Exact D7-brane embedding in the Pilch-Warner background

A new supersymmetric D7-brane embedding in the Pilch-Warner gravitational background is found exactly, by solving the supersymmetric condition. In the dual holographic picture, our setting corresponds to adding a quenched fundamental matter sector to N=2* super Yang-Mills theory, at zero temperature. We show that previous results in the same setting are missing the Wess-Zumino term in the D-brane action, and how our results complete the picture.


Introduction
The holographic principle promises to be a useful framework to tackle strongly coupled gauge theories by means of weakly coupled string theories. Its best known instance is the AdS/CFT duality, conceived by Maldacena in his seminar work [1]. In particular, the duality involves a strongly coupled SU (N ) N = 4 SYM, which is a CFT, and the supergravity in the AdS 5 × S 5 background. This is a very well understood case by now, and despite not describing any real world systems, AdS/CFT is being used as a theoretical laboratory to explore dynamics of real strongly coupled systems.
The most famous example is undoubtedly Quantum Chromodynamics (QCD). To compare N = 4 SYM with QCD, we must first add flavors to AdS/CFT. This means additional hypermultiplet sectors with fields in the fundamental representation of the gauge group SU (N ). The equivalent in the holographic picture is the insertion of probe D7-brane embeddings in AdS 5 × S 5 , see [2]. The fundamental matter fields, or quarks, arise from strings stretched between the D7 and the N coincident D3-branes that generate the supergravity background. When the D7-branes and the D3-branes are separated, the quarks become massive. Significant work has been done in this direction, and we refer readers to the following reviews [3,4].
In this paper, we study the flavor dynamics in a less symmetric theory called N = 2 * SYM, via the holographic correspondence. This theory results from breaking the conformal invariance of N = 4 SYM by adding mass to its adjoint hypermultiplet. Consequently, the supersymmetries are also halved. Its holographic dual is known too, namely the Pilch-Warner supergravity [5,6]. This geometry consists of a product space of a warped AdS 5 and a squashed S 5 , and is asymptotically AdS 5 × S 5 near its boundary. This instance of the holographic duality is a non-conformal extension of AdS/CFT, and has been extensively studied and tested in the last few years [7,8,9,10]. It is therefore a natural step to extend the flavor sector in N = 2 * theory.
This problem was studied in [11], where a perturbative solution to the D7-brane equation of motion was found, and an unexpected logarithmic divergence in the embedding profile was encountered. In their analysis, the authors argued that the Wess-Zumino Lagragian vanishes. In contrast, we show that such divergence does not arise if the D-brane couples to the Ramond-Ramond (RR) fluxes through Wess-Zumino. We provide an exact closed-form solution, obtained by solving the supersymmetric condition imposed on the probe D7-brane embedding.
The present paper starts, in section 2, by reviewing D-branes and outlining the strategy we use to find supersymmetric embeddings. Then, in section 3, we deal with our D7-brane in detail, and find the right configuration. We conclude by describing the implications of our results. The Pilch-Warner background is summarized in the appendix, including explicit forms of the RR potentials that we computed for completeness.

D-branes 2.1 Action
The world-volume action of a single Dp-brane consists of the Dirac-Born-Infeld (DBI) and the Wess-Zumino (WZ) or Chern Simons terms 3 , [13]: which are explicitly where ξ are the coordinates for the worldvolume manifold M, g is the worldvolume metric (in string frame), P [·] denotes the pullback from the target space, Φ is the dilaton, C (n) are the RR forms, and with F being the worldvolume field strength, and B (2) the NSNS 2-form. Finally, the couplings, in terms of the string length l s and the string coupling constant g s , are:

Kappa symmetry projector
For any D-brane configuration there is an associated kappa symmetry projector, which, in Minkowski signature, is given by [14]: where and | Vol indicates projection to the volume form. The operators K and I act on a spinor ψ: We also have built from the pullback of the gamma matrices in the curved target space. The kappa symmetry projector satisfies the traceless and idempotent conditions.

Supersymmetric condition
The condition for the D-brane configuration to be supersymmetric is that the kappa symmetry projector Γ applied to the background Killing spinor fulfills 4 : If we are to impose the supersymmetric condition on an ansatz, it will lead us to first order differential equations for the ansatz, which are easier to solve than the standard second order equations of motion from the Dbrane action. Here we will outline our strategy to solve the supersymmetric condition under certain conditions. Let us consider the Killing spinor with the following structure: where O is an invertible operator and P is a projector satisying so that there exists a complementary projector satisyinḡ Then, the supersymmetric condition implies the conditionP If we find n further projectors on the Killing spinor as necessary conditions for the supersymmetric condition to be fulfilled, then, it means that the Dbrane configuration breaks 1/2 n copies of the background supersymmetry.

D7-brane
The holographic dictionary for N f flavors of quarks in a four-dimensional SU (N ) SYM theory is a set of N f D7-branes in the ten-dimensional supergravity dual. We work in the probe limit, when N f N , meaning the additional branes do not back-react on the background geometry. We study the simplest setting, with only N f = 1 probe brane. Our D7-brane wraps the warped AdS 5 and the three-dimensional ellipsoid of the deformed S 5 of the Pilch-Warner metric. Furthermore, our D7-brane carries no charge, hence no worldvolume gauge field: F = 0. This is the equivalent setting studied in [2] for AdS 5 × S 5 , which our configuration will reduce to, near the boundary.
Let us consider the D7-brane embedding, whose worldvolume is induced from the target space with In this case, the induced metric from (45) with dθ = θ (c)dc and dφ = 0 is:

Action
Our particular choice of φ 0 simplifies our problem, because Then the D7-brane action is simply Let us focus on the P [C (8) ] term, which was considered as vanishing in [11] and [15]. It is important to consider the string frame metric in the Hodge star operation while deriving C (8) , which in our scenario is The dilaton term from the string frame effectively cancels the factor that vanishes at φ 0 , leading to a finite value for the pullback of this potential. We decided to compute it explicitly, and the full result is given in the appendix A.4. We can quickly see that it is non-zero for our ansatz for φ in (16). However, P [C (8) ] term can be much simpler, as we will see now. First, we can show that: The left-hand-side is simply and, via (21), we can integrate the above expression over θ and obtain: The full pullback is obtained by just replacing θ by θ(c). Now we can write the action in a more explicit way:

Kappa symmetry projector
The kappa symmetry projector for our configuration is: with where we used capital gammas to denote the gamma matrices in the local frame; see appendix A.2.
The projector can be further simplified by combining it with the chirality condition, which for the mostly-minus metric convention is: Then, the supersymmetric condition (10) becomes: and where we have applied I = −i and Γ 610 = −i . The latter identity is due to P − = 0, which is straightforward to show, and the projector is defined in (62).

Supersymmetric condition
For the Killing spinor (68), the invertible operator in (15) is: The kappa symmetry projector contains the operator I, which can be replaced as follows (see notation in A.5): where we used P − η = 0 in the last step. Then, (15) reduces to: which, after manipulating the gamma matrices, gives: Therefore, the condition our configuration must satisfy in order to preserve supersymmetry is: One can repeat the analysis of (15) for the other projector P ± , and it will give the same condition (35).
The projector (29) at the solution (35) is simply since ξ = tan α. No more projectors are found, therefore, ours is a 1/2-BPS embedding.

Equation of motion
As a consistency check for our results, the equation of motion from the action (24) is fulfilled with the solution (35). In particular, where EL[·] is the Euler-Lagrange operator: Therefore, (37) is another proof for the non-vanishing WZ term.

Solution
The solution to the differential equation (35) is: where L is an integration constant, which is proportional to the mass of the fundamental matter (or quark) field in the dual field theory, as we explain next. The upper bound of c is set by the maximum of the sine. In the near-boundary limit, c ≈ 1 + z 2 /2, our solution reduces to the exact solution found in the AdS 5 × S 5 background, see [2] and [16], i.e. sin θ(z) = Lz. (40) As [16] explains, in the flat embedding space limit, this embedding describes a planar D-brane located at a constant distance L away from the stack of N D3-branes: and this distance is proportional to the quark mass m: Figures in 1 show the vielbeins of the induced metric at the solution, from which we learn how the geometry of the embedding looks like at different values of c. First, observe the divergence at the horizon c max = √ 1 + M −2 . This is the location of the well-known enhançon locus, at θ = π/2, see [17] and [18]. The spheroid is undeformed at the boundary c = 1, and becomes squashed until it vanishes at the enhançon.

Holographic renormalization
The action evaluated at the solution (39) is: where the integration of c is the range shown in (39). The action is divergent near the boundary since its geometry is asymptotically AdS. The divergent terms are: The method to handle them is the holographic renormalization, and the counterterms for our action are the same ones derived in section 4 of [16]. The chiral condensate evaluated at the background is exactly the one in [16], and it is zero. This has to be the case because a non-zero chiral condensate is prohibited by supersymmetry, [19].

Conclusion
In this paper, we found an exact supersymmetric D7-brane embedding in the Pilch-Warner geometry by solving the supersymmetric condition. In the field theory side, this setting corresponds to having a quark sector in the N = 2 * SYM at zero temperature. Our D7-brane configuration reduces to the known solution in the AdS/CFT case [16], in the near-boundary limit. Our solution also resums the asymptotic series of [11] and simplifies some of their analysis; for example, the quark condensate is zero, a result that is compatible with supersymmetry.
We also demonstrated that the pullback of C (8) for values of φ that are odd fractions of π is non-zero, in contrast to [11] and [15]. Our supersymmetric condition method does not require the D-brane Lagrangian, so that the fulfilment of the equation of motion at the solution provides a strong proof that the WZ term is there, besides our explicit computations of the fluxes. We can conclude also that the string frame is the right metric frame to use in the D-brane analysis, instead of the Einstein frame.
As a possible future work, we could study the fluctuations around our D-brane, corresponding to the meson spectrum on the field theory side. It is also interesting to consider non-vanishing gauge fields on the deformed sphere, as done in [12] for the AdS 5 × S 5 case, where their mesons carry angular momentum. One can also review the thermal case studied in [20], where the geometry was derived in [21]. Another direction is to study various probe branes, for example [22] recently explored the two D7-branes case for AdS 5 × S 5 .

A Pilch-Warner supergravity
The Pilch-Warner solution to the type II supergravity equations was originally found in [5]. In this section, let us review its metric, the background fields and the Killing spinors, the latter first derived in [6].

A.1 Metric
We parametrize the ten dimensional Pilch-Warner metric in the following way: with µ = 1, . . . , 4 and the coordinates: The various coefficients are functions of c, θ and φ, where the dependence on the latter comes only from the dilaton Φ prefactor 5 , with its explicit form shown in the next subsection. The coefficients are given by: and X 1 = cos 2 θ + cA sin 2 θ, X 2 = c cos 2 θ + A sin 2 θ, (47) 5 The dilaton factor comes from the fact that we are using the metric in the string frame. In the Pilch-Warner literature, often the metric in the Einstein frame is shown. Both frames are related by a general conformal transformation, i.e. ds 2 Einstein = e −Φ/2 ds 2 string .
The deformed 3-sphere is parametrized by the SU (2) left invariant forms, i.e. the Maurer-Cartan forms: where τ i are the Pauli matrices and g is a group element of SU (2). The 1-forms satisfy the relation 6 We do not need to explicitly parametrize these forms for the purpose of the present paper; however, for the interested readers, there is an example using Euler angles in the appendix of [8].

A.2 Local frame
The non-coordinate basis, also known as the local frame, is specified by the Minkowski metric η ab . It is related to the curved space metric G M N via vielbeins, according to: G M N = e a M e b N η ab . In our case, the metric (45) is diagonal 7 , hence the vielbeins and the inverse vielbeins are precisely the coefficients (46) and its inverse, respectively.
When we handle the curved-space gamma matrices γ M , we will go to the local frame, in order to use the constant Γ a matrices:

A.3 The near-boundary geometry
The boundary of the Pilch-Warner geometry is located at c = 1. Close to the boundary, c ≈ 1 + z 2 2 , with z small, we recover the AdS 5 × S 5 geometry in Poincare coordinates and the Hopf parametrization for S 5 :

A.4 Background fields
The Pilch-Warner solution has non-trivial background fields. Following the conventions of [17] and [6]. The dilaton Φ and axion C (0) fields are given by: The 2-form potential that is the linear combination of the RR and the NSNS 2-form potentials, A (2) = C (2) + iB (2) , is: with the real functions 8 : And the self-dual 5-form field strengthF (5) is given by: where ω(c, θ) = AX 1 4(c 2 − 1) 2 . (55) Using the following definitions for the field strengths: and the Hodge duality relation * F (n+1) = (−) n(n−1)/2F we could compute all the RR-forms C (n) and the NSNS-form B (2) , up to an exact form.
We would like to comment on the Hodge star operation, which is defined as: where g is the metric, and the Levi-Civita symbol satisfying 1...n = 1. Notice the metric to be used here is the metric in the string frame. This is important since the dilaton term is non-trivial in the Pilch-Warner background, unlike in AdS 5 × S 5 . Although we do not need all the potentials for our problem, we list the explicit solutions below: A((c 2 − 1)A + 4c) cos 4 θ 8c 2 (c 2 − 1) 2 σ 1 ∧ σ 2 ∧ σ 3 ∧ dc ∧ dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 − A 2 sin 3 (2θ) cos(2φ) 2cA + A 2 sin 2 θ + c 2 cos 2 θ 16c(c 2 − 1)X 2
The projectors are and they commute with each other: [Π ± , P ± ] = 0. The exponentials can be written in terms of cosines and sines: where K is the complex conjugation, and the angles are defined as cos α = cos θ √ X 1 , sin α = cA X 1 sin θ, where Ω, X 1,2 , A are functions defined in section A.1. The gamma matrices are in real representations. It is convenient to write the e i φ 2 factor in real representation too. That is achieved by using P − η = 0, which leads to After some straightforward manipulations, the Killing spinor can be rewritten as: