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Advances in High Energy Physics
Volume 2010 (2010), Article ID 196714, 93 pages
http://dx.doi.org/10.1155/2010/196714
Review Article

Unquenched Flavor in the Gauge/Gravity Correspondence

1Department of Physics, University of Swansea, Singleton Park, Swansea SA2 8PP, UK
2Departament de Física Fonamental, ICCUB Institut de Ciències del Cosmos, Universitat de Barcelona, Martí i Franquès 1, 08028, Barcelona, Spain
3Departamento de Física de Partículas, Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain

Received 26 February 2010; Accepted 25 May 2010

Academic Editor: Radoslav C. Rashkov

Copyright © 2010 Carlos Núñez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Within the AdS/CFT correspondence, we review the studies of field theories with a large number of adjoint and fundamental fields, in the Veneziano limit. We concentrate in set-ups where the fundamentals are introduced by a smeared set of D-branes. We make emphasis on the general ideas and then in subsequent chapters that can be read independently and describe particular considerations in various different models. Some new material is presented along the various sections.

1. Introduction, General Idea, and Outline

1.1. Introduction and Outline

The AdS/CFT conjecture originally proposed by Maldacena [1, 2], refined in [3, 4] and reviewed in [5], has been one of the most interesting developments in theoretical physics of the last decades. It has become one of the most powerful analytic tools to deal with strong coupling effects of some particular gauge theories in the planar limit . The most studied and best understood case corresponds to  SYM which is a highly supersymmetric conformal theory and which only contains matter in the adjoint representation of the gauge group. Certainly, there are many interesting field theories which do not share these properties and this fact has lead to an enormous amount of effort devoted to extending the duality along different paths. Consequently, people have constructed gravity duals of nonsupersymmetric, non-conformal gauge theories, in different vacua and with diverse matter contents. One can mention the attempt of building a dual as close as possible to QCD as an aim for these generalizations. However, one should keep in mind that this is just one among many desirable motivations, since understanding gauge theories at strong coupling (or using gauge theories to understand gravity) is a very relevant problem per se, with both theoretical and phenomenological possible implications.

An important development of AdS/CFT has been to generalise the matter content of the gauge theories under consideration and, in particular, to include fields which transform in the fundamental representation of the gauge group, as the QCD quarks do. (With an abuse of language, we will use throughout this review the words quark or flavor to refer to any field, fermionic or bosonic, transforming in the fundamental representation of the gauge group. Accordingly, by mesons we will mean bound states of quarks.) A first possibility is to add the flavors in the quenched approximation. The word quenched comes from the lattice literature and, in that context, it amounts to setting the quark fermion determinant to one. In more physical terms, quenching corresponds to discarding the quark dynamical effects. This means that quantum effects produced by the fundamentals are neglected; the quarks are considered as external nondynamical objects in the sense that they do not run in the loops. (In the lattice, usually, quenching is thought to be a good approximation for heavy quarks whereas for the gauge-gravity examples the relevance of the quenched approximation comes from having parametrically less fundamental than adjoint fields .) From the string side, adding quenched quarks to a given gauge theory corresponds to incorporating a set of brane probes in the dual background, which is not modified with respect to the quark-less case. By analysing the worldvolume physics of these flavor branes (typically using the Dirac-Born-Infeld + Wess-Zumino action) a lot of physically interesting questions can be understood. For instance and just to mention a few, chiral symmetry breaking can be neatly described, phase diagrams can be constructed, and meson spectra can be exactly computed. It is hard to do justice to the huge literature in the subject; so let us just mention the seminal papers [6, 7] and a recent review [8].

Thus, it is fair to say that the study of quenched flavor within the gauge-gravity correspondence has been a very fruitful program. Nonetheless, there are physical features which are intimately related to the quantum effects of the quarks. Examples are the consequences of the presence of fundamentals on the running couplings, which may ultimately lead to conformal points, conformal windows [9], or Seiberg-like dualities [10]. More phenomenologically, multihadron production, the screening of the color charge, or the large mass of the meson are spin-offs of these quantum effects. Let us also mention that the most successful application of string duals towards phenomenology has been the construction of solutions that can be used as toy models for the experimental quark-gluon plasma. Thus, a very interesting program is to build black hole solutions with unquenched flavor which are really dual to quark-gluon plasmas, that is, such that the effect of the dynamical quarks affects the plasma physics, as is expected to be the case in the real world.

These observations largely motivate the study of theories with unquenched quarks from the string theory dual point of view. Unquenching the flavors of the gauge theory has a very precise implication for the dual theory: the gravity background has to be modified by the inclusion of the quarks; namely, one needs to take into account the back-reaction on the geometry produced by the flavor branes. The main goal in the following will be to present methods to compute such back-reacted solutions. This will be done by presenting different examples that, hopefully, will help the reader to gain insight in both the physical questions and the technical tools used to address them.

In this review, we will focus on a specific family of unquenched constructions. Namely, we will discuss at length just solutions of type IIA or type IIB string theory in which the fundamentals come from a smeared set of flavor branes. In Section 1.5, we will try to provide a general understanding of this notion of smearing the flavor and argue why we find it a case of particular interest. As we will see, by considering the case of smeared D-branes we can build a systematic approach applicable to different situations and which typically results in large simplifications as compared to other kinds of flavor D-brane distributions. This smearing procedure referring to flavors was first introduced in [11] in a noncritical string framework and in [12] in a well-controlled ten-dimensional context.

It is important to remark that this smearing is by no means the only possibility to introduce unquenched fundamentals in gauge-gravity duals. Many important works have followed alternative paths to construct different models. We are not able to review them here, but we provide a survey of the literature in Section 1.6.

Outline
We will devote the rest of Section 1 to further clarifying the kind of physical problems we want to address and to give the general methods and notions which are common to all the constructions we will present later.
Then, Sections 26 will analyse different models that are ordered in increasing order of complexity. Each section can be read mostly independently from the rest. The discussion of each model can always be regarded as a two-step process. First, one has to solve the equations for finding the back-reacted solutions of type II supergravity coupled to a set of D-brane sources. Second, one can use these solutions to extract the physics of the conjectured gauge theory duals with unquenched flavors. Readers interested in different aspects of the problem can consult the different parts independently. We would like to stress that, even without making any reference to the gauge-gravity correspondence, the string theory solutions and methods developed to find them are interesting by themselves.
Section 2 deals with the backreaction of D7-branes on spaces, where stands for a Sasaki-Einstein space. As a matter of fact, a large part of the discussion can be carried out without specifying the . Notwithstanding, the two most interesting cases correspond to and . At different points during Section 2, we will refer to these particular examples in order to explain concrete features. We will present supersymmetric solutions for massless and massive quarks and also non-supersymmetric black hole solutions which are dual to theories at finite temperature, in a deconfined plasma phase. All these solutions share the property of having a singularity, associated with a UV Landau pole in the field theory (when quarks are massless and the temperature is zero, there is also a naked IR singularity). We will show how to make well-defined IR predictions from the geometry, even in the presence of the UV singularity (in much the same spirit as in field theory renormalization).
In Section 3, we will discuss a model in which both the color and flavor branes are D5's. It is dual to a (3+1)-dimensional theory with a UV completion. Among several nice features of the model that will be presented, we would like to remark here that it incorporates a geometrical description of a Seiberg-like duality. Section 4 is also built from a D5-D5 intersection and in fact shares several similarities with the previous model. The construction corresponds to color D5's wrapping a compact 3-cycle and therefore the dual field theory is (2+1)-dimensional.
In Section 5 we examine the addition of D7-branes to the conspicuous Klebanov-Strassler model [13]. From the physical point of view, how the unquenched flavors affect a duality cascade is particularly interesting. From a technical point of view, the system is slightly more involved than the previous ones because different RR and NSNS forms are turned on. However, despite this complication, it is remarkable that almost all functions of the ansatz can be integrated in a closed form.
Section 6 reviews a different class of models. The dual gauge theories are built on wrapped supersymmetric D-branes with the peculiarity that some of the adjoint scalars remain massless. As we will explain, it is not sensible to smear the branes in all the transverse directions. The associated main technical difficulty will be the fact that one has to solve partial differential equations to find the background.
Profiting from the experience gained by discussing these examples one by one, in Section 7, we will give a more mathematical viewpoint of the constructions. In particular, we will take some tools of differential geometry to describe in a concise and compact way the distributions of mass and charge due to the presence of the flavor branes.
Finally, in Section 8 we will conclude by summarizing the whole topic trying to give a general perspective of the results obtained and by also providing an outlook of the subject.

1.2. Presentation of the Problem

As anticipated in the introduction, we will discuss the addition of flavors to field theories (mostly focusing on SUSY examples, but this is not mandatory) using AdS/CFT or, more generally, gauge-strings duality.

We hope it is clear to the reader that the addition of flavors (fields transforming in the fundamental representation) is a very interesting exercise from a dynamical point of view. Indeed, in a theory with adjoint fields (let us, for the sake of this discussion, consider the case of a confining field theory) the presence of fields transforming in the fundamental will produce the breaking of the “QCD-string” or screening. Of course, the fundamentals will add a new symmetry, that can be or, like in massless QCD, ; a baryonic symmetry should also appear. Obviously the presence of global symmetries (and their possible spontaneous or explicit breaking) will directly reflect in the spectrum. Apart from this, it will happen that the states before the addition of fundamentals, that is the glueballs, will interact and mix with the mesons, giving place to new diagonal combinations that will be the observed states. Moreover, anomalies will be modified, as fermions that transform in the fundamental will run in the triangles. Also gauge couplings will run differently and finally new dualities (Seiberg-like [10]) may appear. In the rest of this article, we will discuss how all of the above mentioned features are encoded in string backgrounds.

It is clear that we need to add new objects to our string background. These new objects are D-branes, on which a gauge field propagates, encoding the presence of a gauge symmetry (in the bulk), dual to the global in the dual QFT. Also, it is on these D-branes that the meson fields, represented by excitations of the branes, propagate and interact. Following a nomenclature that by now became standard, we will call these D-branes “flavor branes”.

It is then clear that to add flavors to a field theory whose dual we know, we should consider the original (unflavored) string background and add flavor-branes. Now, the point is how to proceed technically to add these new branes. It may be useful to consider two developments of the 1970s, that will turn to illuminate on the answer to this question.

In [14, 15] 't Hooft and Veneziano, respectively, considered the influence of fundamentals when the following scaling is taken: and considered the two possible cases ('t Hooft and Veneziano, resp.) It is very illuminating to see how different diagrams contributing to the same physical process (to fix ideas, an n-point correlator of mesonic currents) scale in these two cases. In this respect, a formula for the kinematical factor of the scattering of mesons was produced in [16], considering diagrams with being internal fermion loops (windows), nonplanar handles, and boundaries: Consider the case of scattering of two mesons . We see that diagrams like the first one in the figure () scale like a constant , the second diagram (with ) scales like , while the third one (with , that is nonplanar) goes like .

So, we see that from this view point, the Veneziano scaling captures more physics, represented here by diagrams like (b) in Figure 1. Nevertheless, there may be some particular problems for which studying things in the 't Hooft scaling may be enough.

fig1
Figure 1: Diagrams for a meson propagator, with two insertions of the meson operator () shown as thick points on the boundaries. The dashed lines are gluons that fill the diagram in the large limit and the thick lines are quarks. (a) Planar diagram with no internal quark loops (, ), the scaling is 1. (b) Planar diagram with an internal quark loop , , the scaling is ~. (c) Nonplanar diagram with no internal quark loops , , , the scaling is .

From the view point of a lattice theorist, working in the 't Hooft scaling, hence neglecting the effects of fundamentals running inside loops, is the same as working in what they would call the “quenched approximation”. We can think of the field theory as being quenched when the fundamental fields do not propagate inside the loops. One natural way of doing this is to consider the case of very massive quarks. Indeed, when quenching, one considers an expansion of the fermionic determinant (for massive quarks) of the form: Keeping only the constant term (or considering a very large mass) is equivalent to saying that fundamentals are very difficult to pair-produce; hence their presence inside loops will be very suppressed. Another way to quench in the field theory is to consider the case in which the quotient is very small. Notice that the quenched theory is not equivalent to a theory with only adjoints, as fundamentals can occur in external lines, like in a correlator of two mesonic currents as exemplified in the diagram (a) of Figure 1. Needless to say, lattice theorists developed techniques to quench fundamental fields with arbitrary mass. Also, while at first sight the quenching as described above is not a good operation as it breaks unitarity (not including all possible diagrams), this kind of troubles will be avoided when working in the 't Hooft scaling, where unitarity problems will be suppressed in (but of course will be present in a lattice version of theories with finite ).

The interesting point to be taken from this, by a physicist working in gauge-string duality, is that both scalings ('t Hooft's and Veneziano's) can be realized with D-branes. Indeed, in both cases we must add D-branes (to realize symmetries and new states as discussed above), but we can add these flavor branes in two ways. (i)We can add flavor branes in such a way that we will only probe the geometry produced by the color branes. In this case the dynamics of the probe-flavor branes (the mesons) will be influenced by the dynamics of the color branes (the glueballs) but not viceversa. This is a good approximation if , which immediately sets us in the 't Hooft scaling limit. Notice, however, that when the contribution to some particular quantity vanishes, the flavor effects may be the leading ones even when .(ii)We can add flavor branes, in such a way that they will deform the already existing geometry, in other words backreacting on the original “color” geometry. In field theory language, we would say that the dynamics of the glueballs and that of the mesons influence each other, leading to new states that will be a mixture of mesons and glueballs. This is surely what we need to do if and doing this will set us in the Veneziano scaling limit.

More technically, in the 't Hooft scaling limit we are studying the Born-Infeld-Wess-Zumino dynamics for a D flavor brane in a background created by color D-branes (we will always work in Einstein frame in the following): where are fields induced by the color branes background on the (few) flavor branes. The “shape” of the flavor branes (induced metric) will then influence the mass spectrum and interactions of the fluctuations of the flavor branes (the mesons), explicitly realizing the picture advocated above. This line of research was initiated by Karch and Katz [6] and substantially clarified in subsequent papers [7, 1721]. This line kept on growing in the last few years, finding numerous applications. See [8] for a comprehensive review.

On the other hand, working in the Veneziano scaling limit implies that we will need to study the action There will be new equations of motion, encoding explicitly the numbers . As discussed above, it is now very clear that proceeding like this will be the only possibility when the number of flavors is comparable with the number of colors. Also, it makes manifest the fact that the dynamics of glueballs (represented on the string side by which is of order ) is influenced and influences back on the dynamics of fundamentals (represented by , of order ). The rest of this review will focus on this second scaling (Veneziano).

Notice that (in both scalings) we are making an explicit difference between the color gauged symmetry and the flavor global symmetry on the field theory side. From the string theory construction, this qualitative difference is connected to the fact that the volume of the flavor branes is infinite, as compared to the volume of the color branes. Indeed, in the bulk, we only need to realize the field theory global symmetry, and we do it with the gauge field present in the Born-Infeld-Wess-Zumino action. Searching for solutions of color branes in interaction with flavor branes in pure IIA/IIB supergravity is an interesting problem but will not represent the physical system we are after, as only flavor singlet states would be included in the dynamics.

Before we proceed studying the formalism and examples to clarify the details, some comments are in order.

1.3. The String Action and the Scaling Limit in and

Let us study a bit more the expression of (1.6), being careful about coefficients. We will consider the case of a set of “color” D-branes and “flavor” D-branes. The action for this system will be, in Einstein frame, where by we have denoted the various RR fields and with CS-terms the possible Chern-Simons terms. We have taken the simplification of writing the action for the set of flavor branes as times that of a single D-brane, which is enough for the large counting we want to undertake here. The gravitational constant and D-brane tension are The typical quantization condition for the color branes reads As a consequence of (1.7), we will have equations of motion, that generically will read for the metric and dilaton: together with the modified (by the CS-terms) Maxwell equations and, importantly, the Bianchi identity for the (magnetic) Ramond-Ramond field that couples to the flavor D-branes: indicating that the flavor branes are localized (all together) at the position . Similarly the contains delta functions with support on the position of the flavor branes. In principle, one will need to solve second-order, nonlinear, partial differential equations.

Instead of directly dealing with the above equations, we want to present here an argument to understand which parameter controls the size of the flavor effects on the action and, therefore, on the solution. We remark that the reasoning below is qualitative and in particular we will just write a background for flat D-branes as considered, for instance, in [22]. This will be enough for understanding the scaling with the parameters, at least in the cases studied in this review. In the following, we just focus on the behaviour with respect to , , and do not care about numerical prefactors. We will use notation similar to [22]. Consider the background associated to a stack of D color branes (in Einstein frame): where is a known numerical constant and is an energy scale. On a background of this kind, we want to introduce D-flavor branes and to know which is the relative importance of the associated terms in the action (1.7) and equations of motion (1.10). With that aim, let us start by computing the coefficient in front of the term coming from the RR-form sourced by the color branes in (1.7), namely, . Using (1.9) and (1.12), we find that the Lagrangian density associated to the color branes goes as: Let us now look at the DBI term. We assume that the flavor D-branes are extended along the Minkowski directions, the radial direction and directions within the sphere. We find where in order to get the last expression we have used (1.8), (1.12), and (1.13) and defined a dimensionless effective coupling as in [22]: Thus, parametrically, the action from the flavor branes as compared with that from the color background is weighed by . We now want to take a low-energy decoupling limit as in [22] (see also [23]); namely, the dimensionless effective coupling and are fixed as . Thus, the Veneziano scaling limit in this framework amounts to where the last relation comes from demanding that the flavor effects are also fixed. Staying in the supergravity limit requires a constraint that limits the range of energy scales for which the supergravity description is valid [22]. Notice that if we further require that the flavor terms do not parametrically dominate over the color ones, this can further restrict , depending on and .

The probe limit, in which the flavor action is negligible as compared to the gravity action, comes from making the last quantity in (1.16) vanishingly small. (In the literature, it is usually written that the probe limit is good when . That is not strictly correct. For instance, in the D3–D7 case, the probe approximation is valid when, parametrically, .) As expected, that term is strictly zero in the 't Hooft limit. We now comment on the values of , that will appear in the following sections.

For the D3–D7 case of Section 2, the parameter weighing the flavor effects is . For the cascading case of Section 5, the result is similar but one has to replace by the number of fractional branes . Getting ahead of the discussion of upcoming sections we notice that, in these cases, it is not enough to take this parameter fixed, but it should also be small. This will be due to positive beta functions, as will be thoroughly discussed.

From (1.16), we see that (Sections 3, 4 and 6) is particularly interesting since it is really what has to be taken fixed. (Even if in all these sections we will deal with wrapped branes and therefore the backgrounds are not that similar to (1.12), the argument above still yields the correct result.) For this reason, only in these cases one can hope to describe—within gravity—phenomena as Seiberg-like dualities (see Section 3.6.2). Loosely speaking, the Klebanov-Strassler duality cascade [13] lies in this class of theories, since it can be understood as the interplay between two sets of D5-branes wrapping vanishing two-cycles.

We close this section by mentioning other brane intersections that will not be discussed further in later sections. In a D2–D6 system, the effective coupling (1.15) decreases at large and therefore the flavor backreaction on the glue fades away in the UV—see (1.16)—as expected in a superrenormalizable theory. This was observed in [24] when studying a solution with localized D6-branes. In a D4–D8 intersection, the opposite happens. The probe brane approach can be valid in an intermediate regime but at a given the fundamentals eventually take over and dominate. Notice that the value of for which the D4-D8 theory loses its validity is parametrically smaller than that for which the unflavored D4-brane theory becomes pathological, which is set by (1.17).

1.4. The Method

Looking back at (1.7), one can appreciate that in general finding the solution describing the backreaction between the type II closed strings and the open strings described by the Born-Infeld action is quite a challenging problem. Indeed, the fact that the flavor branes (BIWZ) are localized in the ten-dimensional space implies that we will have to solve second-order, nonlinear, coupled, and partial differential equations with localized sources. Basically what makes things so difficult are the presence of delta function sources and the fact that the differential equations describing the dynamics are “partial” (in principle depending on all the variables describing the space transverse to the flavor branes). In order to get some intuition of the answer, we may consider the case in which we will “erase” the dependence on these transverse coordinates (this is like considering the “s-wave” of the putative multipole decomposition of the full solution in this transverse space) and delocalize the sources. To achieve this, we will propose to smear the flavor branes over their perpendicular space.

On the field theory side, this will amount to considering systems where the addition of the degrees of freedom transforming in the fundamental does not break any of the global symmetries of the unflavored QFT. Also, it may happen that the original is explicitly broken to as we are separating the flavor branes—see the discussion in [25, Section ] and in [26, Section ]. An intuitive understanding of the smearing procedure will be discussed in Section 1.5, while a more formal approach will be treated in Section 7. For technical reasons, this procedure is cleaner in examples preserving some amount of SUSY, since the force between flavor branes is cancelled and the smearing is at no cost of energy.

In the examples described in the following sections, we will proceed like this. (i)Consider an unflavored string background and find the embedding of flavor branes that will preserve some SUSY, or (in non-SUSY examples) that will be stable and solve the equations of motion for the brane. In the SUSY cases, this can be achieved by considering kappa-symmetric embeddings, that we review generically below. (ii)Consider now flavor branes in that particular embedding and smear them, getting an action in ten dimensions, as will be explained with all generality in Section 7. (iii)Solve the equations derived from (1.7), that will now contain smeared branes and will be ordinary differential equations. In SUSY cases, there will be a set of BPS equations to be solved. In non-SUSY examples one might manage to get a fake superpotential and fake-BPS equations [27].

Moreover, one has to check that the flavor embeddings considered are still a solution in the backreacted geometry.

Let us review briefly the main technical points collected above.

1.4.1. BPS Equations, Kappa Symmetry (SUSY Probes), and Smearing

Let us consider the case of a supersymmetric background, namely, a solution of type II sugra for which the supersymmetry variations of the gravitino and the dilatino vanish . We will not give here details on the form of these expressions, which can be found elsewhere. For instance, the string frame SUSY transformations of both type IIA and type IIB are written down in [28, Appendix ].

Given a background that preserves some amount of SUSY, the idea is to find the hyper-surfaces in which to extend the flavor branes (in other words, finding the embeddings for flavor branes) so that these will preserve all (or a fraction) of the SUSY of the background.

One then writes an eigenvalue problem, imposing that the preserved spinors of the background are eigenspinors of the kappa-symmetry matrix: See [29, 30] for the definition of .

Once we have the kappa-symmetric embeddings as described above, we now proceed to write an action describing the dynamics of the closed and open strings, as in (1.7). We then realize that the problem will lead (unless we are adding D9-branes) to a system of partial differential equations. As discussed above, we then proceed to smear these flavor branes. For this we propose an ansatz for the metric, where the embedding of the flavor branes is clear and distribute them along the directions of their transverse space. This distribution of the flavor branes can be done in a uniform way. In some sense, we are “deconstructing” the transverse space to the flavor branes by adding at each point one of the many flavor branes.

The key point is that once the BPS equations and kappa-symmetry conditions are simultaneously satisfied, the problem is solved. In fact, it is a general result [31] that the SUSY equations , together with the Bianchi identities—and equations of motion—for the different forms modified by calibrated (namely, kappa-symmetric) sources imply the full set of equations of motion.

In the following, we will discuss first an intuitive way of understanding this smearing. Then we will apply this to different examples in Sections 26. Finally, in Section 7, we will present a formal way of implementing the backreaction from smeared sources.

1.5. A Heuristic Viewpoint

In the following sections, we will introduce the necessary mathematical machinery to consistently compute solutions of string theory in which smeared backreacting flavor branes are present. Before that, it is worth to make a digression in order to explain the general set-up in simple, heuristic terms.

Let us make an analogy with electrostatics. Suppose that we want to compute the electric field generated by a point-like charge and a couple of lines of charge, as depicted on Figure 2(a). In order to depict the situation, we show dimension 1 lines of charge in a total space of dimension 2, but clearly the situation can be generalized by changing such dimensions. Since in the left plot there is no particular symmetry in the configuration, the resulting electric field will have a not so simple expression. But let us imagine that we consider a huge number of lines of charge as in the plot of the right and homogeneously distribute them in the angle they form with the horizontal axis. In the limit of many lines, radial symmetry is recovered, and the charge density is “smeared” and will be just given by a single (monotonically decreasing) function . The electric field, accordingly, will also be radially symmetric. Notice that this process of describing a large number of discrete objects by a continuous distribution is ubiquitous in physics: for instance, a “homogeneous” gas is a collection of atoms or the “homogeneous” Universe considered in cosmological models contains a collection of galaxy clusters. Also, solutions with different kinds of smeared sources have been considered many times in string theory contexts not necessarily related to gauge-gravity duality; see, for instance, [32, 33].

fig2
Figure 2: (a) A point-like charge (in red) and two lines of charge at different angles. (b) A configuration with many lines of charge. In the asymptotic limit of an infinite number of lines, they just correspond to a radial charge density. This picture depicts an analogous situation to the case of smeared flavored branes, when the fundamental fields are massless.

When comparing to the string theory set-up, the point-like charge in the center corresponds to the color branes and the lines to the flavor branes (which have to extend to infinity). The limiting radially symmetric configuration corresponds to the kind of smeared situations that we will discuss in this review. (More precisely, it corresponds to the situations analysed in Sections 25. For the cohomogeneity 2 cases analysed in Section 6, the different functions depend on two radial variables. A heuristic picture for such situations is presented in Section 6.) All functions of the ansatz can then be considered to depend on a single radial variable. For flavor branes, the different “angles” correspond to adding fundamental matter which couples differently to the rest of the fields. In some of the cases discussed in the following, we will see how this is reflected in the field theory superpotential (Section 2.3.1).

We can still get further intuition from this simple analogy. In Figure 2(b), we see that all lines intersect at the center. From the string point of view, that means that the flavor branes are stretched down to the bottom of the geometry and the quarks are massless. In this situation, the charge density is highly peaked at . Essentially, that is the reason why for the solutions with massless quarks described in the following sections there is a curvature singularity at the origin, where all flavor branes meet.

Then, a simple way of getting rid of such a singularity is to displace the lines of charge from the origin, while still keeping the radial symmetry. This is depicted in Figure 3. If we dub the distance from any of the lines to the center as , the density of charge will vanish for , while it will asymptote to the “massless” one as . From the brane construction, this displacement typically corresponds to giving a mass to the fundamentals (or, in particular cases, it could correspond to a nontrivial vacuum expectation value). The solution of Section 2.3.2, which indeed is regular in the IR, is a neat example of this notion. Another possibility is to add temperature and to hide the singularity behind a horizon; see Section 2.5.

fig3
Figure 3: These pictures depict analogous situations to the case of flavored branes, when the fundamental fields are massive. (b) Again, we add a large number of lines, such that in the limit radial symmetry is recovered.

Going back to electrostatics for Figure 3(b), we know from Gauss' law that the charge density outside does not affect the central region. The corresponding statement in the field theory is that the massive fields decouple from the IR physics below the scale given by their mass. We find it interesting that, through this heuristic reasoning, Gauss' law is connected to the decoupling of heavy particles (or holomorphic decoupling in the SUSY cases).

Even if the example of electrostatics is useful to qualitatively picture what we will do in the following, the analogy is by no means perfect. We note two differences: first, we will be working with gravity, which is nonlinear and, thus, one cannot find the final solution by superposing the fields generated by different sources (which in the case of electrostatics would make it rather trivial to find the electric field for the configurations depicted on Figures 2(a) and 3(a)). Second, our “lines of charge” (the flavor branes) are dynamical. This means that is it not enough to compute the background fields generated from the sources but one also has to check that the sources are stably embedded in the geometry.

We end this section by summarizing the pros and cons of looking for duals of unquenched theories for which the string solutions include smeared flavor branes, many of which can be inferred from the heuristic discussion above. On the positive side, one has the following.(i)The smearing simplifies the situation allowing us to write ansätze depending on a single radial coordinate, and therefore the problem is eventually reduced to a set of ODEs. (For the cases of Section 6, they depend on two radial coordinates and thus one finds PDEs in terms of two independent variables, but again, without delta-function localized sources.)(ii)Possible issues related to singularities and strong coupling are ameliorated in the same sense as they are washed out in electrostatics when considering a smooth charge density rather than a sum of delta-functions over a large number of electrons.(iii)It allows a simple application of the powerful mathematical tools of calibrated geometry [34]; see Section 7.

On the negative side, one has the following. (i)Obviously, if we require the flavor branes to be smeared, we are limiting ourselves to considering a very particular subset of all the possible flavored theories. In particular, we require the superpotentials to effectively recover (some of) the global symmetries of the theory without flavors.(ii)Related to the previous point, one cannot realize, in general, theories with flavor groups. Since the flavor branes are required to sit at different points in the internal space, the typical string connecting different flavor branes is heavy and the flavor symmetry is typically broken to (one may also interpret the solutions as having flavor symmetry for some ). From the point of view of the field theory, this amounts to having a Veneziano expansion with “one window graphs”, as pointed out in [26]. In principle, this fact can hinder the realization of some interesting physical features in the dual set-ups considered.

1.6. Localized Sources and Other Approaches

As already remarked, this review focuses on solutions of string theory for which there are D-brane sources homogeneously smeared over a given family of possible embeddings and that can be interpreted as duals of strongly coupled gauge theories in the Veneziano limit. As stressed above, this is a very particular subset of all the possible duals of theories with flavor. In a generic case, one should consider the sources to be localized at certain positions, such that the density of charge is given by a sum over Dirac delta functions. Such generic case is technically more challenging. However, remarkable works along these lines have appeared, pursuing solutions with the flavor branes localized at a single point of space (notice this is not the most general case either). We will not review them in any detail here, but the goal of this section is to provide a brief guide to the literature on the subject.

The main ingredient of this approach consists of finding solutions of supergravity which can be interpreted as intersections of branes of different dimension, with each stack of branes localized at a fixed position of space-time. In the context of gauge-gravity duality, the search for such solutions was initiated in [35, 36]. These papers discussed D3-D7 intersections, which have been the most studied in the literature (see below for different set-ups). A lot of progress was reported in subsequent years [3739]. Among other aspects, these papers presented a better understanding of the D3-D7 solutions, the inclusion of fractional branes, and clear matching with field theory issues such as the running of couplings and anomalies. Further work on the D3-D7 localized system was performed in [40] (where the conifold was also addressed) [41] (where D7 brane backreaction on bubbling geometries was considered), and [42], where the solution corresponding to D3-D7 in flat space was completed by providing an expression for the warp factor in closed form. It is also worth mentioning [43] where a flavor D7-brane in a cascading theory was considered and its backreaction introduced as a perturbation. The finite temperature generalization of the background of [43] was discussed in [44, 45].

Let us now outline the literature on D2-D6 localized intersections, which can be interpreted as duals of 2+1 supersymmetric gauge theories coupled to fundamentals introduced by the D6-branes. The construction of the type IIA solutions (and their relation to M-theory) was carried out in [24, 46, 47]. In [48], meson excitations of this background were discussed and, in particular, the holographic dictionary relating meson-like operators to certain (closed string) supergravity modes was presented. On the other hand, the authors of [49] found a finite temperature version of the solution, which was used to discuss the thermodynamics of the system. Very recent progress in the D2-D6 systems, their M-theory uplifts, and the detailed relation to Chern-Simons theories with flavor has been reported in [50, 51].

Regarding D4-D8 intersections, localized solutions in that set-up were constructed in [52] in an early attempt to build a QCD dual. In the context of the Sakai-Sugimoto model [20], backreaction from localized D8- branes was analysed in [53].

It is also worth mentioning recent solutions in heterotic string theory which were argued to be related to flavored theories [54].

Interestingly, there are a few papers in which similar situations were considered in subcritical string theory and therefore defined in dimensions lower than ten. In many of these cases, each flavor brane fills the whole space-time (therefore they are not localized, neither smeared). Some physics can then be extracted by using exact string theory methods but what these models have in common is that it is not possible to handle them within a well-controlled gravity description: gravity-like actions with just two derivatives suffer curvature corrections which cannot be neglected, nor consistently computed. However, there is the hope that the two-derivative actions can nevertheless provide additional nontrivial insights in the physics of the system. This idea was put forward by Klebanov and Maldacena in [55], who considered a D3-D5 system in a six-dimensional background (the cigar). Such a system is dual to 4D  SQCD as was shown using exact worldsheet methods in [56, 57]. For a recent discussion on the dual to the flavor singlet sector of superconformal QCD in a subcritical string framework, see [58]. The set-up of [55] was generalized to different situations in [11, 59, 60]. The finite temperature physics of a model in [11] was analysed in [61]. Bottom-up approaches (in which a high-dimensional gravity theory is proposed to describe some specific features of QCD) with space-time filling flavor branes have been discussed in [62, 63]. Recently, a bottom-up approach to the conformal window along these lines has appeared [64].

Finally, let us mention a recent contribution by Armoni [65], in which a way of departing from the quenched approximation was proposed. The fermion determinant is expanded in terms of Wilson loops. It then turns out that a sum of correlators of an observable with the Wilson loops boils down to an expansion in , which can in principle be computed. It would be nice to further develop possible implications of this observation in holographic set-ups.

2. Flavor Deformations of

Our first concrete application of the procedure described will be the flavor deformation of . This is the simplest possible case and, hopefully, it will neatly illustrate the comments of Section 1.2. In fact, for most of the discussion, the formalism applies to any geometry, being a five-dimensional compact Sasaki-Einstein (SE) space so we will refer to this more general case during this whole section. At some points, we will use the two notable examples or to clarify particular issues.

Let us start with a general comment. Since the theories without flavor are conformal, we expect that once we include extra matter, a positive beta function is generated. This is in fact the case and leads to the appearance of a Landau pole. Nevertheless, as in QED, the theory renders meaningful IR physics even if the UV is ill-defined as long as the IR and UV are well-separated scales. However, this separation does not allow to have and of the same order. As we will see, one can define a parameter which weighs the internal flavor loops and that has to be kept small. The effect of the unquenched quarks can then be computed as an expansion in .

After introducing the framework in Sections 2.1 and 2.2, we present the unquenched supersymmetric ( in 4d) solutions in Section 2.3. In Section 2.4, we present an instance of the effects of the unquenched flavors on a physical quantity, namely, on the mass of a particular meson tower. Then, in Section 2.5, we break supersymmetry by turning on temperature and analyse the physics of the dual quark-gluon plasma. We end in Section 2.6 by discussing the range of validity for the solutions and approximations used.

2.1. The Geometries and Field Theories without Flavors

The models we discuss here are obtained by placing a stack of D3-branes at the origin of the six-dimensional cone over . The corresponding type IIB background reads where we have taken the near horizon limit. The dilaton is constant and all the other fields of type IIB supergravity vanish. In general the metric of the SE space can be written as a Hopf fibration over a four-dimensional Kähler-Einstein (KE) manifold: where is the fiber and is the connection one-form whose exterior derivative gives the Kähler form of the KE base:

Let us first consider the particular case in which is the five-sphere . In this case the KE base is the manifold (with the Fubini-Study metric) and the space transverse to the color branes, with metric , is just . When the coefficient appearing in (2.1) is just . Moreover, as is well known, the field theory dual to the background is SYM in 4d, which, in language, can be written in terms of a vector multiplet and of three chiral superfields () transforming in the adjoint representation of the gauge group and interacting by means of the cubic superpotential: If we represent the transverse of the solution in terms of three complex variables (), one can regard the 's as the geometric realization of the adjoint superfields .

The second prominent example which we will analyze in detail is the one in which is the space with metric: where the range of the angles is , , . Since , the coefficient for this solution is . In this case the space transverse to the color branes is the conifold, which is a 6d Calabi-Yau cone which can also be described as the locus of the solutions of the algebraic equation: where the are four complex coordinates. The relation between these variables and the coordinates used in (2.5) is the following: Notice also that the metric written in (2.5) is of the form (2.2) where the KE base is just the space parameterized by the angles and one should make the following identifications: The field theory dual to the background is the superconformal quiver gauge theory with gauge group and bifundamental matter fields and transforming, respectively, in the () and in the () representations of the gauge group [66], that is, the so-called Klebanov-Witten (KW) model. The matter fields form two doublets and interact through a quartic superpotential: For a single brane the fields and can be related to the coordinates by means of the following relations: which automatically solve the defining conifold equation (2.6).

2.2. Flavor Branes and Smeared Charge Distribution

The flavor branes for the backgrounds just described are D7-branes extended along the four Minkowski directions as well as along a noncompact submanifold of the cone over . The type of flavor that the D7-branes add depends both on the space and on the submanifold they wrap in the transverse space. We first illustrate the situation with the two examples of and and at the end display the general expressions.

The first instance is the case in which . In this case a simple kappa symmetry analysis shows that, in order to preserve eight supersymmetries, the D7-branes must be extended along a codimension two hyperplane in which, in terms of the complex coordinates can be written as with the and being complex constants satisfying . On the field theory side these flavor branes introduce fundamental hypermultiplets ()—nonetheless, a generic collection of branes within the family (2.11) retains just susy. The corresponding superpotential for an embedding such as the one in (2.11) can be written as where the mass is related to the constant in (2.11). Notice that since the embeddings are holomorphic, it is not possible to smear them in a way in which the full isometry is realised. After smearing over the embeddings (2.11), one can recover, at most, , as will be seen directly from the dual solution.

In the case of the background there are two classes of holomorphic embeddings which correspond to different types of flavors in the KW theory. In terms of the coordinates of (2.7) the representative embedding of the first class is given by the equation . This is the so-called Ouyang embedding [43], which has two branches in the massless limit . In each of these branches the D7-brane adds fundamental matter to one of the two nodes of the KW quiver and antifundamental matter to the second. The corresponding superpotential contains cubic couplings between the quark fields and () and the bifundamental fields and . For example, for the massless embedding the superpotential (2.9) is modified as where, here and in the following, traces over color indices and sums over the flavor indices are implied. The second class of D7-brane embeddings is the one giving rise to nonchiral flavors, whose representative element is given by the equation . In this case every D7-brane adds fundamental and antifundamental flavor to one node of the KW quiver and the flavor mass terms do not break the classical symmetry of the massless theory. The corresponding superpotential contains only mass terms and quartic couplings, namely,

In order to develop our program and construct backreacted gravity solutions for smeared distributions of flavor branes following [67], we should be able to find a family of equivalent embeddings for each type of configuration described above. In the case of the background (2.11) provides such a family. Notice that, even if each individual embedding of the form (2.11) preserves , which supersymmetries are preserved depends on the 's. Nevertheless, one can check that all the holomorphic embeddings of the type (2.11) are mutually supersymmetric and, due to the holomorphic nature of the linear equation (2.11), they preserve the same common four supersymmetries () for all values of the constants . Thus, we can use these constants to parameterize the family of different planes that constitute our continuous distribution of flavor branes.

In the case of the background one can generalize the chiral embedding by acting with the symmetry of the conifold. The corresponding family of embeddings takes the following form: with the complex constants spanning a conifold (up to overall complex rescalings): Notice that embeddings like are not in this family. Indeed, the nonchiral embeddings can be generalized as where span a unit 3-sphere; that is, they satisfy .

In spite of the differences among the cases presented above, the charge distribution generated by these families of embeddings can be written in a common form. The reason for this universality is the underlying Sasaki-Einstein structure. In order to illustrate this fact, let us consider the chiral embeddings (2.15) in the case in which the mass parameter is zero. Without loss of generality we can rescale the coefficients and fix . Then, (2.16) fixes and, after using (2.7), the massless embedding equation nicely factorizes as Notice that the vanishing of each of the factors in (2.19) determines a branch in which the branes sit at a fixed point of one of the two two-spheres parameterized by the angles . The constants and determine the particular point at which each brane is sitting in each . Indeed, if and are systems of worldvolume coordinates for the D7-branes, these two branches can be written as In Figure 4 we have represented the two branches for the embedding (2.18). From the field theory side, which particular embedding we choose determines the coupling between the associated quarks and the bifundamentals. Roughly speaking, the contribution to the superpotential of an embedding determined by some , is . Thus, when we smear and sum over all the possible and , both 's (the one rotating the 's and the one rotating the 's) are effectively recovered. Figure 4 is the geometric interpretation of this effect.

196714.fig.004
Figure 4: We see on the left side the two stacks of flavor-branes localized on each of their respective 's (they wrap the other ). The flavor group is clearly . After the smearing on the right side of the figure, this global symmetry is broken to .

It is straightforward to compute the charge density produced by this localized D7-brane configuration. Indeed, taking into account the contribution of the two branches, one gets To produce a homogeneous configuration of D7-branes we should distribute in every branch the branes homogeneously along their transverse two-sphere. In the continuum limit this procedure amounts to performing an integration over each with the corresponding volume element, namely, The integrations over and in (2.22) can be immediately performed, yielding the following expression for the smeared charge distribution of D7-branes: Notice that in (2.22) we have included the normalization factor in such a way that the resulting distribution densities are normalized to when they are integrated over . Notice that, as already pointed out above, the flavor symmetry of the smeared configuration is rather than , since the branes are not placed on top of each other. Interestingly, a similar calculation for the embeddings (2.17) in the massless case gives rise to the same charge density for the smeared configuration as in (2.23) [68, 69]. This is because the form of in (2.23) is determined by the global symmetry which we want to recover after smearing.

Actually, one can rewrite (2.23) in a form which can be easily generalized to any continuous family of equivalent D7-brane massless embeddings in an arbitrary Sasaki-Einstein manifold. Indeed, by using (2.8) one can rewrite the right-hand side of (2.23) in terms of the Kähler form of as For an arbitrary Sasaki-Einstein space , the expression (2.24) generalizes to where is the following constant coefficient In (2.26) is the compact submanifold of wrapped by the D7-brane in a massless embedding. Notice that in this case the D7-brane worldvolume along the space transverse to the color branes is always of the form , where is a noncompact interval along the holographic radial direction. It is worth noticing that the factor appearing on the right-hand side of (2.26) is just the volume transverse to any individual flavor brane, over which we are distributing the D7-branes. For the massless chiral embeddings in the conifold one can readily check, after taking into account the contribution of both branches in (2.20), that . Since , one can easily prove that (2.25) reduces to (2.24). In the case the three-manifold is just a unit and . Therefore, we obtain the following values of for : The charge density determines the ansatz of in the backreacted geometry. Indeed, the WZ part of the D7-brane action contains a term in which the RR eight-form potential is coupled to the D7-brane worldvolume. The continuous limit for this term amounts to performing the following substitution: which leads to the following violation of the Bianchi identity for : Taking into account the general expression of for a massless embedding written in (2.25) as well as the relation (2.3) between the one-form and the Sasaki-Einstein Kähler form , one is led to adopt [67] the following ansatz for : A simple modification of (2.30) for allows us to extend the ansatz to the case in which the quarks are massive [67]. This modification corresponds to introducing a function of the holographic coordinate and performing the substitution in (2.30). The radial coordinate will be conveniently chosen and, in general, will be different from the one we used so far. The function encodes the effects of the nontrivial profile of the D7-branes. Indeed, when the quarks are massive, the brane does not extend along the full range of the radial coordinate and, accordingly, must vanish for , where is the radial location of the tip of the D7-brane. Moreover, the function should approach the value when since in this region the quarks are effectively massless. The form of the function is not universal and depends on the particular embedding of the D7-brane. For the three embeddings in the cases and discussed above, the expressions for are given below. At this point let us simply notice that the charge density is modified with respect to the massless case as where the dot denotes derivative with respect to the radial variable .

2.3. Backreacted Ansatz and Solution

Let us now write an ansatz for the backreacted D3-D7 background for a generic Sasaki-Einstein space [67]. It is clear from the discussion of the previous subsection that, after performing the smearing, the resulting RR one-form introduces a distinction between the directions of the fiber and of the KE base of . Therefore, it seems clear that the effect of the smeared flavor branes on the metric should be an internal deformation of the in the form of a relative squashing between the KE space and the Hopf fiber. (Just in the case when is the sphere , this squashing breaks part of the isometry , where is the isometry of the Kähler-Einstein base .) Accordingly, let us adopt the following ansatz for the metric in Einstein frame: where and are the functions that implement the squashing mentioned above and the function multiplying amounts to choosing a particular radial variable which is convenient for our purposes. Moreover, the dilaton will depend on and the RR forms and have the following form: where is the volume element of and and are written in (2.1) and (2.26), respectively. The function , whose form depends on the D7-brane embedding, takes into account the effects of massive quarks, as explained above.

Given the ansatz (2.32)-(2.33) one can easily study the supersymmetric variations of the dilatino and gravitino in type IIB supergravity and find the corresponding first-order BPS equations, which ensure the preservation of four supersymmetries. The resulting equations are [67] Remarkably, the system (2.34) can be integrated analytically for any function . In order to present this solution, let us define the function as follows: where is the value of the radial coordinate at the tip of the flavor brane (). Then, we can write down quite simple expressions for , namely, where we have introduced a reference scale and we have defined . Notice that the warp factor can be obtained as the integral of as follows from the last equation in the BPS system (2.34). In (2.36) and are integration constants that we now fix. First, if we demand IR regularity of the solution, we need when . Since vanishes at , we need . Moreover, the constant is just some overall scale and has no physical meaning. It is natural to fix it to in order to give appropriate dimensions and to recover the usual expression for the metric when . Therefore, we find Notice that when , we recover the unflavored background. Indeed, in this case and and, after performing the change of the radial variable , we get that and the background (2.32)-(2.33) coincides with the one written in (2.1).

Let us now introduce the following parameter: which, as we will see in a while, controls the effects of quark loops in the backreacted supergravity solution. Indeed, the gauge/gravity dictionary for the type of theories we are studying relates the exponential of the dilaton to the Yang-Mills coupling constant. For example, for the (flavored) theory, dual to the deformed background, the gauge coupling is and, thus, the 't Hooft coupling at the scale is given by For the quiver theories that correspond to different geometries, the gauge groups are of the form . Let us generalize a relation from the orbifold constructions [66, 70, 71] and consider all the gauge couplings to be equal. Then , strictly speaking, gives the 't Hooft coupling at each node of the quiver, divided by . However, with an abuse of language we will simply refer to it as the 't Hooft coupling. Therefore, by using (2.39) and the definition of in (2.26) in (2.38), we get In particular, when this relation becomes Notice that the fact that is not constant in the backreacted solution is simply a reflection, in the gauge theory dual, of the running of the Yang-Mills coupling constant when matter is added to a conformal theory.

In terms of the dilaton and the function of (2.35) take the following form: One of the prominent features of our solution is the fact that, for , the dilaton blows up at some UV scale , determined by the following condition: Clearly, in order to have a well-defined solution, we should restrict the value of the radial coordinate to the range . In view of the relation between the Yang-Mills coupling and the dilaton (), the divergence of implies that blows up at some UV scale, that is, that the gauge theory develops a Landau pole. This UV pathology of our solution was expected on physical grounds since the flavored gauge theory has positive beta function. Indeed, we will check below in some particular case that our solution reproduces the running of the coupling constant of the dual field theory.

2.3.1. The Supersymmetric Solution with Massless Quarks

We now consider the particular case of massless quarks, which corresponds to taking the charge distribution given by (2.25) or simply and . In this case (2.42) simply gives and the solution written in (2.37) reduces to Notice that the location of the Landau pole in this case is just and that the range of for which the solution (2.45) makes sense is . Moreover, by using the definition of in (2.38) one can immediately show that the dilaton can be written as Let us now verify that the dependence on of in (2.46) matches the expectations from field theory. For concreteness we will consider the case of SYM with matter. Similar checks can be done in other cases (see [67] for the case of the Klebanov-Witten theory). By using the relation between the Yang-Mills coupling and the dilaton discussed above as well as the value of for written in (2.27), one gets In order to read the running of the coupling constant from (2.47) we must convert the dependence on the coordinate in (2.47) into a dependence on the energy scale of the corresponding dual field theory. At an energy scale much lower than the Landau pole scale (i.e., for ) the scaling dimensions of the adjoints and fundamentals take their canonical values and the natural radius/energy relation is Plugging this relation in (2.47) we get Therefore, we get a logarithmic scaling of the coupling of the type , with , which matches the one-loop field theory result in which one has that (see, e.g., [72]). (In principle, one could object that, being strongly coupled, the matter fields could get large anomalous dimensions making this result suspicious. However, since we are performing a small perturbative (in ) deformation of the unflavored backgrounds, the anomalous dimensions for the fundamental multiplets cannot differ much from their quenched values. For the case, those anomalous dimensions vanish. We thank F. Bigazzi for stressing this point to us.)

In order to have a clearer understanding of the deformation of the metric introduced by the flavor, it is very convenient to change to a new radial variable , which is defined by requiring that the warp factor takes the same form as in the unflavored case (see (2.1)): By integrating the last equation in (2.45) we can get and thus . We will perform this integration order by order in a series expansion in powers of . The additive integration constant will be fixed by requiring that . One gets It is now straightforward to obtain the functions , and the dilaton as expansions in powers of . Up to second order we have Equation (2.52) neatly displays the effects of quark loops in the deformation of the geometry and in the running of the dilaton (the latter is related to the running of the gauge coupling, as argued above). It is important to point out that the deformed geometry has a curvature singularity at the origin (or ) (this singularity is similar to the one that appears at in a 2-dimensional manifold with metric ). In the same IR limit, runs to . As argued in Section 1.5, the appearance of this singularity can be intuitively understood as due to the fact that, in this massless case, all branes of our smeared distribution pass through the origin and the charge density is highly peaked at that point. From the field theory side, one can think of the singularity as appearing because the theory becomes IR free, as first pointed out in [36]. Consistently with these interpretations and with the heuristic picture of Section 1.5, the IR singularity can be easily cured by giving a mass to the quarks (it is a “good” singularity according to the criteria of [73, 74]). We will explicitly verify this fact in the next subsection.

2.3.2. The Supersymmetric Solution with Massive Quarks

Let us now find the backreacted supergravity solution for massive quarks. As mentioned above, the function entering the ansatz for in this case is not universal and depends on the particular Sasaki-Einstein space and on the family of D7-brane embeddings chosen. For concreteness we first concentrate in discussing the case in which . The calculation of the function in this case was performed in [75, Appendix ]. If , one has When the function is nonvanishing and one has to perform the integrals appearing in (2.42). These integrals can be straightforwardly done in analytic form and yield the following result: As a check, notice that setting one recovers the massless solution of (2.44) and (2.45). We still have to write the solution for . In this case vanishes and the dilaton is constant and, by continuity, it has the value that can be read from (2.54) inserting : The functions and are equal and given by It follows straightforwardly from these results that the IR singularity at of the massless case disappears when since the background reduces to for . Moreover, one can verify that the metric is also regular at . Thus, as stressed in Section 1.5, the smearing of massive flavors allows one to smooth out IR singularities.

Similar calculations can be done for the conifold theories. In this case we redefine the parameter of the embedding equations (2.15) and (2.17) as . The charge distribution for the family (2.15) of chiral embeddings was obtained in [26], with the following result: Similarly, for the nonchiral embeddings (2.17) the function is given by [76] The corresponding supergravity solutions have been written down in [26, 76]. They are regular in the IR, much in the same way as in the case detailed above.

2.4. Screening Effects on the Meson Spectrum

The holographic theories with flavors present mesonic excitations, meaning that there exists a spectrum of colorless physical states created by operators which are bilinears in the fundamental fields. They are associated to normalizable excitations of the flavor branes as was neatly explained in the seminal paper [7]. For a review of this broad subject, see [8]. Notice that the notion of “meson” we use here generalizes that used in QCD. For instance, the “mesons” of [7] are excitations of a nonconfining theory and in this case the dimensionful quantity that sets the meson masses is just the quark mass (divided by a power of the 't Hooft coupling), not a dynamically generated scale.

In the present section, we review how the presence of unquenched flavors can affect the discrete mesonic spectrum. Again, we will restrict ourselves to the smeared set-up and follow [77]. For discussions about screening effects on the spectrum in cases with localized rather than smeared backreacting flavor branes, we refer the reader to [8, 42]. The effect of the smeared flavors on the hydrodynamical transport coefficients (in a finite temperature setting) was studied in [75, 78]. It is also worth mentioning that, within the model we will introduce in Section 3, the screening effects on the glueball spectrum have been recently analyzed in [79].

For the sake of briefness, we will just focus on an example and discuss a particular mesonic excitation in the backreacted Klebanov-Witten model. The analysis and conclusions for different modes and/or different models should be similar; see [77] for some other examples. In particular, we will consider oscillations of a D7-brane which introduces massive nonchiral flavor [80] and just look at the oscillation of the gauge field that gives rise to a vector mode in the dual field theory. Thus, we discuss the physics of a meson whose “constituent quarks” are massive in the presence of many dynamical massless flavors.

We write the gauge field along the Minkowski directions as , where is a constant transverse vector. The equation describing this oscillation was found in [77], building on the formalism introduced in [80]. It reads where , the constant is the minimal value of reached by the D7-brane (related to the quark mass), and are given in (2.45).

Notice that for the meson excitation, we just use a D-brane probe; namely, we consider the oscillation of a single brane in a fixed background. At first sight, this could look contradictory, since our aim is always to take into account the effect of the flavor branes on the geometry. Then, one may think about considering coupled fluctuations of brane and background fields. Nevertheless, this is not necessary: there are flavor branes which are affecting the background, but when we consider a meson, only one (or two) out of this is fluctuating. Therefore, the effect of this oscillation on the background is suppressed by with respect to the contribution of the whole set of branes and therefore is consistently negligible. On the other hand, the existence of the rest of flavors and the associated quantum effects on the spectrum are taken into account through the deformation they have produced in the background geometry.

Following the standard procedure [7, 8], a discrete tower of values for should be found when selecting solutions of (2.59) which are regular and normalizable. Since the background has a Landau pole, some prescription is needed for dealing with the UV limit (large ). Technically, we will just require that the fluctuation vanishes at . Physically, one can check that this is a consistent procedure if : we are interested in some IR physics which should be independent of the UV completion of the theory at , up to corrections suppressed by powers of the UV scale. Namely, we neglect contributions of order and check that the spectrum can be written in terms of IR quantities. The value disappears from the final result, apart from the quoted negligible corrections. See [77] for further discussions on the issue. In Section 2.5, we will see similar examples of how to deal with the Landau pole. In that case, the IR scale, which has to be much smaller than the arbitrary UV scale at , is set by the temperature rather than by the quark mass.

In order to estimate the spectrum from (2.59), we can use a WKB approximation. In [77], using a formalism developed in [81], an expression for the mass tower in terms of the principal quantum number was found. Adapting notation to the one we are using here, Let us evaluate this integral at first order in , by inserting (2.45). We still have to fix the additive constant for , which we can do by requiring (in [77] was used. It is crucial that both prescriptions give the same result, up to quantities in ). We shift to a coordinate such that . Defining as the 't Hooft coupling (2.39) at the quark mass scale, inserting the value of in (2.1), and defining as the tension of a hypothetical fundamental string stretched at constant , we can write the estimate for the meson masses as It is important to stress once again that this expression is written only in terms of IR quantities, once we discard terms of order namely, contributions like have cancelled out. Notice that the upper limit of the integrals can be taken to infinity if we again insist in discarding contributions. The expression (2.61) is a neat example of how, even having a Landau pole, the holographic set-up is able to consistently obtain IR predictions, in exactly the same spirit as in field theory. We can perform numerically the integration in (2.61), and we get [77] where in order to substitute as in (2.40) we have used , . The expression (2.62) is the result quoted in [77], apart from a different factor of 2 in the definition of .

The lesson we want to take from this section is that there is a well-defined method to obtain the shift produced by the flavor quantum effects on the meson spectrum (or, eventually, on any physical observable) as an expansion in the parameter which weighs the flavor loops. Heuristically, it may be useful to think of the computation leading to (2.62) as (partially) a strong coupling analogue of the Lamb shift corrections of QED.

2.5. Black Hole Solutions: D3-D7 Quark-Gluon Plasmas

In this subsection we will review the results in [75]. We start by showing how one can find a black hole solution which includes the backreaction effects due to massless quarks. To perform this analysis it is more convenient to work with a new radial variable such that the metric takes the following form: Notice that we have introduced a new function which parameterizes the breaking of Lorentz invariance induced by the nonzero temperature . All functions appearing in the metric (2.63), as well as the dilaton , depend on . Moreover, the RR field strengths and are given by the ansatz (2.33) with the function . We remind the reader that fixing corresponds to taking massless quarks. (In [75], the more involved case of massive quarks was also discussed. An extra complication is the necessity of finding the nontrivial D7-brane embeddings in the backreacted geometry.)

In this non-supersymmetric case we will not have the first-order BPS equations at our disposal and we will have to deal directly with the second-order equations of motion. Actually, since all the functions we need to compute depend only on the radial coordinate , it is possible to describe the system in terms of a one-dimensional effective action. One can find this effective action by directly substituting the ansatz in the gravity plus branes action (1.7). One gets In (2.64) denotes the (infinite) integral over the Minkowski coordinates. The second derivatives coming from the Ricci scalar have been integrated by parts and, as is customary, only the angular part of is inserted in the term (otherwise the would not enter the effective action since, on-shell, due to the self-duality condition). The last term in (2.64), proportional to , comes from the DBI contribution in (1.5). Notice also that the WZ term does not enter (2.64) because it does not depend on the metric or the dilaton.

The equations of motion stemming from the effective action (2.64) are It is straightforward to check that these equations solve the full set of Einstein equations provided that the following “zero-energy” constraint is also satisfied:

This constraint can be thought of as the component of the Einstein equations or, alternatively, as the Gauss law from the gauge fixing of in the ansatz (2.63).

Let us now find a solution of the system of equations (2.65) and of the “zero-energy” constraint (2.66) that corresponds to a black hole for the backreacted D3-D7 system. We will require that such a solution is regular at the horizon and tends to the supersymmetric one at energy scales much higher than the black hole temperature . Actually, the biggest advantage of the radial variable introduced above is that the equations of motion of and in (2.65) are decoupled from the ones corresponding to the other functions of the ansatz. These decoupled equations can be easily integrated in terms of an integration constant as follows: where . We now define a new radial coordinate by means of the following relation: Then, and take the following form: with . Notice that is given by the same expression as in (2.50). Moreover, it is clear from (2.69) that is the position of the horizon and, thus, the extremal limit is attained by sending to zero. In terms of the metric takes the following form: where we have defined the functions and as follows: In order to determine completely the background we still have to solve (2.65) and (2.66) for , and the dilaton . We will find this solution by introducing a reference UV scale and by expanding the functions in terms of the parameter defined in (2.38). We will impose that the functions , and are equal to the SUSY ones of (2.52) when the extremality parameter vanishes. Moreover, we will also require that these functions coincide with those in (2.52) at the UV scale . These conditions fix uniquely a solution of (2.65) and (2.66). Up to second order in this solution is given by