Contents

On the holographic dual of SQCD with massive flavors


Eduardo Conde 111eduardo@fpaxp1.usc.es, Jérôme Gaillard 222pyjg@swansea.ac.uk and Alfonso V. Ramallo 333alfonso@fpaxp1.usc.es

Departamento de Física de Partículas, Universidade de Santiago de Compostela

and

Instituto Galego de Física de Altas Enerxías (IGFAE)

E-15782, Santiago de Compostela, Spain

Department of Physics, Swansea University

Singleton Park, Swansea SA2 8PP, United Kingdom

Abstract

We construct holographic duals to SQCD with a quartic superpotential and unquenched massive flavors. Our backgrounds are generated by D5-branes wrapping two-dimensional submanifolds of an internal space. The flavor degrees of freedom are introduced by means of D5-branes extended along two-dimensional calibrated surfaces, and act as sources of the different supergravity fields. The backgrounds we get include the backreaction of the flavor branes and generalize the geometries obtained so far to the case in which the fundamental matter is massive. The supergravity solutions we find are regular everywhere and depend on a radial function which can be determined from the distribution of flavor branes used as sources. We also work out the holomorphic structure of the model and explore some of its observable consequences.

1 Introduction

The AdS/CFT correspondence [1] is one of the greatest conceptual developments in the study of the dynamics of gauge theories of the recent years (see [2] for a review). This correspondence, which is in agreement with early ideas of ’t Hooft [3] on the large limit of QCD, has provided a whole set of analytical tools to study gauge theories in the strongly coupled regime. Besides, through the holographic principle of quantum gravity, the correspondence has established a fascinating and far-reaching connection between gauge theories and the physics of black holes.

Although the final goal would be finding a dual to QCD, a less ambitious and more feasible objective is trying to construct gravitational backgrounds dual to minimal supersymmetric models. In this paper we will concentrate on the model [4] dual to super-Yang-Mills (SYM) theory, which is based on the geometry obtained by Chamseddine and Volkov in ref. [5]. This geometry can be regarded as generated by a set of D5-branes wrapping a compact two-cycle of a Calabi-Yau cone. If the size of the cycle is small, the low-energy description of the wrapped D5-branes is effectively (3+1)-dimensional. However, at larger energies the Kaluza-Klein modes on the compact cycle show up and mix with the four-dimensional field theory degrees of freedom. Nevertheless, the model of [4] successfully encodes confinement and chiral symmetry breaking in a geometric setup (see [6] for various reviews).

Another step in the process of approaching the holographic gravitational models to phenomenology is the addition of flavor, i.e. of fields transforming in the fundamental representation of the gauge group (quarks). From the point of view of the gravitational theory, adding quarks to a given gauge theory corresponds to incorporating additional branes to the setup [7]. These flavor branes should extend along the gauge theory directions and wrap a non-compact cycle in the internal manifold, in order to make their worldvolume symmetry a global flavor symmetry. If the number of flavor branes is much smaller than the number of color branes, one can reasonably neglect the effect of the flavor branes on the geometry and treat them as probes. This is the so-called quenched approximation which, on the field theory side, amounts to considering the quarks as external non-dynamical objects that do not run in the loops. For the model dual to SYM the flavor branes are also D5-branes which wrap a non-compact submanifold of the internal space in such a way that is preserved [8] (see [9] for a review of similar studies in several other models).

In this paper we are interested in studying unquenched flavor in the holographic dual of SYM. In this case one has to compute the backreaction by solving the equations of motion of a system of gravity with brane sources. Generically, these sources modify the Einstein equations and the Bianchi identities of some Ramond-Ramond field strengths. We will follow the approach initiated in [10], in which one has a large number of flavor brane sources which are delocalized and one has to deal with a continuous smeared distribution of branes (see [11] for an earlier implementation of this idea in the context of non-critical string theory). In this approach the sources do not contain Dirac -functions, which greatly simplifies the task of solving the equations of motion. On the field theory side this setup corresponds to the so-called Veneziano limit, in which both and are large but their ratio is kept fixed. In refs. [12, 13, 14] different aspects of the supergravity duals of SYM with smeared flavor branes were studied, whereas this approach has been also successfully applied to other types of backgrounds (see [15] for a detailed review).

The flavors added in refs. [10, 12, 13] are massless, which amounts to considering flavor branes extended along the full range of the holographic coordinate . The corresponding supergravity solutions are singular in the IR. This is, actually, a common feature of most massless flavored solutions found so far with the smearing technique (see, for example, refs. [16, 17, 18] for the D3-D7 systems on the conifold). This curvature singularity can be qualitatively understood as due to the fact that, for massless flavors, all branes pass through the origin and, therefore, the brane density is highly peaked at (an exception to this behavior is the solution recently found in [19] for the gravity dual of Chern-Simons-matter theories with flavors).

To remove the IR singularity one can consider massive quarks or, equivalently, a family of flavor branes which do not reach the origin (another possibility is to add temperature and to hide the singularity behind a horizon, as was done in ref. [20]). For the D3-D7 system these regular solutions for massive flavors were found in refs. [21, 22, 23]. As argued in [16], passing from the massless to the massive case in these systems just amounts to substituting in the ansatz by , where is a profile function that interpolates between zero in the IR and one in the UV. To calculate one has to perform a microscopic calculation of the flavor brane charge density, whose result is not universal since it depends both on the characteristics of the unflavored system and on the particular family of flavor brane embeddings.

In this paper we find supergravity backgrounds dual to SYM theories with unquenched massive quarks. The first step in our analysis will be finding the precise deformation of the background which corresponds to the backreaction induced by massive flavors. We will show that the compatibility with the supersymmetry implies a certain type of deformation which is also parameterized by a profile function . When this function is identically equal to one we recover the results of [10] for massless quarks. However it is important to point out that, in this D5-brane case, the massive quark ansatz cannot be recovered by performing the substitution in the massless ansatz of [10].

From our ansatz we will be able to obtain a consistent system of first-order BPS equations which can be partially integrated and reduced to a second-order master equation which is the generalization to this massive case of the equation derived in [13] for massless quarks. To solve this master equation (and the full BPS system) one needs to know first the profile function which, as mentioned above, is not universal and depends on the family of embeddings of the flavor D5-branes. Such families are generated by acting with isometries on a fiducial representative embedding. It turns out that only a particular set of these families produces a backreaction which is compatible with our ansatz. For this reason we must generalize the results of [8] and find new classes of supersymmetric embeddings of flavor D5-branes. In order to carry out this analysis we will introduce a convenient set of complex coordinates suitable to represent the metric and forms of the -structure of our geometry. Employing these variables we will be able to find a family of compatible embeddings and to compute the corresponding profile function .

For massive quarks the function vanishes when is less than a certain value , which is related to the mass of the quarks. For the BPS system coincides with the unflavored one, which corresponds to the fact that the quarks are effectively integrated out in this low-energy region. As shown in refs. [10, 13] there exists a one-parameter family of solutions of the unflavored system which are regular at . Our flavored solutions coincide with these in this region and, although a potential threshold singularity could appear at , we will show how to engineer brane distributions which give rise to geometries that are regular everywhere.

The rest of this paper is organized as follows. In section 2 we review the basic features of the holographic dual to unflavored SYM. In section 3 we study the addition of massive flavor to the background and we present our ansatz for the backreaction induced by a smeared distribution of flavor branes. In this section we will also present the result of the partial integration of the BPS system, as well as the master equation for massive flavors. The holomorphic structure of the model is worked out in section 4. In section 5 we develop a technique to compute the charge distribution function . In this method is obtained by comparing the Wess-Zumino action for the continuous set of branes and that of a single representative embedding. By applying this procedure we will discover that not all the families of embeddings produce a backreaction compatible with our ansatz. In section 6 we find a simple class of compatible embeddings and we compute the corresponding profile function.

The problem of the threshold singularities is analyzed in section 7, where we show how to avoid them and how one can construct regular flavored backgrounds. In section 8 we integrate numerically the master equation and we provide numerical solutions for the different functions of the ansatz. Some observable consequences of our model are analyzed in section 9. In section 10 we summarize our main results and we discuss some further lines of research. The paper is completed with several appendices. In appendix A we study in detail the realization of supersymmetry for our ansatz and we analyze the corresponding BPS system. In appendix B we write in detail the equations of motion satisfied by our solutions. Appendix C contains a microscopic calculation of the charge density for some embeddings. Finally, in appendix D we reconsider the Klebanov-Strassler model with unquenched massive flavors and we apply the new techniques developed in the main text to compute the D7-brane source distribution.

2 The holographic dual of Sym

In this section we will briefly review the supergravity dual to SYM found in ref. [4], which is based on the four-dimensional supergravity solution obtained in [5]. This supergravity background is generated by D5-branes that wrap a compact two-cycle inside a Calabi-Yau threefold. At low energies this supergravity solution is dual to a four-dimensional gauge theory, whereas, at sufficiently high energy, the theory becomes six-dimensional. Moreover, due to a twisting procedure in the compactification, the background preserves four supercharges. The corresponding ten-dimensional metric in Einstein frame is given by:

(2.1)

where is the dilaton. In the remaining of the paper we will use units where . The angles and parameterize a two-sphere which is fibered by the one-forms , which can be regarded as the components of an non-abelian gauge vector field. Their expressions can be written in terms of a function and the angles as follows:

(2.2)

The ’s appearing in eq. (2.1) are the left-invariant one-forms, satisfying , which parameterize a three-sphere and can be represented in terms of three angles , and :

(2.3)

The three angles , and take values in the range , and . For a metric ansatz such as the one written in (2.1) one obtains a supersymmetric solution when the functions , and the dilaton are:

where is the value of the dilaton at . Near the origin the function in (2) behaves as and the metric is non-singular. The solution of the type IIB supergravity includes a Ramond-Ramond three-form given by:

(2.4)

where is the field strength of the gauge field , defined as:

(2.5)

When the ’s are given by (2.2), the different components of are:

(2.6)

where the prime denotes derivative with respect to .

One can readily verify that, due to the relation (2.5), the three-form written in (2.4) is closed, i.e. it satisfies the Bianchi identity . Moreover, the field strength (2.4) satisfies the flux quantization condition corresponding to color D5-branes, namely:

(2.7)

where the three-sphere is the one parameterized by the three angles , and at a fixed value of all the other coordinates and, in order to check (2.7), one should take into account that, in our units, and .

It was argued in [4] that the background written above is dual to SYM in four dimensions plus some Kaluza-Klein (KK) adjoint matter. The four-dimensional theory is obtained by reducing the six-dimensional theory living on the D5-branes with the appropriate topological twist. The latter is necessary to realize the supersymmetry on the curved space [24] . The KK modes in the four-dimensional theory have masses of the order . Since this mass is of the order of the strong coupling scale, the dynamics of the KK modes cannot be decoupled from the dynamics of confinement. A proposal for a concrete lagrangian of the vector multiplet and the different KK modes has been written in [13] (see also [25]). Schematically, this lagrangian has the form :

(2.8)

where and represent the infinite number of massive chiral and vector multiplets and denotes the curvature of the massless vector multiplet .

Let us finish this section by recalling that there exists another solution of type IIB supergravity which is directly related to the one written above. In this solution the metric and the RR three-form are also given by the ansatz (2.1)-(2.6) but, in this case, the function vanishes, and . Actually, these functions are just the UV limit () of the ones written in (2). On the other hand, at this new background has a (bad) singularity that is solved by the turning on of the function in the solution (2), which makes the a non-abelian one-form connection with components along the three directions. This way of resolving the singularity is related, on the field theory side, with the phenomena of confinement and R-symmetry breaking of SYM.

3 Addition of massive flavors

Let us now introduce flavors by means of pairs of chiral multiplets and transforming in the fundamental and antifundamental representations of both the gauge group and the flavor group . The lagrangian for the fields is given by the usual kinetic terms and the Yukawa interaction between the quarks and the KK modes, which can be schematically written as:

(3.1)

In the effective low-energy theory obtained by integrating out the massive modes, the Yukawa coupling between and the gives rise to a quartic term for the quark fields (see [10, 12, 13] for details).

On the gravity side the addition of flavors can be performed by means of flavor branes, which add an open string sector to the unflavored closed string background. For the geometry of section 2 the flavor branes are D5-branes extended along a non-compact cycle of the Calabi-Yau threefold [8]. If the branes reach the origin of the geometry, the corresponding flavor fields are massless. If, on the contrary, the D5’s do not reach , the quark fields are massive (the minimal value of attained by the brane is related to the mass of the quark fields).

In this paper we are interested in getting a holographic dual of the model with unquenched matter, in which the dynamics of fundamentals is encoded in the background. To achieve this goal we must go beyond the probe approximation and find a solution of the equations of motion derived from an action of the type:

(3.2)

where is the action of ten-dimensional type IIB supergravity and denotes the sum of the Dirac-Born-Infeld (DBI) and Wess-Zumino (WZ) actions for the flavor branes. Generically, the branes act as sources for the different supergravity fields. In particular, the WZ term of is a source term for the RR fields which induces a violation of the Bianchi identity of the corresponding RR field strength. In our case, the WZ term of the action of a set of D5-branes is:

(3.3)

where is the RR six-form potential and denotes its pullback to the D5-brane worldvolume. Let us rewrite (3.3) as a ten-dimensional integral, in terms of a charge distribution four-form :

(3.4)

The term (3.4) induces a violation of the Bianchi identity of . In order to determine it, let us write supergravity plus branes action (3.2) in terms of the RR seven-form and its six-form potential . This action contains a contribution of the form:

(3.5)

The equation of motion of derived from (3.5) gives rise to the Maxwell equation for with playing the role of a source, which is just:

(3.6)

Taking into account that, in our units, and that , we get that (3.6) is equivalent to the following violation of Bianchi identity of :

(3.7)

The four-form is just the RR charge distribution due to the presence of the D5-branes. Clearly, is non-zero on the location of the sources. In a localized setup, in which the branes are on top of each other, will contain Dirac -functions and finding the corresponding backreacted geometry is technically a very complicated task. For this reason we will separate the branes and we will distribute them homogeneously along the internal manifold in such a way that, in the limit in which is large, they can be described by a continuous charge distribution .

As we will detail below, the continuous set of flavor branes that we will use in our construction can be generated by acting with the isometries of the background on a representative fiducial embedding and, therefore, all the branes of the continuous set are physically equivalent. Actually, we will not choose an arbitrary distribution of branes. First of all, we will require that all branes are mutually supersymmetric (and thus they will not exert force on each other) and that they preserve the same supercharges as the unflavored background. Moreover, we will also require that the deformation induced on the metric is mild enough, in such a way that it reduces to squashing the unflavored metric (2.1) by means of squashing functions that depend only on the radial coordinate . One can prove that the most general squashing of this type compatible with the supersymmetry of the unflavored background is the one in which the size of one of the fibered directions in the metric (2.1) is different from the other two. Accordingly, we will adopt the following ansatz for the Einstein frame metric of the flavored theory:

(3.8)

Notice that the ansatz (3.8) is exactly the same as the one considered in [10] for the case of massless flavors.

Let us next consider the deformation of the RR three-form . Clearly, due to the modified Bianchi identity (3.7) that must be satisfied in the flavored case, cannot have the same form as in (2.4). Actually, we will slightly modify (2.4) and we will adopt the following ansatz for the RR three-form :

(3.9)

where is an one-form gauge connection and is its two-form field strength, defined as in (2.5), namely:

(3.10)

In (3.9), the are two-forms that parameterize the violation of the Bianchi identity and thus the flavor deformation of the RR three-form. Indeed, when the three-form is closed by construction, due to the relation (3.10) between and . We will take, as in [10], the following ansatz for :

(3.11)

where is different from the fibering function of the metric (they are equal in the background of [4]). By applying the definition (3.10), we get that the different components of the two-form field strength are:

(3.12)

We will adopt for the flavor deformation two-forms an ansatz that parallels , namely:

(3.13)

where and are two functions of the radial variable to be determined. Actually, after a detailed study of the realization of supersymmetry for the metric ansatz (3.8) one can show that (3.13) gives rise to the most general form of . By computing the exterior derivative of (3.9) and applying (3.7), one gets the following expression of the smearing form :

(3.14)

One can now study the realization of supersymmetry in type IIB supergravity for a background with metric and RR three-form given by the ansatz written in (3.8) and (3.9). This analysis is performed in detail in appendix A and leads to a system of first-order BPS equations for the different functions of the ansatz. Combining these equations, a partial integration is possible. Let us summarize in this section the results of this study of the BPS equations. First of all, one can verify that the functions and parameterizing and are not independent. Actually, from the BPS system one can prove that can be written in terms of the derivative of as follows:

(3.15)

Therefore, if we define the function as:

(3.16)

then, the two-forms of (3.13) become:

(3.17)

whereas the smearing form can be written in terms of as:

(3.18)

Moreover, the function parameterizing the one-forms can be written as:

(3.19)

where is defined as the following integral involving :

(3.20)

It follows from these results that the RR three-form in (3.9) is determined in terms of a unique function . Notice that the case of massless flavors studied in [10] is recovered by taking in our formulas. Indeed, in this case only the first term on the right-hand side of (3.18) is non-zero and the charge density distribution is independent of the radial variable. Moreover, by computing the integral in (3.20) one can show that our ansatz for is reduced to the one adopted in [10].

In the case of massive flavors one expects the charge distribution to depend non-trivially on the radial coordinate and, actually, to vanish for values of smaller than a certain scale related to the mass of the quarks. In our approach this non-trivial structure is encoded in the dependence of the function on the radial variable. Notice also that should approach the massless value as since the quarks are effectively massless in the deep UV. The way in which the profile function interpolates between the IR and UV values depends on the particular set of D5-branes that constitutes our delocalized source and should be obtained by means of a microscopic calculation of the charge density (see below).

Another interesting observation is that, contrary to the backgrounds with massive flavors studied in [21, 22, 23], in this case passing from the massless to the massive case is not equivalent to substituting by in the massless ansatz. Indeed, it is immediate to check that making this substitution only the first line in (3.18) is generated, while the last two components of (which are essential for the consistency of the approach) are missing. Notice that these last two terms in are precisely those in (3.14) which are proportional to the function which, according to (3.15), always vanishes when . This means that, in the UV, the two-forms that implement the flavor deformation of are non-vanishing only along the third direction, while the other two components are excited when we move towards the IR. Interestingly, this structure is reminiscent of the way in which the singularity of the particular solution of the unflavored theory reviewed at the end of section 2 is resolved in the full solution (2.1)-(2.6), namely by turning on the function and making the two-form the field strength of a non-abelian magnetic monopole.

Actually, as shown in appendix A, it turns out that one can also integrate partially the BPS system for the functions of the metric in terms of . First of all, the function is related to the dilaton as:

(3.21)

Moreover, the dilaton can be related to the other functions , and as:

(3.22)

where is a constant. In order to represent the remaining functions of the metric (3.8) let us define, following [13], the functions and in terms of and as:

(3.23)

The inverse of this relation is:

(3.24)

It is demonstrated in appendix A that, from the BPS system, one can express and in terms , and , namely:

(3.25)

It follows from eqs. (3.21)-(3.25) that the dilaton and the functions of the metric are determined in terms of , and . Actually, the function can be integrated in terms of the profile , namely:

(3.26)

where is a constant of integration. Moreover, as in [13], one can find a master equation:

(3.27)

One can first notice that in the case eq. (3.27) reduces to the equation found in [13]. Otherwise, knowing the function (from a microscopic description of the smearing), one can get from (3.26) and solve the second-order master equation (3.27) for . As argued above, each solution of this equation will give a complete solution of the problem. Moreover, in appendix B we have explicitly written the equations of motion derived from the type IIB supergravity plus sources action. One can check that any solution of the BPS system also solves the second-order equations of motion written in appendix B.

Finding an analytic solution of this master equation is probably not possible, but we will be able to find numerical solutions, and their asymptotics. In order to achieve this goal we will have first to identify a family of supersymmetric embeddings whose backreaction on the background is compatible with our ansatz and then we must be able to compute the corresponding profile function . In the next section we will start to develop the machinery necessary to carry out this calculation.

4 Holomorphic structure

As stated at the end of section 3, to find the profile function we must analyze the families of supersymmetric embeddings of the flavor D5-branes. This problem was addressed in ref. [8] by looking at the realization of kappa symmetry for probe D5-branes in the unflavored background described in section 2. The analysis of [8] was performed in terms of the angular coordinates of the metric (2.1) and some particularly interesting embeddings were found. For our present purposes we clearly need a more systematic approach, which could allow us to study different families of embeddings and to determine whether or not their backreaction is consistent with our ansatz (3.8)-(3.13). As the internal manifold of our background is complex, it is quite natural to work in a system of holomorphic coordinates. The purpose of this section is to define these complex coordinates and to uncover the holomorphic structure of our background.

Let us begin by introducing a set of four complex variables () parameterizing a deformed conifold, i.e. satisfying the following quadratic equation:

(4.1)

We will also introduce a radial variable , related to the as:

(4.2)

In order to find a useful parameterization of the ’s, let us arrange them as the following complex matrix :

(4.3)

Then, the defining equations (4.1) and (4.2) can be written in matrix form as:

(4.4)

It is immediate to verify that the matrix

(4.5)

is a particular solution of (4.4). The general solution of this equation can be found by realizing that the equations in (4.4) exhibit the following symmetry:

(4.6)

A generic point in the conifold can be obtained by acting with isometries on the point (4.5). Actually, if we parameterize the matrices above in terms of Euler angles as:

(4.7)

then, the four complex variables that solve (4.4) are given by:

(4.8)

where . We will show below that these holomorphic coordinates are very convenient to analyze the supersymmetric embeddings in our flavored backgrounds. It is also useful to introduce a new set of complex variables , related to the by means of the following linear combinations:

(4.9)

These variables satisfy:

(4.10)

and there is an obvious invariance that is obtained by rotating the ’s. The so-called -invariant (1,1)-forms are defined as [26]:

(4.11)

In terms of the radial and angular coordinates these forms are given by:

(4.12)

The fundamental two-form of the -structure can be written in terms of the forms, which will be very useful in what follows. In order to find the corresponding expression, let us notice that has been written in (A.4) in terms of the angle that rotates the projections of the Killing spinors. By using the value of written in (A.15) and the relations (A.16), one easily proves that can be written as:

(4.13)

In terms of the ’s, one can rewrite as:

(4.14)

Let us now check that the complex variables defined in (4.8) are good holomorphic coordinates for the internal manifold. Indeed, since our six-dimensional internal manifold is a complex manifold, we can write its metric in terms of the -form . Actually, if one writes as:

(4.15)

which is allowed thanks to the fact that is a -form, then one can prove that the metric of the internal space can be written as:

(4.16)

where we have split the ten-dimensional metric (3.8) as . The coefficients appearing in (4.15) and (4.16) can be read from (4.14) by using the relation between the ’s and the coordinates (see (4.11) and (4.9)). Moreover, by using again (A.15) and (A.16), one can write the three-form of (A.4) as:

(4.17)

Furthermore, taking into account (3.22), we can write as:

(4.18)

which shows that is, indeed, a holomorphic -form for the complex structure corresponding to the coordinates (4.8).

The RR six-form potential , defined as , can also be written in terms of the one-forms. In fact, it follows from the first of the BPS conditions in (A.5) that can be written in terms of the form as:

(4.19)

where . Obviously, since can be written in terms of the variables, the six-form can also be written as a form in the internal space. Notice that is related to the calibration form of a D5-brane, whose pullback to the worldvolume determines if the embedding is supersymmetric or not. Having written in complex coordinates is very convenient from the technical point of view since it will allow us to analyze the different supersymmetric embeddings by employing the full machinery of the complex variables.

Another relevant quantity that should be invariant under the isometry is the smearing form in (3.18), since it is giving us the charge distribution of the system. It is a (2,2)-form which can be cast in terms of (1,1)-forms as follows:

(4.20)

4.1 Supersymmetric embeddings

It is now straightforward to show that any embedding defined with holomorphic functions of the complex coordinates is supersymmetric. Let us study the case of an embedding extended in the Minkowski directions, and defined in the internal space in the following way:

(4.21)

where, for definiteness, we have chosen and as worldvolume coordinates in the internal space. Recall that . The calibration form for a D5-brane in Einstein frame is given by:

(4.22)

By using (4.15) one can easily get the pullback of this calibration form on the worldvolume of the embedding, namely:

(4.23)

where we have defined the function as:

(4.24)

Now, we look at the induced metric on the worldvolume of the embedding. We get from (4.16):

(4.25)

Therefore, , and one has

(4.26)

This means that the embedding is supersymmetric, proving explicitly that all holomorphic embeddings are supersymmetric.

5 Charge distributions

The supersymmetric D5-brane embeddings we are looking at are characterized by two algebraic equations of the type:

(5.1)

which define a non-compact two-cycle in the internal six-dimensional manifold. As argued above, the preservation of supersymmetry is ensured if the two functions in (5.1) are holomorphic. However, in the brane setup we are considering we will not deal with a particular embedding of the flavor D5-branes but, instead, with a family of equivalent embeddings. This family can be generated from a particular representative of the form (5.1) by acting with the isometries of the conifold. Let us recall how these symmetries act on the holomorphic coordinates. Under the holomorphic coordinates transform as , where

(5.2)

with . Similarly, the transformation is:

(5.3)

where the complex constants and satisfy the condition . We want now to determine the charge distribution four-form (parameterized by the profile function ) for a given family of embeddings. Actually, we will employ a procedure which does not require performing the detailed analysis of the whole family and that allows to extract the function by studying one single particular embedding belonging to the family [27]. This method is based on the comparison between the action for the whole set of flavor branes and the one corresponding to a representative embedding. We can choose to compare either the DBI or the WZ part of the actions, since supersymmetry guarantees that they are the same. The WZ term of the action of the full set of D5-branes is given by the following ten-dimensional integral:

(5.4)

whereas the action of one of the embeddings is just:

(5.5)

with being the worldvolume of the representative embedding chosen and