Contents
###### Abstract

We construct a holographic dual to the three-dimensional ABJM Chern-Simons matter theory with unquenched massive flavors. The flavor degrees of freedom are introduced by means of D6-branes extended along the gauge theory directions and delocalized in the internal space. To find the solution we have to solve the supergravity equations of motion with the source terms introduced by the D6-branes. The background we get is a running solution representing the renormalization group flow between two fixed points, at the IR and the UV, in both of which the geometry is of the form , where is a six-dimensional compact manifold. Along the flow, supersymmetry is preserved and the flavor group is Abelian. The flow is generated by changing the quark mass . When we recover the original unflavored ABJM solution, while for our solution becomes asymptotically equivalent to the one found recently for massless smeared flavors. We study the effects of the dynamical quarks as their mass is varied on different observables, such as the holographic entanglement entropy, the quark-antiquark potential, the two-point functions of high dimension bulk operators, and the mass spectrum of mesons.

Unquenched massive flavors and flows

in Chern-Simons matter theories

Yago Bea,***yago.bea@fpaxp1.es Eduardo Conde,econdepe@ulb.ac.be Niko Jokela,niko.jokela@usc.es and Alfonso V. Ramallo§§§alfonso@fpaxp1.usc.es

Departamento de Física de Partículas

and

Instituto Galego de Física de Altas Enerxías (IGFAE)

E-15782 Santiago de Compostela, Spain

Physique Théorique et Mathématique and International Solvay Institutes

Université Libre de Bruxelles

Campus Plaine - CP 231, B-1050 Bruxelles, Belgium

## 1 Introduction

Three-dimensional Chern-Simons matter theories have been studied extensively in the last few years due to their rich mathematical structure and their connection with different systems of condensed matter physics. In particular, the Aharony-Bergman-Jafferis-Maldacena (ABJM) theory [1] has provided a highly non-trivial example of the AdS/CFT correspondence [2, 3]. The ABJM theory is an supersymmetric gauge theory with Chern-Simons levels and , coupled to matter fields which transform in the bifundamental representations and of the gauge group. The ABJM construction was based on the analysis of [4, 5], in which the supersymmetric Chern-Simons theories were proposed as the low energy theories of multiple M2-branes. When and are large the ABJM theory admits a gravity dual in type IIA supergravity in ten dimensions. The corresponding background is a geometry of the form with fluxes (see refs. [6, 7, 8, 9] for reviews of different aspects of the ABJM theory).

One of the possible generalizations of the ABJM theory is the addition of flavor fields transforming in the fundamental representations and of the gauge group. In the supergravity description these flavors can be added by considering D6-branes extended along the directions and wrapping a three-dimensional submanifold of . By imposing the preservation of supersymmetry one finds that the D6-brane must wrap a submanifold of the internal space [10, 11]. When the number of flavors is small one can treat the D6-branes as probes, which is equivalent to the quenched approximation on the field theory side. This is the approach followed in refs. [12, 13, 14, 15] (see also [16]).

In order to go beyond the quenched approximation, one must be able to solve the supergravity equations of motion including the backreaction induced by the source terms generated by the flavor branes. The sources modify the Bianchi identities satisfied by the forms and the Einstein equations satisfied by the metric. These equations with sources are, in general, very difficult to solve, since they contain Dirac -functions whose support is the worldvolume of the branes. In order to bypass this difficulty we will follow here the approach proposed in [17] in the context of non-critical holography, which consists of considering a continuous distribution of flavor branes. When the branes are smeared in this way there are no -function sources in the equations of motion and they become more tractable. Substituting a discrete set of branes by a continuous distribution of them is only accurate if the number of flavors is very large. Therefore, this approach is valid in the so-called Veneziano limit [18], in which both and are large and their ratio is fixed. The smearing procedure was successfully applied to obtain supergravity solutions that include flavor backreaction in several systems [19, 20, 21] (see [22] for a detailed review and further references).

A holographic dual to ABJM with unquenched massless flavors in the Veneziano limit was found in [23]. In this setup the flavor branes fill the and are smeared in the internal space in such a way that supersymmetry is preserved. Notice, that since the flavor branes are not coincident, the flavor symmetry is rather than . A remarkable feature of the solution found in [23] is its simplicity and the fact that the ten-dimensional geometry is of the form , where is a compact six-dimensional manifold whose metric is a squashed version of the unflavored Fubini-Study metric of . The radii and squashing factors of this metric depend non-linearly on the flavor deformation parameter , where is the ’t Hooft coupling of the theory. Moreover, the dilaton is also constant and, since the metric contains an factor, the background is the gravity dual of a three-dimensional conformal field theory with flavor. Actually, it was checked in perturbation theory in [24] that the ABJM theory has conformal fixed points even after the addition of flavor. This solution captures rather well many of the effects due to loops of the fundamentals in several observables [23]. Its generalization at non-zero temperature in [16] leads to thermodynamics which pass several non-trivial tests required to a flavored black hole.

Contrary to other backgrounds with unquenched flavors, the supergravity solutions dual to ABJM with smeared sources are free of pathologies, both at the IR and the UV. This fact offers us a unique opportunity to study different flavor effects holographically in a well-controlled setup. In this paper, we will study such effects when massive flavors are considered. The addition of massive flavors breaks conformal invariance explicitly and, therefore, the corresponding dual geometry should not contain an Anti-de Sitter factor anymore. Actually, for massive flavors the quark mass is an additional parameter at our disposal which we can vary and see what is the effect on the geometry and observables. Indeed, let denote the quark mass. In the IR limit in which is very large we expect the quarks to be integrated out and their effects to disappear from the different observable quantities. Thus, in the IR limit we expect to find a geometry which reduces to the unflavored ABJM background. On the contrary, when , we are in the UV regime and we should recover the deformed Anti-de Sitter background of [23]. The important point to stress here is that the quark mass triggers a non-trivial renormalization group flow between two fixed points and that we can vary to enhance or suppress the effects due to the loops of the fundamentals.

To find the supergravity solutions along the flow, we will adopt an ansatz with brane sources in which the metric and forms are squashed as in [23]. By imposing the preservation of supersymmetry, the different functions of the ansatz must satisfy a system of first-order BPS equations, which reduce to a single second-order master equation. The full background can be reconstructed from the solution to the master equation.

The flavor branes corresponding to massive flavors do not extend over the full range of the holographic coordinate. Indeed, their tip should lie at a finite distance (related to the quark mass) from the IR end of the geometry. Moreover, in the asymptotic UV region, the geometry we are looking for should reduce to the one in [23], since the quarks should be effectively massless in that region. Therefore, we have to solve the BPS equations without sources at the IR and match this solution with another one in which the D6-brane charge is non-vanishing and such that it reduces to the massless flavored solution of [23] in the deep UV. Amazingly, we have been able to find an analytic solution in the region without sources which contains a free parameter which can be tuned in such a way that the background reduces to the massless flavored geometry in the asymptotic UV. This semi-analytic solution interpolates between two different conformal geometries and contains the quark mass and the number of flavors as control parameters.

With the supergravity dual at our disposal, we can study the holographic flow for different observables. The general picture we get from this analysis is the following. Let be a length scale characterizing the observable. Then, the relevant parameter to explore the flow is the dimensionless quantity . When is very large (small) the observable is dominated by the IR unflavored (UV massless flavored) conformal geometry, whereas for intermediate values of we move away from the fixed points. We will put a special emphasis on the study of the holographic entanglement entropy, following the prescription of [25]. In particular, we study the refined entanglement entropy for a disk proposed in [26], which can be used as a central function for the F-theorem [27]. We check the monotonicity of the refined entropy along the flow (see [28] for a general proof of this monotonic character in three-dimensional theories). Other observables we analyze are the Wilson loop and quark-antiquark potential, the two-point functions of high-dimension bulk operators, and the mass spectrum of quark-antiquark bound states.

The rest of this paper is divided into two parts. The first part starts in Section 2 with a brief review of the ABJM solution. In Section 3 we introduce the squashed ansatz, write the master equation for the BPS geometries with sources, and classify its solutions according to their UV behavior. In Section 4 we write the analytic solution of the unflavored system that was mentioned above while, in Section 5 we construct solutions which interpolate between an unflavored IR region and a UV domain with D6-brane sources. The backgrounds corresponding to ABJM flavors with a given mass are studied in Section 6.

In the second part of the paper we study the different observables. In Section 7 we analyze the holographic entanglement entropy for a disk. Section 8 is devoted to the calculation of the quark-antiquark potential from the Wilson loop. In Section 9 we study the two-point functions of bulk operators with high mass, while the meson spectrum is obtained in Section 10. Section 11 contains a summary of our results and some conclusions. The paper is completed with several appendices with detailed calculations and extensions of the results of the main text.

## 2 Review of the ABJM solution

The ten-dimensional metric of the ABJM solution in string frame is given by:

where and are respectively the and metrics. The former, in Poincaré coordinates, is given by:

where is the Minkowski metric in 2+1 dimensions. In (2.1) is the radius of the part of the metric and is given, in string units, by:

 L4ABJM=2π2Nk, (2.3)

where and are two integers which correspond, in the gauge theory dual, to the rank of the gauge groups and the Chern-Simons level, respectively. The ABJM background contains a constant dilaton, which can be written in terms of and as:

 eϕABJM=2LABJMk=2√π(2Nk5)14. (2.4)

Apart from the metric and the dilaton written above, the ABJM solution of type IIA supergravity contains a RR two-form and a RR four-form , whose expressions can be written as:

where is the Kähler form of and is the volume element of the metric (2.2). It follows from (2.5) that and are closed forms (i.e., ).

The metric of the manifold in (2.1) is the canonical Fubini-Study metric. Following the approach of [23], we will regard as an -bundle over , where the fibration is constructed by using the self-dual instanton on the four-sphere. This representation of is the one obtained when it is constructed as the twistor space of the four-sphere. As in [23], this - representation will allow us to deform the ABJM background by squashing appropriately the metric and forms, while keeping some amount of supersymmetry. More explicitly, we will write as:

 ds2CP3=14[ds2S4+(dxi+ϵijkAjxk)2], (2.6)

where is the standard metric for the unit four-sphere, () are Cartesian coordinates that parameterize the unit two-sphere () and are the components of the non-Abelian one-form connection corresponding to the instanton. Let us now introduce a specific system of coordinates to represent the metric (2.6) and the two-form . First of all, let () be a set of left-invariant one-forms satisfying . Together with a new coordinate , the ’s can be used to parameterize the metric of the four-sphere as:

 ds2S4=4(1+ξ2)2[dξ2+ξ24((ω1)2+(ω2)2+(ω3)2)], (2.7)

where is a non-compact coordinate. The instanton one-forms can be written in these coordinates as:

 Ai=−ξ21+ξ2ωi. (2.8)

Let us next parameterize the coordinates of the unit by two angles and (, ),

 x1=sinθcosφ,x2=sinθsinφ,x3=cosθ. (2.9)

Then, it is straightforward to demonstrate that the part of the Fubini-Study metric can be written as:

 (dxi+ϵijkAjxk)2=(E1)2+(E2)2, (2.10)

where and are the following one-forms:

 E1=dθ+ξ21+ξ2(sinφω1−cosφω2), E2=sinθ(dφ−ξ21+ξ2ω3)+ξ21+ξ2cosθ(cosφω1+sinφω2). (2.11)

Therefore, the metric can be written in terms of the one-forms defined above as:

 ds2CP3=14[ds2S4+(E1)2+(E2)2]. (2.12)

We will now write the expression of in such a way that the - split structure is manifest. Accordingly, we define three new one-forms as:

 S1=sinφω1−cosφω2, S2=sinθω3−cosθ(cosφω1+sinφω2), S3=−cosθω3−sinθ(cosφω1+sinφω2). (2.13)

Notice that the are just the rotated by the two angles and . In terms of the forms defined in (2.13) the line element of the four-sphere is obtained by substituting in (2.7). Let us next define the one-forms and as:

 Sξ=21+ξ2dξ,Si=ξ1+ξ2Si,(i=1,2,3), (2.14)

in terms of which the metric of the four-sphere is . Moreover, the RR two-form in (2.5) can be written in terms of the one-forms defined in (2.11) and (2.14) as:

 F2=k2(E1∧E2−(Sξ∧S3+S1∧S2)). (2.15)

The solution of type IIA supergravity reviewed above is a good gravity dual of the ABJM field theory when the radius is large in string units and when the string coupling constant is small. From (2.3) and (2.4) it is straightforward to prove that these conditions are satisfied if and are in the range .

## 3 Squashed solutions

Let us consider the deformations of the ABJM background which preserve the - splitting. These deformed backgrounds will solve the equations of motion of type IIA supergravity (with sources) and will preserve at least two supercharges. We will argue below that some of these backgrounds are dual to Chern-Simons matter theories with fundamental massive flavors.

The general ansatz for the ten-dimensional metric of our solutions in string frame takes the form:

 ds210=h−1/2dx21,2+h1/2[dr2+e2fds2S4+e2g((E1)2+(E2)2)], (3.1)

where the warp factor and the functions and depend on the holographic coordinate . Notice that and determine the sizes of the and within the internal manifold. Actually, their difference determines the squashing of the and will play an important role in characterizing our solutions. We will measure this squashing by means of the function , defined as:

 q≡e2f−2g. (3.2)

Clearly, the ABJM solution has . A departure from this value would signal a non-trivial deformation of the metric. Similarly, the RR two- and four-forms will be given by:

 F4 =Kd3x∧dr, (3.3) F2 =k2(E1∧E2−η(Sξ∧S3+S1∧S2)), (3.4)

where is a constant and , are new functions. The background is also endowed with a dilaton . As compared with the ABJM value (2.15), the expression of in our ansatz contains the function which generically introduces an asymmetry between the and terms. Moreover, when the two-form is no longer closed and the corresponding Bianchi indentity is violated. Indeed, one can check that:

 (3.5)

The violation of the Bianchi identity of means that we have D6-brane sources in our model. Indeed, since , if then the Maxwell equation of contains a source term, which is due to the presence of D6-branes since the latter are electrically charged with respect to . The charge distribution of the D6-brane sources is determined by the function , which we will call the profile function.

The function of the RR four-form can be related to the other functions of the ansatz by using its equation of motion and the flux quantization condition for the integral of over the internal manifold. The result is [23]:

 K=3π2Nh−2e−4f−2g, (3.6)

where the integer is identified with the ranks of the gauge groups in the gauge theory dual (i.e., with the number of colors).

It is convenient to introduce a new radial variable , related to through the differential equation:

 xdrdx=eg. (3.7)

From now on, all functions of the holographic variable are considered as functions of , unless otherwise specified. The ten-dimensional metric in this new variable takes the form:

 ds210=h−12dx21,2+h12[e2gdx2x2+e2fds2S4+e2g((E1)2+(E2)2)]. (3.8)

It was shown in [23] that the background given by the ansatz written above preserves supersymmetry in three dimensions if the functions satisfy a system of first-order differential equations. It turns out that this BPS system can be reduced to a unique second-order differential equation for a particular combination of the functions of the ansatz. The details of this reduction are given in Appendix A. Here we will just present the final result of this analysis. First of all, let us define the function as:

 W(x)≡4kh14e2f−g−ϕx. (3.9)

Then, the BPS system can be reduced to the following second-order non-linear differential equation for :

 W′′+4η′+(W′+4η)[W′+10η3W−W′+4η+6x(W′+4η)]=0. (3.10)

We will refer to (3.10) as the master equation and to as the master function. Interestingly, the BPS equations do not constrain the profile function . Therefore, we can choose (which will fix the type of supersymmetric sources of our system) and afterwards we can solve (3.10) for . Given and one can obtain the other functions that appear in the metric. Indeed, as proved in Appendix A, and are given by:

 eg(x)=xW13exp[23∫xη(ξ)dξW(ξ)], ef(x)=√3xW′+4ηW16exp[23∫xη(ξ)dξW(ξ)], (3.11)

while the warp factor can be written as:

 h(x)=4π2Nke−g(W′+4η)[∫∞xξe−3g(ξ)W(ξ)2dξ+β], (3.12)

where is a constant that determines the behavior of as ( if we impose that as ). Finally, the dilaton is given by:

 eϕ(x)=12kxh14W13(W′+4η)exp[23∫xη(ξ)dξW(ξ)]. (3.13)

From the expression of and in (3.11) it follows that the squashing function can be written in terms of the master function and its derivative as:

 q=3Wx(W′+4η). (3.14)

### 3.1 Classification of solutions

Let us study the behavior of the solutions of the master equation in the UV region . This analysis will allow us to have a classification of the different solutions. We will assume that the profile function reaches a constant value as , and we will denote:

 limx→∞η(x)=η0. (3.15)

Let us restrict ourselves to the case in which . We will assume that behaves for large as:

 W(x)≈A0xα,x→∞, (3.16)

where and are constants. It is easy to check that this type of behavior is consistent only when the exponent or, in other words, when grows at least as a linear function of when .

We will also characterize the different solutions by the asymptotic value of the squashing function , which determines the deformation of the internal manifold in the UV. Let us denote

 q0=limx→∞q(x). (3.17)

It follows from (3.14) that the asymptotic value of the squashing function and that of the profile function are closely related. Actually, this relation depends on whether the exponent in (3.16) is strictly greater or equal to one. Indeed, plugging the asymptotic behavior (3.16) in (3.14) one immediately proves that:

 q0=⎧⎪ ⎪⎨⎪ ⎪⎩3α,forα>1,3A0A0+4η0,forα=1. (3.18)

This result indicates that we have to study separately the cases and . As we show in the next two subsections these two different asymptotics correspond to two qualitatively different types of solutions.

#### 3.1.1 The asymptotic G2 cone

Let us assume that the master function behaves as in (3.16) for some . By plugging this asymptotic form in the master equation (3.10) and keeping the leading terms as , one readily verifies that the coefficient is not constrained and that the exponent takes the value:

 α=32. (3.19)

Therefore, it follows from (3.18) that the asymptotic squashing is:

 q0=2. (3.20)

Let us evaluate the asymptotic form of all the functions of the metric. From (3.11), we get, at leading order:

 eg≈C√x, (3.21)

where is a constant of integration. Moreover, since , the asymptotic value of the function is:

 ef≈√2C√x. (3.22)

Let us now evaluate the warp factor from (3.12). Clearly, we have to compute the integral:

 ∫∞xξe−3g(ξ)W(ξ)2dξ∼x−52, (3.23)

which vanishes when . Therefore, by choosing the constant in (3.12) to be non-vanishing we can neglect the integral (3.23) and, since , then the warp factor becomes also a constant when . To clarify the nature of the asymptotic metric, let us change variables, from to a new radial variable , defined as . Then, after some constant rescalings of the coordinates the metric becomes:

 ds210≈dx21,2+ds27, (3.24)

where is:

 ds27=dρ2+ρ24[2ds2S4+(E1)2+(E2)2]. (3.25)

The metric (3.25) is a Ricci flat cone with holonomy, whose principal orbits at fixed are manifolds with a squashed Einstein metric. In the asymptotic region of large the line element (3.25) coincides with the metric of the resolved Ricci flat cone found in [29], which was constructed from the bundle of self-dual two-forms over and is topologically (see [30] for applications of this manifold to the study of the dynamics of M-theory).

#### 3.1.2 The asymptotic AdS metric

Let us now explore the second possibility for the exponent in (3.16), namely . In this case the coefficient cannot be arbitrary. Indeed, by analyzing the master equation as we find that and must be related as:

 A20+(9−η0)A0−20η20=0. (3.26)

On the other hand, should be related to the asymptotic squashing as in (3.18), which we now write as:

 A0=4q03−q0η0. (3.27)

By plugging (3.27) into (3.26) we arrive at the following quadratic relation between and :

 q20−3(1+η0)q0+5η0=0. (3.28)

Using this equation we can re-express as:

 A0=q0(η0+q0)2−q0. (3.29)

Moreover, we can solve (3.28) for and obtain the following two possible asymptotic squashings in terms of :

 q±0=12[3+3η0∓√9η20−2η0+9]. (3.30)

Thus, there are two possible branches in this case, corresponding to the two signs in (3.30). In this paper we will only consider the case, since this is the one which has the same asymptotics as the ABJM solution when there are no D6-brane sources. Indeed, (3.30) gives when , which means that the internal manifold in the deep UV is just the un-squashed (when there are no D6-brane sources in the UV, see (3.5)).

Let us now study in detail the asymptotic metric in the UV corresponding to the squashing (which from now on we simply denote as ). By substituting and in (3.11) and performing the integral, we get:

 eg(x)≈Cx23(1+η0A0), (3.31)

where is a constant of integration. Using (3.27) this expression can be rewritten as:

 eg(x)≈Cx1b, (3.32)

where is given by:

 b=2q0q0+1. (3.33)

The remaining functions of the metric can be found in a similar way. We get for and the following asymptotic expressions:

 ef(x)≈C√q0x1b,h(x)≈4π2Nk2−bC4A01x4b. (3.34)

Let us write the above expressions in terms of the original variable, which can be related to by integrating the equation:

 xdrdx=eg≈Cx1b. (3.35)

For large we get:

 r≈bCx1b, (3.36)

and the functions , , and can be written in terms of as:

 eg≈rb,ef≈√q0br,h≈L40r4, (3.37)

where is given by:

 L40=4π2Nk(2−b)b4A0. (3.38)

In terms of the asymptotic values and , can be written as:

 L40=128π2Nk(2−q0)q30(η0+q0)(q0+1)5. (3.39)

Using these results we find that the asymptotic metric takes the form:

where we have rescaled the Minkowski coordinates as . The metric (3.40) corresponds to the product of space with radius and a squashed . The parameter will play an important role in the following. Its interpretation is rather clear from (3.40): it represents the relative squashing of the part of the asymptotic metric with respect to the part.

It is now straightforward to show that in the UV the dilaton reaches a constant value , related to and as:

 eϕ0≈4√2π[(2−q0)5q0(q0+1)(η0+q0)5]14(2Nk5)14, (3.41)

while the RR four-form approaches the value:

where is the volume element of .

Interestingly, when the profile function is constant and equal to , the metric, dilaton, and forms written above solve the BPS equations not only in the UV, but also in the full domain of the holographic coordinate. Equivalently, is an exact solution to the master equation (3.10) if is constant and equal to and is given by (3.27). Actually, when one can check that and the asymptotic background becomes the ABJM solution ( for this case). Moreover, when the background corresponds111 Notice that the expression for written in (3.33) is equivalent to the one obtained in [23], namely: In order to check this equivalence it is convenient to use the following relation between and :   to the one found in [23] for the ABJM model with unquenched massless flavors, if one identifies with , where is the number of flavors.

The main objective of this paper is the construction of solutions which interpolate between the ABJM background in the IR and the asymptotics with in the UV. Equivalently, we are looking for backgrounds such that the squashing function varies from the value when to for . These backgrounds naturally correspond to gravity duals of Chern-Simons matter models with massive unquenched flavors. Indeed, in such models, when the energy scale is well below the quark mass the fundamentals are effectively integrated out and one should recover the unflavored ABJM model. On the contrary, if the energy scale is large enough the quarks can be taken to be massless and the corresponding gravity dual should match the one found in [23]. In the next section we present a one-parameter family of analytic unflavored solutions which coincide with the ABJM background in the deep IR and that have a squashing function which grows as we move towards the UV. In Sections 5 and 6 we show that these running solutions can be used to construct the gravity duals to massive flavor that we are looking for.

## 4 The unflavored system

In this section we will consider the particular case in which the profile is . In this case and there are no flavor sources. It turns out that one can find a particular analytic solution of the BPS system written in Appendix A. This solution was found in [23] in a power series expansion around the IR. Amazingly, this series can be summed exactly and a closed analytic form can be written for all functions. Let us first write them in the coordinate . The functions and are given by:

 ef=r√1+cr1+2cr,eg=r1+cr1+2cr, (4.1)

where is a constant. For this solution is ABJM without flavor (i.e., with fluxes), while for it is a running background which reduces to ABJM in the IR, . The squashing function can be immediately obtained from (4.1):

 q=1+2cr1+cr. (4.2)

For the squashing function interpolates between the ABJM value in the IR and the UV value:

 q0=2. (4.3)

The warp factor for this solution is:

 h(r)=2π2Nk(1+2cr)2r4(1+cr)2[1+2cr(3cr(1+2cr)−1)+ +12c3(1+cr)r3(log[cr1+cr]+α)], (4.4)

where is a constant which has to be fixed by adjusting the behavior of the metric in the UV. Finally, the dilaton can be related to the warp factor as:

 eϕ=2k1+cr(1+2cr)2rh14. (4.5)

Let us now re-express this running analytic solution in terms of the variable , related to by (3.7), which in the present case becomes:

 1xdxdr=1+2cr(1+cr)r. (4.6)

This equation can be easily integrated:

 γx=cr(1+cr), (4.7)

where is a constant of integration which parameterizes the freedom from passing to the variable. By solving (4.7) for we get:

 r=12c[√1+4γx−1]. (4.8)

It is straightforward to write the functions and in terms of :

 ef=γcx[2√1+4γx(√1+4γx+1)]12, eg=γcx√1+4γx, (4.9)

while the squashing function is:

 q=2√1+4γx1+√1+4γx. (4.10)

The warp factor in terms of the variable is:

 +24log[√4γx√1+4γx+1]+α]. (4.11)

By choosing appropriately the constant in (4.11) this running solution behaves as the -cone in the UV region . The dilaton as a function of is:

 eϕ=2kγcx1+4γxh14. (4.12)

Working in the variable , it is very interesting to find the function . For the solution described above, can be found by plugging the different functions in the definition (3.9). We find:

 W(x)=4(1+4γx)x1+√1+4γx. (4.13)

One can readily check that the function written in (4.13) solves the master equation (3.10) for . For large , the function behaves as:

 W∼8√γx32, (4.14)

which corresponds to an exponent in (3.16). This is consistent with the asymptotic value of the squashing found above.

Let us finally point out that we have checked explicitly that the geometry discussed in this section is free of curvature singularities.

## 5 Interpolating solutions

Let us now construct solutions to the BPS equations which interpolate between an IR region in which there are no D6-brane sources (i.e., with ) and a UV region in which and, therefore, the Bianchi identity of is violated. In the variable the profile will be such that for , while for . In the region our interpolating solutions will reduce to the unflavored running solution of Section 4 for some value of the constant . In order to match this solution with the one in the region it is convenient to work in the coordinate (4.7). The point will correspond to some . Notice, however, that we have some freedom in performing the change of variables. This freedom is parameterized by the constant in (4.7). We will fix this freedom by requiring that , i.e., that the transition between the unflavored and flavored region takes place at the point . Then, (4.7) immediately implies that is given in terms of and :

 γ=crq(1+crq). (5.1)

We will use (5.1) to eliminate the constant in favor of and . Actually, if we define as:

 ^γ≡√1+4γ, (5.2)

then is given by

 c=^γ−12rq. (5.3)

In this running solution the squashing factor is equal to one in the deep IR at . When the function grows monotonically until it reaches a certain value at , which is related to the parameter as:

 ^γ=^q2−^q. (5.4)

In the region we have to solve the master equation (3.10) with and initial conditions given by the values of and attained by the unflavored running solution at . These values depend on the parameter . They can be straightforwardly found by taking in the function (4.13) and in its derivative. We find:

 W(x=1)=4^γ21+^γ,W′(x=1)=6^γ−4. (5.5)

Let us now write the different functions of these interpolating solutions in the two regions and . For we have to rewrite (4.9) after eliminating the constant by using (5.3) (which implies that ). For the functions , , and the dilaton we get:

 ef=rq^γ+1√2x[√1+4γx(√1+4γx+1)]12, eg=rq^γ+12x√1+4γx,(x≤1), eϕ=rq^γ+1kx1+4γxh14, (5.6)

where is the function written in (4.11) for . By using the general equations of Section 3, the solution for can be written in terms of , which can be obtained by numerical integration of the master equation with initial conditions (5.5). This defines a solution in the full range of for every and . Notice that (3.11), (3.12), and (3.13) contain arbitrary multiplicative constants, which we will fix by imposing continuity of , , and at . We get for , and :

 eg=rq[(^γ+1)22^γ]13xW13exp[23∫x1η(ξ)dξW(ξ)],(x≥1), eϕ=rq[(^γ+1)22^γ]1312kxh14W13(W′+4η)exp[23∫x1η(ξ)dξW(ξ)]. (5.7)

The warp factor for is given by (3.12), where the integration constant is related to the constant of (4.11) by the following matching condition at :

 limx→1−h(x)=limx→1+h(x). (5.8)

For a given profile function , the solution described above depends on the parameter , which determines for through (4.13) and sets the initial conditions (5.5) needed to integrate the master equation in the region. The solution obtained numerically in this way grows generically as for large which, according to our analysis in Section 3.1.1, gives rise to the geometry of the -cone in the UV. We are, actually, interested in obtaining solutions with the asymptotics discussed in Section 3.1.2, for a set of profiles that correspond to flavor D6-branes with a non-zero quark mass. In order to get these geometries we have to fine-tune the parameter to some precise value which depends on the number of flavors. This analysis is presented in the next section.

## 6 Massive flavor

We now apply the formalism developed so far to find supergravity backgrounds representing massive flavors in ABJM. These solutions will depend on a deformation parameter , related to the total number of flavors and the Chern-Simons level as:

 ^ϵ≡3Nf4k, (6.1)

where the factor is introduced for convenience and is just