Gravity dual of a multilayer system

JHEP03(2019)064 Published for SISSA by Springer Received:January 23, 2019 Accepted:March 6, 2019 Published:March 12, 2019 Gravity dual of a multilayer system Niko Jokela,a,b Jos´e Manuel Pen´ın,c,d Alfonso V. Ramalloc,d and Dimitrios Zoakose aDepartment of Physics, University of Helsinki, P.O. Box 64, Helsinki FIN-00014, Finland bHelsinki Institute of Physics, P.O. Box 64, Helsinki FIN-00014, Finland cDepartamento de F´ısica de Part´ıculas, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain dInstituto Galego de F´ısica de Altas Enerx´ıas (IGFAE), Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain eDepartment of Physics, National and Kapodistrian University of Athens, 15784 Athens, Greece E-mail: [email protected],[email protected], [email protected],[email protected] Abstract: We construct a gravity dual to a system with multiple (2+1)-dimensional layers in a (3 + 1)-dimensional ambient theory. Following a top-down approach, we generate a geometry corresponding to the intersection of D3- and D5-branes along 2+1 dimensions. The D5-branes create a codimension one defect in the worldvolume of the D3-branes and are homogeneously distributed along the directions orthogonal to the defect. We solve the fully backreacted ten-dimensional supergravity equations of motion with smeared D5- brane sources. The solution is supersymmetric, has an intrinsic mass scale, and exhibits anisotropy at short distances in the gauge theory directions. We illustrate the running behavior in several observables, such as Wilson loops, entanglement entropy, and within thermodynamics of probe branes. Keywords: AdS-CFT Correspondence, D-branes, Gauge-gravity correspondence, Holography and condensed matter physics (AdS/CMT) ArXiv ePrint: 1901.02020 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP03(2019)064 JHEP03(2019)064 Contents 1 Introduction 1 2 Supergravity ansatz 3 2.1 Unflavored solution 6 2.2 Massless flavored solution 6 2.3 Massive flavors 7 3 Integration of the master equation 8 4 Wilson loops 13 4.1 Intralayer potential 14 4.1.1 UV limit 16 4.2 Interlayer potential 17 4.2.1 UV limit 19 5 Entanglement entropy 19 5.1 Parallel slab 20 5.1.1 UV limit 22 5.2 Transverse slab 22 5.2.1 UV limit 24 5.3 Flow of mutual information 25 6 Thermodynamics of a massless probe brane 26 7 Summary and conclusions 30 A Details of the background 32 B Probe analysis and profile function 36 B.1 General integration of the BPS equation 40 B.2 The profile function 40 C Equations of motion of the probe 43 C.1 The BPS solution 45 C.2 The massless solution 45 D Probes in the Higgs branch 46 D.1 Kappa symmetry 48 – i – JHEP03(2019)064 1 Introduction The holographic AdS/CFT correspondence relates strongly interacting quantum field theories (QFT) to classical gravity theories in higher dimensions [1]. Apart from being a conceptual breakthrough, this duality has become an important and versatile tool in the study of the possible states of matter (for reviews see [2–5]). There are basically two types of approaches to implement the holographic idea. In the so-called bottom-up approach, in order to model a d-dimensional QFT, one considers a gravity system in d+1 dimensions. This is somehow the minimal version of the correspondence and has been very successful in describing many interesting phenomena with models that evade the rigid constraints of string theory. On the contrary, the top-down models use the full machinery of string theory. They employ ten-dimensional gravity solutions of the corresponding equations of motion of type II supergravity. These ten-dimensional solutions are typically more difficult to obtain. However, the extra internal directions encode precious information which is lost in many phenomenological bottom-up approaches. Moreover, knowing the brane configuration which gives rise to the supergravity background allows to determine its field theory dual. Actually, one can engineer such holographic duals from the corresponding brane setup. In this paper we will employ string theory techniques to find gravity duals of systems composed by multiple parallel (2 + 1)-dimensional layers in a (3 + 1)-dimensional ambient theory (see figure 1). The ambient 4d theory will be N= 4 super Yang-Mills theory with gauge group SU(Nc), which is realized in a stack of coincident Nccolor branes. The multiple layers are obtained by adding parallel D5-branes sharing two spatial directions with the stack of color branes, according to the array: 0123456789 (Nc)D3 : × × × × (Nf)D5 : × × × × × × (1.1) The field theory corresponding to the array (1.1) is well-known [6–9]. It consists of a supersymmetric theory with matter hypermultiplets (flavors) living on the (2 + 1)-dimensional defect and coupled to the ambient N= 4 theory. To construct a multilayer structure of the type shown in figure 1we consider a continuous distribution of D5-branes along the third direction in (1.1). In the absence of D5-branes, the geometry generated by the D3-branes is AdS5×S5. We want to obtain a supergravity solution which includes the backreaction of the flavor D5-branes. This backreaction can be obtained by using the techniques developed to study unquenched holographic flavor (see [10] for a review and references). In some cases the solutions found with these techniques are analytic and preserve some amount of supersymmetry. In this approach the D5-branes of the array (1.1) are smeared both in the cartesian direction orthogonal to the defect and on the internal directions. The dual gravity background can be obtained by solving the supergravity equations of motion with D5-brane sources. These backgrounds are dual to anisotropic systems, since there is a distinct field theory direction, the direction orthogonal to the defect. Let us also comment on the novelty of our approach. There are several previous holographic works which – 1 – JHEP03(2019)064 x3 martes, 16 de octubre de 18 Figure 1. Our system has a bulk (3 + 1)-dimensional theory together with multiple (2 + 1)- dimensional layers. The direction of the coordinate x3is perpendicular to the layers. provide anisotropy, see, e.g., [11–22]. The main difference between our model and other anisotropic models in the literature is that in our case the anisotropy is produced by the presence of dynamical objects (the D5-branes of the multiple layers) and not by fluxes or fields depending anisotropically on the coordinates. The D3- and D5-branes of the array (1.1) can be separated in the directions 789 transverse to both types of branes. When this separation is zero the mass of the hypermultiplets living on the defect vanishes, i.e. we have massless flavors. This D3-D5 massless flavor case was considered in [23], where an analytic supersymmetric solution was found that displays a Lifshitz-like anisotropic scaling invariance. The non-zero temperature generalization of the scaling solution was found and studied in [24]. In this paper, we study the massive flavor case of the D3-D5 intersection (1.1). The solutions we present here preserve the same supersymmetry as the massless flavor case, but do not possess the scaling invariance of the latter. The gravity duals of theories with massive flavors are running solutions which naturally represent a renormalization group flow. This flow is generated by changing the quark mass (see [25] for an example of these massive flavored backgrounds in the ABJM theory). When the mass of the quarks is very large, the flavors decouple and we expect to recover the unflavored solution (AdS5×S5in our case). On the other hand, when the flavors are massless we obtain the anisotropic scaling solution of [23]. For a finite non-vanishing value of the quark mass we expect to get a background interpolating between these two solutions: unflavored in the IR and massless flavored in the UV. Once we have the background at our disposal we can study the effects of the flow on different observables. In general, we expect to obtain the results corresponding to the isotropic AdS5×S5solution in the IR and to be able to tune the amount of anisotropy by changing some of the parameters of our solutions. In our analysis of several observables we will find that, indeed, the UV behavior is determined by the scaling solution of [23], whereas the long distance IR behavior depends on a free parameter of our geometry. The rest of this paper is organized as follows. In section 2we present our ansatz for the metric, for the dilaton, and for the forms. All functions of the model depend on a master function, which in turn satisfies a second order differential equation. This equation follows from supersymmetry analysis detailed in appendix A. A crucial ingredient entering the equations is the so-called profile function, which encodes the distribution of sources – 2 – JHEP03(2019)064 along the holographic coordinate. To determine this function for D5-brane sources one needs to analyze in detail the embeddings of the D5-branes and their kappa symmetry. This analysis is deferred to appendix B. In section 3we tackle the problem of integrating the master equation. This task requires the redefinition of some of the functions and a change of variables. In its final form our solution depends on a constant parameter which characterizes the IR deformation of the metric. In section 4we begin our study of the observables in our background. In this section we study the Wilson loops and the potentials for quark-antiquark pairs, when these particles are in the same layer or separated in the direction orthogonal to the layers. In section 5we do a similar analysis for the entanglement entropy of slabs. In section 6we explore our supergravity solution with a probe D5-brane with a worldvolume gauge field dual to a chemical potential. We analyze the zero temperature thermodynamics of the probe and, in particular, the UV-IR flow of the speed of sound. This is not the only interesting configuration of probe branes with non-zero worldvolume gauge field. An interested reader is invited to appendix Dwhere we consider D5-branes with worldvolume flux along the internal directions. This internal flux induces a bending of the probe brane along the direction x3in the array (1.1), which can be interpreted as a recombination of the flavor D5-branes and the color D3-branes, realizing the Higgs branch of the theory [26,27]. Finally, in section 7we summarize our results and discuss some research lines for the future. 2 Supergravity ansatz In this section we review the supergravity setup of [23], corresponding to the array (1.1) of D3- and D5-branes. More details are given in appendix A. The D3-branes are color branes which generate an AdS5×S5space, whereas the flavor D5-branes create a codimension one defect in the (3 + 1)-dimensional gauge theory and, when the backreaction is included, the original AdS5×S5metric gets deformed. The D5-branes are homogeneously distributed in the internal space in such a way that some amount of supersymmetry is preserved. When the S5space is represented as a U(1) bundle over CP2, the deformation of the five-sphere depends on a single radial function, which measures the relative squashing between the fiber and the base of the deformed S5. Choosing a convenient radial coordinate ζ(with boundary corresponding to ζ=∞), the ten-dimensional Einstein frame metric takes the form: ds2 10 =h−1 2−(dx0)2+ (dx1)2+ (dx2)2+e−2φ(dx3)2 +h1 2hζ2e−2fdζ2+ζ2ds2 CP2+e2f(dτ +A)2i,(2.1) where φis the dilaton of type IIB supergravity, his the warp factor, and fis the squashing function. These functions are assumed to depend only on ζ. Moreover, Ais a one-form on CP2which implements the non-trivial U(1) bundle. The Minkowski directions x1and x2 are parallel to the defect, whereas x3is orthogonal to it. Besides the metric and the dilaton, the type IIB supergravity solution contains a RR five-form F5and a RR three-form F3. The former is self-dual and given by the standard – 3 – JHEP03(2019)064 ansatz in terms of the dilaton φand warp factor h: F5=∂ζe−φh−11 + ∗d4x∧dζ . (2.2) Clearly, d F5= 0, since the D3-branes have been replaced by a flux in the supergravity solution. On the contrary, the D5-branes are dynamical and are governed by the standard DBI+WZ action which, in particular, contains the term: SWZ =T5 Nf XZM6 ˆ C6,(2.3) where 1/T5= (2π)5gsα03and C6is the six-form potential for F7=−eφ∗F3(the hat over C6denotes its pullback to the D5-brane worldvolume M6). Therefore, SWZ contributes to the equation of motion of C6or, equivalently, to the Bianchi identity of F3. Indeed, let us write the six-dimensional integral in (2.3) as a ten-dimensional integral: Nf XZM6 ˆ C6=ZM10 Ξ∧C6,(2.4) where Ξ is a four-form with support on the worldvolume of the D5’s and with legs along the directions orthogonal to M6, which is just the RR D5-brane charge distribution. The equation of motion for C6is: dF3= 2 κ2 10 T5Ξ,(2.5) where 2 κ2 10 = (2π)7g2 sα04. In the smearing approach Ξ does not contain Dirac δ-function singularities. Its form can be obtained once the ansatz of F3compatible with supersymmetry is fixed. This has been done in [23], a result which we now review. The CP2manifold is a K¨ahler-Einstein space endowed with a K¨ahler two-form J= dA/2, which can be canonically written as J=e1∧e2+e3∧e4, where e1, . . . , e4are vielbein one-forms of CP2, whose explicit coordinate expression can be found in appendix A. Let us introduce the complex two-form ˆ Ω2as: ˆ Ω2=e3iτ (e1+ie2)∧(e3+ie4).(2.6) Then, F3is given by: F3=Qfp(ζ)dx3∧Im ˆ Ω2,(2.7) where Qfis a constant and p(ζ) is an arbitrary function of the holographic coordinate ζ. By computing the exterior derivative of F3we get its modified Bianchi identity: dF3=−Qf3p(ζ)dx3∧Re ˆ Ω2∧(dτ +A) + p0(ζ)dx3∧dζ ∧Im ˆ Ω2.(2.8) Comparing (2.8) and (2.5) we can extract the D5-brane charge distribution Ξ which, in what follows, we will refer to as the smearing form. Clearly, Ξ does not depend on the x3 coordinate, although it contains dx3in its expression. This means that we are continuously distributing our D5-branes along x3, giving rise to a system of multiple (2+1)-dimensional parallel layers. Moreover, the function p(ζ) introduces a profile of the charge distribution – 4 – JHEP03(2019)064 in the holographic coordinate. Notice that the D3- and D5-branes in the array (1.1) can be separated in the 789 directions. In principle, we could have an arbitrary distribution of D5-branes in these transverse coordinates, which is reflected in the fact that the profile function p(ζ) is arbitrary. However, for a stack of flavor D5-branes with the same quark mass the function p(ζ) has a well-defined form (see appendix B) and Qfis related to the density of smeared branes along the direction x3. As shown in appendix B, if we distribute NfD5-branes along a distance L3in the third cartesian direction, then Qf=4π gsα0Nf 9√3L3 (see (B.59)). The preservation of two supercharges for our ansatz imposes a system of first-order differential equations in the radial variable. These BPS equations are reviewed in appendix A, where it is shown that they can be reduced to a single second-order differential equation for a master function W(ζ). This equation is: d dζ ζdW dζ + 6 dW dζ =−6Qfp(ζ) ζ2√W.(2.9) From Wwe can reconstruct the full solution. The squashing function f(ζ) is given by: e2f=6ζ2W 6W+ζdW dζ ,(2.10) while the dilaton is: e−φ=W+1 6ζdW dζ .(2.11) A nice way of measuring the deformation of the metric (2.1) with respect to the AdS5×S5 geometry is obtained by considering a squashing factor, which we define as follows q=ef ζ.(2.12) It is clear from our ansatz (2.1) that qrepresents the relative size of the U(1) fiber with respect to the CP2base. In terms of the master function W,qis given by: q=√6W q6W+ζdW dζ .(2.13) Once fand φare known, the warp factor hcan be obtained by integrating the following first-order differential equation: dh dζ +Qf e3φ 2−fp ζh=−Qc ζ3e−2f,(2.14) where Qcis related to the number Ncof D3-branes as Qc= 16 πgsα02Nc. In what follows we will study several solutions of the master equation (2.9). – 5 – JHEP03(2019)064 2.1 Unflavored solution Let us consider the master equation in the case in which there are no flavor brane sources. It turns out that the general solution of (2.9) can be analytically found in this case. Indeed, when Qf= 0 the master equation (2.9) can be trivially integrated once as: ζdW dζ + 6 W= constant .(2.15) A further integration gives: W=C1−b6 ζ6,(2.16) where Cand bare constants. Plugging this expression of Winto the right-hand-side of (2.11) one readily verifies that the dilaton is constant and given by: e−φ=C . (2.17) Moreover, the function fand the squashing function qare given by: e2f=ζ21−b6 ζ6, q =1−b6 ζ6 1 2 ,(2.18) and the warp factor hcan be obtained by integrating (2.14). This solution coincides with the general unflavored one found in [23]. For b= 0 this geometry is just AdS5×S5. When b6= 0 the solution approaches AdS5×S5in the UV. If we take bto be real and positive, then the minimal value of ζis ζ=band the metric has a blown-up CP2cycle at ζ=b. It was argued in [28] that this b6= 0 background is dual to the superconformal N= 4 field theory deformed by the VEV of a dimension 6 operator. 2.2 Massless flavored solution Let us now consider the massless flavor case with Qf6= 0 and p= 1. In this case it is possible to find a special solution of (2.9): W=A ζα.(2.19) Indeed, by plugging this ansatz into the master equation we readily get that the exponent αis given by: α=−2 3.(2.20) Similarly, we can obtain the value of the constant A. The final formula for Wis: W=9 8√2Qf2 3ζ−2 3.(2.21) Using this expression in (2.10) and (2.11) we arrive at the following values of fand φ: e2f=9 8ζ2, e−φ=√2Qf2 3ζ−2 3.(2.22) – 6 – JHEP03(2019)064 D5 D3 ⇣q miércoles, 7 de noviembre de 18 ⇣q miércoles, 7 de noviembre de 18 Figure 2. On the left we depict a localized embedding of a D5-brane with a separation ζqfrom the stack of color D3-branes. On the right, several D5-branes with different orientations and the same distance ζqgenerate a cavity ζ≤ζqwhich does not contain flavor sources. It is straightforward to verify that this solution coincides with the one found in [23] for massless flavors.1The warp factor for this solution is: h=¯ R4 ζ4,(2.23) where ¯ Ris the same as in [23], ¯ R4=4 15 Qc.(2.24) Since the profile function pis constant, this solution represents massless smeared flavors extending all the way down to ζ= 0. As shown in [23], the background corresponding to the master function (2.21) is invariant under a set of anisotropic scale transformations in which the x3coordinate transforms with an anomalous scaling dimension. Moreover, the squashing factor (2.12)qis constant and equal to 3 2√2≈1.06, see (2.22). The purpose of this paper is to find solutions corresponding to massive flavors, which should interpolate between the unflavored solution of subsection 2.1 at the IR and the scaling solution studied in this subsection at the UV. We start to discuss these solutions in the next subsection. 2.3 Massive flavors In the holographic approach the mass of the fundamentals is related to the distance between the color and flavor branes (in our case D3’s and D5’s, respectively). When these two sets of branes are separated, the fundamentals are massive and there is an IR region of the geometry which is not occupied by the flavor brane and, as a consequence, the D5- brane charge density vanishes there. In the smearing setup we have many D5-branes with different orientations in the internal space which, if they correspond to flavors with the same mass, should have the same separation from the D3-branes. As illustrated in figure 2, the sourceless region has a ζcoordinate less or equal to some value ζq, a region we will call 1The relation between ζand the radial variable rused in [23] and in the ansatz (A.1) is ζ=3 2√2r. – 7 – JHEP03(2019)064 where g2is the induced metric on the worldsheet of the string. We will calculate the q¯q potential in two cases. First we will consider the intralayer case, in which the quark and the antiquark are in the same layer and have the same value of the coordinate x3. After this we will consider the interlayer configuration, where the quarks are separated in the anisotropic direction. 4.1 Intralayer potential We take (t, ˆx1) as worldvolume coordinates of a fundamental string and we will consider an ansatz with x=x(ˆx1) with the other cartesian coordinates being constant. The induced metric on the two-dimensional worldsheet is: ds2 2=−ˆ h−1 2(dˆx0)2+ˆ h−1 2 1 + ˆ h Z2 q q2(x0)2!(dˆx1)2,(4.2) where the prime denotes derivative with respect to ˆx1and we have denoted Zq≡Z(xq). The Nambu-Goto action of the string is: S ˆ T=1 2πZdˆx1eφ 2√−g2≡Zdˆx1L , (4.3) where ˆ T=Rdˆx0and the Lagrangian density Lis: L=1 2π√C ˆ h−1 2q pˆ Ws1 + Z2 q ˆ h q2(x0)2.(4.4) As Ldoes not depend explicitly on the holographic coordinate x, one has the following first integral: x0∂L ∂x0−L= constant ,(4.5) from which we get: q pˆ hˆ W 1 q1 + Z2 q ˆ h q2(x0)2 =q0 pˆ h0ˆ W0 ,(4.6) where q0,ˆ h0, and ˆ W0are the values of the functions q,ˆ h, and ˆ W, respectively, at the turning point x=x0. From this relation we obtain x0as: x0=±q Zqpˆ hsˆ h0ˆ W0q2 ˆ hˆ W q2 0−1,(4.7) which can be straightforwardly integrated to give: ˆx1(x) = ±ZqZx x0qˆ h(¯x) q(¯x) d¯x rˆ h0ˆ W0q2(¯x) ˆ h(¯x)ˆ W(¯x)q2 0−1 .(4.8) – 14 – JHEP03(2019)064 The (hatted) quark-antiquark distance at the boundary is: ˆ dk= 2 ZqZ∞ x0qˆ h(x) q(x) dx rˆ h0ˆ W0q2(x) ˆ h(x)ˆ W(x)q2 0−1 .(4.9) Let us now calculate the on-shell action of the fundamental string. Plugging the solution into the action, we get: Son-shell ˆ T=Zq π√CZxmax x0 dx pˆ Wr1−q2 0ˆ hˆ W q2ˆ h0ˆ W0 .(4.10) As usual, this on-shell action is divergent and must be regulated. We do it by subtracting the action of two straight fundamental strings stretched from the origin x= 0 to x=xmax: Sreg on-shell ˆ T=Son-shell ˆ T−2Zq 2πZxmax 0 dx eφ 2 q=Son-shell ˆ T−Zq π√CZxmax 0 dx pˆ W .(4.11) The quark-antiquark potential is then given by: Vq¯q=Sreg on-shell T=Qf C3 2 Sreg on-shell ˆ T,(4.12) where we have used the relation between Tand ˆ T: ˆ T=Qf C3 2 T . (4.13) More explicitly: Vq¯q=ZqQf π C2   Z∞ x0 dx pˆ W    1 r1−q2 0ˆ hˆ W q2ˆ h0ˆ W0 −1   −Zx0 0 dx pˆ W    .(4.14) We have numerically evaluated the potential Vq¯qas a function of ˆ dk. The results are presented in figure 7for different values of xq. We notice that all curves become coincident for small ˆ dk. As ˆ dk∼m 3 4 qdk(see eq. (3.20)), one expects to recover the massless scaling solution in the UV domain ˆ dk→0. Indeed, we prove below that Vq¯q∼ˆ d−4 3 kin the UV region of small ˆ dk. This behavior matches the one obtained numerically, as shown in figure 7. As we move towards the IR by decreasing the turning point coordinate x0and increasing ˆ dk, we obtain that there is a maximal value of ˆ dk(corresponding to a minimal value xmin 0< xqof the turning point coordinate). For x0< xmin 0the dominant configuration is a disconnected one, in which the two ends of the string go straight from the boundary to the origin. This behavior has been obtained previously in backgrounds dual to unquenched flavors [31–34] (in other types of backgrounds, see also [35–39]). Indeed, dynamical quarks produce string breaking and a maximal length due to pair creation. In our case this breaking is not – 15 – JHEP03(2019)064 C2 Qf V ˆ dk pQc 1 2 3 4 5 -0.8 -0.6 -0.4 -0.2 0.0 jueves, 20 de diciembre de 18 Figure 7. We depict the ¯qq intralayer potential (4.14) versus ˆ dk∝m 3 4 qdkfor xq= 0.1 (red), xq= 0.5 (blue), and xq= 0.9 (brown). The dashed curve is the UV limit (4.18). Notice that the maximal separation for ¯qq increases with xq. produced in the scaling solution with mq= 0. Moreover, the critical distance at which the string breaks grows with xq, as is also evident in figure 7, and becomes very large when xq∼1. This is easy to understand since the breaking occurs when the string penetrates deeply in the cavity, whose size is maximal when xq∼1, and the integrals (4.9) and (4.14) get their main contribution from the sourceless region inside the cavity. For large enough values of ˆ dkthe dominant configuration is the disconnected one with zero energy, which means that the external quarks are completely screened by dynamical quarks popping out from the vacuum. A recent interesting work [22], in a seemingly unrelated context of holographic QCD, parallels our findings. The authors of [22] demonstrated that large amounts of anisotropy will completely screen the interactions between quarks and antiquarks, while in the absence of anisotropy the model would otherwise be confining. 4.1.1 UV limit Let us now evaluate the potential in the UV limit in which dkis small (large x0) and our embedding is close to the boundary. In this limit we can use that, at leading order in the UV, the squashing function qis constant and that ˆ W∼x−1 3and ˆ h∼x−4(see eqs. (3.17), (A.20), and (A.21)). We will use these values to calculate the integrals for ˆ dk and Vq¯qin (4.9) and (4.14). At leading order we get the following relation between ˆ dkand x0: ˆ dk≈8√2 3√15 √Qc Zq √π Γ5 7 Γ3 14  1 x0 .(4.15) Similarly, we approximate the potential Vq¯qby the following integral: Vq¯q≈2√2 3·21 6π Z 4 3 qQf C2x 4 3 0 Z∞ 1 dz z 1 3  z7 3 qz14 3−1−1 −3 4 .(4.16) – 16 – JHEP03(2019)064 Performing the integral, we get: Vq¯q≈ − 1 21 6√2π Z 4 3 qQf C2 Γ5 7 Γ3 14 x 4 3 0.(4.17) By using the relation (4.15) we can eliminate x0in favor of the q¯qdistance ˆ dk. After some calculation we get: C2 Qf Vq¯q≈ −βk √Qc ˆ dk!4 3 , βk=16π1 6 9·52 3  Γ5 7 Γ3 14   7 3 ,(4.18) which coincides with the result found in [24] for the massless scaling background. In figure 7 we show that the potential (4.18) does indeed coincide with the numerical results in the UV domain ˆ dk→0 for all values of the parameter xq. 4.2 Interlayer potential We now consider a Wilson loop that extends in the x3direction with x1and x2constant, which corresponds to two fundamentals located at different layers. Accordingly, we take (t, ˆx3) as worldvolume coordinates and consider an ansatz in which x=x(ˆx3). The twodimensional induced metric is now: ds2 2=−ˆ h−1 2(dˆx0)2+ˆ h−1 2 q4ˆ W2+Z2 qˆ h q2(x0)2(dˆx3)2,(4.19) where the prime now denotes derivative with respect to ˆx3. The Nambu-Goto action is: S ˆ T=1 2πZdˆx3eφ 2√−g2≡Zdˆx3L , (4.20) with L=1 2π√C ˆ h−1 2 qpˆ Wqˆ W2+Z2 qˆ h q2(x0)2.(4.21) The first integral derived from Lis: pˆ W qpˆ h 1 r1 + Z2 qq2ˆ h ˆ W2(x0)2 =pˆ W0 q0pˆ h0 ,(4.22) where again q0,ˆ h0, and ˆ W0are the values of the functions q,ˆ h, and ˆ W, respectively, at the turning point x=x0. Then: x0=±ˆ W Zqqpˆ hsˆ h0q2 0ˆ W ˆ h q2ˆ W0−1.(4.23) – 17 – JHEP03(2019)064 C2 Qf V ˆ d? pQc 1 2 3 4 5 -0.8 -0.6 -0.4 -0.2 0.0 jueves, 20 de diciembre de 18 C2 Qf V ˆ d pQc 1 2 3 4 5 -0.8 -0.6 -0.4 -0.2 jueves, 20 de diciembre de 18 Figure 8. On the left we plot the ¯qq interlayer potential (4.27) versus ˆ d⊥∝m 1 4 qd⊥for xq= 0.1 (red), xq= 0.5 (blue), and xq= 0.9 (brown). The dashed curve is the UV potential (4.31). On the right we compare the intralayer and interlayer potentials for xq= 0.1 (red), xq= 0.5 (blue), and xq= 0.9 (brown). The continuous (dashed) curves correspond to the intralayer (interlayer) potentials. Integrating this equation we get: ˆx3(x) = ±ZqZx x0 q(¯x)qˆ h(¯x) ˆ W(¯x) d¯x rˆ h0q2 0ˆ W(¯x) ˆ h(¯x)q2(¯x)ˆ W0−1 .(4.24) Therefore, the quark-antiquark distance along the direction transverse to the layers is: ˆ d⊥= 2 ZqZ∞ x0 q(x)qˆ h(x) ˆ W(x) dx rˆ h0q2 0ˆ W(x) ˆ h(x)q2(x)ˆ W0−1 .(4.25) The unregulated on-shell action for this case is: Son-shell ˆ T=Zq π√CZxmax x0 dx pˆ Wr1−q2ˆ hˆ W0 q2 0ˆ h0ˆ W .(4.26) Proceeding as in the intralayer case to regulate this action, we arrive at the following quark-antiquark potential: Vq¯q=ZqQf π C2   Z∞ x0 dx pˆ W    1 r1−q2ˆ hˆ W0 q2 0ˆ h0ˆ W −1   −Zx0 0 dx pˆ W    .(4.27) The numerical results for the interlayer potential have been plotted in figure 8. They are qualitatively similar to the intralayer case of figure 7. In the UV region ˆ d⊥∝m 1 4 qd⊥→0, the potential decays as Vq¯q∼d−4 ⊥, in agreement with our analytic calculation of section 4.2.1. – 18 – JHEP03(2019)064 In this case there is also a maximal length which increases with xq. We have also compared the intralayer and interlayer potentials for the same value of xq. These potentials are plotted together on the right panel of figure 8, where we notice that they have very different behavior in the UV but they become very similar in the IR (cf. also [22]). This IR similarity increases as xqapproaches its maximal value xq= 1, which is consistent with the fact that for large values of xqthe isotropic unflavored limit is rapidly attained in the IR. 4.2.1 UV limit Let us consider the UV limit in which ˆ d⊥is small and we have the following approximate relation between d⊥and x0: ˆ d⊥≈8·21 6 √15 √Qc Z 1 3 q √π Γ3 5 Γ1 10  1 x 1 3 0 .(4.28) In this limit the potential can be approximated as: Vq¯q≈2√2 3·21 6π Z 4 3 qQf C2x 4 3 0 Z∞ 1 dz z 1 3  z5 3 qz10 3−1−1 −3 4 .(4.29) The integral can be computed analytically and yields the following result for the potential: Vq¯q≈ − 1 21 6√2π Z 4 3 qQf C2 Γ3 5 Γ1 10 x 4 3 0.(4.30) In terms of ˆ d⊥we get: C2 Qf Vq¯q≈ −β⊥ √Qc ˆ d⊥!4 , β⊥=212 π3 2 32·52  Γ3 5 Γ1 10   5 ,(4.31) which is equivalent to the result obtained in [24] for the massless background. As illustrated in figure 8, the potential (4.31) nicely matches the numerical results. 5 Entanglement entropy The entanglement entropy of a region and its complement is a good measure of the quantum correlations of the system. In holography the entanglement entropy is obtained by minimizing an area functional for an eight-dimensional surface embedded in ten-dimensional spacetime [40,41]. Let Abe a spatial region in the gauge theory. The holographic entanglement entropy between Aand its complement is: SA=1 4G10 ZΣ d8ξpdet g8,(5.1) where Σ is the eight-dimensional spatial surface whose boundary is Aand minimizes SA, G10 is the ten-dimensional Newton constant (G10 = 8π6in our units) and g8is the induced – 19 – JHEP03(2019)064 metric on Σ in the Einstein frame. In this section we will apply this prescription when Ais a slab of infinite extent in the two cartesian directions and having a finite width in the remaining cartesian coordinate. Clearly, there are two cases to study, namely parallel and transverse slabs, which we analyze separately. We note that again we find striking similarity with the results in [22]. 5.1 Parallel slab First we consider the case in which Ais an infinite slab with a finite width parallel to the layers, namely: A=(− ˆ lk 2≤ˆx1≤ ˆ lk 2,−∞ <ˆx2,ˆx3<+∞).(5.2) We will parameterize Σ by a function x=x(ˆx1). The eight-dimensional induced metric on Σ is: ds2 8=ˆ h−1 2"1 + Z2 q ˆ h q2(x0)2#(dˆx1)2+ˆ h−1 2"(dˆx2)2+ˆ W2 q4(dˆx3)2# +ˆ h1 2Z2 q(1 + x−xq)2ds2 CP2+q2(dτ +A)2,(5.3) where the prime now denotes derivative with respect to ˆx1. If we integrate over all the coordinates except x, we get: Sk ˆ L2ˆ L3 =Z5 q 32 π3Zdˆx1(1 + x−xq)5ˆ h1 2ˆ W qs1 + Z2 q ˆ h q2(x0)2.(5.4) The first integral derived from Skis: ˆ h1 2ˆ W q (1 + x−xq)5 q1 + Z2 q ˆ h q2(x0)2 =ˆ h 1 2 0ˆ W0 q0 (1 + x0−xq)5,(5.5) where the subscript nought denotes that the corresponding quantity is evaluated at the minimal value x0of x. From this last equation we get: x0=±q Zqpˆ hs(1 + x−xq)10 (1 + x0−xq)10 ˆ hˆ W2q2 0 ˆ h0ˆ W2 0q2−1,(5.6) which can be integrated to give ˆ lk: ˆ lk= 2ZqZ∞ x0pˆ h q dx r(1+x−xq)10 (1+x0−xq)10 ˆ hˆ W2q2 0 ˆ h0ˆ W2 0q2−1 .(5.7) The entanglement entropy for this configuration is given by the divergent integral: Sk ˆ L2ˆ L3 =Z6 q 16 π3Zxmax x0 ˆ hˆ W q2 (1 + x−xq)5 r1−(1+x0−xq)10 (1+x−xq)10 ˆ h0ˆ W2 0q2 ˆ hˆ W2q2 0 dx . (5.8) – 20 – JHEP03(2019)064 Sk Qc ˆ L2 ˆ L3 ˆ lk pQc 0.5 1.0 1.5 2.0 2.5 3.0 3.5 -0.0020 -0.0015 -0.0010 -0.0005 0.0000 domingo, 23 de diciembre de 18 Figure 9. Plot of the entanglement entropy for a parallel slab as a function of its rescaled width. The continuous lines are the results of the numerical integration of (5.11) and (5.7) for xq= 0.1 (red), xq= 0.5 (blue) and xq= 0.9 (brown). The dashed curve is the UV result (5.15). We will regularize Skby subtracting the entropy of a configuration that we call the flat surface in the following. To conform with the homology constraint, the surface is not disconnected, but it is connected at the bottom x= 0. The full flat surface consists of three constant pieces: two straight ˆx1=±lk/2 ones and a horizontal one x=const. = 0, each individually being solutions to the equation of motion. It turns out that the contribution of the horizontal surface to the area integral vanishes. The entropy for the flat embedding is thus Sflat k ˆ L2ˆ L3 =Z6 q 16 π3Zxmax 0 ˆ hˆ W q2(1 + x−xq)5dx . (5.9) Thus, we define the finite entanglement entropy as: Sfinite k ˆ L2ˆ L3 =Sk−Sflat k ˆ L2ˆ L3 .(5.10) After some calculations, we get: Sfinite k ˆ L2ˆ L3 =−Z6 q 16 π3   Z∞ x0 ˆ hˆ W q2(1 + x−xq)5    1−1 r1−(1+x0−xq)10 (1+x−xq)10 ˆ h0ˆ W2 0q2 ˆ hˆ W2q2 0     dx +Zx0 0 ˆ hˆ W q2(1 + x−xq)5dx    .(5.11) The numerical results for Skversus ˆ lkare presented in figure 9for several values of the parameter xq. In this plot we notice that Skbecomes positive when ˆ lkis large enough. According to our regularization procedure (5.10) this means that the disconnected surface is dominant with respect to the connected one when ˆ lk>ˆ lc k, where ˆ lc kis a critical length – 21 – JHEP03(2019)064 which grows with xq. On the contrary, for small ˆ lkthe entropy behaves as Sk∼ˆ l−4 3 k, with a coefficient independent of xq. As pointed out in [24], this universal behavior is the one corresponding to an effective D2-brane and can be obtained analytically, as we show in the next subsection. A similar behavior of the EE has previously been obtained in backgrounds dual to confining theories and unquenched flavors, see [42–44]. 5.1.1 UV limit When ˆ lkis small, the minimal value x0of xis large and we can use the expansion of the functions of the background valid for large x. At leading order, we get: ˆ lk≈8√2 3√15 √Qc ZqZ∞ x0 dx x2q(x/x0)14 3−1 =8√2π 3√15 Γ5 7 Γ3 14  √Qc Zq 1 x0 .(5.12) Similarly, the regulated entanglement entropy for the parallel slab is: Sfinite k ˆ L2ˆ L3≈ − Z 4 3 qQc 15 ·25 3π3x 4 3 0  3 4−Z∞ 1 dz z 1 3  z7 3 qz14 3−1−1  ,(5.13) which can be integrated analytically with the result: Sfinite k ˆ L2ˆ L3≈ − Z 4 3 qQc 5·211 3π5 2 Γ5 7 Γ3 14 x 4 3 0.(5.14) If we eliminate x0and write the entanglement entropy in terms of ˆ lk, we get: Sfinite k Qcˆ L2ˆ L3≈ −γk √Qc ˆ lk!4 3 , γk=2 45 ·52 3π11 6  Γ5 7 Γ3 14   7 3 ,(5.15) which is the same result as in [24] and, as shown in figure 9, matches perfectly the numerical results when ˆ lkis small. 5.2 Transverse slab We now consider a slab with finite width in the direction of ˆx3. The region Ain this case is: A=(−∞ <ˆx1,ˆx2<+∞,−ˆ l⊥ 2≤ˆx3≤ˆ l⊥ 2),(5.16) and the induced metric on Σ becomes: ds2 8=ˆ h−1 2h(dˆx1)2+ (dˆx2)2i+ˆ h−1 2 ˆ W2 q4"1 + Z2 q ˆ h q2 ˆ W2(x0)2#(dˆx3)2 +ˆ h1 2Z2 q(1 + x−xq)2ds2 CP2+q2(dτ +A)2,(5.17) – 22 – JHEP03(2019)064 where now x=x(ˆx3). The transverse entropy functional is: S⊥ ˆ L1ˆ L2 =Z5 q 32 π3Zdˆx3(1 + x−xq)5ˆ h1 2ˆ W qs1 + Z2 q ˆ h q2 ˆ W2(x0)2.(5.18) Now the first integral is: ˆ h1 2ˆ W q (1 + x−xq)5 q1 + Z2 q ˆ h q2 ˆ W2(x0)2 =ˆ h 1 2 0ˆ W0 q0 (1 + x0−xq)5,(5.19) from which it follows that: x0=±ˆ W Zqqpˆ hs(1 + x−xq)10 (1 + x0−xq)10 ˆ hˆ W2q2 0 ˆ h0ˆ W2 0q2−1.(5.20) Therefore, the transverse length ˆ l⊥is: ˆ l⊥= 2ZqZ∞ x0 qpˆ h ˆ W dx r(1+x−xq)10 (1+x0−xq)10 ˆ hˆ W2q2 0 ˆ h0ˆ W2 0q2−1 .(5.21) The entropy functional evaluated on the minimal surface for this configuration is: S⊥ ˆ L1ˆ L2 =Z6 q 16 π3Zxmax x0 ˆ h(1 + x−xq)5 r1−(1+x0−xq)10 (1+x−xq)10 ˆ h0ˆ W2 0q2 ˆ hˆ W2q2 0 dx , (5.22) whose divergent part is: Sdiv ⊥ ˆ L1ˆ L2 =Z6 q 16 π3Zxmax 0 ˆ h(1 + x−xq)5dx . (5.23) We define Sfinite ⊥as: Sfinite ⊥ ˆ L1ˆ L2 =S⊥−Sdiv ⊥ ˆ L1ˆ L2 .(5.24) After some calculations one can demonstrate that: Sfinite ⊥ ˆ L1ˆ L2 =−Z6 q 16 π3   Z∞ x0 ˆ h(1 + x−xq)5    1−1 r1−(1+x0−xq)10 (1+x−xq)10 ˆ h0ˆ W2 0q2 ˆ hˆ W2q2 0     +Zx0 0 ˆ h(1 + x−xq)5dx    .(5.25) – 23 – JHEP03(2019)064 5 10 15 20 25 30 d 0.5 0.6 0.7 0.8 0.9 1.0 us 2 Figure 11. Speed of sound as a function of the density ˆ dfor xq= 0.005 (black,top), xq= 0.5 (blue,second from top), xq= 0.9 (brown,third from top), and xq= 0.995 (orange,bottom). The numerical values of u2 sobtained from (6.22) as a function of ˆ dfor different values of ˆxqhave been plotted in figure 11. In this plot we notice that the UV asymptotic result (6.29) is satisfied for all values of xqand for xq∼1 there is a minimum at low ˆ d, in which u2 sapproaches the conformal value u2 s= 1/2. This is consistent with the behavior found in the analysis of other observables: the UV behavior is independent of xqand given by the scaling solution, whereas the IR is controlled by xq. By taking xqclose to one, the long distance behavior of our system becomes more isotropic. 7 Summary and conclusions The goal of this paper has been the construction of a gravity dual to a system containing multiple (2 + 1)-dimensional layers in a (3 + 1)-dimensional ambient theory. In order to deal with this problem we adopted a top-down holographic approach and used brane engineering to generate the corresponding gravity dual. We considered the setup (1.1), in which D3- and D5-branes intersect along 2 + 1 dimensions and a codimension-one defect is created along the worldvolume of the D3-branes. Moreover, the D5-branes are distributed homogeneously along the gauge theory directions orthogonal to the defect, giving rise in this way to a multilayer system. To find the corresponding supergravity solution, we regarded the D5-branes as flavor branes and used the techniques developed to find the backgrounds dual to unquenched smeared flavor. The background found is supersymmetric and solves the supergravity equations of motion with D5-brane sources. It is given in terms of the master function W, which can be obtained by solving the master equation (2.9). This master equation contains the profile function which can, in principle, be arbitrarily chosen and depends on the particular distribution of the D5-brane charge along the holographic coordinate. The solutions corresponding to a constant profile function pwere obtained in [23]. In the flavor language, the solutions of [23] are dual to models with massless flavors living on the defect (the corresponding black hole was constructed and analyzed in [24]). The corresponding field theories display anisotropy in the third direction which, by construction, is produced by the multiple (2 + 1)-dimensional layers. – 30 – JHEP03(2019)064 Here we generalized the scaling solution found in [23] to the case in which the quarks are massive and there is a region in the bulk, which we called the cavity, in which the flavor sources vanish. The size xqof this cavity provides us with a parameter which determines the long distance behavior of the model. Indeed, as we tuned xqto its maximal value xq= 1, the IR behavior of several observables we analyzed approaches the one of the un-layered theory, whereas the short distance behavior is independent of xqand given by the scaling solution. Thus, as xq→1 the theory in the IR becomes more isotropic and effectively retains its (3 + 1)-dimensional character. We studied the running anisotropic behavior described above in several quantities, but it is clear that we have not exhausted the list of observables to analyze. Let us mention some possible extensions of our work. In section 6we explored our background with probe D5-branes with a worldvolume gauge field dual to a chemical potential. For simplicity we considered embeddings of the probe with trivial embedding function χ, corresponding to massless flavors. The more general case of massive embeddings can be readily obtained by considering a general function χ(r). It would be interesting to see how the quantum phase transition studied in [49–51] between Minkowski and black hole embeddings is modified by the backreaction as has been seen in other (2 + 1)-dimensional systems [52]. It is clearly an interesting task to find the condensed matter system for which our holographic model could offer a framework for concrete calculations. Each layer effectively mimics graphene, when some of the D5-branes blow up into D7’-branes [53–62], so the full multilayer system could perhaps be understood as a (gapped) holographic graphite. More study is needed to make this identification precise. Nevertheless, some of the recent multilayered systems are known to have strong coupling dynamics [63], so in an ideal scenario, we would love to engineer the holographic geometry to mimic the physics in these settings. This is, however, a highly non-linear task which requires novel ideas. Fortunately, we have a rather straightforward avenue ahead of us to find a killer application. In [64] it was demonstrated that the original D3-brane background (the dual to N= 4 SYM theory), supplemented with flavor D7-brane probes (introducing quenched quark matter) at finite chemical potential, provides a realistic equation of state for cold and dense quark matter. This realization paved the road for further investigations [65–68]. How much then can holography help in understanding the deep cores of neutron stars where deconfined quark matter could reside? In the current context we would like to ask the following question. Black holes are known to forget about their past and are characterized by a handful of parameters (mass, rotation, charges). Neutron stars, on the other hand, seem to depend on a variety of parameters through their complicated internal composition. Surprisingly, however, certain (dimensionless) macroscopic properties (tidal deformability, quadrupole moment, moment of inertia) were found to obey universal relations to quite high degree of accuracy [69] independent of the underlying equation of state.2A recent interesting paper [71] drew attention to the resemblance of the neutron star universal relations and the no-hair relations of black holes. In order to formally study the limit of 2If there is a strong first order phase transition close to the surface then the universal relations are known to be violated up to 20% [67,70]. – 31 – JHEP03(2019)064 strong gravity and the black hole formation, the equation of state has to be anisotropic due to Buchdahl bound [71]. We believe that the rather theoretical construction laid out in this paper (smeared D5-brane providing the necessary anisotropy) is a key step towards this direction and we hope to report progress on this front in the near future. Acknowledgments We are grateful to Carlos Hoyos and Daniele Musso for discussions and comments on a draft version of this paper. J.M.P. and A.V.R. are funded by the Spanish grants FPA2014- 52218-P and FPA2017-84436-P by Xunta de Galicia (GRC2013-024), by FEDER and by the Maria de Maeztu Unit of Excellence MDM-2016-0692. J.M.P. is supported by the Spanish FPU fellowship FPU14/06300. A Details of the background The form of the ten-dimensional metric in Einstein frame has been written in (2.1). In this appendix we give further details and an explicit coordinate representation. Let χbe an angular coordinate taking values in the range 0 ≤χ≤πand let ωi(i= 1,2,3) be a set of left-invariant SU(2) one-forms satisfying dωi=1 2ijkωj∧ωk. Then, we can write ds2 10 as: ds2 10 =h−1 2−(dx0)2+(dx1)2+(dx2)2+e−2φ(dx3)2+h1 2"dr2 +e2g 4dχ2+cos2χ 2((ω1)2+(ω2)2)+cos2χ 2sin2χ 2(ω3)2+e2fdτ +1 2cos2χ 2ω32#, (A.1) where the fiber τtakes values in the range 0 ≤τ≤2π. Notice that we are using in (A.1) a radial variable rwhich is different from the one in (2.1) (see below for the relation between rand ζ). The one-form Ain (2.1) is given by: A=1 2cos2χ 2ω3.(A.2) The one-forms ωican be represented in terms of three angles (θ, ϕ, ψ) as: ω1= cos ψ dθ + sin ψsin θ dϕ ω2= sin ψ dθ −cos ψsin θ dϕ ω3=dψ + cos θ dϕ , (A.3) where 0 ≤θ≤π, 0 ≤ϕ < 2π, and 0 ≤ψ < 4π. The coordinate representation of the canonical vielbein basis of C P2is: e1=1 2cos χ 2ω1, e2=1 2cos χ 2ω2, e3=1 2cos χ 2sin χ 2ω3, e4=1 2dχ . (A.4) – 32 – JHEP03(2019)064 By plugging these expressions into (2.6) and (2.7) we get the value of the RR three-form F3 written in (2.7), where we already used the new radial variable ζdefined in (A.10) below. Our solution preserves some amount of supersymmetry. Indeed, let us choose the following vielbein basis for the ten-dimensional metric (A.1): Exµ=h−1 4dxµ,(µ= 0,1,2), Ex3=h−1 4emdx3, Er=h1 4dr , Ei=1 2h1 4egcos χ 2ωi,(i= 1,2) , E3=1 2h1 4egcos χ 2sin χ 2ω3, E4=1 2h1 4egdχ , E5=h1 4efdτ +1 2cos2χ 2ω3.(A.5) Then, in this basis of one-forms, the Killing spinor of the background can be written as: =h−1 8e3 2Γ12 τη , (A.6) where ηis a constant spinor satisfying some projection conditions. If we represent the Killing spinors of type IIB supergravity as a Majorana-Weyl doublet, then ηsatisfies the projections: Γrx314 σ1η=η , Γ12 η= Γ34 η= Γr5η=iσ2η , (A.7) which means that our background preserves two supercharges. Actually, the different functions of our ansatz must satisfy the following system of first-order BPS equations [23]: φ0=Qfp(r)e3φ 2−2g g0=ef−2g f0= 3 e−f−2ef−2g+Qfp(r) 2e3φ 2−2g h0=−Qce−4g−f−Qfp(r)e3φ 2−2gh , (A.8) where p(r) is the profile function of (2.7), the prime denotes derivative with respect to the radial coordinate rand Qfand Qcare related to the number of flavors Nfand colors Ncas: Qf=4π gsα0Nf 9√3L3 , Qc= 16 πgsα02Nc,(A.9) where L3=Rdx3. Let us now write the BPS system in terms of a new radial variable ζ, related to ras: dζ dr =ef−g.(A.10) Then, the equations for φ,g, and ftake the form: dφ dζ =Qfp(ζ)e3φ 2−f−g dg dζ =e−g df dζ = 3 eg−2f−2e−g+Qfp(ζ) 2e3φ 2−f−g.(A.11) – 33 – JHEP03(2019)064 In this new variable ζ, the BPS equation for gin (A.11) can be immediately integrated: eg=ζ , (A.12) and we can rewrite the remaining BPS equations as: dφ dζ =Qfp(ζ) ζe3φ 2−f df dζ = 3 ζ e−2f−2 ζ+Qfp(ζ) 2ζe3φ 2−f.(A.13) Notice that our ansatz for the metric in the ζvariable is precisely the one written in (2.1). We now introduce the new master function Was: W≡e2f−φ ζ2.(A.14) One can easily show that the BPS system (A.13) implies that Wsatisfies the following first-order differential equation: dW dζ =6 ζe−φ−W.(A.15) Moreover, one can demonstrate that the equation for φin (A.13) can be written as: d dζ e−φ=−Qfp(ζ) ζ2√W.(A.16) One can combine these last two equations in the following second-order master equation for W:d dζ ζdW dζ + 6 dW dζ =−6Qfp(ζ) ζ2√W,(A.17) which coincides with (2.9). It is also straightforward to verify that the function fand the dilaton φare given in terms of Was in (2.10) and (2.11), respectively. As mentioned in subsection 2.3 the master equation can be solved in powers of ζq/ζ for large ζ. The result for Wwas written in (2.27). The function fis readily computed using (2.10): ef=3 2√2ζ"1−1 16 ζ4 q ζ4+3 32 ζ6 q ζ6+7 1024 ζ8 q ζ8+O ζ10 q ζ10 !#.(A.18) The dilaton in this expansion is: eφ= ζ √2Qf!2 3"1 + 1 24 ζ4 q ζ4+1 48 ζ6 q ζ6+43 4608 ζ8 q ζ8+O ζ10 q ζ10 !#,(A.19) and the warp factor becomes: h≈Qc ζ4"4 15 +1 110 ζ4 q ζ4−3 140 ζ6 q ζ6−45 23936 ζ8 q ζ8#+ +C1 ζ2 3"1−1 24 ζ4 q ζ4−1 48 ζ6 q ζ6−35 4608 ζ8 q ζ8#,(A.20) – 34 – JHEP03(2019)064 where C1is a constant of integration. The squashing factor q=ef/ζ for this solution can be obtained from (2.13) and takes the form: q=3 2√2"1−1 16 ζ4 q ζ4+3 32 ζ6 q ζ6+7 1024 ζ8 q ζ8#+O ζ10 q ζ10 !.(A.21) We can also solve the master equation (2.9) perturbatively close to the cavity for ζ≥ζq. In order to do that, let us substitute in the master equation an expansion of the form: Wapprox IR =C0+C2ζ ζq−1+X i=4 βiζ ζq−1i 2 .(A.22) Identifying the above approximate solution (A.22) and the analytic solution (2.16) at the edge of the cavity ζ=ζqwe get the following two constraints C0=C1−b6 ζ6 q, C2= 6 Cb6 ζ6 q .(A.23) The first non-vanishing coefficients are β4=−7 2C2, β5=−12 √2 5√C0 Qf ζq , β6=28 3C2, β7=3√2 (109 C0+ 6 C2) 35 C3/2 0 Qf ζq , β8=−21 C2.(A.24) The function fcorresponding to (A.22) is: ef ζq1 + 1 6 C2 C0−1 2 = 1 + 1 + 1 2 C2 C0ζ ζq−1+1 √2C 3 2 01 + 1 6 C2 C0−1Qf ζqζ ζq−13 2 −5 4 C2 C01 + 1 10 C2 C0ζ ζq−12 +. . . , (A.25) and the dilaton can be expanded as: eφ=C−1 01 + 1 6 C2 C0−1 +√2 C 5 2 01 + 1 6 C2 C0−2Qf ζqζ ζq−13 2 −41 C0+ 6 C2 10 √2C 7 2 0 ×1 + 1 6 C2 C0−2Qf ζqζ ζq−15 2 +2 C4 01 + 1 6 C2 C0−3Q2 f ζ2 qζ ζq−13 +. . . . (A.26) Finally, plugging these expansions into (2.14) we can solve for the warp factor h, which is given by h=C1−1 + 1 6 C2 C0Qc ζ4 qζ ζq−1−√2C1 C 3 2 01 + 1 6 C2 C0−1Qf ζqζ ζq−13 2 +5 21 + 1 5 C2 C01 + 1 6 C2 C0Qc ζ4 qζ ζq−12 +. . . . (A.27) The integration constant C1is connected with the integration constant coming from solving the warp factor differential equation inside the cavity. – 35 – JHEP03(2019)064 B Probe analysis and profile function In this appendix we study the embeddings of a D5-brane probe which preserves the supersymmetry of a background given by our ansatz. Once these embeddings are characterized we will be able to obtain the corresponding profile function p(ζ) which, in turn, is a necessary ingredient to solve the BPS equations and determine completely the different functions of the ansatz. The supersymmetric embeddings we are looking for must satisfy the kappa symmetry condition: Γκ=± , (B.1) where is a Killing spinor of the background. For a D5-brane without any worldvolume gauge field, Γκis given by: Γκ=1 6!√−g6 α1···α6σ1γα1···α6,(B.2) where g6is the determinant of the induced worldvolume metric, γα1···α6is the antisymmetrized product of induced Dirac matrices and we are representing the Killing spinors of type IIB supergravity as a Majorana-Weyl doublet. We will take ξα= (x0, x1, x2, r, θ, ψ) as our set of worldvolume coordinates. In this case the kappa symmetry matrix takes the form: Γκ=1 √−g6 σ1γx0x1x2r θ ψ .(B.3) Recall that the Killing spinor of the background can be written as in (A.6) in terms of a constant spinor ηwhich satisfies the projections (A.7). We can cast the kappa symmetry condition (B.1) as the following condition on η: ˜ Γκη=±η , (B.4) where ˜ Γκis defined as the matrix: ˜ Γκ≡e−3 2Γ12 τΓκe3 2Γ12 τ.(B.5) We will consider an embedding in which x3and ϕare constant, while χ=χ(r), τ =τ(ψ).(B.6) The induced γ-matrices on the worldvolume for this embedding ansatz are: γxµ=h−1 4Γxµ,(µ= 0,1,2) γr=h1 4Γr+1 2h1 4egχ0Γ4 γθ=1 2h1 4egcos χ 2cos ψΓ1+1 2h1 4egcos χ 2sin ψΓ2 γψ=1 2h1 4egcos χ 2sin χ 2Γ3+h1 4ef1 2cos2χ 2+ ˙τΓ5,(B.7) – 36 – JHEP03(2019)064 where we have denoted χ0=dχ dr ,˙τ=dτ dψ ,(B.8) and the Γ’s are constant ten-dimensional Dirac matrices. From these induced matrices we can compute the antisymmetrized product γrθψ and get an expression of the type: γrθψ =h3 4eg 2cos χ 2hc1Γr13+c2Γr23+c3Γ413+c4Γ423+c5Γr15+c6Γr25+c7Γ415+c8Γ425i, (B.9) where the different coefficients are given by c1=eg 2cos χ 2sin χ 2cos ψ , c2=eg 2cos χ 2sin χ 2sin ψ , c3=e2g 4cos χ 2sin χ 2cos ψ χ0, c4=e2g 4cos χ 2sin χ 2sin ψ χ0, c5=ef1 2cos2χ 2+ ˙τcos ψ , c6=ef1 2cos2χ 2+ ˙τsin ψ , c7=ef+g 21 2cos2χ 2+ ˙τcos ψ χ0, c8=ef+g 21 2cos2χ 2+ ˙τsin ψ χ0.(B.10) This expression can be rewritten as: γrθψ =h3 4eg 2cos χ 2e−ψΓ12 hd1Γr13 +d2Γ413 +d3Γr15 +d4Γ415i,(B.11) where the coefficients diare: d1=eg 2cos χ 2sin χ 2, d2=e2g 4cos χ 2sin χ 2χ0, d3=ef1 2cos2χ 2+ ˙τ, d4=ef+g 21 2cos2χ 2+ ˙τχ0.(B.12) In order to compute the form of Γκ, we use that γx0x1x2=h−3 4Γx0x1x2.(B.13) We find Γκ=1 2 egcos χ 2 √−g6 e−ψΓ12 σ1Γx0x1x2hd1Γr13 +d2Γ413 +d3Γr15 +d4Γ415i(B.14) Let us now calculate the matrix ˜ Γκ. As {Γ12,Γr13}={Γ12,Γ413}={Γ12,Γr15}={Γ12,Γ415}= 0 ,(B.15) we have that Γκe3 2Γ12 τ=e−3 2Γ12 τΓκ(B.16) and, therefore: ˜ Γκ=1 2 egcos χ 2 √−g6 e−(ψ+3τ)Γ12 σ1Γx0x1x2hd1Γr13 +d2Γ413 +d3Γr15 +d4Γ415i.(B.17) – 37 – JHEP03(2019)064 We now study how the different terms of ˜ Γκact on η. We begin by considering the action of the first term on the right-hand-side of (B.17). First of all, we notice that, using the ten-dimensional chirality condition satisfied by and the projections (A.7), one can show that ηsatisfies [23]: Γx0x1x2η=iσ2Γx3η . (B.18) Using (B.18) and the first projection in (A.7), we get: σ1Γx0x1x2Γr13 η=σ1Γ43 Γ14rx3η=iσ2Γ34 η . (B.19) Finally, the last projection in (A.7) allows us to write: σ1Γx0x1x2Γr13 η=−η . (B.20) Let us next consider the second term in ˜ Γκ. We first use: Γ413 =−Γr4Γr13 (B.21) from which it immediately follows that: σ1Γx0x1x2Γ413 η= Γr4η . (B.22) Similarly, since Γr15 =−Γ35 Γr13, we get that the third term of (B.17) acts on ηas: σ1Γx0x1x2Γr15 η= Γ35 η . (B.23) But, according to the last set of projections in (A.7), we have Γ35η=−Γr4η. Therefore: σ1Γx0x1x2Γr15 η=−Γr4η . (B.24) Finally, as Γ415 =−Γr4Γr15, we get: σ1Γx0x1x2Γ415 η=−η . (B.25) Thus, collecting all these results, we have: ˜ Γκη=1 2 egcos χ 2 √−g6 e−(ψ+3τ)Γ12 h−d1−d4+ (d2−d3) Γr4iη . (B.26) As ˜ Γκshould act as (plus or minus) the identity matrix on η, we must have: sin(ψ+ 3τ) = 0 , d3=d2.(B.27) The first equation fixes that ψ+ 3τ=n π , (B.28) with n∈Z. This implies that ˙τ=−1 3.(B.29) – 38 – JHEP03(2019)064 In order to have τhaving values within the range [0,2π], we choose n= 4 in (B.28). Therefore, the function τ=τ(ψ) is: τ(ψ) = 4π−ψ 3,(B.30) and the values of τfor the embedding range from τ= 0 to τ=4π 3. Moreover, the second equation in (B.27) implies the following first-order equation for χ: χ0= 2 ef−2gcos2χ 2+ 2 ˙τ sin χ 2cos χ 2 .(B.31) By making use of (B.29), this equation can be written as: χ0=2 3ef−2g3 cos χ−1 sin χ.(B.32) The induced metric on the D5-brane worldvolume for our ansatz is: ds2 6=h−1 2−(dx0)2+ (dx1)2+ (dx2)2+h1 21 + e2g 4(χ0)2dr2+h1 2e2g 4cos2χ 2(dθ)2 +h1 2e2g 4sin2χ 2cos2χ 2+e2f−2gcos2χ 2+ 2 ˙τ2(dψ)2,(B.33) whose determinant is equal to: √−g6=e2gcos χ 2 41 + e2g 4(χ0)2 1 2sin2χ 2cos2χ 2+e2f−2gcos2χ 2+ 2 ˙τ21 2 .(B.34) Let us compute this determinant for the embeddings satisfying the BPS equations obtained by imposing kappa symmetry. First of all we notice that: 1 + e2g 4(χ0)2BPS =sin2χ 2cos2χ 2+e2f−2g(cos2χ 2+ 2 ˙τ)2 sin2χ 2cos2χ 2 ,(B.35) from which it follows that: √−g6BPS =e2g 4 sin χ 2sin2χ 2cos2χ 2+e2f−2gcos2χ 2+ 2 ˙τ2.(B.36) Moreover, one can easily verify that: √−g6BPS =egcos χ 2 2d1+d4BPS .(B.37) Plugging this result and the BPS equations (B.27) into (B.26), we get that: ˜ Γκη=−cos(ψ+ 3τ)η=−η , (B.38) which shows that our brane configuration preserves the same amount of SUSY as the background. – 39 – JHEP03(2019)064 It follows that: J∗=4 3√3.(C.18) It remains to satisfy the equation of motion for A0. The Lagrangian density for the gauge field is: L=−T eφ 2+2ghq1−e−φA02 0−1i,(C.19) where the effective tension Tis given by: T=T5 8J∗=T5 6√3.(C.20) Since A0is a cyclic variable in the Lagrangian density (C.19), the equation of motion for A0can be integrated once, giving: e2g−φ 2A0 0 p1−e−φA02 0 =d , (C.21) where dis a constant proportional to the charge density. From this equation we get: A0 0=eφ 2d √d2+e4g.(C.22) The chemical potential µis just the value of A0at the boundary, and is given by the following integral: µ=dZ∞ 0 eφ 2 √d2+e4gdr . (C.23) D Probes in the Higgs branch We now study embeddings of a probe D5-brane in which the coordinate x3is not constant but instead bends as the holographic coordinate changes. This bending can be interpreted as a recombination between the D3- and D5-branes, realizing the Higgs branch of the dual theory, see, e.g., [72]. In order to find this configuration let us consider the following set of worldvolume coordinates ξα= (x0, x1, x2, r, θ, ψ) and the following embedding ansatz: χ=χ(r), τ =τ(ψ), x3=z(r).(D.1) In addition, the probe D5-brane will have a worldvolume flux, given by: F=Q dθ ∧dψ , (D.2) with Qbeing a constant. We show below that, if certain BPS equations are satisfied, this configuration preserves the supersymmetries of the background. The induced metric for the ansatz (D.1) is: ds2 6=h−1 2−(dx0)2+ (dx1)2+ (dx2)2+h1 21 + e2g 4(χ0)2+h−1e−2φ(z0)2dr2 +h1 2e2g 4cos2χ 2(dθ)2+h1 2e2g 4sin2χ 2cos2χ 2+e2f−2gcos2χ 2+ 2 ˙τ2(dψ)2. (D.3) – 46 – JHEP03(2019)064 The DBI action in Einstein frame is given by (C.1). In the present case the DBI determinant is: −det(g6+e−φ 2F) = e4g 64 J2 DBI(χ)+64 h−1e−4g−φQ21 + e2g 4(χ0)2+h−1e−2φ(z0)2, (D.4) where JDBI(χ) is the function of χdefined in (C.8). The DBI Lagrangian density is: LDBI =−T5 8eφ 2+2gqJ2 DBI(χ)+64h−1e−4g−φQ2r1+ e2g 4(χ0)2+h−1e−2φ(z0)2.(D.5) The WZ action now takes the form: SWZ =T5Zˆ C6+T5Zˆ C4∧F≡Zd6ξLWZ ,(D.6) where the WZ Lagrangian density is given by: LWZ =T5 16 eφ 2+2gcos(3τ+ψ)JWZ(χ, χ0) + T5Q e−φh−1z0,(D.7) and JWZ(χ, χ0) has been defined in (C.8). Let us now study the equations of motion for z(r). By inspecting LDBI and LWZ we immediately conclude that they do not depend on z(only on z0) and, therefore, z(r) is a cyclic variable. Thus, the equation of motion for z(r) can be integrated once as: ∂LDBI ∂z0+∂LWZ ∂z0= constant .(D.8) We will consider the case in which the constant on the right-hand side of (D.8) is zero which, as we will verify below, is the supersymmetric configuration. Therefore, we must have: −∂LDBI ∂z0=∂LWZ ∂z0.(D.9) Moreover, the derivatives of LDBI and LWZ with respect to z0are: −∂LDBI ∂z0=T5 8e−3φ 2+2gh−1qJ2 DBI(χ) + 64 h−1e−4g−φQ2 q1 + e2g 4(χ0)2+h−1e−2φ(z0)2 z0 ∂LWZ ∂z0=T5Q e−φh0.(D.10) Plugging these values into (D.9) and solving for z0, we get: z0= 8 Qeφ 2−2g JDBI(χ)r1 + e2g 4(χ0)2.(D.11) A useful relation that can be derived from this last equation is: r1+ e2g 4(χ0)2+h−1e−2φ(z0)2=r1+ e2g 4(χ0)2qJ2 DBI(χ)+64h−1e−4g−φQ2 JDBI(χ).(D.12) – 47 – JHEP03(2019)064 Let us now derive the equations of motion for the embedding function χ(r). First of all, we calculate the derivatives of the DBI Lagrangian density with respect to χ0and χ, which are given by: ∂LDBI ∂χ0=−T5 32 eφ 2+4gqJ2 DBI(χ) + 64 h−1e−4g−φQ2 q1 + e2g 4(χ0)2+h−1e−2φ(z0)2 χ0 ∂LDBI ∂χ =−T5 8eφ 2+2gq1 + e2g 4(χ0)2+h−1e−2φ(z0)2 qJ2 DBI(χ) + 64 h−1e−4g−φQ2JDBI(χ)∂JDBI(χ) ∂χ .(D.13) Let us now use (D.12) to compute the square root of the right-hand-side of (D.13) involving χ0. We get: ∂LDBI ∂χ0=−T5 32 eφ 2+4gJDBI(χ) q1 + e2g 4(χ0)2 ∂LDBI ∂χ =−T5 8eφ 2+2gr1 + e2g 4(χ0)2∂JDBI(χ) ∂χ ,(D.14) which are just the derivatives corresponding to the case Q=z0= 0 with A0= 0 (see (C.7) and (C.9)). As the derivatives of LWZ with respect to χ0and χare the same as in the Q=z0= 0 case, we immediately conclude that the equation of χis satisfied if τ(ψ) and χ(r) fulfill (B.28) and (B.32) respectively. Thus, the addition of internal flux, related to the bending of the brane in the x3direction as in (D.11), does not modify the BPS solution for χ(r) and τ(ψ). Let us now write an explicit equation for the bending function z(r). This equation can be obtained by plugging into (D.11) the relation:5 r1 + e2g 4(χ0)2=JDBI sin χcos χ 2 .(D.16) We get: z0=8Q eφ 2−2g sin χcos χ 2 .(D.17) D.1 Kappa symmetry In this subsection we will derive the first-order equations for τ(ψ), χ(r), and z(r) from the kappa symmetry condition of the probe. We begin by computing the induced Dirac 5Another useful relation is JDBI ∂JDBI(χ) ∂χ =1 2sin χcos χ 2 ∂JWZ ∂χ .(D.15) – 48 – JHEP03(2019)064 matrices for the ansatz (D.1): γxµ=h−1 4Γxµ,(µ= 0,1,2) γr=h1 4Γr+1 2h1 4egχ0Γ4+h−1 4e−φz0Γx3 γθ=1 2h1 4egcos χ 2cos ψΓ1+1 2h1 4egcos χ 2sin ψΓ2 γψ=1 2h1 4egcos χ 2sin χ 2Γ3+h1 4ef1 2cos2χ 2+ ˙τΓ5.(D.18) The kappa symmetry matrix with Fθψ flux in our conventions is: Γκ=1 q−det(g6+e−φ 2F)hσ1γx0x1x2rθψ +e−φ 2Fθψ (iσ2)γθψ γx0x1x2rθψi,(D.19) with Fθψ =Q. In the second term in (D.19) we need to compute γθψ γθψ =gθθ gψψ (γθψ)2,(D.20) with gθθ and gψψ being elements of the inverse induced metric (D.3). In order to calculate the square of γθψ, let us write this matrix as: γθψ =A e−ψΓ12 BΓ13 +CΓ15,(D.21) with A,B, and Cbeing given by: A=eg 2h1 2cos χ 2, B =eg 2cos χ 2sin χ 2, C =eg 2cos2χ 2+ 2 ˙τ.(D.22) As {Γ13,Γ12}={Γ15,Γ12}= 0, we can write: (γθψ)2=A2BΓ13 +CΓ152=−A2(B2+C2),(D.23) where, in the last step, we have used that (Γ13)2= (Γ15)2=−1 and that {Γ13,Γ15}= 0. Moreover, by inspecting (D.3) we can write gθθ and gψψ as: gθθ =gθθ−1=h1 2A−2, gψψ =gψψ−1=h−1 2B2+C2)−1.(D.24) It follows that: γθψ γθψ =−1.(D.25) Thus, Γκcan be written as the sum of two terms: Γκ= Γ(1) κ+ Γ(2) κ,(D.26) where Γ(1) κand Γ(2) κare given by: Γ(1) κ=1 q−det(g6+e−φ 2F) σ1γx0x1x2rθψ Γ(2) κ=1 q−det(g6+e−φ 2F) (−iσ2)γx0x1x2r.(D.27) – 49 – JHEP03(2019)064 Let us now proceed as in appendix Band compute the antisymmetrized product γrθψ by using (D.18). We get: γrθψ =h3 4eg 2cos χ 2e−ψΓ12 hd1Γr13+d2Γ413 +d3Γr15 +d4Γ415+d5Γx313+d6Γx315i,(D.28) where the coefficients d1,d2,d3, and d4are given by the same expression as in (B.12) and the new coefficients d5and d6are: d5=h−1 2eg−φ 2cos χ 2sin χ 2z0, d6=h−1 2ef−φ 2cos2χ 2+ 2 ˙τz0.(D.29) Let us next define the rotated kappa symmetry matrix ˜ Γκas in (B.5). Then, the kappa symmetry condition is the one written in (B.4). Moreover, we can write ˜ Γκ=˜ Γ(1) κ+˜ Γ(2) κ, with ˜ Γ(i) κ=e−3 2Γ12 τΓ(i) κe3 2Γ12 τ.(D.30) The rotated matrix ˜ Γ(1) κcan be written as: ˜ Γ(1) κ=1 2 egcos χ 2 q−det(g6+e−φ 2F) e−(ψ+3τ)Γ12 σ1Γx0x1x2hd1Γr13 +d2Γ413 +d3Γr15 +d4Γ415 +d5Γx313 +d6Γx315i,(D.31) whereas ˜ Γ(2) κis given by: ˜ Γ(2) κ=Q e−φ 2h−1 2 q−det(g6+e−φ 2F) (−iσ2) Γx0x1x2Γr+eg 2χ0Γ4+h−1 2e−φz0Γx3.(D.32) Let us now calculate ˜ Γ(1) κη. This calculation is similar to the one performed to derive (B.26). In order to compute the additional terms, we need to use the projections: σ1Γx0x1x2Γx313η=σ3Γ13η , σ1Γx0x1x2Γx315η=−σ3Γr4Γ13η . (D.33) Using these results, we arrive at: ˜ Γ(1) κη=1 2 egcos χ 2 q−det(g6+e−φ 2F) e−(ψ+3τ)Γ12 h−d1−d4+(d2−d3)Γr4+d5σ3Γ13 −d6σ3Γr4Γ13iη. (D.34) Next, we compute ˜ Γ(2) κη. We need the projections: (−iσ2) Γx0x1x2rη=−σ3Γ13 η , (−iσ2) Γx0x1x24η=σ3Γr4Γ13 η , (−iσ2) Γx0x1x2x3η=−η , (D.35) and we obtain: ˜ Γ(2) κη=Q e−φ 2h−1 2 q−det(g6+e−φ 2F)−σ3Γ13 +eg 2χ0σ3Γr4Γ13 −h−1 2e−φz0.(D.36) – 50 – JHEP03(2019)064 Apart from the two equations written in (B.27), we get two extra conditions by imposing that ˜ Γκacts as the identity on η. From the terms containing σ3Γ13 in (D.34) and (D.36), we have: eg 2cos χ 2d5=Q e−φ 2h−1 2.(D.37) Moreover, the vanishing of the terms with σ3Γr4Γ13 yields: cos χ 2d6=Q e−φ 2h−1 2χ0.(D.38) Taking into account the expression of d5in (D.29), it is straightforward to check that (D.37) is equivalent to the bending equation (D.11). Moreover, (D.38) is equivalent to: Q χ0=ef−φ 2 2cos χ 2cos2χ 2+ 2 ˙τz0,(D.39) and one can easily show that it is a consequence of the BPS equations for χand z. Therefore, we can write: ˜ ΓκηBPS =−1 q−det(g6+e−φ 2F)eg 2cos χ 2(d1+d4) + Q e−3φ 2h−1z0ηBPS .(D.40) Let us now compute the terms containing the unit matrix in ˜ Γκηwhen the BPS equations are satisfied. We get: (d1+d4)BPS =eg 8 sin χ 2cos3χ 2JDBI Q e−3φ 2h−1z0BPS =4Q2h−1e−2g−φ sin χ 2cos2χ 2 q−det(g6+e−φ 2F)BPS =e2g 8 sin χ 2cos2χ 2J2 DBI + 64 h−1e−4g−φQ2.(D.41) Using these results it is straightforward to verify that ˜ Γκη=−η. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. References [1] J.M. Maldacena, The Large Nlimit of superconformal field theories and supergravity,Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE]. [2] J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal and U.A. Wiedemann, Gauge/String Duality, Hot QCD and Heavy Ion Collisions,arXiv:1101.0618 [INSPIRE]. [3] J. McGreevy, Holographic duality with a view toward many-body physics,Adv. High Energy Phys. 2010 (2010) 723105 [arXiv:0909.0518] [INSPIRE]. [4] A.V. Ramallo, Introduction to the AdS/CFT correspondence,Springer Proc. Phys. 161 (2015) 411 [arXiv:1310.4319] [INSPIRE]. – 51 – JHEP03(2019)064 [5] J.D. Edelstein, J.P. Shock and D. Zoakos, The AdS/CFT Correspondence and Non-perturbative QCD,AIP Conf. Proc. 1116 (2009) 265 [arXiv:0901.2534] [INSPIRE]. [6] A. Karch and E. Katz, Adding flavor to AdS/CFT,JHEP 06 (2002) 043 [hep-th/0205236] [INSPIRE]. [7] O. DeWolfe, D.Z. Freedman and H. Ooguri, Holography and defect conformal field theories, Phys. Rev. D 66 (2002) 025009 [hep-th/0111135] [INSPIRE]. [8] J. Erdmenger, Z. Guralnik and I. Kirsch, Four-dimensional superconformal theories with interacting boundaries or defects,Phys. Rev. D 66 (2002) 025020 [hep-th/0203020] [INSPIRE]. [9] K. Skenderis and M. Taylor, Branes in AdS and p p wave space-times,JHEP 06 (2002) 025 [hep-th/0204054] [INSPIRE]. [10] C. N´u˜nez, A. Paredes and A.V. Ramallo, Unquenched Flavor in the Gauge/Gravity Correspondence,Adv. High Energy Phys. 2010 (2010) 196714 [arXiv:1002.1088] [INSPIRE]. [11] T. Azeyanagi, W. Li and T. Takayanagi, On String Theory Duals of Lifshitz-like Fixed Points,JHEP 06 (2009) 084 [arXiv:0905.0688] [INSPIRE]. [12] E. Kiritsis and V. Niarchos, Josephson Junctions and AdS/CFT Networks,JHEP 07 (2011) 112 [Erratum ibid. 10 (2011) 095] [arXiv:1105.6100] [INSPIRE]. [13] D. Mateos and D. Trancanelli, Thermodynamics and Instabilities of a Strongly Coupled Anisotropic Plasma,JHEP 07 (2011) 054 [arXiv:1106.1637] [INSPIRE]. [14] M. Ammon, V.G. Filev, J. Tarrio and D. Zoakos, D3/D7quark-gluon Plasma with Magnetically Induced Anisotropy,JHEP 09 (2012) 039 [arXiv:1207.1047] [INSPIRE]. [15] S. Jain, N. Kundu, K. Sen, A. Sinha and S.P. Trivedi, A Strongly Coupled Anisotropic Fluid From Dilaton Driven Holography,JHEP 01 (2015) 005 [arXiv:1406.4874] [INSPIRE]. [16] L. Cheng, X.-H. Ge and S.-J. Sin, Anisotropic plasma at finite U(1) chemical potential, JHEP 07 (2014) 083 [arXiv:1404.5027] [INSPIRE]. [17] E. Banks and J.P. Gauntlett, A new phase for the anisotropic N= 4 super Yang-Mills plasma,JHEP 09 (2015) 126 [arXiv:1506.07176] [INSPIRE]. [18] D. Roychowdhury, On anisotropic black branes with Lifshitz scaling,Phys. Lett. B 759 (2016) 410 [arXiv:1509.05229] [INSPIRE]. [19] U. G¨ursoy, I. Iatrakis, M. J¨arvinen and G. Nijs, Inverse Magnetic Catalysis from improved Holographic QCD in the Veneziano limit,JHEP 03 (2017) 053 [arXiv:1611.06339] [INSPIRE]. [20] D. Giataganas, U. G¨ursoy and J.F. Pedraza, Strongly-coupled anisotropic gauge theories and holography,Phys. Rev. Lett. 121 (2018) 121601 [arXiv:1708.05691] [INSPIRE]. [21] G. Itsios, N. Jokela, J. J¨arvel¨a and A.V. Ramallo, Low-energy modes in anisotropic holographic fluids,Nucl. Phys. B 940 (2019) 264 [arXiv:1808.07035] [INSPIRE]. [22] U. G¨ursoy, M. J¨arvinen, G. Nijs and J.F. Pedraza, Inverse Anisotropic Catalysis in Holographic QCD,arXiv:1811.11724 [INSPIRE]. [23] E. Conde, H. Lin, J.M. Penin, A.V. Ramallo and D. Zoakos, D3-D5 theories with unquenched flavors,Nucl. Phys. B 914 (2017) 599 [arXiv:1607.04998] [INSPIRE]. – 52 – JHEP03(2019)064 [24] J.M. Penin, A.V. Ramallo and D. Zoakos, Anisotropic D3-D5 black holes with unquenched flavors,JHEP 02 (2018) 139 [arXiv:1710.00548] [INSPIRE]. [25] Y. Bea, E. Conde, N. Jokela and A.V. Ramallo, Unquenched massive flavors and flows in Chern-Simons matter theories,JHEP 12 (2013) 033 [arXiv:1309.4453] [INSPIRE]. [26] D. Arean, A.V. Ramallo and D. Rodriguez-Gomez, Mesons and Higgs branch in defect theories,Phys. Lett. B 641 (2006) 393 [hep-th/0609010] [INSPIRE]. [27] D. Arean, A.V. Ramallo and D. Rodriguez-Gomez, Holographic flavor on the Higgs branch, JHEP 05 (2007) 044 [hep-th/0703094] [INSPIRE]. [28] S. Benvenuti, M. Mahato, L.A. Pando Zayas and Y. Tachikawa, The Gauge/gravity theory of blown up four cycles,hep-th/0512061 [INSPIRE]. [29] J.M. Maldacena, Wilson loops in large Nfield theories,Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE]. [30] S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large Ngauge theory and anti-de Sitter supergravity,Eur. Phys. J. C 22 (2001) 379 [hep-th/9803001] [INSPIRE]. [31] F. Bigazzi, A.L. Cotrone, C. N´u˜nez and A. Paredes, Heavy quark potential with dynamical flavors: A First order transition,Phys. Rev. D 78 (2008) 114012 [arXiv:0806.1741] [INSPIRE]. [32] F. Bigazzi, A.L. Cotrone and A. Paredes, Klebanov-Witten theory with massive dynamical flavors,JHEP 09 (2008) 048 [arXiv:0807.0298] [INSPIRE]. [33] F. Bigazzi, A.L. Cotrone, A. Paredes and A.V. Ramallo, The Klebanov-Strassler model with massive dynamical flavors,JHEP 03 (2009) 153 [arXiv:0812.3399] [INSPIRE]. [34] A.V. Ramallo, J.P. Shock and D. Zoakos, Holographic flavor in N= 4 gauge theories in 3d from wrapped branes,JHEP 02 (2009) 001 [arXiv:0812.1975] [INSPIRE]. [35] S.D. Avramis, K. Sfetsos and K. Siampos, Stability of strings dual to flux tubes between static quarks in N= 4 SYM,Nucl. Phys. B 769 (2007) 44 [hep-th/0612139] [INSPIRE]. [36] S.D. Avramis, K. Sfetsos and K. Siampos, Stability of string configurations dual to quarkonium states in AdS/CFT,Nucl. Phys. B 793 (2008) 1 [arXiv:0706.2655] [INSPIRE]. [37] S.D. Avramis, K. Sfetsos and D. Zoakos, On the velocity and chemical-potential dependence of the heavy-quark interaction in N= 4 SYM plasmas,Phys. Rev. D 75 (2007) 025009 [hep-th/0609079] [INSPIRE]. [38] S.D. Avramis, K. Sfetsos and D. Zoakos, Complex marginal deformations of D3-brane geometries, their Penrose limits and giant gravitons,Nucl. Phys. B 787 (2007) 55 [arXiv:0704.2067] [INSPIRE]. [39] Y. Bea, N. Jokela, A. P¨onni and A.V. Ramallo, Noncommutative massive unquenched ABJM,Int. J. Mod. Phys. A 33 (2018) 1850078 [arXiv:1712.03285] [INSPIRE]. [40] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE]. [41] S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy,JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE]. [42] I.R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of confinement,Nucl. Phys. B 796 (2008) 274 [arXiv:0709.2140] [INSPIRE]. – 53 – JHEP03(2019)064 [43] U. Kol, C. N´u˜nez, D. Schofield, J. Sonnenschein and M. Warschawski, Confinement, Phase Transitions and non-Locality in the Entanglement Entropy,JHEP 06 (2014) 005 [arXiv:1403.2721] [INSPIRE]. [44] G. Georgiou and D. Zoakos, Entanglement entropy of the Klebanov-Strassler model with dynamical flavors,JHEP 07 (2015) 003 [arXiv:1505.01453] [INSPIRE]. [45] M. Headrick, Entanglement Renyi entropies in holographic theories,Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE]. [46] O. Ben-Ami, D. Carmi and J. Sonnenschein, Holographic Entanglement Entropy of Multiple Strips,JHEP 11 (2014) 144 [arXiv:1409.6305] [INSPIRE]. [47] V.W. de Spinadel, The metallic means family and renormalization group techniques,Trudy Inst. Mat. i Mekh. UrO RAN 6(2000) 173. [48] V. Balasubramanian, N. Jokela, A. P¨onni and A.V. Ramallo, Information flows in strongly coupled ABJM theory,JHEP 01 (2019) 232 [arXiv:1811.09500] [INSPIRE]. [49] A. Karch and A. O’Bannon, Holographic thermodynamics at finite baryon density: Some exact results,JHEP 11 (2007) 074 [arXiv:0709.0570] [INSPIRE]. [50] M. Ammon, M. Kaminski and A. Karch, Hyperscaling-Violation on Probe D-branes,JHEP 11 (2012) 028 [arXiv:1207.1726] [INSPIRE]. [51] G. Itsios, N. Jokela and A.V. Ramallo, Collective excitations of massive flavor branes,Nucl. Phys. B 909 (2016) 677 [arXiv:1602.06106] [INSPIRE]. [52] Y. Bea, N. Jokela and A.V. Ramallo, Quantum phase transitions with dynamical flavors, Phys. Rev. D 94 (2016) 026003 [arXiv:1604.03665] [INSPIRE]. [53] O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, Quantum Hall Effect in a Holographic Model,JHEP 10 (2010) 063 [arXiv:1003.4965] [INSPIRE]. [54] O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, Striped instability of a holographic Fermi-like liquid,JHEP 10 (2011) 034 [arXiv:1106.3883] [INSPIRE]. [55] G.W. Semenoff, Engineering holographic graphene,AIP Conf. Proc. 1483 (2012) 305 [INSPIRE]. [56] H. Omid and G.W. Semenoff, D3-D7 Holographic dual of a perturbed 3D CFT,Phys. Rev. D 88 (2013) 026006 [arXiv:1208.5176] [INSPIRE]. [57] C. Kristjansen and G.W. Semenoff, Giant D5 Brane Holographic Hall State,JHEP 06 (2013) 048 [arXiv:1212.5609] [INSPIRE]. [58] N. Jokela, G. Lifschytz and M. Lippert, Magnetic effects in a holographic Fermi-like liquid, JHEP 05 (2012) 105 [arXiv:1204.3914] [INSPIRE]. [59] N. Jokela, G. Lifschytz and M. Lippert, Magneto-roton excitation in a holographic quantum Hall fluid,JHEP 02 (2011) 104 [arXiv:1012.1230] [INSPIRE]. [60] N. Jokela, G. Lifschytz and M. Lippert, Holographic anyonic superfluidity,JHEP 10 (2013) 014 [arXiv:1307.6336] [INSPIRE]. [61] N. Jokela, G. Lifschytz and M. Lippert, Flowing holographic anyonic superfluid,JHEP 10 (2014) 21 [arXiv:1407.3794] [INSPIRE]. [62] A.G. Grau, C. Kristjansen, M. Volk and M. Wilhelm, A Quantum Check of Non-Supersymmetric AdS/dCFT,JHEP 01 (2019) 007 [arXiv:1810.11463] [INSPIRE]. – 54 – JHEP03(2019)064 [63] M. Geisler et al., Single-crystalline gold nanodisks on WS2 mono- and multilayers: Strong coupling at room temperature,arXiv:1812.09495. [64] C. Hoyos, D. Rodr´ıguez Fern´andez, N. Jokela and A. Vuorinen, Holographic quark matter and neutron stars,Phys. Rev. Lett. 117 (2016) 032501 [arXiv:1603.02943] [INSPIRE]. [65] C. Hoyos, N. Jokela, D. Rodr´ıguez Fern´andez and A. Vuorinen, Breaking the sound barrier in AdS/CFT,Phys. Rev. D 94 (2016) 106008 [arXiv:1609.03480] [INSPIRE]. [66] C. Ecker, C. Hoyos, N. Jokela, D. Rodr´ıguez Fern´andez and A. Vuorinen, Stiff phases in strongly coupled gauge theories with holographic duals,JHEP 11 (2017) 031 [arXiv:1707.00521] [INSPIRE]. [67] E. Annala, C. Ecker, C. Hoyos, N. Jokela, D. Rodr´ıguez Fern´andez and A. Vuorinen, Holographic compact stars meet gravitational wave constraints,JHEP 12 (2018) 078 [arXiv:1711.06244] [INSPIRE]. [68] N. Jokela, M. J¨arvinen and J. Remes, Holographic QCD in the Veneziano limit and neutron stars,arXiv:1809.07770 [INSPIRE]. [69] K. Yagi and N. Yunes, I-Love-Q,Science 341 (2013) 365 [arXiv:1302.4499] [INSPIRE]. [70] M. Sieniawska, W. Turczanski, M. Bejger and J.L. Zdunik, Tidal deformability and other global parameters of compact stars with phase transitions,arXiv:1807.11581 [INSPIRE]. [71] S.H. Alexander, K. Yagi and N. Yunes, An Entropy-Area Law for Neutron Stars Near the Black Hole Threshold,Class. Quant. Grav. 36 (2019) 015010 [arXiv:1810.01313] [INSPIRE]. [72] G. Itsios, N. Jokela and A.V. Ramallo, Cold holographic matter in the Higgs branch,Phys. Lett. B 747 (2015) 229 [arXiv:1505.02629] [INSPIRE]. – 55 –