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Holographic Duals of Flavored N=1 Super Yang-Mills: Beyond the Probe Approximation

We construct backreacted D3/D7 supergravity backgrounds which are dual to four-dimensional N=1 and N=2 supersymmetric Yang-Mills at large N_c with flavor quarks in the fundamental representation of SU(N_c). We take into account the backreaction of D7-branes on either AdS(5) x S(5) or AdS(5) x T^{1,1}, or more generically on backgrounds where the space transverse to the D3-branes is Kaehler. The construction of the backreacted geometry splits into two stages. First we determine the modification of the six-dimensional space transverse to the D3 due to the D7, and then we compute the warp factor due to the D3. The N=2 background corresponds to placing a single stack of N_f D7-branes in AdS(5) x S(5). Here the Kaehler potential is known exactly, while the warp factor is obtained in certain limits as a perturbative expansion. By placing another D7'probe in the backreacted D3/D7 background, we derive the effect of the D7-branes on the spectrum of the scalar fluctuations to first order in N_f. The two systems with N=1 supersymmetry that we discuss are D3/D7/D7' and D3/D7 on the conifold. In both cases, the Kaehler potential is obtained perturbatively in the number of D7-branes. We provide all the ingredients necessary for the computation of each term in the expansion, and in each case give the first few terms explicitly. Finally, we comment on some aspects of the dual gauge theories.


Introduction
In the framework of the gauge/gravity correspondence [1] a complete holographic description of QCD is still beyond our current reach. Nevertheless, substantial progress has been achieved in the holographic understanding of pure N = 1 super Yang-Mills (SYM) at large N c [2,3]. The holographic understanding of flavor dynamics at the quantitative level has developed slower, in great measure due to the lack of a fully backreacted supergravity background describing gauge theories with fundamental flavor. In this paper we address the construction of fully backreacted supergravity backgrounds dual to N = 1 SYM with fundamental flavor.
Why go beyond the probe approximation? One of the major stepping stones in a holographic understanding of QCD is a control of flavor dynamics. For example, dynamical mesons and baryons play a central rôle in obtaining a complete understanding of the QCD phase diagram. In particular, in the large N c limit the critical temperature should behave very differently in the presence of baryons. On the other hand, it is known that baryons play a subleading rôle at large N c , which leads to the question of how precisely they change the critical temperature. On a more formal level, the quantum moduli space of SU(N c ) supersymmetric gauge theories with N f flavors crucially depends on the relative value of N c and N f . There is a host of phenomena in flavor dynamics of supersymmetric theories, such as the conformal window, that can be explored only in the limit of comparable N c and N f . Having N f ≪ N c excludes, in principle, the possibility of a holographic description of such phenomena. Clearly, unless a fully backreacted solution becomes available, a quantitative understanding of several physical questions of gauge theories will remain elusive in the holographic approach.
In this paper we address the question of backreaction in the context of supergravity backgrounds dual to N = 1 supersymmetric gauge theories. While the general class of supergravity backgrounds dual to N = 1 gauge theories is wide, we concentrate only on backgrounds containing D3 and D7-branes. In this case the backreacted metric of the space transverse to the stack of D3-branes remains Kähler. For Kähler metrics a remarkable simplification of the Einstein equations takes place. In fact, solving the Einstein equations is equivalent to solving an equation that in many cases becomes a standard Monge-Ampère equation. Therefore, the problem of finding backreacted solutions splits into two main steps. In the first one, after finding a suitable Ansatz for the dilaton, we determine the Kähler potential of the resulting six-dimensional background transverse to the D3 worldvolume. In the second step we find the warp factor due to the presence of the D3-branes.
The paper is organized as follows. In section 2 we review the construction of the D3/D7 system to introduce notation and comment on some of the choices that we make further in the paper. Section 3 contains a description of the D3/D7/D7 ′ system, that is, two perpendicular stacks of the D7-branes that are parallel to the D3. This system has N = 1 supersymmetry, and we discuss it in some detail. Section 4 presents a probe calculation of the scalar (meson) spectrum of the N = 1 gauge theory. We provide the spectrum as an expansion in the number of D7-branes of the D3/D7 geometry. We then turn to the conifold picture by first reviewing the structure of the conifold as a Kähler manifold in section 5, and then describing a solution with D7-branes holomorphically embedded in the conifold in section 6. Section 7 contains some comments on the gauge theory side. We conclude in section 8 by enumerating a number of open problems.
Appendix A contains some useful results pertaining to properties of the Monge-Ampère equation.

D3/D7 Systems
In this section we review the basic setup for examining D3/D7 systems. The D3brane geometry by itself breaks the 10-dimensional Poincaré invariance into manifest SO(3, 1) × SO (6). This may be represented by a metric of the form for the case of a single stack of D7-branes.
D7-brane configurations have been effectively studied in [19] in the context of cosmic strings, and D3/D7 systems have been examined in [20,21]. The starting point for exploration of D3/D7 geometries preserving some fraction of supersymmetries is the set of Killing spinor equations obtained from the IIB supersymmetry transformations [22] are the SL(2, R) scalar kinetic and composite connection terms for the dilaton/axion The Killing spinor equations simplify in the absence of G 3 flux to yield Corresponding to the 4 + 6 coordinate split of (2.1), we decompose the Dirac matrices according to γ µ = (Γ µ ⊗ 1), γ m = (Γ 5 ⊗ Γ m ), where Γ represents either the SO(1, 3) or SO(6) Dirac matrices as appropriate. This allows us to define the chirality matrices The complex IIB spinor ǫ has definite chirality, γ 11 ǫ = ǫ. Hence the four and sixdimensional chiralities are correlated, so that γ 5 (4) ǫ = γ 7 (6) ǫ = sǫ where s = ±1. In addition, the coordinate split also allows us to write the most general five form consistent with four-dimensional Poincaré symmetry and self-duality: where F M is a function only of the y m coordinates, and is zero for M = 0 . . .

To insure
Poincaré symmetry we must also take τ to be a function only of the y m coordinates.
Using the above, the Killing equations, (2.4), can now be written as where ∇ (6) is covariant with respect to the metric on the six dimensional space transverse to the D3 branes. Taking ǫ to be only a function of y m , the second equation can immediately be solved by taking γ 5 (4) ǫ = ǫ (so that s = 1) and at the same time setting This is a remarkable result, as it demonstrates that the usual relation between the D3brane warp factor h(y m ) and the five-form remains unchanged, even in the presence of D7-branes. Note that the s = 1 condition on Killing spinors yields the familiar result that the D3-branes by themselves are half-BPS objects.
Taking the warp factor as above, the Killing spinor equations (2.7) reduce to where η = h 1/8 ǫ. These equations may be used to determine the structure of the D7-branes. Since the second equation above implies that η is covariantly constant, the connection ∇ m is in U(3), and the transverse space to the D3-branes is complex Kähler.
The first of the above equations is a familiar one for D7-branes, namely (γ m ∂ m τ )η * = 0, and can be satisfied by taking τ to be holomorphic on the complex 3-fold. This yields the further conditions γ m η * = 0 on the Killing spinors whenever ∂ m τ = 0 (for the generic case), where z m (m = 1, 2, 3) is now a complex coordinate. As a result, the D3/D7 system preserves 1/4, 1/8 and 1/8 of the IIB supersymmetries for τ depending holomorphically on one, two and three of the complex z m coordinates, respectively.
Note that there is no further reduction of supersymmetry for the last case, as the product of the three independent D7 projections gives precisely the D3 projection.
The second equation in (2.9) has a useful integrability condition, R mn = P m P * n + P * m P n . This has a particularly nice form for Kähler geometry, namely R where m andn denote complex indices. Taking τ (z m ) to be holomorphic and using P m from (2.3) results in −∂ m ∂n log(det(g (6) )) = −∂ m ∂n log(τ − τ * ), (2.10) which may be integrated to give det(∂ m ∂nF ) = ΩΩ Im (τ ). (2.11) Here F is the Kähler potential and Ω is an arbitrary holomorphic function accounting for the integration factors. The same Ω is used for the holomorphic and antiholomorphic integration functions to insure that the Kähler potential is real. In cases with 'radial' symmetry, Eq. (2.11) can be reduced to a real Monge Ampère equation (see Appendix A for further comments on this equation).
Finally, returning to the D3-brane warp factor, the integrability condition combined with the Einstein equations gives the result that the function h is harmonic As a result, solving for the D3/D7 geometry can be reduced to several steps. First, we take a (noncompact) Calabi-Yau background geometry, and then turn on the D7branes by choosing a holomorphic τ with appropriate monodromies. Then we must solve the complex Monge Ampère equation (2.11) for the Kähler potential, yielding the geometry of the six-dimensional transverse space. Finally, to include the D3branes, we must solve the harmonic function equation (2.12). This gives the warp factor and corresponding five-form flux through (2.8). Unlike for most intersecting brane configurations, at least in principle this procedure yields fully localized D3/D7 solutions, although the resulting equations are often quite intractable in practice. Note that the possibility of constructing localized intersecting solutions for the D3/D7 system is a feature shared with the D2/D6 system [23].

D3 and a single stack of D7 in flat space
We now consider the simplest case where the six-dimensional transverse space to the D3-branes is flat before turning on any D7-branes. In particular ds 6, background = dz 1 dz 1 +dz 2 dz 2 +dz 3 dz 3 , F flat, background = z 1z1 +z 2z2 +z 3z3 . (2.13) In the presence of D7-branes, the Kähler function may be obtained by solving (2.11).
However, before doing so, we must understand the form of the dilaton/axion field, For a single stack of D7-branes transverse to z 3 , we may follow the procedure given in [19], which we outline here. If one requires modular invariance for τ , holomorphicity and the proper monodromy around the D7 branes, the basic solution can be written in terms of the modular invariant j-function as [19,21] j(τ ) = f (z 3 ), (2.14) where for now f is an arbitrary holomorphic function. In [19] it was observed that poles in f correspond to D7-branes, in the sense of decompactification at the cores of the branes (as will be indicated below). Solving (2.11) with this form of the dilaton/axion is quite involved; hence we only consider the neighborhood close to one of the poles The rôle of the j-function in the above expression is to provide a map from the fundamental domain into the whole complex plane which is then related to N f coverings of the z 3 -plane.
We have limited ourselves to z 3 close to z 0 which requires that τ 2 is large. In this case the j function can be approximated as exp(−2πiτ ), so that the above equation where we have shifted z 3 to remove z 0 . This is known as the "decompactification limit" from F-theory because it corresponds to the modulus of the compactified torus going to infinity. We will work in this limit for the remainder of the paper. (Note that Eq. (2.16) is no longer appropriate for z 3 1, where instead τ approaches a constant value.) It is also natural to consider scalings of z 3 so that b may be removed as well.
Two scalings that are natural are b = 1 and b = e 2π , making the constant log(b)/2π either 0 or 1, respectively.
In [19], Ω is fixed by considering modular invariance of the metric, imposing regularity of the metric, and noting that g 33 is proportional to τ 2 . This gives Near the core of the D7-brane, where τ 2 is large, this has the simple form In particular, the additional factor |η(τ )| 4 |z 3 | −N f /6 in the metric simply approaches a constant near the cores of the D7-branes.
We now turn to the question of the Kähler potential. We have chosen to stack the D7-branes together so that they are extended along z 1 and z 2 , and are transverse in z 3 . Making use of the background symmetry, and noting that τ 2 is a function of the real combination z 3z3 suggests that we take The background Kähler potential is obtained from (2.11) by taking Imτ = 1 and Ω = 1, so that when f = z 3z3 we simply get flat space. We now take b = e 2π and turn on non-zero N f , so that (2.11) becomes By making a change of variables, y 3 = log(z 3z3 ), we arrive at This may be straightforwardly integrated to obtain the Kähler function for a single stack of D7-branes located at z 3 = 0 The constants C 1 and C 2 do not affect the metric; we will therefore set them to zero (we may always add a holomorphic plus antiholomorphic function to a Kähler potential).
Of course, the transverse space metric is given by prove essential when considering multiple stacks of D7-branes as well as D7-branes in other backgrounds.

The warp factor
Now that we have found the D7-brane contribution to the transverse metric, (2.23), we may turn to the D3-brane warp factor. To determine the warp factor, we must solve the transverse Laplacian (2.12). For a single stack of D7-branes, we first rewrite the metric (2.23) in the generic form 24) in which case the transverse Laplacian is more explicitly This expression for the warp factor was examined in [24], where it was solved to a first order approximation away from the D7-branes, where the metric is approximately flat, with a deficit angle.
In the absence of D7-branes, it is natural to place the stack of D3-branes at the origin, z m = 0, in which case the SO(6) R-symmetry is preserved, so that h = L 4 /R 4 where R is the radial coordinate, In orienting the D7-branes to lie fully in the z 1 and z 2 -planes, we preserve a natural SO(4) × SO(2) ⊂ SO (6) subgroup. Note, however, that separating the D3 and D7-branes in the z 3 -plane will break the SO (2). With this symmetry in mind, we introduce real coordinates so that If the D7-branes are separated by a distance d along the real z 3 -axis, the metric function e Ψ has the form Although no straightforward solutions to (2.27) appear to exist in the general case, we may find approximate solutions in appropriate limits. For example, when d = 0 the D3-branes overlap the D7-branes, and the remaining SO(2) symmetry is restored.
In this case, the Laplacian reduces to one with only two radial variables, ρ 3 and r.
Furthermore, we may consider solving for the warp factor near the core of the D7branes, where e Ψ ≫ 1. This is the decompactification region of the complex z 3 -plane, and corresponds to the region where the backreaction is strong.
It can be checked that the equation 1 2.27 allows separation of variables in the form: The equation for the function f (r) can be solved in terms of Bessel functions. The equation for g can be further separated in variables, after Fourier expanding in an appropriate angular variable, resulting in an equation for the radial variable along.
The latter is a more involved second order differential equation which we will not discuss here in detail.

Dand two stacks of D7-branes
The D3 geometry in the presence of a single stack of D7-branes is perhaps the most straightforward to investigate, as there is little ambiguity in how the D7-branes are turned on. However, we are generally more interested in configurations admitting N = 1 duals with flavor. This may be achieved by either starting with a transverse space to the D3-branes with reduced supersymmetry or by turning on multiple stacks of D7-branes. In this section we focus on the latter possibility by turning on two stacks of mutually perpendicular D7-branes.
In particular, we can construct a solution representing two stacks of D7-branes by placing, as before, N 3 D7-branes perpendicular to the z 3 -plane and another set of N 2 D7-branes transverse to the z 2 -plane. This configuration may be represented as follows 0 1 2 3 4 5 6 7 8 9 Note that turning on a third stack of D7-branes transverse to z 1 would not further reduce any supersymmetry. However for simplicity we avoid this case as well as other possible non-perpendicular configurations.
In order to have two stacks of D7-branes, located at z 2 → 0 and z 3 → 0, we take τ (z 1 , z 2 , z 3 ) to have the appropriate monodromies around the respective cycles. More precisely, near the D7-branes we have As a result, it is natural to take an Ansatz for the Kähler function in the form Since both F D7/D7 ′ and Imτ depend only on the magnitudes of the complex coordinates, we make a similar change of variables as in the single stack case so that the equation determining the Kähler potential, (2.11), becomes where subscripts on f denote differentiation with respect to the appropriate y 2 or y 3 variable. This is a non-linear second order differential equation, and to our knowledge A crucial rôle in understanding the solutions of (3.4) is played by the choice of boundary conditions. In the following subsection we present a solution that is perturbative in the number of D7-branes. The boundary conditions that we impose imply that to lowest order the solution behave like two independent superposed stacks of D7-branes.

Perturbative solutions for two stacks of D7-branes
In order to develop a series solution to (3.4) where both N 2 and N 3 are small, we introduce an expansion parameter ǫ and define We have also absorbed the factors of 4π to simplify the subsequent expressions. Note that this expansion presupposes that both N 2 and N 3 are of the same order. In this case, (3.4) takes the form We now proceed to look for a perturbative solution of the form The zeroth order solution corresponds to flat space with no D7-branes, and the corresponding part of the Kähler potential is simply given by At the first order, we find In general, solutions to this equation are only determined up to choice of boundary conditions. However, at the linear order, this can be viewed as a sum of two equationsone for each D7 stack. Hence we choose a simple solution of the form This may be viewed as defining our boundary conditions (at first order) corresponding to the desired two stack configuration.
We now demonstrate how this series expansion may be developed to higher orders.
Although we have started with a non-linear equation, the perturbative expansion yields linear equations at each order. In particular, for n ≥ 2, f (n) is determined by solving Note that the linear operator L = e −y 2 ∂ 2 2 + e −y 3 ∂ 2 3 separates, allowing a solution to be obtained through separation of variables. However we will not pursue this method here, but will instead develop the next few orders by inspection. The second order equation becomes demonstrating that y 2 and y 3 become mixed at this order. This equation has a solution of the form Of course, this solution is not unique. However, it maintains the structure of the lower order terms, namely f ∼ e y 2 f 2 (y 2 , y 3 ) + e y 3 f 3 (y 2 , y 3 ). In order to maintain symmetry between the two stacks, we take a = b = −(n 2 n 3 )/2. Similarly, we find a solution to third order which maintains interchange symmetry between y 2 ↔ y 3 (with corresponding n 2 ↔ n 3 ).
Thus the above mentioned structure of f as a sum of two independent terms persists to third order. It is worth remarking, however, that the solution necessarily has a different structure at fourth order in perturbation theory. The reason is that precisely at this order the term f 23 2 starts contributing, and therefore breaks the symmetry of the first three orders.
Collecting the above terms in the expansion, we find that the Kähler potential takes the form Given the Kähler potential, we can easily read off the metric. We present a few entries; the rest can be recovered using the symmetries under z 2 and z 3 : (3. 16) We see that, to first order in the number of D7-branes, the metric looks like a simple superposition of two stacks of D7. This validates our implicit choice of boundary conditions imposed above. However, already at second order we see a mixing between the z 2 and z 3 planes. Note that g 32 signals a highly nontrivial dependence on the coordinates z 2 and z 3 through the presence of ratios of complex coordinates.

A perturbation in the relative number of D7-branes
While in the above we have focused on N 2 and N 3 both of the same order, we may alternatively consider the case where one of the stacks of D7-branes is much smaller than the other. To be concrete, we take N 2 ≪ N 3 , and consider an expansion in terms of the small parameter N 2 /N 3 . In this case, we define so that the determining equation, (3.4), becomes Clearly this equation may be expanded in the same manner as (3.6). However, the source terms will be rearranged at the first two orders.
At zeroth order, we take the exact solution for a single stack of N 3 coincident D3-branes At first order, we find that f (1) must satisfy This has a particularly simple solution of the form For higher orders, the perturbative expansion has the form which is again amenable to separation of variables.
Collecting the terms in the Kähler potential computed above, we obtain This Kähler potential results in the following metric: Note that the main effect of turning on a second stack of D7-branes transverse to z 2 is to change g 33 by an expected amount proportional to log z 2z2 . What is slightly unexpected, however, is the appearance of non-diagonal terms g 23 at first order in the The above expression can be obtained from (2.15) by considering an expansion near |z| → ∞. Note that the radius of convergence of the above expression excludes z = 0.
The leading term is nothing but a constant dilaton thus allowing us to recover flat space in this limit.
The general solution to (2.11) in the case of one stack of D7-branes implies that the metric entry g zz is simply proportional to τ 2 . The arbitrary function Ω of (2.11) was fixed in [19] by imposing modular invariance and regularity of the metric, leading to In the large |z| limit that we are interested in in this section, the metric becomes that of flat space with a deficit angle. Namely, for large |z| the dilaton is constant to first order and therefore we have for the relevant part of the metric (defining z = re iφ ) where ρ = r 1−N f /12 . Thus the effect of the D7 is to create a deficit angle at asymptotic infinity. Due to this deficit angle we are forced to consider N f ≤ 12 (where ρ = log r must be taken for N f = 12) 2 .
To be specific, we take the orientation of the probe D7 ′ brane as indicated in the The D3/D7 background is In the static gauge, with an Ansatz such that angular coordinate φ 1 in the z 1 -plane is constant and the radial coordinate has a profile ρ 1 = ρ 1 (ρ 3 ), the pull-back metric on becomes (up to some infinite volume factor 2πV ol(R 6 )) S = −µ 7 dρ 3 e −Φ e Ψ/2 ρ 3 e Ψ +ρ 1 2 . (4.7) The equation of motion for ρ 1 determines the profile of the probe brane d dρ 3 e −Φ e Ψ/2 ρ 3ρ 1 e Ψ +ρ 1 2 = 0, (4.8) where in the asymptotic region e Ψ = ρ −N f /6 3 and e Φ = const 3 . The most general solution in the asymptotic region is given by where C 1 , C 2 are integration constants.
Since ρ 1 = const. is a solution, the probe can be placed at a fixed position in the z 1 plane and therefore the probe has a flat profile. A consequence of this fact will be the absence of a quark condensate, i.e. QQ = 0.

The spectrum of the scalar fields
By expanding the probe action in fluctuations around the flat profile solution ρ 1 = d, φ 1 = const., which is a solution even in the presence of a running dilaton, and identifying the transverse fluctuations X, Y with the scalar fields of the dual gauge theory, we extract their action (to leading order) as where . Furthermore, upon integration over φ 3 , we remain only with the constant dilaton factor in the effective action of the D7 ′ probe.

The equation of motion for either of the two scalar fields (collectively denoted by X)
is where we recall that the warp factor h obeys and the corresponding Laplacian can be approximated by The effect of the D7-branes on the background, to first order, is to generate a conical deficit in the otherwise flat z 3 directions.
One way to proceed is to take full advantage of the observation that, to first order in N f , the geometry of the six-dimensional space transverse to the D3-branes is (conformally) flat, with a conical deficit in the z 3 -plane. The warp factor is then simply the Green's function [25] G( z, 0) associated with this six-dimensional geometry: (4.16) Thus to first order in N f the effect of the D7-branes on the warp factor is a mere rescaling with (1 − N f /2). This will carry through when investigating the meson spectrum.
As we discussed earlier, the equation of motion for the two scalar fields corresponding to fluctuations in the transverse directions z 1 , can be written as By substituting into the equation of motion, we find The next step is to make yet another change of variables, r 3 = R cos α, ρ 2 = R sin α, and to further expand X m,n (R, α) into the eigenfunctions of the differential operator where λ = −l(l + 2) and l = |m| + |n|ν. We have thus reduced the equation of motion of the scalar fluctuations to an ordinary differential equation for X(R): The square normalizable solution is where we have to impose the additional constraint that the hypergeometric series terminates: with N a positive integer. To first order in the number of background D7-branes, the mass spectrum of the D7 ′ probe scalar fluctuations is M 2 = 4(N + |m| + |n| + 2)(N + |m| + |n| + 1) +N f 2(N + |m| + |n| + 2)(N + |m| + |n| + 1) − (3 + 2(|m| + |n|) + 2N)(n 2 + |n| + |mn| − 1) π(|m| + |n| + 1) .

(4.25)
Notice that by setting N f to zero we recover the spectrum of the scalar fluctuations of a D7 ′ probe in the AdS 5 × S 5 derived in [9]. Holographically, the scalar fluctuations in [9] correspond to the spinless mesons of an N = 2 SYM gauge theory, whereas in our case, the mesons belong to a theory with reduced supersymmetry (N = 1) and flavor.

Review of the conifold as a Kähler manifold
After exploring the basic D3/D7 and D3/D7/D7 ′ systems, we now turn to the conifold.
Here, we review the construction of the Ricci-flat metric on the conifold. We anticipate that the level of detail included here will provide a better understanding of the choices that we make in the next section when considering the inclusion of D7-branes on the conifold. We follow the presentation of [26] but attempt to make some of the computations more explicit.
We begin by noting that the conifold is defined as the following quadric in C 4 : The symmetries and general form of the conifold can be made manifest by writing the defining equation as: In this case, the radial coordinate is given by We now clarify the Kähler structure, with an eye toward generalizations that include non Ricci-flat metrics. If a metric on the conifold is Kähler it can be written as where F is the Kähler potential. A Kähler potential invariant under SU(2) × SU (2) will be a function of r 2 only. For such Kähler potentials, the metric can be written as where prime means differentiation with respect to r 2 . In terms of W the metric takes the form The Ricci tensor for a Kähler manifold is given by where g = det g µν . The Ricci flatness condition is easily obtained from computing the determinant of the metric (5.7). In particular As not all coordinates are independent, one of the coordinates, say w 4 , may be eliminated in favor of the other three complex coordinates, i.e. w 2 4 = −w 2 1 − w 2 2 − w 2 3 . The question of finding the Ricci flatness condition can now be rephrased as finding the Kähler potential such that (5.10) is a product of a holomorphic and antiholomorphic function (e.g. |ω 4 | 2 ). The simplest way to achieve this is by introducing γ = r 2 F ′ . The determinant can now be written as It now follows that choosing γ ∝ r 4/3 will remove all r dependence. This yields the Kähler potential F = (r 2 ) 2/3 = (w 1w1 + w 2w2 + w 3w3 + w 4w4 ) 2/3 . (5.12) After introducing ρ = 3 2 r 2/3 , we obtain the standard Ricci-flat metric on the conifold: The angular variables arise from the natural SU(2) × SU(2) symmetries of the conifold.
Furthermore, they allow us to write the general solution of the defining equation (5.3) where the L i 's are SU(2) matrices and Z 0 is a particular solution to the defining relation; for example Note that, as a result of the coset structure of the conifold, only the combination ψ = ψ 1 + ψ 2 appears in the metric (5.13).

D3/D7 on the conifold
The idea of considering a stack of D3-branes at the apex of the conifold was originally considered in [27] as a way to obtain N = 1 superconformal gauge theories. This model is very attractive from a variety of viewpoints, and has been subsequently developed into one of the trademark examples of the gauge/gravity correspondence [3].
In this section we consider adding D7-branes to the Klebanov-Witten background.
We explicitly develop two ways of adding D7-branes which are characterized by the method of embedding of the D7-branes.
Our approach to including backreacted D7-branes in the geometry of the conifold is rooted in preserving the Kähler structure. The supersymmetry and integrability conditions give, as before, the same condition presented in (2.11).
We now seek a solution to Eq. (2.11) with Imτ = 1 − n f log t, where n f = N f /(4π): The denominator and constant factor on the right hand side are to ensure that the Ricci-flat solution F = (r 2 ) 2/3 works for n f = 0. With the Kähler potential taken to be a function of both s and t, this equation becomes We look for a linearized solution of the form Note that the zeroth order solution naturally corresponds to the conifold in the absence of D7-branes.
We now observe that the linearization of (6.3) splits the perturbation problem into an infinite set of equations that can be schematically written as . . . = . . . L F (n) = S n F (1) , . . . , F (n−1) , (6.5) where S i are functions that can be explicitly determined. The most important property is that at any order i, the function S i depends on already determined functions of lower order in the perturbation expansion. In addition, the same second order differential operator L = (3s 2 + 2st − 2t 2 )∂ 2 s + (2st + t 2 )∂ 2 t + (2st + 4t 2 )∂ s ∂ t +(9s + 6t)∂ s + (2s + 5t)∂ t + 4 = (s + t) −2/3 ∂ s ∂ t (s + t) 2/3 3s 2 + 2st − 2t 2 st + 2t 2 st + 2t 2 2st + t 2 ∂ s ∂ t + 4 (6.6) appears at every level in the perturbation expansion. Given that the operator L is linear, and given the general form of the sources, we conclude that the above system is consistent. Moreover, one can, in principle, find a solution for any given set of boundary conditions.
In general, the functions F (n) depend on both s and t. However, it turns out that for F (1) we can find a solution of the form At second order in perturbation theory the situation is radically different, as there is no solution depending only on t. The equation at second order is To solve the second (and higher) order equation in general, we would like to find a Green's function for the linear operator L. To do so, it is convenient to perform the following change of variables: In this case the operator L in (6.6) becomes L = 3ρ 2 ∂ 2 ρ + 11ρ∂ ρ + 4 + (1 − 1 2 sec 2 θ)∂ 2 θ + 1 2 (cot θ − tan θ(3 + sec 2 θ))∂ θ . (6.10) A general solution to the homogeneous equation can be constructed using separation of variables. Namely we take The θ equation, Lχ(θ) = λχ(θ), is solved by Legendre polynomials Note that, because of the way s and t are defined in (6.1) the angular variable is restricted to 1 2 ≤ cos 2 θ ≤ 1. Thus the argument of the Legendre polynomial lies in the range [−1, 1]. The remaining R(ρ) equation is then homogeneous in powers of ρ; hence it may be solved by taking R ∼ ρ λ . This gives the general solution to the homogeneous equation of the form: 1+3n(n+1)) P n (−3 + 4 cos 2 θ), (6.13) where the c ± n are coefficients that must be determined by boundary conditions. Of course, to obtain the actual solution to the inhomogeneous equation (6.8), we would have to proceed with an actual construction of the Green's function. Nevertheless, even without doing so, we can discern some general features of the solution. For example, the rôle of n in (6.13) can be understood as follows. Given that ρ = s + t is the distance from the D3-brane, it is natural to conclude that higher terms in n correspond to deforming the gauge theory by operators with higher conformal dimension.
From the Kähler potential we can easily derive the metric to first order in the number of D7-branes, N f . "Turning on" the D7-branes alters the g 0 mn conifold geometry in two ways: First, there is the rescaling of the background metric with a factor (1+N f (log t−1)/3).

A P 1 -inspired holomorphic embedding
Here we consider a different embedding which is inspired by the small resolution of the conifold [26,28]. In this embedding the D7-branes are perpendicular to a P 1 submanifold inside the conifold. The resolution of the conifold can naturally be described in terms of the (z 1 , z 2 , z 3 , z 4 ) of (5.4). In particular, resolving the conifold means replacing the equation z 1 z 2 − z 3 z 4 = 0 by the pair of equations where the product λ 1 λ 2 = 0. Note that (λ 1 , λ 2 ) ∈ CP 1 (any pair obtained from a given one by multiplication by a nonzero complex number is also a solution). Thus (λ 1 , λ 2 ) is uniquely characterized by the ratio λ = λ 2 /λ 1 in the region where λ 1 = 0. Working on this patch, a solution to (6.16) takes the form 4 Thus (z 2 , z 3 , λ) are the three complex coordinates characterizing the resolved conifold in the patch where λ 1 = 0.
As usual, the conifold metric is g mn = ∂ m ∂nF , where F is the Kähler potential. In this case, the radial coordinate defined by (5.5) takes the form The Kähler potential for the conifold is simply 5 In this parametrization it is natural to consider a modification of the Kähler potential that depends on the two variables: In the region where λ 1 is allowed to be zero we have instead λ 2 = 0. In this case the general solution can be written as W = z 1 −z 1 µ 5 Note that in this specific patch the matrix (6.17) satisfies the defining equation for the conifold.
One can think of this matrix as a "solution" with z 1 = −λz 3 and z 4 = −λz 2 .
Doing so, we find that the determinant of the metric takes the form As explained previously, our task is to find a solution F (s, t) of equation (2.11) with the condition that when the number of D7-branes goes to zero we recover the Kähler potential of the conifold.
We now observe that the above equation is homogeneous in powers of s (as well as in powers of t). Hence, although the equation is nonlinear, in principle all s dependence could be removed by a judicious choice of F ∼ s n for some power n.
This suggests a separation of variables in s and t of the form F = s n f (t). Eq. (6.21) becomes: Taking n = 2/3 in the above gives an s independent determinant, and leaves a nonlinear ODE to solve: (1 − n f log(t)). The standard conifold case can be recovered by taking l = (2 cosh( y 2 )) 2/3 . Interestingly, the combination l = (2A cosh( y−y 0 2A )) 2/3 also gives a Ricci flat metric where the parameters are related to those discussed at length in [29]; however we take the simplest case with A = 1, y 0 = 0.
At linear order in n f , we find Lg 1 = −y/2, which admits a simple solution Taking this form of g 1 , the next order equation has source Lg 2 = −y 2 /4, and may be solved to yield g 2 = −2 log(1 + e y ) + y(4 − y)e y 2(1 + e y ) + 1 − e y 1 + e y Li 2 (−e y ), (6.32) up to the possible addition of a solution to the homogeneous equation.
We have therefore managed, in this simple case, to find a modification to the Kähler potential of the conifold accommodating the presence of D7-branes up to second order (and in principle extendible to arbitrary order) in the number of D7-branes.
Interestingly, however, as we will explain in the next section, this embedding of the D7-branes seems to be in a different class from the w 4 = 0 one.

Gauge theory
Let us start by commenting on the decoupling limit. To make connection with gauge theories we first show that the supergravity solutions presented here allow for a region where only the low energy D-brane degrees of freedom are described. This is a necessary requirement to claim a duality between our supergravity backgrounds and the gauge theories appearing in the worldvolumes of branes.
The Yang-Mills coupling of the gauge theory living on a Dp-brane is We want to take the string scale to zero l s → 0 keeping the D3 SYM coupling fixed g 2 D3 = g s . This automatically means that g 2 D7 → 0 which implies that the D7-brane dynamics decouple, leaving us with a flavor symmetry.
There are other quantities that we will keep fixed in the limit l s → 0 such as the mass of the 7-7, 3-7 and 3-3 strings. This translates into a rescaling of the radial coordinate perpendicular to the D3-brane as u = r/l 2 s . This decoupling limit provides us with a consistency check on the solutions we find. In particular, we will require that when turning off the number of D7-branes the metric becomes either AdS 5 × S 5 for the D7 in flat space or AdS 5 × T 1,1 for D7's on the conifold.
The gauge theory of a stack of D7's on a D3 background has been discussed in [21]. A complete account of the degrees of freedom of this system (and in general of Dp/D(p+4)) is given in the textbook [30]. One interesting fact about the D3/D7 system is that the condition for the dilaton to be small, that is for the string loop corrections to be small requires r ≪ e 2π/(gsN f ) . The condition to suppress the α ′ corrections, on the other hand, implies in the case of D3/D7 the radial distance from the D7 must satisfy r ≫ e −2πNc/N f . These two conditions have to be taken into consideration when interpreting the range of validity of the supergravity calculation. Generically, for large g s N f we must consider only N c ≫ N f . However, for very small g s N f , there is a range in which, in principle we could relax the relation between N c and N f .

Matching Pathologies: Naked singularities and Landau Poles
If one restricts to the form of the dilaton given in 2.16 and rewrites the metric using it becomes clear that the metric has a naked singularity when vanishes, that is, at: A simple way to realize the relation to the Landau pole is through the identification (7.5) Thus, the metric singularity coincides with the point at which g Y M → ∞. It is, therefore, natural to consider only the region with ρ 3 ≤ ρ L .
Gauge theory dual of D3/D7/D7 ′ The gauge theory living on this brane configuration has been described in the literature, for example in [24,4]. The analysis is closely related to a configuration of intersecting M5's discussed in [31]. The theory includes the N = 4 supermultiplet plus N f hypermultiplets Q and N ′ f hypermultiplets T transforming in the fundamental representation of SU(N). A simple way to describe this theory is via the superpotential where Z 2 and Z 3 are two of the three chiral superfields of the N = 4 multiplet and we are explicitly using N = 1 notation.
The results of our supergravity analysis point to a generic appearance of nondiagonal terms in the metric. The explicit form of the metric (3.16) shows a complicated pattern of mixing the z 2 and the z 3 coordinates. This seems to suggest that, at strong coupling, the two flavor sectors mix in a complicated way due to interactions.
As the example of N = 4 shows, the structure of the deformation of the superpotential in the gauge theory is dictated by the geometry in the supergravity background.
In particular, note that the superpotential for the conifold is intimately related to the defining equation for the conifold (5.3) under the identification To write down the superpotential corresponding to the embedding defined by w 4 = 0, recall that w 4 ∼ z 1 + z 2 and therefore the natural deformation of the above superpotential is of the form In [7] a deformation of this type was motivated using the theory of D3-branes on C 2 /Z 2 .

Comments on the β-functions
The exact β function for the canonical gauge coupling of N = 1 with gauge group SU(N) and F flavors is [32] β where γ is the anomalous dimension of the matter fields. One of the most attractive features of the solutions describing branes on the apex of the conifold is the precise geometrical description of the NSVZ β-function [27,33].
One very important feature of (7.10) is that it serves as a guide in determining the boundary conditions of the supergravity solution describing flavor. Namely, as describe by equation (2.14) the dilaton can be any holomorphic function of the coordinates. In particular the sign of the coefficient of the term containing the logarithm is arbitrary in the supergravity solution. In this case equation (7.10) dictates the sign.
Another important remark about (7.10) is that it also allows to understand the region of the radial coordinates where the theory is describing the flavor sector. Note that in general the dilaton goes from a constant at infinity to a logarithmic term near the D7-branes. As we will see in what follows for the case of the conifold, only the logarithmic region (decompactification limit) contributes to the β-function in the expected way.
For the Klebanov-Witten theory the assignment of anomalous dimension of γ = −1/2 to the operators TrA i B j guarantees that the beta function is zero, which corresponds to a superconformal fixed point [27] d d log(Λ/µ) The two gauge couplings are related to the supergravity background as [27,3]: The case with probe D7-branes was considered in [7]. In the absence of B 2 field we obtain that the two couplings are the same. To read off their value we need the explicit expression for w 4 which can be obtained from (5.3), (5.14) and (5.15) Assume, that the renormalization energy scale is proportional to the radius Λ ∼ ρ (the perpendicular distance from the D3-branes) we have the following expression for the running of the coupling ∂ ∂Λ 14) The value of γ Q can be found, similarly to a computation of [7], by power counting.
Given that the dimension of the superpotential (7.9) is three, and that the dimension of the fields A's and B's is 3/4 as argued in [27], we conclude that γ Q = 1/4.
For the case of the P 1 parametrization of the conifold we note that τ ∼ log λ and λ = tan θ 2 2 e −iφ 2 . (7.15) This combination is independent of the radial coordinate ρ, therefore, keeping the natural identification of ρ with the RG energy scale Λ we conclude that this deformation of the gauge theory does not affect the β-function and therefore does not truly amount to adding flavor to the KW gauge theory.

Conclusions
We conclude with a list of open problems which we believe would be interesting to address. The first, and perhaps most haunting problem in the gauge/gravity correspondence is how to achieve asymptotic freedom, or in other words how to follow the theory from strong coupling where supergravity is a good approximation into the weak coupling regime. This problem is well beyond the scope of the present paper. But it must be kept in mind in order to understand the limitations of the present work. Some direct open problems resulting from our investigation are: • Application of our techniques to other Kähler spaces. Most prominently, the resolved and deformed conifolds.
• The inclusion of 3-form fluxes. In the case of the conifold we refer to G 3 which corresponds to including fractional branes which change the relative rank of the two gauge groups.
• An interesting open problem would be to understand the backreaction for theories that do not necessarily admit a formulation exploiting Kähler geometry. Such seems to be the case for the Maldacena-Núñez solution and the D4/D6 bound state.
• Ideally, it would be desirable to obtain the spectrum of the different modes for the class of solutions we have presented. We should point out that some analysis of the spectrum (for example of mesons) has been presented in the probe approximation [11,9,10,8]. But an analysis of the fully backreacted case will certainly yield a more robust picture. In this paper we took a step further by showing how the backreacted D3/D7 solution could be used to derive the scalar (meson) spectrum of a N = 1 gauge theory realized on the worldvolume of a D7 ′ probe.
To conclude, we believe we have explicitly demonstrated a viable way to construct fully backreacted supergravity backgrounds dual to four-dimensional N = 1 SYM with flavor in the fundamental representation. We hope to return to many of the fascinating issues in the gauge/gravity correspondence with flavor in future work.

A Comments on the Monge-Ampère equation
In this appendix we collect some known results about the Monge-Ampère equation that are relevant for the analysis in the main text. The literature on this subject is vast [34][35][36] and we refer the reader to those references for further details.
We start with some motivation of why Monge-Ampère equations are relevant for our analysis. We assume (four-dimensional) Poincaré symmetry and (for simplicity) turn off 3-form flux. As in section 2, one arrives at equation (2.11) for type IIB supergravity det(∂ µ ∂νF ) = ΩΩ Im(τ ). (A.1) To place this in the proper context of the Monge-Ampère equation, we must change the differentiation appearing above into differentiation with respect to real variables.
First, we consider the "decompactification" limit of seven branes [19], where τ ∼ iC + ia log(z). We will then place our D3's close enough to the D7's so that this approximation is valid. Thus Im(τ ) ∼ log(zz) where we have chosen our origin so that b = 0. We then make the Ansatz that the functional dependence of the Kähler potential is F (log(z 1z1 ), . . . , log(z nzn )), where the log's have been inserted for later convenience. This requires that Ω has the form Ω = Πz The Dirichlet problem for Monge-Ampère equations is formulated as follows. Let Ω be a bounded strictly convex domain in R n (n ≥ 2) defined by a C ∞ strictly convex function h onΩ satisfying h| ∂Ω = 0. Given u(x), a C ∞ function on ∂Ω which is the restriction to ∂Ω of a C ∞ function γ onΩ, we consider the equation: where f (x, t) ∈ C ∞ (Ω, R). The main result that we will use is an existence theorem which can be stated as follows [34][35][36]. and if f ′ t (x, t) ≥ 0 for all x ∈ Ω and t ≤ sup ∂Ω u.