 We are not aware  whether there exist any
 physical reasons for the difference between these two holographic
  contributions
  to the superconformal anomaly. Since the essence of the holographic anomaly
  is the anomaly inflow from the bulk theory \cite{callan},\footnote{
  We thank  A. Kobakhidze for discussion on this point.}
  this is the reason why one can extract out the anomaly  from a
  higher-dimensional bulk theory,
  Thus it might be relevant to the difference between the
  anomaly inflows contributed by the gravitational and gauge field
  backgrounds.

Finally, it is worth to emphasize another way of calculating the
holographic Weyl anomaly proposed in Ref.\,\cite{imbi}. This approach
is independent of the concerete form of the classical bulk gravitational
action, and depends  purely on a special bulk diffeomorphism (called
the `` PBH '' transformation \cite{pbh}), which keeps the form of
the Fefferman-Graham metric \cite{feff} of an arbitrary $d+1$-dimensional
manifold with the boundary topologically isomorphic to $S^d$
invariant and reduces to a Weyl transformation on the boundary.
With a choice on the holomorphic dimensional regularization, there emerges
no logarithmic  function of the radial coordinate in the
near-boundary asymptotic expansion. Further, the invariance of a
 general bulk gravitational action, which admits an $AdS_{d+1}$
 classical solution,
 under this particular diffeomorphism  yields a
Wess-Zumino-like consistency condition satisfied by the generating
functional for the Weyl anomaly on the boundary ($d$ being
an even integer).  Hence the hologrpahic Weyl anomaly can be extracted
out and no holographic renormalization procedure is necessary \cite{imbi}.
The advantage of this approach is that one can avoid the
complicated near-boundary analysis
and the subsequent holographic renormalization procedure. One may consider
to use  this approach to calculate the super-Weyl
anomaly of a supersymmetry current. However, there is one crucial obstacle
to overcome. That is, one needs to find a supersymmetric generalization of
above special bulk diffeomorphism, which should keep both the supersymmetry
in the bulk and on the boundary. In such an approach, one must
employ the holographic dimensional regularization to prevent the
logarithmic dependence from emerging
in the near-boundary asymptotic expansion. However, the dimensional
regularization in general does not preserve the supersymmetry.
This fact has cast a shadow on the application of the above approach to the
evaluation of the holographic supercurrent anomaly.



\bigskip

\acknowledgments
\noindent We are indebted to M. Grisaru,
 A. Kobakhidze, M. Nishimura and A. Schwimmer
for discussions.
We also would like to thank G. Kunstatter, R.G. Leigh,
 C. Montonen and M. Sheikh-Jabbari for useful remarks.
 This work is supported by
the Academy of Finland under the Project No. 163394.


\begin{references}


\bibitem{mald} J. Maldacena, Adv. Theor. Math. Phys. {\bf 2} (1998) 231.
%The Large N limit of superconformal field theories and supergravity
%hep-th/9711200

\bibitem{gkp} S.S. Gubser, I.R. Klebanov and A.M. Polyakov,
Phys. Lett. {\bf B428} (1998) 105.
%GAUGE THEORY CORRELATORS FROM NONCRITICAL STRING THEORY
%hep-th/9802109

\bibitem{witt1} E. Witten,  Adv. Theor. Math. Phys. {\bf 2} (1998) 253.
%Anti-de Sitter space and holography
%hep-th/9802150


\bibitem{gst} M. G\"{u}naydin, G. Sierra and P.K. Townsend,
Nucl. Phys. {\bf B242} (1984) 244; Nucl. Phys. {\bf B253} (1985)
573.

\bibitem{awada} M. Awada and P.K. Townsend,
Nucl. Phys. {\bf 255} (1985) 617;
L.J. Romans, Nucl. Phys. {\bf B267} (1986) 433.


\bibitem{guna2}M. G\"{u}naydin, L. J. Romans and N.P. Warner,
Phys. Lett. {\bf B154} (1985) 268; Nucl. Phys. {\bf 272} (1986) 598.


\bibitem{ferr} S. Ferrara, C. Fronsdal and A. Zaffaroni,
Nucl. Phys. {\bf B532} (1998) 153;  S. Ferrara and A. Zaffaroni,
Phys. Lett. {\bf B431} (1998) 49.
%hep-th/9803060.
%hep-th/98020203.

\bibitem{agmo} For a review, see O. Aharony, S.S. Gubser, J. Maldacena,
H. Ooguri and Y. Oz, Phys. Rept. {\bf 323} (2000) 183.
%LARGE N FIELD THEORIES, STRING THEORY AND GRAVITY.
%hep-th/9905111.

\bibitem{freu} P.G.O. Freund and M.A. Rubin,
Phys. Lett. {\bf B97} (1980) 233.

\bibitem{duff} For a review, see M. Duff, B.E.W. Nilsson and
C.N. Pope, Phys. Rep. {\bf 130} (1986) 1.

\bibitem{marcus} M. G\"{u}naydin and N. Marcus,
Class. Quant. Grav. {\bf 2} (1985) L11.

\bibitem{kim} H.J. Kim, L.J. Romans and P. Van Nieuwenhuizen,
Phys. Rev. {\bf D32} (1985) 389.

\bibitem{roman2} L.J. Romans, Phys. Lett. {\bf B153} (1984) 401.

\bibitem{kw2} I.R. Klebanov and E. Witten, Nucl. Phys. {\bf B536}
(1998) 536.


\bibitem{kach} S. Kachru and E. Silverstein,  Phys. Rev. Lett. {\bf 80}
(1998) 4855.

\bibitem{law} A. Lawrence, N. Nekrasov and C. Vafa, Nucl. Phys.
 {\bf B533} (1998) 199.

\bibitem{shif} N.A. Novikov, M.A. Shifman, A.I. Vainshtein
and V.I. Zakharov, Nucl. Phys. {\bf B229} (1983) 381.

\bibitem{seib} N. Seiberg, Nucl. Phys. {\bf B435} (1995) 129.

\bibitem{lei2}R.G. Leigh and M.J. Strassler, Nucl. Phys.
 {\bf B447} (1995) 95.


\bibitem{early1} For earlier works discussing on the ${\cal N}=1$
  superconformal symmetry based on the perturbative $\beta$-function,
  see P. West, Phys. Lett. {\bf B137} (1984) 371;
  A. Parkes and P. West,  Phys. Lett. {\bf B138} (1984) 99;
  D.R.T. Jones and L. Mezincescu, Phys. Lett. {\bf B138} (1984) 293;
  S. Hamidi, J. Patera and J. Schwarz, Phys. Lett. {\bf B141} (1984) 349;
  W. Lucha and H. Neufeld, Phys. Lett. {\bf B174} (1986) 186;
  P. Piguet, K. Sibold, Phys. Lett. {\bf B177} (1986) 373; Int. J. Mod. Phys.
  {\bf A1} (1986) 913; Phys. Lett. {\bf B201} (1988) 241; C. Lucchesi,
  O. Piguet and K. Sibold, Helv. Phys. Acta, {\bf 61} (1988) 321.
  D.R. T. Jones, Nucl. Phys. {\bf B277} (1986) 153; A.V. Ermushev,
  D.I. Kazakov and O.V. Tarasov, Nucl. Phys. {\bf B281} (1987) 72;
  X.D. Jiang and X.J. Zhou,  Phys. Lett. {\bf B197} (1987) 156;
  Phys. Lett. {\bf B216} (1989) 160; Phys. Rev. {\bf D42} (1990) 2109.
  C. Lucchesi and  G. Zoupanos, Fortsch. Phys. {\bf 45} (1997) 129.


\bibitem{early2} For earlier works discussing on the ${\cal N}=2$
  superconformal symmetry based on the perturbative $\beta$-function, see
  P. Howe, K. Stelle and P. West, Phys. Lett. {\bf B124} (1983) 55;
  I.G. Koh and S. Rajpoot, Phys. Lett. {\bf B135} (1984) 397;
  F.X. Dong, T.S. Tu, P.Y. Xue and X.J. Zhou,
  Phys. Lett. {\bf B140} (1984) 333;
  J.P. Derendinger, S. Ferrara and A. Masiero,
  Phys. Lett. {\bf B143} (1984) 133.


\bibitem{liu} H. Liu and A.A. Tseytlin, Nucl. Phys. {\bf B533}
(1998) 88.
%hep-th/9804083

\bibitem{ilio}J. Iliopoulos and B. Zumino, Nucl. Phys. {\bf B76}
(1974) 310; S. Ferrara and B. Zumino, Nucl. Phys. {\bf B87}
(1975) 207; P. Howe, K.S. Stelle and P.K. Townsend, Nucl. Phys. {\bf B192}
(1981) 332.



\bibitem{anm} For reviews, see M.F. Sohnius, Phys. Rep. {\bf 128} (1985)
39; P. West, {\it Introduction to Supersymmertry and Supergravity}
(World Scientific, Singapore, 1990); M. Chaichian, W.F. Chen and C. Montonen,
Phys. Rep. {\bf 346} (2001) 89.


\bibitem{sibold} T.E. Clark, O. Piguet and K. Sibold,
Nucl. Phys. {\bf B143} (1978) 445; O. Piguet and K. Sibold,
Nucl. Phys. {\bf B196} (1982) 428; {\it ibid} (1982) 447.


\bibitem{grisaru} For a brief review, see  M.T. Grisaru,
{\it Anomalies in Supersymmetric Theories},
Invited talk given at NATO Advanced Study Institute
on Gravitation: {\it Recent Developments}, Cargese, France, Jul 10-29, 1978;
 D. Anselmi, D.Z. Freedman, M.T. Grisaru and  A.A. Johansen,
Nucl. Phys. {\bf B526} (1998) 543.

\bibitem{kaku} S. Ferrara, M. Kaku, P.K. Townsend, P. van Nieuwenhuizen,
Nucl. Phys. {\bf B129} (1977) 125; M. Kaku, P.K. Townsend,
P. van Nieuwenhuizen, Phys. Rev. {\bf D17} (1978) 3179; Phys. Lett. {\bf
B69} (1977) 304; J. Crispim-Romao, Nucl. Phys. {\bf B145}
(1978) 535; T. Kugo and S. Uehara, Nucl. Phys. {\bf B226} (1983) 49.


\bibitem{fra}For a review, see
E.S. Fradkin and A. A. Tseytlin, Phys. Rep. {\bf 119} (1985) 233.


\bibitem{hesk} M. Henningson and K. Skenderis,
JHEP {\bf 9807} (1998) 023.
% THE HOLOGRAPHIC WEYL ANOMALY, hep-th/9806087
%Fortsch. Phys. {\bf 48} (2000) 125.
%HOLOGRAPHY AND THE WEYL ANOMALY, hep-th/9812032

\bibitem{bian1} M. Bianchi, D.Z. Freedman and K. Skenderis,
 Nucl. Phys. {\bf B631} (2002) 159;
 %{\tt hep-th/0105276}.
 %\bibitem{bian2} M. Bianchi, D.Z. Freedman and K. Skenderis,
 JHEP: {\bf 0108} (2001) 041.
 %{\tt hep-th/0112119};


\bibitem{imbi} C. Imbimbo, A. Schwimmer, S. Theisen and S.
Yankielowicz, Class. Quant. Grav. {\bf 17} (2000) 1129.
%{\it Diffeomorphisms and Holographic Anomalies},
%{\tt hep-th/9910267}.

\bibitem{feff} C. Fefferman and C.R. Graham, `` Conformal Invariants ''
in {\it Elie Cartan et les Math\'{e}matiques d'aujourd'hui}
(Ast\'{e}risque, 1985) P. 95.

\bibitem{noji} S. Nojiri and S.D. Odintsov,
Phys. Lett. {\bf B444} (1998) 92; Int. J. Mod. Phys. {\bf A15}
(2000) 413;  Mod. Phys. Lett. {\bf A15} (2000) 1043;
S. Nojiri, S.D. Odintsov, S. Ogushi, A. Sugamoto
and M. Yamamoto, Phys. Lett. {\bf B465} (1999) 128.
%hep-th/9810008;  9908066; 9903033; 9910113

\bibitem{kraus} V. Balasubranian and P. Kraus, Comm. Math. Phys. {\bf 208}
 (1999) 3731; A.M. Awad and C.V. Johnson, Phys. Rev. {\bf D61} (2000) 084025.
 %hep-th/9902121

\bibitem{ahar} O. Aharony, J. Pawelczyk, S. Theisen and
S. Yankielowicz,  Phys. Rev. {\bf D60} (1999) 066001;
M. Blau, E. Gava and K.S. Narain, JHEP 9909 (1999)  010;
P. Mansfield and D. Nolland, Phys. Lett. {\bf B495} (2000) 435;
S.G. Naculich, H.J. Schnitzer and N. Wyllard,
 Int. J. Mod. Phys. {\bf A17} (2002) 2567.
% hep-th/9901134, 9904179, 0005224, 0106020

\bibitem{fuku} J. Kalkkinen and D. Martelli,
Nucl. Phys. {\bf B596} (2001) 215; J. Kalkkinen, D. Martelli
and W. M\"{u}ck, JHEP 0104 (2001) 036;
M. Fukuma, S. Matsuura and  T. Sakai, Prog. Theor. Phys.
{\bf 104} (2000) 1089.
% hep-th/0103187, 0007234, 0103111, 0007062

\bibitem{crem1} E. Cremmer, B. Julia and J. Scherk,
Phys. Lett. {\bf B76} (1978) 409.

\bibitem{crem2} E. Cremmer, in {\it Superspace and Supergravity}, ed. by
S.W. Hawking and M. Roc\v{e}k (Cambridge University Press, 1981).


\bibitem{guna} For a recent discussion relevant to five-dimensional $N=2$
gauged supergravity, see M. G\"{u}naydin and M. Zagermann,
Nucl. Phys. {\bf B572} (2000) 131;
Phys. Rev. {\bf D62} (2000) 044028; {\it ibid} {\bf D63} (2001) 064023.
% {\tt hep-th/9912027}; {\tt hep-th/0002228};
%{\tt hep-th/0004117}
A. Ceresole and G. Dall'Agata, Nucl. Phys. {\bf B585} (2000) 143.
%{\tt hep-th/0004111}.


\bibitem{nish} M. Nishimura and Y. Tanii, Phys. Lett. {\bf B446} (1999)
37; Int. J. Mod. Phys. {\bf A14} (1999) 3731; Mod. Phys. Lett. {\bf A14}
(1999) 2709. M. Nishimura, Nucl. Phys. {\bf B588} (2000) 471.
% hep-th/9810148, 9904010, 9910192, 0004179.

\bibitem{bala} V. Balasubramanian, E. Gimon, D. Minic and J. Rahmfeld,
 Phys. Rev. {\bf D63} (2001) 104009.
 %hep-th/0007211.

\bibitem{abb} L.F. Abbott, M.T. Grisaru and H.J. Schnitzer,
Phys. Lett. {\bf B73} (1978) 71.


\bibitem{callan}C.G. Callan and J.A. Harvey,
Nucl. Phys. {\bf B250} (1985) 427;
S. Naculich, Nucl. Phys. {\bf B296} (1988) 837.


\bibitem{pbh} R. Penrose and W. Rindler, {\it Spinors and Spacetime},
Chapter 9, Volume 2 (Cambridge University Press, 1986);
J.D. Brown and M. Henneaux, Commun. Math. Phys. {\bf 104} (1986) 207.


\end{references}

\end{document}




