\vspace*{-1.8cm} \begin{flushright} \normalsize{CERN-PH-TH/2009-159}\\ \end{flushright} \vspace{1cm} {\hspace{-1cm} Brane Embeddings in $AdS_4 \times \C\cP^3$\\~~\\~~\\ }
~~
We construct $D$-brane embeddings in $AdS_4 \times \C\cP^3$ by studying the consistency conditions following from the pull back of target space equations of motion. We explicitly discuss the supersymmetry preserved by these embeddings by analyzing the compatibility of kappa symmetry projections with the target space Killing spinors in each case. The embeddings correspond to AdS/dCFT dualities involving a CFT with a defect. %We identify an asymptotically %$AdS_3 \times S^2$ embedding that should correspond to a holographic RG-flow %on the defect.
0909.3061.tex
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\begin{document}
\title{ \vspace*{-1.8cm}
\begin{flushright}
\normalsize{CERN-PH-TH/2009-159}\\ 
\end{flushright}
\vspace{1cm}
\bf{\hspace{-1cm} Brane Embeddings in $AdS_4 \times \C\cP^3$\\~~\\~~\\ }}



\author{B.~Chandrasekhar$^{1}$\footnote{bcsekhar@fma.if.usp.br} and Binata Panda $^{2}$\footnote{ binata@iopb.res.in}~~\\~~\\~~\\}
\affiliation{$^1$Instituto de F\'{i}sica, Universidade de S\~{a}o Paulo,
C. Postal 05315-970, S\~{a}o Paulo, Brasil \\~~\\
$^2$ Institute of Physics, Bhubaneswar 751 005, India \\~~\\~~\\~~\\~~\\~~\\}

%\date{\today}



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


\begin{abstract}


We construct $D$-brane embeddings in $AdS_4 \times \C\cP^3$ by studying the consistency
conditions following from the pull back of target space equations of motion.
We explicitly discuss the supersymmetry preserved by these embeddings by analyzing the 
compatibility of kappa symmetry projections with the target space Killing spinors in
each case. The embeddings correspond to AdS/dCFT dualities 
involving a CFT with a defect. 
%We identify an asymptotically 
%$AdS_3 \times S^2$ embedding that should correspond to a holographic RG-flow
%on the defect. 


\end{abstract}
%\pacs
\maketitle

\section{Introduction}


Recently, string theory on $AdS_4\times\C\cP^3$ has enjoyed a special study, due to its appearance in a new example of the AdS/CFT duality~\cite{Maldacena:1997re}. The example involves, ${\cal N}=6$ superconformal ${\rm SU}(N)\times
{\rm SU}(N)$ Chern-Simons theory in three dimensions and
M-theory on ${\rm AdS}_4\times {\rm S}^7/{\mathbb Z}_k$, where $k$
is the level of the Chern-Simons action~\cite{Aharony:2008ug}. The new duality is motivated by the conjecture
that Superconformal Chern-Simons theories describe the low-energy world-volume dynamics of
multiple M2-branes~\cite{Schwarz:2004yj}. The ABJM model is characterized by two parameters -- the rank $N$
of the two gauge groups ${\rm SU}(N)$ and the integer level $k$
which is opposite for  the  gauge groups. Remarkably, there exists
an analogue of the 't Hooft limit, where $N,k\to\infty$ with the
ratio $\lambda=2\pi^2 N/k$ kept fixed. In this limit $\lambda$
becomes continuous allowing therefore for application of standard
perturbative techniques. In particular~\cite{Aharony:2008ug}, at strong coupling, i.e,
when $\lambda$ becomes large, the M-theory on ${\rm AdS}_4\times
{\rm S}^7/{\mathbb Z}_k$ can be effectively described by type IIA
superstring theory on the ${\rm AdS}_4\times \C\cP^3$ background. 
The connection between M-theory and type IIA string theory has been an exciting source
of information, especially, since the fundamental strings and D-branes of type IIA are on a unified footing in eleven dimensions. Consequently, starting from M-theory, it is possible to have an alternative description as a weakly coupled type IIA string theory, giving us the handle to bridge the non-perturbative
and perturbative features of these theories.  Low energy theory on the
M2-branes governed by non-dynamical CS-theory can have interesting relation to the dynamical Yang-Mills
living on the world-volume of $D$-branes in type IIA~\cite{Mukhi:2008ux,Ezhuthachan:2008ch}. Construction of
string and brane actions for IIA strings in $AdS_4\times \C\cP^3$ is a very interesting subject and has been
discussed extensively in~\cite{Arutyunov:2008if}-\cite{Grassi:2009yj}.\\

\smallskip

\noindent
An intriguing direction to explore for an extension of the new duality~\cite{Aharony:2008ug}, is to
consider CFT's with a defect, corresponding to $D$-branes in the bulk. Such theories have several 
applications in a variety of situations as discussed extensively in the context of $AdS_5 \times S^5$
~\cite{Mirabelli:1997aj}-\cite{Myers:2006qr}. One of the models  proposed in~\cite{randall} corresponds to that of a 3-brane living inside an $AdS_5$ can give rise to a warped geometry of
the form $AdS_4$. Such a brane is expected to
be a viable candidate for proposals of localized gravity~\cite{Randall:1999vf}. Interestingly, the metric
for such curved branes in AdS can be written down, allowing for a detailed analysis. Thus,
in this work, we study the embeddings of $D$-branes in $AdS_4\times \C\cP^3$.  These embeddings can actually lead to dualities between supersymmetric AdS embeddings and conformal field theories with a defect, AdS/dCFT duality~\cite{bachas2}-\cite{johanna}. In fact, such situations leading to $AdS4/CFT_3$ with flavors have been considered 
in~\cite{Fujita:2009kw}-\cite{Gaiotto:2009tk} and D-branes in various $AdS_4$ models have been discussed 
in~\cite{Koerber:2007jb,Koerber:2009he}. In particular, kappa-symmetry gauge fixings of the superstring, $D0$ and $D2$-brane actions in the complete $AdS_4\times \C\cP^3$ superspace have been discussed in~\cite{Grassi:2009yj}. Studying branes in $AdS_4\times \C\cP^3$ is also important from the point of view understanding local and non-local operators in the holographic dual theory~\cite{Chen:2008bp}
A discussion of particle like branes was presented in~\cite{Aharony:2008ug},
both from the M-theory and IIA point of view. Here, the brane embeddings we discuss are non compact. Furthermore, our discussion of embeddings of branes in $AdS_4\times \C\cP^3$ is slightly different from the analysis considered above. 
We take the lead of~\cite{Skenderis:2002vf} , where a very general study of possible supersymmetric embeddings of $D$-branes in $AdS_5 \times S^5$ and its penrose limit, was undertaken. The most general $D$-brane field equations following from the pull over of
the equations of motion in the target space geometry were written down and explicit embeddings were 
constructed~\cite{Skenderis:2002vf}.  This type of analysis is better to understand the classification of embeddings based on the symmetry properties associated to a particular geometry, which is in this case, the
$AdS_4\times \C\cP^3$ background. The stability analysis involves finding out the number of 
target space supersymmetries compatible with the kappa symmetry projection on to the brane world
volume. This $D$-brane picture gives a nice view of the dynamics of a defect on the boundary CFT. 
It is thus interesting to consider an AdS brane in the present context, as it also lends
itself to a very natural holographic interpretation. To realize this situation, one adds additional structure on both sides of the duality: a D-brane in the bulk 
and a defect in the boundary theory. The theory on the defect captures holographically the physics of the D-brane in the bulk, and the interactions between the bulk and the D-brane
modes are encoded in the couplings between the boundary 
and the defect fields. \\



%Such a relationship has been nicely formulated in~\cite{Skenderis:2002vf} for the
%case of  $D$-brane embeddings in $AdS_5 \times S^5$ and analyzing their supersymmetry.
\smallskip

\noindent
The new $AdS_4/CFT_3$ duality shares many common features with the well studied $AdS_5 \times S^5$-Super Yang-Mills
correspondence. In fact, using the insights from the later, many results can be obtained for the former.
In tune with the earlier AdS/dCFT dualities~\cite{KR}, in the following, we  
consider a $D4$-brane wrapping an $AdS_3 \times \C\cP^1$
submanifold of $AdS_4 \times \C\cP^3$. This configuration may be considered
as the near-horizon limit of a certain $D2-D4$ system, and the AdS/CFT 
duality is considered to act twice: both in the bulk and on the worldvolume. 
In the limit discussed in~\cite{ooguri1}, 
the bulk description can be taken to be in terms of supergravity coupled 
to a probe $D4$-brane. The dual theory is the ABJM model~\cite{Aharony:2008ug} coupled to 
a two dimensional defect. The defect theory may be associated with
the boundary of $AdS_3$ and  it should be a conformal field theory, following the
logic established for earlier brane embeddings~\cite{KR}. In fact, the existence of solutions
we present in this work, gives enough evidence for the existence of dCFT's in the case of
$AdS_4/CFT_3$ duality. \\

\smallskip

\noindent
The plan of the rest of the paper is  as follows. In Section \ref{dbranembd},
we discuss the D-brane embeddings into the $AdS_4 \times \C\cP^3$ IIA background. In Section \ref{Ap:Killing}, we use the kappa symmetry projector to determine the supersymmetry preserved by these D-brane embeddings and
conclude in Section \ref{conclusion}. Finally, in Appendix A we discuss the D-brane field equations used in Section~\ref{dbranembd}.\\

\smallskip

\noindent
Note : After submission of this work, we were informed of $D$-branes and defect ABJM models considered in~\cite{Fujita:2009kw}, which has some overlap. 
%The construction of $D$-brane
%embeddings which are both supersymmetric and non-supersymmetric, and the stability analysis
%involving checking the consistency conditions following from the pull back of target space equations, 
%is explicit in our work. 
More recently, an extensive discussion of branes in $AdS_4\times \C\cP^3$, including the construction of defect CFT's appeared in~\cite{Ammon:2009wc}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{D-brane Embeddings} \label{dbranembd}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section, we start by giving a heuristic discussion of brane embeddings in $ AdS_4\times \C\cP^3$
background and present some general features. The issue of stability of the embeddings is dealt with in more detail in the later sections when analyzing the supersymmetry. The $ AdS_4\times \C\cP^3$ IIA background geometry we start with has the form~\cite{Aharony:2008ug,Nishioka:2008gz},
\be \label{adscp3}
ds_{IIA}^2=\tilde{R}^2(ds_{AdS4}^2+4ds_{CP3}^2), 
\ee 
where
\be
\tilde{R}^2=\frac{R^3}{4k}=\pi\sqrt{\frac{2N}{k}}.  
\ee
\bea 
ds_{AdS4}^2 & = & \frac{du^2}{u^2} + u^2 (dx \cdot dx)_3 \\
&=&\frac{du^2}{u^2} + u^2 [ -(dx^0)^2 + (dx^1)^2 + (dx^2)^2 ]
\eea

\bea \label{cp3earlier}
ds^2_{CP^3}&=&d\xi^2+\cos^2\xi\sin^2\xi\left(d\psi+\frac{\cos\theta_1}{2}d\varphi_1-
\frac{\cos\theta_2}{2}d\varphi_2\right)^2   \\ \nonumber
&& +\frac{1}{4}\cos^2\xi\left(d\theta_1^2+\sin^2\theta_1
d\varphi_1^2\right)+\frac{1}{4}\sin^2\xi(d\theta_2^2+\sin^2\theta_2
d\varphi_2^2). \label{cp}
\eea
Now, we wish to introduce a $D4$-brane in this background which wraps an $\C\cP^1$ inside the 
$\C\cP^3$. This can be taken to be defined as:
\be
\xi =0
\ee
The radius of the $S^2$ which the brane wraps is ${\tilde R}$. The brane world volume lies inside
the $AdS_4$ and stretches along the direction $x^2 = x =0$. Thus, it fills the $AdS_3$ defined by
the coordinates $u,x_0,x_1$ and wrapping the $\C\cP^1$ parameterized by $\theta_1,\phi_1$.
The isometry group of the metric (\ref{adscp3}), preserved by the D4-brane is $SO(2,2) \times
SU(3) \times U(1)$.  From a field theory viewpoint $SU(3)
\times U(1)$ is the unbroken R-symmetry and $SO(2,2)$ is the 2D
conformal group, suggesting that the dual field theory must be exactly
conformal. In the near horizon limit, one expects superconformal enhancement to twelve 
supercharges.\\

\noindent
Using a general ansatz for the embedding
surface as:
\be \label{x}
x = \frac{C}{u} \, \, ,
\ee
the induced metric can be seen to be $AdS_3$:
\be
ds^2  = (1 + C^2)\frac{du^2}{u^2} + u^2 [ -(dx^0)^2 + (dx^1)^2 ]
\ee
This leads to a shift in the curvature radius to $r^2 = \tilde{R}^2 (1 + C^2)$. Here, $C$ denotes
the minimum distance of the $AdS_3$ brane from the center. This distance is controlled depending on
whether some of the $D2$-branes end on a $D4$-brane and related the ratio of their
tensions~\cite{Karch:2000gx}. Similar to the various cases
of brane embeddings considered in~\cite{Karch:2000gx}, one can check that the situation where $q$ of the $N$ $D2$-branes end on the $D4$-brane, lead to a nonzero value for $C$. This is essentially because, as a reaction to being pulled on by the $D2$-branes, the $D4$-brane position along $x$ becomes a function of $u$.
Since this is also a co dimension 3 defect, the bending is of the form given in eqn.(\ref{x}). \\

\noindent
Notice that we are considering the case of an $M5$-brane in the AdS$_4$ $\times$ $S^7/\bZ_k$ geometry coming from $N$ $M2$ branes at a $\bZ_k$ singularity.  The intersecting
case, where none of the branes are intersecting corresponding to with a $C=0$ AdS$_3$ inside the AdS$_4$
times an equatorial $S^3$ inside the $S^7$. When there is an additional world-volume flux, the bending was argued in~\cite{Karch:2000gx} to go as ${ x=C/u^2}$, describing 
the embedding inside the AdS$_4$. It was further pointed out that this behavior is only
adequate far away from the intersection and that in the $M2$ near-horizon
region the $M5$-brane bending is different. From the type IIA picture, we see from the above analysis that this is indeed true and the bending is actually as given in eqn.(\ref{x}). 
The naive analysis above was all in the probe approximation and it is important to see how the situation
changes when the back reaction effects are considered. In particular, the embedded branes are sitting at 
the top of the potential, signaling a tachyonic instability. However, the mass of the mode is still above
the Breitenlohner-Freedman bound~\cite{Breitenlohner:1982bm} and hence is stable.\\


\noindent
The DBI part of the action of the embedded $D$-brane on $S^2$, the case where $\xi = 0$ takes the form
$\sqrt{1 + q^2}$, in the units $\tilde R = 1$. Similarly, the contribution of the DBI action from $AdS_3$
and the WZ-terms can be calculated. The RR 4-form reads 
\bea
 F^{(4)} &=&\frac{3R^3}{8}\epsilon_{AdS_4} \\
&=& \frac{3R^3}{8} u^2 dx^0 \wedge dx^1 \wedge dx^2\wedge du
\eea
where $ \epsilon_{AdS_4} $ is the unit volume form of the  ${AdS_4} $  space.
The RR 2-form $F^{(2)}=d\tilde{A}$
in the type IIA string is explicitly given by
\bea \label{2form}
F^{(2)}&=&
k( -\cos\xi\sin\xi d\xi \wedge 
(2d\psi+\cos\theta_1d\varphi_1-\cos\theta_2 d\varphi_2) \\ \nonumber 
&&-\frac{1}{2}\cos^2\xi\sin\theta_1 d\theta_1\wedge  d\varphi_1 -\frac{1}{2}\sin^2\xi\sin\theta_2 d\theta_2 \wedge  d\varphi_2 )  
\eea 
Using these, we get the full result:
\be
\mathcal L \approx   -T_{D4} \; \left ( R^2 \sqrt{1 + { \pi^2
q^2} } \; u \; \sqrt{u^4 + (u')^2}  - \;  q
\; u^3 \right )
\ee
Both the cases $q = 0$ and $q \neq 0$ exist and the difference is in the AdS curvatures. One can derive the equations of motion and check the validity of solutions. We will not try to do this, as we do it in full generality it the following section. 
%Here, we wish to
%see whether there is any tachyonic instability by putting the brane on at the $\theta_1 = \frac{\pi}{2}$ in a $\C\cP^1$ inside $\C\cP^3$. This can be checked as follows. We choose a $\C\cP^1 \subset \C\cP^3$ 
%and consider perturbing as
%$\xi=\pi/2+\delta(t)$. Evaluating the Born-Infeld action det$(g +{\cal F})$, where { \cal F} is the flux ${ {\cal F} = q  \; \sin\theta \;
%d\theta
%d \phi,}$
%and the metric is of the form 
%$ ds^2= {\tilde R}^2 \sin^2 \xi \left (d \theta_1^2+\sin^2\theta_1 d\Phi_1^2 \right)
%- {\tilde R}^2 \frac{C^2}{x^2} dt^2 + .... $. The relevant terms go as
%\be
%\sqrt{\det (g +{\cal F})} = {\tilde R}^3 \sqrt{1 + q^2} \left ( \frac{x}{2 C} \dot{\delta}^2 + 
%\frac{C}{x}\frac{\delta^2}{1+ q^2} \right )
%\ee
%Recognizing that the analysis is same as in~\cite{Karch:2000gx}, mass of the scalar one gets 
%is again $m^2 = -2/r^2$ above the Breitenlohner-Freedman bound~\cite{Breitenlohner:1982bm}.
%\be
%  {x^2 \over L^2 C^2} \dot{\delta}^2 + 2 {\delta^2 / L^2
%\over 1 + {\pi^2 q^2 \over k_N^2} } 
%\ee
Since the remaining part of spacetime is an $AdS_3$, one does not need to be
bothered by the tachyonic mode corresponding to fluctuations in $S^2$.\\

%Recognizing ${x^2 \over L^2 C^2}$ as $g^{00}$
%and
%using our result that
%$$l^2 = L^2 (1 + {\pi^2 q^2 \over k_N^2} )$$
%we see that $\delta$
%describes an excitation of
%\eqn{mass}{m^2 = -{2 \over l^2} }
%which is above the bound of ${- 9 \over 4 l^2}$ \cite{BF} 
%independent of $q$. 
\noindent
Let us now look at the full set of equations of motion specifically for a D4-brane embedding into a $AdS_4 \times \C\cP^3$ background.
Following the general procedure given in~\cite{Skenderis:2002vf}, for the world volume analysis of 
$Dp$-brane field equations in $AdS_n \times S^m$ space times, we discuss the embeddings of branes in
$CP^3$. A summary of the D-brane field equations is given in Appendix A.
Using (\ref{finfe}),  the $D4$-brane field equations in $AdS_4 \times \C\cP^3$ background, reduces to:
\bea 
e^{-\Phi}\partial_{i} (\sqrt{-M} \theta^{i i_1}) &=& \frac{1}{(2!)^2} 
\epsilon^{i_1 i_2 i_3 i_4 i_5 } F_{i_2 i_3}f_{ i_4 i_5 } + \frac{1}{4!} 
\epsilon^{i_1 i_2 i_3 i_4 i_5 } f_{i_2 i_3 i_4 i_5 }. \label{krfe1} 
\eea
\bea
&&\hspace{-25mm}\frac{1}{(2!)^3} 
\epsilon^{i_1 i_2 i_3 i_4 i_5 } (F_{i_1 i_2} \wedge F_{i_3 i_4})f_{i_5  m} +
\frac{1}{2! 3!} 
\epsilon^{i_1 i_2 i_3 i_4 i_5} F_{i_1 i_2} f_{i_3 i_4 i_5 m} \label{krfe2}   \\ \nonumber
&&= e^{-\Phi}\Big[- \partial_{i} (\sqrt{-M} G^{ij} \partial_{j} X^{n} g_{mn})  + 
\frac{1}{2} \sqrt{-M} (G^{ij} \partial_{i} X^{n} \partial_{j} X^{p} g_{np,m})\Big] . 
\eea
where $f_{i_3 i_4 i_5 m}$  denotes the pullback of $f$ on the first three indices, i.e. $f_{i_3 i_4 i_5 m} = \del_{i_3} X^{m_3} \del_{i_4} X^{m_4}
\del_{i_5} X^{m_5} f_{m_3 m_4 m_5 m}$ and in a similar manner for  $f_{i_5  m}$.
The solution set of these equations describes all possible
embeddings of D4-branes into the target space. Here we describe D4-branes which wrap on $\C\cP^1$ in the $\C\cP^3$.
%and the remaining world volume directions preserve the Poincare invarince.
Such embeddings can be realized from the following
ansatz: split the embedding coordinates $X^{m}$ into $\{ \eta^{i},
X^{\lambda} (\eta^{i}) \}$, where the worldvolume coordinates are
\be
\eta^{i} = \{ x^0, x^1,  u, \theta_1, \varphi_1 \} \label{ans1}
\ee
and the transverse scalars are 
\be
X^{\lambda} = \{ x^2(u) \equiv x(u), \xi, \psi , \theta_2, \varphi_2 \}, 
\label{ans2}
\ee
where  we relabel $x^2$ as $x$ and we assume
that the only dependence of the transverse scalars on the worldvolume
coordinates is in $x(u)$. We  switch on a worldvolume flux
\be  \label{flux}
F_{\theta_1 \varphi_1} = q \sin \theta_1.
\ee
Notice that this 2-form flux is proportional to the RR 2-form in eqn. (\ref{2form}) at the point $\xi =0$. 
With this ansatz, we proceed to  calculate all the
quantities appearing in (\ref{krfe1}) and (\ref{krfe2}); for example,
\be
\sqrt{-M} =\tilde{R}^3 u (1 + u^4 (x')^2)^{\frac{1}{2}} L_{(\theta_1 \xi)}
 \label{m1}
\ee
where prime denotes the derivative with respect to $u$ and we define,
\be
L_{(\theta_1 \xi)} =\Big(\tilde{R}^4 \cos^4\xi( \sin^2\xi \cos^2\theta_1 + \sin^2\theta_1 ) + q^2 \sin^2\theta_1  \Big)^{\frac{1}{2}}.
\label{lthetaxi}
\ee
Substituting the ansatz into (\ref{krfe2}), we find that the equations derived  from the $X^{m} = \{x^0, x^1,\psi, \theta_2, \varphi_1,\varphi_2\}$ equations are satisfied automatically by the ansatz. And the rest of the equations are discussed below.\\

\noindent
The $X^{m} = x^2 = x $ equation gives :
\bea
\partial_{u} 
\left ( \frac{L_{(\theta_1 \xi})} {(1 + u^4 (x')^2)^{\frac{1}{2}}} 
u^5 x' - q u^3 \sin \theta _1 \right ) = 0;
\eea
The $X^{m} = \xi $ equation leads to 
\bea
L^{-1}_{(\theta_1 \xi)} u (1 + u^4 (x')^2)^{\frac{1}{2}} \Big[ \cos^3\xi \sin\xi ( \cos^2\xi \cos^2\theta_1 - 2 \sin^2 \xi\cos^2\theta_1  - 2 \sin^2 \theta_1) \Big] = 0
\eea
 And the $X^{m} = \theta_1 $ equation leads to 
\bea
L^{-3}_{(\theta_1 \xi)} u (1 + u^4 (x')^2)^{\frac{1}{2}} \cos^4\xi \sin^2\xi \cos \theta_1 \sin \theta_1= 0
\eea
Moreover, the gauge field equation (\ref{krfe1}) 
%\bea \nonumber
%e^{-\Phi}\partial_{i} (\sqrt{-M} \theta^{i i_1}) &=& \frac{1}{(2!)^2} 
%\epsilon^{i_1 i_2 i_3 i_4 i_5 } F_{i_2 i_3}f_{ i_4 i_5 } + \frac{1}{4!} 
%\epsilon^{i_1 i_2 i_3 i_4 i_5 } f_{i_2 i_3 i_4 i_5 }
%\eea
leads to : %(for $ i_1 = \varphi _1 ,i = \theta_1$) (RHS is automatically  0)
\bea
 L^{-3}_{(\theta_1 \xi)} u (1 + u^4 (x')^2)^{\frac{1}{2}} \cos^4\xi \sin^2\xi \cos \theta_1 = 0
\eea
The  equation deriving from $u$ follows from the $x$-equation,
and that from $\theta_1 $ follows from the gauge field equation.
So, for the above ansatz, we find that the
only independent equations are the ones deriving from 
the $X^{m} = \{x, \xi\}$ equations along with the gauge field equation
(This is expected as worldvolume diffeomorphisms can be used
to eliminate $p+1 = 5$ equations. In addition the metric does not depend upon 3 coordinates $ \psi , \varphi_1, \varphi_2$.). \\
%[I kept them inside brackket as i am not very sure. since one equation is eliminated by the gauge field eqn.]

\noindent
Let us summarize the independent equations  
\bea
%x &:& \hspace{5mm} 
&&\partial_{u} 
\left ( \frac{L_{(\theta_1 \xi})} {(1 + u^4 (x')^2)^{\frac{1}{2}}} 
u^5 x' - q u^3 \sin \theta _1 \right ) = 0; \label{set1} \\
%\xi &:& \hspace{5mm}  
&&L^{-1}_{(\theta_1 \xi)} u (1 + u^4 (x')^2)^{\frac{1}{2}} \Big[ \cos^3\xi \sin\xi ( \cos^2\xi \cos^2\theta_1 - 2 \sin^2 \xi\cos^2\theta_1  - 2 \sin^2 \theta_1) \Big] = 0 ;\nonumber \\
\label{set2}
&& L^{-3}_{(\theta_1 \xi)} u (1 + u^4 (x')^2)^{\frac{1}{2}} \cos^4\xi \sin^2\xi \cos \theta_1 = 0
\label{gaugefield}
\eea
To solve (\ref{set1})-(\ref{gaugefield}) simultaneously, we first note that the $\xi$-equation (\ref{set2}) and the gauge field equation (\ref{gaugefield}) together can be solved  either when (i) $\xi= 0 $, which we will refer to as a maximal, as the radius of the $CP_1$ over which the brane is wrapped $\sim$  $\cos^2\xi$, or when (ii) $\xi= \frac{\pi}{2} $, which we will refer to as a minimal.
The $x$-equation  (\ref{set1}) yields
\be
x' = \frac{(qu^3 \sin \theta_1 -c)}{u^2\sqrt{ u^6 L_{(\theta_1 \xi)}^2 
- (qu^3\sin \theta _1 -c)^2}}, \label{xpm}
\ee
where $c$ is an integration constant.\\

\begin{flushleft}
{Branes wrapping CP: {\em Case A}}
\end{flushleft}

\noindent
Let us now substitute solutions of the $\xi$-equation and gauge field equation
into (\ref{xpm}). We focus first on the case of the brane wrapping are maximal
 i.e., the case when $\xi= 0 $.
In this case  (\ref{lthetaxi}) simplifies to,
\be
L_{(\theta_1 \xi)} =(\tilde{R}^4 + q^2)^{\frac{1}{2}} \sin \theta_1
\label{lthetaxi-max}
\ee
Using (\ref{lthetaxi-max}), (\ref{xpm}) reduces to 
\be
x' = \frac{(qu^3-c)}{u^2(\tilde{R}^4 u^6 + 2cqu^3-c^2)^{\frac{1}{2}} }, \label{xppm}
\ee
and the induced metric on the brane is
\be
ds^2 = \tilde{R}^2 \Big[ u^2 (dx \cdot dx)_2 + \frac{u^4 (1+q^2)}{(\tilde{R}^4 u^6 + 2cqu^3-c^2) }du^2 + (d\theta_1^2 + \sin^2 \theta_1 d\varphi_1^2) \Big],
\label{indmetric1}
\ee
Setting $\tilde{R}= 1$, the induced metric takes a form,
\be
ds^2 =  u^2 (dx \cdot dx)_2 + \frac{u^4 (1+q^2)}{(u^3 - u_{+}^3)
(u^3 + u_{-}^3) }du^2 + (d\theta_1^2 + \sin^2 \theta_1 d\varphi_1^2) ,
\label{indmetric2}
\ee
where we write
\be
(u^6 + 2cq u^3 - c^2) = (u^3 - u_{+}^3)(u^3 + u_{-}^3),
\label{poly}
\ee
with $u_{+}^3, u_{-}^3 \geq 0$. The explicit form for the roots
of (\ref{poly}) are
\bea 
u^3_{+} &=& - cq + \left | c \right | \sqrt{1+q^2}; \\
u^3_{-} &=& cq + \left | c \right | \sqrt{1+q^2}. \nonumber
\eea
\\
\noindent
Now, the $AdS_3 \times S^2$ embeddings can be found as follows. When $c = 0 $, (\ref{xppm}) integrates to the simple
expression
\be
x = x_{0} - \frac{q}{u}. \label{xq0} \, ,
\ee
This solution corresponds to the Karch-Randall embedding~\cite{KR}.
In this limit ,the induced metric is,
\be
ds^2 =  u^2 (dx \cdot dx)_2 + \frac{ (1+q^2)}{u^2 }du^2 + (d\theta_1^2 + \sin^2 \theta_1 d\varphi_1^2) ,
\ee
The embedded geometry is $AdS_3 \times S^2$ when $c=0$. Also note that the embeddings exist even in the zero  flux ($q=0$) limit. The zero flux embedding must satisfy the 
zero extrinsic curvature trace condition (\ref{kcon}) 
since for this solution $J_{m}$ vanishes.\\
\\


\noindent
An explicit calculation of  the components of the second fundamental form given in (\ref{secondf}) leads to the following nonzero $ {\cal K}^{m}_{ij}$ components while all other components vanish.
\bea
&&{\cal K}^{x^2}_{x^0x^0} = x'u^3 \, , \, {\cal K}^{x^2}_{x^1x^1} = -x'u^3\, , \,
{\cal K}^{x^2}_{uu} = \frac{-3x'}{u} - x'' \, , \, {\cal K}^{u}_{uu} = x'^2u^3 \\ \nonumber
&&{\cal K}^{\xi}_{\theta_1 \theta_1} = \frac{-1}{4} \cos\xi \sin\xi \, , \,
{\cal K}^{\xi}_{\theta_1 \varphi_1} = \frac{1}{4} \cos\xi \sin\xi [  \cos^2 \theta_1 \cos 2 \xi - \sin^2 \theta_1] \\ \nonumber
&&{\cal K}^{\psi}_{\theta_1 \varphi_1} = \frac{-1}{4} \cos^2 \xi \sin \theta_1 - \frac{\sin^2 \xi}{2 \sin \theta_1}- \frac{\cos^2 \theta_1}{\sin \theta_1} \cos^2 \xi \\ \nonumber
&& {\cal K}^{\varphi_1}_{\theta_1 \varphi_1} = \cot \theta_1 - \frac{1}{4}[3 + \cos 2\xi]  \cot \theta_1 \, , \,
{\cal K}^{\theta_1}_{ \varphi_1\varphi_1} = \cos \theta_1 \sin \theta_1 [\cos^2 \xi -1]
\eea
For consistency,
one can check that the trace of the second fundamental form of the embedding vanishes.
Note that when $q=0$ from (\ref{xq0}), $x$ being a constant, those $ {\cal K}^{m}_{ij}$ components vanish which are functions of $x'$ or $x''$. Again, for the specific embedding using $\xi = 0$, rest of the above listed components vanish except ${\cal K}^{\psi}_{\theta_1 \varphi_1}$. Thus, unlike the similar embeddings considered in~\cite{Skenderis:2002vf}, the ones
corresponding to $D4$-branes wrapping an $S^2$ in the present context are not totally
geodesic~\cite{Bachas:1999um}. This is due to the marked difference in the sphere and $\C\cP^3$ geometries. The second fundamental form is an important quantity when studying the possible corrections
to the DBI action at higher orders in $\alpha'$. 
Since, the second fundamental form does not vanish, it modifies the pull back of the ambient curvature
tensor and contributes to processes involving the scattering of closed and open strings.
\\


\noindent
We now see how the asymptotically $AdS_3 \times S^2$ embeddings look like.
For the general case when $c \neq 0$, the induced geometry is asymptotically $AdS_3 \times S^2$  for $u \gg u_{+}$. When $u < u_{+}$, (\ref{xppm}) implies that the brane ends at $u = u_{+}$. To get rid of singularity, changing variables to $u = u_{+} + \rho^2$ with $\rho \ll 1$ gives,
\be
ds^2 = u_{+}^2 (dx \cdot dx)_{2} + \frac{u_{+}^2 (1+q^2)}
{(u_{+}^3 + u_{-}^3)} d\rho^2 + (d\theta_{1}^2 
+ \sin^2 \theta_{1} d\varphi_1^2),
\ee
In fact, one can make another change of variables to see that the radial coordinate has a range as in~\cite{Skenderis:2002vf}, indicating a
mass gap in the dual theory.\\

\begin{flushleft}
{Branes wrapping CP: {\em Case B}}
\end{flushleft}

\noindent
Let us now discuss embeddings in which the brane wrapping are minimal i.e $\xi= \frac{\pi}{2}$.
 In this case (\ref{xpm}) reduces to
\be
x' = \frac{(qu^3-c)}{u^2 (2cq u^3 - c^2)^{\frac{1}{2}}}.
\ee
It is useful to rescale the parameter $c$ such that $c = C q$;
this removes all $q$ dependence in $x'$:
\be
x' = \frac{(u^3-C)}{u^2 (2C u^3 - C^2)^{\frac{1}{2}}}. \label{dif}
\ee
and the induced metric on the brane is then
\be
ds^2 = \tilde{R}^2 \Big[u^2 ( dx \cdot dx)_2+ \frac{u^4 du^2}{2 C (u^3 - \frac{C}{2})} \Big].
\ee
Note that $x'$ becomes imaginary for  $u^3 < \frac{1}{2} C = u_c^3$, which implies that the brane ends
at $u_{c}$. The induced geometry is non-singular at $u_{c}$
and the embedded hyper surface is  incomplete. \\

\noindent
The induced metric on the $\C\cP^1$ is degenerate. In order to interpret such embeddings in which the $S^2$ is minimal, let us look for D2-brane embedded in the $AdS_4$ and lie at a point in the $\C\cP^3$. In fact the D4-brane  has effectively collapsed to a D2-brane embedded in $AdS_4$, which can be verified deriving explicitly the solution of the D2-brane equations of motion.
The D2-brane field equations in the $AdS_4 \times \C\cP^3 $ target
space are
\be
\frac{1}{3!} \ep^{i_1i_2i_3} f_{i_1 i_2 i_3  m} =e^{-\Phi}\Big[
- \del_{i} (\sqrt{-M} G^{ij} \del_{j} X^{n} g_{mn})
+ \half \sqrt{-M} (G^{ij} \del_{i} X^{n} \del_{j} 
X^{p} g_{np,m})\Big].
\ee
Our ansatz for the worldvolume coordinates is
\be
\eta^{i} = \{ x^0, x^1,  u  \} 
\ee
while the transverse scalars are 
\be
X^{\lambda} = \{ x^2(u) \equiv x(u), \xi, \psi ,\varphi_1,\varphi_2,\theta_1, \theta_2  \}, 
\ee
Then the only equation of motion which is not trivially satisfied by the anatz is
\be
\del_{u} \left ( \frac{u^5 x'}{\sqrt{1+ u^4 (x')^2}} - u^3
\right ) = 0. \label{d3fe}
\ee
Note that this is precisely the $x$ field equation whereas the $u$ field equation follows from it as the previous case.
Since the general solution of (\ref{d3fe}) is (\ref{dif}), 
this implies that the  D4-brane wrapping
a minimal sphere can be interpreted as a D2-brane. When the flux on the D4-brane is positive we get a D2-brane, whereas negative flux corresponds to anti-D2 brane. These embeddings will be shown to break all
supersymmetry. A new ansatz describing the $D4$-branes wrapping the $\C\cP^1$ and whose worldvolume lie along $x$ is helpful to preserve supersymmetry. Let us consider,
\bea \label{new}
&&\eta^{i} = \{ x^0, x^1, x , \theta_1, \varphi_1 \}; \\ \nonumber
&&X^{\lambda} = \{u, \xi, \psi , \theta_2, \varphi_2 \}; \\ \nonumber
&&F_{\theta_1 \varphi_1} = q \sin \theta_1.
\eea
where all transverse scalars are constant.
Now we have :
\be
\sqrt{-M} =\tilde{R}^3 u^3 L_{(\theta_1 \xi)}
\ee
\be
L_{(\theta_1 \xi)} =\Big(\tilde{R}^4 \cos^4\xi( \sin^2\xi \cos^2\theta_1 + \sin^2\theta_1 ) + q^2 \sin^2\theta_1  \Big)^{\frac{1}{2}}.
\ee
The only field equations which are not already satisfied by the anasatz are 
\bea
u&:& \hspace{5mm} 
u^3 (L_{(\theta_1 \xi)} - q \sin \theta_1) = 0;\\
\xi &:& \hspace{5mm}  
L^{-1}_{(\theta_1 \xi)} u^3 \Big[ \cos^3\xi \sin\xi ( \cos^2\xi \cos^2\theta_1 - 2 \sin^2 \xi\cos^2\theta_1  - 2 \sin^2 \theta_1) \Big] = 0 ;\nonumber
\eea
And the gauge field equation,
\bea
L^{-3}_{(\theta_1 \xi)} u^3 \cos^4\xi \sin^2\xi \cos \theta_1 = 0 \, .
\eea
The  equation deriving from $\theta_1 $ follows from the gauge field equation.
As before the guage field equation and the the $\xi$ equation  can be solved  either when $(i)$ $\xi= 0 $  or when $(ii)$ $\xi= \frac{\pi}{2} $. In  case $(i)$, 
\be
L_{(\theta_1 \xi)} =(\tilde{R}^4 + q^2)^{\frac{1}{2}} \sin \theta_1 \, ,
\ee
which simplifies the $u$-equation as follows,
\be
u^3 ( (\tilde{R}^4 + q^2)^{\frac{1}{2}} - q ) \sin \theta_1 = 0 \, .
\ee
It has  a solution only for $u=0$ or $ \theta_1 = 0$, which signifies
that the only non-generate solution is  for minimal case. In case $(ii)$,
\be
L_{(\theta_1 \xi)} = q \sin \theta_1
\ee
So the $u$- equation is automatically satisfied. So there exists a solution  with non-zero flux $q$ for any $u_0$.
We use kappa symmetry projections to check the supersymmetry preserved by these embeddings in the next section. It
should be mentioned that kappa symmetry gauge fixing of the $D4$ brane embeddings considered in this paper from the world volume point of view is a subject which requires further study. For instance, for the case of fundamental strings and D2 branes wrapping 
$AdS_2 \times S^1$ and at the minkowski boundary of $AdS_4$ interesting gauges were discussed 
in~\cite{Grassi:2009yj}. 






\section{Supersymmetry of Embeddings}\label{Ap:Killing}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
In this section, we find BPS configurations of D-branes that are wrapped on compact
portions of our background, and are point like in the AdS. In order for the D-brane to be
supersymmetric, we only need to check that the kappa-symmetry conditions \cite{Cederwall:1996pv,Aganagic:1996pe,Cederwall:1996ri,
Bergshoeff:1996tu, Aganagic:1996nn, Bergshoeff:1997kr} 
\be
\Gamma \ep = \ep. \label{projc}
\ee
is satisfied, where $\ep$ is the Killing spinor corresponding to the unbroken supersymmetry
%(more precisely, the pullback onto the brane world volume). 
The projection matrix is
given by
\be
d^{p+1} \eta \Gamma = - e^{-\Phi} {\cal{L}}_{DBI}^{-1} 
e^{\cal F} \wedge X |_{vol}, \label{kap1} 
\ee
with $X = \bigoplus_{n} \G_{(2n+1)} \G_{11}^{n+1} \id $ and ${\cal{L}}_{DBI}^{-1}$  is to be 
evaluated on the background. Also, $\G_{(n)} = \frac{1}{n!} d\eta^{i_{n}} \wedge ... \wedge d\eta^{i_1} 
\G_{i_1...i_n} $ where $\G_{i_1...i_n}$ is the pullback for the target space
gamma matrices $\G_{i_1...i_n} =\del_{i_1} X^{m_1} ... \del_{i_n} X^{m_n} \G_{m_1...m_n}$.
$\G$ has the special property that it squares to one and 
is traceless. It follows that one can use $\G$ to project out 
half of the worldvolume fermions, thus preserving supersymmetry.
In type IIA case, the two 16-component Majorana�Weyl spinorial coordinates of opposite
chirality form a 32-component Majorana spinor $\psi^{\alpha}$, $\alpha = 1, .., 32$. 
In the Weyl basis, the corresponding gamma matrices are given by real block-off-diagonal matrices with a diagonal $\G_{11}$-matrix.
To find the relevant Killing spinor equation for this background we look at 
the supersymmetry transformation of the gravitino
\be
\delta\Psi_\mu=D_\mu\epsilon-\frac{1}{288}\Big(
\Gamma_\mu^{\,\n\l\r\s}
-8\delta_\mu^\nu\Gamma^{\lambda\r\sigma}\Big)F_{\n\l\r\s}\epsilon\,,\qquad
D_\mu\epsilon=\partial_\mu\epsilon+\frac{1}{4}\omega_\mu^{ab}\gamma_{ab}\epsilon\,.
\ee
%
The 4-form corresponding to the $AdS_4 \times S^7$ solution is
$F_{\nu\lambda\rho\sigma} = 6\, \varepsilon_{\nu\lambda\rho\sigma}$, 
where the epsilon symbol is the volume form on $AdS_4$ (so the indices take 
the values $0,1,2,3$). Plugging this into the variation
above one finds the Killing spinor equation
%
\be
D_\mu \epsilon = \frac{1}{2} \hat \gamma \gamma_\mu \epsilon
\ee
%
where $\m$ runs over all 11 coordinates, and $\hat \g =\g^{0123}$. Calculating the spin-connection from
$\omega^{\hat a \hat b}_{\mu} =e^{\hat a}_{\mu} ( \partial_{\mu} e^{\nu\hat b} + e^{\tau\hat b} \Gamma^{\nu}_{\tau \mu} ) $ we find for the $AdS_4$,
\bea
  &&\omega^{\hat u\hat u}_{u}= 0,\qquad\omega^{\hat u\hat0}_{0}= -u dx_0,\qquad
  \omega^{\hat u\hat1}_{1}=-u dx_1,\\
  &&\omega^{\hat u\hat2}_{2}=-u dx_2,
\eea
admits a full compliment of thirty-two independent solutions. 
We denote by $\g_a=e_a^m \G_m$ the tangent space gamma matrices.\\

\noindent
To proceed, here we will construct the Killing spinors preserved in the $AdS_4
\times \C\cP^3$ background starting from $AdS_4
\times S^7$ background. Some relevant calculations have been performed in
~\cite{Claus:1998,Lu:1998,Skenderis:2002vf}. We however repeat the calculations in our basis and also introduce
projection operators in this basis. We take the following form for the $AdS_4$
\bea
ds_{AdS4}^2 & = & \frac{du^2}{u^2} + u^2 (dx \cdot dx)_3 \\
&=&\frac{du^2}{u^2} + u^2 [ -(dx^0)^2 + (dx^1)^2 + (dx^2)^2 ]
\eea
The three-dimensional $N=6$ CS theory is conjectured to be dual to 
M-theory on $AdS_4\times S^7/\bZ_k$. To understand the action of the 
$\bZ_k$ orbifold, it is instructive to write the $S^7$ as a circle fibration over 
$\C\cP^3$, where the orbifold acts on the fiber.  
For large $k$ the radius of this ``M-theory circle'' becomes small, so the 
theory can be described in terms of type IIA string theory 
on $AdS_4\times \C\cP^3$ with the metric
\be
ds^2=\frac{R^3}{4k}\left(ds^2_{AdS_4}+4ds^2_{\C\cP^3}\right)\,.
\label{metric}
\ee
The metric on $\C\cP^3$ can be written in terms of four complex projective
coordinates $z_i$ as
\be
ds_{\C\cP^3}^2=\frac{1}{\rho^2}\sum_{i=1}^4 dz_i\,d\bar z_i
-\frac{1}{\rho^4}\bigg|\sum_{i=1}^4 z_i\,d\bar z_i\bigg|^2\,,\qquad
\rho^2=\sum_{i=1}^4|z_i|^2\,.
\ee
A specific representations in terms of angular 
coordinates is obtained by parameterizing $S^7/Z_k$ as~\cite{Cvetic:2000yp,Nishioka:2008gz},
$
z_1=\cos\frac{\xi}{2}\cos\frac{\vartheta_1}{2}\,e^{i(2\varphi_1+\psi+\zeta)/4}\,,
z_2=\cos\frac{\xi}{2}\sin\frac{\vartheta_1}{2}\,e^{i(-2\varphi_1+\psi+\zeta)/4}\,,
z_3=\sin\frac{\xi}{2}\cos\frac{\vartheta_2}{2}\,e^{i(2\varphi_2-\psi+\zeta)/4}\,,
z_4=\sin\frac{\xi}{2}\sin\frac{\vartheta_2}{2}\,e^{i(-2\varphi_2-\psi+\zeta)/4}\,
$
The metric on $S^7$ is then given by
\bea
ds^2_{S^7}&=&\frac{1}{4}\Bigg[
d\xi^2
+\cos^2\frac{\xi}{2}(d\vartheta_1^2+\sin^2\vartheta_1^2\,d\varphi_1^2)
+\sin^2\frac{\xi}{2}(d\vartheta_2^2+\sin^2\vartheta_2^2\,d\varphi_2^2)
\nonumber\\ \hskip1cm
&+&\sin^2\frac{\xi}{2}\cos^2\frac{\xi}{2}
(d\psi+\cos\vartheta_1\,d\varphi_1-\cos\vartheta_2\,d\varphi_2)^2
+\frac{1}{4}(d\zeta+A)^2\, \Bigg]\,,
\label{S7-metric}\\ \nonumber
A&=&\cos\xi\,d\psi+2\cos^2\frac{\alpha}{2}\cos\vartheta_1\,d\varphi_1
+2\sin^2\frac{\xi}{2}\cos\vartheta_2\,d\varphi_2\,.
\label{S7-form}
\eea
The angle $\zeta$ appears only in the last term and if we drop it 
we end up with the metric on $\C\cP^3$,
\bea
ds^2_{\C\cP^3}&=&\frac{1}{4}\bigg[
d\xi^2
+\cos^2\frac{\xi}{2}(d\vartheta_1^2+\sin^2\vartheta_1^2\,d\varphi_1^2)
+\sin^2\frac{\xi}{2}(d\vartheta_2^2+\sin^2\vartheta_2^2\,d\varphi_2^2) \nonumber
\\
&+&\sin^2\frac{\xi}{2}\cos^2\frac{\xi}{2}
(d\chi+\cos\vartheta_1\,d\varphi_1-\cos\vartheta_2\,d\varphi_2)^2\bigg].
\eea
%The ranges of the angles are
%$0\leq\xi,\vartheta_1,\vartheta_2\leq\pi$, 
%$0\leq\varphi_1,\varphi_2\leq2\pi$ and $0\leq\psi\leq4\pi$.
Although, the above metric is slightly different from the one of eqn. (\ref{cp3earlier}) used earlier  in obtaining
solutions, it can be seen later that it does not effect the analysis in this section.

\noindent
The vielbeins coming from $g_{\mu\nu} = e^{\hat a}_{\mu}e^{\hat a}_{\mu}\eta_{{\hat a}{\hat b}}$ for
$AdS_4$ are
\be
  e^{\hat u}_{u}=\frac{R}{2u}\,du,\qquad
  e^{\hat 0}_{0}= \frac{R}{2}u\,dx_0,\qquad e^{\hat 1}_{1}= \frac{Ru}{2}\,dx_1,\qquad
  e^{\hat 2}_{2}=\frac{Ru}{2}\,dx_2,
\ee
and for $S^7$,
\bea
e^4 &=&\frac{R}{2}  d\xi, \qquad 
e^5 =\frac{R}{2}  \cos\frac{\xi}{2}\,d\vartheta_1,\qquad 
e^6 =\frac{R}{2}  \sin\frac{\xi}{2} \, d\vartheta_2, \nonumber \\
e^7 &=& \frac{R}{2}  \cos\frac{\xi}{2}\sin\frac{\xi}{2} \Bigl( \cos \vartheta_1 \,
d\varphi_1 - \cos\vartheta_2\,d\varphi_2 + d\psi \Bigr),\nonumber \\
e^8 &=& \frac{R}{2}  \cos\frac{\xi}{2} \sin\vartheta_1\,d\varphi_1,\quad
e^9 =\frac{R}{2}  \sin\frac{\xi}{2}\sin\vartheta_2 \, d\varphi_2\,,\nonumber\\
e^\natural &=&-\frac{R}{4}\left(d\zeta+2\cos^2\frac{\xi}{2}\cos\vartheta_1\,d\varphi_1
+2\sin^2\frac{\xi}{2}\cos\vartheta_2\,d\varphi_2+\cos\xi\,d\psi\right).
\eea
Using the change of coordinates $u = e^r$, the killing spinor equation takes the form:
\be
\partial_{r}\epsilon -
\frac{1}{2} \hat \g \, \gamma_{r}\epsilon=0\, , \quad 
\partial_{p}\epsilon +
\frac{1}{2} \gamma_p(\hat \g + \gamma_{r})\epsilon=0\, ,
\ee
where $p$ runs over $0,1,2$. The solution of these equations is:
\be
\epsilon=e^{\frac{1}{2} r  \hat \g \gamma_r} \Big( 1+\frac{1}{2}\, \hat \g x^p \gamma_p
(1- \hat \g \gamma_r)\Big)\epsilon_0\ ,
\ee
where $\epsilon_0$ is an arbitrary constant spinor. 
%let us introduce the projections~\cite{Claus:1998}
%$\ep_{\pm} = {\cal P}_{\pm} \ep = \half (1 \mp \g^{0123} ) \ep.$ 
%Using these projectors we can rewrite the Killing spinor equations
%(\ref{eq1}) as 
%\be
%\del_{p} \ep_{\pm} \pm \frac{u}{2} \g_{p}(1 \pm \g_{3} ) \ep_{\pm} = 0,  \quad \del_{3} \ep_{\pm} \pm \half u^{-1} \g_3 \ep_{\pm} = 0 ,\nonumber \\
%\ee
%where $p$ runs over $0,1,2$ and $u=3$.
%The solution to the Killing spinor equation for $\ep_+$ is, 
%\be
%\ep_{+} =  u^{\frac{1}{2}}  h(\C\cP^3) \eta_2, \quad {\rm or} \quad
%\ep_{+} =  h(\C\cP^3) (u^{-\frac{1}{2}} - u^{\frac{1}{2}} x \cdot \gamma 
%)\eta_1 ,  \label{epp1}
%\ee
%where $\eta_1$ and $\eta_2$ are constant spinors, satisfying
%$ \gamma_3 \eta_1 =  \eta_1, \gamma_3 \eta_2 = - \eta_2 $. Similar solutions 
%can be obtained for $\ep_-$ as well with two constant spinors $\lambda_1$ 
%and $\lambda_2$.
%That is,
%we can choose
%$ \eta_{1} = \lambda -  \g^{012} \lambda $ and
%$\eta_2 = \eta +  \g^{012} \eta, $
%Here, $\lambda$ and $\eta$ are real spinors of negative and 
%positive chirality respectively,
%with $16$ independent components. So, this gives us $32$ independent
%real spinors.
% Such a choice
%makes the complex conjugation of the spinors manifest. 
The full solution for the killing spinor also includes the function
$ h(\C\cP^3)$, which appears from solving the Killing
equation on $ \C\cP^3$. A quicker way to obtain this, is
the equation on the sphere given as
\be
(D_{\alpha} + \half  \Gamma_{\alpha} )\ep = 0,
\ee
where $D_{\alpha}$ is the covariant derivative on $S^7$,
and the solution is given by
\bea
h(\C\cP^3) =  e^{\frac{\xi}{4} ( \hat \g \g_4 - \g_{7\natural}   ) }
e^{\frac{\vartheta_1}{4} (  \hat \g \g_5 - \g_{8\natural}  ) }
e^{\frac{\vartheta_2}{4} ( \g_{79} + \g_{46} ) }
e^{-\frac{\xi_1}{2} \hat \g \g_\natural}
e^{-\frac{\xi_2}{2} \g_{58}}
e^{-\frac{\xi_3}{2} \g_{47}}
e^{-\frac{\xi_4}{2} \g_{69}}
\label{killcp}
\eea
%
where the $\xi_i$ are given by
$\xi_1=\frac{2\varphi_1+\chi+\zeta}{4}\,,\qquad
\xi_2=\frac{-2\varphi_1+\chi+\zeta}{4}\,,\qquad
\xi_3=\frac{2\varphi_2-\chi+\zeta}{4}\,,\qquad
\xi_4=\frac{-2\varphi_2-\chi+\zeta}{4}\,$.
The Dirac matrices were chosen such that
$\gamma_{012345678 9\natural}=1$. Similar calculations in different coordinate systems were done in 
\cite{Nishioka:2008ib,Drukker:2008zx}. 
To see which Killing spinors survive the orbifolding, one writes the spinor 
$\epsilon_0$ in a basis which diagonalizes
$i\hat\gamma\gamma_\natural\epsilon_0=s_1\epsilon_0\,,\qquad
i\gamma_{58}\epsilon_0=s_2\epsilon_0\,,\qquad
i\gamma_{47}\epsilon_0=s_3\epsilon_0\,,\qquad
i\gamma_{69}\epsilon_0=s_4\epsilon_0\,$. 
All the $s_i$ take values $\pm1$ and by our conventions on the product 
of all the Dirac matrices, the number of negative eigenvalues is even. 
Now consider a shift along the $\zeta$ circle, which changes all the 
angles by $\xi_i\to\xi_i+\delta/4$, the Killing spinors transform 
as
\be
{\mathcal M}\epsilon_0\to {\mathcal M} e^{i\frac{\delta}{8}
(s_1+s_2+s_3+s_4)}\epsilon_0\,.
\ee
This transformation is a symmetry of the Killing spinor when two of the $s_i$ eigenvalues 
are positive and two negative and not when they all have the same sign (unless $\delta$ 
is an integer multiple of $4\pi$). Note that on $S^7$ the radius of the $\zeta$ circle 
is $8\pi$, so the $\bZ_k$ orbifold of $S^7$ is given by taking $\delta=8\pi/k$. 
The allowed values of the $s_i$ are therefore
\be
(s_1,s_2,s_3,s_4)\in\left\{
\begin{matrix}
(+,+,-,-),\ (+,-,+,-),\ (+,-,-,+),\\
(-,+,+,-),\ (-,+,-,+),\ (-,-,+,+)
\end{matrix}
\right\}
\label{signs}
\ee
Each configuration represents four supercharges, so the orbifolding 
breaks $1/4$ of the supercharges (except for $k=1,2$) and leaves 
24 unbroken supersymmetries.\\
\noindent
The supersymmetry of the $AdS_3 \times 
CP^1$ brane embeddings discussed in the previous section is checked as follows. The explicit form of the kappa symmetry projection is
\be \label{kappacon}
\ep = -\frac{1}{u^3 (1+q^2)} \g^{01}  
\left ( (qu^3-c) \g^2 + \sqrt{(u^6 + 2 c q u^3 - c^2)} \g^3 \right ) 
\left ( -\g_{58\natural} + q \right)\ep \, . 
\ee
As in~\cite{Skenderis:2002vf} the projector $\Gamma$ involving the flux on the worldvolume of the brane
is considered for convenience. Preservation of supersymmetry requires that this condition must
be satisfied for some subset of the background Killing spinors
at all points on the brane worldvolume. In particular, it
must hold at all values of $x^p = (x^0,x^1,x^2)$. Below, we explicitly calculate the supersymmetry preserved
by the $c=0$ embeddings and argue that $c \neq 0$ ones break all supersymmetry.
Before proceeding, it is necessary to project the solutions for the killing spinors on to the
brane world volume. This is done as follows. Using the solution in eqn. (\ref{xq0}) and after a few manipulations, it is convenient to write the killing spinors as:
\be \label{kill}
\epsilon= \Big( e^{\frac{1}{2} r  \hat \g \gamma_r} + \frac{1}{2}e^{\frac{1}{2} r  \hat \g \gamma_r} \, 
(\g_r + \hat \g) (x^l \gamma_l + \tilde x \gamma_2)
- \frac{q}{2} e^{-r}(\g_r + \hat \g) \g_2 \Big)\,  h(\C\cP^3)\, \epsilon_0\ ,
\ee
where $l=0,1$. From the terms in
the Killing spinors which are linear in $x^l$ we find
the following condition:
\bea \label{k1}
&&e^{\frac{1}{2} r  \hat \g \gamma_r}(\g_r + \hat \g)\,\g_l\, h(\C\cP^3)\, \epsilon_0 \nonumber \\
&&=  - \frac{1}{ (1+q^2)} \g_{01}  \,\Big( e^{-\frac{1}{2} r  \hat \g \gamma_r} q( \g_3 - \g_{258\natural}) +
e^{\frac{1}{2} r  \hat \g \gamma_r} (-\g_{358\natural} + q^2 \g_2)
\Big)\, (\g_r + \hat \g)\,\g_l\, h(\C\cP^3)\, \epsilon_0 \, .
\eea
To proceed further, we start by noticing that terms in the function $h(\C\cP^3)$ in (\ref{killcp}), which do not contain gamma 
matrices cancel generally. For the rest of the terms in  $h(\C\cP^3)$, one checks that $\Gamma$ commutes
with $\hat\g \g_5$ and $\g_{8\natural}$. Further, one can multiply by the inverse of $h(\C\cP^3)$ as,
\be
{h(\C\cP^3)}^{-1} (1- \Gamma) h(\C\cP^3) \epsilon_0 = 0 \, ,
\ee
to eliminate $h(\C\cP^3)$ completely from the equations. The eqn. (\ref{k1}) gives two sets of conditions:
\be \label{k4a}
 \frac{1}{ (1+q^2)} \g_{01} (\g_3 + \g_{258\natural}) (\g_r + \hat \g) \epsilon_0= 0 \, , 
\ee
and 
\be \label{k4b}
 \frac{1}{ (1+q^2)} \g_{01}  ( \g_{358\natural} + q^2 \g_2)(\g_r + \hat \g)\epsilon_0  = (\g_r + \hat \g)\epsilon_0 \, .
\ee
To solve eqns. (\ref{k4a}) and  (\ref{k4b}), let us introduce the projections,
\be \label{proj}
 \g_{012} \epsilon_0 = \pm \epsilon_0^{\pm} \, .
\ee
Using eqn. (\ref{proj}), both the conditions (\ref{k4a}) and  (\ref{k4b}) are solved by introducing the
projection,
\be \label{r1}
(1 + \g_{3258\natural})\, \epsilon_0^{+} = 0 \,
\ee
where, $\epsilon_0^{-}$ disappears from the equations. Thus, if one introduces the projections
$\g_{3258\natural} \epsilon_0^{+} = \pm \eta_{\pm}$, then eqn. (\ref{r1}) means that $\eta_+$ 
is eliminated and $\eta_-$ remains undetermined at this level.  Eqn. (\ref{r1}) will be analyzed further below. \\

\noindent
The remaining terms from (\ref{kill}) give the following kappa symmetry projection:
\bea \label{k2}
&&\left[e^{\frac{1}{2} r  \hat \g \gamma_r}( 1 + \frac{1}{2}\tilde x (\g_r + \hat \g) \g_2 ) 
- e^{-\frac{1}{2} r  \hat \g \gamma_r}\frac{q}{2}(\g_r + \hat \g) \g_2\right]\, h(\C\cP^3)\, \epsilon_0 \\ \nonumber 
&&= - \frac{1}{ (1+q^2)} \g_{01} \left[ \{e^{-\frac{1}{2} r  \hat \g \gamma_r} q( \g_3 - \g_{258\natural})
+ e^{\frac{1}{2} r  \hat \g \gamma_r} (-\g_{358\natural} + q^2 \g_2)\} ( 1 + \frac{1}{2}\tilde x (\g_r + \hat \g) \g_2 ) \right. \\ \nonumber
&& \left. - \{ e^{\frac{1}{2} r  \hat \g \gamma_r} q( \g_3 - \g_{258\natural}) + e^{-\frac{1}{2} r  \hat \g \gamma_r} (-\g_{358\natural} + q^2 \g_2)\} \frac{q}{2}(\g_r + \hat \g) \g_2\right]\, h(\C\cP^3)\, \epsilon_0 \, .
\eea
The kappa-symmetry projection condition (\ref{kappacon}) has to be satisfied at all points on the
brane world-volume. Commuting  $h(\C\cP^3)$ as before, thus eqn. (\ref{k2}) gives rise to two independent conditions,
%i.e., the condition coming from terms proportional to $e^{-r/2}$ takes the form:
%\be \label{k2}
%(\g_r + \hat \g)\, h(\C\cP^3)\, \epsilon_0 =  - \frac{1}{u^3 (1+q^2)} \g^{01}  \,
%(-\g_3 + q\g_2)(\g_{58\natural} + q)\, (\g_r + \hat \g)\, h(\C\cP^3)\, \epsilon_0
%\ee
%Now, we specialize to the case $\xi =0, \theta_2=0$on the brane worldvolume to simplify the
%analysis. 
\bea \label{k3a}
&&\epsilon_0^{-} + ( 1 + \tilde x \g_{32} )\epsilon_0^{+}  \\ \nonumber
&&= - \frac{1}{ (1+q^2)} \g_{01} \left((-\g_{358\natural} + q^2 \g_2) (\epsilon_0^{-} + ( 1 + \tilde x \g_{32} )\epsilon_0^{+}) - q^2  (\g_3 - \g_{258\natural}) \g_{32} \epsilon_0^{+}
\right)
\eea
and
\be \label{k3b}
q \g_{32} \epsilon_0^{+} = \frac{1}{ (1+q^2)} \g_{01} \left( q  (\g_3 - \g_{258\natural})(\epsilon_0^{-} + ( 1 + \tilde x \g_{32} )\epsilon_0^{+}) - q (-\g_{358\natural} + q^2 \g_2) \g_{32} \epsilon_0^{+}
\right)
\ee
If one introduces the projections,
\be \label{r2}
 \g_{3258\natural}\, \epsilon_0^{-} = \pm \lambda_{\pm} \, ,
\ee
then both the eqns. (\ref{k3a}) and (\ref{k3b}) can be solved by using the relation,
\be
2 \lambda_{-} = - \tilde x\, \g_{32} \eta_- \, .
\ee
$\lambda_+$ is undetermined as well.  Now, let us analyze the conditions (\ref{r1})
and (\ref{r2}) further. If  $s_1 = s_2$, then
\be \label{s}
\g_{58\natural} \epsilon_0 = \hat \g \epsilon_0 \, ,
\ee
turning the projection condition (\ref{r1}) in to,
\be \label{r3}
\g_{01} \epsilon_0 =   \epsilon_0 \, .
\ee
On the other hand, if $s_1 = - s_2$, the condition is,
\be \label{r4}
\g_{01} \epsilon_0 =  - \epsilon_0 \, .
\ee
Notice that $s_1 = s_2$ is satisfied by eight supersymmetries and they satisfy the projection in eqn. (\ref{r3}).
When $s_1 = - s_2$, there are sixteen supersymmetries and they satisfy the projection conditions (\ref{r4}).
Thus, one sees that twelve of the supersymmetries are 
preserved and half of the target space supersymmetry is broken by these embeddings. As in the
$AdS_5 \times S^5$, the projections do not depend on the flux, but, on the value of $\tilde x$. Thus,
each extra defect breaks the supersymmetry further. \\

\noindent
For $c \neq 0$, one has different type of conditions. For instance, one ends up with conditions of the form:
\be
 -\frac{1}{u^3 (1+q^2)} \g^{01}  
\left ( - (qu^3-c) \g^{258\natural} + q \sqrt{(u^6 + 2 c q u^3 - c^2)} \g^3 \right )h(\C\cP^3) \, \epsilon_0 = 0
\ee
From inspection, one sees that these conditions cannot hold at all points on the brane world volume. Thus,
the $c\neq 0$ embeddings corresponding to asymptotically $AdS_3 \times S^2$ branes, break all supersymmetry. \\
\noindent
For the embeddings corresponding to {\em case B}, the kappa-symmetry projection leads to:
\be
\epsilon = - \g_{01}\Big( \frac{\sqrt{2Cu^3 - C^2}}{u^3} \g_3 + (u^3-C)\g_2  \Big) \epsilon
\ee
Using the form of the killing spinors in eqn. (\ref{kill}), we end up with the following conditions:
\be
\frac{\sqrt{2Cu^3 -C^2}}{u^3} \g_{013} \, h(\C\cP^3) \, \epsilon_0 = 0\, , \qquad
(u^3 - C)\g_{012} h(\C\cP^3) \, \epsilon_0 = 0
\ee
Clearly, these conditions cannot be satisfied for any non zero $C$ and  $\epsilon_0$. Thus, there are no
non zero solutions to the kappa symmetry projections on the world volume and hence these embeddings break
all supersymmetry as well, as in the cases considered in~\cite{Skenderis:2002vf}. Even in the present
case, one attribute the breaking of all supersymmetries to the misaligned branes in the $AdS_4\times \C\cP^3$ background. For instance, if one had considered an embedding of the form given in eqn. (\ref{new}),
the induced metric in this case is very simple:
\be
ds^2 = u^2 (dx.dx) \, .
\ee
In this case, the kappa symmetry projection is straightforward, giving:
\be \label{sgn}
\ep = - sgn(q) \, \g_{012} \ep \, .
\ee
Note that for the sign of $\Gamma$ we have chosen, only embeddings which satisfy $(1-\Gamma)\epsilon = 0$ respect supersymmetry and those with $(1+\Gamma)\epsilon = 0$ break all supersymmetries. From 
eqn. (\ref{sgn}), one
can check that, for the embeddings with $q = +1$, the condition in eqn. (\ref{r1}) is enough and hence, half of the supersymmetry is preserved. The ones with $q = -1$ break
all supersymmetry, as there is an over all negative sign, incompatible with our choice of $\Gamma$.
Although, one expects that in the plane wave limit, both the signs of charges are compatible with
the kappa symmetry projection. It would be interesting to further the analysis of this work to 
the plane wave limit and study the possible brane embeddings in that context.
\\


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}\label{conclusion}

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


In this paper, we discussed various brane embeddings in $AdS_4\times \C\cP^3$ and the supersymmetry preserved 
by them. In the case of duality between Type IIB string theory on $AdS_5 \times S^5$ and $N=4$ super Yang-Mils system, there exists a good notion of renormalization group flow on the defect CFT, coming from
the embedded branes.  The defect theory should be associated with the boundary of the $AdS_3$ of the $AdS_3 \times S^2$ D4-brane. We expect such RG flows to hold for the present case as well, when the defect theory
is conformal. As in the examples considered in~\cite{Skenderis:2002vf}, the D4-brane theory does not back react on the bulk. Thus, the deformations in which the boundary theory
remains conformal but the defect theory runs are possible. An example is the
asymptotically $AdS_3 \times S^2$ embeddings.\\

\noindent
Since, the active scalar in our embeddings behaves at large $u$ as:
\be
u x \approx u x_0 + \frac{c}{4u^3} \, ,
\ee
one expects that the scalar is dual to an operator $O_{x}$ of conformal dimension $3$ in the defect theory.
Since, the operator $O_{x}$ is sourced by the scalar $x$ with vev $c$, it is expected to break supersymmetry.\\

\noindent
Thus far in our discussion, the branes were considered to be probes. It would be interesting to consider the
back reaction of the $D4$-brane in the target space geometry. In this case, one also has to find fully localized brane solutions in $AdS \times CP$ backgrounds. A fully localized solution is also important to understand the back reaction effects of the embeddings considered. So far we considered a single $D4$-brane, whose backreaction on the near-horizon geometry
can be neglected in the 't Hooft limit, allowing it to be treated as a probe hosting open
strings. Thus, it is not possible to get insight
about the dynamics of boundary versus bulk modes in the CFT coming from gravity fluctuations.
Further, intersecting brane solutions in $AdS \times CP$ backgrounds could be constructed to get 
insights in to the spectrum of defect conformal field theory. It would be interesting to construct
the defect CFT and understand the correlation functions along the lines of~\cite{Sethi:1997zz}-\cite{ooguri2}.
Although we considered specific embeddings in the 
$AdS_4\times \C\cP^3$ background, it is possible to consider and classify other embeddings.
The classification of embeddings can be done following the analysis in~\cite{Skenderis:2002vf},
using the number of intersection directions of Neumann and Dirichlet directions.
Supersymmetry preserved by the brane configurations depends on the coordinate splitting. When they are
sitting at arbitrary positions of the transverse space they preserve one quarter of
supersymmetry, but they preserve twelve supercharges when located at the origin of the transverse
space.


\noindent
\acknowledgments
We thank Alok Kumar for useful discussions and the organizers of String Theory and Fundamental Physics, Kanha, where some part of this work was done. B.C. thanks IFT, Sao Paulo for support, IOP, Bhubaneswar, HRI, Allahabad and MPI, Halle, for hospitality, during the course of this work. B.C. is supported by the CNPQ grant 150417/2009-8. B.C. would also like to thank D. Sorokin for useful discussions. B. P. would like to thank the Theory Division of CERN physics department for their hospitalities during the course of the work. The work of B. P. is supported by the European Commision under the ERC Advanced Grant $226371$. 

\medskip

\appendix

\section{D-brane Field equations}
In this appendix, we summarize the D-brane field equations derived in all generality and for all Dp-branes in~\cite{Skenderis:2002vf}. 
The worldvolume action for a single $Dp$-brane is given by
\bea
I_{p} &=& I_{DBI} +  I_{WZ}  \label{dbiact} \\
I_{DBI} &=&
=- T_{p} \int_{M} d^{p+1}\eta e^{-\Phi} 
\sqrt{-\det \left (g_{ij} + {\cal{F}}_{ij} \right )}, \nonumber \qquad
I_{WZ} = T_{p} \int_{M} e^{\cal{F}} \wedge C, \nonumber
\eea
with $T_p$ the Dp-brane tension. 
Here $\eta^i$ are the coordinates of the $(p+1)$-dimensional
worldvolume with metric $g_{ij} = g_{mn} \del_{i} X^{m} \del_{j} X^{n}$, following from
the space-time string frame metric $g_{mn}$. Worldvolume field strength ${\cal{F}} = F - B$ is the gauge invariant two-form with 
$B_{ij}=\del_i X^m \del_j X^n B_{mn}$ the pullback of the target 
space NS-NS 2-form.
%The gauge field equation is 
%\be
%J^{j}\equiv \d I_{WZ}/\delta A_j = \del_{i} (e^{-\Phi} \sqrt{-M} \theta^{ij}), \label{gfe}
%\ee
Finally, we summarize the D-brane field equations
\bea
&& \del_{i} (e^{-\Phi} \sqrt{-M} \theta^{ii_1}) =  
 \ep^{i_1..i_{p+1}} \sum_{n \ge 0} \frac{1}{n! (2!)^n (q-1)!} 
({\cal F})^{n}_{i_2...i_{2n+1}} 
\bar{F}_{i_{2n+2}... i_{p+1}}. \label{finfe} \\
&&\hspace{-10mm}\sum_{n \ge 0} \frac{1}{n! (2!)^n q!} 
\ep^{i_1..i_{p+1}}
({\cal F})^{n}_{i_1..i_{2n}} \bar{F}_{m i_{2n+1}... i_{p+1}}=
e^{-\Phi} \left( 
\sqrt{-M} \left ( G^{ij} \del_{i} X^{p} \del_{j} X^{n} g_{mn} 
\Phi_{,p} - \Phi_{,m} \right ) - {\cal K}_m \right). \nonumber 
\eea
where 
\be
{\cal K}_m=-\del_{i} (\sqrt{-M} G^{ij}) \del_{j} X^{n} g_{mn} 
-\sqrt{-M} M^{ij} \left ( (\del_{i} \del_{j} X^{n}) g_{mn} 
   + \tilde{\Gamma}_{mnp} \del_{i} X^{n} \del_{j} X^{p} \right ) 
\ee
\be
\bar{F}_{m_1..m_{q+1}} = f_{m_1..m_{q+1}} - \frac{(q+1)!}{3! (q-2)!}
H_{[m_1..m_3} C_{m_4..m_{q+1}]}.
\label{rrfe}
\ee
where the following notations are introduced,
\be
M_{ij} = (\del_{i} X^{m} \del_{j} X^{n} g_{mn} - 
\del_{i} X^{m} \del_{j} X^{n} B_{mn} + F_{ij}).
\ee
and the inverse as $M^{ij}$ such that $M^{ij} M_{jk} = \delta^{i}_{\hspace{2mm} k}$.
Moreover, $G^{ij} \equiv M^{(ij)}$ 
and $\theta^{ij} \equiv M^{[ij]}$.
And we have $\tilde{\Gamma} = \G - \half H$, with  $\G_{mnp}$ is the Levi-Civita connection of the target space
metric and $H_{mnp}$ is the field strength of the
NS-NS two form.\\
For the special case, when $F_{ij} = B_{mn} = \Phi = 0$, one has,
\be
J^{m}=- \sqrt{-g} g^{ij} {\cal K}^{m}_{ij}
\ee
where 
\be \label{secondf}
{\cal K}^{m}_{ij}=\gamma^{k}_{ij} \del_{k} X^{m}
- (\del_{i} \del_{j} X^{m}) - \Gamma^{m}_{np} \del_{i}X^{n} \del_{j} X^{p}
\ee
is the second fundamental form ($\gamma^k_{ij}$ is the Levi-Civita 
connection of the induced worldvolume metric).
If in addition $J_{m} = 0$, the field equation becomes
\be
g^{ij} {\cal K}^{m}_{ij} = 0, \label{kcon}
\ee
that is, the trace of the second fundamental form of the embedding
must be zero.

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\end{thebibliography}




\end{document}



















































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\bea
ds^2_{S^7}&=&\frac{1}{4}\left[
d\alpha^2 + \cos^2\alpha d\beta^2 +\cos^2\alpha\cos^2\beta d\gamma^2 + \sin^2\alpha d\xi_1^2 
\cos^2\alpha\sin^2\beta d\xi_2^2 \right. \\ \nonumber
&+& \left.  \cos^2\alpha\cos^2\beta\sin^2\gamma d\xi_3^2 +  \cos^2\alpha\cos^2\beta\cos^2\gamma d\xi_4^2 \right]
\label{S7-form}
\eea
The vielbeins coming from $g_{\mu\nu} = e^{\hat a}_{\mu}e^{\hat a}_{\mu}\eta_{{\hat a}{\hat b}}$ for
$AdS_4$ are
\be
  e^{\hat u}_{u}=\frac{R}{2u}\,du,\qquad
  e^{\hat 0}_{0}= \frac{R}{2}u\,dx_0,\qquad e^{\hat 1}_{1}= \frac{Ru}{2}\,dx_1,\qquad
  e^{\hat 2}_{2}=\frac{Ru}{2}\,dx_2,
\ee
and for $S^7$,
\bea
&&e^{\hat \alpha}_{\alpha}=R d\alpha,\quad e^{\hat \beta}_{\beta}=R\cos\alpha d\alpha,\\ \nonumber
&&e^{\hat \gamma}_{\gamma} =R\cos\alpha \cos\alpha d\gamma, \qquad e^{\hat \xi_1}_{\xi_1} =R\sin\alpha d\xi_1,\qquad e^{\hat \xi_2}_{\xi_2}=R\cos\alpha\sin\beta d\xi_2,\\ \nonumber
&&e^{\hat \xi_3}_{\xi_3}=R\cos\alpha\cos\beta\sin\gamma d\xi_3,\qquad
e^{\hat \xi_4}_{\xi_4}=R\cos\alpha\cos\beta\cos\gamma d\xi_4,
\eea
and the spin connections calculated from 
$\omega^{\hat a \hat b}_{\mu} =e^{\hat a}_{\mu} ( \partial_{\mu} e^{\nu\hat b} + e^{\tau\hat b} \Gamma^{\nu}_{\tau \mu} ) $ becomes
\bea
  &&\omega^{\hat u\hat u}_{u}= 0,\qquad\omega^{\hat u\hat0}_{0}= -u dx_0,\qquad
  \omega^{\hat u\hat1}_{1}=-u dx_1,\\
  &&\omega^{\hat u\hat2}_{2}=-u dx_2,\qquad  \omega^{\hat \alpha\hat\beta}_{\beta}=\sin\alpha d\beta,\qquad
  \omega^{\hat \alpha\hat\gamma}_{\gamma}=\sin\alpha\cos\beta d\gamma,\qquad \omega^{\hat \beta\hat\gamma}_{\gamma}=\sin\beta
  d\gamma, \\
  &&\omega^{\hat \alpha\hat\xi_1}_{\xi_1}=-\cos\alpha d\xi_1,\qquad \omega^{\hat \alpha\hat\xi_2}_{\xi_2}=\sin\alpha\sin\beta
  d\xi_2,\qquad \omega^{\hat \beta\hat\xi_2}_{\xi_2}=-\cos\beta d\xi_2,\qquad \\
  &&\omega^{\hat \alpha\hat\xi_3}_{\xi_3}=\sin\alpha\cos\beta\sin\gamma d\xi_3,\qquad
 \omega^{\hat \beta\hat\xi_3}_{\xi_3}=\sin\beta\sin\gamma d\xi_3,\qquad \omega^{\hat \gamma\hat\xi_3}_{\xi_3}=-\cos\gamma
  d\xi_3,\\
  &&\omega^{\hat \alpha\hat\xi_4}_{\xi_4}=\sin\alpha\cos\beta\cos\gamma d\xi_4,\qquad
  \omega^{\hat \beta\hat\xi_4}_{\xi_4}=\sin\beta\cos\gamma d\xi_4,\qquad 
  \omega^{\hat \gamma\hat\xi_4}_{\xi_4}=\sin\gamma d\xi_4.
\eea

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\end{thebibliography}