On the gravity dual of Chern-Simons-matter theories with unquenched flavor
aa r X i v : . [ h e p - t h ] N ov On the gravity dual of Chern-Simons-matter theorieswith unquenched flavor
Eduardo Conde and Alfonso V. Ramallo Departamento de F´ısica de Part´ıculas, Universidade de Santiago de CompostelaandInstituto Galego de F´ısica de Altas Enerx´ıas (IGFAE)E-15782, Santiago de Compostela, Spain
Abstract
We find solutions of type IIA supergravity which are dual to three-dimensional Chern-Simons-matter theories with unquenched fields in the fundamental representation of thegauge group (flavors). In the holographic dual the addition of flavor is performed by meansof D6-branes that are extended along the Minkowski gauge theory directions and are delo-calized in the internal space in such a way that the system is N = 1 supersymmetric and theflavor group is abelian. For massless flavors the corresponding geometry has the form of aproduct space AdS × M , where M is a six-dimensional compact manifold whose metric isobtained by squashing the Fubini-Study metric of CP with suitable constant factors whichdepend on the number of flavors. We compute the effect of dynamical quarks in severalobservables and, in some cases, we compare our results with the ones corresponding to the N = 3 supergravity solutions generated by localized flavor branes. We also show how togeneralize our results to include massive flavors. [email protected] [email protected] ontents B.1 SUSY for the Anti-de Sitter solutions . . . . . . . . . . . . . . . . . . . . . . 43B.2 SUSY for the unflavored ABJM solution . . . . . . . . . . . . . . . . . . . . 45
C A consistent truncation 46
C.1
AdS solution with squashing . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
D Kappa symmetry 51
E Equations of motion 55
The recent results on the
AdS /CF T correspondence constitute a rich framework in whichfundamental questions about the holographic correspondence [1] can be posed and, in somecases, highly non-trivial answers can be obtained. The crucial breakthrough in this subjecttook place with the results of Bagger and Lambert [2] and Gustavsson [3], who proposedthat the low energy theory on multiple M2-branes is given by a new class of maximallysupersymmetric Chern-Simons-matter theories. Inspired by these results, Aharony et al.(ABJM) [4] constructed an N = 6 supersymmetric Chern-Simons-matter theory which isnow believed to describe the dynamics of multiple M2-branes at a C / Z k singularity.The ABJM theory is a U ( N ) × U ( N ) Chern-Simons gauge theory with levels ( k, − k )and bifundamental matter fields. In the large N limit this theory admits a supergravitydescription in M-theory in terms of the AdS × S / Z k geometry. If we represent S as a U (1) Hopf bundle over CP , the Z k orbifold acts by quotienting the S fiber. When theChern-Simons level k is large the size of the fiber is small and the system is better describedin terms of type IIA supergravity by performing a dimensional reduction to ten dimensionsalong the Hopf fiber of S / Z k . In this ten-dimensional description the geometry is of theform AdS × CP with fluxes and preserves 24 supersymmetries.The precise knowledge of the field theory dual to a system of multiple M2-branes on C / Z k has allowed to test some of the non-trivial predictions of the AdS/CFT correspondence. Inparticular, in [5] it was checked by means of a purely field theoretic calculation, using matrixmodel techniques and localization, that the number of low energy degrees of freedom of N coincident M2-branes scales as N for large N , as predicted by the gravity dual (see also[6]). Moreover, the ABJM model has been generalized in several directions. By addingfractional M2-branes, the gravity dual of U ( M ) × U ( N ) Chern-Simons-matter theories with M = N was constructed in [7] . If the sum of Chern-Simons levels for the two gauge groupsis non-zero, the corresponding gravity dual can be found in massive type IIA supergravity byconsidering solutions in which the Romans mass parameter is non-vanishing [8] (see also [9]).The ABJM construction has been extended to include different quivers and gauge groupswith several amount of supersymmetry in refs. [10]-[15].In this paper we will study the generalization of the ABJM model which is obtained byadding flavors, i.e. fields transforming in the fundamental representations ( N,
1) and (1 , N )of the U ( N ) × U ( N ) gauge group. The holographic dual of such a system was proposedin refs. [16, 17]. In the type IIA description the addition of massless flavor is achieved byconsidering D6-branes that fill the AdS space and wrap an RP submanifold inside the2 P , while preserving N = 3 supersymmetry (see also [18]-[21] for different setups withD6-branes in Chern-Simons-matter theories). When the number of flavors is small one canstudy the system in the quenched approximation, in which the D6-branes are considered asprobes in the AdS × CP background. This quenched approach has been adopted in refs.[22]-[24], where different observables of the Chern-Simons-matter theory with flavor havebeen analyzed.In the present article we will study the holographic dual of the ABJM theory with un-quenched flavor. For a system of localized and coincident D6-branes, the correspondinggravity dual in M-theory is a purely geometric background which, in the near horizon limit,is a space of the type AdS × M , where M is a tri-Sasakian seven-dimensional manifoldwhose cone is an eight-dimensional hyperk¨ahler manifold [25]. Notice that the backreactedmetric always has an Anti-de-Sitter factor. This is related to the fact that the dual Chern-Simons theory remains conformally invariant after the addition of flavor (see ref. [26] for aperturbative calculation of the beta functions and a study of the corresponding fixed points).However, the tri-Sasakian metric of M has, in general, a complicated structure which makesdifficult to use it for many purposes. For this reason, in this paper we study the backreactioninduced by a smeared continuous distribution of a large number N f of flavor branes. Thisapproach was initiated in [27] in the context of non-critical strings and has been successfullyapplied to study unquenched flavor in several other setups (see [28] for an extensive reviewand references to the original works).In order to obtain the gravity dual of a field theory with unquenched flavor one has tosolve the equations of motion of supergravity with brane sources. These sources typicallymodify the Bianchi identities of the forms and, as they contribute to the energy-momentumtensor of the system, they also modify the Einstein equations. If the flavor branes arelocalized the sources contain Dirac δ -functions and, as a consequence, solving the equationsof motion is, in general, a difficult task. On the contrary, if the sources are delocalized thereare no δ -function terms in the equations of motion and finding explicit analytic solutions ismuch simpler. Notice that when the branes are smeared, they are not coincident anymoreand, therefore, the flavor symmetry for N f flavors is U (1) N f rather than U ( N f ). Moreover,the solutions with smeared unquenched flavor are generically less supersymmetric than theones with localized flavor, due to the fact that we are superposing branes with differentorientations in the internal space. Indeed, in our flavored ABJM case the solutions will be N = 1 SUSY instead of being N = 3.The backreaction of the flavor branes induces a deformation of the unquenched solutionwhich, in particular, results in a suitable squashing of the metric. In order to determineprecisely this flavor deformation one has to write the metric in a way such that it can besquashed without breaking all supersymmetry. We will argue below that for the ABJM casein the type IIA description the convenient way of writing the CP metric is as S -bundleover S . After representing the CP metric in this way, the flavor deformation just amounts3o squashing the S fiber with respect to the S base, as well as to changing the radii of both AdS and CP factors of the metric (similar squashed deformations of CP were consideredrecently in [29, 30, 15]). In our solutions, the squashing factors are constants which havea precise dependence on the number of flavors N f and encode the effects due to loops offundamentals. Indeed, we will be able to determine the effects due to dynamical flavors inseveral observables such as, among others, the free energy on the sphere, the quark-antiquarkpotential energy or the dimension of meson operators.The organization of the rest of this paper is the following. In section 2 we review theABJM AdS × CP solution and we rewrite it in terms of the S fibration over S . Insection 3 we study the deformations of the ABJM unflavored background with N = 1supersymmetry which preserve the S -bundle structure. The corresponding supergravitysolutions are obtained by solving a system of first-order BPS equations which we partiallyintegrate in general. In particular we find two Anti-de-Sitter solutions which correspond tothe original unflavored ABJM model and to the squashed N = 1 model of reference [15]. Insection 3 we also find running solutions which approach the ABJM background in the IR.In section 4 we study the supersymmetric embeddings of flavor D6-brane probes in thedeformed geometries. The backreaction for branes corresponding to massless quarks is ana-lyzed in section 5, where an ansatz for the background deformed by the massless fundamentalfields is proposed and a set of first-order BPS equations is obtained. Section 6 is devotedto studying the solutions of the flavored BPS system that are Anti-de-Sitter. In section 7we analyze the effect of the unquenched flavor in several observables. In section 8 we studythe backreaction of unquenched massive flavors. Finally, in section 9 we summarize ourresults and discuss some directions for future work. The article is completed with severalappendices, where some detailed calculations not included in the main text are performedand some other aspects of our work are explored. The metric of the ABJM background is given by: ds = L ds AdS + 4 L ds CP , (2.1)where ds AdS and ds CP are respectively the AdS and CP metrics. The former, in Poincarecoordinates, is given by: ds AdS = r dx , + dr r , (2.2)with dx , being the Minkowski metric in 2+1 dimensions. This solution depends on twointegers N and k which represent, in the gauge theory dual, the rank of the gauge groupsand the Chern-Simons level, respectively. In string units, the AdS radius L can be written4n terms of N and k as: L = 2 π Nk . (2.3)Moreover, for this background the dilaton is constant and given by: e φ = 2 Lk = 2 √ π (cid:16) Nk (cid:17) . (2.4)This solution of type IIA supergravity is also endowed with a RR two-form F and a RRfour-form F whose expression can be written as: F = 2 k J , F = 32 k L Ω AdS = 3 π √ (cid:0) kN (cid:1) Ω AdS , (2.5)with J being the K¨ahler form of CP and Ω AdS is the volume element of the AdS metric(2.2). This solution is a good gravity dual of the Chern-Simons-matter theory when the AdS radius is large in string units and the string coupling e φ is small. By looking at eqs. (2.3)and (2.4) this amounts to the condition N << k << N .The CP metric in (2.1) is the canonical Fubini-Study metric. In the context of the ABJMsolution the CP space is usually represented as foliated by the T , ∼ S × S manifold.Here we will write the CP metric in a form which is more convenient for our purposes. Wewill regard CP as an S -bundle over S , with the fibration constructed by using the self-dual SU (2) instanton on the four-sphere. Explicitly, ds CP will be written as: ds CP = 14 h ds S + (cid:0) dx i + ǫ ijk A j x k (cid:1) i , (2.6)where ds S is the standard metric for the unit four-sphere, x i ( i = 1 , ,
3) are cartesian co-ordinates that parametrize the unit two-sphere ( P i ( x i ) = 1) and A i are the componentsof the non-abelian one-form connection corresponding to the SU (2) instanton. Mathemati-cally, the representation (2.6) is obtained when CP is constructed as the twistor space of thefour-sphere. We shall now introduce a specific system of coordinates to represent the metric(2.6). First of all, let ω i ( i = 1 , ,
3) be the SU (2) left-invariant one-forms which satisfy dω i = ǫ ijk ω j ∧ ω k (for an explicit representation of the ω i ’s in terms of angular coordinates,see (4.1)). Together with a new coordinate ξ , the ω i ’s can be used to parameterize the metricof a four-sphere S as: ds S = 4(1 + ξ ) (cid:20) dξ + ξ (cid:0) ( ω ) + ( ω ) + ( ω ) (cid:1)(cid:21) , (2.7)where 0 ≤ ξ < + ∞ is a non-compact coordinate. The SU (2) instanton one-forms A i can bewritten in these coordinates as: A i = − ξ ξ ω i . (2.8)5et us next parametrize the x i coordinates of the S by means of two angles θ and ϕ (0 ≤ θ < π , 0 ≤ ϕ < π ), namely: x = sin θ cos ϕ , x = sin θ sin ϕ , x = cos θ . (2.9)Then, one can easily prove that: (cid:0) dx i + ǫ ijk A j x k (cid:1) = ( E ) + ( E ) , (2.10)where E and E are the following one-forms: E = dθ + ξ ξ (cid:0) sin ϕ ω − cos ϕ ω (cid:1) ,E = sin θ (cid:18) dϕ − ξ ξ ω (cid:19) + ξ ξ cos θ (cid:0) cos ϕ ω + sin ϕ ω (cid:1) . (2.11)Therefore, the canonical Fubini-Study metric of CP can be written in terms of the one-formsdefined above as: ds CP = 14 h ds S + ( E ) + ( E ) i . (2.12)As a check, one can verify that the volume of CP obtained from the above metric is π /
6. Weshall now consider a rotated version of the forms ω i by the two angles θ and ϕ . Accordingly,we define three new one-forms S i ( i = 1 , ,
3) as: S = sin ϕ ω − cos ϕ ω ,S = sin θ ω − cos θ (cid:0) cos ϕ ω + sin ϕ ω (cid:1) ,S = − cos θ ω − sin θ (cid:0) cos ϕ ω + sin ϕ ω (cid:1) . (2.13)In terms of the forms defined in (2.13) the line element of the four-sphere is just obtainedby substituting ω i → S i in (2.7). Let us next define the one-forms S ξ and S i as: S ξ = 21 + ξ dξ , S i = ξ ξ S i , ( i = 1 , , , (2.14)in terms of which the metric of the four-sphere is just ds S = ( S ξ ) + P i ( S i ) . The RRtwo-form F can be written in terms of the one-forms defined in (2.11) and (2.14) as: F = k (cid:16) E ∧ E − (cid:0) S ξ ∧ S + S ∧ S (cid:1) (cid:17) . (2.15)6otice that F is a closed two-form due to the relation: d (cid:0) E ∧ E (cid:1) = d (cid:0) S ξ ∧ S + S ∧ S (cid:1) == E ∧ ( S ξ ∧ S − S ∧ S (cid:1) + E ∧ ( S ξ ∧ S + S ∧ S (cid:1) . (2.16)Notice that there is a non-trivial CP = S in the geometry, which in our coordinates isparametrized by the angles θ and ϕ at a fixed point in the base S . As one can readily verifyby a simple calculation from (2.15), the flux of the RR two-form F through this CP is givenby: 12 π Z CP F = k . (2.17)Eq. (2.17) is essential to understand the meaning of k as the Chern-Simons level of thegauge theory. Indeed, let us consider a fractional D2-brane, i.e. a D4-brane wrapping a CP two-cycle and extended along the Minkowski directions. For such a brane there is a couplingto the worldvolume gauge field A of the type R Min , AdA R CP F which, taking into account(2.17), clearly induces a Chern-Simons coupling for the gauge field A with level k .Some basic facts of the geometry of the ABJM solution in our coordinates are compiledin appendix A. In particular, it is shown how the uplifted solution in M-theory correspondsto the space AdS × S / Z k , where the S is realized as an S -bundle over S . Also, thenon-trivial cycles of CP are displayed. We will now analyze the generalization of the ABJM background obtained by performing acertain deformation of the metric which preserves some amount of supersymmetry. Specifi-cally, we shall consider a ten-dimensional string frame metric of the form: ds = h − dx , + h ds , (3.1)where h is a warp factor and ds is a seven-dimensional metric containing an S fibered overan S in the same way as in CP , namely: ds = dr + e f ds S + e g h ( E ) + ( E ) i , (3.2)with h , f and g being functions of the radial variable r . Notice that f and g determine thesizes of the S and S of the internal CP manifold. If f = g we will say that the CP issquashed. We will verify below that this squashing is compatible with supersymmetry whenthe functions of the ansatz satisfy certain first-order BPS equations.7he type IIA supergravity solutions we are looking for are endowed with a RR two-form F , for which we will adopt the same ansatz as in (2.15). In addition, as it is always the casefor the solutions associated to D2-branes, there is a RR four-form F given by: F = K ( r ) d x ∧ dr , (3.3)where K ( r ) is a function of the radial coordinate r . Moreover, we will assume that thedilaton φ depends only on r .Notice that the Bianchi identities dF = dF = 0 are automatically satisfied. Moreover,the Hodge dual of F is equal to: ∗ F = − K h e f +2 g Vol( S ) ∧ E ∧ E , (3.4)and, thus, the equation of motion of the four-form F (namely d ∗ F = 0), leads to: K h e f +2 g = constant ≡ β , (3.5)where the constant β should be determined from a quantization condition. Thus, it followsthat K ( r ) can be written in terms of the other functions of the ansatz, namely: K = β h − e − f − g . (3.6)Notice that F is subjected to the following flux quantization condition:12 κ Z M ∗ F = ± N T D , (3.7)where M is the six-dimensional angular manifold enclosing the D2-brane. Using our ansatzthis quantization condition is converted into:12 κ T D Z M ∗ F = − π ) Z M Kh e f +2 g Vol( S ) ∧ Vol( S ) = − β π , (3.8)where Vol( S n ) denotes the volume form of a unit n -sphere and we are using string units.Therefore, the coefficient β should be related to the number of D2-branes as: β = 3 π N . (3.9)and the function K is related to the other functions in the ansatz as: K = 3 π N h − e − f − g . (3.10)We will determine the functions entering our ansatz by requiring that our backgroundpreserves (at least) two supersymmetries. As shown in detail in appendix B, this requirement8s fulfilled if the dilaton φ , the warp factor h and the functions f and g satisfy the followingsystem of first-order BPS equations: φ ′ = − k e φ h − (cid:0) e − g − e − f (cid:1) − e φ K h ,h ′ = k e φ h (cid:0) e − g − e − f (cid:1) − e φ K h ,f ′ = k h − e φ (cid:2) e − f − e − g (cid:3) + e − f + g ,g ′ = k e φ h − e − f + e − g − e − f + g . (3.11)In the first two equations of the system (3.11) the function K should be understood as givenby (3.10). This BPS system is obtained after imposing the following projections on theKilling spinors: Γ ǫ = Γ ǫ = Γ ǫ , Γ ǫ = − ǫ , Γ ǫ = − ǫ , (3.12)where the Γ a ··· a n denote antisymmetrized products of flat Dirac matrices in the basis ofone-forms written in (B.2).The BPS system (3.11) can be rewritten in a compact fashion in terms of two calibrationforms. In order to recast (3.11) in this way, let us next define the calibration seven-form K as: K = 17! K a ··· a e a ··· a , (3.13)where the components K a ··· a are the fermionic bilinears: K a ··· a = e φ h ǫ † Γ a ··· a ǫ , (3.14)with ǫ being a Killing spinor of the background and the prefactor in (3.14) is included toaccount for the proper normalization of ǫ (see (B.22)). By using the projections satisfied by ǫ , one can verify that K is given by: K = − e ∧ (cid:0) e − e + e + e + e + e + e (cid:1) . (3.15)In a background generated by D2-branes, it is natural to have also a calibration three-form.Accordingly, we also define the three-form ˜ K , as:˜ K = 13! ˜ K a a a e a a a , ˜ K a a a = e φ h ǫ † Γ a a a ǫ . (3.16)9sing again the projections satisfied by the spinor ǫ , one can show that:˜ K = e . (3.17)One can now verify that the BPS equations (3.11) can be compactly recast as: ∗ F = − d (cid:16) e − φ K (cid:17) , d (cid:16) e − φ h − ∗ K (cid:17) = 0 , F = − d (cid:0) e − φ ˜ K (cid:1) . (3.18) Let us now carry out some simple manipulations of the BPS system (3.11), which will allowus to perform a partial integration. First of all, let us define the function Λ as follows: e Λ ≡ e φ h − . (3.19)Clearly, from this definition one has: Λ ′ = φ ′ − h ′ h . (3.20)Moreover, it is easy to prove that Λ, f and g close the following system of first-order differ-ential equations: Λ ′ = k e Λ − f − k e Λ − g ,f ′ = k e Λ − f − k e Λ − g + e − f + g ,g ′ = k e Λ − f + e − g − e − f + g . (3.21)Notice that the function K has disappeared from the system (3.21) and that φ and h onlyappear through the combination Λ. Actually, by combining the first two equations in (3.11)one proves that K can be written as: K = ddr (cid:16) e − φ h − (cid:17) . (3.22)The warp factor h and the dilaton φ can be obtained from the solution of the system(3.21). Indeed, by using again the first two equations in (3.11) together with (3.6) , onearrives at: h ′ + 43 h φ ′ = − π N e Λ − f − g , (3.23)where we have used the definition (3.19). Eliminating φ ′ between (3.20) and (3.23), we get: h ′ + h Λ ′ = − π N e Λ − f − g . (3.24)10q. (3.24) is a first-order differential equation for h that can be solved by the method ofvariation of constants. The result is: h ( r ) = e − Λ( r ) h α − π N Z r e z ) − f ( z ) − g ( z ) dz i , (3.25)where α is a constant of integration that should be adjusted appropriately. We have notbeen able to integrate the BPS system (3.21) in general. However, we have found someimportant particular solutions which we will discuss in the next two subsections and inappendix C. In some of these solutions the supersymmetry is enhanced with respect to thetwo supersymmetries preserved by the generic solution of (3.21). We will be mostly interested in backgrounds with the Anti-de-Sitter geometry and in theircorresponding deformations. In order to find these solutions systematically, let us nowintroduce a new radial variable τ , related to r as follows: e f ddr = ddτ . (3.26)If the dot denotes derivative with respect to τ , the system of equations (3.21) reduces to:˙Λ = k e Λ − f − k e Λ+ f − g , ˙ f = k e Λ − f − k e Λ+ f − g + e − f + g , ˙ g = k e Λ − f + e f − g − e − f + g . (3.27)Let us next define the following combination of functions:Σ ≡ Λ − f , ∆ ≡ f − g . (3.28)Notice that ∆ measures the squashing of the S and S internal directions. Actually, theright-hand-side of the equations in (3.27) depends only on Σ and ∆ and it is straightforwardto find the following system of two equations involving Σ and ∆:˙Σ = k e Σ (cid:16) − e (cid:17) − e − ∆ , ˙∆ = − k e Σ (cid:16) e (cid:17) − e ∆ + 2 e − ∆ . (3.29)11ne can take Σ, ∆ and (say) g as independent functions. In fact, g can be obtained bysimple integration once Σ and ∆ are known, due to the equation:˙ g = k e Σ + e ∆ − e − ∆ . (3.30)We have thus reduced the full BPS system to a set of two coupled differential equationsfor the functions Σ and ∆.We will now obtain some particular solutions of (3.30) in which the squashing factor ∆ isconstant, as expected for an AdS background. It follows from the second equation in (3.29)that, in this case, Σ must be also constant. Actually, we can eliminate Σ from the equations˙Σ = ˙∆ = 0 and get a simple algebraic equation for ∆. In order to express this equation insimple terms, let us define the quantity q as: q ≡ e = e f − g . (3.31)Then, ˙Σ = ˙∆ = 0 implies the following quadratic equation for q : q − q + 5 = 0 , (3.32)which has two solutions, namely: q = 1 , q = 5 . (3.33)Notice that q = 1 corresponds to the N = 6 ABJM background, while q = 5 shouldcorrespond to a background of reduced SUSY of the form AdS × CP , with CP beinga squashed version of CP . This AdS × CP background was proposed in ref. [15] tobe the gravity dual of an N = 1 superconformal Chern-Simons-matter gauge theory with Sp (2) × U (1) ∼ = SO (5) × U (1) global symmetry. We will describe this solution and somegeneralizations in appendix C. In the remaining of this subsection we will concentrate inshowing how the ABJM background reviewed in section 2 is obtained in this formalism fromthe q = 1 solution. First of all, we notice that when q = e = 1, the system (3.29) leads tothe following solution for Σ: e Σ = 2 k . (3.34)Using these values of Σ and ∆ in (3.30) one readily gets that f = g = τ and, by using (3.26)one shows that the radial variables τ and r are related as r = e τ . Therefore, in the originalvariable r , one has: e f = e g = r . (3.35)Taking into account that the function Λ defined in (3.19) is Λ = Σ + ∆ + g , we get that e Λ = 2 r/k and we can obtain h , φ and K from eqs. (3.25), (3.19) and (3.22) respectively.12he dilaton obtained in this way is constant and is just the one written in (2.4), while h and K are given by: h = 2 π Nk r , K = 3 k π N r . (3.36)By rescaling the Minkowski coordinates as x µ → ¯ λx µ with ¯ λ = π q N k , one can verify that,indeed, the metric and RR four-form for this solution coincide with the ones in (2.1) and(2.5). We will now solve the BPS system (3.21) in a power series expansion in the radial coordinate r . The aim is to find new solutions that approach the AdS × CP background in the IRlimit r →
0. We begin by rewriting the system (3.21) in a more convenient form. Let usdefine the new function F as: F = k e Λ . (3.37)Then, one can recast (3.21) as: F ′ = F h e − f − e − g i , (cid:0) e f (cid:1) ′ = F h e − f − e f − g i + e g − f , (cid:0) e g (cid:1) ′ = F e g − f + 1 − e g − f . (3.38)The N = 6 ABJM solution (without squashing ) can be simply written as F = e f = e g = r .We will now solve (3.38) in a series expansion in powers of r around this solution. We willlook for a solution in which F , e f and e g take the form: F = r (cid:2) a r + a r + · · · (cid:3) , e f = r (cid:2) b r + b r + · · · (cid:3) , e g = r (cid:2) c r + c r + · · · (cid:3) . (3.39)By substituting this ansatz on the system (3.38) and solving for the different powers of r upto third order, one can find the following solution: F = r (cid:2) c r + 8 c r + 20 c r + · · · ] ,e f = r (cid:2) c r + 78 c r + 2516 c r + · · · (cid:3) ,e g = r (cid:2) c r + 2 c r + 4 c r + · · · (cid:3) , (3.40)13here c is an arbitrary constant (if c = 0 we come back to the AdS × CP solution).Plugging the expansions (3.40) into the right-hand side of (3.25) and adjusting the integrationconstants in such a way that h vanishes at r = ∞ , one gets the following expression of thewarp factor h : h = 2 π Nk h r + c r − c + 12 c log( r ) r + · · · i . (3.41)Similarly, the dilaton runs as: e φ = 2 √ π (cid:16) Nk (cid:17) h c r + 33 c r c (cid:16) − log( r ) (cid:17) r + · · · i , (3.42)whereas the function K is given by: K = 3 k π N r (cid:16) − cr + 2 c (cid:0)
29 + 12 log( r ) (cid:1) r + · · · (cid:17) . (3.43)Notice that, when the constant c is non-vanishing, the geometry is not Anti-de-Sitter andthe internal space is squashed by an r -dependent function. In this section we will study the addition of flavor D6-branes to a background of the typestudied in section 3. We will analyze certain configurations in which the D6-branes preservesome amount of supersymmetry of the background. In the present section we will work inthe probe approximation, corresponding to having quenched quarks on the field theory side,in which the background supergravity solution is not affected by the presence of the flavorD6-branes. The effect of the backreaction will be considered in detail in sections 5 and 8.Generically, we will consider configurations corresponding to massless quarks which extendalong the three Minkowski directions x µ , the radial coordinate r and that wrap a three-dimensional cycle of CP . On general grounds [16, 17] it is expected that this three-cycle of CP is a special lagrangian cycle which can be identified with RP . Let us show how this RP arises in our coordinates. With this purpose let us parametrize the SU (2) left invariantone-forms ω i of the four-sphere metric (2.7) in terms of three angles ˆ θ , ˆ ϕ and ˆ ψ , namely: ω = cos ˆ ψ d ˆ θ + sin ˆ ψ sin ˆ θ d ˆ ϕ ,ω = sin ˆ ψ d ˆ θ − cos ˆ ψ sin ˆ θ d ˆ ϕ ,ω = d ˆ ψ + cos ˆ θ d ˆ ϕ , (4.1)with 0 ≤ ˆ θ ≤ π , 0 ≤ ˆ ϕ < π , 0 ≤ ˆ ψ ≤ π . In order to write down a coordinate descriptionof the D6-brane configuration, let us choose the following set of worldvolume coordinates: ζ α = ( x µ , r, ξ, ˆ ψ, ϕ ) . (4.2)14n these coordinates our embedding is defined by the conditions:ˆ θ , ˆ ϕ = constant , θ = π . (4.3)Notice that ξ and ˆ ψ vary inside the S , whereas ϕ varies inside the S . Actually, ξ and ˆ ψ parametrize an S ⊂ S , while θ = π is a maximal S ⊂ S . The induced worldvolume metricis: ds = h − dx , + h dr + ds , (4.4)where the metric ds of the three-cycle is given by: ds = 4 h e g h q (cid:16) dξ (1 + ξ ) + ξ ξ ) ( d ˆ ψ ) (cid:17) + 14 (cid:0) dϕ − ξ ξ d ˆ ψ (cid:1) i , (4.5)with q being the squashing factor defined in (3.31). Let us verify that this three-dimensionalmetric corresponds to (a squashed) RP = S / Z . We first perform the following change ofvariable from ξ to a new angular variable α , defined as: ξ = tan (cid:16) α (cid:17) , ≤ α < π . (4.6)In terms of α the metric ds becomes: ds = h e g h q ( dα ) + q sin α (cid:16) d ˆ ψ (cid:17) + (cid:0) dϕ + 12 (cid:0) cos α − (cid:1) d ˆ ψ (cid:17) . (4.7)Let us next define new angles β and ψ as: β = ˆ ψ , ψ = ϕ − ˆ ψ . (4.8)Then, the metric (4.7) becomes: ds = h e g h q ( dα ) + q sin α ( dβ ) + (cid:0) dψ + cos α dβ (cid:1) i . (4.9)It is clear from (4.8) that 0 ≤ β < π . Moreover, by comparing the volume obtained withthe metric (4.7) with the one obtained with (4.9), one concludes that 0 ≤ ψ < π and thatour three-cycle is indeed a squashed RP manifold inside CP .Let us now verify that the cycle (4.3) preserves some amount of supersymmetry. Firstof all, let us note that the embedding (4.3) can be characterized by the three differentialconditions: S = S = E = 0 , (4.10)which are integrable due to Frobenius theorem since d S = d S = dE = 0 when (4.10)holds and θ = π/
2. From this result one can verify that the cycle is calibrated by the form K K (denoted by ˆ K ) is the one containing e in (3.15) and one has:ˆ K = 2 ξ (1 + ξ ) h e f + g d x ∧ dr ∧ dξ ∧ dψ ∧ dϕ , (4.11)which one can easily show that coincides with the volume form derived from the worldvolumemetric (4.4). In appendix D we have confiormed, by making use of kappa symmetry, thatthese embeddings preserve the supersymmetry of the background. Actually, in that appendixwe generalize the embeddings (4.3) to the case in which θ is not constant. If θ = θ ( r ), thesupersymmetric configurations are those where the following first-order BPS equation issatisfied: e g dθdr = cot θ . (4.12)Notice that, indeed, the solutions of (4.12) with constant θ must necessarily have θ = π/ θ is not constant, the radialcoordinate reaches a minimal value r ∗ in the corresponding brane embedding. Therefore, onecan interpret these D6-brane configurations with varying θ as flavor branes that add massiveflavors to the Chern-Simons-matter theory. The corresponding quark mass is related tothe minimal distance r ∗ . Notice also that one can generate a whole continuous family ofembeddings equivalent to the ones studied so far by acting with the different isometries ofthe metric. In the next sections we will construct supergravity solutions that incorporatethe deformation of the geometry due to the presence of such a continuous set of branes. Let us now study the backreaction of the flavor branes on the backgrounds of the ABJMtype. With this purpose in mind we first analyze the modification of the Bianchi identityintroduced by D6-brane sources. This modification is determined by the WZ term of theD6-branes, which for a collection of N f of them is given by: S W Z = T D N f X i =1 Z M ( i )7 ˆ C , (5.1)where the hat denotes the pullback to the D6-brane worldvolume. Let us rewrite this ex-pression in terms of a charge distribution three-form Ω as: S W Z = T D Z M C ∧ Ω , (5.2)with Ω being only non-vanishing at the location of the D6-branes. The term (5.2) inducesa violation of the Bianchi identity of F . Indeed, let us write the supergravity plus branes16ction in terms of the RR eight-form F and its seven-form potential C . This action containsa contribution of the form: − κ Z M F ∧ ∗ F + T D Z M C ∧ Ω . (5.3)It is manifest from (5.3) that the D6-branes act as a source for the RR seven-form potential C . Actually, the equation of motion of C derived from (5.3) gives rise to the followingMaxwell equation for F : 12 κ d ∗ F = T D Ω , (5.4)which, as F = ∗ F , is equivalent, as claimed, to the violation of the Bianchi identity of F ,namely: dF = 2 π Ω , (5.5)where we have used the fact that, in our units, we have 2 κ T D = 2 π .Following the general procedure reviewed in [28], we will consider a large number N f offlavor branes and we will substitute the discrete set of branes by a continuous distributioncharacterized by the three-form Ω. We will assume that the flavor branes are delocalized insuch a way that there are no Dirac δ -functions in the expression of Ω. In this way we will beable to find solutions of the different field equations of supergravity with sources. Actually,it is easy to find an expression of Ω which preserves both the two supersymmetries and theform of the metric of the deformed unflavored solutions. Notice, however, that the ansatz for F must be modified in order to satisfy (5.5). In fact, by looking at (2.16) it is easy to findthe appropriate modification of our ansatz (2.15). Indeed, the two-form F written in (2.15)is closed because there is a precise balance between the term E ∧ E (along the fibered S ) and the two other terms with components along the S . Clearly, to get a non-closedtwo-form F without distorting much the S - S structure of our ansatz, one should squashthe two type of terms in (2.15) by means of some squashing factor η . Accordingly, we willadopt the following ansatz for F in this flavored case: F = k h E ∧ E − η (cid:0) S ξ ∧ S + S ∧ S (cid:1) i . (5.6)Notice that eq. (2.17) is still satisfied and, therefore, the constant k continues to be theChern-Simons level of the gauge theory. In this section we will consider the case of masslessflavors, which corresponds to taking η constant (see section 8 for the case of massive flavors).The precise relation between η and the number of flavors is obtained below. The violationof the Bianchi identity for F can be verified by computing the exterior derivative of thetwo-form written in (5.6) which, in turn, yields the expression of the smearing form Ω,namely:Ω = k π (cid:0) − η (cid:1) h E ∧ ( S ξ ∧ S − S ∧ S (cid:1) + E ∧ ( S ξ ∧ S + S ∧ S (cid:1) i . (5.7)17et us argue that this is the correct smearing form for a collection of flavor branes, embeddedas in section 4, giving rise to massless flavors . Notice that the smearing form should containthe volume element of the space transverse to the brane worldvolume. In the case of the setof embeddings (4.3) this space should be spanned by the three one-forms S , S and E (seeeq. (4.10)). Thus, one expects to have a term of the type E ∧ S ∧ S in Ω, which is indeedcontained in our ansatz (5.7). It is also easy to find embeddings with E = S = S = 0,which contribute to the E ∧S ∧S component of Ω. Presumably, there are other embeddingsgenerating the other components of the charge-density three-form written in (5.7).In order to relate the squashing coefficient η to the number of flavors N f , let us comparethe smeared DBI action with the DBI action of a single brane. The former is given by: S smearedDBI = − T D Z M e − φ K ∧ Ω , (5.8)where we have taken into account that K is a calibration form for the D6-brane worldvolume.By using the explicit expression of K (eq. (3.15)) and our ansatz for Ω, we get: K ∧
Ω = k (1 − η ) π h e f + g d x ∧ dr ∧ Vol( S ) ∧ Vol( S ) , (5.9)with Vol( S ) being the volume form of the metric (2.7) and Vol( S ) = sin θ dθ ∧ dϕ . In-tegrating over S and S gives a factor 32 π / S smearedDBI = Z d x dr L smearedDBI , (5.10)where the DBI lagrangian density of the smeared set of flavor branes is: L smearedDBI = − π k (1 − η )3 T D e − φ h e f + g . (5.11)Let us now compare the action (5.10) for the whole set of smeared branes with the onecorresponding to a single representative brane, which we will choose to be the one writtenin (4.3). In terms of the angular coordinates defined in (4.6) and (4.8) the DBI action forthe embedding can be written as: S singleDBI = − T D Z d x dr dα dβ dψ e − φ p − det g ≡ Z d x dr L singleDBI , (5.12)where g is the induced metric written in (4.4) and (4.9). By integrating over the angularvariables one easily gets the effective lagrangian density for the representative embedding,namely: L singleDBI = − π T D e − φ h e f + g . (5.13)Since all the embeddings of the family of D6-branes are related by isometries, they areequivalent and they should give the same action. Thus we should have: L smearedDBI = N f L singleDBI . (5.14)18t is now straightforward to use the lagrangian densities (5.11) and (5.13) in (5.14) to findthe precise relation between the squashing factor η and the number of flavors N f . One gets: η = 1 + 3 N f k . (5.15)Notice that η depends linearly on the deformation parameter ǫ , defined as: ǫ ≡ N f k . (5.16)Indeed, it follows from (5.15) that η = 1 + 3 ǫ/
4. Interestingly, ǫ can be rewritten in termsof gauge theory quantities as: ǫ = N f N λ , (5.17)where λ = N/k is the ’t Hooft coupling of the Chern-Simons-matter theory. As we willshow below, the deformations of the metric, dilaton and RR four-form will also depend on ǫ , similarly to what happens in other flavored backgrounds such as the D3-D7 system (see[28] for a review and further references). We are now in the position of addressing the central problem of this paper, namely findingthe backgrounds dual to Chern-Simons-matter theories with flavors. We will adopt an ansatzfor the metric as the one written in (3.1) and (3.2), in which the line element is parametrizedby a warp factor h and two squashing functions f and g . Moreover, the RR four-form F will depend on the function K as in (3.3), while F will be given by (5.6). By imposing onthe Killing spinors the projection conditions (3.12), one can find a system of first-order BPSequations for the different functions of the ansatz. Actually, these BPS equations imposedby supersymmetry are readily obtained from (3.11) by performing the substitution: k e − f → k η e − f . (5.18)In this way, one gets: φ ′ = − k e φ h − (cid:0) e − g − η e − f (cid:1) − e φ K h ,h ′ = k e φ h (cid:0) e − g − η e − f (cid:1) − e φ K h ,f ′ = k h − e φ (cid:2) η e − f − e − g (cid:3) + e − f + g ,g ′ = k e φ h − η e − f + e − g − e − f + g . (5.19)19he fulfillment of the system (5.19) guarantees the preservation of two supercharges, whichcorresponds to N = 1 supersymmetry in three dimensions. Moreover, one can show thatthe BPS system can be rewritten as in (3.18), in terms of the calibration forms K and ˜ K ,which are written in (3.15) and (3.17) in the frame basis. Furthermore, in appendix E weshow that (5.19) implies the fulfillment of the equations of motion of ten-dimensional typeIIA supergravity with sources corresponding to delocalized branes.The system (5.19) is very similar to the unflavored one in (3.11). Therefore, we canfollow the same procedure as in section 3 to perform a partial integration of the system ofdifferential equations and to find some of its particular solutions. First of all, we define thefunction Λ as in (3.19), in terms of which the system (5.19) becomes:Λ ′ = k η e Λ − f − k e Λ − g ,f ′ = k η e Λ − f − k e Λ − g + e − f + g ,g ′ = k η e Λ − f + e − g − e − f + g . (5.20)As in (3.21), the function K does not appear anymore in the system (5.20) and can bedetermined in terms of the functions appearing in (5.20). Indeed, one can immediately showthat K can be obtained from φ and h by means of the expression written in (3.22). Moreover,eqs. (3.23) and (3.24) also hold in this unflavored case and one can integrate the warp factorin terms of Λ as in (3.25).The system (5.20) can be further reduced by introducing the new radial variable τ definedin (3.26) and by defining the functions Σ and Λ as in (3.28). The resulting system ofequations is: ˙Σ = k e Σ (cid:16) η − e (cid:17) − e − ∆ , ˙∆ = − k e Σ (cid:16) η + e (cid:17) − e ∆ + 2 e − ∆ . (5.21)In the next section we will concentrate on studying a particular solution of the system (5.21)which leads to the Anti-de-Sitter geometry in this flavored case. In close analogy with the study carried out in section 3.2, let us consider solutions of thereduced system (5.21) in which the functions Σ and ∆ are constant. Notice that, accordingto the definition in (3.31), constant ∆ implies that the squashing parameter q = e is also20onstant. Actually, by imposing ˙Σ = ˙∆ = 0 in (5.21) one can straightforwardly prove that q must satisfy the following quadratic equation: q − η ) q + 5 η = 0 , (6.1)which reduces to (3.32) when η = 1. The relation (6.1) can be regarded as the relationbetween the deformation η of the RR two-form and the internal deformation of the CP metric, parametrized by the squashing factor q . Both parameters are related to the numberof flavors or, to be more precise, to the deformation parameter ǫ defined in (5.16). Actually,by solving (6.1) as a quadratic equation in q , one gets: q = 3(1 + η ) ± p η − η + 92 . (6.2)By using (5.15) one can obtain the squashing factor q in terms of N f and k , namely: q = 3 + 98 N f k ± r N f k + (cid:16) (cid:17) (cid:16) N f k (cid:17) . (6.3)The two signs in (6.2) give rise to the two possible branches. The minus sign in (6.2)corresponds to the flavored ABJM model (it reduces to q = 1 when η = 1), while the plussign corresponds to the squashed model with SO (5) global symmetry which is discussedin appendix C. Notice that the discriminant in (6.2) is never negative and, therefore, theparameter η can be arbitrary. Actually, when η → ∞ one has the following behavior in thetwo branches: lim η →∞ q = / , for the - branch , ∞ , for the + branch . (6.4)Similarly, one can compute the squashing factor for the case in which N f is small. At secondorder in N f /k , one gets: q ≈ N f k − (cid:0) N f k (cid:1) , for the - branch , N f k + (cid:0) N f k (cid:1) , for the + branch . (6.5)It is interesting to point out that, in the two branches in (6.3), the squashing factor q takesvalues in ranges that are disjoint. In the flavored ABJM case 1 ≤ q ≤ /
3, whereas q ≥ q should be understood as the right-hand-side of (6.3) with the minus sign. Let uswrite the complete supergravity solution in this case. First of all, it follows from the system(5.21) that Σ is given by: k e Σ = 2 √ q − qη + q . (6.6)21oreover, the equation for g in (5.20) can be rewritten as: ddr (cid:0) e g (cid:1) = 1 √ q h kη e Σ + √ q − √ q i . (6.7)Since the right-hand side of (6.7) is constant, it follows that e g is a linear function of r .Actually, by making use of (6.6) one can prove that g and f can be written as: e g = rb , e f = √ qb r , (6.8)with the coefficient b being given in terms of the squashing parameters η and q by: b = q ( η + q )2( q + ηq − η ) . (6.9)Let us next compute the warp factor h by using the general expression (3.25). A glance atthe right-hand side of (3.25) reveals that we have to compute the function Λ first. However,it follows from the definition (3.28) that e Λ = e Σ e f and, therefore, we can obtain e Λ from(6.6) and (6.8). One gets: e Λ = 8 k − qq ( η + q ) (cid:2) q + ηq − η ] r . (6.10)Taking into account this result, we can write the warp factor h as: h = β k q ( η + q ) (2 − q ) (cid:2) q + ηq − η ] r , (6.11)where we have adjusted the integration constant in (3.25) by requiring that h vanishes at r → ∞ . Notice that, as h ∼ r − , this solution does indeed lead to an Anti-de-Sitter metric.Actually, the AdS radius L is related to the warp factor h as: L = r h . (6.12)From (6.11) we can extract the expression of L , which can be written as: L = 2 π Nk (2 − q ) b q ( q + ηq − η ) . (6.13)Using these results we can represent the ten-dimensional metric for this solution as: ds = L ds AdS + L b h q ds S + ( E ) + ( E ) i . (6.14)In order to write the AdS part of the metric as in (2.2) we have to rescale the Minkowskicoordinates as x µ → L x µ , where L is the same as in (6.13). From (6.14) the interpretation22f the parameter b is rather clear: it represents the relative squashing, due to the flavor, ofthe CP part of the metric with respect to the AdS part. It is also interesting to rewritethe metric (6.14) in terms of the variables used in (2.7). One has: ds = L ds AdS + ds , (6.15)where ds is the metric of the deformed internal six-dimensional manifold, written in termsof the SU (2) instanton on the four-sphere, as: ds = L b h q ds S + (cid:0) dx i + ǫ ijk A j x k (cid:1) i . (6.16)Let us now obtain the remaining non-vanishing fields for this solution. First of all, theconstant dilaton can be found by using (3.19). One gets: e φ = 4 √ πη + q (2 − q ) (cid:2) q ( q + ηq − η ) (cid:3) Nk ! . (6.17)This expression can be rewritten in a more compact form as: e − φ = b η + q − q kL . (6.18)Finally, let us write the four-form F for these solutions. After rescaling the Minkowskicoordinates as before, we can write F as proportional to the volume element Ω AdS of AdS ,namely: F = L Kr Ω AdS . (6.19)The function K was written in (3.6) in terms of h , f and g . Taking into account that e f = q e g and the relation (6.12), one can rewrite F as: F = βq r e − g L Ω AdS . (6.20)Using the expressions of g and L in (6.8) and (6.13) and the one for β written in (3.9), wearrive at: F = 3 π √ q ( η + q ) ( q + ηq − η ) √ − q √ k N Ω AdS . (6.21)We can rewrite this result more compactly as: F = 3 k η + q ) b − q L Ω AdS . (6.22)Eqs. (6.15) and (6.16) are the flavored generalization of the ABJM metric written in (2.1) and(2.6). Notice that the radius L is not the same in both cases (compare (2.3) and (6.13)) and,23n addition, the flavored metric is deformed by the parameters b and q . The RR two-form F for the flavored solution was written in (5.6) (it was our starting point) and, together withthe F written in (6.21), generalize (2.5). Finally, the constant dilaton also gets correctedby the effect of the matter fields, as one can see by comparing eqs. (6.17) and (2.4).Let us finish this section by discussing the regime of validity of our supergravity dual.On general grounds we must require that the curvature of the space is small in string units(or, equivalently, that the curvature radius is large) and that the string coupling e φ is small(otherwise we should describe the system in eleven-dimensional supergravity). Thus, thetwo conditions that make our type IIA supergravity approximation valid are: L >> , e φ << . (6.23)Let us analyze the two conditions in (6.23) in two different regimes of the deformationparameter ǫ = N f /k . If ǫ is of the order one or less, the squashing parameters are also of thissame order and they do not modify the order of magnitude of L and e φ . Therefore, (6.23)leads to the same conditions as in the unflavored case, namely: N << k << N , (6.24)with N f being, at most, of the order of the Chern-Simons level k . Let us consider next theopposite limit, namely when N f >> k . In this case, as q remains finite (see (6.4)) and η islarge, it follows from (6.13) that: L ≈ π Nk (2 − q ) b q ( q −
1) 1 η . (6.25)Moreover, since η ∼ N f k in this limit, one has: L ∼ NN f . (6.26)Similarly, for N f >> k the dilaton behaves as: e φ ≈ √ π (cid:16) Nk (cid:17) (2 − q ) q ( q − η − ∼ (cid:16) NN f (cid:17) . (6.27)Thus, the conditions (6.23) for N f >> k are equivalent to: N << N f << N . (6.28)Notice that conditions similar to (6.24) and (6.28) were found in ref. [17] for general tri-Sasakian manifolds. 24et us now discuss the regime of validity of the DBI+WZ action used to describe theflavor branes . In principle the DBI action is considered to be valid when g s N f is small [31].Indeed, g s N f is the effective coupling for the process in which an open string ends on the N f branes. However, as argued in refs. [32, 33], when the flavor branes are smeared the situationis more subtle and the effective coupling g s N f is further suppressed due to the fact that thebranes are separated a large distance in √ α ′ units and only a small fraction of the N f braneswill be available for an open string process. In general, if R denotes the typical radius of aninternal dimension of the geometry in √ α ′ units, the number of flavor branes involved in atypical process will be of order N f /R d , where d is the codimension of the flavor branes in theinternal space. Thus, we should require that g s N f /R d be small. In our case d = 3 and R isjust the radius L . Therefore, we should require that e φ N f /L be small. When ǫ = N f /k issmall this condition is satisfied if (6.24) holds since e φ N f /L ∼ ǫ p k/N in this case. In theopposite large ǫ regime, L and e φ behave as in (6.26) and (6.27) respectively and one has: e φ N f L ∼ r N f N . (6.29)Thus, we should require that N f << N , which is the same condition obtained by imposingthat the curvature is small in string units. Notice also that the typical separation scalebetween two D6-branes is: D ∼ LN f . (6.30)For consistency we should require that the distribution of branes is dense enough to bedescribed by a continuos charge density. This condition amounts to require that D << L ,which is clearly satisfied if N f is large. On the other hand, in accordance with our discussionabove, we should also require that D must be greater than 1 (in √ α ′ units) which, for large N f , leads to the condition N f << N . Notice that this requirement is more restrictive thanthe one written in (6.28). Having a simple supergravity solution with no evident pathologies for the unquenched flavorgives us a great opportunity to explore the effects of dynamical matter in several observables.Moreover, since our solution is an Anti-de-Sitter background, we have at our disposal sev-eral techniques and holographic prescriptions to evaluate the observables of the unquenchedtheory in a neat form. Furthermore, some of these observables can also be computed forthe localized flavor solutions, which gives us a unique chance to compare with our resultsand to explore the effects of the smearing technique. In the next subsections we will ana-lyze these flavor effects for some of these observables. We will show that, although the two We are grateful to A. Cotrone for discussions on what follows. N = 3 → N = 1 breakingintroduced by the smearing is, in fact, rather mild. Let us consider the euclidean version of the conformal field theory formulated in a three-sphere. The corresponding free energy is given by: F ( S ) = − log | Z S | , (7.1)where Z S is the euclidean path integral. The holographic calculation of this quantity in AdS gives [34]: F ( S ) = πL G N , (7.2)where L is the AdS radius and G N is the effective four-dimensional Newton constant. In ourcase, G N is related to the ten-dimensional Newton constant G by means of the equation:1 G N = 1 G e − φ Vol( M ) , (7.3)where M is the internal manifold and Vol( M ) its volume. For our flavored Anti-de-Sittersolutions this volume can be readily computed from the metric of M written in (6.16) andis equal to: Vol( M ) = 32 π q L b . (7.4)Taking into account that, in our units, the ten-dimensional Newton constant G is given by G = 8 π , and using the value of the dilaton for our solution (eq. (6.18)), we get: F ( S ) = k π q ( η + q ) (2 − q ) b L . (7.5)Using the value of the AdS radius L for our geometry written in (6.13), we get that F ( S )can be represented as: F ( S ) = π √ k N ξ (cid:16) N f k (cid:17) , (7.6)where ξ (cid:16) N f k (cid:17) is given by: ξ (cid:16) N f k (cid:17) ≡ q ( η + q ) (2 − q ) ( q + ηq − η ) . (7.7)26otice that ξ = 1 for the unflavored case and we recover the ABJM result, namely: F ABJM ( S ) = π √ k N = π √ N √ λ , (7.8)where, in the last step, we have written the result in terms of the ’t Hooft coupling λ = N/k .For small values of N f /k we can expand ξ as: ξ = 1 + 34 N f k − (cid:16) N f k (cid:17) + O (cid:16)(cid:16) N f k (cid:17) (cid:17) . (7.9)Thus, the free energy of the flavored theory can be expanded in powers of N f /k as: F ( S ) = π √ N √ λ + π √ N f N √ λ − π √ N f λ + · · · . (7.10)If, on the contrary, N f /k is large, one can verify that, at leading order, ξ behaves as: ξ ∼ r r N f k ≈ . r N f k , (7.11)and, therefore, the free energy in this large N f case becomes: F ( S ) ∼ √ π N f N . (7.12)Let us next compare our results with the ones obtained with the N = 3 tri-Sasakiangeometry that corresponds to a localized stack of flavor D6-branes. In M-theory these ge-ometries are obtained as the base X ( t ) of a hyperk¨ahler cone M ( t ), labelled by threenatural numbers t = ( t , t , t ). These cones are obtained as hyperk¨ahler quotients of theform H ///U (1), where the U (1) action is characterized by the three charges t . The dual tothe N = 3 Chern-Simons-matter theories with N f fundamentals has charges t = ( N f , N f , k )(see [17]). The volume of X ( t ) has been computed in [35] by using localization techniques.From this result one can obtain the corresponding free energy F ( S ) [17], which matches thematrix model field theory calculation [36] . One gets an expression like (7.6) with a flavorcorrection factor ξ simply given by: ξ − S = 1 + N f k q N f k . (7.13)Let us expand ξ − S in powers of N f /k , namely: ξ − S = 1 + 34 N f k − (cid:16) N f k (cid:17) + O (cid:16)(cid:16) N f k (cid:17) (cid:17) . (7.14)27his result is indeed very similar to our result (7.9). For large N f /k one has: ξ − S ∼ √ r N f k , (7.15)which is again amazingly close to the value we have found in (7.11). Actually, one can plottogether the two functions ξ written in (7.7) and (7.13) and check that the two curves are,indeed, almost identical (see figure 1).
10 20 30 40 N f k Ξ H N f k L Tri - Sasakian Smeared
Figure 1: Comparison of the flavor correction factor ξ obtained with our smeared setup(lower red curve) and the one corresponding to the tri-Sasakian geometry (upper blue curve). Most of the results derived for Wilson loops in N = 4 super-Yang-Mills in four dimensionscan be adapted to our flavored setup. To illustrate this fact let us consider the calculationof the quark-antiquark static potential. Due to conformal invariance, the quark-antiquarkpotential must be Coulombic, namely of the form: V q ¯ q = − Qd , (7.16)where d is the distance between the quark and the antiquark and the coefficient Q measuresthe strength of the Coulombic potential. The holographic calculation of Q is just the same28s the one performed in [37] and yields the result: Q = 4 π L (cid:2) Γ (cid:0) (cid:1) (cid:3) . (7.17)Using the value of the AdS radius L (see (6.13)), one can write Q as: Q = 4 π √ λ (cid:2) Γ (cid:0) (cid:1) (cid:3) σ , (7.18)where σ parametrizes the screening effects due to dynamical quarks, and is given by: σ = s − qq ( q + ηq − η ) b = 14 q ( η + q ) (2 − q ) ( q + ηq − η ) . (7.19)For small N f /k , one can expand σ as: σ = 1 − N f k + 964 (cid:16) N f k (cid:17) + · · · . (7.20)Clearly, the fact that the first correction is negative means that the screening makes theCoulombic attraction between the quark and the antiquark smaller, as expected on physicalgrounds. In the opposite limit, when N f /k is large, σ is small. Actually, one gets: σ → q (2 − q ) ( q − η − = r s kN f , for N f k >> . (7.21)Similarly, the calculation of the circular Wilson loop can be done by applying the sametechniques as in the AdS × S background (see [38]). The result depends on the AdS radius L , namely: < W > ∼ e L . (7.22)Using our value of L , we get: < W > ∼ e π √ λ σ , (7.23)and, as before, the screening effects are encoded in σ . It is interesting to compare againour results with the ones found in refs. [17, 36] from the tri-Sasakian geometry. The corre-sponding screening factor σ is given by: σ − S = 1 q N f k ≈ − N f k , for N f << k , √ q kN f , for N f >> k . (7.24)By comparing the right-hand side of (7.24) with (7.20) and (7.21) we conclude that ourresult is qualitatively similar to the one of [17, 36], although our smeared setup gives rise toa larger screening effect. 29 .3 Dimensions of scalar meson operators Let us analyze how the dimensions of the meson operators (bilinears in the fundamentalfields) change when the effect of dynamical unquenched matter is taken into account. Withthis purpose, let us consider a D6-brane probe in the flavored background which fluctuatesaround the static BPS configuration (4.3). The induced metric on the D6-brane worldvolumefor this static configuration is given by: G αβ dζ α dζ β = L ds AdS + L b h q ( dα ) + q sin α ( dβ ) + (cid:0) dψ + cos α dβ (cid:1) i . (7.25)For simplicity we will concentrate on the case in which only the angle θ varies with respectto the value written in (4.3) and we will consider a perturbed configuration in which theangle θ is given by: θ ( r ) = π λ ( r ) , (7.26)with λ ( r ) being a small fluctuation of the transverse scalars of the BPS embedding (4.3) (itshould not be confused with the ’t Hooft coupling). In order to study the equation of motionfor λ let us compute the worldvolume induced metric g at second order in λ . We represent g as: g = G + g ′ , (7.27)where G is the metric (7.25). At second order in λ the metric perturbation g ′ is the following: g ′ αβ dζ α dζ β = L b h λ (cid:0) dψ + cos α dβ (cid:1) + ( λ ′ ) ( dr ) i , (7.28)with λ ′ = dλ/dr . The DBI lagrangian density for the flavor D6-brane is just: L DBI = − T D e − φ p − det g . (7.29)By plugging the results written above for g , one gets the following second-order result: L DBI = − T D e − φ √− det G h r b ( λ ′ ) − λ i . (7.30)Similarly, the WZ term of the lagrangian is: S W Z = T D Z M e − φ ˆ K , (7.31)with K being the calibration form and ˆ K its pullback to the D6-brane worldvolume M .One can easily show that, at second order, one has:ˆ K = (cid:16) − λ − rb λ λ ′ (cid:17) Vol (cid:0) M (cid:1) , (7.32)30here Vol (cid:0) M (cid:1) is the volume form of the metric G . After integrating by parts, we can writethe langrangian density for the WZ part of the action as: L W Z = − T D e − φ √− det G h (cid:16) − b (cid:17) λ i . (7.33)The total lagrangian density is thus: L = − T D e − φ √− det G h r b ( λ ′ ) + 12 (cid:16) − b (cid:17) λ i , (7.34)and the corresponding equation of motion for λ is:1 r ∂ r (cid:2) r ∂ r λ (cid:3) + b (3 − b ) λ = 0 . (7.35)Let us assume that the fluctuation λ ( r ) in AdS behaves for large r as: λ ∼ c r − a + c r − a , a > a , (7.36)where a ( a ) corresponds to the non-normalizable (normalizable) mode. The associatedconformal dimension of the dual operator ¯ ψ ψ is:∆ = dim( ¯ ψ ψ ) = 32 + a − a . (7.37)In order to find the values of the exponents a and a in our case, let us assume that thereis a solution of the fluctuation equation (7.35) in the form: λ ∼ r α . (7.38)It is now straightforward to show that the exponent α can take the values α = − b, b −
3. Itfollows that a and a are given by: a = b , a = 3 − b . (7.39)Thus, it follows from (7.37) that the dimension ∆ is just given by:dim( ¯ ψψ ) = 3 − b . (7.40)Let us expand this result in the number of flavors. As the parameter b written in (6.9) isgiven by the following power series in N f /k : b = 1 + 316 N f k − (cid:16) N f k (cid:17) + · · · , (7.41)the dimension of the dual operator is:dim( ¯ ψψ ) = 2 − N f k + 63512 (cid:16) N f k (cid:17) + · · · . (7.42)31his equation shows how the canonical dimension dim( ¯ ψψ ) = 2 is corrected by the additionof dynamical quarks in the regime in which N f /k is not large. Let us consider next theopposite limit in which N f /k is large. In this case one can easily verify that: b → , (cid:16) N f k → ∞ (cid:17) , (7.43)and therefore dim( ¯ ψψ ) → , (cid:16) N f k → ∞ (cid:17) . (7.44) A high spin operator can be holographically realized as a rotating string [39]. The anomalousdimension ∆ of such operator can be computed in the large λ limit. Indeed, the calculationis just the same as in [39] and the result for the difference between the dimension ∆ and thespin S can be written as: ∆ − S = f ( λ, ǫ ) log S , (7.45)with f ( λ, ǫ ) being the so-called cusp anomalous dimension which depends on the ’t Hooftcoupling λ and on the flavor deformation parameter ǫ (the unflavored result was obtained in[4]). From the calculations in [39], we have: f ( λ, ǫ ) = L π . (7.46)In terms of the gauge theory parameters the cusp anomalous dimension can be written as: f ( λ, ǫ ) = √ λ σ , (7.47)where σ is the screening factor of the quark-antiquark potential defined in (7.19), whichalso encodes the effects of the unquenched quarks on the anomalous dimensions of high spinoperators. As our final example let us analyze how quark loops effects change the dimensions of the op-erators dual to some particle-like brane configurations. In general, the conformal dimensionof the operator dual to an object of mass m in the AdS space of radius L is given by:∆ = m L . (7.48)First of all, let us consider the case of D0-branes which, according to [4], are dual to di-monopole operators with charges (1 ,
1) under the two gauge groups. These operators are32quivalent to Wilson line operators carrying k fundamental indices of one group and k anti-fundamental indices of the other group. From the value of the dilaton in (6.17) weimmediately obtain the mass of a D0-brane ( m D = 1 /g s ): m D = η + q √ π q ( q + ηq − η ) (2 − q ) (cid:16) k N (cid:17) . (7.49)The conformal dimension of the gauge dual is just obtained by applying (7.48). We get:∆ D = 18 q ( η + q ) ( q + ηq − η ) (2 − q ) k . (7.50)As a check, we notice that (7.50) reduces to k/ η = q = 1, which is the value expectedfor an operator which is the product of k bi-fundamentals of dimension 1 / N = 3 Chern-Simons-matter theory is corrected by the funda-mentals as ∆ D → ∆ D + N f . In order to compare this result with our expression (7.50),let us evaluate ∆ D for N f small and large with respect to the Chern-Simons-Level k . Onegets: ∆ D ≈ k + N f for N f << k , N f for N f >> k . (7.51)Notice that (7.51) is not very different from the result obtained in [17], specially for small N f (although both results refer to theories with different amount of SUSY and flavor group).We now consider di-baryons in the flavored geometry. They should correspond to D4-branes wrapped on (deformed) CP , which we will take to be given by the same angularembedding as in the unflavored case in (A.11), namely it will be defined by ϕ = θ = π/
2. Interms of the angle χ defined in (A.14), the induced metric in the four-cycle is given by thefollowing deformation of the Fubini-Study metric: L b h ( dχ ) + cos χ (cid:16)(cid:0) q −
1) sin χ (cid:1) (cid:0) ( ω ) + ( ω ) (cid:1) + q sin χ ω ) (cid:17) i . (7.52)The volume V of this cycle can be immediately obtained by integration, namely: V = 8 π q ) L b , (7.53)and the mass of the wrapped D4-brane is: m D = e − φ (2 π ) V = L π b (2 + q )( η + q )2 − q k . (7.54)33he corresponding conformal dimension ∆ D = m D L is just:∆ D = 196 q ( η + q ) (2 + η )( q + ηq − η ) N . (7.55)As a check of the formula (7.55) one can verify that its right-hand side reduces to theunflavored value N/ N f = 0. Actually, the dimension ∆ D does not vary much withthe number of flavors. For small N f /k one can expand∆ D = (cid:16) N f k − (cid:16) N f k (cid:17) + · · · (cid:17) N . (7.56)Moreover, for large N f /k the dimension approaches the following constant asymptotic limit:∆ D → N . (7.57)which is very close to the unflavored value N/
2. Actually, it was argued in [35, 18] from theanalysis of the tri-Sasakian geometry dual to N = 3 theories that ∆ D should not be changedwhen fundamentals are added. Again, we see that the results obtained with our smearedgeometry are not very different from the ones found with the localized N = 3 backgrounds. Let us write an ansatz for the backreacted background in the case that the quarks introducedby the flavor D6-branes are massive. According to what happens in other setups [28] analyzedwith the smearing technique, we will modify the ansatz of F by substituting N f by N f p ( r ),where p ( r ) is a function of the radial coordinate to be determined. This new function shouldsatisfy the following conditions: p ( r ) = 0 if r < r ∗ , lim r →∞ p ( r ) = 1 , (8.1)where r ∗ is related to the mass of the quarks. Therefore, the new ansatz for F is: F = k E ∧ E − (cid:16) k + 3 N f p ( r ) (cid:17) (cid:16) S ξ ∧ S + S ∧ S (cid:17) . (8.2)Now, the smearing form Ω = dF / π takes the form:Ω = − N f π p ( r ) h E ∧ ( S ξ ∧ S − S ∧ S (cid:1) + E ∧ ( S ξ ∧ S + S ∧ S (cid:1) i −− N f π p ′ ( r ) dr ∧ (cid:16) S ξ ∧ S + S ∧ S (cid:17) , (8.3)34nd has new components (the last line in (8.3)) which were not present in the massless case.Notice, however, that the BPS equations for this massive case can be simply obtained bychanging: kη → k + 3 N f p ( r ) , (8.4)in the system for massless matter (5.20) .Let us see how one can obtain the function p ( r ) by comparing the smeared WZ action ofthe D6-brane with the one corresponding to a single massive embedding, which was studiedin appendix D. First of all we notice that, with our new expression (8.3) for the smearingform Ω, we get: K ∧
Ω = 3 N f π h p ( r ) + 12 e g p ′ ( r ) i h e f + g d x ∧ dr ∧ Vol( S ) ∧ Vol( S ) . (8.5)By integrating over the S and S (which amounts to multiplying by 32 π /
3) and using thisresult in (5.2) (with C as in (D.26)), one gets the effective WZ lagrangian density in the x µ and r variables, namely: L W Z = 8 π N f T D e − φ h e f + g h p ( r ) + 12 e g p ′ ( r ) i . (8.6)By comparing with the lagrangian (D.29) multiplied by N f , we get that one can identify p ( r ) with: p ( r ) = (cid:0) sin θ ( r ) (cid:1) Θ( r − r ∗ ) . (8.7)By using the expression (D.17) of θ ( r ), one gets: p ( r ) = h − exp (cid:2) − Z rr ∗ e − g ( z ) dz (cid:3) i Θ( r − r ∗ ) , (8.8)where we used the fact that θ = 0 when r = r ∗ . One can also get this same result bycomparing the smeared and localized DBI actions. Notice that, when the kappa symmetryequation (D.11) for the embedding holds, the DBI lagrangian density (D.24) for a massiveembedding reduces to: L DBI ( BP S ) = − π T D e − φ h e f + g . (8.9)The smeared DBI lagrangian is just the same as in (8.6) with opposite sign. Then, it followsthat p ( r ) must satisfy the following differential equation: e g p ′ ( r ) + 2 p ( r ) = 2 . (8.10)It is straightforward to verify that (8.8) is the most general solution of the ODE (8.10)satisfying the required conditions (8.1). Moreover, by inspecting (8.10) one realizes that, inthis massive case, it is quite convenient to introduce a new radial variable ρ such that: dρdr = e − g . (8.11)35enoting by a dot the derivative with respect to ρ , one finds that the first-order differentialequation (8.10) becomes: ˙ p + 2 p = 2 , ( ρ > ρ ∗ ) , (8.12)where ρ ∗ is the value of the ρ coordinate that corresponds to the minimal value r ∗ of r . Eq.(8.12) can be solved as: p ( ρ ) = h − e ρ ∗ − ρ ) i Θ( ρ − ρ ∗ ) , (8.13)where we have required the continuity of p ( ρ ) at ρ = ρ ∗ . Therefore, in the ρ variable thefunction p ( ρ ) is known and, in order to determine the background, one has to integrate thesystem: ˙Λ = (cid:16) k + 3 N f p ( ρ ) (cid:17) e Λ − f + g − k e Λ − g , ˙ f = 14 (cid:16) k + 3 N f p ( ρ ) (cid:17) e Λ − f + g − k e Λ − g + e − f +2 g , ˙ g = 12 (cid:16) k + 3 N f p ( ρ ) (cid:17) e Λ − f + g + 1 − e − f +2 g . (8.14)We will not attempt here to solve the system (8.14). Let us just mention that, when themass of the quarks is non-zero, there should be solutions of (8.14) that interpolate betweentwo AdS spaces, namely, the original unflavored ABJM geometry in the IR and our flavored
AdS space in the deep UV. We leave the verification of this fact and the detailed analysis ofthe system (8.14) for a future work.
Let us recapitulate our main results. We have studied the holographic dual of unquenchedflavor in the ABJM Chern-Simons-matter theory in the type IIA description. We have founda geometry of the type
AdS × M , where M is a compact six-dimensional manifold whosemetric we have explicitly determined. In order to get this result we have considered a setupin which the transverse internal space is filled with a continuous set of D6-branes which actas sources for the violation of the Bianchi identity of the RR two-form F and deform theoriginal unflavored AdS × C P geometry. To describe this deformation we have writtenthe C P manifold as an S -bundle over S and we have argued that the flavor induces asquashing of the S base with respect to the S fiber and, at the same time, changes the sizeof both the Anti-de-Sitter and the internal space. These squashing factors induced by thebackreaction of the flavor branes are constant and depend non-linearly on the deformationparameter ǫ defined in (5.16). 36ontrary to what happens to other flavored backgrounds obtained with the smearingmethod (see, for example, those of refs. [40, 41]) our supergravity solution has a good UV.Actually, since the metric has an Anti-de-Sitter factor, the holographic methods are firmlyestablished and it is possible to apply a whole battery of techniques to perform a cleananalysis of the flavor screening effects in different observables. This good behavior of theflavored solutions is a reflection of the fact that the D6-branes lift to pure geometry in elevendimensions and, as a consequence, it is possible to have a nice geometrical description ofmatter in these systems. We have carried out this study for some of these observables insection 7 and, in some cases, we have compared our results with the ones obtained from thelocalized tri-Sasakian geometry, in order to determine how these quantities change with thesmearing of the flavor branes. Although a more detailed understanding of the field theorydual to our background would be desirable, we showed that the flavor effects on severalquantities match the ones expected for a background in which dynamical quarks have beenincorporated.Our results can be generalized in several directions which we now briefly sketch. Firstof all, it is clear that the running solutions such as the ones of section 3.3 and appendixC deserve further analysis. In particular, it would be very interesting to determine if thereare non-trivial flows that connect different fixed points (see [42]). Another point that wouldbe worth to investigate is the integration of the BPS system (8.14) for massive flavors. Inthis case one expects to find solutions which interpolate between the AdS × CP unflavoredgeometry in the IR and our massless flavored Anti-de-Sitter solutions in the UV.Our analysis of the flavor effects in section 7 is clearly incomplete. One could also studyWilson loops in representations different from the fundamental such as, for example, Wilsonloops in the antisymmetric representations. Moreover, one could also compute the spectraof meson masses (and not just the dimension of the meson operators). In the unflavoredcase this analysis was performed in [23]. The expectation is that the screening effects wouldreduce the binding energy and, as a consequence, the masses would increase with respect tothe unflavored values (similar results for the theories on the conifold were obtained in [43]).Another possible direction for future research could be trying to generalize the flavoredsupergravity solutions found here. One of these generalizations could be constructing a blackhole solution with flavor (with finite temperature and, eventually, finite chemical potential)along the lines of [44] and then studying its corresponding thermodynamical and hydrody-namical properties. Another possibility could be adding fractional branes to our solution, insuch a way that the ranks of the two gauge groups is different [7], which would amount tointroduce a NSNS B field. Notice that, according to the analysis of [45], even the unflavoredABJM solution should have a non-vanishing flat B field due to the so-called Freed-Wittenanomaly. It would be interesting to explore how this issue changes when the dynamicalquarks are added to the background. Another topic of interest would by trying to finda flavored solution of type IIA supergravity with non-zero Romans mass which, according37o [8], would correspond to a theory in which the sum of the two Chern-Simons levels arenon-vanishing. Finally, one could also try to apply the same methodology to add flavor tobackgrounds with reduced supersymmetry, whose internal space is not CP .We are already working along some of these lines and we hope to report on these topicselsewhere. Acknowledgments
We are grateful to F. Benini, F. Bigazzi, A. Cotrone, J. Gaillard, N. Jokela, C. N´u˜nez,A. Paredes, D. Rodriguez-Gomez, S. Cremonesi, J. Tarrio and D. Zoakos for very usefuldiscussions. This work was funded in part by MICINN under grant FPA2008-01838, by theSpanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), by Xunta de Galicia(Conseller´ıa de Educaci´on and grant INCITE09 206 121 PR) and by FEDER. AVR thanksthe Galileo Galilei Institute for Theoretical Physics for the hospitality and the INFN forpartial support during the completion of this work. E.C. is supported by a Spanish FPUfellowship, and thanks the FRont Of Galician-speaking Scientists for unconditional support.
A ABJM geometry
Let us study the uplift of the ABJM metric (2.1) to eleven-dimensional supergravity. Thecorresponding uplifting formula for the metric is: ds = e − φ ds + e φ (cid:0) dx − A (cid:1) , (A.1)where we take the eleven-dimensional coordinate x to take values in the range x ∈ [0 , π )and A is the one-form potential for the type IIA field strength F : F = dA . (A.2)For the ABJM solution of section 2, the actual value of A is: A = − k (cid:16) cos θ dϕ + ξ S (cid:17) , (A.3)where S has been defined in (2.14). Let us next define a new angular variable ψ as: ψ ≡ k x , ψ ∈ (cid:2) , πk (cid:1) , (A.4)as well as a new one-form E as: E ≡ dψ + cos θ dϕ + ξ S . (A.5)38hen, the uplifted metric (A.1) can be written as the one corresponding to the product space AdS × S / Z k , namely : ds = R ds AdS + R ds S / Z k , (A.6)where R is given by R = 2 π N k , (A.7)and the S / Z k metric is: ds S / Z k = 14 h ds S + ( E ) + ( E ) + ( E ) i , (A.8)with ds S being the metric of the four-sphere written in (2.7). As a check one can verify thatthe eight-dimensional cone with metric dr + r ds S / Z k is locally flat. Moreover, the metric(A.8) can be written as an S -bundle over S . Indeed, let ˜ ω i ( i = 1 , ,
3) be a second set ofleft-invariant one-forms, defined in terms of the angles θ , ϕ and ψ as:˜ ω = − sin ϕ dθ + cos ϕ sin θ dψ , ˜ ω = cos ϕ dθ + sin ϕ sin θ dψ , ˜ ω = dϕ + cos θ dψ . (A.9)These forms satisfy d ˜ ω i = ǫ ijk ˜ ω j ∧ ˜ ω k . In terms of the ˜ ω i the metric (A.8) takes the form: ds S / Z k = 14 h ds S + X i (cid:0) ˜ ω i + A i (cid:1) i , (A.10)where the one-forms A i are the components of the SU (2) instanton connection written in(2.8).Let us come back to the CP metric, written in (2.12) as an S -bundle over S . Asmentioned at the end of section 2, there is a non-trivial CP ⊂ CP , which is spanned bythe angles θ and ϕ for a fixed point of the S . Let us now check that, in our coordinates,the embedding corresponding to a CP is obtained by taking the S angles θ and ϕ to beconstant and given by: ϕ = θ = π . (A.11)In this case the pullbacks of E and E are given by:ˆ E = ξ ξ ω , ˆ E = − ξ ξ ω . (A.12)Then, the pullback of the CP metric (2.12) is just: dξ (1 + ξ ) + ξ ξ ) h ( ω ) + ( ω ) i + ξ ξ ) ( ω ) . (A.13)39ntroducing the new angle χ as: ξ = cot (cid:16) χ (cid:17) , ≤ χ ≤ π , (A.14)we can rewrite the above metric as:14 h (cid:0) dχ (cid:1) + cos χ h ( ω ) + ( ω ) + sin χ ω ) i i , (A.15)which is, indeed, the Fubini-Study metric of CP . Notice that the total volume for thismetric is just π /
2. As an application of this result, let us consider now a D4-brane wrappedon the CP . Its mass is given by: m D = T D V CP , (A.16)where T D and V CP are: T D = 1(2 π ) e φ , V CP = π L , (A.17)where the dilaton has been written in (2.4) and L is the AdS radius for the unflavoredABJM solution displayed in (2.3). Using these values of g s and L , we get: m D = 12 √ π (cid:16) kN (cid:17) . (A.18)The resulting conformal dimension ∆ D is just m D L , namely:∆ D = N , (A.19)which is just the expected result since these branes are dual to dibaryon operators which areproducts of N bifundamental fields, each of them of dimension 1 / B SUSY analysis
In this appendix we will find the system of BPS first-order equations satisfied by our super-gravity solutions, as well as the corresponding Killing spinors. To find the supersymmetricconfigurations, we will use the SUSY variations of the dilatino λ and the gravitino ψ µ of thetype IIA SUGRA in string frame (which we take from [46]): δ λ = h
12 Γ µ ∂ µ φ + 38 e φ F (2) µν Γ µν Γ − e φ F (4) µνρσ Γ µνρσ i ǫ ,δψ µ = h ∇ µ − e φ F (2) ρσ Γ ρσ Γ Γ µ − e φ F (4) µνρσ Γ µνρσ Γ µ i ǫ . (B.1)40n order to study the supersymmetric metrics of form (3.1), let us choose the following basisof frame one-forms: e µ = h − dx µ , ( µ = 0 , , , e = h dr , e = h e f S ξ ,e = h e f S , e = h e f S , e = h e f S ,e = h e g E , e = h e g E . (B.2)Let us first compute the dilatino variation. One gets: δλ = h h − Γ φ ′ + 3 k e φ h − (cid:16) e − g Γ − η e − f Γ − η e − f Γ (cid:17) Γ ++ e φ K h Γ Γ i ǫ . (B.3)We will first impose the following projection conditions:Γ ǫ = Γ ǫ = Γ ǫ . (B.4)Then, the vanishing of the dilatino variation, δλ = 0, leads to the following equation: φ ′ ǫ + 3 k e φ h − (cid:0) e − g − η e − f (cid:1) Γ Γ ǫ − e φ K h Γ ǫ = 0 . (B.5)Let us next impose the following projection on ǫ :Γ ǫ = − ǫ . (B.6)Notice that, as Γ is defined as: Γ = Γ ··· , (B.7)the two matrices appearing on (B.5) are related, namely:Γ Γ = Γ Γ . (B.8)Since Γ ǫ = − ǫ (see eq. (B.4)), one has that the projection (B.6) implies that:Γ Γ ǫ = ǫ . (B.9)Using these projections, the dilatino equation becomes the following first-order differentialequation: φ ′ = − k e φ h − (cid:0) e − g − η e − f (cid:1) − e φ K h . (B.10)41he variation of the components of the gravitino along the Minkowski directions leads tothe equations: h ∂ x µ ǫ − h ′ µ ǫ − h k e φ h (cid:16) e − g − η e − f (cid:17) − e φ K h i Γ µ Γ ǫ = 0 . (B.11)When the projection (B.6) is imposed, (B.11) can be solved by means of a spinor which doesnot depend on the cartesian coordinates x µ . Indeed, if ∂ x µ ǫ = 0 eq. (B.11) leads to thefollowing differential equation for h : h ′ = k e φ h (cid:0) e − g − η e − f (cid:1) − e φ K h . (B.12)Let us now consider the equation obtained from the SUSY variation of the component ofthe gravitino along the direction 4. After using the projections (B.4) and imposing that thespinor does not depend on the internal coordinates, one arrives at the following equation:( h ′ + 4 hf ′ ) ǫ + 4 h e − f + g Γ ǫ − h e φ (cid:16) k e − g + 2 hK (cid:17) Γ ǫ = 0 . (B.13)In order to solve this equation, let us impose a new projection:Γ ǫ = − ǫ . (B.14)Using this projection, together with the one in (B.6), leads to the differential equation: h ′ + 4 hf ′ = − k h e φ e − g − e φ K h + 4 h e − f + g . (B.15)By combining eqs. (B.12) and (B.15) one can easily prove that: f ′ = k h − e φ (cid:2) η e − f − e − g (cid:3) + e − f + g . (B.16)Let us next consider the equation obtained from δψ = 0. Again, after imposing (B.4) andthe independence of the spinor on the internal coordinates, one arrives at: (cid:0) h ′ + 4 h g ′ (cid:1) ǫ + 4 h (cid:0) e − g − e − f + g (cid:1) Γ ǫ ++ 12 h e φ (cid:0) kη e − f + k e − g − Kh (cid:1) Γ ǫ = 0 . (B.17)By using again the projections (B.6) and (B.14), we get: h ′ + 4 h g ′ = k e φ h (cid:0) e − g + 2 η e − f (cid:1) − e φ K h + 4 h (cid:0) e − g − e − f + g (cid:1) . (B.18)42liminating h ′ from (B.12) we arrive at: g ′ = k e φ h − η e − f + e − g − e − f + g . (B.19)Let us finally analyze the supersymmetry variation of the gravitino component along theradial direction. After imposing (B.4) one arrives at the following equation: ∂ r ǫ = − e φ h K Γ ǫ . (B.20)This equation can be easily integrated. First of all we impose (B.6). Secondly, as shown insection 3.1, from the equations derived above the function K can be written in terms of φ and h as in (3.22) and one can show that: e φ h K = − ddr log (cid:16) e φ h (cid:17) . (B.21)Therefore, the Killing spinor equation (B.20) can be integrated as: ǫ = e − φ h − ǫ , (B.22)where ǫ is a constant spinor satisfying the same projections as ǫ .Eqs. (B.10), (B.12), (B.16) and (B.19) constitute the system of first-order BPS equa-tions (5.19). They have been obtained by imposing the projections (3.12) and ensure thepreservation of two supercharges, both in the unflavored and flavored theories. As we willshow in the next subsection, the actual number of supersymmetries is increased for certainparticular solutions of the BPS equations due to the fact that some of the projections whichare imposed in the generic case are not needed in these special solutions. In particular, forthe case of AdS solutions of sections 3.2 and 5.1, the projection (B.6) is not needed and thereare four Killing spinors (as it corresponds to N = 1 superconformal supersymmetry in threedimensions). Moreover, for the unflavored ABJM solution one can solve the BPS equationswithout imposing any of the projections written in (3.12) and, after a detailed study, onecan show that there are 24 Killing spinors, as it corresponds to N = 6 in 3d. B.1 SUSY for the Anti-de Sitter solutions
Let us consider the particular solution of the BPS equations which leads to the
AdS metric.Since the dilaton φ is constant in this case, it follows from (B.10) that the following relationholds: 2 η e − f − e − g = 2 Kh k . (B.23)Actually, by using (B.23) and (B.8) one can show that (B.5) is satisfied by imposing onlythe projections (B.4), without requiring the condition (B.6). Moreover, by plugging (B.23)43nto (B.11) one gets: ∂ x µ ǫ = 18 h − h ′ Γ µ ǫ − h e φ K Γ µ Γ ǫ . (B.24)Furthermore, when φ is constant (B.21) can be used to relate K to h ′ . This relation can bewritten as: e φ K = − h − h ′ . (B.25)By eliminating K on the right-hand side of (B.24), one gets: ∂ x µ ǫ = 18 h − h ′ Γ µ (cid:0) (cid:1) ǫ . (B.26)Since for these solutions h = L /r , where L is the AdS radius, we can rewrite (B.26) as: ∂ x µ ǫ = − r L Γ µ (cid:0) (cid:1) ǫ . (B.27)We can now combine this equation with (B.20) to obtain the dependence of the Killingspinors on the AdS coordinates. Indeed, by using (B.25) in (B.20) it is straightforward toprove that: ∂ r ǫ = − r Γ ǫ . (B.28)It is now easy to integrate (B.27) and (B.28) following [47]. One gets: ǫ = r − Γ0122 (cid:16) L x µ Γ Γ µ (cid:0) (cid:1) (cid:17) ǫ , (B.29)where ǫ is a constant spinor satisfying the projection conditions (B.4). Notice that thespinors ǫ in (B.29) with Γ ǫ = − ǫ satisfy (B.6) and are independent of the cartesiancoordinates. On the contrary, if we choose an ǫ such that Γ ǫ = ǫ , the resulting Killingspinors ǫ do depend on the cartesian coordinates and do not have a well-defined eigenvalue ofΓ . Moreover, since for these AdS solutions h ′ + 4 hf ′ = h ′ + 4 hg ′ = 0, one can easily verifythat the equations obtained from the variation of the gravitino along the internal directions( i.e. eqs. (B.13) and (B.17)) are satisfied if the following projection:Γ Γ ǫ = ǫ , (B.30)is imposed on ǫ . Notice that the matrix on the left-hand side of (B.30) commutes with theone multiplying the constant spinor ǫ in (B.29). Thus, ǫ must also satisfy (B.30) and these AdS backgrounds preserve four supercharges, as claimed.Interestingly, the BPS equations for the
AdS solutions can be recast as the ones corre-sponding to a compactification with fluxes in an internal manifold with an SU (3)-structure(see [48] for a review). To verify this fact, let us define the fundamental two-form J as: J = h − e − f (cid:16) e ∧ e + e ∧ e + e ∧ e (cid:17) , (B.31)44here the one-forms e , · · · e are the ones written in (B.2). Moreover, let Ω hol be theholomorphic three-form defined as: i Ω hol = h − e − f (cid:0) e + ie (cid:1) ∧ (cid:0) e + ie (cid:1) ∧ (cid:0) e + ie (cid:1) . (B.32)One can check that these forms satisfy: d J = 32 W Im (cid:0) Ω hol (cid:1) ,d Ω hol = −W J ∧ J − W ∧ J , (B.33)where W and W are the so-called torsion classes which, for our solutions with e g = e f / √ q ,are given by: W = 23 q + 1 √ q ,h e f W = 23 2 − q √ q (cid:16) e ∧ e + e ∧ e − e ∧ e (cid:17) . (B.34)Notice that, in terms of the one-forms in (B.2), our ansatz (5.6) for F can be written as: h e f F = − k h η (cid:0) e ∧ e + e ∧ e (cid:1) − √ q e ∧ e i . (B.35)Then, one can check that, if the squashing factors q and η are related as in (6.1), the two-form F is also given in terms of the torsion classes and of the fundamental form by: F = − √ q − q η k h W J − ∗ (cid:0) W ∧ J (cid:1) i , (B.36)where the W i are given in (B.34) and ∗ denotes the Hodge dual with respect to the six-dimensional internal metric h − / e f (cid:0) ( e ) + · · · + ( e ) (cid:1) . B.2 SUSY for the unflavored ABJM solution
For the ABJM unflavored solution the squashing factors q and η are equal to one. Moreover,it follows from (3.35) and (3.36) that K h e g = 3 k/ δλ = 0. Indeed, it isstraightforward to show from (B.3) that the equation for the SUSY variation of the dilatinoleads to: ǫ = (cid:0) Γ − Γ − Γ (cid:1) ǫ . (B.37)In order to study the solutions of this equation let us work on a representation of theDirac algebra in which the spinors are characterized by their ± { Γ , i Γ , i Γ , i Γ , i Γ } . In this representationthe matrices on the right-hand side of (B.37) will also act diagonally. Let us parametrizetheir eigenvalues as:Γ ǫ = s ǫ , Γ ǫ = s ǫ , Γ ǫ = − s s ǫ . (B.38)One immediately shows that (B.37) is equivalent to the following condition on s and s : s − s + s s = 1 . (B.39)Since only three of the four possible values of ( s , s ) satisfy (B.39), the projection (B.37)preserves 3/4 of the supercharges, i.e.
24 of them. This is, indeed, the amount of super-symmetry of an N = 6 supersymmetric theory in 3d. Moreover, one can show that theremaining equations for ǫ can also be solved without imposing any additional projection. C A consistent truncation
Let us see that there exists a consistent truncation of the reduced unflavored BPS system(3.29) that allows to find some non-trivial solutions. In these truncations Σ and ∆ are relatedas: e Σ = A e − ∆ , (C.1)with A being a constant. To have a natural interpretation of this truncation, let us look atthe metric obtained after uplifting to eleven dimensions. This metric is given by (A.1). Forour explicit ansatz (3.1)-(3.2) the uplifted eleven-dimensional metric has the form: ds = e − φ h − dx , + e − φ h dr + e − φ h e f ds S ++ e − φ h e g h ( E ) + ( E ) i + e φ ( dx − A ) . (C.2)Notice that the relative coefficient between the U (1) fiber and the E E parts of ds (thelast two terms in (C.2)) is given by e φ h − e − g = e , which is constant if the truncationcondition (C.1) holds. Therefore, when (C.1) is satisfied these two parts of the metric cangive rise to the metric of a three sphere.Substituting the relation (C.1) on the right-hand side of the system (3.29), we get thefollowing two equations: ˙Σ = (cid:16) kA − (cid:17) e − ∆ − kA e ∆ , ˙∆ = (cid:16) − kA (cid:17) e − ∆ − (cid:16) kA (cid:17) e ∆ . (C.3)46he truncation condition (C.1) implies that ˙Σ = − ˙∆ which in turn leads to: (cid:16) kA (cid:17) (cid:16) e − ∆ − e ∆ (cid:17) = 0 , (C.4)There are two possible ways to solve (C.4). The first one is by imposing that e ∆ = e − ∆ ,which would lead to e = 1 or ∆ = 0. This solution with constant squashing correspondsto the unflavored ABJM solution described in section 2. Indeed, from (C.1) it follows thatΣ is constant and, by taking ∆ = ˙∆ = ˙Σ = 0 in (C.3) one discovers that A = 2 /k = e Σ forthis solution. Notice that this implies that k must be positive in this case.The other possibility to solve (C.4) consists in taking kA = −
2, namely: A = − k , (C.5)which implies that e Σ = − k e − ∆ . (C.6)Notice that k must be negative in this case. In this solution the squashing can vary with theradial coordinate. To continue with the analysis of this case it is much more convenient tocome back to the original system (3.21) in terms of the radial variable r and the functionsΛ, f and g . Actually, from the definitions of Σ and ∆, the truncation (C.6) is equivalent to: e Λ = − k e g . (C.7)After using (C.7), the equations for f and g in (3.21) become: f ′ = 12 e g − f + 12 e − g ,g ′ = − e g − f + e − g . (C.8)It follows by combining these equations that: g ′ − f ′ = − e g − f , (C.9)which can be immediately integrated, namely: e f − g = 3 r + c , (C.10)with c being a constant. Let us use this result in the equation for g ′ in (C.8). If we define: y ≡ e g , (C.11)47hen, the equation for y is: y ′ + 23 r + c y = 1 . (C.12)By using the method of variation of constants, we get the general solution of (C.12), namely: y = 35 (cid:0) r + c (cid:1) + 35 a (cid:16) r + c (cid:17) , (C.13)with a being a constant. By defining a new constant b as: b ≡ a , (C.14)we get the following solution for f and g : e f = (3 r + c ) h b (3 r + c ) i ,e g = (3 r + c ) h b (3 r + c ) i . (C.15)Let us rewrite this solution in terms of the new radial variable ρ defined as:3 ρ = 3 r + c . (C.16)Introducing a new constant µ , related to b and a as: µ = − b = − a , (C.17)the functions f and g can be written as: e f = 95 ρ h − (cid:16) µρ (cid:17) i , e g = 925 ρ h − (cid:16) µρ (cid:17) i . (C.18)The squashing corresponding to this solution is: e = 51 − (cid:16) µρ (cid:17) , (C.19)which varies with the radial coordinate except when the constant µ is chosen to vanish. Inthis last case one gets the gravity dual of the N = 1 model [15] with squashed CP and SO (5) × U (1) global symmetry (see below). Notice that, in this variable ρ , the function Λis given by: e Λ = 65 | k | ρ h − (cid:16) µρ (cid:17) i , (C.20)48here we have taken into account that k should be negative for this solution and that | k | = − k . Let us now compute the warp factor h from the integral (3.25). Let us expressthe result in terms of the quantity ¯ R defined as:¯ R = 2 | k | β . (C.21)By using the relation β = 3 π N , one can express ¯ R in terms of the rank N of the gaugegroup as: ¯ R = 2 π N | k | . (C.22)From (3.25) one can now compute the warp factor h . One gets: h ( ρ ) = − R k ρ h − (cid:0) µρ (cid:1) i Z ρρ dξξ h − (cid:0) µξ (cid:1) i , (C.23)where ρ is a constant that should be determined. The dilaton φ for this solution can beobtained simply from the relation e φ = e Λ h , which follows from the definition of Λ in(3.19). Similarly, the function K which parametrizes the RR four-form F can be obtainedfrom (3.6). C.1
AdS solution with squashing
Let us consider now the squashed solution obtained above in the case in which the constant µ vanishes. It follows from (C.19) that e = 5 and that the functions f and g are given by: e f = 95 ρ , e g = 925 ρ . (C.24)The integral (C.23) giving the warp factor can be straightforwardly computed. If one chooses ρ = −∞ , one ends up with the result: h ( ρ ) = 9400 ¯ R | k | ρ , (C.25)where ¯ R has been defined in (C.21). Moreover, the dilaton is constant with this election of ρ and given by: e φ = 27125 ¯ R | k | . (C.26)Let us now write the metric corresponding to this solution. By rescaling the Minkowskicoordinates as x µ = λ ¯ x µ one can show that the coordinates x µ and r parametrize and AdS space. The value of λ that one has to choose is: λ = 320 ¯ R | k | . (C.27)49n order to have an interpretation of the parameter ¯ R , let us look at our solution upliftedto M-theory. By using eq. (C.2) and redefining the coordinate x as in (A.4), we get: ds = ¯ R ds AdS + ¯ R h ds S + 9100 (cid:2) ( E ) + ( E ) + ( E ) (cid:3) i , (C.28)where the one-form E has been defined in (A.5). The metric (C.28) corresponds to a spacewhich is the product of AdS with radius ¯ R/ S /Z k , with ˜ S being thesquashed seven-sphere with metric ds S /Z k = 14 h ds S + 925 X i (cid:0) ˜ ω i + A i (cid:1) i , (C.29)where the ˜ ω i are the one-forms defined in (A.9). The resulting ten-dimensional metric takesthe form: ds = ¯ L ds AdS + 3625 ¯ L ds CP , (C.30)where ds CP is the metric of a squashed CP , given by: ds CP = 54 ds S + 14 (cid:2) ( E ) + ( E ) (cid:3) , (C.31)and the Anti-de-Sitter radius ¯ L is given by:¯ L = 9400 ¯ R | k | = 250243 π N | k | . (C.32)The RR four-form F for this solution is given by: F = 52 | k | ¯ L Ω AdS = 25 π r (cid:16) | k | N (cid:17) Ω AdS . (C.33)As an application of the results found above, let us compute the conformal dimension of thegauge dual of a D0-brane. First of all we notice that the string coupling constant in thissquashed case is given by: g s = e φ = 2 √ π h N | k | i . (C.34)Using this result we can compute the conformal dimension of the D0-brane by means of theformula ∆ D = ¯ L m D = ¯ L/g s . One gets:∆ D = 56 | k | , (C.35)which suggests that the some of the scalar bifundamental fields have dimension 5 / Kappa symmetry
Let us consider a flavor D6-brane embedded in the ten-dimensional background (3.1) andlet us choose the following set of worldvolume coordinates: ζ α = ( x µ , r, ξ, ˆ ψ, ϕ ) . (D.1)The embedding is then defined by the equations:ˆ θ , ˆ ϕ = constant , θ = θ ( r ) . (D.2)We want to determine, by using kappa symmetry, the function θ ( r ) that makes the embeddingsupersymmetric. First of all we compute the pullback of the frame one-forms:ˆ e µ = h − dx µ , ˆ e = h dr , ˆ e = 21 + ξ h e f dξ , ˆ e = 0 , ˆ e = ξ ξ h e f sin θ d ˆ ψ , ˆ e = − ξ ξ h e f cos θ d ˆ ψ , ˆ e = h e g θ ′ dr , ˆ e = h e g sin θ (cid:16) dϕ − ξ ξ d ˆ ψ (cid:17) . (D.3)The induced gamma matrices are: γ x µ = h − Γ µ , γ r = h (cid:16) Γ + e g θ ′ Γ (cid:17) , γ ξ = 21 + ξ h e f Γ ,γ ˆ ψ = ξ ξ h e f sin θ h Γ − cot θ Γ − ξ e g − f Γ i , γ ϕ = h e g sin θ Γ . (D.4)Let γ ∗ be the antisymmetrized product of all induced gamma matrices, namely: γ ∗ ≡ γ x x x r ξ ˆ ψ ϕ . (D.5)In terms of the flat 10d Dirac matrices γ ∗ is: γ ∗ = 2 ξ (1 + ξ ) h e f + g ( sin θ ) Γ (cid:16) Γ + e g θ ′ Γ (cid:17) Γ (cid:16) Γ − cot θ Γ (cid:17) Γ . (D.6)Notice that the kappa symmetry matrix Γ κ is just:Γ κ = 1 √− det ˆ g γ ∗ , (D.7)where ˆ g is the induced metric on the worldvolume. With our notations the supersymmetricembeddings are those that satisfy Γ κ ǫ = − ǫ . Using the projection Γ ǫ = − ǫ (see eq.51B.6)), one obtains that γ ∗ acts on the Killing spinors as: γ ∗ ǫ = − ξ (1 + ξ ) h e f + g ( sin θ ) ×× " Γ ǫ − cot θ Γ ǫ + e g θ ′ Γ ǫ + e g θ ′ cot θ Γ ǫ . (D.8)From the projections satisfied by the Killing spinors it follows that:Γ ǫ = − Γ ǫ = ǫ , Γ ǫ = Γ ǫ = − Γ ǫ . (D.9)Therefore: γ ∗ ǫ = − ξ (1 + ξ ) h e f + g ( sin θ ) h e g θ ′ cot θ + (cid:0) cot θ − e g θ ′ (cid:1) Γ i ǫ . (D.10)In order to fulfill the condition Γ κ ǫ = − ǫ we should require that the terms not containingthe unit matrix vanish. Thus, we are led to the following differential equation for θ ( r ): dθdr = e − g cot θ . (D.11)The induced metric for an embedding like the one in our ansatz is given by: ds wv = h − ( dx µ ) + h (cid:0) e g ( θ ′ ) (cid:1) ( dr ) + h e f ξ ) ( dξ ) ++ h e f ξ (1 + ξ ) ( d ˆ ψ ) + h e g sin θ (cid:16) dϕ − ξ ξ d ˆ ψ (cid:17) , (D.12)whose determinant is just: p − det ˆ g = 2 ξ (1 + ξ ) h e f + g sin θ p e g ( θ ′ ) . (D.13)If the BPS equation (D.11) holds, the determinant of the induced metric becomes: p − det ˆ g | BP S = 2 ξ (1 + ξ ) h e f + g . (D.14)Therefore, we have:Γ κ ǫ | BP S = − ( sin θ ) h e g θ ′ cot θ i ǫ | BP S = − ǫ , (D.15)which proves the supersymmetric character of the embeddings satisfying (D.11).52et us now integrate the BPS equation (D.11). The integration of this equation is imme-diate, namely: log (cid:0) cos θ (cid:1) = − Z r e − g ( z ) dz + constant . (D.16)This result can be rewritten as:cos θ = C exp h − Z r e − g ( z ) dz i , (D.17)where C is a constant. Let us suppose that the function g is given by: e g = rb , (D.18)with b being a positive constant. This is the form of e g for the different AdS solutions (see,for example, (6.8)). Using this expression of e g in (D.17) and defining r ∗ as C = r b ∗ , one gets:cos θ = (cid:16) r ∗ r (cid:17) b . (D.19)Notice that r ∗ is the minimal value of the coordinate r (which occurs at θ = 0) and that θ → π/ r → ∞ . Moreover, when r ∗ = 0, which corresponds to the massless case, theangle θ takes the constant value θ = π/ b is equal to one and the embedding is givenby: cos θ = r ∗ r . (D.20)Furthermore, in the squashed ABJM model b = 5 / θ ( r ) is:cos θ = (cid:16) r ∗ r (cid:17) . (D.21) D.1 Equations of motion of the probe
The dynamics of the D6-brane probe is governed by the DBI+WZ action. The first of thesetwo terms is given by: S DBI = − T D Z e − φ p − det ˆ g d ζ . (D.22)For our ansatz, the determinant of the induced metric has been computed in (D.13). Let usintegrate over the coordinates ξ , ˆ ψ and ϕ . This integration generates the following constantmultiplicative factor: Z ∞ ξ (1 + ξ ) dξ Z π d ˆ ψ Z π dϕ = 8 π . (D.23)53herefore, the DBI lagrangian density in the remaining coordinates is given by: L DBI = − π T D e − φ h e f + g sin θ p e g ( θ ′ ) . (D.24)Let us next consider the WZ term of the action, which is given by: S W Z = T D Z ˆ C , (D.25)with ˆ C being the pullback of the RR seven-form potential of F ≡ − ∗ F ( i.e. F = dC ). Itis clear from the calibration condition (3.18) that K provides such a potential, namely onecan choose C as: C = e − φ K . (D.26)Thus, the WZ action takes the form: S W Z = T D Z e − φ ˆ K . (D.27)By looking at the expression of the calibration form K in (3.15) and the pullbacks of the basisone-forms in (D.3), one easily concludes that only the terms with e and e contribute.The result for ˆ K is the following:ˆ K = 2 ξ (1 + ξ ) h e f + g sin θ h sin θ + cos θ e g θ ′ i d ζ . (D.28)Integrating again over ξ , ˆ ψ and ϕ , we get the following WZ lagrangian density: L W Z = 8 π T D e − φ h e f + g sin θ h sin θ + cos θ e g θ ′ i . (D.29)Thus, the total lagrangian density is: L = − π T D e − φ h e f + g sin θ h p e g ( θ ′ ) − (cid:16) sin θ + cos θ e g θ ′ (cid:17) i . (D.30)From this expression we compute the two derivatives that enter the equations of motion ofthe probe, namely: ∂ L ∂θ = − π T D e − φ h e f + g h cos θ p e g ( θ ′ ) − θ cos θ − (cos θ − sin θ ) e g θ ′ i ,∂ L ∂θ ′ = − π T D e − φ h e f + g sin θ h e g θ ′ p e g ( θ ′ ) − cos θ e g i . (D.31)One can verify straightforwardly that the terms in parentheses on the right-hand side of(D.31) vanish when the BPS equation for the embedding (D.11) holds. This shows that,indeed, the equations of motion of the probe are satisfied for the kappa symmetric configu-ration. 54 Equations of motion
In this appendix we write the equations of motion for the supergravity plus branes systemand we verify that the first-order BPS system (5.19) implies the second-order equations ofmotion for the different fields. In what follows we will work in Einstein frame. The totalaction is given by: S = S IIA + S sources , (E.1)with the type IIA term being given by: S IIA = 12 κ " Z √− g (cid:16) R − ∂ µ φ ∂ µ φ (cid:17) − Z h e φ ∗ F ∧ F + e φ ∗ F ∧ F i , (E.2)and the source contribution is just the sum of the DBI and WZ action of the D6-branes,which can be written as: S sources = − T D Z (cid:16) e φ K − C (cid:17) ∧ Ω , (E.3)where the calibration form K is given by (3.15), with the e a ’s being the one-forms writtenin (B.2) multiplied by e − φ/ , as it corresponds to the Einstein frame.The Maxwell equations for the forms F and F derived from (E.1) are just: d (cid:16) e φ ∗ F (cid:17) = 0 , d (cid:16) e φ ∗ F (cid:17) = 0 , (E.4)and one can show that they are satisfied for our ansatz as a consequence of the BPS equations(5.19). Similarly, the equation for the dilaton φ derived from (E.2) is just: d ∗ dφ = 34 e φ ∗ F ∧ F + 14 e φ ∗ F ∧ F + 32 κ T D e φ K ∧ Ω , (E.5)and it is also fulfilled as a consequence of the first-order equations (5.19). Let us now studyEinstein equations, which read: R µν − g µν R = 12 ∂ µ φ ∂ ν φ − g µν ∂ ρ φ ∂ ρ φ + 14 e φ h F (2) µρ F (2) ρν − g µν F i ++ 148 e φ h F (4) µρσλ F (4) ρσλν − g µν F i + T sources µν , (E.6)where T sources µν is just the stress-energy tensor for the flavor branes, defined as: T sources µν = − κ √− g δS sources δg µν . (E.7)55n order to write the explicit expression of T sources µν derived from (E.7), let us define thefollowing operation for any two p -forms ω ( p ) and λ ( p ) : ω p y λ ( p ) = 1 p ! ω µ ...µ p λ µ ...µ p . (E.8)Then, by computing explicitly the derivative of the action (E.3) with respect to the metric,one can check that: T sources µν = κ T D e φ h g µν ∗ K y Ω −
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