Abstract

We review recent progress on holographic renormalization in the context of the gauge-gravity correspondence when the bulk geometry is not asymptotically AdS. The prime example is the Klebanov-Strassler background, whose dual gauge theory has logarithmically running couplings at all energy scales. The presented formalism provides the counterterms necessary for obtaining finite two-point functions of the scalar operators in the corresponding dual gauge theories. The presentation is self-contained and reviews all the relevant background material concerning a gauge-invariant description of the fluctuations around holographic renormalization group backgrounds.

1. Introduction

Gauge/string duality offers an alternative approach to aspects of nonAbelian gauge theories that are hard to describe with conventional techniques. For example, at strong coupling, many nonAbelian gauge theories exhibit confinement, the familiar yet still somewhat mysterious phenomenon that the only finite-energy states are singlets under the color gauge group: in colliders, we never see quarks directly, only colorless hadrons. The details of confinement, and other phenomena such as chiral symmetry breaking, are difficult to capture with conventional gauge theory methods—they are inherently nonperturbative. Instead, in the dual string picture, the nonperturbative gauge theory regime is typically described by weakly coupled closed strings propagating on a space of higher dimensionality (the “bulk”), and their dynamics can be approximated by classical supergravity. In this case, one also speaks of gauge/gravity duality.

One of the most powerful applications of gauge/gravity duality is the calculation of field theory correlation functions from the dual bulk dynamics. This idea was developed in [1–3] for superconformal gauge theories, whose gravity duals are Anti-de Sitter (AdS) spaces. Let us briefly illustrate with a simple example how such a calculation is done. The starting point is the correspondence formula [3] where, on the left-hand side, denotes the renormalized bulk on-shell action evaluated as a functional of suitably defined boundary values of the various bulk fields. The right-hand side represents the QFT generating functional for connected correlation functions of the operators , and the play the role of sources. It is often more practical to consider the exact one-point functions of the QFT operators, which are obtained from (1.1) as

Let us consider a massless scalar field in imensional AdS space, with a bulk action where the bulk metric is taken to be This is AdS space with characteristic length (or “radius”) . The boundary is located at . The equation of motion for is where . In momentum space, the general solution of (1.5), which is regular in the AdS bulk space (in particular, for ), can be expressed as where , and is a modified Bessel function. The overall constant has been chosen such that, for , one has [4] so that is the boundary value, which plays the role of the source, , in the dual QFT.

To calculate the on-shell action, we introduce a cutoff boundary at , integrate (1.3) by parts and use the equation of motion in the bulk part, leading to Then, substituting (1.7) and switching to momentum space yields where the ellipses denote terms that vanish in the near-horizon limit . The two divergent terms in the near-horizon limit are removed by adding counterterms, which must take the form of local, boundary covariant functionals of the full fields (not of the boundary values ). Hence, we add where is the metric on the cutoff boundary, and is an arbitrary constant of dimension mass. Changing is, therefore, the same as adding some finite counterterms. After putting everything together and taking the near-horizon limit, we end up with the renormalized on-shell action where , a constant of dimension mass that depends on , represents the renormalization scale. In (1.11), one can recognize the generating functional for the CFT 2-point function of operators of dimension 4.

This short example illustrates the general philosophy behind holographic renormalization (HR) and shows that the procedure has certain features in common with the usual renormalization procedure in QFT. First, the bare, regularized generating functional diverges when removing the cutoff. Second, counterterms are added at the cutoff, after which the cutoff can be removed. Third, different renormalization schemes differ from each other by finite (and local) counterterms. Last, the necessity to introduce a dimensionful constant (renormalization scale) gives rise to a scale anomaly. For a more detailed account of HR in AdS/CFT, valid in any asymptotically locally AdS (aAdS) bulk space-time and including also gravity, fermion and form fields, we refer to the extensive original literature and reviews [5–15]. The little “a” in aAdS is particularly important, because it allows to calculate correlators in QFTs with running couplings, where the flow away from the UV fixed point is driven either by an operator insertion or a spontaneous breaking of the scale symmetry. These are known in AdS/CFT as holographic RG flows.

There are many gauge/gravity dualities, which do not involve aAdS bulk spaces. For example, a field that has attracted much interest recently is nonrelativistic AdS/CFT, because of its potential applications to condensed matter physics. Let us just note that HR has been formulated for backgrounds with Schrödinger [16, 17] and Lifshitz-like scale symmetries [18, 19]. These and other cases share the fact that their bulk backgrounds possess some anisotropic conformal infinity, which turns out to be crucial for HR [20]. Other nonconformal cases are given by with a nonvanishing dilaton. These cases exhibit a generalized conformal structure and imply couplings that run with a power law in the UV. HR has been carried out in [21, 22] following the standard method as reviewed in [12].

In this review, we will be concerned with relativistic QFTs, whose gravity duals are formally given by so-called “fake” SUGRAs [23]. These theories describe gravity coupled to an arbitrary number of dynamical scalars, with the condition that the scalar potential can be expressed via a “superpotential”. (Hence, “fake” just means that the theory is not necessarily supersymmetric, because this condition is weaker than supersymmetry. The relation between supergravity and fake supergravity was analyzed in [24, 25].) This implies the existence of BPS domain wall background solutions, which are the holographic duals of RG flows. AdS/CFT systems (with aAdS geometry) typically fall into this class of theories. What we drop, therefore, is only the requirement that the background has an asymptotic AdS region. On the QFT side of the duality, this implies to give up the existence of a UV conformal fixed point. (For this reason, we can speak of “nonAdS/nonCFT” dualities. These play a prominent role in holographic models of QCD-like theories showing features such as mesons, chiral symmetry breaking and confinement. Further details on this subject can be found in the reviews [26–28].) In contrast to the nonconformal systems mentioned earlier, in which one can perform HR based on the general asymptotic structure of the bulk space-time and fields, we will make no specific assumptions on the asymptotic geometry of the bulk. Thus, a general treatment of HR along the lines of the standard method [12] appears to be possible only on a case-by-case basis. What we would like to answer is the question whether the fake SUGRA structure of the bulk theories can be exploited to perform HR for the QFTs dual to the BPS domain wall backgrounds. The answer to this question is positive, and we review recent progress on a perturbative, or order-by-order, approach to HR. In this approach, one considers the fluctuations around the exact BPS background flow and removes the divergences order by order in the fluctuations. So far, the second-order counterterms have been constructed, which are sufficient for the calculation of two-point functions in flat backgrounds [29]. In a sense, the approach is inspired by [13, 14], where the philosophy was put forward to concentrate on the part of the counterterm action which is really necessary to calculate -point functions for a given , that is, the terms of order in the fluctuations. In this spirit, [29] considered the case . It might be possible to derive the counterterms from a fully covariant expression (similar to [30]), but this has not been achieved yet. The approach is based on the gauge-invariant formalism for the dynamics of the bulk fluctuations [31–33], which we include in this review for completeness. This formalism has turned out to be very useful for the holographic calculation of correlation functions, mass spectra and scattering amplitudes, both in aAdS and nonaAdS backgrounds.

The prime example of a nonaAdS/nonCFT fake SUGRA system, which exhibits a logarithmically warped AdS geometry in the asymptotic region, is the Klebanov-Strassler (KS) solution [34], which is well approximated in the UV by the Klebanov-Tseytlin (KT) solution [35]. The fake SUGRA systems can be obtained by a consistent truncation of type-IIB SUGRA [33, 36–38] and are the gravity duals of an gauge theory undergoing a series of Seiberg dualities, . Whereas the KT solution correctly describes the duality cascade, its bulk singularity makes it unreliable in the IR. The resolution of the singularity in the KS background provides a geometrical description of chiral symmetry breaking and confinement. Calculating correlation functions in these cases is much more involved, also because the procedure of HR has not been worked out yet in a systematic way similar to the aAdS case. As a result, only a few attempts to calculate correlators using the KT background have been made, compare for example, [30, 39–41], and only in [30, 41] the program of HR, as reviewed in [12], was applied. A generalization of this procedure to include flavor degrees of freedom was discussed in [42]. Furthermore, calculations of mass spectra in the KS background [39, 43–47] have been done using a pragmatic approach assuming that a consistent method of HR in nonaAdS backgrounds exists.

Other systems of recent interest are derived from the Maldacena-Nunez background [48], the gravity dual of  SYM describing, in the UV, the exact NSVZ -function [49] and a nonzero gluino condensate in the IR that breaks the chiral symmetry. In particular, so-called walking solutions have been considered [50–52], where in a certain regime the gauge coupling changes very little—it walks as opposed to runs. The walking region imitates, in many respects, a conformal fixed point, and the slightly broken scale symmetry implies the existence of a light particle. Walking Technicolor [53] is a phenomenologically interesting alternative to the Higgs sector of the Standard Model. Glueball mass spectra in backgrounds with walking couplings have been calculated in [54, 55].

There is, therefore, continued interest in understanding better the various aspects of HR in nonaAdS backgrounds. As far as the calculation of correlation functions is concerned, the state of the art are the mass spectra that have been mentioned above. To put these calculations on a more solid footing and to prepare the ground for further investigations (fully momentum-dependent propagators, three-point functions, etc.), a consistent and systematic procedure of HR should be constructed.

Let us now state the main idea of the order-by-order approach. The starting point of the holographic calculation of correlation functions in AdS/CFT, which we generalize to nonaAdS configurations, is the correspondence formula (1.1). In AdS/CFT, the sources are identified with asymptotically rescaled boundary values of the on-shell bulk fields. When trying to generalize this identification, one has to be aware of some implicit assumptions that one usually uses in AdS/CFT [33]. The existence of a fixed point, where the bulk geometry would be global AdS, allows to parameterize the scalars in such a way that their origin is located at the fixed point, and that the individual scalars are the duals of boundary operators with certain scaling dimensions. The fixed point corresponds to some conformally invariant QFT (CFT). Therefore, holographic RG backgrounds that flow away from the fixed point can be considered either as the duals of deformations of the CFT by relevant operators or as the duals of vacua with spontaneously broken conformal symmetry, with finite couplings or vacuum expectation values (VEVs), respectively, which are related to the background scalars. In the nonaAdS case, such an interpretation is not directly possible. The absence of a fixed point does not allow to fix an origin or to choose some otherwise preferred parameterization of the scalar fields. On the field theory side, it is not possible to say that the RG flow is due to some operator deformation or a ground state with a VEV of some special QFT. All we seem to be able to say is that the holographic RG background is the dual of some QFT that has running couplings. (In AdS/CFT, one would simply say that the QFT under consideration is the CFT + deformations or in a spontaneously broken phase.) Just as in AdS/CFT, however, correlation functions of this QFT can be obtained by studying the dynamics of the fluctuations of the bulk fields around the holographic RG flow. For two-point functions, which we will be concerned with, it is sufficient to solve the linearized equations of motion, which have a useful form in the gauge-invariant formalism. The sources will be given by certain asymptotic boundary values of the on-shell bulk fluctuations.

Scheme dependence of correlation functions (but not of physical amplitudes) is a general feature of renormalization in QFT. When adding counterterms that cancel the cutoff dependent divergences, one is free to add finite counterterms, which are normally chosen such that certain renormalization conditions are satisfied at some renormalization scale. The same holds in HR in AdS/CFT. Also the order-by-order approach presented here includes scheme dependence, although in a more indirect way. The choice of scheme is reflected in certain ambiguities in the choice of the asymptotic basis functions for the fluctuations.

The rest of the review is structured as follows. In Section 2, we review the linearized bulk dynamics of a fake SUGRA system around BPS domain wall backgrounds, expressed in the language of gauge-invariant variables introduced in [31–33], and we introduce the asymptotic decomposition of the on-shell fields into dominant and subdominant basis solutions. The holographic renormalization for two-point functions following the order-by-order approach of [29] is presented in Section 3. We restrict our attention to the scalar sector, but the recipe can be extended easily to the traceless transversal fluctuations of the metric. Because the results of this part yield new insights into the pole structure of the two-point functions, we will also include the relevant updated material from [45]. Then, it will be discussed how scheme dependence is incorporated in the order-by-order approach. Section 4 is dedicated to examples with aAdS backgrounds, that serve to verify the agreement of the results of Section 3 with known results from AdS/CFT. Apart from the pure AdS case, we will consider the GPPZ [56] and the Coulomb branch (CB) flows [57, 58], which have been extensively studied in the literature. The GPPZ flow represents a massive deformation of  SYM, while the CB flow is dual to a vacuum of  SYM, in which the conformal invariance is spontaneously broken by a VEV. This gives us enough confidence to carry on and consider the KS case in Section 5. Finally, Section 6 contains our conclusions.

2. Bulk Dynamics

Let us start by reviewing the equations governing the dynamics of the bulk fields [32, 33], which encode the information about two-point functions in holographic renormalization group flows.

The systems we consider are fake SUGRAs in dimensions with actions of the form with , and where the potential is given in terms of a superpotential by We will not specify at this point the boundary terms in (2.1), as they do not affect the bulk dynamics, although they are important for holographic renormalization. We will come back to them in Section 3. In (2.1) and (2.2), is the metric on the sigma-model target space and is its inverse. Moreover, we used the notation .

Holographic renormalization group flows are described by domain wall backgrounds of the form with , satisfying the BPS equations Linearized fluctuations around such a domain wall background are best described in a gauge invariant fashion, which we review next.

2.1. The Sigma-Model Covariant Field Expansion

It is our aim to study the dynamics of the fake supergravity system (2.1), (2.2) on some known backgrounds of the form (2.3), (2.4). In this section, we will expand the fields around the background, exploiting the geometric nature of the physical variables to achieve a gauge-invariant formulation of the fluctuation dynamics.

As is well known in gravity, reparametrization invariance of space-time comes at the price of dragging along redundant metric variables together with the physical degrees of freedom. One usually attempts to reduce redundancy by gauge fixing, but such an approach causes problems for the study of fluctuations in holographic RG flows, due to the coupling between the metric and scalar fluctuations [59–61]. Thus, following [31], we will start from a clean slate keeping all metric degrees of freedom and describe in the next subsection how to isolate the physical ones.

Using the sigma-model metric , one can define the sigma-model connection, and its curvature tensor, We also define the covariant field derivative as usual, for example, All indices after a bar “” are intended as covariant field derivatives according to (2.7). Moreover, field indices are lowered and raised with and , respectively.

Armed with this notation, it is straightforward to expand the scalar fields in a sigma-model covariant fashion. The naive ansatz , introducing simply as the coordinate difference between the points and in field space, leads to noncovariant expressions at quadratic and higher-orders, because these do not form a vector in (tangent) field space. In other words, the coordinate difference is not a geometric object. However, it is well known that a covariant expansion is provided by the exponential map, see for instance [62, 63], where the higher-order terms have been omitted, and the connection is evaluated at . Geometrically, represents the tangent vector at of the geodesic curve connecting the points and , and its length is equal to the geodesic distance between and ; see Figure 1.

Figure 1 

Illustration of the exponential map.

It is also a standard result that the components coincide with the Riemann normal coordinates (RNCs) (with origin at ) of the point (see, e.g., [63]). This fact can be used to simplify the task of writing equations in a manifestly sigma-model covariant form. Namely, given a background point , we can use RNCs to describe some neighborhood of it and then employ the following properties at the origin of the RNC system: in order to express everything in terms of tensors. Because the background fields depend on , we must be careful to use (2.9) only outside -derivatives, but the simplifications are still significant.

Finally, let us also define a “background-covariant” derivative , which acts on sigma-model tensors as, for example, If a tensor depends on only implicitly through its background dependence, then we find the identity The background-covariant derivative will be important in our presentation of the field equations in Section 2.4.

2.2. Gauge Transformations and Invariants

The form of the background solution (2.3) lends itself well to the ADM (or time-slicing) formalism for parametrizing the metric degrees of freedom, compare for example, [62, 64]. Instead of slicing in time, we will write a general bulk metric in the radially-sliced form where is the induced metric on the hypersurfaces of constant , and and are the lapse function and shift vector, respectively. It will be important to us that the objects , and transform properly under coordinate transformations of the radial-slice hypersurfaces.

We can now expand the radially-sliced metric around the background configuration: where , and denote small fluctuations. Henceforth, we will adopt the notation that the indices of metric fluctuations, as well as of derivatives , are raised and lowered using the flat (Minkowski/Euclidean) metric, .

Now let us turn to the question of isolating the physical degrees of freedom from the set of fluctuations which we have introduced so far. In the earlier AdS/CFT literature one usually removed the redundancy following from diffeomorphism invariance by partial gauge fixing, that is, by placing conditions on certain components of the metric, such as , . And indeed, it is always possible to perform a change of coordinates which transforms the metric into a form that satisfies the gauge conditions.

Alas, as mentioned above, partial gauge fixing can create problems in coupled systems. Instead, we will obtain the equations of motion in gauge-invariant form. Let us start by considering the effect of diffeomorphisms on the fluctuation fields. We consider a diffeomorphism of the form where is infinitesimal. Notice that we found it convenient to apply the diffeomorphism inversely, that is, we have expressed the old coordinates in terms of the new coordinates . The use of the exponential map implies that also the transformation laws for the fields can be written covariantly (the functions are thought of as the components of a vector field). For example, a scalar field transforms as whereas a covariant tensor of rank two transforms as For the metric tensor , (2.16) is simplified to Equations (2.15) and (2.16) are most easily derived using RNCs around and exploiting (2.9). The second-order terms in have been included here in order to illustrate the covariance of the transformation laws. For our purposes, the linear terms will suffice.

Splitting the fake supergravity fields into background and fluctuations, as defined in (2.13) and (2.8), the transformations (2.15) and (2.17) become gauge transformations for the fluctuations, to lowest order: By we mean terms of order in the fluctuations . Moreover, let us decompose as follows (in the following we will always assume ): where denotes the traceless transverse part, and is a transverse vector. It is straightforward to obtain from (2.18) The symbol denotes the transverse projector,

The main idea of our approach is to construct gauge-invariant combinations from the fields . Using the transformation laws (2.18) and (2.20), this is straightforward, and to lowest order, one finds the gauge-invariant fields (The choice of gauge-invariant variables is, of course, not unique, as any combination of them will be gauge-invariant as well.) The variables and both arise from , which has been split into its longitudinal and transverse parts.

Although we have carried out the construction of gauge-invariant variables only to lowest order, and as we will see below, this is all that is needed, it is necessary for consistency that the preceding analysis can, in principle, be extended to higher-orders, which is indeed the case. In this context it becomes clear that the geometric nature of the field expansions, which is implied by the exponential map, is a crucial ingredient of the method.

Finally, let us prepare the ground for the arguments of the next subsection, where we will analyze the implications of gauge-invariance on the equations of motion. To be concise, we continue in a symbolic fashion. Let us consider the set of gauge-invariant fields, . From the definitions (2.22)–(2.26) we see that there is a one-to-one correspondence between and a subset of the fluctuation fields, . We collect also the remaining fluctuation variables into a set, . Henceforth, the symbols , and will be used also to denote the members of the corresponding sets.

One can better understand the correspondence between and by noting that (2.22)–(2.26) can be rewritten as where is a linear functional of the fields . Going to quadratic order in the fluctuations, one would find where and are bi-linear in their arguments. Terms of the form do not appear, as they can be absorbed into .

We interpret the gauge-invariant variables as the physical degrees of freedom, whereas the variables represent the redundant metric variables. This can be seen by observing that one can solve the transformation laws (2.20) for the generators , which yields equations of the form with being a linear functional.

2.3. Einstein's Equations and Gauge Invariance

It is our aim to cast the equations of motion into an explicitly gauge-invariant form. This means that the final equations should contain only the variables and make no reference to and . Reparametrization invariance suggests that this should be possible, and we will establish the precise details in this subsection.

Let us consider Einstein's equations, symbolically written as but it is clear that the arguments given below hold also for the equations of motion for the scalar fields. To start, let us expand the left-hand side of (2.30) around the background solution, which yields, symbolically, Here, and denote linear and bilinear terms, respectively. The background equations are satisfied identically. Substituting for using (2.28) yields Notice that the functionals and are unchanged ( is just replaced by ), whereas the others are modified by the -dependent terms of (2.28), which we indicate by adorning them with a tilde. For example, receives contributions from , and .

In order to simplify (2.32), we consider its transformation under the diffeomorphism (2.14). On the one hand, from the general transformation law of tensors (2.16) we find, using also (2.29), that it should transform as On the other hand, the variation of (2.32) is Let us compare (2.33) and (2.34) order by order. The absence of first-order terms on the right-hand side of (2.33) implies that It can easily be checked that this is indeed the case. Then, substituting into the right-hand side of (2.33) yields Comparing (2.36) with the second-order terms of (2.34), we obtain Hence, we find that a simple expansion of Einstein's equations yields gauge-dependent second-order terms, but they contain the (gauge-independent) first order equation, and so can consistently be dropped. Happily, we arrive at the following equation, which involves only : The argument generalizes recursively to higher-orders. One will find that the gauge-dependent terms of any given order can be consistently dropped, because they contain the equation of motion of lower orders.

Equation (2.38) and its higher-order generalizations are obtained using the following recipe:

Expand the equations of motion to the desired order dropping the fields and replacing every field by its gauge-invariant counterpart .

This rule is summarized by the following substitutions: Since is traceless and transverse, the calculational simplifications arising from (2.39) are considerable.

Let us conclude with the remark that, although the rules (2.39) can be interpreted as the gauge choice , the equations we found are truly gauge invariant.

2.4. Equations of Motion

In this section, we will put into practice what we have just learned. The equations of motion that follow from the action (2.1) are for the scalar fields, and Einstein's equations Notice that we use the opposite sign convention for the curvature with respect to [31, 32].

We are interested in the physical, gauge-invariant content of (2.40) and (2.41) to quadratic order in the fluctuations around an RG flow background of the form (2.3), (2.4). As we have learned in the last section, it is obtained by expanding the fields according to (2.13) and (2.8) and then applying the substitution rules (2.39). Since we defined the expansion (2.8) geometrically, it is assured that we will obtain sigma-model covariant expressions. To carry out this calculation in practice, it is easiest to use RNCs at a given point in field space, so that one can use the relations (2.9) outside -derivatives.

In the following, we will present the linearized equations of motion. We just collect the results without going into details of the derivation. For intermediate steps we refer the reader to the appendices B and C of [33]. (Note that we use a different notation for the indices here than in [33]. The dimensional indices are denoted by here and by there, whereas the dimensional indices are denoted by here and by there.) Let us start with the equation of motion for the scalar fields (2.40), which gives rise to the following fluctuation equation: Second, the normal component of Einstein's equations gives rise to Third, the mixed components of (2.41) yield

The appearance of the fields , , and on the left-hand sides of (2.42)–(2.44) seems to indicate the coupling between the fluctuations of active scalars (nonzero ) to those of the metric, which is familiar from the AdS/CFT calculation of two-point functions in the literature. However, the gauge-invariant formalism resolves this issue, because (2.43) and (2.44) can be solved algebraically (in momentum space) for the metric fluctuations , and , so that the coupling of metric and scalar fluctuations at linear order is completely disentangled. One easily obtains (using our assumption )

We proceed by substituting (2.45) and (2.46) into (2.42), using also the identities which follow from (2.2) and (2.4), and we end up with the second-order differential equation where we introduced the matrix Equation (2.49) is the main result of the gauge-invariant approach and governs the dynamics of scalar fluctuations around generic Poincaré-sliced domain wall backgrounds.

Let us also consider the tangential components of (2.41). Because of Bianchi's identity, their trace and divergence are implied by (2.42), (2.43), and (2.44), which is easily checked at linear order. Thus, we can use the traceless transverse projector, in order to obtain the independent components. This yields As expected, the physical fluctuations of the metric satisfy the equation of motion of a massless scalar field.

In the following, we focus on the scalar field equation (2.49). Let be the number of scalar fields (components of ). As in [45], we will assume the existence of a set of independent solutions of (2.49), which are defined as power series in (in momentum space), with -dependent coefficients that are more and more suppressed with increasing powers of . (This is tantamount to demanding that the warp function grows without limit for , so that in (2.49) can be regarded as a correction in the asymptotic region.) Moreover, the leading term (for large ) in each solution should be independent of . In position space, simply translates to the operator . Amongst these solutions, one can distinguish between asymptotically dominant solutions () and subdominant solutions with respect to their behaviour at large . Including the field index, we will interpret and as matrices. A regularity condition in the bulk interior, that is, allows only for independent regular combinations of the asymptotic basis solutions. Hence, we will decompose a general regular solution of (2.49) into where and are called the source and response coefficients, respectively, and . The bulk regularity condition uniquely determines the (functional) dependence of the responses on the sources and gives rise to the nonlocal information for the two-point functions of the dual operators.

Throughout the paper, we will consider mostly the analogue of (2.54) in momentum space, sometimes omitting the dependence on . Moreover, a will be used to denote the inner product in field space, or the contraction of field space indices, for example, .

3. Perturbative Holographic Renormalization

3.1. Scalar Two-Point Functions

In this section, we will present the general formalism for obtaining finite, renormalized two-point functions for the QFT operators that are dual to the bulk scalar fields. Our starting point is the following action, which is quadratic in the fluctuations and encodes the bulk field equations (2.49), with some symmetric counterterm matrix , which is a local operator that will be specified in a moment. The bulk integral in (3.1) is to be understood with a cutoff , where also the boundary counterterm is evaluated. It follows that the variation of the on-shell action with respect to a variation of the boundary value is given by where the right-hand side is evaluated at . Let us define the counterterm matrix as where is the inverse of the matrix , defined in momentum space as a series in , or equivalently, in position space as a series in . We will see momentarily that this definition leads to finite one- and two-point functions. We also note the following subtlety. The counterterm in (3.1) needs to be local in the fields, which means that should be a polynomial in (in momentum space) or (in position space). The assumptions made in Section 2 imply that is a series in . However, we also assumed that the coefficients of the series with increasing powers of are suppressed for large due to the factor , so that we can truncate the series in (3.3) to some polynomial, because the terms thus neglected vanish in the large- limit. Hence, strictly speaking, the counterterm operator in (3.1) is a polynomial truncation of (3.3).