Riccati equations for holographic 2-point functions

Any second order homogeneous linear ordinary differential equation can be transformed into a first order non-linear Riccati equation. We argue that the Riccati form of the linearized fluctuation equations that determine the holographic 2-point functions simplifies considerably the numerical computation of such 2-point functions and of the corresponding transport coefficients by computing directly the response functions, eliminating the arbitrary source from the start. Moreover, it provides a neat criterion for the infrared regularity of the fluctuations. In particular, it is shown that the infrared regularity conditions for scalar and tensor fluctuations coincide, and hence they are either both regular or both singular. We demonstrate our numerical recipe based on the Riccati equations by computing the holographic 2-point functions for the stress tensor and a scalar operator in a number of asymptotically anti de Sitter backgrounds of bottom up scalar-gravity models. Analytical results are obtained for the 2-point function of the transverse traceless part of the stress tensor in two confining geometries, including a geometry that belongs to the class of IHQCD. We find that in this background the spin-2 spectrum is linear and, as expected, the position space 2-point function decays exponentially at large distances at a rate proportional to the confinement scale.


Introduction
The AdS/CFT correspondence [1] has found fruitful ground in building holographic models following top down/bottom up approaches. Such models have applications in cosmology, QCD and condensed matter physics and often the computation of the 2-point functions [2,3] and the transport coefficients in the dual QFT is required.
The holographic computation of 2-point functions involves solving second order linear differential equations in an asymptotically locally anti de Sitter background. The general solution of such equations contains two arbitrary functions of the boundary coordinates or momenta, one of which is generically identified with the source of the dual gauge invariant operator while the other is identified with its 1-point function. Requiring that the solution of the linear equations be regular in the interior of the bulk imposes a relation between the two arbitrary functions that parameterize the solution, thus leaving one arbitrary function. The fact that the desired solution of the second order linear equations contains an arbitrary function renders the numerical computation of the holographic 2-point functions tedious and inefficient. However, homogeneous second order linear differential equations can always be transformed to a first order non-linear Riccati equation. Namely, starting from the second order linear equation α(x)y ′′ + β(x)y ′ + γ(x)y = 0, (1.1) and defining w = − y ′ αy , (1.2) one obtains the Riccati equation so we will make no attempt to embed the scalar potential in some gauged supergravity here. Of course, our techniques and analysis apply equally well to embeddable models.
Moreover, we will allow the domain wall backgrounds to contain mild singularities. In particular we will allow for "good singularities" according to [4], since these can be in principle resolved by stringy effects and/or by a horizon. In fact, such singular backgrounds have been extensively used as toy holographic models for confinement (see e.g. [5] and references therein.) The paper is organized as follows. In section 2 we present the model we are going to use as well as the Poincaré domain wall backgrounds in a form that will be most suitable for our subsequent analysis. In section 3 we derive the linearized fluctuation equations in both second order and first order Riccati forms, closely following the analysis of [6]. Section 4 deals with the ultraviolet divergences of the 2-point functions, again using the techniques of [6], while in section 5 we classify confining infrared geometries that allow for non-singular 2-point functions.
In particular, we provide a general criterion for the infrared regularity of scalar and tensor fluctuations and in lemma 5.1 we show that the assumptions of the holographic c-theorem ensure that such fluctuations are either both regular or both singular at the infrared. In section 6 we describe our numerical strategy while in section 7 we provide indicative exact and numerical results for three different backgrounds, namely exact AdS and two asymptotically AdS geometries that are confining in the infrared. We end with some discussion and a summary of our results in section 8. Some technical results are presented in the appendices.

RG flows and models of confinement in AdS/CFT
In order to demonstrate the use of the Riccati equation for the computation of holographic 2-point functions we consider a bottom up gravity-scalar action where κ 2 = 8πG d+1 is the gravitational constant in d + 1 dimensions and the potential is left unspecified at this point, except for the requirement that the equations of motion admit AdS d+1 as a solution. The equations of motion take the form where the subscript φ denotes derivative with respect to φ.
We will compute the holographic 2-point functions in backgrounds that preserve Poincaré invariance in d dimensions, namely where 1 η = diag (−, +...+) is the Minkowski metric in d dimensions. Such backgrounds describe renormalization group flows of the dual CFT. They have also been considered as toy models for the holographic study of QCD [5,7,8,9,10], of heavy ion collisions [11,12,13,14,15] and thermalization [16,17,18,19] in the AdS/CFT context. Although the radial coordinate r is most suitable for setting up the holographic dictionary, it is often useful to use instead a 'conformal' radial coordinate z such that One can easily switch between the two radial coordinates using and we will freely do so in the following. In particular, from now on dots denote derivatives with respect to r, while primes indicate derivatives with respect to z.
Inserting the ansatz (2.3) in the equations of motion leads to the set of coupled equationṡ 1 Occasionally we will consider Euclidean signature instead below.
It is well known that these equations are automatically solved provided a function W (φ) can be found such thatȦ and W (φ) is related to the scalar potential as These relations follow most naturally from a Hamilton-Jacobi analysis of the domain wall backgrounds (2.3) [20,21,22,6].
The domain wall solutions we are interested in here are asymptotically AdS, which means that as r → ∞, A(r) ∼ r/L, where L is the AdS radius. Moreover, the weaker energy condition requires thatÄ ≤ 0 and soȦ is monotonically increasing along the RG flow, starting with its minimum value, 1/L, at the far UV. This leads to the holographic c-theorem [23]. As we approach the IR there are three mutually exclusive possibilities [5]. Namely, either another AdS of a different radius is reached or a curvature singularity is found at a finite value r o of the radial coordinate or at r → −∞. The curvature singularity can be good or bad according to the criteria of [4]. Moreover, some singularities give rise to confinement according to the Wilson loop test [5,24,25,26]. Two of the examples we will consider below belong to this class.
The requirement that the equations of motion admit AdS of radius L as a solution implies that the scalar potential takes the form 2 as φ → 0, where m is the mass of the scalar field and the dots stand for higher powers of the scalar field. Stability with respect to scalar perturbations requires that the mass satisfies the Breitenlohner-Freedman bound m 2 L 2 ≥ −(d/2) 2 [27]. We will assume additionally that m 2 < 0 so that φ → 0 in the UV and the scalar field is dual to a relevant operator.
Since the scalar potential is a priori arbitrary in our bottom up model except for the asymptotic behavior (2.9), we can parameterize the backgrounds (2.3) in terms of the 'superpotential' The warp factor A(r), the scalar φ B (r) and the potential V (φ) can then be obtained via (2.7) and (2.8). The asymptotic form (2.9) of the scalar potential implies that W (φ B ) asymptotically takes the form where the scaling dimension ∆ of the dual scalar operator is related to the scalar mass as . Assuming ∆ > d/2, if the superpotential behaves asymptotically as in the 2 One can always redefine the scalar such that AdS of radius L corresponds to φ = 0.
first line of (2.10) then the background describes an RG flow due to a single-trace deformation of the dual CFT by a relevant operator of dimension ∆, while if W (φ B ) asymptotes to the second line of (2.10) then the background describes an RG flow due to a vacuum expectation value of such an operator [6]. For ∆ = d/2 the corresponding asymptotic forms of the superpotential are [28] W (2.11) The backgrounds (2.3) with generic scalar potential can alternatively be parameterized by specifying the wrap factor A. This parameterization is particularly useful for studying the infrared behavior of the background and of linear perturbations around it, as well as for numerical calculations. All other quantities can be easily obtained from the warp factor through the following relations in conformal coordinates: In this section we provide a general derivation of these fluctuation equations in the Riccati form closely following the analysis of [6].

Linearized fluctuation equations in Riccati form
Without loss of generality we will consider linearized metric fluctuations that preserve the partial gauge-fixing of the metric where γ ij is the induced metric on the constant r hypersurfaces. In this gauge the radial canonical momenta π ij and π φ are given by where K ij = 1 2γ ij is the extrinsic curvature of γ ij and S[γ, φ] is the on-shell action as a function of the induced fields γ ij and φ. The AdS/CFT dictionary relates the on-shell action to the generating function of connected correlation functions and the radial canonical momenta to the corresponding 1-point functions. For our present analysis the relations (3.2) suffice, but for a general radial Hamiltonian analysis in the context of holographic renormalization we refer the reader to [22,6] and [21,29,30,31,32,33,34] for related work.
The most general linear fluctuations around the backgrounds described in section 2 that preserve the partially gauge-fixed metric (3.1) are of the form The extrinsic curvature can be expressed to linear order in the fluctuations as where S i j ≡ γ ik B h kj can be decomposed into irreducible components as with ∂ i e i j = e i i = ∂ i ǫ i = 0 and indices are raised with the inverse background metric e −2A η ij . Conversely, all irreducible components can be expressed in terms of S i j as via the projection operators and .
The modes e i j and ξ satisfy decoupled second order linear equations. The solution of these second order equations captures all the non-trivial physics contained in the 2-point functions of the stress tensor and of the scalar operator O(x) dual to φ. However, these linear second order equations can be transformed into first order non-linear equations of Riccati type, which are much more amenable to numerical analysis.
To derive the Riccati form of these equations we note that to linear order in the fluctuations we can writeė where E(A, φ B ) and Ω(A, φ B ) are the response functions and depend only on the background.
Inserting these expressions in the second order fluctuation equations for e i j and ξ leads to first order Riccati equations for the response functions E(A, φ B ) and Ω(A, φ B ), which in momentum Contrary to the second order equations for the modes e i j and w, these first order equations only require one boundary condition each. As we will discuss in section 5, this boundary condition is provided by imposing regularity of the fluctuations e i j and w in the infrared. The fact that the Riccati equations (3.12) directly compute the response functions bypassing the arbitrary sources provides a much more efficient strategy for computing the 2-point functions numerically.
Moreover, as we now show, these 2-point functions can be directly expressed in terms of the response functions E(A, φ B ) and Ω(A, φ B ) so that the 2-point functions can be simply red off from the solutions of (3.12).
The connection between the response functions E(A, φ B ) and Ω(A, φ B ) and the 2-point functions can be easily shown by using the canonical momenta (3.2). Namely, given the response functions the radial velocities can be written aṡ Moreover, expanding the canonical momenta (3.2) to linear order in the fluctuations we get Inserting the radial velocities in these expansions of the momenta and isolating the terms linear in the fluctuations gives and According to (3.2) these canonical momenta can be expressed as gradients of the quadratic in fluctuations part of the on-shell action, which we will denote by S (2) . Indeed, these expressions for the momenta can be integrated straightforwardly to obtain the generating functional for all 2-point functions, namely This expression encodes all 2-point functions between the stress tensor and the operator O(x), but it suffers from ultraviolet divergences which must be consistently removed. Moreover, many terms in (3.17) are trivial contact terms that can be removed by finite local counterterms. We address both these issues in the subsequent section.

Renormalized 2-point functions
In order to obtain the renormalized 2-point functions from the generating functional (3.17) we need to add local covariant boundary terms to remove the ultraviolet divergences. Moreover, we are free to add any finite local counterterms we find convenient, which reflects the usual renormalization scheme choice. This freedom can be utilized to greatly simplify (3.17). Namely, we observe that all terms that don't contain E and Ω in (3.17) are local covariant terms that can be simply removed by local counterterms. 4 The generating functional (3.17) can therefore be simplified to More correctly, here we are ignoring for simplicity all non-analytic terms in the 2-point functions. In the case of backgrounds describing RG flows due to a vacuum expectation value there are generically contact terms that are proportional to the VEVs and these should not be removed by local counterterms as they are physical. But such terms can be recovered easily by a careful analysis of the counterterms. We refer the reader to [6] for a more complete analysis of this issue.
Note that now only the decoupled dynamical modes e i j and w survive after trivial contact terms are removed.
However, this simplified generating function still suffers from ultraviolet divergences. The local covariant counterterms that are required to remove these divergences depend on the particular theory, i.e. the scalar potential V (φ), and can be systematically obtained directly at the linearized level by the algorithm described in [6]. Namely, the response functions E and Ω can be expanded in eigenfunctions of the dilatation operator where the subscript indicates the eigenvalue under the dilation operator, e.g.
The terms up to E (d) and Ω (2∆−d) are local, contribute to the ultraviolet divergences, and can be determined by inserting these expansions in the equations (3.12) for the response functions and using the fact that and Ω (2∆−d) defined so that they are independent of the radial coordinate and correspond to the renormalized response functions that contain all the physical information of the renormalized 2-point functions. Terms of higher order than E (d) and Ω (2∆−d) drop out when the UV cutoff is removed and so we need not consider them. We will see explicit examples of these expansions in section 7.
The final outcome of this analysis is that the renormalized generating functional for the 2-point functions can be written in the form All renormalized 2-point functions can be simply red off this expression. Namely,

Infrared regularity conditions
In this section we classify various IR behaviors of the warp factor A(z) and we determine the corresponding IR behavior of the response functions E and Ω following from (3.12). The definitions (3.11) of the response functions imply that the linear fluctuations e i j and w are regular in the IR providedˆr We will now determine the appropriate IR asymptotic solutions of E and Ω for various choices of warp factors such that these conditions are satisfied. Noting that the quantity W ∂ 2 φ log W can be expressed in terms of the warp factor as the equations (3.12) in Poincaré coordinates take the form 5 where the primes denote differentiation w.r.t. z. This form of the equations for the response functions allows us to prove the following Lemma 5.1 For any asymptotically AdS Poincaré domain wall that fulfills the conditions of the holographic c-theorem [23], the asymptotic behaviors of the response functions E and Ω in the IR are identical provided the IR geometry is not another AdS, in which case the answer depends on how fast AdS is approached.
The coefficient of Ω in (5.3) can be written as Hence, given that A → −∞ in the IR, and excluding the case that the IR geometry is another Hence,Ȧ is monotonically decreasing towards the UV. Since at the far UVȦ → 1/L > 0, it follows thatȦ > 0 along the entire flow, or equivalently A ′ ≤ 0. Since −1/A ′ ≥ 0 is bounded from below, ∂ z (−1/A ′ ) can only possibly tend to −∞ at a finite value z 0 of z and not asymptotically as z → ∞. In that case −1/A ′ has a branch cut at z = z 0 such that This lemma implies that it suffices to examine the IR behavior of E only, since that of Ω is identical. Moreover, we will work in Euclidean signature in this section so that p 2 ≥ 0. The equation for E in (3.12) in the IR becomes Trying a solution E ∼ E o z ǫ we get The only solutions are The positive solution for E o satisfies the regularity condition (5.1) for the fluctuations and so there exists a regular mode.
(a) For α < 1 we have E o = p 2 , which as in the previous example leads to a regular solution at least in Euclidean space. In Lorentzian space, according to [5], such a geometry is not confining independently of the coefficient c.
(b) For α = 1 the two solutions for E o are According to [5], the corresponding geometry yields a mixed quantized and continuum spectrum. In the Euclidean case, only the coefficient with the positive sign leads to a regular solution.
(c) Finally, for α > 1 there is no regular solution with the above ansatz for E but there is one for the slightly more general ansatz E ∼ E o z a e cz α . Namely, the choice a = 1 − α leads to regular fluctuations according to (5.1). This background is confining according to the Wilson loop test [5].
In all, for every α > 0 there exists a regular solution.
This admits a solution k = 1 + a and The recipe involves the following steps.

Background:
Specification of the background by providing the warp factor A(z) in conformal coordinates.
All other quantities of the background can be deduced from the warp factor via (2.12).

Dimensionless parameters:
The response functions E and Ω are determined by the first order equations (5.3). Since E and Ω have dimensions of mass, it is convenient to introduce the dimensionless quantities

IR solution:
In order to solve

Examples
In this section we consider three particular backgrounds, exact AdS and two confining geometries, in order to demonstrate the use of the Riccati equations for the computation of the 2-point functions.

Unbroken conformal symmetry
Our first example is exact AdS space, i.e.
where L is the AdS radius and the boundary is at z = 0. This implies that φ B = 0 and hence The fluctuation equation for E (3.12) takes the form where m 2 is the scalar mass defined in (2.9). Writing agaiṅ ϕ = F ϕ, (7.5) and changing to the z coordinate leads to the Riccati equation where F := F L. Note that (7.3) is a special case of (7.6) corresponding to m 2 = 0. Introducing the variable u = p 2 z 2 (7.6) becomes identical to equation (3.1.24) in [35]. The general solution takes the form where k = ∆−d/2 > 0, c(p) is an integration constant, and the primes here denote differentiation with respect to the argument of the Bessel functions and not z. Moreover we have defined p := p 2 , working for simplicity in Euclidean signature so that p 2 ≥ 0. It follows that and hence only F − leads to a solution of the second order equation for ϕ. Requiring that ϕ remains finite in the IR forces us to set c = 0 so that the desired solution for F is In this case the expansion of F asymptotically in eigenfunctions of the dilatation operator can be simply obtained by expanding the Bessel functions for small z. Taking for concreteness d to be even this leads to the renormalized response functions [35] LF where µ is an arbitrary energy scale.

A toy model for confinement
Next we consider a warp factor of the form as well as and Moreover, the UV asymptotic expansion of the background scalar is which determines the asymptotic form of the superpotential and the scalar potential to be (7.16) and Comparing these with (2.10) and (2.9) we conclude that ∆ = d/2 and so the mass saturates the BF bound. Moreover this background describes a VEV and not a deformation. This is therefore exactly analogous to the Coulomb branch flow of N = 4 super Yang-Mills [36,37], which was analyzed using the Riccati form of the fluctuation equations in [6].
Riccati equations 6 A closed form expression for φB(z) can be obtained in terms of Appel functions.
The Riccati equations (3.12) for the response functions take the form 7 where we have introduced w := Λz, and E := EL and Ω := ΩL as before.

IR asymptotic solutions
The IR is located at w = 1. The IR behavior of the response functions E and Ω can therefore be obtained from the form of (7.18) in the vicinity of w = 1, namely Ω satisfies an identical in the IR, as ensured by lemma 5.1. The IR solutions that satisfy the regularity conditions (5.1) are (7.20)

UV asymptotic solutions
To determine the UV behavior of the response functions we can, without loss of generality, drop all terms containing w d in (7.18), which gives The first equation is identical to (7.3) above and so the covariant asymptotic expansion for E, taking for concreteness d even, is given by [6,35] where µ is an arbitrary dimensionful constant. Moreover, the asymptotic form of Ω is For d = 4 these asymptotic expansions are in agreement with those for the Coulomb branch flow given in [6].

Exact solution for E
The equation for E in (7.18) can be solved analytically. The general solution is 25) and c(p) is an integration constant. As in the first example above, the criterion (5.1) for the IR regularity of the fluctuations requires that we set c = 0. Expanding the solution with c = 0 in the UV and subtracting the asymptotic form (7.22) we obtain the renormalized response function where µ is a dimensionful constant. It follows that the non analytic part of the 2-point function of the transverse traceless part of the stress tensor in the background (7.11) is identical to that of empty AdS.

Numerical solution for Ω in d=3 and d=4
In this subsection we follow step 4 in section 6, in order to compute Ω(w; p/Λ) and from there, through (7.23), we show how to extract the renormalized response function Ω (0) (p). We choose units such that Λ = 1 and we numerically solve equation (7.18) for Ω as a function of w as p varies. We do that for the cases d = 3 and d = 4. The numerical analysis for these two examples is not performed in complete detail as the purpose here is to outline the method through a simple but non-trivial example. A complete analysis 8 is carried out in the next and more interesting example, in section 7.3, which concerns a geometry that belongs to the class of IHQCD.
The numerical results for Ω (0) (p) are summarized in figures 1 and 2 where we show the IR and the UV behavior of Ω and extract the coefficient Ω (0) (p) for a few values of the momentum p. In particular, for large values of p such that p ≫ Λ, we find The same numerical solution is plotted in the far UV region (i.e. w ≪ 1), and is compared with the UV asymptotic solution (7.23) (blue curve) with Ω (0) (p 0 ) = log(0.707p 0 ) ≈ 1.96. In fact, it has been checked that for large momenta p ≫ Λ, the UV fitting is always achieved by Ω (0) (p) = log(0.707p) (see (7.27) and right panel in fig. 2). Thus, as expected at large momenta, the numerical result reduces to the conformal limit (see (7.10) for k = ∆ − d/2 = 0).

Toy holographic QCD
Our last example is a background with warp factor where again Λ is a constant and we consider only d = 4 in this example. As the previous example, this background is confining according to the Wilson loop test, with confinement scale Λ [5]. From the relations (2.12) we determine φ ′ B = ±Λ 6(3 + Λ 2 z 2 ), (7.29) together with and V (z) = − 3e Λ 2 z 2 L 2 4 + 5Λ 2 z 2 + 3Λ 4 z 4 . and p Λ = 100 is compared with the IR asymptotic solution (blue curve), which provides the IR boundary condition for the numerical shooting. The geometry belongs to the class of confining geometries with a finite radial range (w ∈ [0, 1]). The d = 3 case is special as, according to (7.20), in the IR it tends to the constant p 2 9Λ in contrast to the d = 4 case where it tends to zero (see fig. 1). Right panel: The renormalized response function Ω (0) (p) extracted from the numerical solution is plotted for large momenta p ≫ Λ. The red dots refer to p Λ = 100, 200, ..., 1000 and they are determined numerically by an appropriate UV fitting analogous to the right panel of fig.   1. The blue dashed curve is given by equation Ω (0) (p) = log(0.707p) (see (7.27)) and it evidently fits the numerically obtained Ω (0) (p) very accurately. Thus, as expected at large momenta, the numerical result reduces to the conformal limit (see (7.10) for k = ∆ − d/2 = 0).
Moreover, the asymptotic form of the background scalar field takes the form This expansion can be used to determine the asymptotic form of the superpotential and of the scalar potential Comparing these with (2.10) and (2.9) respectively we conclude that this background describes a deformation of the dual CFT by a dimension 3 scalar operator and with deformation parameter proportional to Λ. This is therefore exactly analogous to the GPPZ flow of N = 4 super Yang-Mills [38], which was analyzed using the Riccati form of the fluctuation equations in [6].

IR asymptotic solutions
In the IR the Riccati equation (7.35) for E becomes and again Ω satisfies an identical equation. The regular IR asymptotic solution is given by

UV asymptotic solutions
The asymptotic UV solutions of (7.35) are easily found to be of the form , 38) or equivalently These expansions for E and Ω are identical to those for GPPZ flow derived in [6] (See eqs. 3.57-3.58).

Exact solution for E
The equation for E in (7.35) can again be solved exactly. The general solution can be written in the form where ψ is the digamma function. In appendix A we derive the analogous result for arbitrary even dimension d. As a crosscheck, we note that this expression should approach the empty AdS result given in equation ( implies that the right limit is indeed recovered. The physical content of (7.43) can be extracted by using the identity Such a linear spin-2 spectrum is in agreement with what was found in e.g. [5] (see formulas (6.12) and (6.17) for a = 2) and [39] (see formulas (6.22) and (6.23)).

Numerical solution for Ω
In this subsection we follow step 4 in section 6, in order to extract Ω(w; p/Λ). We choose units such that Λ = 1 and we introduce a convenient dimensionless function g by rescaling Ω as Ω = p 2 3Λ 2 g(p, w). (7.50) From (7.35) we see that g(w) satisfies the equation w∂ w g − p 2 3Λ 2 g 2 − 4 + 3w 2 − 4 1 + w 2 + 6 3 + w 2 g + 3w 2 = 0, (7.51) and it is subject to the IR boundary condition (7.37) which in the g variable implies g → 1 as w → ∞. (7.52) The utility for g is that it has the same IR boundary condition for any momentum p and hence checking its asymptotics becomes easier. In order to extract the QFT information, we need to specify numerically the coefficient Ω (2) appearing in (7.39). In terms of the g variable, the UV asymptotics are where Ω (2) (p) is the renormalized response function to be determined. The procedure is now straightforward. Shooting from the IR using the regularity condition (7.52) for several p's, we fit the solution at small w using (7.53) and extract Ω (2) . The numerical results for g(w; p) are summarized in figures 3 and 4 where we show the IR and UV behaviors of g(w; p) and we extract the coefficient Ω (2) for various values of the momentum p.
Given the numerical solution for g(w; p), the renormalized response function Ω (2) (p) can be extracted by comparison with the asymptotic UV solution (7.39). The result is depicted in the two plots of figure 5. The plots zoom in the small and large momentum regions, relative to the

Conclusions & summary of results
In this paper we have demonstrated the utility of the Riccati form of the fluctuation equations that determine the holographic 2-point functions in the context of bottom up scalar-gravity models.
• Using the Riccati form of the fluctuation equations we were able to provide a general criterion, 5.1, for the infrared regularity of the scalar and tensor fluctuations around asymptotically AdS Poincaré domain wall backgrounds, and to prove in general (lemma 5.1) that provided the conditions of the holographic c-theorem hold, scalar and tensor fluctuations are either both singular or both regular in the infrared. These results together greatly simplify the classification of backgrounds according to their IR singularities. Four classes of IR geometries were studied in detail is section 5. for the scalar fluctuations is obtained numerically as a function of the momentum following the recipe outlined in section 6. It is found that at large momenta, the response functions Ω (2∆−d) (p) behaves as in the empty AdS case with the right conformal weight ∆ for each case. In the two confining cases, Ω (2∆−d) (p) is found to deviate from the AdS result at small momenta (see fig. 5), as it should be expected. The numerically computed response function Ω (2∆−d) (p) determines all renormalized 2-point functions, except from the transverse traceless 2-point function of the stress tensor, through (4.5). The 2-point function of the transverse traceless part of the stress tensor is determined by the response function E (d) (p), which we were able to compute analytically in all three examples. In the first confining background which corresponds to a VEV by a dimension d/2 operator we find that the non-analytic part of E (d) (p) is identical to the AdS result, which implies that the transverse traceless 2-point function of the stress tensor only differs from the AdS result by contact terms proportional to the scalar VEV. In the IHQCD background, however, the non-analytic part of the response function E (d) (p) differs from the AdS result. From the response function we extract the spin-2 spectrum, which turns out to be linear (see (7.49)).
Moreover, in appendix B we evaluate the large distance behavior of the response function in position space, and it is found that as expected it falls exponentially at a rate proportional to the confinement scale.
In a follow up work we plan to apply our method to a wider class of models and backgrounds, to include gauge fields, an axion, and finite temperature. We believe our algorithm greatly simplifies the calculation of holographic 2-point functions in general, but especially when the latter can only be obtained numerically. We hope that this program will be useful for applications of AdS/CFT to QCD, condensed matter and other areas of physics.