The AdS/CFT partition function, AdS as a lift of a CFT, and holographic RG flow from conformal deformations
TThe AdS/CFT partition function, AdS as a lift of aCFT, and holographic RG flow from conformal deformations
Sean Cantrell
Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD 21218
Abstract
Conformal deformations manifest in the AdS/CFT correspondence as boundary conditions onthe AdS field. Heretofore, double-trace deformations have been the primary focus in this context.To better understand multitrace deformations, we revisit the relationship between the generatingAdS partition function for a free bulk theory and the boundary CFT partition function subject toarbitrary conformal deformations. The procedure leads us to a formalism that constructs bulk fieldsfrom boundary operators. Using this formalism, we independently replicate the holographic RG flownarrative to go on to interpret the brane used to regulate the AdS theory as a renormalization scale.The scale-dependence of the dilatation spectrum of a boundary theory in the presence of generaldeformations can be thus understood on the AdS side using this formalism. a r X i v : . [ h e p - t h ] J a n ontents The AdS/CFT correspondence has emerged over the last two decades as our best means of under-standing quantum gravity. Originally born as a mapping between states in a string theory livingin AdS with compactified extra dimensions and SUSY Yang-Mills theories, the correspondence isnow understood broadly as a duality between theories in an AdS background and local conformaltheories. The AdS/CFT correspondence has been understood via two dictionaries:1. Taking AdS correlation functions to the boundary and extracting the leading order behaviorto recover correlators of dual operators constructed from a local CFT [1, 2], (cid:104)OO . . . (cid:105) = lim z → (cid:104) z − ∆ φ ( z ) z − ∆ φ ( z ) . . . (cid:105) and2. Evaluating the on-shell AdS partition function as a functional of a boundary source, φ b , andcomputing CFT correlators in the usual way [3, 4], Z CFT [ φ b ] = Z AdS [ φ b ] . The dictionaries have been shown to be equivalent in the presence of bulk interactions [5], theintuition being that interactions turn off near the boundary [6], rendering the on-shell descriptionadequate.The AdS/CFT correspondence also provides a holographic means to understand the renormal-ization group flow of CFTs as the classical evolution of dual bulk theories in the radial direction ofAdS [7–19]. Explicitly, the radial coordinate is interpreted as the renormalization scale. Efforts toapproach RG flow via entropy-esque quantities and H-theorem type constraints have lead to the a-,F-, and c- theorems [17, 20–22], and their associated geometric formulations [15, 18, 19]. hile few conformal theories appear in nature, deformations can be added to certain ones tomore closely reproduce physical theories. Naturally, we must consider what becomes of the AdS/CFTcorrespondence when conformality at the boundary is spoiled since, by construction, interactions donot turn off near the boundary. Conformal deformations were first examined in the context ofAdS/CFT in [23], and recent work in conformal dominance [24, 25] invites their continuing presence.The role of boundary conditions in AdS theories was examined by [26], and the connection betweenboundary conditions and CFT deformations was made explicit by [27, 28].Double-trace deformations have been the predominant focus in the context of AdS/CFT [23, 27–32]. While demonstrative of many salient features of the deformed correspondence, most double-trace techniques, in which the relationship between the bulk and boundary field remains linear whileapproaching the boundary, do not manifestly apply to more general deformations. It is also unclearfrom the literature whether we are instructed to evaluate the AdS partition function on-shell inthe usual manner when employing the second dictionary to compute correlators. By this we meancomputing the bulk field as the classical solution to the field equations in the presence of a boundarysource, φ b . Intuition says ‘no’ as this would omit quantum effects from our correlators.In the first part of this paper, § §
4, we aim to clarify the ambiguities in handling multi-tracedeformations and establish a framework that makes manifest the equivalency of the two dictionariessubject to these deformations. This is achieved by deriving the explicit relationship between thegenerating bulk partition function and the dual CFT partition function with deforming Lagrangian W [ O ].This framework then leads to what we call a lift formalism in § § d +1 with ds = 1 z (cid:32) dz + d (cid:88) i =1 dx i (cid:33) (1.1)and R AdS = 1. We consider only CFTs dual to free theories in the bulk. We employ the compactnotation ¯ d = d π for integral measures. We will consider only scalar fields herein, and, where applica-ble, we will consider a general scaling dimension, ∆ ≥ d −
1, of the operator dual to the bulk field;however, with the goal of examining RG flow in mind, we will usually restrict the scaling dimensionin the UV to ∆ = ∆ − ≤ d .We first offer a convenient summary of the results of this paper. Summary of results
The generating partition function from which bulk correlators can be computed is Z AdS [ J ] = exp (cid:20)(cid:90) d d x (cid:90) dz J ( x, z ) φ cl ( x, z ) (cid:21) × (cid:90) D α D β exp (cid:20) (cid:90) z = (cid:15) d d x (cid:0) − ν(cid:15) ν β ( x ) + W [ α ( x ) + (cid:15) ν β ( x ) + (cid:15) − ∆ − φ cl ( x, (cid:15) )] (cid:1) + S ∂ [ α ] (cid:21) . (2.1)AdS/CFT correlators are generated by functionally differentiating Eq.(2.1) with respect to to thesource J . Bulk fields scale to the boundary as φ → z → αz d/ − ν + βz d/ ν , and the classical, particularsolution to the field equations is given by φ cl ( x, z ) = (cid:82) d d x (cid:48) (cid:82) ∞ z (cid:48) =0 G ( x − x (cid:48) ; z, z (cid:48) ) J ( x (cid:48) , z (cid:48) ), where G is the bulk-bulk propagator. W [ O ] is the deforming Lagrangian for the dual CFT ( α → O ) with∆ − = d − ν, < ν < O . From the perspective of the bulk, W isjust a boundary term, rendering the bulk theory free. z = (cid:15) (cid:28) S ∂ is a generic conformalaction that generates dynamics for O . The functional integral over α and the written dependence of S ∂ on α are formalities that simply instruct us to evaluate α as O in the CFT. β is an auxiliary field, and integrating it out reproduces Witten’s prescription for the boundaryconditions: β = 12 ν W (cid:48) [ α ] . (2.2)The partition function given in Eq.(2.1) differs from what usually appears in the literature, wherethe “on-shell” action is taken to have the form S ∝ (cid:82) z = (cid:15) d d x αβ to leading order in (cid:15) . In this paper,we argue that these leading terms are canceled and that second order effects must thus be consideredto correctly yield Eq.(2.2) from the on-shell behavior for β .It additionally follows from Eq.(2.1) that setting (cid:15) → J →
0, and φ b α ⊂ W [ α ] yields Z AdS [ φ b ] = Z CF T [ φ b ] , (2.3)establishing the second line in Eq.(2.1) as a modified form of the CFT partition function and con-firming the second dictionary in the presence of boundary deformations.Constructing the AdS partition function from the CFT partition function as above is reminis-cent of the use of smearing functions to construct AdS operators from their CFT duals. Smearingfunctions typically map a tower of operators to a bulk field. We offer a similar procedure that mapsthe operators in a different way, but recovers the ultimate goal of constructing AdS correlators fromboundary ones. At the level of operators, we write φ ( x, z ) = (cid:90) d d x (cid:48) L α ( x, x (cid:48) ; z ) α ( x (cid:48) ) + 2 ν (cid:90) d d x (cid:48) L β ( x, x (cid:48) ; z ) (cid:0) β ( x (cid:48) ) + β ( x (cid:48) ) (cid:1) , (2.4)where the lift kernels are given by L α ( x, x (cid:48) ; z ) = (cid:90) ¯ d d p Γ(1 − ν ) (cid:16) p (cid:17) ν z d/ I − ν ( pz ) e ip · ( x − x (cid:48) ) (2.5) L β ( x, x (cid:48) ; z ) = (cid:90) ¯ d d p (cid:20)
12 Γ( ν ) I − ν ( pz ) − − ν ) K ν ( pz ) (cid:21) (cid:16) p (cid:17) − ν z d/ e ip · ( x − x (cid:48) ) . (2.6) acts as a functional derivative when inserted into correlators: β ( x ) = 12 ν δδα ( x ) , (2.7)while β takes its usual form of Eq.(2.2).These results indeed confirm that the bulk field cannot be computed directly as a classicalfunctional of φ b when using the second dictionary in the presence of general deformations. It mustbe computed in terms of α , which itself can only be computed as a classical functional of φ b whencomputing correlators in the presence of, at most, double-trace deformations.Using the formalism, we find two interesting results for particular boundary correlators in mo-mentum space: (cid:104) W (cid:48) [ α ]( − p ) W (cid:48) [ α ]( p ) + W (cid:48)(cid:48) [ α ]( p ) (cid:105) = Σ( p )1 − g ( p )Σ( p ) , (2.8)and (cid:104) αW (cid:48) [ α ] (cid:105) ( p ) = g ( p )Σ( p )1 − g ( p )Σ( p ) . (2.9)Here, g is the free boundary propagator and Σ is the sum over two-point function 1PI diagrams at theboundary. Evidently, there is a strong connection between β terms in the AdS/CFT dictionary and1PI diagrams at the boundary. The generally non-vanishing value of the RHS of Eq.(2.9) indicatesthat the normal ordering one would naively apply to W (cid:48) when treating it as a multitrace operator isinstead applied to the full αW (cid:48) , meaning that W (cid:48) generally contains both multitrace operators andadditional, non-normal ordered operators. The connection to 1PI diagrams is thus not surprisingsince W (cid:48) involves operators that appear in the OPE spectrum of the theory.We go on to find the conformal dimension of a dual boundary operator depends on RG scale, µ ,as ∆ = µ →∞ − µ∂ µ [ (cid:104) φα (cid:105) ( p, µ )][ (cid:104) φα (cid:105) ( p, µ )] (cid:12)(cid:12)(cid:12)(cid:12) | p | = µ . (2.10)We are instructed to evaluate the correlators with z = µ − as a UV brane. Specifically, we find thebulk-boundary propagator and pull it to the UV brane on which the CFT sits.It is useful to demonstrate the procedure by computing ∆ for the well-understand example of adouble-trace deformation, λ O , with UV scaling dimension ∆ − . We find in momentum space nearthe boundary (cid:104) φα (cid:105) ( p, µ − ) (cid:12)(cid:12) | p | = µ = µ →∞ −
11 + Γ( ν )Γ(1 − ν ) (cid:0) µ (cid:1) − ν λ Γ( ν )Γ(1 − ν ) (cid:16) µ (cid:17) d/ − ν . (2.11)Inserting Eq.(2.11) into Eq.(2.10) reproduces the known result for double-trace deformations:∆( µ ) = ∆ − + λµ ν + λ ν . (2.12)Eq.(2.12) has the simple interpretation that operators in a theory with double-trace deformationswith scaling dimension ∆ − in the UV ( µ → ∞ ) flow to operators with scaling dimensions ∆ + =∆ − + 2 ν in the IR ( µ → Boundary conditions and classical bulk fields
Conformal deformations manifest as boundary conditions in AdS. Witten originally proposed thata deformation of the form S W [ O ] = (cid:90) d d x W [ O ] , (3.1)where O is a generalized free field with scaling dimension ∆, should be interpreted as a boundaryaction in AdS. For dual bulk fields that scale to the boundary as φ ( x, z ) → z → α ( x ) z ∆ + β ( x ) z d − ∆ , (3.2)the boundary term becomes S W [ α ]. This leads to the following constraint on the asymptotic behav-ior : β ( x ) = 1 d − δS W [ α ] δα ( x ) . (3.3)This condition was argued to arise classically from an on-shell bulk theory ending on a UV braneat z = (cid:15) (cid:28) S [ φ ] = S bulk [ φ ] + S ct [ φ ] + S W [ φ ] = (cid:90) d d x (cid:90) dz √ g (cid:40) z ( ∂φ ) − (cid:34)(cid:18) d (cid:19) − ν (cid:35) φ (cid:41) + (cid:90) z = (cid:15) d d x √ g L ct + (cid:90) z = (cid:15) d d x W [ (cid:15) − ∆ φ ] , (3.4)where S ct = (cid:90) z = (cid:15) d d x (cid:15) ∆ φ (3.5)is a boundary counter term that ensures the convergence of the on-shell action for ∆ ≥ d − ≤ d , the counter term additionally specifies which of the two viable dual conformal theoriessits at the boundary, ∆ = ∆ ± = d ± ν, ≤ ν ≤
1. The mass-squared has been written in terms of ν ≡ (cid:113)(cid:0) d (cid:1) − m for later convenience.Classically, Eq.(3.4) with L ct = (cid:15) ∆ φ leads to the differential equations (cid:2) D z + D ∂ (cid:3) φ =0 (3.6)[ B + δB ] φ =0 , (3.7)where D z = z d +1 ∂ z (cid:104) z − d +1 ∂ z (cid:105) + (cid:34)(cid:18) d (cid:19) − ν (cid:35) , (3.8) D ∂ = z ∂ µ ∂ µ , (3.9) B = (cid:15)∂ z − ∆ | z = (cid:15) (3.10) δB = − (cid:15) d − ∆ W (cid:48) [ (cid:15) − ∆ φ ] (cid:12)(cid:12)(cid:12) z = (cid:15) . (3.11)Note, the µ index runs over only the boundary coordinates. Eqs.(3.7)&(3.2) then imply, to leadingorder in (cid:15) , ( d − β = W (cid:48) [ α ] , (3.12) The coefficient d − is actually dependent on the normalization of O . Here, we take α = (cid:104)O(cid:105) . hich is, as promised, Eq.(3.3). For now, the classical arguments will be kept to leading order in (cid:15) ;it will be shown later than we must consider second order effects to properly recover the conditionin the quantum theory.We employ the above cumbersome notation to provide a simple, general solution scheme to theclassical bulk equations. If we solve the easier problem given by D z Z =0 (3.13) B Z = − δBφ, (3.14)and write ϕ ≡ φ − Z , we can reduce our task to solving a theory with better known boundaryconditions: (cid:2) D z + D ∂ (cid:3) ϕ = − D ∂ Z (3.15) B ϕ =0 . (3.16)The particular solution is readily apparent: ϕ ( x, z ) = − (cid:90) d d x (cid:48) (cid:90) ∞ dz (cid:48) (cid:112) g (cid:48) D ∂ (cid:48) G ( x − x (cid:48) ; z, z (cid:48) ) Z ( x (cid:48) , z (cid:48) ) , (3.17)where G is the AdS propagator, modulo factors of − − . With this choice, the propagator is G ( x ; z, z (cid:48) ) = − (cid:90) ¯ d d p (cid:90) ∞ dm ( zz (cid:48) ) d J − ν ( mz ) J − ν ( mz (cid:48) ) mp + m e ip · x . (3.18)The homogenous solution, by construction, must be the same as the φ solution under the unmodifiedboundary conditions, and will thus be denoted φ . From this, it follows that φ ( x, z ) = φ ( x, z ) + (cid:90) d d x (cid:48) (cid:90) ∞ dz (cid:48) (cid:112) g (cid:48) D z (cid:48) G ( x − x (cid:48) ; z, z (cid:48) ) Z ( x (cid:48) , z (cid:48) ) . (3.19)To verify the procedure, let us consider a boundary source and determine the bulk-boundarypropagator. Specifying W = (cid:15) d − ∆+1 φ b φ implies δB = − (cid:15) d − ∆ φ b . From Eqs.(3.13)&(3.14) we find, toleading order in (cid:15) , respectively, Z = a ( x ) z ∆ + b ( x ) z d − ∆ (3.20) b ( x ) = 12 ν φ b ( x ) . (3.21)The action of the integral kernel in Eq.(3.19) on z (cid:48) ∆ is trivial and, consequently, only b ( x ) matters.Since we are interested in the particular solution, we discard φ , leaving, in momentum space, φ ( p, z ) = (cid:20) − − ν ) (cid:16) p (cid:17) − ν z d K ν ( pz ) (cid:21) φ b ( p ) . (3.22)As expected, the factor in brackets is indeed the momentum space incarnation of the bulk-boundarypropagator, K ( x, x (cid:48) ; z ) = − ν Γ(1 − ν ) Γ(∆ − ) π d z ∆ − ( z + | x − x (cid:48) | ) ∆ − . (3.23)Next, let us seek the modification to the bulk-boundary propagator with a double trace defor-mation: L b = (cid:15) d − ∆+1 φ b φ + 12 λ(cid:15) d − φ = ⇒ δB [ φ ] = − (cid:15) d − ∆ φ b − λ(cid:15) d − ∆ α. (3.24) his time, Eq.(3.14) implies b ( x ) = 12 ν [ φ b ( x ) + λα ( x )] (3.25) a ( x ) =0 , (3.26)and, consequently, φ ( p, z ) = − (cid:20) − ν ) (cid:16) p (cid:17) − ν z d K ν ( pz ) (cid:21) [ φ b ( p ) + λα ( p )] . (3.27)Since we are interested in the particular solution to the boundary source equation, α is not arbitraryand we must solve for it. Insisting Eq.(3.27) match Eq.(3.2) and solving for α results in: φ ( p, z ) = −
11 + Γ( ν )Γ(1 − ν ) (cid:0) p (cid:1) − ν λ − ν ) (cid:16) p (cid:17) − ν z d K ν ( pz ) φ b ( p ) . (3.28)These results are so far known [29, 45], but careful attention should be paid to the procedureof throwing out φ and, more precisely, solving for α . It is only with a priori knowledge that theparticular solution gives us the bulk-boundary propagator that we have the luxury of demanding α arise only from the source φ b . Indeed, had φ not been discarded, it would scale to the boundary as φ ( x, z ) → z → z − ∆ A ( x ) , (3.29)with B ( x ) = 0 ensured by the boundary counter-term.Restricting to the K solution has its roots in demanding the bulk field be regular for z → ∞ [3],but, since we are permitting the boundary fields to be dynamical, we should reconsider the origin ofthis restriction.For general α and β , the field in the bulk takes the form φ ( x, z ) = (cid:90) d d x (cid:48) L α ( x, x (cid:48) ; z ) α ( x (cid:48) ) + 2 ν (cid:90) d d x (cid:48) L β ( x, x (cid:48) ; z ) β ( x (cid:48) ) , (3.30)where the lift kernels L α,β are given by L α ( x, x (cid:48) ; z ) = (cid:90) ¯ d d p Γ(1 − ν ) (cid:16) p (cid:17) ν z d/ I − ν ( pz ) e ip · ( x − x (cid:48) ) (3.31) L β ( x, x (cid:48) ; z ) = (cid:90) ¯ d d p (cid:20)
12 Γ( ν ) I − ν ( pz ) − − ν ) K ν ( pz ) (cid:21) (cid:16) p (cid:17) − ν z d/ e ip · ( x − x (cid:48) ) . (3.32)Demanding β = W (cid:48) [ α ] / ν customarily serve as a source for α requires α ( x ) =2 ν (cid:90) d d x (cid:48) g ( x − x (cid:48) ) W (cid:48) [ α ] , (3.33)where the undeformed boundary propagator is given by g ( x − x (cid:48) ) = − (cid:90) ¯ d d p Γ( ν )2Γ(1 − ν ) (cid:16) p (cid:17) − ν e ip · ( x − x (cid:48) ) ∝ | ∆ x | . (3.34)Inserting Eq.(3.33) into Eq.(3.30) yields φ ( x, z ) = 2 ν (cid:90) d d x (cid:48) K ( x − x (cid:48) ; z ) W (cid:48) [ α ] , (3.35) hich is precisely the form achieved by discarding φ in the above formalism.This indicates that including S ct and S W in the AdS action cannot be the entire story. An actionfor α , denoted herein as S ∂ , that classically results in Eq.(3.33) must either be generated or appearby explicit insertion. Given the nature of the constraint, S ∂ should include terms that lead to adynamical α , such as what would be contained in a generalized free theory in the large N limit. Weargue in the following section that S ∂ is generated by integrating out the bulk.As will be shown in the following section, multi-trace deformations require a proper quantumtreatment, and so our chief classical analysis ends here; however, we consider multi-trace deformationsand bulk wave functions and demonstrate that using the double-trace techniques of this sectionrequires α to be classical in Appendix A. Conformal deformations generate interactions and, generally, quantum corrections. Indeed, quarticinteractions anomalously break conformal invariance precisely due to the appearance of a renor-malization scale arising from loop corrections. Since Witten diagrams will include vertices at theboundary, we should expect the bulk theory to inherit certain quantum effects. To demonstrate theappearance of such quantum corrections to bulk correlation functions at the level of the partitionfunction, consider a multi-trace deformation constrained to a bulk UV brane at z = (cid:15) for a field φ dual to a conformal operator with scaling dimension ∆ − , W [ φ ] = 1 n λ(cid:15) − n ∆ φ n ( x, (cid:15) ) (4.1) L ct = 12 (cid:15) ∆ − φ ( x, (cid:15) ) . (4.2)The generating partition function for bulk correlators is given by Z [ J ] = (cid:90) D φ exp (cid:34)(cid:32) (cid:90) d d x (cid:90) dz √ g (cid:40) z ( ∂φ ) − (cid:34)(cid:18) d (cid:19) − ν (cid:35) φ (cid:41) + J φ + (cid:90) z = (cid:15) d d x (cid:15) − d − (cid:26) (cid:15) ∆ − φ + 1 n λ(cid:15) d − n ∆ − +1 φ n (cid:27) (cid:33) + S ∂ [ φ ] (cid:35) . (4.3)We can separate the quantum effects from the classical solutions by changing the integral measure: φ = φ cl + θ (4.4) J = √ g (cid:40) ∇ + (cid:34)(cid:18) d (cid:19) − ν (cid:35)(cid:41) φ cl , (4.5)where, additionally, φ cl ’s boundary conditions are chosen to minimize the on-shell action, (cid:104) − (cid:15)∂ z φ cl + ∆ − φ cl + λ(cid:15) d − n ∆ − φ n − cl (cid:105)(cid:12)(cid:12)(cid:12) z = (cid:15) =0 , (4.6)which leads to Eq.(3.3). The regular mode of the classical solution, α , is additionally forced by S ∂ to reproduce only the particular solution from Eq.(4.5) Here, the source is given its conventional symbol, J , which should not be mistaken for the Bessel functions appearingin the propagator. ntegrating the first term in Eq.(4.3) by parts and inserting Eqs.(4.5)&(4.6) into the result yields Z [ J ] = exp (cid:34) (cid:90) d d x (cid:90) dz J φ cl [ J ] (cid:35) × (cid:90) D θ exp [ S [ θ, J = 0]] exp (cid:34)(cid:90) ∂ d d x n λ(cid:15) − n ∆ − n − (cid:88) m =2 (cid:18) nm (cid:19) θ m φ n − mcl [ J ] + S ∂ ( θ ) (cid:35) , (4.7)where the functional dependence of φ cl on J has been emphasized. Bulk correlators are specified bythe integral kernels in the functional expansion of Eq.(4.7) in terms of J . Evidently, the coupling inthe second line vanishes for n ≤ n >
2, however,loop effects begin appearing, revealing the short coming of the usual approach taken for doubletrace deformations. For example, cubic interactions generate loop corrections to the bulk two pointfunction at order λ . Explicitly, (cid:104) φ ( x (cid:48) , z (cid:48) ) φ ( x, z ) (cid:105) = δ δJ ( x (cid:48) , z (cid:48) ) δJ ( x, z ) Z [ J ] (cid:12)(cid:12)(cid:12)(cid:12) J =0 = (cid:34) δφ cl ( x, z ) δJ ( x (cid:48) , z (cid:48) ) + λ(cid:15) − − (cid:90) d d y (cid:104) θ ( y, (cid:15) ) (cid:105) δ φ cl ( y, (cid:15) ) δJ ( x (cid:48) , z (cid:48) ) δJ ( x, z )+ λ (cid:15) − − (cid:90) d d y (cid:90) d d y (cid:48) (cid:104) θ ( y (cid:48) , (cid:15) ) θ ( y, (cid:15) ) (cid:105) φ cl ( y (cid:48) , (cid:15) ) δJ ( x (cid:48) , z (cid:48) ) δφ cl ( y, (cid:15) ) δJ ( x, z ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J =0 . (4.8)The factors of (cid:15) − − divide out the vanishing parts of the classical bulk correlators as they are takento the boundary. The term naively proportional to λ vanishes since the vacuum specified by action S does not support vevs of θ or its composites; nonetheless, the functional derivatives of φ cl arenon-zero and arise from the second harmonic term of φ cl , which itself is proportional to λ , ensuringthe quantum corrections are indeed of order λ . Higher order corrections are implicitly containedin the usual loop term (cid:104) θ ( y (cid:48) ) θ ( y ) (cid:105) ; as addressed in Appendix A, the bulk-bulk and bulk-boundarypropagators do not contribute larger λ corrections as they are non-vanishing only for non-vanishing J . It follows that pulling correlators computed according to Eq.(4.7) to the boundary and comput-ing correlators by evaluating the bulk partition function classically after pulling the source to theboundary are inequivalent. This is not to say the bulk and CFT partition functions are inequivalent,only that we should forgo computing the bulk fields classically. While the interior of the AdS actionmay be treated classically in the absence of bulk interactions, quantum effects appear through theboundary conditions. This suggests that the bulk theory should be understood as a lift of a quantumboundary theory. To clarify, this simply means the bulk is a classical boundary-value problem withquantum dynamics governing the behavior of the boundary conditions. To formalize this notion, wewish to express the generating bulk partition function as a functional of the boundary fields. We fol-low the procedure of dividing the generating path integral into IR ( (cid:15) + δ < z ) and UV ( (cid:15) ≤ z ≤ (cid:15) + δ )contributions [5, 46], Z [ J ] = (cid:90) IR D φ exp (cid:20) S bulk [ φ ] + (cid:90) d d x (cid:90) ∞ (cid:15) + δ dz J φ (cid:21) × (cid:90) UV D φ exp (cid:20) S [ φ ] + (cid:90) d d x (cid:90) (cid:15) + δ(cid:15) dz J φ (cid:21) , (4.9)and let δ → S and S bulk are given by Eq.(3.4) and we will choose ∆ = ∆ − as the scaling dimensionof the undeformed boundary theory. As before, we separate the classical and quantum effects, but reak down the quantum corrections in the IR differently: φ ( x, z ) = φ ( x, z ) + φ cl ( x, z ) + θ ( x, z ) , (4.10) φ ( x, z ) = 12 ν (cid:15) − ∆ − (cid:90) d d x (cid:48) L α ( x, x (cid:48) ; z ) [∆ + θ ( x, (cid:15) ) − (cid:15)∂ z θ ( x, (cid:15) )] − (cid:15) − ∆ + (cid:90) d d x (cid:48) L β ( x, x (cid:48) ; z ) [∆ − (cid:15)θ ( x, (cid:15) ) − (cid:15)∂ z θ ( x, (cid:15) )] , (4.11) φ cl ( x, z ) = (cid:90) d d x (cid:48) (cid:90) dz (cid:48) G ( x − x (cid:48) ; z, z (cid:48) ) J ( x (cid:48) , z (cid:48) ) (4.12)The φ term contained in the IR bulk field is, explicitly, a lift of the homogeneous boundary (UV)fields. As before, θ is the quantum perturbation. With this ansatz, the generating partition functionbecomes Z [ J ] = δ → exp (cid:20)(cid:90) d d x (cid:90) dz J φ cl (cid:21) (cid:90) IR D θ exp [ S bulk [ θ ]] × (cid:90) D θ ( (cid:15) + δ ) D θ ( (cid:15) ) (cid:20) (cid:90) z = (cid:15) d d x (cid:18) − (cid:15) − d +1 φ ∂ z φ + 12 (cid:15) − d ∆ − φ (cid:19) + S W [ φ cl + φ ] (cid:21) . (4.13)The path integral variable θ ( (cid:15) + δ ) appears only through the derivative ∂ z φ = θ ( (cid:15) + δ ) − θ ( (cid:15) ) δ . It ismore convenient to work in terms of the usual functions α and β that parameterize the asymptoticbehavior of φ . Writing (cid:18) φ ( (cid:15) ) φ ( (cid:15) + δ ) (cid:19) = (cid:18) (cid:15) ∆ − (cid:15) ∆ + ( (cid:15) + δ ) ∆ − ( (cid:15) + δ ) ∆ + (cid:19) (cid:18) αβ (cid:19) , (4.14)the generating partition function becomes Z [ J ] = exp (cid:20)(cid:90) d d x (cid:90) dz J φ cl (cid:21) (cid:90) IR D θ exp [ S bulk [ θ ]] × (cid:90) D α D β exp (cid:20)(cid:90) d d x (cid:0) − ναβ − ν(cid:15) ν β + W [ α + (cid:15) ν β + (cid:15) − ∆ − φ cl ] (cid:1)(cid:21) , (4.15)modulo irrelevant factors of δ and (cid:15) that arise from the Jacobian from our change of integral measure.In the absence of a bulk source and as (cid:15) →
0, we expect to recover the correct CFT partitionfunction with the appropriate S ∂ . The derivative terms in the IR partition function may be integratedby parts, leaving only boundary terms, and the bulk fields can be integrated out resulting inexp (cid:20) (cid:90) d d xdz √ g (cid:90) d d x (cid:48) dz (cid:48) (cid:112) g (cid:48) G ( x − x (cid:48) ; z, z (cid:48) ) δδθ ( x, z ) δδθ ( x (cid:48) , z (cid:48) ) (cid:21) × exp (cid:20) − (cid:90) z = (cid:15) d d y(cid:15) − d +1 θ∂ z θ (cid:21) . (4.16)We do not offer a rigorous proof that this generates the necessary terms, but point out that for theAdS/CFT dictionary to hold in the absence of deformations and for the counter terms currentlyused in the literature to be correct, Eq.(4.16) must evaluate to exp (cid:20)(cid:90) d d x ( ναβ + S ∂ [ α ]) (cid:21) . (4.17) The S ∂ term should be expected from the AdS/CFT story; the ναβ term is necessary to counter the − ναβ term onewould obtain by evaluating the bulk action on-shell and integrating by parts. Without canceling it out, the boundarycorrelators would not be evaluated as expected in the dual CFT. nserting this into Eq.(4.15) finally yields Z AdS [ J ] = exp (cid:20)(cid:90) d d x (cid:90) dz J φ cl (cid:21) × (cid:90) D α D β exp (cid:20)(cid:90) d d x (cid:0) − ν(cid:15) ν β + W [ α + (cid:15) ν β + (cid:15) − ∆ − φ cl ] (cid:1) + S ∂ [ α ] (cid:21) . (4.18)Evidently, β is an auxiliary field. Integrating it out sets, to leading order in (cid:15) , β = 12 ν W (cid:48) [ α ] , (4.19)thus recovering the usual boundary conditions.Meanwhile, the functional integral over α and the written dependence of S ∂ on α are formalitiesthat simply instruct us to evaluate α as O (up to normalization factors) given the appropriateboundary CFT.It is immediately apparent from Eq.(4.18) thatlim (cid:15) → Z AdS [ φ b ] = Z CF T [ φ b ] . (4.20) With Eq.(4.18) establishing an appropriate dictionary for multi-trace deformations, we may computebulk correlators and compare them to expectations from Witten diagrams.The expression of the bulk partition function in terms of a boundary partition function throughthe construction of the bulk fields from boundary fields in the previous section encourages the literalinterpretation of the bulk theory as a theory of quantum boundary conditions. This leads us toconsider the use of smearing functions to compute bulk correlators from boundary correlators. Wereview the use of smearing functions, and go on to develop an alternative, but related, formalism.
It immediately follows from Eq.(4.18) that bulk correlation functions take the form (cid:104) φ ( x , z ) φ ( x , z ) . . . φ ( x n − , z n − ) φ ( x n , z n ) (cid:105) = G ( x − x ; z , z ) . . . G ( x n − x n − ; z n , z n − ) + { perm } + n (cid:89) i =1 (cid:20)(cid:90) d d y i K ( x i − y i ; z i ) (cid:21) Γ n ( y , y , . . . , y n ) , (5.1)where the vertex function, Γ n , arises from derivatives of W . This is in agreement with the form thatfollows from Witten diagrams.Computed via diagrams, the two-point function, shown in Fig. 1, for instance, is given inmomentum space by (cid:104) φ ( − p, z ) φ ( p, z ) (cid:105) = G ( p ; z , z ) + K ( p, z ) Σ( p )1 − g ( p )Σ( p ) K ( p, z ) , (5.2)where Σ( p ) is the usual sum of 1PI diagrams at the boundary. Using Eq.(4.18), we find, schematically,Γ = (cid:104) W (cid:48) [ α ] W (cid:48) [ α ] + W (cid:48)(cid:48) [ α ] (cid:105) . (5.3) dentifying this with Eq.(5.2) requiresΣ1 − g Σ = (cid:104) W (cid:48) [ α ] W (cid:48) [ α ] + W (cid:48)(cid:48) [ α ] (cid:105) . (5.4) Figure 1:
Witten diagram for the bulk two-point function. The sum over 1PI parts at the boundary is absorbedinto the vertex.For a double-trace deformation, W = λα , we find (cid:104) W (cid:48) [ α ] W (cid:48) [ α ] + W (cid:48)(cid:48) [ α ] (cid:105) = λ − g ( p ) λ , (5.5)where the boundary correlators are computed according to the usual CFT rules, including summingover all double trace insertions. This predicts Σ = λ , as would be expected diagrammatically fromadding a mass term.From the cubic deformation W [ α ] = λα , we find (cid:104) W (cid:48) [ α ]( y ) W (cid:48) [ α ]( y ) = (cid:18) λ (cid:19) (cid:104) α ( y ) α ( y ) (cid:105) . (5.6)The boundary correlator, which is represented diagrammatically as the bracketed factor in Fig. 2,evaluates to (cid:18) λ (cid:19) (cid:104) α ( − p ) α ( p ) (cid:105) = Σ( p )1 − g ( p )Σ( p ) , (5.7)with the usual cubic 1PI diagram for Σ,Σ( p ) ∝ λ (cid:90) ¯ d d l l − ν ( p − l ) − ν + . . . , (5.8)manifestly agreeing with the Witten diagram. Figure 2:
A diagrammatic representation of the vertex function (cid:104) α α (cid:105) . The 1PI diagrams in brackets are indeedamputated. A factor of 2 appears at each vertex due to symmetry. he bulk three-point function arising from the cubic deformation is similarly easy to assess usingEq.(4.18): (cid:104) φ ( x , z ) φ ( x , z ) φ ( x , z ) (cid:105) = (cid:89) i =1 (cid:20)(cid:90) d d y i K ( x i − y i ; z ) (cid:21) × (cid:34) λ (cid:90) d d y δ d ( y − y ) δ d ( y − y ) δ d ( y − y )+ (cid:18) λ (cid:19) (cid:104) α ( y ) α ( y ) α ( y ) (cid:105) (cid:35) . (5.9)As should be expected, loop effects enter through the boundary correlator (cid:104) α ( y ) α ( y ) α ( y ) (cid:105) .The agreement between bulk correlators computed via diagrams and those computed usingEq.(4.18) confirms Eq.(4.18) as the appropriate AdS/CFT partition function to compute bulk cor-relators with boundary deformations.To this end, given the construction of the bulk φ in the previous section as a lift of the boundaryfields, we should equivalently be able to compute bulk correlators by using Eq.(3.30) and computingthe resulting correlators of α and β using the boundary path integral in Eq.(4.18). This is reminiscentof the effort to construct bulk observables from boundary operators using smearing functions. Beforedeveloping our lift formalism, it is worthwhile to first very briefly review the program and status ofsmearing functions. The goal of the smearing program is to construct bulk operators from their CFT duals through a linear integral operation: φ ( B ) = (cid:90) dbK ( B ; b ) O ( b ) , (5.10)where the integral kernel K ( B ; b ) is the smearing function that integrates over a boundary coordinate b to generate a field at bulk coordinate B .Without interactions, K ( B ; b ) was found in global coordinates in [33] through both a Greenfunction approach and mode function expansion. A review of the mode expansion approach is givein [37] and is sketched here.A free bulk field may be generically expanded as φ ( B ) = (cid:88) n f n ( B ) a n + h.c. (5.11)where f n denotes the wave function (eigenfunction) with quantum numbers (eigenvalues) n satisfyingthe classical bulk equations of motion, and a n ( a † n ) is the associated annihilation (creation) operator.With an appropriate normalization, { f n } forms an orthonormal set and a n consequently satisfies theappropriate algebra, [ a n , a † m ] = δ nm .Carrying φ ( B ) to the boundary, B → b , maps to O ( b ) in the usual way, implying O then inheritsa related mode expansion: O ( b ) = (cid:88) n f ∂,n ( b ) a n + h.c. (5.12) rovided an appropriate foliation of AdS, f n can be defined to be orthonormal along radial sliceson AdS and thus remain orthonormal at the boundary, implying (cid:104) f ∂,n | f † ∂,m (cid:105) = δ nm . With this,Eq.(5.11) becomes φ ( B ) = (cid:88) n f n ( B ) (cid:104) f † ∂,n O(cid:105) + h.c. (5.13)From this, the schematic form of the smearing function can be immediately extracted: K ( B ; b ) = (cid:88) n f n ( B ) f † ∂,n ( b ) + h.c. (5.14)From Eq.(5.14), it immediately follows that K ( B ; b ) satisfies the classical equations of motion in thebulk.The existence of K ( B ; b ) is predicated on the convergence and support of Eq.(5.14) for nontrivial B . For certain backgrounds, such as in the presence of a black hole, the sum is non-convergent or lackssupport [37], and certain constructions result in a non-causal map (lim B → b K ( B ; b (cid:48) ) (cid:54) = δ ( b − b (cid:48) )) [38].Nonetheless, smearing provides a powerful means of probing conformal theories dual to free AdStheories. We aim to develop an alternative construction of bulk fields in the same spirit as smearing. Instead of seeking a linear operation that maps O to φ , we employ Eq.(3.30) with a caveat onthe form of β to be lifted that will be derived here. To develop this formalism, consider first theundeformed bulk-boundary two point function: (cid:15) − ∆ − (cid:104) φ ( y, (cid:15) ) φ ( x, z ) (cid:105) = (cid:90) d d x (cid:48) (cid:2) L α ( x, x (cid:48) ; z ) (cid:104) α ( y ) α ( x (cid:48) ) (cid:105) + 2 νL β ( x, x (cid:48) ; z ) (cid:104) α ( y ) β ( x (cid:48) ) (cid:105) (cid:3) . (5.15)The first boundary correlator is (cid:104) α ( y ) α ( x (cid:48) ) (cid:105) = g ( y − x (cid:48) ) (the boundary propagator); the secondcorrelator must, perhaps surprisingly, evaluate to a local term, (cid:104) α ( y ) β ( x (cid:48) ) (cid:105) = ν δ d ( y − x (cid:48) ), to obtainthe correct bulk-boundary propagator. However, integrating β out in the absence of a deformationin Eq.(4.18) sets β = 0. This suggests that, at the level of operators, we must make the substitution β → β + β (5.16)in Eq.(3.30). The action of this undeformed β ( x ) on f [ α ] can be viewed as the functional derivativewhen evaluating correlators: β ( x ) = 12 ν δδα ( x ) . (5.17)Next, we evaluate the undeformed bulk-bulk two-point function using the same technique to find (cid:104) φ ( − p, z ) φ ( p, z (cid:48) ) (cid:105) = − ( zz (cid:48) ) d/ (cid:2) I − ν ( pz ) K ν ( pz (cid:48) ) + I − ν ( pz (cid:48) ) K ν ( pz ) + I − ν ( pz ) I − ν ( pz (cid:48) ) (cid:3) + (2 ν ) L β . (cid:104) ( β + β )( β + β ) (cid:105) .L β , (5.18)where the ‘ . ’ binary operator denotes the integration over the common boundary coordinates of theadjacent objects. Since there are no interactions, the second line vanishes. The first of the threeremaining terms is the two-point function for z (cid:48) > z , the second is for z > z (cid:48) , and the third isfor when z = z (cid:48) . We have encountered a problem with treating the bulk as the lift of a CFT: theboundary has no knowledge of the relative z − positions of our correlators in the bulk. This issuecan be fixed by inserting a z − ordering operator, Z , into correlators: (cid:104) φφ (cid:105) → (cid:104) Zφφ (cid:105) . The operatorannihilates β for the field φ with the smaller z in Wick-contracted φφ pairs. It should be noted hat the operator does only affects the non-interacting Witten diagrams as the diagrams containsboundary interactions are tautologically ordered.With boundary deformations turned on, β takes its usual form: β = 12 ν W (cid:48) [ α ] . (5.19)With this feature, we find (cid:104) Zφ ( x, z ) φ ( x (cid:48) , z (cid:48) ) (cid:105) = G ( x − x (cid:48) ; z, z (cid:48) ) + L α . (cid:104) αα − α α (cid:105) .L α + L α . (cid:104) αW (cid:48) [ α ] (cid:105) .L β + L β . (cid:104) W (cid:48) [ α ] α (cid:105) .L α + L β . (cid:104) W (cid:48) [ α ]( y ) W (cid:48) [ α ]( y (cid:48) ) + W (cid:48)(cid:48) [ α ] δ d ( y − y (cid:48) ) (cid:105) .L β , (5.20)where α is the undeformed α . It follows from pulling Eq.(5.2) to the boundary, identifying thenear-boundary bulk-boundary two-point function with (cid:15) − ∆ − (cid:104) α ( x ) φ ( x (cid:48) , (cid:15) ) (cid:105) ≈(cid:104) α ( x )( α ( x (cid:48) ) + β ( x (cid:48) ) (cid:15) ν ) (cid:105) , (5.21)and demanding consistency among already confirmed results that, in momentum space, (cid:104) αW (cid:48) [ α ] (cid:105) = g Σ1 − g Σ . (5.22)With this identification, Eq.(5.20) reproduces Eq.(5.2): (cid:104) Zφ ( x, z ) φ ( x (cid:48) , z (cid:48) ) (cid:105) = G ( x − x (cid:48) ; z, z (cid:48) ) + K ( x − y ; z ) . (cid:104) W (cid:48) [ α ]( y ) W (cid:48) [ α ]( y (cid:48) ) + W (cid:48)(cid:48) [ α ] δ d ( y − y (cid:48) ) (cid:105) . K ( y (cid:48) − x (cid:48) ; z (cid:48) ) . (5.23)All rules for computing bulk correlators from boundary data are now in place, completing theformalism.It is worthwhile noting the subtle distinctions between lifting and smearing. While the liftand smear kernels both satisfy the linear classical bulk equations of motion in the absence of bulkinteractions, lifting generally provides a nonlinear map from CFT operators to bulk fields in thepresence of boundary interactions as a consequence of the boundary conditions and an affine map in the absence of deformations. The mapping also provides a means of reproducing the resultsof Witten diagrams at the level of correlators without the need to adjust a tower of coefficientsto cancel noncausal effects in the presence of (boundary) interactions [35]. The lift kernel alsomanifestly approaches a delta function when taken to the boundary, ensuring bulk fields constructedfrom boundary operators map correctly when taken to the boundary.It is also worthwhile to recapitulate the results of this and the last few sections: • The computation of bulk correlators in the presence of multi-trace boundary deformationsshould be carried out with Eq.(4.18) as the generating partition function. • Pulling bulk correlators to the boundary in the usual way returns the same results as computingCFT correlators with conformal deformations. That is to say, Z CF T [ φ b ] = lim (cid:15) → Z AdS [ φ b , (cid:15) ]. • Bulk observables can be constructed from CFT observables via the lift provided by Eq.(3.30).The formalism makes the identification β = ν (cid:2) W (cid:48) [ α ] + δδα (cid:3) when computing correlators, andthe operator Z was introduced to order the lift operation by z -coordinate. • There is a strong connection between the β term in the bulk and the 1PI diagram whencomputing correlators. In particular, the results Eq.(5.4) and Eq.(5.22) are interesting anduseful. A linear map is one of the form y = mx , while an affine map takes the form y = mx + b . Dilatation spectrum and RG flow
We now wish to use the results of the lift formalism to find a generic form for the conformal dimensionof a dual operator O as a function of energy scale in the presence of multi-trace deformations. Theresults are the first steps to the multi-trace generalization of [47].In the absence of deformations, the dilatation spectrum is dual to the bulk mass spectrum via themapping m = ∆(∆ − d ). The inclusion of conformal deformations triggers RG flow such that theIR spectrum can often be extracted from the UV spectrum. For instance, in the presence of double-trace deformations, operators with dimension ∆ − (= d − ν ≤ d ) in the UV flow to a ∆ + (= d + ν )fixed point in the IR [45]. The conformal dominance program exploits the UV conformal basis toconstruct IR mass states for certain deformations [24, 25]. In what follows we restrict the scalingdimension to ∆ = ∆ − . Many results can be immediately extended to ∆ = ∆ + , however we arechiefly concerned with the RG flow of the CFTs between potential fixed points.Dilatation eigenstates in the undeformed CFT are created by placing an operator at the origin: | (cid:105) = O (0) | Ω (cid:105) . It follows from the identity [ D, O ( x )] = ( x · ∂ + ∆ − ) O ( x ) that (cid:104) x | D | (cid:105) = ∆ − (cid:104) x | (cid:105) = − ( x · ∂ + ∆ − ) (cid:104) x | (cid:105) . (6.1)Evidently, the CFT two-point functions are eigenfunctions of the differential representation of thedilatation operator.When lifting to the bulk, the scaling dimension is replaced by differentiation with respect to z :( x · ∂ + ∆ − ) O → ( x · ∂ + z∂ z ) φ . With this replacement, the bulk-boundary propagator is found tobe an eigenfunction of the dilatation operator: (cid:104) φ ( x, z ) | D | (cid:105) = − ( x · ∂ + z∂ z ) K ( x ; z ) = ∆ − K ( x ; z ) . (6.2)Even more readily, and perhaps crucially, the classical field φ is an eigenfunction of z∂ z as z → − .When deformations are introduced, the story becomes more complex. In momentum space, theboundary two-point function remains an eigenfunction of the dilatation operator, but the eigenvaluegains a momentum-dependence: (cid:18) p · ∂∂p + ∆ − (cid:19) (cid:104) p | (cid:105) = (cid:20) ∆ − + g ( p )1 − g ( p )Σ( p ) (cid:18) p · ∂∂p − ν (cid:19) Σ( p ) (cid:21) (cid:104) p | (cid:105) . (6.3)As expected for a double trace deformation (Σ = const. ) in the UV, we find g Σ (cid:28)
1, which leads to∆ = ∆ − ; in the IR, we find g Σ (cid:29)
1, which leads to ∆ = ∆ + .The bulk-boundary two point function also remains an eigenfunction of the dilatation operatorwith the same eigenvalue as its boundary counterpart: (cid:18) p · ∂∂p + z∂ z (cid:19) − g ( p )Σ( p ) K ( p ; z ) = (cid:20) ∆ − + g ( p )1 − g ( p )Σ( p ) (cid:18) p · ∂∂p − ν (cid:19) Σ( p ) (cid:21) − g ( p )Σ( p ) K ( p ; z ) . (6.4)The relation between the bulk operator z∂ z and the conformal dimension becomes more obscure inthe presence of deformations. We could attempt to demand the field φ remain the eigenfunction ofthe operator at the boundary designated by the z = (cid:15) cutoff and identify the eigenvalue with ∆.Carrying through with the procedure with a double-trace deformation and keeping next to leadingorder terms in (cid:15) for φ results in ∆ = φ − (cid:15)∂ (cid:15) φ = ∆ − + λ(cid:15) ν λ ν (cid:15) ν . (6.5) f we interpret the UV brane on which the bulk theory terminates as the inverse of the renormalizationscale, (cid:15) ∝ µ − , the RG flow in Eq.(6.5) matches exactly Eq.(6.3) for double trace deformations. Theprocedure as presented is serendipitous for double-trace deformations since β is classically linear in α , allowing the field to completely divide out; this does not occur for more complicated deformations.However, the promising connection between the z -direction in the bulk and RG flow at the boundarybegs for the procedure to be generalized.When transitioning to the quantum formulation, we should expect to deal with correlators ofthe fields instead of the fields themselves. Classically, we may multiply and divide Eq.(6.5) by α , sotransitioning suggests we compute ∆ in momentum space by pulling the bulk-boundary propagatorto the UV cutoff and evaluating the momentum at this UV scale:∆ = (cid:15) → (cid:15)∂ (cid:15) [ (cid:104) φα (cid:105) ( p, (cid:15) )][ (cid:104) φα (cid:105) ( p, (cid:15) )] (cid:12)(cid:12)(cid:12)(cid:12) | p | = (cid:15) − (6.6)for (cid:15) →
0. This procedure actually trivially follows from Eq.(6.4) by simply demanding the mo-mentum be evaluated at the UV cutoff. Physically, Eq.(6.6) says that the scaling dimension is ameasure of how the bulk-boundary propagator changes as the location of the UV brane is shiftedwhile keeping the energy near the renormalization scale.Using Eqs.(3.30)&(4.19) to expand the bulk fields in Eq.(6.6) in terms of α , inserting Eq.(5.22)into the result, and setting (cid:15) → µ − yields∆ =∆ − + µ − ν ν [2 ν − µ∂ µ ] Σ( µ )1 + µ − ν ν Σ( µ ) . (6.7)Once again, the correct RG flow has been recovered. This indicates that the running of a deformedCFT from the UV to the IR may be studied in AdS by ending the theory on a brane at z = (cid:15) andcomputing appropriate quantities by setting the renormalization scale µ → (cid:15) − . This procedure isin independent agreement with the holographic RG program of interpreting the classical evolutionof fields in the radial direction in AdS as RG flow at the boundary. The dilatation spectrum canbe explicitly computed in the bulk using Eq.(6.6), and we wish to emphasize the necessity of the β piece defined via the lift formalism in computing the bulk correlators. It is not difficult to imaginea similar analytical procedure should hold for the mass-squared spectrum, although the details arenot immediately apparent. We have constructed rules utilizing modified boundary conditions in AdS to compute bulk correlatorsthrough the use of an appropriate AdS/CFT partition function and via a lift formalism for theoriessubject to a conformal deformation. The partition function explicitly relates the AdS theory tothe CFT theory and establishes the AdS side in the absence of bulk interactions as a theory thatevolves classically in the z - (radial-) direction with boundary conditions subject to quantum effects.The lift formalism, as an alternative to smearing, provides the inverse of the usual boundary-scalingdictionary that relates the bulk and boundary operators ( O = lim z → z − ∆ φ ( z )). Utilizing the resultsof the lift formalism, a formula to compute the conformal dimension of CFT operators as a functionof energy scale using AdS correlators was derived.We have not considered the obstructions that may hinder obtaining an appropriate smearingkernel. While we expect the lift formalism to fail or require modification when bulk interactions areturned on as the bulk would no longer evolve classically in the z-direction, it is our hope that itremains valid for asymptotically AdS spaces so that semiclassical gravity may be studied.For now, only the running of the scaling dimension with the renormalization scale was consideredsince its fundamental role in the AdS/CFT correspondence makes it easy to handle. A similar trategy of finding an appropriate differential operator in the bulk and writing down a ratio ofcorrelators may likely be employed to compute the dependence of mass-squared elements on therenormalization scale to approach conformal dominance from a bulk perspective. It would be ofinterest to explore this approach in the context of the c-theorem. The author would like to thank Christopher Brust, Nikhil Anand, and Jared Kaplan for usefuldiscussions. The author was partially supported by NSF grant PHY-1316665.
A Classical fields with multi-trace deformations
We can extend the formalism of § L b = (cid:15) d − ∆+1 φ b φ + 1 n λ(cid:15) d − n ∆+1 φ n = ⇒ δB [ φ ] = − (cid:15) d − ∆ φ b − λ(cid:15) d − ∆ α n − . (A.1)The usual procedure results in b ( x ) = 12 ν (cid:2) φ b ( x ) + λα n − ( x ) (cid:3) , (A.2)and we face solving α ( x ) = (cid:90) d d x (cid:48) (cid:90) ¯ d d p Γ( ν )Γ(1 − ν ) (cid:16) p (cid:17) − ν (cid:2) φ b ( x (cid:48) ) + λα n − ( x (cid:48) ) (cid:3) e ip · ( x − x (cid:48) ) (A.3)for α in terms of φ b . Determining α exactly seems like a hopeless endeavor, but we can still computethe contribution of the deformation to the bulk-boundary propagator. Given the nonlinearity weshould expect these deformations to generate interaction terms in higher-point correlation functions.Classically, the interacting piece of N -point functions can be computed as contributions to the N thharmonic of the source. This follows explicitly from (cid:104) α ( x ) . . . α ( x N ) (cid:105) ∝ δδφ b ( x ) . . . δδφ b ( x N ) e (cid:82) d d y α ( y ) φ b ( y ) (cid:12)(cid:12)(cid:12)(cid:12) φ b =0 , (A.4)the purely interacting piece of which is δδφ b ( x ) . . . δδφ b ( x N ) α ( x ) + · · · + δδφ b ( x ) . . . δδφ b ( x N − ) α ( x N ) (cid:12)(cid:12)(cid:12)(cid:12) φ b =0 . (A.5)We can build an expansion of Eq.(A.3) in nested functionals of φ b , which poises us perfectly to com-pute N -point functions as sums of products of integral kernels in accordance with Eqs.(A.4)&(A.5).The bulk-boundary propagator would then simply be the first functional derivative of φ with respectto φ b . Since α [ φ b = 0] = 0, there are no classical contributions to the bulk-boundary propagator for n > N -point function in the bulk should involve an N -point function atthe boundary, and, from Eq.(A.3), we expect a non-vanishing tree-level correlator only for N = n .There is more in the details, but at the level of the bulk-boundary propagator, we can stop here. n the absence of boundary deformations, the wave functions are well known [48, 49]: f ( (cid:126)p, m ; x, z ) = (cid:115) im p m z d J − ν ( mz ) e ip m · x , (A.6)where p m = i (cid:112) (cid:126)p + m . As was the case with the bulk-boundary propagator, we can use thisknowledge to compute the modified wave functions in the presence of boundary deformations.The procedure works as before with φ → f , except we exclude boundary source terms from ourLagrangian and boundary conditions, L b = 1 n λ(cid:15) d − n ∆+1 φ n = ⇒ δB [ φ ] = − λ(cid:15) d − ∆ α n − , (A.7)and we seek homogeneous solutions. Now, Eq.(A.3) becomes α ( x ) = (cid:115) im p m − ν ) (cid:16) m (cid:17) − ν e ip m · x + (cid:90) d d x (cid:48) (cid:90) ¯ d d p Γ( ν )Γ(1 − ν ) (cid:16) p (cid:17) − ν λ α n − ( x (cid:48) ) e ip · ( x − x (cid:48) ) . (A.8)For n = 2, this is straight forward: α ( x ) = 11 − Γ( ν )Γ(1 − ν ) (cid:0) p m (cid:1) − ν λ (cid:115) im p m − ν ) (cid:16) m (cid:17) − ν e ip m · x . (A.9)And so f ( x, z ) = (cid:115) im p m z d (cid:34) J − ν ( mz )+ λ − Γ( ν )Γ(1 − ν ) (cid:0) p m (cid:1) − ν λ − ν ) (cid:16) m (cid:17) − ν (cid:16) p m (cid:17) − ν K ν ( p m z ) (cid:35) e ip m · x . (A.10)It is worthwhile to point out that Eq.(A.8) requires α to be classical at the boundary, demon-strating that the classical methods sufficient for double-trace deformations are inapplicable for moregeneral deformations. References [1] L. Susskind and E. Witten, “The Holographic bound in anti-de Sitter space,” arXiv:hep-th/9805114 [hep-th] .[2] T. Banks, M. R. Douglas, G. T. Horowitz, and E. J. Martinec, “AdS dynamics from conformalfield theory,” arXiv:hep-th/9808016 [hep-th] .[3] E. Witten, “Anti-de Sitter space and holography,” Adv.Theor.Math.Phys. (1998) 253–291, arXiv:hep-th/9802150 [hep-th] .[4] S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from noncriticalstring theory,” Phys.Lett. B428 (1998) 105–114, arXiv:hep-th/9802109 [hep-th] .[5] D. Harlow and D. Stanford, “Operator Dictionaries and Wave Functions in AdS/CFT anddS/CFT,” arXiv:1104.2621 [hep-th] .[6] S. B. Giddings, “The Boundary S matrix and the AdS to CFT dictionary,” Phys.Rev.Lett. (1999) 2707–2710, arXiv:hep-th/9903048 [hep-th] .
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