Intersecting branes and adding flavors to the MaldacenaNŭnez background
Abstract:
Following a proposal in ref.[3], we study adding flavors into the MaldacenaNŭnez background. It is achieved by introducing spacetime filling D9branes or intersecting D5branes into the background with a wrapping D5brane. Both D9branes and D5branes can be spacetime filling from the 5D bulk point of view. At the probe limit it corresponds to introducing nonchiral fundamental flavors into the dual SYM. We propose a method to twist the fundamental flavor which has to involve open string charge. It reflects the fact that coupling fundamental matter to SYM in the dual string theory corresponds to adding an open string sector.
1 Introduction
It is expected that 4d pure YangMills theory, with gluon as degrees of freedom, is mapped to an entirely 5d noncritical closed string theory background[1]. This expectation has been realized in many supersymmetric examples starting from a proposal on AdS/CFT correspondence[2]. In these examples we knew the geometry of the background. Although it is difficult to quantize string theory on the background, it was conjectured that at the large limit, the super YangMills theories (SYM) are dual of the low energy effective theories of a closed string theory, i.e., supergravity(SUGRA), on various backgrounds. When we add massless fundamental flavors to SYM, however, it is very difficult to get the dual background. Adding fundamental flavors effectively introduces an open string sector to the string theory. Light open string states inevitably modify the original closed string background. In other words, Dbranes, which generate nontrivial background, have to carry open string charge besides the closed string charge (RR charge).
It was noted by Karch and Katz[3] that this difficulty can be partially overcome at certain probe limit. They proposed an effective method to add flavors into AdS/CFT correspondence via introducing few D7brane probe into . The key points of their proposal can be summarized as follows:

Adding fundamental flavors in the gauge theory is equivalent to adding spacetime filling Dbranes into the 5d bulk theory.

It is not necessary to introduce orientifold in order to cancel the RR tadpole of spacetime filling Dbranes. Spacetime filling Dbranes can wrap on a topological trivial cycle inside the original background. This way they may not carry any charge and avoid a RR tadpole.

At the probe limit, the effect of Dbranes on the bulk geometry can be ignored. The bulk geometry is still AdS but the dual field theory is now CFT with a defect or boundary (dCFT). When fundamental flavors get mass, it corresponds to bulk geometry receiving supergravity fluctuations.
It is interesting to extend their proposal because it provides a possible way to add fundamental flavors in gauge/string duality on known backgrounds. Authors of ref.[6] has incorporated massive fundamental quarks in supergravity dual of SYM by introducing D7 brane probes in KlebanovStrassler background. In ref.[7] it was shown how to realize chiral symmetry breaking and production of pseudoscalar mesons in some nonsupersymmetric gauge/gravity duals. The purpose of the present paper is to consider adding fundamental flavors into SYM which is dual to the wellknown MaldacenaNŭnez (MN) background[8].
The MN background is described by N D5branes wrapping on a topological nontrivial 2cycle inside a CY threefold. The normal bundle within the CY space has to be twisted in order to preserve some supersymmetry. This configuration is much complicated than the one considered by Karch and Katz, in which the D3branes background are flat. Similar to adding D7branes into the D3brane background in the AdS case, we may consider to add finite number of D9branes or D5branes into the MN background at the large N limit respectively. In the following we shall discuss two different cases respectively.

Adding M D9branes: In the flat case, D5D9 system supports supersymmetry. Adding D9branes is effectively to add M flavor hypermultiplets into the 6d gauge theory. When D5branes wrap on a supersymmetric 2cycle, the 6d vector multiplet on the worldvolume of D5branes reduces to 4d or vector multiplet. The background geometry is . Therefore, D9branes have to wrap on a CY threefold so that supersymmetry is still kept. This configuration, however, seems to receive some unexpected features. The first problem should be asked whether D9branes can be treated as a probe since CY threefold is open. If it is not true, MN solution is not a welldefined background to describe this system. The second problem is that the D9branes shall always carry induced RR charge from the background, whether they are treated as probe or not. This is different from the case discussed in [3]. At rigid probe limit the effects of RR tadpole can be ignored. The original MN background is still good description on this configuration. When we study this system beyond probe limit, we encounter dangerous RR tadpoles. They have to be cancelled by either introducing orientifold planes or antiD9s. The latter is in general unstable, and the former destroy MN background description.
Although there are those unpleasant problems, it is helpful to count spectrum of the dual field theory. The twist on vector multiplet by D5branes is wellknown. The twist on hypermultiplet, however, should be checked carefully. In general, the original twist on 6d hypermultiplet is no longer to leave any massless states, otherwise D9branes can not be felt by the field theory. As discussed above, when massless fundamental flavors are introduced, D5branes will carry open string charge. A natural suggestion is that these charges should play a role in twisting supermultiplet. We will show that it indeed works.

Adding M D5branes: The D5branes intersect with the original 5branes, and have four dimensions in common. In the flat case, this system supports a gauge theory. Adding D5branes is to add M flavor hypermultiplet into the 4d gauge theory. When D5branes wrap on a supersymmetric 2cycle inside a CY 3fold, the D5branes should wrap another orthogonal supersymmetric 2cycle inside the CY space. If the 2cycle is topological nontrivial, D5branes carry RR charge from the background. The charge, however, does not bring dangerous tadpole problem since the CY space is open while the 2cycle is compact. If the 2cycle is topological trivial, D5branes do not get RR charge and we don’t have tadpole problem again. Thus adding D5branes is a nice choice: D5branes can be treated as a probe in the original D5brane background and it does not cause the tadpole problem.
The twist on various supermultiplet, similar to the D5D9 case, should be checked carefully. The open string charge will play a role in twisting in the hypermultiplet.
When D5branes does not carry induced charge from the background, stability of Dbranes should be considered. The stability of Dbranes is ensured by the similar mechanism in[3, 9, 10]: there are negative mass modes in 5d world which control the slipping of the Dbranes off the cycle they wrap, but the mass is not negative enough to lead to an instability in the curved 5d geometry. We will show that similar mechanism works in our cases. The negative mass modes in 5d bulk spacetime are generated by KaluzaKlein spectrum on a compact 5cycle inside a CY threefold, and have a lower bound (similar to BF bound in AdS space).
The paper is organized as follows: In section 2 we consider adding D9branes to the MN background. By a careful twist, the flat part of the system supports SYM with fundamental matter. In section 3 we show how to add D5branes to the MN background and twist it. The flat part of this system again supports SYM with fundamental matter. We give a brief conclusion in section 4. The appendix of this paper devotes to computing KK modes on the compact part of CY threefold which is described by the MN solution.
2 Adding D9branes into the MN background
2.1 The flat spectrum
The massless spectrum for flat D5D9 system has been given in ref.[11]. In the following we review some details. Assuming D5branes spread in {0, 1, 2, 3, 4, 5} directions, the massless spectrum of this system is as follows,

55 open strings generate vector supermultiplet with Rsymmetry .

99 open strings generate vector supermultiplet.

59 open strings: In the NSsector four periodic worldsheet fermions , namely in ND directions , generate four massless bosonic states. We label their spins in the (6,7) and (8,9) planes, with taking values . The GSO projection requires so that two massless bosons survive. In the Rsector four transverse worldsheet fermions with are periodic. They again generate four massless fermionic states, and are labelled by . The GSO projection requires so that again two massless fermions survive. Then massless content of the 59 spectrum amounts to a halfhypermultiplet. The other half comes from strings of opposite orientation, 95 strings. From the point of view of D9brane world volume, 59 and 95 carry opposite charges. In other words, they associate to a global symmetry, where denotes orientation ^{1}^{1}1This global symmetry, however, is gauge symmetry on D9 worldvolume from view point of worldvolume field on D9branes.. For the sake of convenience we split 10d Lorentz group into
(1) The massless states lie in the following representation of the group
bosons: (0, 1, 1, 2, 1, )
fermions: (, 1, , 1, 1, )
2.2 Wrapping D5branes on a 2cycle
Wrapping worldvolume of N D5branes on a topological nontrivial 2cycle leads to a nonsupersymmetric 3branes. In order to preserve supersymmetry one need to twist the normal bundle of this 2cycle, i.e., to identify charge on a 2cycle (denotes by ) with an external gauge field. For D5branes there are two choices: to identify the charge with the charge of diagonal sub group of Rsymmetry group , or with the charge of subgroup of . Some massless states become massive under the twist and decouple with other massless states. The flat part of D5branes supports a (3+1)dimensional twist gauge theory with or supersymmetry corresponding to the previous two choices.
When M D9branes are introduced, the twist on vector supermultiplet on D5branes is the same as the case without D9branes. The key point is to focus on twisting hypermultiplet generated by 59 and 95 string. When an D5brane wrap on , the 6d hypermultiplet in the previous subsection reduces to:

Bosons: .

Fermions:
where is the spin basis in 4d spacetime, subscript denotes the charge, and denotes the representation of .
Now let us consider twisting on the above states. Naively we may consider two choices mentioned above:

.

Bosons: .

Fermions:
Here subscript denotes the total charge. Hence there are no massless states surviving and the resulted field theory is still pure SYM.


. This choice is the same as the first one, but the resulted field theory is pure SYM.

. Denoting the total charge by we have

Bosons: .

Fermions:
Then this choice breaks supersymmetry.

We see that all of the above choices do not generate massless fundamental matters in the framework of 4d supersymmetric gauge theory. It can be understood naturally: The fundamental matter carries open string charges. The charges should play a role in twisting on fundamental matter. Consequently we can say that the 4d field theory gets effects from the open string sector. Therefore, we propose the following twist,

. The total charge is of .

Bosons: .

Fermions:
Two real scalars (one complex scalar) and one Majorana spinor survive under the twist. They form a chiral multiplet of superalgebra. Because the supermultiplet generated by 55 string does not carry charge, the twist on vector multiplet on D5branes is not changed. Then we obtain gauge theory which contains a vector multiplet, a chiral multiplet in the adjoint representation and M chiral multiplets in the fundamental representation of the gauge group.


. Again two real scalars and one Majorana spinor survive under the twist. We obtain a gauge theory which contains a vector multiplet and M chiral multiplets in the fundamental representation of the gauge group.

. No massless states survive at this case.
Our proposal is indeed valid for 1) and 2). The charges play essential role in twisting on various string spectrum. Because the symmetry is gauge symmetry on D9brane worldvolume, it looks like that some worldvolume field are turned on. This is true since endpoints of each 59 and 95 string pair carry opposite charges. Then a nonvanishing flux crosses on a pair of endpoints and lies within D9brane worldvolume. The fluxes act as background gauge field strength on worldvolume. Similar to role of field background discussed in [12], the worldvolume field compensates breaking of supersymmetry when introducing D9brane in wrapping D5brane background. The same conclusion can be obtained from the following supergravity argument.
Because endpoints of an open string couple to NS 1form potential, NS 1form potential should be introduced into supergravity when we introduce fundamental flavor in the dual field theory. To make reduction, RR 2form potential induces gauge field[13] in the 7d gauged supergravity while NS 1form is still kept. Then the twist on normal bundle of the 2cycle now becomes
(2) 
where is the spin connection of the 2cycle. The connection on normal bundle is now cancelled by RR field generated from the close string sector together with NS 1form generated by the open string sector.
2.3 Discussions
Several open questions should be asked here. The first question is whether an D9brane can be treated as a probe in the wrapped D5brane background for fixed. If we treat an D9brane as a probe, i.e., the effect of D9branes is ignored, D9branes wrap on the CY threefold described by the wrapping D5brane background. The energy of D9branes, however, is still infinity since CY space is open. The same puzzle appears in introducing a D7branes probe into a D3brane background proposed by Karch and Katz[3]. An effective treatment is to introduce a cutoff near the boundary. Then energy of D7branes will be of order to the energy of a D3brane probe locating at the boundary of the background geometry. Consequently D7branes can be treated as a probe as same as a D3branes probe. For D9branes, however, this treatment does not work well. For example, in the MN background the ratio of the energy of D9branes to one of the D5brane background locating at the boundary is of order where is a cutoff and is the definite scale. Therefore, it is decided by fine tuning between and whether D9branes can be treated as a probe.
Whether D9branes are treated as probe or not, D9branes carry RR charge induced from wrapped D5brane background. At rigid probe limit the analysis is fine. Beyond probe limit, however, we encounter unexpect RR tadpole problem. This correction is order , and is also implied by fundamental matter spectrum found in previous subsection that the spectrum is anomalous due to lack of antifundamental matter. The anomaly (or RR tadpole) makes theories be inconsistent. They should be cancelled via introducing orientifold planes or antiD9s. The former will destroy original wrapped D5brane background even though D9branes are treated as probe. The introduction on antiD9branes looks like strange since braneantibrane system is in general nonsupersymmetric and unstable. This is not fact when background worldvolume field is turned on. It was shown in refs.[14, 15] that, the braneantibrane system will be 1/4 BPS states when background magnetic fields take opposite directions on worldvolume of branes and antibranes respectively. In addition, the specific worldvolume background electric field makes the usual tachyonic degrees disappear. Thus the system will be stable. Both of these conditions can be satisfied here since a nonvanishing background field is turned on D9brane worldvolume. Then introduction on antiD9s to cancel RR tadpole is good choice. The open strings stretching between D5branes and antiD9s give massless antifundamental flavors which cancels anomaly in massless spectrum.
When D9branes are treated as a probe, it is particularly interesting for the second choice on twist. For this twist supergravity on the MN background is dual to a (nonchrial) SQCD with M fundamental flavors at the probe limit and the large limit. It reflects the fact that at the large limit the role of gluons is more important than the role of quarks (without chiral symmetry spontaneously breaking effects). The effect of fundamental flavors is reflected by DBI action on D9brane worldvolume as in ref.[16, 17]. In order to manifest the quantum number of fundamental flavors, NS gauge field should be turned on in supergravity, i.e., we have to consider type I string fluctuations on the type IIB background. It corresponds to fluctuation on the MN background. The leading order correction is expected to be order of but cancelled by antiD9s.
3 Adding intersecting D5branes
We consider () D5branes intersecting with ( fixed) D5branes. The D5branes spread in {0,1,2,3,4,5} directions while the D5branes spread in {0,1,2,3,6,7} directions. It is different from the D5D9 system. We can naturally treat D5branes as a background and D5brane as a probe on this background.
3.1 Flat spectrum
The flat spectrum on intersecting D5branes was presented in ref.[18]. Here we list the main results for the sake of convenience as follows. Massless spectrum of 55 string and 55 string are 6d (1,1) vector supermultiplet on D5brane and D5brane worldvolume respectively. The new ingredient is the 55 string. We divide 10d spacetime into three sectors: the NN sector with NewmannNewmann boundary conditions for (), the DD sector with DirichletDirichlet boundary condition for () and the ND sector with DirichletNewmann boundary condition for (). In the NS sector, two real scalars survive by GSO projection, namely with where and denotes spins in (4,5) and (6,7) planes. In the R sector, two fermions survive by GSO projection. That is with , where and denotes spins in (2,3) and (8,9) planes respectively. Together with massless states generated by 55 string, they form a hypermultiplet of superalgebra.
Tendimensional Lorentz symmetry now breaks down to
where “ND” denotes ND directions on D5brane worldvolume and “ND” denotes ND directions on D5brane worldvolume. Together with 4d Rsymmetry, the total symmetry now is
.
The representations of various massless field are as follows,

Sixdimensional vector multiplet generated by 55 string:

:

:

:

:


Fourdimensional hypermultiplet generated by 55 string:

Scalars :

Spinors :
where denotes spinor in (3+1)dimension spacetime.

3.2 Twisting on spectrum
Now we consider D5branes wrapping on a supersymmetric 2cycle. It twist the above spectrum and consequently some states become massive and decouple from others. As for D5D9 system, open string charge plays an important role in twisting the fundamental hypermultiplet. We claim that the following two choices are valid for our purpose.

. Total charge is defined by . Then under symmetry various states reduce to

:

:

:

:

:

:
Therefore, we end with the gauge theory whose massless spectrum includes a vector multiplet, a chiral multiplet in the adjoint representation and M chiral multiplets in the fundamental representation of the gauge group. When D5branes are absent, this choice reduces to , and the resulted field theory is pure SYM.


. Total charge is defined by . Under symmetry various states reduce to

:

:

:

:

:

:
We again end with the gauge theory but whose massless spectrum now includes a vector multiplet and M chiral multiplets in the fundamental representation of the gauge group. When D5branes are absent, this choice reduces to , and the resulted field theory is pure SYM.

Most remarks on D5D9 system in the previous section is still applicable here. The main advantage for D5D5 system is D5branes can be treated as a probe in the D5 background without any fine tuning like in a D5D9 system. Hence at the probe limit the supergravity background on wrapping D5branes can be used to study the dual gauge theory with fundamental flavors. In particular, supergravity on the MN background is dual to the large N SQCD with (nonchiral) fundamental quarks. The effect of fundamental flavors is reflected by the DBI action on a D5brane probe. In order to manifest the quantum number of fundamental flavors, NS gauge field should be turned on in supergravity. It corresponds to fluctuation on the MN background and is expected to be of order .
Different aspect arises from considering a 2cycle (denotes 2cycle) wrapped by D5branes. It is orthogonal to the 2cycle wrapped by D5branes and inside a CY threefold. The 2cycle can be either topological trivial or nontrivial. If it is topological nontrivial, D5branes will carry induced charge from the D5brane background. Because D5branes are not spacetime filling now, there is no dangerous tadpole problem. Meanwhile, D5branes are “real” probes on background. They are probe interaction not only of fundamental flavors but also of gauge bosons on D5branes. If the 2cycle is topological trivial, D5branes do not carry any induced charge. Then there is no RR tadpole excitation even for spacetime filling D5branes. The stability of D5branes is ensured by negative mass modes propogating in 5d spacetime. In addition, absence of RR tadpole implies that there should be no anomaly in massless spectrum. In other words, antifundamental flavors should appear besides of fundamental flavors found previously. This is achieved because although total induced RR charge carried by D5branes vanishes, the RR potential does not vanish in general. When D5branes wrap on topological trivial 2cycle inside CY 3fold, the remainder part of CY space can be represented as product form of two topological equivalent . The RR field strength carried by D5branes can spread along two different transverse directions in order to get vanishing total charge. A special case is that one component spread on a wrapped by D5branes, and another component spread on another . The flux consequently splits into two disconnect pieces since is compact, i.e., D5branes are also split into two pieces. The two pieces carry opposite charge since they possess opposite orientation. The open strings stretching between D5branes and one piece of D5branes give massless fundamental flavors, while one stretching between D5branes and another piece of D5branes give massless antifundamental flavors. In other words, the flavors are doubled when D5branes wrap on topological trivial 2cycle^{2}^{2}2The splitting, however, does not affect gauged vector supermultiplet because that multiplet is supported by flat part of D5branes, contrary to hypermultiplet which associated to wrapped directions of 5branes.. This point will be manifested via geometry analysis in the next subsection.
3.3 Geometry analysis
We are interested in the MN background. At the probe limit the effect of D5branes is ignored. The background reads[8]
(3) 
with ,
(4)  
where is an integral constant, parameterize the 3sphere,
(5)  
and gauge field are written as
(6) 
Then the D5brane charge is given by
(7) 
D5branes wrapping on is parameterized by which shrinks to zero for . Then D5branes disappear in resolution and fractional D3branes are created. D5branes, meanwhile, may wrap on another orthogonal parameterized by which carries induce charge. This is with finite radius for . Consequently no fractional D3branes are created in resolution. The finite size of implies that gauge theory on probing D5branes flow to a conformal point in IR. Hence effects of fundamental flavors are manifested in IR even at the probe limit. If we go beyond the probe limit, both wrapped by D5 and D5branes no longer shrink to zero because there is a new flux go through them. This flux, however, does not yield any singularity. It just reflects the effect of fundamental matter in background and shifts charge carried by the D5brane by M.
Another interesting possibility is that D5branes wrap on a topological trivial 2cycle inside a CY threefold, namely directions. Thus they are now spacetime filling branes. To introduce a cutoff in UV (large ), the energy ratio of D5branes to a D5brane probe locating at UV is about . It means that D5brane can be consistently treated as a probe. The induced magnetic and electric charge carried by D5branes are now
(8) 
Charge quantization condition requires
(9) 
In other words, D5branes do not carry any induced charge. The stability of D5branes is ensured by the mechanism mentioned above.
The eq. (3.3) implies that induced magnetic and electric flux carried by D5branes do not vanish although total charges vanish. In particular, the electric field strength is proportional to
(10) 
It indicates one components of electric flux spread in direction which splits wrapped by D5branes into two pieces. It is consistent with argument in previous subsection. Consequently, we obtain not only M fundamental flavors, but also M antifundamental flavors.
There is an additional remark on spacetime filling D5branes because it is similar to the case described by Karch and Katz[3]. The simple induced form of the worldvolume metric on D5branes is modified when the mass term for chiral multiplet is turned on. It is achieved by separating D5branes and D5branes with a small distance c in direction. Using parameterization of eq. (3.3), in unit leads to
(11) 
where on the D5 worldvolume takes values between and . For we recover the topological trivial 2cycle defined by . Then fluctuation of worldvolume scalar corresponds to the mass perturbation of the chiral multiplet. The ingredient different from ref.[3] is that D5 is till spacetime filling instead of ending in the middle of nowhere.
4 Summary and more discussions
We have shown that a few fundamental flavors can be introduced into the closed string dual of large nonchiral SQCD by adding a few D9branes or orthogonal D5branes to a wrapped D5brane background. The twist on the resulted spectrum is much subtler than without probing Dbranes. The essential point is that an open string sector is introduced in the string dual of the field theory. Consequently open string charge plays a role in twisting with the closed string charge. All resulted field theory are gauge theory with fundamental chiral multiplet and with (or without) adjoint chiral multiplet. At the probe limit the known supergravity solution needs not to be modified. The effect of fundamental flavors is manifested by the induced DBI action on worldvolume of probing branes. If probing branes do not carry any induced charges, their stability is ensured by the negative mass modes in 5d spacetime. In particular, adding spacetime filling D5branes in the MN background makes the background supergravity dual to a large SQCD with nonchiral quarks.
We can introduce two distinct sets of probing D5branes into the original wrapped D5 background. The three sets of D5branes are orthogonal to each other. Then this configuration gives rise to a global symmetry, as in QCD. It is a nonchiral global symmetry. When we introduce mass deformation mixing two distinct flavors, for the global symmetry in the field theory gets broken to the diagonal one. It corresponds to two orthogonal 5branes merging into one smooth 5branes wrapping on a 2cycle[19, 20]. A unsolved difficulty is how to incorporate chiral symmetry via the above approach. The usual treatment is to use HananyWitten setups[21] in brane setups[22]. The setups, however, have to introduce NS5 branes and a background with singularity such as orbifold[23] and consequently it destroies the original wrapped D5brane background.
We have pointed out that the worldvolume theories of probe branes include effects of fundamental flavor. Authors of refs.[16, 17] argued that, for a spacetime filling brane probe, the worldvolume theory is just a meson theory. The meson masses and mass gap has been studied in the AdS case. In principle it can be extended to the MN background. The difficulty is that the MN background are much more complicated than the AdS background, and there is no explicit Rsymmetry to classify the meson spectrum.
Appendix A Harmonic function on a 5cycle inside a CY 3fold
A negative mass mode in the AdS space does not lead to instability as long as the mass is above the BF bound[4], with curvature radius . In scenario of AdS/CFT correspondence, the negative mass mode is from KaluzaKlein spectrum of the IIB supergravity on AdS[24, 25]. For example, eigenvalues of the scalar harmonic function on are and they are exactly related to the spectrum in AdS as . In the MN background, the masses of particles propagated in 5d bulk spacetime are also determined by the KaluzaKlein spectrum of the IIB supergravity on a compact 5cycle inside a CY 3fold. In other words, we need to evaluate eigenvalues of harmonic functions on the 5cycle. Because this 5cycle possesses normal bundle structure, it is very difficult to evaluate all eigenvalues of harmonic functions. In this paper we only focus on the existence of positive eigenvalues (negative masses). Precisely, there is no usual eigenvalue for harmonic functions on the 5cycle since metrics on a 5cycle depends on the radius parameter . Thus we should call them as eigenvaluefunction of . It works as a potential in 5d bulk spacetime. We will show, however, there is a constant part in the eigenvaluefunction of scalar harmonic function, which generate masses of particles propagated in 5d bulk spacetime.
Let us consider IIB equation of motion for the scalar field in 10d,
(12) 
with
(13) 
is the Laplace operator defined by bulk metric in eq. (3.3) and
(14) 
is defined by 5cycle metric
(15) 
where is the determinant of this metric.
At UV limit () the metric (15) reduces to
(16)  
It has the product form of . Since radius of goes to infinity, harmonic functions on give continuous spectrum. The main contribution is from with constant radius. The eigenvalue of scalar harmonic function on is . It indeed contains a positive eigenvalue 1.
At IR limit () the metric (16) becomes
(17)  
At the shrinks to zero and it is with constant radius. At the leading order of expansion, the explicit form of the Laplace operator on 5cycle is
(18)  
Here we focus on the maximal nonzero eigenvalue of the above operator. Two quantum number and are generated by two symmetries associating to the shift of and . The maximal eigenvalue of the Laplace corresponds to . Then harmonic function gives maximal eigenvalue . Hence at the leading order we yield an attractive potential
(19) 
It is obtained by shrinking wrapped by D5branes to zero.
The negative mass mode in 5d bulk spacetime comes from the subleading order of whose eigenvalues are constant. In fact, explicit expression of in subleading order is precisely the same as the Laplace on . It agrees with our first glimpse that the 5cycle is equivalent to at the IR limit. The eigenvalue of scalar harmonic function on a 5cycle, therefore, indeed contains a positive constant eigenvalue 1 at the subleading order.
Matching UV result with IR result, we conclude that eigenvaluefunction of scalar harmonic function has the form
(20) 
where
(21) 
This result indicates that there are indeed negative mass modes in the 5d bulk parameterized by the MN background. They associate to the KaluzaKlein spectrum on in a 5cycle. When we introduce spacetime filling Dbranes probe in the MN background, they cancels the NSNS tadpole and consequently it ensure the stability of spacetime filling branes.
The authors greatly thank Prof. J.X. Lu and M. Li for useful discussions. The work is partially supported by the NSF of china, 10231050.
References
 [1] A.M. Polyakov, The wall of cave, Int. J, Mod. Phys. A14 (1999) 645, hepth/9809057.
 [2] J. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231, hepth/9711200.
 [3] A. Karch and E. Katz, Adding flavor to AdS/CFT, JHEP 06 (2002) 043, hepth/0205236.
 [4] P. Breitenlohner and D.Z. Freedman, Stability in gauged extended supergravity, Ann. Phys. 144 (1982) 249.
 [5] P. Breitenlohner and D.Z. Freedman, Positive energy in Antide Sitter backgrounds and gauged extended supergravity, Phys. Lett. B115 (1982) 197.
 [6] T. Sakai and J. Sonnenschein, Probing Flavored Mesons of Confining Gauge Theories by Supergravity, hepth/0305049.
 [7] J.Babington, J.Erdmenger, N.Evans, Z.Guralnik and I.Kirsch, Chiral Symmetry Breaking and Pions in NonSupersymmetric Gauge/Gravity Duals, hepth/0306018.
 [8] J. Maldacena and C. Nŭnez, Towards the Large N Limit of Super YangMills, Phys. Rev. Lett, 86 (2001) 588.
 [9] A. Karch and L. Randall, Open and closed string interpretation of SUSY CFT’s on branes with boundaries, JHEP 06 (2001) 063.
 [10] C. Bachas and M. Petropoulos, AntideSitter Dbranes, JHEP 02 (2001) 025.
 [11] J. Polchinski, String theory II, Page 162, Cambridge University Press, 1998.
 [12] E. Witten, BPS bound states of D0D6 and D0D8 systems in a Bfield, JHEP 04 (2002) 012.
 [13] M. Cvetic, H. Lu and C.N. Pope, Consistent KaluzaKlein Sphere Reduction, Phys. Rev. D62 (2000) 064028.
 [14] D. Bak and A. Karch, Supersymmetric brane antibrane configurations, Nucl. Phys. B626 (2002) 165.
 [15] D. Bak, N. Ohta and M.M. SheikhJabbari, Supersymmetric brane antibrane systems: Matrix model description, stablity and decoupling limits, JHEP 09 (2002) 048.
 [16] A. Karch, E. Katz and N. Weiner, Hadron masses and screening from AdS Wilson loops, Phys. Rev. Lett. 90 (2003) 091601.
 [17] M. Kruczenski, D. Mateos, R.C. Myers and D.J. Winters, Meson spectroscopy in AdS/CFT with flavour, hepth/0304032.
 [18] M. Berkooz, M.R. Douglas and R.G. Leigh, Branes intersecting at angles, Nucl. Phys. B480 (1996) 265, hepth/9606139.
 [19] M. Aganagic, A. Karch, D. Lust and A. Miemiec, Mirror symmetries for brane configurations and branes at singularities, Nucl. Phys. B569 (2000) 277, hepth/9903093.
 [20] M. Aganagic and A. Karch, CalabiYau mirror symmetry as a gauge theory duality, Class. Quant. Grav. 17 (2000) 919, hepth/9910184.
 [21] A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles, and threedimensional gauge dynamics, Nucl. Phys. B492 (1997) 152.
 [22] J.H. Brodie and A. Hanany, Type IIA superstrings, chiral symmetry, and 4D gauge theory dualities, Nucl. Phys. B506 (1997) 157.
 [23] J. Park, R. Rabadan and A.M. Uranga, type IIA brane configuration, chirality and Tduality, Nucl. Phys. B570 (2000) 3.
 [24] M. Gunaydin and N. Marcus, The spectrum of the compactification of the chiral supergravity and the unitary supermultiplets of , Class. Quant. Grav. 2 (1985) L11.
 [25] H.J. Kim, L.J. Roman and P.van Nieuwenhuizen, The mass spectrum of chiral chiral supergravity on , Phys. Rev. D32 (1985) 389.