Contents

The supergravity dual of 3d supersymmetric gauge theories with unquenched flavors


Felipe Canoura111canoura@fpaxp1.usc.es, Paolo Merlatti 222merlatti@fpaxp1.usc.es and Alfonso V. Ramallo333alfonso@fpaxp1.usc.es

Departamento de Fisica de Particulas, Universidade de Santiago de Compostela

and

Instituto Galego de Fisica de Altas Enerxias (IGFAE)

E-15782, Santiago de Compostela, Spain

Abstract

We obtain the supergravity dual of gauge theory in 2+1 dimensions with a large number of unquenched massless flavors. The geometries found are obtained by solving the equations of motion of supergravity coupled to a suitable continuous distribution of flavor branes. The background obtained preserves two supersymmetries. We find that when the behavior of the solutions is compatible with having an asymptotically free dual gauge theory with dynamical quarks. On the contrary, when the theory develops a Landau pole in the UV. We also find a new family of (unflavored) backgrounds generated by D5-branes that wrap a three-cycle of a cone with holonomy.

1 Introduction

The gauge/gravity correspondence [1, 2] has provided us with a very powerful tool to explore the dynamics of gauge theories at strong coupling. In its original formulation the correspondence is a duality between the background of type IIB supergravity and super Yang-Mills theory, in which all fields transform in the adjoint representation of the gauge group. Clearly, to extend this duality to systems closer to the particle physics phenomenology we should be able to add matter fields transforming in the fundamental representation of the gauge group. This is equivalent to adding open string degrees of freedom to the supergravity side of the correspondence.

It was originally proposed in [3] that such an open string sector can be obtained by adding certain D-branes to the supergravity background. If the number of such flavor branes is small compared with the number of colors, we can treat the flavor branes as probes in the background created by the color branes. This is the so-called quenched approximation, which corresponds, in the field theory side, to suppressing quark loops by factors in the ’t Hooft large expansion. The fluctuation modes of the probe D-brane in the supergravity background provide a holographic description of the flavor sector of the gauge theory and one can extract the corresponding meson spectrum by analyzing the normalizable fluctuations of the probe [4] (for a review and a list of references, see [5]).

When the number of flavors is of the same order as the number of colors, the backreaction of the flavor branes on the metric can no longer be neglected. On the field theory side the inclusion of the backreaction is equivalent to considering the so-called Veneziano limit, in which and are large and their ratio is fixed. In this limit quark loops are no longer suppressed. In the last few years there have been several attempts to construct supergravity duals of these unquenched systems, both for four-dimensional [6] and three-dimensional gauge theories [7], by using solutions of supergravity generated by localized intersections of branes.

Recently, a different approach has been proposed in ref. [8]. Instead of solving the equations of pure supergravity, the authors of [8] considered the full gravity plus (flavor) branes system. The action of such a system contains the Dirac-Born-Infeld action of the flavor branes, which governs their worldvolume dynamics and their coupling to the different supergravity fields. Notice that this is consistent with the fact that color branes undergo a geometric transition and are converted into fluxes, whereas, on the contrary, flavor branes are still present after the geometric transition. Thus, from a conceptual point of view, color and flavor branes are not equivalent and, therefore, should be treated differently. By considering a suitable continuous distribution of flavor branes the authors of [8] were able to find a set of BPS equations and to solve them numerically (see [9] for a similar approach in the context of non-critical string theory). The resulting solution is the flavored backreacted version of the background found in [10] and proposed in [11] as the supergravity dual of super Yang-Mills theory in four dimensions. Further developments of this approach can be found in refs. [12]-[17]. In this paper we will apply this circle of ideas to the case of gauge theories in 2+1 dimensions. The corresponding unflavored supergravity dual was found in ref. [18], and it was interpreted as being generated by D5-branes wrapped on a three-cycle of a manifold of holonomy in [19] (see also [20]-[23]).

The low number of supersymmetries preserved by the solution of [18] (just two real supercharges) is a nice feature and makes it appealing also from the perspective of its dual field theory. As pointed out in ref. [19], this theory reduces in the IR to 2+1 dimensional supersymmetric Yang-Mills theory with a level Chern-Simons interaction. Such theory coupled to an adjoint massive scalar field should arise on the domain walls separating the different vacua of pure super-Yang-Mills in 3+1 dimensions. For the theory has a mass gap, at least classically, with mass of order . This implies that for , i.e. when we can trust the classical result, there are no Goldstone fermions and, therefore, supersymmetry is unbroken. Actually, the Witten index for such a theory was computed in [24], where it has been shown that for supersymmetry is unbroken, while it is broken for . In the borderline case () there is just one supersymmetric vacuum. Being the supergravity solution of [18] supersymmetric and without parameters that could label different vacua, it is reasonable to expect that the dual field theory is the one describing the case. It was shown in ref. [19] that this is actually the case.

We will start our analysis by generalizing the ansatz of [18] for the unflavored solutions. This generalization will allow us to find a new class of solutions in which, in the UV, the metric becomes asymptotically the direct product of a cone and a three-dimensional Minkowski space, while the dilaton becomes constant. This is in contrast to the background of [18], in which the dilaton grows linearly with the holographic coordinate. For this generalized ansatz we will be able to find a system of first-order BPS equations which ensure that our solutions preserve two supersymmetries. We will perform a careful analysis of the regularity conditions to be imposed on the functions of our ansatz, which will allow us to fix some parameters of our solutions and to determine the appropriate initial conditions needed to solve the BPS differential equations. The new solutions, which are found numerically, can be naturally interpreted as non-near horizon versions of the one of [18].

After completing the analysis of the unflavored backgrounds, we will study the addition of flavor D5-branes. First of all, we will use kappa symmetry [25] to determine a continuous family of embeddings of probes that preserve all the supersymmetries of the background and which can be used as flavor branes for massless quarks. These embeddings have the topology of a cylinder and are very similar to the ones found in [26] (and used in [8]) in the case of the supergravity dual of gauge theories in four dimensions. It turns out that the embeddings we will find can be straightforwardly smeared in their transverse directions without breaking supersymmetry. Moreover, one can combine them in a way compatible with our generalized metric ansatz. We will use this fact to compute the backreacted geometry.

As the flavor branes act as a source of the RR forms in the backreacted solution, we will have to modify the ansatz of the RR three-form to include the violation of its Bianchi identity in a very precise form. After this modification of the ansatz, we will look again at the BPS equations that enforce supersymmetry and we will get a system of differential equations that generalizes the one found for the unflavored system. These equations depend now both on and and can be solved by imposing regularity conditions that are similar to the ones used for the unflavored case. By solving the BPS equations for different numbers of colors and flavors we will discover that the system behaves differently depending on whether is larger or smaller than . The most interesting case occurs when . In this regime the behavior of the solution is compatible with having an asymptotically free gauge theory with dynamical massless quarks. We will confirm this result by computing, from our solution, the beta function and the quark-antiquark potential energy. We will get the expected linear confining potential and the dual description of the confining string breaking due to pair creation. On the contrary, when the solution ceases to exist beyond some value of the holographic coordinate. This behavior is compatible with having a Landau pole in the UV.

This paper is organized as follows. In section 2 we will formulate our generalized ansatz for the unflavored case. The corresponding BPS equations are obtained in appendix A. It turns out that these equations admit a truncation, in which some functions of the ansatz are fixed to some particular values and, as a consequence, the system of BPS equations greatly simplifies. Due to this simplification we will first analyze this truncated system in subsection 2.1. In subsection 2.2 we will consider the full system which, in general, presents a better IR behavior. In this subsection we carefully examine the regularity conditions to be imposed on the solutions of the BPS equations.

In section 3 we consider the addition of flavor branes. We first determine the kappa symmetric cylinder embeddings and then we find the particular distribution of them that preserves supersymmetry and is compatible with our metric ansatz. This distribution dictates the modification of the ansatz of the RR three-form needed to encompass the modification of the Bianchi identity induced by the flavor branes. The corresponding BPS equations for this case are also found in appendix A while, in appendix B we verify that, quite remarkably, the first-order equations derived from supersymmetry imply the fulfillment of the second order equations of motion for the coupled gravity plus branes theory. It turns out that the BPS system with flavor admits the same truncation as in the case. We study this truncated system in section 4. The full system for is analyzed in section 5, whereas section 6 is devoted to the study of this same system when .

Finally, in section 7 we recapitulate our results and discuss some possible extensions of our work.

2 Deforming the unflavored solution

Let us begin by describing in detail the ansatz that we will adopt for the unflavored backgrounds we are interested in. As a particular case the family of our solutions will include the one found originally in [18] and interpreted in [19] as a supergravity dual of super Yang Mills theory in 2+1 dimensions. More concretely, let and be two sets of SU(2) left-invariant one forms, obeying:

(2.1)

The forms and parameterize two three-spheres. In the geometries we will be dealing with, these spheres are fibered by a one-form . The corresponding ten-dimensional metric of the type IIB theory in the Einstein frame is given by:

(2.2)

where is the Minkowski metric in 2+1 dimensions, is a radial (holographic) coordinate and , and are functions of . In addition, the one-form will be taken as:

(2.3)

with being a new function of . The backgrounds considered here are also endowed with a non-trivial dilaton and an RR three-form , which we will take as:

(2.4)

where is a new one-form and are the components of its field strength, given by:

(2.5)

In (2.4) is a three-form that is determined by imposing the Bianchi identity for , namely:

(2.6)

By using (2.1) one can easily check from the explicit expression written in (2.4) that, in order to fulfill (2.6), the three-form must satisfy the equation:

(2.7)

In what follows we shall adopt the following ansatz for :

(2.8)

where is a new function. After plugging the ansatz of written in (2.8) into (2.5), one gets the expression of in terms of , i.e.:

(2.9)

where the prime denotes the derivative with respect to the radial variable . Using this result for in (2.7) one can easily determine the three-form in terms of . Let us parameterize as:

(2.10)

Then, by solving (2.7) for , one can verify that is the following function of the radial variable:

(2.11)

with being an integration constant.

In the particular case in which the function is constant and the fibering functions and are equal our ansatz reduces to the one considered in refs. [18, 19]. Actually, we will verify that the BPS equations fix, in this case, the constant value of to be . Moreover, this type of solution is naturally obtained by considering a fivebrane wrapped on a three-sphere in seven dimensional gauged supergravity. This three-sphere of the seven dimensional solution is just the one parameterized by the ’s, while is the gauge field of the gauged supergravity and the corresponding three-form. The expressions (2.2) and (2.4) for the metric and RR three-form of our ansatz are just the ones that are obtained naturally upon uplifting the solution from seven to ten dimensions. Notice that characterizes the flux of the RR three-form and it corresponds to the number of colors on the gauge theory side, whereas the constant of (2.11) is, in the analysis of [19], related to the coefficient of the Chern-Simons term in the 2+1 dimensional gauge theory.

By requiring that our background preserves some fraction of supersymmetry we arrive at a system of first-order BPS equations for the different functions of our ansatz. In its full generality this analysis is rather involved and it is presented in detail in appendix A. Let us mention here that the number of supersymmetries preserved by our solutions is equal to two, which is the right amount of SUSY expected for an gauge theory in 2+1 dimensions. Moreover, supersymmetry imposes the following relation between the dilaton and the function appearing in the metric (2.2):

(2.12)

2.1 The truncated system

As mentioned above the equations imposed by supersymmetry on the functions of our ansatz are obtained in appendix A. By inspecting these equations one can check that they admit solutions in which the fibering functions and vanish, as well as the integration constant , namely:

(2.13)

By performing the truncation (2.13) the first-order BPS system simplifies drastically. Actually, one can verify that it reduces to the following three equations for , and :

(2.14)

To integrate this system, let us consider first the possibility of having solutions with constant. It follows from the equation for written in (2.14) that must be such that:

(2.15)

Plugging this value of in the second equation in (2.14) one easily shows that the equation for becomes:

(2.16)

which can be integrated immediately as:

(2.17)

Using these values of and the dilaton can be readily obtained from the first equation in (2.14), namely:

(2.18)

where is a constant. The solution given by eqs. (2.15), (2.17) and (2.18) was obtained in [21] as the background generated by fivebranes wrapped on a three-cycle of a manifold of holonomy. The corresponding metric is singular at . Notice also that the dilaton (2.18) grows linearly with the holographic coordinate for , as it should for a background created by fivebranes in the near-horizon limit.

In order to study the system (2.14) in general and find other classes of solutions, let us define a new radial variable as:

(2.19)

and a new function as:

(2.20)

We will consider as a function of . From the equations for and written in (2.14) we get the following equation for :

(2.21)

while the equation for the dilaton is:

(2.22)

From the equation for in the system (2.14) it is straightforward to verify that the jacobian of the change of radial variable is:

(2.23)

and, thus, one can write the metric as:

(2.24)

By inspecting (2.21) we recognize our special solution (2.15)-(2.18) as the one that is obtained by taking in (2.21) and (2.22). Another case in which the BPS equations can be solved analytically is when . Indeed, in this case the RR three-form vanishes, the dilaton is constant and (2.21) becomes:

(2.25)

The general solution of (2.25) can be found easily:

(2.26)

where is an integration constant. Let us write the form of this solution in a more suitable form. For this purpose it is convenient to perform a new change in the radial variable, namely:

(2.27)

and to define the constant , related to the integration constant in (2.26) as . In terms of these quantities the metric (2.24) becomes:

(2.28)

where is the constant value of the dilaton. Notice that (2.28) is just the metric of the direct product of a 2+1 Minkowski space and a manifold of holonomy. This metric of holonomy is just the well-known Bryant-Salamon metric [27], which has the topology of and asymptotes to a cone for large values of the radial coordinate . Notice that in (2.28) and as one of the two three-spheres shrinks to a point while the other remains finite. When this manifold is singular at the origin . This singularity is cured by switching on a non-zero value of the parameter , in a way very similar to that which happens to the resolved conifold.

Having obtained the previous solutions for near-horizon fivebranes and (resolved) cones without branes, it is quite natural to look at solutions with RR three-form whose metric becomes in the UV the direct product of 2+1 Minkowski space and a cone. In a sense these solutions would correspond to going beyond the near-horizon region of the fivebrane background. In terms of the variables and it is clear that we are looking for solutions such that:

(2.29)

Notice that, when (2.29) holds, for and, therefore, eq. (2.22) shows that the dilaton is stabilized in the UV, i.e.  for large , in contrast to what happens in (2.18). Actually, one can show that for large the solution of the differential equation (2.21) that behaves as in (2.29) can be approximated as:

(2.30)

By plugging this expansion in the equation (2.22) for the dilaton, one gets the following UV expansion:

(2.31)

which can be integrated as:

(2.32)

where is the UV value of the dilaton. For small one gets two possible consistent behaviors, namely . Notice that in one case diverges at small , while in the other it remains finite. Actually, when diverges at one can show that can be expanded in powers of as follows:

(2.33)

where is a non-zero constant that must be taken to be positive if we want to ensure that . Plugging the expansion (2.33) into the right-hand side of (2.22), one can get the IR expansion of the dilaton :

(2.34)

Notice that is regular as , although diverges. In figure 1 we have plotted the numerical results for and for two different values of the constant .

Figure 1: and for two different values of the constant of eqs. (2.33) and (2.34) ( and ) and .

Let us now consider the case in which is regular as . Apart from the solution in which for all , there are other solutions where is not constant and can be expanded near as:

(2.35)

Notice that in this case vanishes at . By taking one can make positive for small values of . However, one can check by numerical integration that, after having a maximum the function starts to decrease and becomes negative as increases. Due to this pathological behavior we will consider this solution as unphysical.

2.2 Analysis of the general system

Let us now come back to the general ansatz (2.2)-(2.11) and perform an analysis of this system by using the new radial variable and the function defined in (2.19) and (2.20). The corresponding BPS equations are written in appendix A. From eqs. (A.27) and (A.28) it is easy to verify that the equation that determines the function is given by:

(2.36)

where the functions , , and are:

(2.37)

with being the following function of , and :

(2.38)

The quantities and appearing in (2.36) characterize the dependence of the Killing spinors on the holographic coordinate (see appendix A). They can be written in terms of an angle as , . Alternatively, one can write as in (A.25). The explicit expressions of and are given in (A.24). In terms of the variables and , they are:

(2.39)

Similarly, the equations that determine the functions and can be easily obtained from (A.26) and (LABEL:phi-gamma). Let us write them as:

(2.40)

where and are the same as in (2.37) and the new functions , , and are:

(2.41)

Moreover, from (A.27) one easily gets that the jacobian for the change of the radial variable is:

(2.42)

In terms of these quantities, the metric can be written as:

(2.43)

Similarly, from (LABEL:phi-gamma) one can obtain the differential equation that determines the dependence of the dilaton on the variable , namely:

(2.44)

where the new functions and are given by:

(2.45)

As a check of eqs. (2.36)-(2.45) one can verify that they reduce to the ones of the truncated system when . Notice that in this case and and, as a consequence and vanish and eq. (2.40) is solved by the truncated values (2.13). Moreover, one easily demonstrates that, in this case, (2.36) and (2.44) reduces to (2.21) and (2.22) respectively.

2.2.1 Initial conditions

Given a set of initial conditions for the functions , , and , and a value of the integration constant , the system of equations (2.36), (2.40) and (2.44) can be numerically integrated. Let us see how one can determine these initial data in a meaningful way. First of all, let us fix the value of the function at . Recall (see (2.3)) that parameterizes the one-form which, in turn, determines the mixing of the two three-spheres in the ten-dimensional fibered geometry. The curvature of the gauge connection (defined as in (2.5) with ) determines the non-triviality of this mixing. Indeed, if it vanishes the one-forms are a pure gauge connection that can be taken to vanish after a suitable gauge transformation. In this case one can choose a new set of three one-forms in which the two three-spheres are disentangled in a manifest way. On the other hand, from the wrapped brane origin of our solutions, one naturally expects such an un-mixing of the two ’s to occur in the IR limit of the metric, where it should be possible to factorize the directions parallel and orthogonal to the brane worldvolume in a well-defined way. Moreover, by a direct calculation using (2.1) it is easy to verify that for the curvature of the one-form vanishes and, thus, is pure gauge. Thus, it follows that the natural initial condition for is:

(2.46)

Let us now fix the value of the constant by adapting the procedure employed in ref. [19] in the case of backgrounds that are obtained by uplifting from seven-dimensional gauged supergravity. In this reference the authors determined by imposing the vanishing at the origin of the pullback of the RR three-form on the three-cycle of the seven dimensional geometry which, in our notations, is the one parameterized by the one-forms . In the seven dimensional approach this three cycle shrinks at the origin and can be naturally interpreted as the one on which the fivebranes are wrapped. This procedure is possibly ambiguous when one tries to apply it in the ten-dimensional geometry, where actual D5-branes live. Moreover, the solutions studied here cannot be obtained, in general, by uplifting from seven dimensions. Therefore, it is convenient to search for a way to fix directly in ten dimensions.

We start by noting that the seven dimensional cycle, parametrized by the one-forms , does not shrink in the ten dimensional geometry and, thus, it does not look strictly necessary that the RR three-form flux vanishes on it. Indeed, it does not shrink even in the solutions found in [19]. We think that the relevant cycle, which should also be the cycle on which the branes are wrapped, is:

(2.47)

To understand this, let us begin by pointing out that, even if the seven dimensional gauge field is pure gauge at the origin when the initial condition (2.46) holds, it is not vanishing there. This non-vanishing of the gauge connection is the origin of the mixing among the two three cycles in the ten-dimensional fibered geometry. As we are going to argue, this mixing is taken into account if one considers the cycle (2.47) 111Alternatively, by performing a gauge transformation to one can get a new gauge connection , where is a new set of left-invariant one-forms. In this new gauge the condition (2.46) implies that vanishes at the origin and the analogue of the cycle is just the cycle parameterized by the ’s with . . It is indeed easy to see that that cycle is actually shrinking in the full ten-dimensional geometry if some regularity conditions are satisfied. Let us require that the metric function approaches a constant finite value as , namely:

(2.48)

The induced metric on is:

(2.49)

Obviously, due to the factor in brackets in (2.49), as if eqs. (2.46) and (2.48) hold and the dilaton is finite at the origin. Moreover, in order to have a non-singular RR flux at the origin, one should require that vanishes on when . We take this condition as a general criterium to fix the value of . Remarkably, as can be easily verified from our ansatz, the pullback of on is independent of and given by:

(2.50)

Therefore, it is clear that we must fix the value of the constant to the value:

(2.51)

Notice that this is exactly the value of used in [19]. Let us see how one can reobtain this same value of by requiring that the dilaton is finite at . Let be the value of the function defined in (2.38) at . Let us assume that and that and satisfy the initial conditions (2.46) and (2.48). Then, by inspecting (2.39) one concludes that diverges at :

(2.52)

while remains finite at . This means that as and, therefore, the differential equation (2.44) for the dilaton reduces approximately to:

(2.53)

Moreover, from (2.45) and (2.37) we get that the leading behavior of the coefficients and as is:

(2.54)

Therefore, the first-order equation (2.53) for the dilaton becomes:

(2.55)

which, upon integration, gives rise to the divergent IR behaviour:

(2.56)

The only way to escape this conclusion is by requiring the vanishing of , namely:

(2.57)

But, from the expression for in (2.38), we get that:

(2.58)

and, thus, the condition (2.57) fixes again the value of the constant to that written in eq. (2.51). Notice that, contrary to what happens in (2.52), does not diverge at when . Actually, in this case and, therefore, the only possibility of having for , as is required to deduce (2.53), is by imposing that vanishes faster than as which, in particular, implies that we must require:

(2.59)

If, on the contrary, (2.59) is not satisfied, one has that as and , which, again, gives rise to the undesired behavior near . Thus, in order to have a regular dilaton at , we should impose the condition (2.59). Actually, from the expression of in (2.39), as well as the initial conditions (2.46) and (2.48), it is immediately possible to conclude that (2.59) implies that the IR value of should be fine-tuned to the value:

(2.60)

If (2.60) holds, equation (2.53) is still valid and one can check that, indeed, the dilaton remains finite in the IR.

It is also interesting to look at the IR form of the metric (2.43) when the initial conditions just found are satisfied. Since in this case , only the behavior of near is relevant. One has:

(2.61)

Using this result in (2.43), one gets that the part of the metric near takes the form:

(2.62)

Let us now change the radial variable to:

(2.63)

The resulting metric in the sector is:

(2.64)

which is just the metric of flat four-dimensional Euclidean space. Thus, one expects that the metric for these solutions is regular at . We have verified this fact by explicitly computing the scalar curvature for our solutions and by checking that it remains finite at .

2.2.2 Explicit solution

Let us now solve the BPS equations in a series expansion around . For this purpose, let us suppose that is given by the series:

(2.65)

Then, by plugging this expansion into the BPS equations one can get the coefficients for in terms of . The corresponding expression of and is:

(2.66)

Interestingly, one can verify that when the coefficients vanish for and the exact solution is as in the background studied in [19]. Similarly, for the initial conditions at displayed in (2.46) and (2.60), the functions and can be written as:

(2.67)

with the first two coefficients given by:

(2.68)
Figure 2: and for three different values of the initial condition ( and ) for .

One can verify that for one has for all values of the index . Indeed, in this case our generalized solution collapses to the solution studied in [19]. Moreover, the functions and behave near as:

(2.69)

while, for the dilaton can be expanded as:

(2.70)

where is an integration constant. In particular this result implies that is regular at , as claimed above.

The system of BPS equations can be solved numerically with the initial conditions just found. From this numerical analysis we notice that, in addition to the solutions analyzed in [19] (for which and ) there are others which, for , behave as:

(2.71)

where is a finite value. In figure 2 we have plotted the function and the dilaton for several values of the constant . These curves should be compared with the ones in figure 1. The main differences are in the IR behavior of , which is now finite at . In all these solutions the dilaton is asymptotically constant in the UV, in contrast with the ones of [19], for which the dilaton grows linearly with the holographic coordinate. In figure 3 we have represented the functions and for the same set of values of as in figure 2.

Figure 3: The functions and for three different values of the initial condition ( and ) and .

The behaviour (2.71) is easy to reproduce analytically by studying the system of the BPS equations. Indeed, if , and behave as in (2.71) then one readily gets from (2.39) that, at leading order, and for large and, as a consequence, and as . Moreover, one can straightforwardly demonstrate that equation (2.36) determining reduces to the one found in the truncated system in (2.25). From the general solution written in (2.26) we see that for . Furthermore, one can verify that it is consistent to take the following behavior of as :

(2.72)

with being a constant to be determined. Then, at leading order, one gets from (2.39):

(2.73)

where is the asymptotic value of at . Taking into account the expression of (eq. (2.38)), this value can be written in terms of the asymptotic value of as follows:

(2.74)

where we have momentarily considered a general value of the integration constant . Notice also that and behave as:

(2.75)

Using this result one can write asymptotically the differential equation for as:

(2.76)

Consistency at leading order requires that and must be related by:

(2.77)

Taking into account the value of written in (2.74), one gets the value of in terms of the constant , namely:

(2.78)

Notice then that the asymptotic value of is:

(2.79)

Let us now calculate the coefficient that determines the asymptotic behavior of the function . With this purpose, we notice that the functions and defined in (2.41) behave as:

(2.80)

Also taking into account that , as well as eqs. (2.72) and (2.75), one gets that:

(2.81)

which, for consistency with (2.72), implies that:

(2.82)

Finally, one can verify from (2.44) that the dilaton reaches a constant value when .

Taking into account that the regularity conditions in the IR fix to be equal to , one gets that the actual values of and are:

(2.83)

a result which is confirmed by our numerical solutions.

It is interesting to compare the background found here with the one obtained in [19]. The latter corresponds to D5-branes wrapped on a three-cycle of a manifold of holonomy in the near-horizon limit which, as it should, has a dilaton which grows linearly with the holographic coordinate in the UV. In our case the dilaton is asymptotically constant and the metric approaches that of a cone as we move towards the UV, while in the IR region our solution is qualitatively similar to the one analyzed in [19]. It is thus natural to regard our solution as corresponding to D5-branes wrapped on a three-cycle of a cone, in which the near horizon limit has not been taken and, thus, as we move towards the large region the effect of the branes on the metric becomes asymptotically negligible and we recover the geometry of the cone where the branes are wrapped. Notice that in reference [8] the authors found similar backgrounds for the case of D5-branes wrapped on a two-cycle. In this case the solutions asymptotically approach the conifold geometry.

3 Addition of flavor

Our main motivation to study a generalized ansatz of the form (2.2)-(2.4) was to explore the addition of unquenched flavors to the supergravity duals of supersymmetric gauge theories in 2+1 dimensions. Indeed, we will show below that the backreacted flavored metrics that we will find can be represented in the form (2.2), i.e. their deformation with respect to the unflavored ones of [19] is of just the type studied in section 2. We will achieve this conclusion in three steps. First of all, we will study the problem in the approximation in which the flavor brane is considered as a probe in the unflavored background.

The appropriate flavor branes for our case are wrapped D5-branes that fill the Minkowski spacetime and are extended in the holographic direction. By using kappa symmetry [25] of the probe we will be able to find some simple configurations that preserve all supersymmetries of the background. In these configurations the D5-branes are extended along a submanifold of the internal space that has the topology of a cylinder and reaches the origin of the holographic coordinate. They can be used to add massless flavors to the gravity dual of [19]. Actually there is a continuous family of such supersymmetry preserving embeddings. In a second step we will determine how to combine these embeddings to produce a distribution of them that produces a backreaction on the background such that the metric is still of the form (2.2). In general the flavor branes act as sources for the RR fields and also modify the energy-momentum tensor. Due to the fact that we will consider a continuous distribution of D5-branes, these extra terms are not localized and, as we will see, their influence in the background can be obtained.

The main modification of the backreacted ansatz with respect to the one studied in section 2 is that in the unquenched case the new RR source terms give rise to a violation of the Bianchi identity for . In a third step we will determine the appropriate modification of that gives rise to the desired violation of the Bianchi identity. Moreover, once is known we can use it in the supersymmetry variations and obtain the BPS equations of the flavored backgrounds, exactly in the same way as in the unflavored system. This analysis is performed in appendix A, whereas the study of the different solutions of the BPS system will be carried out in the remaining sections of this paper. In appendix B we show that the backgrounds obtained in this way solve the second order equations of motion of the supergravity plus brane system.

3.1 Supersymmetric probes

Let us consider a D5-brane probe in some of the backgrounds studied in section 2 and let () be a set of worldvolume coordinates. If denote ten-dimensional coordinates, the D5-brane embedding will be characterized by a set of functions . The induced metric on the worldvolume is:

(3.1)

where is the ten-dimensional metric. The embeddings of the D5-brane probe that preserve the supersymmetry of the background are those that satisfy the kappa symmetry condition [28]:

(3.2)

where is a matrix that depends on the embedding of the probe and is a Killing spinor of the background. Acting on spinors such that (as the ones of our background, see (A.6)) and assuming that there is no worldvolume gauge field, the matrix for a D5-brane probe is [25, 28]: