Witten diagrams revisited: the AdS geometry of conformal blocks

Journal of High Energy Physics, Jan 2016

We develop a new method for decomposing Witten diagrams into conformal blocks. The steps involved are elementary, requiring no explicit integration, and operate directly in position space. Central to this construction is an appealingly simple answer to the question: what object in AdS computes a conformal block? The answer is a “geodesic Witten diagram”, which is essentially an ordinary exchange Witten diagram, except that the cubic vertices are not integrated over all of AdS, but only over bulk geodesics connecting the boundary operators. In particular, we consider the case of four-point functions of scalar operators, and show how to easily reproduce existing results for the relevant conformal blocks in arbitrary dimension.

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Witten diagrams revisited: the AdS geometry of conformal blocks

JHE Witten diagrams revisited: the AdS geometry of Eliot Hijano 0 1 3 Per Kraus 0 1 3 Eric Perlmutter 0 1 2 River Snively 0 1 3 0 Princeton , NJ 08544 , U.S.A 1 Los Angeles , CA 90095 , U.S.A 2 Department of Physics, Princeton University 3 Department of Physics and Astronomy, University of California We develop a new method for decomposing Witten diagrams into conformal blocks. The steps involved are elementary, requiring no explicit integration, and operate directly in position space. Central to this construction is an appealingly simple answer to the question: what object in AdS computes a conformal block? The answer is a \geodesic Witten diagram", which is essentially an ordinary exchange Witten diagram, except that the cubic vertices are not integrated over all of AdS, but only over bulk geodesics connecting the boundary operators. In particular, we consider the case of four-point functions of scalar operators, and show how to easily reproduce existing results for the relevant conformal blocks in arbitrary dimension. AdS-CFT Correspondence; Conformal and W Symmetry - HJEP01(26)4 1 Introduction 2 Conformal blocks, holographic CFTs and Witten diagrams CFT four-point functions and holography A Witten diagrams primer 2.2.1 2.2.2 Mellin space Looking ahead 3 The holographic dual of a scalar conformal block 3.3 Comments 3.2.1 3.2.2 3.2.3 3.3.1 3.3.2 3.3.3 Proof by direct computation Proof by conformal Casimir equation The Casimir equation Embedding space Geodesic Witten diagrams satisfy the Casimir equation Geodesic versus ordinary Witten diagrams Simpli cation of propagators and blocks Relation to Mellin space 2.1 2.2 2.3 2.4 3.1 3.2 4.1 4.2 4.3 4.4 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Known results Geodesic Witten diagrams with spin-` exchange: generalities Evaluation of geodesic Witten diagram: spin-1 Evaluation of geodesic Witten diagram: spin-2 General `: proof via conformal Casimir equation Comparison to double integral expression of Ferrara et al. Decomposition of spin-1 Witten diagram into conformal blocks 5.7.1 5.7.2 AAA AA and A A { i { Logarithmic singularities and anomalous dimensions What has been computed? HJEP01(26)4 The conformal block decomposition of scalar Witten diagrams An AdS propagator identity Four-point contact diagram Four-point exchange diagram Further analysis 4.4.1 4.4.2 OPE factorization Recovering logarithmic singularities 4.5 Taking stock 5 Spinning exchanges and conformal blocks 5.7.3 5.7.4 Summary 6 Discussion and future work 1 Introduction The conformal block decomposition of correlation functions in conformal eld theory is a HJEP01(26)4 powerful way of disentangling the universal information dictated by conformal symmetry from the \dynamical" information that depends on the particular theory under study; see e.g. [1{7]. The latter is expressed as a list of primary operators and the OPE coe cients amongst them. The use of conformal blocks in the study of CFT correlation functions therefore eliminates redundancy, as heavily utilized, for instance, in recent progress made in the conformal bootstrap program, e.g. [8, 9]. In the AdS/CFT correspondence [10{12], the role of conformal blocks has been somewhat neglected. The extraction of spectral and OPE data of the dual CFT from a holographic correlation function, as computed by Witten diagrams [12], was addressed early on in the development of the subject [13{20], and has been re ned in recent years through the introduction of Mellin space technology [21{27]. In examining this body of work, however, one sees that a systematic method of decomposing Witten diagrams into conformal blocks is missing. A rather natural question appears to have gone unanswered: namely, what object in AdS computes a conformal block? A geometric bulk description of a conformal block would greatly aid in the comparison of correlators between AdS and CFT, and presumably allow for a more e cient implementation of the dual conformal block decomposition, as it would remove the necessity of actually computing the full Witten diagram explicitly. The absence of such a simpler method would indicate a surprising failure of our understanding of AdS/CFT: after all, conformal blocks are determined by conformal symmetry, the matching of which is literally the most basic element in the holographic dictionary. In this paper we present an appealingly simple answer to the above question, and demonstrate its utility via streamlined computations of Witten diagrams. More precisely, we will answer this question in the case of four-point correlation functions of scalar operators, but we expect a similar story to hold in general. The answer is that conformal blocks are computed by \geodesic Witten diagrams". The main feature of a geodesic Witten diagram that distinguishes it from a standard exchange Witten diagram is that in the former, the bulk vertices are not integrated over all of AdS, but only over geodesics connecting points on the boundary hosting the external operators. This representation of conformal blocks in terms of geodesic Witten diagrams is valid in all spacetime dimensions, and holds for all conformal blocks that arise in four-point functions of scalar operators belonging to arbitrary CFTs (and probably more generally). To be explicit, consider four scalar operators Oi with respective conformal dimensions i. The conformal blocks that appear in their correlators correspond to the exchange of { 1 { Z d 12 Z 34 This computes the conformal partial wave for the exchange of a CFTd primary operator of spin ` and dimension . primaries carrying dimension and transforming as symmetric traceless tensors of rank `; we refer to these as spin-` operators. Up to normalization, the conformal partial wave1 in CFTd is given by the following object in AdSd+1: Gbb y( ); y( 0); ; ` (1.1) ij denotes the bulk geodesic connecting boundary points xi and xj , with and 0 denoting the corresponding proper length parameters. Gb@ (y; x) are standard scalar bulk-toboundary propagators connecting a bulk point y to a boundary point x. Gbb y( ); y( 0); ; ` is the bulk-to-bulk propagator for a spin-` eld, whose mass squared in AdS units is m2 = ( d) `, pulled back to the geodesics. The above computes the s-channel partial wave, corresponding to using the OPE on the pairs of operators O1O2 and O3O4. As noted earlier, the expression (1.1) looks essentially like an exchange Witten diagram composed of two cubic vertices, except that the vertices are only integrated over geodesics. See gure 1. Note that although geodesics sometimes appear as an approximation used in the case of high dimension operators, here there is no approximation: the geodesic Witten diagram computes the exact conformal block for any operator dimension. As we will show, geodesic Witten diagrams arise very naturally upon dismantling a full Witten diagram into constituents, and this leads to an e cient implementation of the conformal block decomposition. Mellin space techniques also provide powerful methods, but it is useful to have an approach that can be carried out directly in position space, and that provides an explicit and intuitive picture for the individual conformal blocks. For the cases that we consider, the conformal blocks are already known, and so one of our tasks is to demonstrate that (1.1) reproduces these results. One route is by explicit computation. Here, the most direct comparison to existing results is to the original work of Ferrara, Gatto, Grillo, and Parisi [1{3], who provided integral representations for conformal blocks. In hindsight, these integral expressions can be recognized as geodesic Witten 1Conformal partial waves and conformal blocks are related by simple overall factors as we review below. { 2 { diagrams. Later work by Dolan and Osborn [4{6] provided closed-form expressions for some even-d blocks in terms of hypergeometric functions. Dolan and Osborn employed the very useful fact that conformal partial waves are eigenfunctions of the conformal Casimir operator. The most e cient way to prove that geodesic Witten diagrams compute conformal partial waves is to establish that they are the correct eigenfunctions. This turns out to be quite easy using embedding space techniques, as we will discuss. Having established that geodesic Witten diagrams compute conformal partial waves, we turn to showing how to decompose a Witten diagram into geodesic Witten diagrams. We do not attempt an exhaustive demonstration here, mostly focusing on tree-level contact and exchange diagrams with four external lines. The procedure turns out to be quite economical and elegant; in particular, we do not need to carry out the technically complicated step of integrating bulk vertices over AdS. Indeed, the method requires no integration at all, as all integrals are transmuted into the de nition of the conformal partial waves. The steps that are required are all elementary. We carry out this decomposition completely explicitly for scalar contact and exchange diagrams, verifying that we recover known results. These include certain hallmark features, such as the presence of logarithmic singularities due to anomalous dimensions of double-trace operators. We also treat the vector exchange diagram, again recovering the correct structure of CFT exchanges. Let us brie y mention how the analysis goes. The key step is to use a formula expressing the product of two bulk-to-boundary propagators sharing a common bulk point as a sum of bulk solutions sourced on a geodesic connecting the two boundary points. The elds appearing in the sum turn out to be dual to the double-trace operators appearing in the OPE of the corresponding external operators, and the coe cients in the sum are closely related to the OPE coe cients. See equation (4.1). With this result in hand, all that is needed are a few elementary properties of AdS propagators to arrive at the conformal block decomposition. This procedure reveals the generalized free eld nature of the dual CFT. The results presented here hopefully lay the foundation for further exploration of the use of geodesic Witten diagrams. We believe they will prove to be very useful, both conceptually and computationally, in AdS/CFT and in CFT more generally. The remainder of this paper is organized as follows. In section 2 we review relevant aspects of conformal blocks, Witten diagrams, and their relation. Geodesic Witten diagrams for scalar exchange are introduced in section 3, and we show by direct calculation and via the conformal Casimir equation that they compute conformal blocks. In section 4 we turn to the conformal block decomposition of Witten diagrams involving just scalar elds. We describe in detail how single and double trace operator exchanges arise in this framework. Section 5 is devoted to generalizing all of this to the case of spinning exchange processes. We conclude in section 6 with a discussion of some open problems and future prospects. The ideas developed in this paper originated by thinking about the bulk representation of Virasoro conformal blocks in AdS3/CFT2, based on recent results in this direction [28{ 33]. The extra feature associated with a bulk representation of Virasoro blocks is that the bulk metric is deformed in a nontrivial way; essentially, the geodesics backreact on the geometry. In this paper we focus on global conformal blocks (Virasoro blocks are of course special to CFT2), deferring the Virasoro case to a companion paper [34]. { 3 { u = x212x2324 ; after using conformal invariance to x three positions at 0; 1; 1. g(u; v) can be decomposed into conformal blocks, G ;`(u; v), as where O is a primary operator of dimension be written compactly as a sum of conformal partial waves, W ;`(xi): and spin `.2 Accordingly, the correlator can Let us rst establish some basic facts about four-point correlation functions in conformal eld theories, and their computation in AdSd+1/CFTd. Both subjects are immense, of course; the reader is referred to [35, 36] and references therein for foundational material. CFT four-point functions and holography We consider vacuum four-point functions of local scalar operators O(x) living in d Euclidean dimensions. Conformal invariance constrains these to take the form ; (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) hO1(x1)O2(x2)O3(x3)O4(x4)i = X C12O CO34W ;`(xi) O where as the insertion of a projector onto the conformal family of O, normalized by the OPE coe cients: 2In this paper we only consider scalar correlators, in which only symmetric, traceless tensor exchanges can appear. More generally, ` would stand for the full set of angular momenta under the d-dimensional little group. hO1(x1)O2(x2)O3(x3)O4(x4)i = xj . g(u; v) is a function of the two independent conformal cross-ratios, One can also de ne complex coordinates z; z, which obey 2 x13 v = where X n P ;` j P nOihP nOj and P nO is shorthand for all descendants of O made from n raising operators P . We will sometimes refer to conformal blocks and conformal partial waves interchangeably, with the understanding that they di er by the power law prefactor in (2.6). Conformal blocks admit double power series expansions in u and 1 v, in any spacetime dimension [4]; for ` = 0, for instance, G ;0(u; v) = u =2 X 1 m;n=0 d relations [6, 9]. Especially relevant for our purposes are integral representations of the conformal blocks [1{3]. For ` = 0, G ;0(u; v) = 2 1 34 u =2Z 1 d 0 where we have de ned a coe cient 2F1 34 The blocks can also be expressed as in nite sums over poles in associated with null states of SO(d; 2), in analogy with Zamolodchikov's recursion relations in d = 2 [37{39]; these provide excellent rational approximations to the blocks that are used in numerical work. Finally, as we revisit later, in even d the conformal blocks can be written in terms of hypergeometric functions. Conformal eld theories with weakly coupled AdS duals obey further necessary conditions on their spectra.3 In addition to having a large number of degrees of freedom, which we will label4 N 2, there must be a nite density of states below any xed energy as N ! 1; e.g. [43{46]. For theories with Einstein-like gravity duals, this set of parametrically light operators must consist entirely of primaries of spins ` 2 and their descendants. The \single-trace" operators populating the gap are generalized free elds: given any set of such primaries Oi, there necessarily exist \multi-trace" primaries comprised of conglomerations of these with some number of derivatives (distributed appropriately to make a primary). Altogether, the single-trace operators and their multi-trace composites comprise the full set of primary elds dual to non-black hole states in the bulk. In a four-point 3Finding a set of su cient conditions for a CFT to have a weakly coupled holographic dual remains an unsolved problem. More recent work has related holographic behavior to polynomial boundedness of Mellin amplitudes [40, 41], and to the onset of chaos in thermal quantum systems [42]. 4We are agnostic about the precise exponent: vector models and 6d CFTs are welcome here. More function of Oi, all multi-trace composites necessarily run in the intermediate channel at some order in 1=N . Focusing on the double-trace operators, these are schematically of the form These have spin-` and conformal dimensions (ij)(n; `) = i + j + 2n + ` + (ij)(n; `) ; where (ij)(n; `) is an anomalous dimension. The expansion of a correlator in the s-channel includes the double-trace terms5 hO1(x1)O2(x2)O3(x3)O4(x4)i X P (12)(m; `)W (12)(m;`);`(xi) + X P (34)(n; `)W (34)(n;`);`(xi) m;` n;` Following [26, 43], we have de ned a notation for squared OPE coe cients, P (ij)(n; `) C12OCO34 ; where O = [OiOj ]n;` : The 1=N expansion of the OPE data, 1 r=0 1 r=1 P (ij)(n; `) = X N 2rP (ij)(n; `) ; r (ij)(n; `) = X N 2r (ij)(n; `) ; r induces a 1=N expansion of the four-point function. Order-by-order in 1=N , the generalized free elds and their composites must furnish crossing-symmetric correlators. This is precisely the physical content captured by the loop expansion of Witten diagrams in AdS, to which we now turn. 2.2 A Witten diagrams primer See [36] for background. We work in Euclidean AdSd+1, with RAdS 1. In Poincare coordinates y = fu; xig, the metric is ds2 = du2 + dxidxi u2 : The ingredients for computing Witten diagrams are the set of bulk vertices, which are read o from a Lagrangian, and the AdS propagators for the bulk elds. A scalar eld of mass m2 = ( d) in AdSd+1 has bulk-to-bulk propagator Gbb(y; y0; ) = e (y;y0)2F1 5Unless otherwise noted, all sums over m; n and ` run from 0 to 1 henceforth. { 6 { (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) (y; y0) = log u2 + jx xij2 : Gbb(y; y0; ) is a normalizable solution of the AdS wave equation with a delta-function source,6 (r2 m2)Gbb(y; y0; ) = The bulk-to-boundary propagator is There are also exchange-type diagrams, which involve \virtual" elds propagating between points in the interior of AdS. The simplest tree-level Witten diagrams are shown in gure 2. We focus henceforth on tree-level four-point functions of scalar elds i dual to scalar CFT operators Oi. For a non-derivative interaction 1 2 3 4, and up to an overall quartic coupling that we set to one, the contact diagram equals A4 Contact(xi) = D 1 2 3 4 (xi) = (2.24) We will introduce higher spin propagators in due course. A holographic CFT n-point function, which we denote An, receives contributions from all possible n-point Witten diagrams. The loop-counting parameter is GN O(1=N 2), only tree-level diagrams contribute. The simplest such diagrams are contact 1=N 2. At diagrams, which integrate over a single n-point vertex. Every local vertex in the bulk Lagrangian gives rise to a contact diagram: schematically, L n where Gb@ (y; xi) are bulk-to-boundary propagators for elds with quantum numbers ( i; `i), and pi count derivatives. We abbreviate where (y; y0) is the geodesic distance between points y; y0. In Poincare AdS, D 1 2 3 4 (xi) is the D-function, which is de ned by the above integral. For generic i, this integral cannot be performed for arbitrary xi. There exists a bevy of identities relating various D 1 2 3 4 (xi) via permutations of the i, spatial derivatives, and/or shifts in the i [18, 47]. Derivative vertices, which appear in the axio-dilaton sector of 6We use this normalization for later convenience. Our propagator is 2 d=2 d 2 2 = ( ) times the common normalization found in, e.g., equation (6.12) of [36]. { 7 { contact diagram. On the right is an exchange diagram for a symmetric traceless spin-` tensor eld of dual conformal dimension . Here and throughout this work, orange dots denote vertices integrated over all of AdS. type IIB supergravity, for instance, de ne D-functions with shifted parameters. When i = 1 for all i, 2x213x224 2 d 2 d=2 D1111(xi) = 1 z z 2Li2(z) 2Li2(z) + log(zz) log : (2.25) 1 1 z z This actually de nes the D-bar function, D1111(z; z). For various sets of ing (2.25) with e cient use of D-function identities leads to polylogarithmic representations i 2 Z, combinof contact diagrams. The other class of tree-level diagrams consists of exchange diagrams. For external scalars, one can consider exchanges of symmetric, traceless tensor elds of arbitrary spin `. These are computed, roughly, as A4 Exch(xi) = Z Z y y0 Gbb(y; y0; ; `) (2.26) Gbb(y; y0; ; `) is shorthand for the bulk-to-bulk propagator for the spin-` eld of dimension , which is really a bitensor [Gbb(y; y0; )] 1::: `; 1::: ` . We have likewise suppressed all derivatives acting on the external scalar propagators, whose indices are contracted with those of [Gbb(y; y0; )] 1::: `; 1::: ` . Due to the double integral, brute force methods of simplifying exchange diagrams are quite challenging, even for ` = 0, without employing some form of asymptotic expansion. The key fact about these tree-level Witten diagrams relevant for a dual CFT interpretation is as follows. For contact diagrams (2.24), their decomposition into conformal blocks contains the in nite towers of double-trace operators in (2.14), and only these. This is true in any channel. For exchange diagrams (2.26), the s-channel decomposition includes a single-trace contribution from the operator dual to the exchanged bulk eld, in addition to in nite towers of double-trace exchanges (2.14). In the t- and u-channels, only double-trace exchanges are present. The precise set of double-trace operators that appears is determined by the spin associated to the bulk vertices. { 8 { Higher-loop Witten diagrams are formed similarly, although the degree of di culty increases rapidly with the loop order. No systematic method has been developed to compute these. Logarithmic singularities and anomalous dimensions When the external operator dimensions are non-generic, logarithms can appear in tree-level Witten diagrams [14, 15, 18]. These signify the presence of perturbatively small anomalous dimensions, of order 1=N 2, for intermediate states appearing in the CFT correlator. Let us review some basic facts about this. In general, if any operator of free dimension 0 develops an anomalous dimension , so 0 + , a small- expansion of its contribution to correlators yields an in nite series of logs: G 0+ ;`(u; v) In the holographic context, the double-trace composites [OiOj ]n;` have anomalous dimensions at O(1=N 2). Combining (2.13), (2.14) and (2.16) leads to double-trace contributions to holographic four-point functions of the form (2.27) (2.28) A4(xi) 1=N2 X m;` + X n;` 1 P (12)(m; `) + 1 P (34)(n; `) + 1 P (12)(m; `) 1(12)(m; `)@m 1 P (34)(n; `) 1(34)(n; `)@n W 1+ 2+2m+`;`(xi) W 3+ 4+2n+`;`(xi) where @mW 1+ 2+2m+`;` / log u and likewise for the (34) terms. These logarithmic singularities should therefore be visible in tree-level Witten diagrams. In top-down examples of AdS/CFT, the supergravity elds are dual to protected operators, so the (ij)(n; `) are the only (perturbative) anomalous dimensions that appear, and hence are responsible for all logs. For generic operator dimensions, the double-trace operators do not appear in both the O1O2 and O3O4 OPEs at O(N 0), so P (ij)(n; `) = 0. On the other hand, when the 0 i are related by the integrality condition 1 + 2 3 4 2 2Z, one has P (ij)(n; `) 6= 0 [14]. 0 2.2.2 What has been computed? In a foundational series of papers [13, 15{18, 48{51], methods of direct computation were developed for scalar four-point functions, in particular for scalar, vector and graviton exchanges. Much of the focus was on the axio-dilaton sector of type IIB supergravity on AdS5 S5 in the context of duality with N = 4 super-Yang Mills (SYM), but the methods were gradually generalized to arbitrary operator and spacetime dimensions. This e ort largely culminated in [17, 51] and [18]. [17] collected the results from all channels contributing to axio-dilaton correlators in N = 4 SYM, yielding the full correlator at O(1=N 2). In [51], a more e cient method of computation was developed for exchange diagrams. It was shown that scalar, vector and graviton exchange diagrams can generically { 9 { be written as in nite sums over contact diagrams for external elds of variable dimensions. These truncate to nite sums if certain relations among the dimensions are obeyed.7 These calculations were translated in [18] into CFT data, where it was established that logarithmic singularities appear precisely at the order determined by the analysis of the previous subsection. This laid the foundation for the modern perspective on generalized free elds. Further analysis of implications of four-point Witten diagrammatics for holographic CFTs (e.g. crossing symmetry, non-renormalization), and for N = 4 SYM in particular, was performed in [4, 47, 52{62]. A momentum space-based approach can be found in [63, 64]. More recent work has computed Witten diagrams for higher spin exchanges [65, 66]. These works develop the split representation of massive spin-` symmetric traceless tensor elds, for arbitrary integer `. There is a considerable jump in technical di culty, but the results are all consistent with AdS/CFT. 2.3 Mellin space An elegant alternative approach to computing correlators, especially holographic ones, has been developed in Mellin space [21, 67]. The analytic structure of Mellin amplitudes neatly encodes the CFT data and follows a close analogy with the momentum space representation of at space scattering amplitudes. We will not make further use of Mellin space in this paper, but it should be included in any discussion on Witten diagrams; we only brie y review its main properties with respect to holographic four-point functions, and further aspects and details may be found in e.g. [22{27, 68{70]. Given a four-point function as in (2.1), its Mellin representation may be de ned by the integral transform Z i1 i1 The integration runs parallel to the imaginary axis and to one side of all poles of the integrand. The Mellin amplitude is M (s; t). Assuming it formally exists, M (s; t) can be de ned for any correlator, holographic [21] or otherwise [71, 72]. M (s; t) is believed to be meromorphic in any compact CFT. Written as a sum over poles in t, each pole sits at a xed twist = `, capturing the exchange of twist- operators in the intermediate channel. Given rhe exchange of a primary O of twist O, its descendants of twist = O + 2m contribute a pole M (s; t) C12OCO34 t Q`;m(s) O 2m (2.29) (2.30) where m = 0; 1; 2; : : :. Q`;m(s) is a certain degree-` (Mack) polynomial that can be found in [ 25 ]. Note that an in nite number of descendants contributes at a given m. n-point Mellin amplitudes may be likewise de ned in terms of n(n 3)=2 parameters, and are known to factorize onto lower-point amplitudes [27]. 7For instance, an s-channel scalar exchange is written as a nite linear combination of D-functions if 1 + 2 is a positive even integer [51]. Specifying now to holographic correlators at tree-level,8 the convention of including explicit Gamma functions in (2.29) has particular appeal: their poles encode the doubletrace exchanges of [O1O2]m;` and [O3O4]n;`. Poles in M (s; t) only capture the singletrace exchanges, if any, associated with a Witten diagram. In particular, all local AdS interactions give rise to contact diagrams whose Mellin amplitudes are mere polynomials in the Mellin variables. In this language, the counting of solutions to crossing symmetry in sparse large N CFTs performed in [43] becomes manifestly identical on both sides of the duality. Exchange Witten diagrams have meromorphic Mellin amplitudes that capture the lone single-trace exchange: they take the form9 M (s; t) = C12OCO34 1 X m=0 t Q`;m(s) O 2m + Pol(s; t) : (2.31) Pol(s; t) stands for a possible polynomial in s; t. The polynomial boundedness is a signature of local AdS dynamics [ 25, 40 ]. Anomalous dimensions appear when poles of the integrand collide to make double poles. A considerable amount of work has led to a quantitative understanding of the above picture. These include formulas for extraction of the one-loop OPE coe cients P (ij)(n; `) 1 and anomalous dimensions 1 (ij)(n; `) from a given Mellin amplitude (section 2.3 of [26]); and the graviton exchange amplitude between pairwise identical operators in arbitrary spacetime dimension (section 6 of [65]). 2.4 Looking ahead Having reviewed much of what has been accomplished, let us highlight some of what has not. issues here. First, we note that no approach to computing holographic correlators has systematically deconstructed loop diagrams, nor have arbitrary external spins been e ciently incorporated. Save for some concrete proposals in section 6, we will not address these While Mellin space is home to a fruitful approach to studying holographic CFTs in particular, it comes with a fair amount of technical complication. Nor does it answer the natural question of how to represent a single conformal block in the bulk. One is, in any case, left to wonder whether a truly e cient approach exists in position space. Examining the position space computations reviewed in subsection 2.2, one is led to wonder: where are the conformal blocks? In particular, the extraction of dual CFT spectral data and OPE coe cients in the many works cited earlier utilized a double OPE expansion. 8This is the setting that is known to be especially amenable to a Mellin treatment. Like other approaches to Witten diagrams, the Mellin program has not been systematically extended to loop level (except for certain classes of diagrams; see section 6). Because higher-trace operators appear at higher orders in 1=N , some of the elegance of the tree-level story is likely to disappear. The addition of arbitrary external spin in a manner which retains the original simplicity has also not been done, although see [22]. 9For certain non-generic operator dimensions, the sum over poles actually truncates [21, 23]. The precise mechanism for this is not fully understood from a CFT perspective. We thank Liam Fitzpatrick and Joao Penedones for discussions on this topic. More recent computations of exchange diagrams [65, 66] using the split representation do make the conformal block decomposition manifest, in a contour integral form [73]: integration runs over the imaginary axis in the space of complexi ed conformal dimensions, and the residues of poles in the integrand contain the OPE data. This is closely related to the shadow formalism. However, this approach is technically quite involved, does not apply to contact diagrams, and does not answer the question of what bulk object computes a single conformal block. diagrams. Let us turn to this latter question now, as a segue to our computations of Witten 3 The holographic dual of a scalar conformal block What is the holographic dual of a conformal block? This is to say, what is the geometric representation of a conformal block in AdS? In this section we answer this question for the case of scalar exchanges between scalar operators, for generic operator and spacetime dimensions. In section 5, we will tackle higher spin exchanges. At this stage, these operators need not belong to a holographic CFT, since the form of a conformal block is xed solely by symmetry. What follows may seem an inspired guess, but as we show in the next section, it emerges very naturally as an ingredient in the computation of Witten diagrams. Let us state the main result. We want to compute the scalar conformal partial wave W ;0(xi), de ned in (2.6), corresponding to exchange of an operator O of dimension between two pairs of external operators O1; O2 and O3; O4. Let us think of the external operators as sitting on the boundary of AdSd+1 at positions x1;2;3;4, respectively. Denote the geodesic running between two boundary points xi and xj as ij . Now consider the scalar geodesic Witten diagram, which we denote W ;0(xi), rst introduced in section 1 and drawn in gure 1: Gbb y( ); y( 0); W ;0(xi) Z Z where (3.1) (3.2) (3.3) HJEP01(26)4 Z 12 Z 1 1 d ; Z 34 Z 1 1 d 0 W ;0(xi) = 12 34W ;0(xi) : denote integration over proper time coordinates and 0 along the respective geodesics. Then W ;0(xi) is related to the conformal partial wave W ;0(xi) by The proportionality constant 34 is de ned in equation (2.11) and 12 is de ned analogously. The object W ;0(xi) looks quite like the expression for a scalar exchange Witten diagram for the bulk eld dual to O. Indeed, the form is identical, except that the bulk vertices are not integrated over all of AdS, but rather over the geodesics connecting the pairs of boundary points. This explains our nomenclature. Looking ahead to conformal partial waves for exchanged operators with spin, it is useful to think of the bulk-to-bulk propagator in (3.1) as pulled back to the two geodesics. Equation (3.3) is a rigorous equality. We now prove it in two ways. Consider the piece of (3.1) that depends on the geodesic 12, which we denote '12: : HJEP01(26)4 In terms of '12 y( 0) , the formula for W ;0 becomes It is useful to think of '12(y), for general y, as a cubic vertex along 12 between a bulk eld at y and two boundary elds anchored at x1 and x2. We may then solve for '12(y) as a normalizable solution of the Klein-Gordon equation with a source concentrated on 12. The symmetries of the problem turn out to specify this function uniquely, up to a multiplicative constant. We then pull this back to 34, which reduces W ;0(xi) to a single one-dimensional integral along 34. This integral can then be compared to the well-known integral representation for G ;0(u; v) in (2.10), which establishes (3.3). To make life simpler, we will use conformal symmetry to compute the geodesic Witten diagram with operators at the following positions: 1 solving the wave equation rst in global AdS, then moving to Poincare coordinates and comparing with CFT. We work with the global AdS metric We implement the above strategy by Z 12 Z The relation between mass and conformal dimension is and so the wave equation for '12(y) away from 12 is 1 ds2 = cos2 (d 2 + dt2 + sin2 d 2d 1) : m2 = ( d) r 2 ( d) '12(y) = 0 ; or 2 t ( d) '12(y) = 0 : (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) At 12, there is a source. To compute (3.6), we take t1 ! 1, t2 ! 1, in which limit the geodesic becomes a line at = 0, the center of AdS. This simpli es matters because this source is rotationally symmetric. Its time-dependence is found by evaluating the product of bulk-to-boundary propagators on a xed spatial slice, 12t: Therefore, we are looking for a rotationally-symmetric, normalizable solution to the following radial equation: tion (3.4), is '12( ; t) = 12 e 1t1 2t2 2F1 relation between coordinates is + 2 12 ; 2 12 ; d 2 2 Now we need to transform this to Poincare coordinates, and pull it back to 34. The 2 12 ( d) '12 = 0 : (3.12) The full solution, including the time-dependence and with the normalization xed by equae 2t = u2 + jxj2; cos2 = u 2 u2 + jxj2 : Although the eld '12 is a scalar, it transforms nontrivially under the map from global to Poincare AdS because the bulk-to-boundary propagators in its de nition (3.4) transform as After stripping o the power of x1 needed to de ne the operator at in nity, the eld in In terms of the proper length parameter 0, z( 0) = z( 0) = u( 0) = jz3 z4j : z3 + z4 + z3 + z4 + 2 2 2 cosh 0 z3 z3 2 2 z4 tanh 0; z4 tanh 0; (3.11) 12t: (3.13) (3.14) (3.15) (3.17) (3.18) Poincare coordinates is '12(u; xi) = 12 2F1 where on the right hand side and t are to be viewed as functions of the Poincare coordinates u; xi via (3.14). Now we want to evaluate our Poincare coordinates on the geodesic 34. With our choice of positions in (3.6), 34 is a geodesic in an AdS3 slice through AdSd+1. Geodesics in Poincare AdS3 are semi-circles: for general boundary points z3; z4, 2u2 + (z z4)(z z3) + (z z4)(z z3) = 0 : + 2 12 ; 2 12 ; d 2 2 Therefore, the pullback of '12 to 34 is '12 u( 0); xi( 0) = 12(2 cosh 0) 12 2 2F1 + 2 12 ; into the propagator (2.21), we get z Gb@ y( 0); 1 in (3.5). Plugging (3.18) We may now assemble all pieces of (3.5){(3.6): plugging (3.20) and (3.21) into (3.5) gives the geodesic Witten diagram. Trading z; z for u; v we nd, in the parameterization (3.6), W ;0(u; v) = De ning a new integration variable, we have W ;0(u; v) = 2 d 0 2 Comparing (3.24) to the integral representation (2.10), one recovers (3.3) evaluated at the appropriate values of x1; : : : ; x4. Validity of the relation (3.3) for general operator positions is then assured by the identical conformal transformation properties of the two sides. 3.2 In the previous section we evaluated a geodesic Witten diagram W ;0 and matched the result to a known integral expression for the corresponding conformal partial wave W ;0, thereby showing that W ;0 = W ;0 up to a multiplicative constant. In this section we give a direct argument for the equivalence, starting from the de nition of conformal partial waves. Section 3.2.1 reviews the de nition of conformal partial waves W ;` as eigenfunctions of the conformal Casimir operator. We prove this de nition to be satis ed by geodesic Witten diagrams W ;0 in section 3.2.3, after a small detour (section 3.2.2) to de ne the basic embedding space language used in the proof. The arguments of the present section are generalized in section 5.5 to show that W ;` = W ;` (again, up to a factor) for arbitrary exchanged spin `. The generators of the d-dimensional conformal group SO(d + 1; 1) can be taken to be the Lorentz generators LAB of d + 2 dimensional Minkowski space (with LAB antisymmetric in A and B as usual). The quadratic combination L2 12 LABLAB is a Casimir of the algebra, i.e. it commutes with all the generators LAB. As a result, L2 takes a constant value on any irreducible representation of the conformal group, which means all states jP n conformal family of a primary state jOi are eigenstates of L2 with the same eigenvalue. Oi in the The eigenvalue depends on the dimension and spin ` of jOi, and can be shown to be [4] The SO(d + 1; 1) generators are represented on conformal elds by C2( ; `) = ( d) `(` + d 2) : where L1AB is a di erential operator built out of the position x1 of O1 and derivatives with respect to that position. The form of the L1AB depends on the conformal quantum numbers of O1. Equation (3.26) together with conformal invariance of the vacuum imply the following identity, which holds for any state j i: (L1AB + L2AB)2h0jO1(x1)O2(x2)j i = h0jO1(x1)O2(x2)L2j i : Consistent with the notation for L2, we have de ned 1 2 (L1AB + L2AB) 2 (L1AB + L2AB)(L1 AB + L2 AB) : As discussed in section 2.1, one obtains a conformal partial wave W ;` by inserting into a four-point function the projection operator P ;` onto the conformal family of a primary O with quantum numbers ; `: Xh0jO1(x1)O2(x2)jP nOihP nOjO3(x3)O4(x4)j0i : (3.25) (3.26) (3.27) (3.28) (3.29) One can take this second-order di erential equation, plus the corresponding one with 1; 2 $ 3; 4, supplemented with appropriate boundary conditions, as one's de nition of W ;` [74]. Regarding boundary conditions, it is su cient to require that W ;` have the correct leading behavior in the x2 ! x1 and x4 ! x3 limits. The correct behavior in both limits is dictated by the fact that the contribution to W ;` of the primary O dominates that of its descendants since those enter the OPE with higher powers of x12 and x34. We will prove that geodesic Witten diagrams W ;0 are indeed proportional to conformal partial waves W ;0 by showing that W ;0 satis es the Casimir equation (3.30) and has the correct behavior in the x2 ! x1 and x4 ! x3 limits. The proof is very transparent in the embedding space formalism, which we proceed now to introduce. The embedding space formalism has been reviewed in e.g. [35, 65, 74]. The idea is to embed the d-dimensional CFT and the d + 1 dimensional AdS on which lives the geodesic Witten diagram both into d + 2 dimensional Minkowski space. We give this embedding space the metric ds2 = (dY 1)2 + (dY 0)2 + X(dY i)2: The CFT will live on the projective null cone of embedding space, which is the Lorentzinvariant d-dimensional space de ned as the set of nonzero null vectors X with scalar multiples identi ed: X aX. We will use null vectors X to represent points in the projective null cone with the understanding that X and aX signify the same point. The plane Rd can be mapped into the projective null cone via X+(x) = ajxj2; X (x) = a ; Xi(x) = axi where we have introduced light cone coordinates X choice of the parameter a de nes the same map. Conformal transformations on the plane are implemented by Lorentz transformations in embedding space. As a speci c example, we may consider a boost in the 0 direction with rapidity . This leaves the Xi coordinates unchanged, and transforms X according to = X 1 X0. Of course, any nonzero d i=1 (3.31) (3.32) (3.33) Applying the identity (3.27) to the equation above and recalling that each state jP n an eigenstate of L2 with the same eigenvalue C2( ; `), we arrive at the Casimir equation (L1AB + L2AB)2W ;`(xi) = C2( ; `)W ;`(xi) : (3.30) ! e X+; X ! e X : A point X(x) = (jxj2; 1; xi) gets mapped to (e jx2j; e ; xi) which is projectively equivalent to X(e xi). Thus boosts in the 0 direction of embedding space induce dilatations in the plane. it as Any eld O^ on the null cone de nes a eld O on the plane via restriction: O(x) O^ X(x) . Since O^ is a scalar eld in embedding space, the SO(d + 1; 1) generators act on [LAB; O^(X)] = (XA@B (3.34) and only if O^ satis es the homogeneity condition The induced transformation law for O is the correct one for a primary of dimension if O^(aX) = a O^(X) : Thus in the embedding space formalism a primary scalar eld O(x) of dimension represented by a eld O^(X) satisfying (3.35). Below, we drop the hats on embedding space is elds. It should be clear from a eld's argument whether it lives on the null cone (as O(X)) or on the plane (as O(x)). Capital letters will always denote points in embedding space. Meanwhile, AdSd+1 admits an embedding into d + 2 dimensional Minkowski space, as the hyperboloid Y 2 = 1. Poincare coordinates (u; xi) can be de ned on AdS via The induced metric for these coordinates is the standard one, (2.17). The AdS hyperboloid sits inside the null cone and asymptotes toward it. As one takes u ! 0, the image of a point (u; xi) in AdS approaches (Y +; Y ; Y i) = u 1 (jx2j; 1; xi) which is projectively equivalent to X(xi). In this way, the image on the projective null cone of the point xi 2 R d marks the limit u ! 0 of the embedding space image of a bulk point (u; xi). so are generated by Isometries of AdS are implemented by embedding space Lorentz transformations, and as long as Y is on the AdS slice. This fact, which is not surprising given that L2 is a second-order di erential operator invariant under all the isometries of AdS, can be checked directly from (3.37). 3.2.3 Geodesic Witten diagrams satisfy the Casimir equation The geodesic Witten diagram W ;0(xi) lifts to a function W ;0(Xi) on the null cone of embedding space via a lift of each of the four bulk-to-boundary propagators with the appropriate homogeneity condition i Gb@ (y; Xi) ; i = 1; 2; 3; 4 : The geodesics in AdS connecting the boundary points X1 to X2 and X3 to X4 lift to curves in embedding space which can be parameterized by Y Y = 1. That is, for Y belonging to the AdS slice, L2f (Y ) depends only on the values of f on the slice. In fact, applied to scalar functions on AdS the operator L2 is simply the negative of the Laplacian of AdS: L2f (Y ) = r2Y f (Y ) (3.35) (3.36) (3.37) (3.38) (3.39) (3.40) we see that terms for which m2m = m2n give rise to derivatives of conformal blocks, and hence to logarithms. This is equivalent to the condition d 2 Z, both of these are equivalent to integrality condition stated above. Identical structure is visible in (4.15): logarithms will appear when any of m2m, m2n, m2 coincide. As an explicit example, let us consider D (xi). Then (4.11) can be split into m 6= n and m = n terms, the latter of which yield logarithms: D (xi) = 1 X 2an n=0 X m6=n am m2n m2m (an ) W2 +2n;0(xi) + HJEP01(26)4 This takes the form of the ` = 0 terms in (2.28), with P1(n; 0) = 2 (22 +2n) a n X m6=n am m2n m2m + (an ) 2 and As an aside, we note the conjecture of [43], proven in [26], that 1 2 P0(n; 0) 1(n; 0) = (an ) 2 2 : P1(n; `) = 1 (4.24) (4.25) (4.26) (4.27) We have checked in several examples that this is obeyed by (4.25){(4.26). It would be interesting to prove it using generalized hypergeometric identities. 4.5 Taking stock We close this section with some perspective. Whereas traditional methods of computing Witten diagrams are technically involved and require explicit bulk integration [16] and/or solution of di erential equations [51], the present method skips these steps with a minimum of technical complexity. It is remarkable that for neither the contact nor exchange diagrams have we performed any integration: the integrals have instead been absorbed into sums over, and de nitions of, conformal partial waves. For the contact diagram/D-function, we have presented an e cient algorithm for its decomposition into spin-0 conformal blocks in position space. Speci c cases of such decompositions have appeared in previous works [43, 45], although no systematic treatment had been given. Moreover, perhaps the main virtue of our approach is that exchange diagrams are no more di cult to evaluate than contact diagrams. D-functions also appear elsewhere in CFT, including in weak coupling perturbation theory. For example, the four-point function of the 20' operator in planar N = 4 SYM at weak coupling is given, at order , by [76] hO200 (x1)O200 (x2)O200 (x3)O200 (x4)i / D1111(z; z) (4.28) where D1111(z; z) was de ned in (2.25). The ubiquity of D-functions at weak coupling may be related to constraints of crossing symmetry in the neighborhood of free xed points [71]. The OPE of two scalar primary operators yields not just other scalar primaries but also primaries transforming in symmetric traceless tensor representations of the Lorentz group. We refer to such a rank-` tensor as a spin-` operator. Thus, for the full conformal block decomposition of a correlator of scalar primaries we need to include blocks describing spin-` exchange. The expression for such blocks as geodesic Witten diagrams turns out to be the natural extension of the scalar exchange case. The exchanged operator is now described by a massive spin-` eld in the bulk, which couples via its pullback to the geodesics connecting the external operator insertion points. This was drawn in gure 1. In this section we do the following. We give a fairly complete account of the spin-1 case, showing how to decompose a Witten diagram involving the exchange of a massive vector eld, and establishing that the geodesic diagrams reproduce known results for spin-1 conformal blocks. We also give an explicit treatment of the spin-2 geodesic diagram, again checking that we reproduce known results for the spin-2 conformal blocks. More generally, we use the conformal Casimir equation to prove that our construction yields the correct blocks for arbitrary `. Conformal blocks with external scalars and internal spin-` operators were studied in the early work of Ferrara et al. [1]. They obtained expressions for these blocks as double integrals. It is easy to verify that their form for the scalar exchange block precisely coincides with our geodesic Witten diagram expression (3.1). We thus recognize the double integrals as integrals over pairs of geodesics. Based on this, we expect agreement for general `, although we have not so far succeeded in showing this due to the somewhat complicated form for the general spin-` bulk-to-bulk propagator [65, 66]. Some more discussion is in section 5.6. We will instead use other arguments to establish the validity of our results. Dolan and Osborn [5] studied these blocks using the conformal Casimir equation. Closed-form expressions in terms of hypergeometric functions were obtained in dimensions d = 2; 4; 6. For example, in d = 2 we have G ;`(z; z) = jzj and in d = 4 we have G ;`(z; z) = jzj ` z`2F1 2F1 z 1 z z`+12F1 2F1 12 + ` 2 2 12 + + 34 + ` 2 2 34 + `; z `; z + (z $ z) ` ; ; ; ` 2 12 2 ` 12 + ` + 34 + ` + `; z 2 + 34 2 1; ` 2; z (z $ z) ; ; 1; (5.1) (5.2) The d = 6 result is also available, taking the same general form, but it is more complicated. Note that the d = 2 result is actually a sum of two irreducible blocks, chosen so as to be even under parity. The irreducible d = 2 blocks factorize holomorphically, since the global conformal algebra splits up as sl(2; R) sl(2; R). An intriguing fact is that the d = 4 block is expressed as a sum of two terms, each of which \almost" factorizes holomorphically. Results in arbitrary dimension are available in series form. Since the results of Dolan and Osborn are obtained as solutions of the conformal Casimir equation, and we will show that our geodesic Witten diagrams are solutions of the same equation with the same boundary conditions, this will constitute exact agreement. Note, though, that the geodesic approach produces the solution in an integral representation. It is not obvious by inspection that these results agree with those in [5], but we will verify this in various cases to assuage any doubts that our general arguments are valid. As noted above, in principle a more direct comparison is to the formulas of Ferrara et al. [1]. Geodesic Witten diagrams with spin-` exchange: generalities Consider a CFTd primary operator which carries scaling dimension and transforms in the rank-` symmetric traceless tensor representation of the (Euclidean) Lorentz group. The AdSd+1 bulk dual to such an operator is a symmetric traceless tensor eld h 1::: ` obeying the eld equations Our proposal is that the conformal partial wave W ;`(xi) is given by the same expression as in (3.1) except that now the bulk-to-bulk propagator is that of the spin-` eld pulled back to the geodesics. The latter de nes the spin-` version of the geodesic Witten diagram, W ;`(xi): its precise de nition is W ;`(xi) Z Z (5.3) (5.4) : (5.5) Gbb y( ); y( 0); ; ` and Gbb y( ); y( 0); ; ` is the pulled-back spin-` propagator, Gbb y( ); y( 0); ; ` [Gbb(y; y0; )] 1::: `; 1::: ` d dy 1 : : : dy ` dy0 1 d d 0 : : : dy0 ` d 0 y=y( ); y0=y( 0) To explicitly evaluate this we will use the same technique as in section 3.1. Namely, the integration over one geodesic can be expressed as a normalizable spin-` solution of the equations (5.3) with a geodesic source. Inserting this result, we obtain an expression for the geodesic Witten diagram as an integral over the remaining geodesic. If we call the above normalizable solution h 1::: ` , then the analog of (3.5) is Z 34 h 1::: ` y( 0) dy0 1 d 0 : : : dy0 ` d 0 (5.6) As in section 3.1, we will speci cally compute = jzj 1 3 now written in terms of (z; z) instead of (u; v) to facilitate easier comparison with (5.1) and (5.2). We recall that this reduces 12 to a straight line at the origin of global AdS. The form of 34 is given in (3.18), from which the pullback is computed using cos2 e2t e2i 34 34 34 = = = 1 ; Carrying out this procedure for all dimensions d at once presents no particular complications. However, it does not seem easy to nd the solution h 1::: ` for all ` at once. For this reason, below we just consider the two simplest cases of ` = 1; 2, which su ce for illustrating the general procedure. Evaluation of geodesic Witten diagram: spin-1 In the global AdSd+1 metric we seek a normalizable solution of 1 ds2 = cos2 (d 2 + dt2 + sin2 d 2d 1 ) r2A d) = 0 ; r A = 0 A dx = At( ; t)dt + A ( ; t)d : 12 tand 1 At = 0 which is spherically symmetric and has time dependence e 12t. A suitable ansatz is Assuming the time dependence e 12t, the divergence free condition implies and the components of the wave equation are cosd 1 sind 1 cosd 1 sind 1 212 cos2 212 cos2 d 1 sin2 1)( d+1) At 2 12 cos sin A = 0 1)( d+1) A +2 12 cos sin At = 0 : (5.7) (5.8) (5.9) (5.10) (5.11) (5.12) (5.13) (5.14) where we have inserted a factor of 12 in A to ensure a smooth particular, setting 12 = 0 we have A = 0 and 12 ! 0 limit. In At = (cos ) 12F1 + 1 2 ; 2 1 ; 2 It is now straightforward to plug into (5.6) to obtain an integral expression for the conformal block. Because the general formula is rather lengthy we will only write it out explicitly 12 = 0. In this case we nd (not paying attention to overall normalization W ;1(z; z) = jzj Z 1 0 3 4 1(1 + 34 1 2 (1 2F1 + 1 2 ; j 1 zj2) ) 2 1 ; 2 34 1 1 2 2 1 zj2) j 1 jzj2 (1 (1 j 1 +1 2 ) zj2) : Setting d = 2; 4, it is straightforward to verify that the series expansion of this integral reproduces the known d = 2; 4 results in (5.1), (5.2) for 12 = 0. We have also veri ed agreement for 12 6= 0. 5.4 Evaluation of geodesic Witten diagram: spin-2 In this section we set 12 = 0 to simplify formulas a bit. We need to solve 2 r h d) 2]h = 0 ; = 0 ; h = 0 : (5.18) The normalizable solution is A = At = 12 sin (cos ) 2F1 1 12 + 1 2 ; 2 ; 2 e 12t r h 1 d 1 h = f + ftt + f : f = f ftt : should be static and spherically symmetric, which implies the general ansatz h dx dx = f ( )g d 2 + ftt( )gttdt2 + f ( ) tan2 d 2d 1 : We rst impose the divergence free and tracelessness conditions. We have We use this to eliminate f , Moving to the divergence, only the component r h is not automatically zero. We nd r h = f 0 + d + 1 cos sin f cos sin cos sin f + ftt = 0 (5.15) (5.16) (5.17) (5.19) (5.20) (5.21) (5.22) which we solve as ftt = tan f 0 + 1 We then work out the component of the eld equation, 2 r h d) Setting this to zero, the normalizable solution is d + 1 cos2 f : 2 cot f 0 2) cos2 = (cos ) +22F1 2 ; + 2 2 ; 2 : This completely speci es the solution, and we now have all we need to plug into (5.6). We refrain from writing out the somewhat lengthy formulas. The series expansion of the result matches up with (5.1) and (5.2) as expected. General `: proof via conformal Casimir equation As in the case of scalar exchange, the most e cient way to verify that a geodesic Witten diagram yields a conformal partial wave is to check that it is an eigenfunction of the conformal Casimir operator with the correct eigenvalue and asymptotics. We start from the general expression (5.4). A rank-n tensor on AdS is related to a tensor on the embedding space via The bulk-to-boundary propagator lifted to the embedding space is Y ( ) = p e X1 + e X2 : i : (5.23) f : (5.24) (5.25) (5.26) (5.27) (5.28) (5.29) (5.30) In particular, this holds for the bulk-to-bulk propagator of the spin-` eld, and so we can write Gbb(y; y0; ; `) = [Gbb(Y; Y 0; )]M1:::M`;N1:::N` d : : : d d 0 dY M1 dY M` dY 0N1 : : : dY 0N` d 0 : Now, [Gbb(Y; Y 0; )]M1:::M`;N1:::N` only depends on Y and Y 0. Since Y M dY M = 12 dd (Y Y ) = d 0, when pulled back to the geodesics the only contributing structure is T 1::: n = TM1:::Mn : [Gbb(Y; Y 0; )]M1:::M`;N1:::N` = f (Y Y 0)Y M01 : : : Y M0` YN1 : : : YN` : We also recall a few other useful facts. Lifted to the embedding space, the geodesic connecting boundary points X1 and X2 is We follow the same strategy as in the case of scalar exchange. We start by isolating the part of the diagram that contains all the dependence on X1;2, FM1:::M` (X1; X2; Y 0; ) = Z 12 M1:::M`;N1:::N` d dY M1 : : : dY M` : Here, Y ( ) lives on 12, but Y 0 is left arbitrary. This is the spin-` generalization of '12(y) de ned in (3.4), lifted to embedding space. We now argue that this is annihilated by the SO(d + 1; 1) generators L1AB + L2AB + LYAB0. This generator is the sum of three generators in the scalar representation, plus a \spin" term acting on the free indices N1 : : : N`. This operator annihilates any expression of the form g(X1 X2; X1 Y 0; X2 Y 0)XN1 : : : XN` , where each X stands for either X1 or X2. To show this, we just note the SO(d + 1; 1) invariance of the dot products, along with the fact that XN is the normal vector to the X2 = 0 surface and so is also SO(d + 1; 1) invariant. From (5.28){(5.30) we see that FN1:::N` (X1; X2; Y 0; ) is of this form, and so is annihilated by L1AB + L2AB + LYAB0. We can therefore write (L1AB + L2AB)2FN1:::N` (X1; X2; Y 0; ) = (LYAB0)2FN1:::N` (X1; X2; Y 0; ) = C2( ; `)FN1:::N` (X1; X2; Y 0; ) (5.32) (5.31) where we used that (LYAB0)2 is acting on the spin-` bulk-to-bulk propagator, which is an eigenfunction of the conformal Casimir operator13 with eigenvalue (3.27). The relation (5.32) holds for all Y 0, and hence holds upon integrating Y 0 over 34 with any weight. Hence we arrive at the conclusion (L1AB + L2AB)2W ;`(xi) = C2( ; `)W ;`(xi) (5.33) which is the same eigenvalue equation obeyed by the spin-` conformal partial wave, W ;`(xi). The short distance behavior as dictated by the OPE is easily seen to match in the two cases, establishing that we have the same eigenfunction. We conclude that the spin-` geodesic Witten diagram is, up to normalization, equal to the spin-` conformal partial wave. 5.6 Comparison to double integral expression of Ferrara et al. It is illuminating to compare our expression (5.4) to equation (50) in [2], which gives the general result (in d = 4) for the scalar conformal partial wave with spin-` exchange, written as a double integral. We will rewrite the result in [2] in a form permitting easy comparison to our formulas. First, it will be useful to rewrite the scalar bulk-to-bulk propagator (2.18) by applying a quadratic transformation to the hypergeometric function, Gbb(y; y0; ) = 2F1 2 ; + 1 2 ; + 1 ; 2 : d 2 (5.34) 13Note that the conformal Casimir is equal to the spin-` Laplacian up to a constant shift: (LYAB0)2 = r`2 + `(` + d 1) [77]. Next, recall that in embedding space the geodesics are given by (3.40), from which we compute the quantity with one point on each geodesic 1 = Y ( ) Y ( 0) = d 0 Comparing to [2], we have + and e 1 = the form 2 2 1 = Z Z 12 34 x12x34 . x12x34 With these de nitions in hand, it is not hard to show that the result of [2] takes Here Gbb y( ); y( 0); is the scalar bulk-to-bulk propagator (5.34), and C`0(x) is a Gegenbauer polynomial. This obviously looks very similar to our expression (5.4), and indeed agrees with it for ` = 0. The two results must be equal (up to normalization) since they are both expressions for the same conformal partial wave. If we assume that equality holds for the integrand, then we nd the interesting result that the pullback of the spin` propagator, as written in (5.5), is equal to C`0(2 1)Gbb y( ); y( 0); . The general spin-` propagator is very complicated (see [65, 66]), but apparently has a simple relation to the scalar propagator when pulled back to geodesics. It would be interesting to verify this. 5.7 Decomposition of spin-1 Witten diagram into conformal blocks In the case of scalar exchange diagrams, we previously showed how to decompose a Witten diagram into a sum of geodesic Witten diagrams, the latter being identi ed with conformal partial waves of both single- and double-trace exchanges. We now wish to extend this to the case of higher spin exchange; we focus here on the case of spin-1 exchange for simplicity. A picture of the nal result is given in gure 6. As discussed in section 2, given two scalar operators in a generalized free eld theory, we can form scalar double trace primaries with schematic form [O1O2]m;0 dimension (12)(m; 0) = 1 + 2 + 2m + O(1=N 2), and vector primaries [O1O2]m;1 (12)(m; 1) = 1 + 2 + 1 + 2m + O(1=N 2). The analysis of [43], and later [26, 65, 66] demonstrated that these conformal blocks, and their cousins [O3O4]n;0 and [O3O4]n;1, should appear in the decomposition of the vector exchange Witten diagram, together with the exchange of a single-trace vector operator. The computations below will con rm this expectation. The basic approach is the same as in the scalar case, although the details are more complicated. Before diving in, let us note the main new features. In the scalar case a basic step was to write, in (4.1), the product of two bulk-to-boundary propagators partial waves. The term in the upper right captures the single-trace exchange of the dual vector operator. The second line captures the CFT exchanges of the ` = 0 double-trace operators [O1O2]m;0 and [O3O4]n;0. Likewise, the operators [O1O2]m;1 and [O3O4]n;1. nal line captures the CFT exchanges of the ` = 1 double-trace Gb@ (y; x1)Gb@ (y; x2) as a sum over solutions '12(y) of the scalar wave equation sourced on the 12 geodesic. Here, we will similarly need a decomposition of Gb@ (y; x1)r Gb@ (y; x2), where r is a covariant derivative with respect to bulk coordinates y. It turns out that this can be expressed as a sum over massive spin-1 solutions and derivatives of massive scalar solutions. This translates into the statement that the spin-1 exchange Witten diagram decomposes as a sum of spin-1 and spin-0 conformal blocks, as noted above. Our rst task is to establish the expansion cmAm; (y) + bmr 'm(y) X m where Am; (y) and 'm(y) denote the solutions to the massive spin-1 and spin-0 equations sourced on 12, found earlier in sections 5.3 and 4.1, respectively.14 m labels the masses of the bulk elds, to be determined shortly. We will not attempt to compute the coe cients cm and bm, which is straightforward but involved, contenting ourselves to determining the spectrum of conformal dimensions appearing in the expansion, and showing how the expansion coe cients can be obtained if desired. Following the scalar case, we work in global AdS and send t1 ! 1, t2 ! Dropping normalizations, as we shall do throughout this section, we have 12t: 12t (`) denote the dimension of the corresponding spin, we have, from (5.15) m Now to the computation. We consider a theory of massive scalars coupled to a massive vector eld via couplings ir j A . The Witten diagram with vector exchange is then Z Z y y0 Gbb (y; y0; ) 12 ; (0) m (m1) + 12 +1 2 d 2 2 ; cos2 (1) m 2 (cos ) m e (0) 12t 12 +1 ; (1) m d 2 2 e 12t Letting and (3.13), 'm = 2F1 (m0) + 12 ; 2 m 2 Am; = 12 sin (cos ) m 2F1 (1) Am;t = 1 The various terms have the following powers (cos2 )k in an expansion in powers of cos2 , Am; : k = r 'm : k = Am;t : k = rt'm : k = + q 1 + (1) m 2 (0) m 1 + 2 2 2 (1) m (0) m 2 2 2 + q + q 1 2 1 + q + q 14'm is just '1m2, whose superscript we suppress for clarity, and likewise for 'n and '3n4. (5.38) (5.39) (5.40) (5.41) (5.42) where q = 0; 1; 2; : : :. Comparing, we see that we have the right number of free coe cients for (5.39) to hold, provided we have the following spectrum of dimensions appearing (m0) = (m1) = 1 + 1 + 2 + 2m 2 + 1 + 2m with m = 0; 1; 2; : : :. The formulas above can be used to work out the explicit coe cients cm and bm. We noted at the beginning of this subsection that this spectrum of dimensions coincides with the expected spectrum of double-trace scalar and vector operators appearing in the OPE, at leading order in large N . We may now rewrite (5.38) as15 We expand this out in an obvious fashion as cmAm; (y) + bmr 'm(y) Gbb (y; y0; ) cnAn; (y0) + bnr 'n(y0) : A4Vec(xi) = AAA(xi) + AA (xi) + A A(xi) + A (xi) : The next step is to relate each term to geodesic Witten diagrams, which we now do in turn. The solution Am; (y) can be expressed as X cmcn m;n Z Z y y0 Am; (y)Gbb (y; y0; )An; (y0) : Z 12 2 Z 12 dy ( ) Gbb y( ); y; (1) : m (5.47) which is easily veri ed for a straight line geodesic at the center of global AdS, and hence dy ( ) d is true in general. Using this we obtain (dropping the normalization, as usual) AAA = X cmcn m;n dy ( ) d Gbb y( ); y; (1) Gbb; (y; y0; )Gbb y0; y( 0); (n1) m dy0 ( 0) : (5.48) 15Following the precedent of section 4, all quantities with an m subscript refer to the double-trace operators appearing in the O1O2 OPE, and those with an n subscript refer to the double-trace operators appearing in the O3O4 OPE. (5.43) (5.44) (5.45) (5.46) The bulk-to-bulk propagator for the vector eld obeys m2)Gbb (y; y0; ) = where y0) denotes a linear combination of g this, and the fact that the propagator is divergence free at non-coincident points, we can y0) and r r (y y0). Using verify the composition law Z y0 Gbb (y; y0; )Gbb; (y0; y00; 0) = 1 (m0)2 Gbb (y; y00; ) Gbb (y; y00; 0) : We use this relation twice within (5.48) to obtain a sum of three terms, each with a single vector bulk-to-bulk propagator. Note also that these propagators appear pulled back to the geodesics. Each term is thus a geodesic Witten diagram with an exchanged vector, that is, a spin-1 conformal partial wave. The spectrum of spin-1 operators that appears is 1 + 2 + 1 + 2m ; 3 + 4 + 1 + 2n ; m; n = 0; 1; 2; : : : (5.51) So the contribution of AAA is a sum of spin-1 conformal blocks with internal dimensions corresponding to the original exchanged eld, along with the expected spin-1 double trace operators built out of the external scalars. Next we integrate by parts in y, use r Gbb (y; y0; ) / r y0), and integrate by parts again, to get A A = X bmcn m;n Z y0 r 'm(y0)An(y0) : Now we write An(y0) as an integral over 34 as in (5.47) and then again remove the bulkto-bulk propagator by integrating by parts. This yields A A = X bmcn m;n Z 34 r 'm y( 0) : (5.54) dy ( 0) d 0 Writing 'm as an integral sourced on 12 we obtain A A = X bmcn m;n Z Z 5.7.2 AA and A A We start with A A = X cmbn m;n Z Z y y0 r 'm(y)Gbb (y; y0; )An; (y0) : Integrating by parts and using Gb@ y( 0); x3 Gb@ y( 0); x4 we see that A A decomposes into a sum of spin-0 exchange geodesic Witten diagrams. That is, A A contributes a sum of spin-0 blocks with conformal dimensions 1 + m = 0; 1; 2; : : : (5.49) (5.50) (5.52) (5.53) (5.55) (5.56) By the same token AA yields a sum of spin-0 blocks with conformal dimensions 3 + 4 + 2n ; n = 0; 1; 2; : : : Integration by parts reduces this to A X bmbn Z Z y y0 r 'm(y)Gbb (y; y0; )r 'n(y0) : A X bmbn Z y r 'm(y)r 'n(y) : Now rewrite the scalar solutions as integrals over the respective geodesic sources, X bmbn Z Z Z y0 12 34 r Gbb y( ); y0; (0) m r 0 Gbb y0; y( 0); (n0) : The composition law analogous to (5.50) is easily worked out to be r 0 Gbb y( ); y0; (0) m r 0 Gbb y0; y( 0); (n0) = cmnGbb y( ); y( 0); (0) m A Z y0 + dmnGbb y( ); y( 0); (n0) with some coe cients cmn and dmn that we do not bother to display here. Inserting this in (5.60) we see that A decomposes into a sum of scalar blocks with conformal dimensions 1 + 3 + 4 + 2n ; m; n = 0; 1; 2; : : : (5.62) 5.7.4 We have shown that the Witten diagram involving the exchange of a spin-1 eld of dimension decomposes into a sum of spin-1 and spin-0 conformal blocks. The full spectrum of conformal blocks appearing in the decomposition is scalar: vector: 1 + 2 + 2n ; 3 + 1 + 4 + 2n 2 + 1 + 2n ; 3 + 4 + 1 + 2n (5.63) where n = 0; 1; 2; : : :. This matches the spectrum expected from 1=N counting, including single- and double-trace operator contributions. With some patience, the formulas above can be used to extract the coe cient of each conformal block, but we have not carried this out in full detail here. While we have not explored this in any detail, it seems likely that the above method can be directly generalized to the case of arbitrary spin-` exchange. The split (5.45) will still be natural, and a higher spin version of (5.50) should hold. (5.57) (5.58) (5.59) (5.60) (5.61) HJEP01(26)4 diagrams, and hence decomposed into conformal blocks using our methods. 6 Discussion and future work In this paper, we have shed new light on the underlying structure of tree-level scattering amplitudes in AdS. Four-point scalar amplitudes naturally organize themselves into geodesic Witten diagrams; recognizing these as CFT conformal partial waves signals the end of the computation, and reveals a transparency between bulk and boundary with little technical e ort required. We are optimistic that this reformulation extends, in some manner, to computations of generic holographic correlation functions in AdS/CFT. To that end, we close with some concrete observations and proposals, as well as a handful of future directions. Adding loops. It is clearly of interest to try to generalize our techniques to loop level. We rst note that there is a special class of loop diagrams that we can compute already using these methods: namely, those that can be written as an in nite sum of tree-level exchange diagrams [21]. For the same reason, this is the only class of loop diagrams whose Mellin amplitudes are known [21]. These diagrams only involve bulk-to-bulk propagators that all start and end at the same points; see gure 7 for examples. Careful study of the resulting sums would be useful. More generally, though, we do not yet know how to decompose generic diagrams into geodesic objects. This would seem to require a \geodesic identity" analogous to (4.1) that applies to a pair of bulk-to-bulk propagators, rather than bulk-to-boundary propagators. It would be very interesting to nd these, if they exist. Such identities would also help to decompose an exchange Witten diagram in the crossed channel. Decomposition of Witten diagrams in the crossed channel. The present work studies the decomposition of an s-channel exchange Witten diagram into s-channel partial waves. It is clearly of interest to understand, using the language of geodesic Witten diagrams, how the same diagram decomposes into t-channel partial waves, which corresponds to using the OPE on the pairs of operators O1O3 and O2O4. As explored in [45], the statement is that the basic scalar exchange Witten diagram decomposes into t-channel partial waves involving only scalar double trace operators [O1O3]n;0 and [O2O4]n;0. As we mention ve-point tree-level Witten diagram. However, it is not equal to the ve-point conformal partial wave, as discussed in the text. above, this decomposition seems to require new geodesic identities involving bulk-to-bulk propagators. Derivative interactions. We have primarily focused on decomposing Witten diagrams with non-derivative interactions. For example, the contact diagram of gure 4 is based on the interaction 1 2 3 4 . We would like to be able to e ciently decompose Witten diagrams with derivative interactions too, like 1r 1 r 2 : : : r k 2 3r 1 r 2 : : : r k 4; a precise version of the identity in equation (5.39) would be su cient to treat the k = 1 case, but we would like to generalize that to add more Lorentz indices. We anticipate that this identity exists and involves propagators of massive elds of spin l k. Adding legs. Consider for example a ve-point correlator of scalar operators hO1(x1) : : : O5(x5)i. We can de ne associated conformal partial waves by inserting projection opera W ;`; 0;`0 (xi) = hO1(x1)O2(x2)P ;`O5(x5)P 0;`0 O3(x3)O4(x4)i : (6.1) Using the OPE on O1O2 and O3O4, reduces this to three-point functions. The question is, can we represent W ;`; 0;`0 (xi) as a geodesic Witten diagram? Suppose we try to dismantle a tree level ve-point Witten diagram. For de niteness, we take `0 = ` = 0. All tree level ve-point diagrams will lead to the same structures upon using our geodesic identities: namely, they can be written as sums over geodesic-type diagrams, each as in gure 8, which we label Wc a;0; b;0(xi). This is easiest to see starting from a 5 contact diagram, and using (4.1) on the pairs of propagators (12) and (34). In that case, a 2 f mg and b 2 f ng. As an equation, gure 8 reads Wc a;0; b;0(xi) = Gbb y( ); y5; a Gb@ (y5; x5)Gbb y5; y( 0); b (6.2) Z Z 12 Z y5 34 { 42 { Note that the vertex at y5, indicated by the orange dot in the gure, must be integrated over all of AdS. Could these diagrams be computing W a;0; b;0(xi) as de ned above? The answer is no, as a simple argument shows. Suppose we set 5 = 0 in (6.2), which requires a = b . From (6.1) it is clear that we must recover the four-point conformal partial wave with the exchanged primary ( ; 0). So we should ask whether (6.2) reduces to the expression for the four-point geodesic Witten diagram, W ;0. Using Gb@ (y5; x5)j 5=0 / 1, the integral over y5 becomes Z y5 Gbb y( ); y5; Gbb y5; y( 0); : Therefore, the 5 = 0 limit of (6.2) does not give back the four-point partial wave, but rather its derivative with respect to m2 , which is a di erent object. We conclude that although we can decompose a ve-point Witten diagram into a sum of diagrams of the type in gure 8, this is not the conformal block decomposition. This raises two questions: what is the meaning of this decomposition in CFT terms, and (our original question) what diagram computes the ve-point partial wave? External operators with spin. Another obvious direction in which to generalize is to consider correlation functions of operators carrying spin. As far as the conformal blocks go, partial information is available. In particular, [7] obtained expressions for such blocks as di erential operators acting on blocks with external scalars, but this approach is limited to the case in which the exchanged operator is a symmetric traceless tensor, since only such operators appear in the OPE of two scalar operators. The same approach was taken in [78]. Explicit examples of mixed symmetry exchange blocks were given in [79]. Our formulation in terms of geodesic Witten diagrams suggests an obvious proposal for the AdS computation of an arbitrary conformal partial wave: take our usual expression (1.1), now with the bulk-to-boundary and bulk-to-bulk propagators corresponding to the elds dual to the respective operators. Of course, there are many indices here which have to be contracted, and there will be inequivalent ways of doing so. But this is to be expected, as in the general case there are multiple conformal blocks for a given set of operators, corresponding to the multiplicity of ways in which one spinning primary can appear in the OPE of two other spinning primaries. It will be interesting to see whether this proposal turns out to be valid. As motivation, we note that it would be quite useful for bootstrap purposes to know all the conformal blocks that arise in the four-point function A related pursuit would be to decompose all four-point scalar contact diagrams, including any number of derivatives at the vertices. This would involve a generalization of (5.39) to include more derivatives. Virasoro blocks and AdS3/CFT2. Our calculations give a new perspective on how to construct the dual of a generic Virasoro conformal block: starting with the geodesic Witten diagram, we dress it with gravitons. Because Virasoro blocks depend on c, a computation in semiclassical AdS gravity would utilize a perturbative 1=c GN expansion. In [34], we put the geodesic approach to use in constructing the holographic dual of the heavy-light Virasoro blocks of [30], where one geodesic essentially backreacts on AdS to generate a conical defect or black hole geometry. It would be worthwhile to pursue a 1=c expansion around the geodesic Witten diagrams more generally. A closely related question is how to decompose an AdS3 Witten diagram into Virasoro, rather than global, blocks. For a tree-level diagram involving light external operators like those considered here, there is no di erence, because the large c Virasoro block with light external operators reduces to the global block [38]. It will be interesting to see whether loop diagrams in AdS3 are easier to analyze using Virasoro symmetry. Assorted comments. The geodesic approach to conformal blocks should be useful in deriving various CFT results, not only mixed symmetry exchange conformal blocks. For example, the conformal blocks in the limits of large , ` or d [80{86], and subleading corrections to these, should be derivable using properties of AdS propagators. One can also ask whether there are similar structures present in bulk spacetimes besides AdS. For instance, an analog of the geodesic Witten diagram in a thermal spacetime would suggest a useful ingredient for parameterizing holographic thermal correlators. Perhaps the existence of a dS/CFT correspondence suggests similar structures in de Sitter space as well. It is natural to wonder whether there are analogous techniques to those presented here that are relevant for holographic correlators of nonlocal operators like Wilson loops or surface operators, perhaps involving bulk minimal surfaces. Let us close by noting a basic fact of our construction: even though a conformal block is not a semiclassical object per se, we have given it a representation in terms of classical elds propagating in a smooth spacetime geometry. In a bulk theory of quantum gravity putatively dual to a nite N CFT, we do not yet know how to compute amplitudes. Whatever the prescription, there is, evidently, a way to write the answer using geodesic Witten diagrams. It would be interesting to understand how this structure emerges. Acknowledgments We thank Eric D'Hoker, Liam Fitzpatrick, Tom Hartman, Daniel Ja eris, Juan Maldacena, Joao Penedones and Sasha Zhiboedov for helpful discussions. E.P. wishes to thank the KITP and Strings 2015 for hospitality during this project. P.K. is supported in part by NSF grant PHY-1313986. This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. E.P. is supported by the Department of Energy under Grant No. DE-FG02-91ER40671. 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Eliot Hijano, Per Kraus, Eric Perlmutter, River Snively. Witten diagrams revisited: the AdS geometry of conformal blocks, Journal of High Energy Physics, 2016, 146, DOI: 10.1007/JHEP01(2016)146