AdS Maps and Diagrams of Bi-local Holography

We present in detail the basic ingredients contained in bi-local holography, representing a constructive scheme for reconstructing AdS bulk theories in Vectorial/AdS duality. Explicit Mapping to bulk AdS and higher spin fields is seen to be given by a double Fourier transform. All order interactions are explicitly specified through the collective action. This generates bulk Feynman (Witten) diagrams (at tree and loop level). We give details of the four-point case evaluation. It is noted that the bi-local construction goes beyond the assumptions in various discussions of non-locality.


Conclusion 23
A

Introduction
The approach to Holography based on collective fields represents one constructive scheme to understand the appearance of emergent space-time degrees of freedom and of specifying their bulk interactions [1][2][3][4]. Collective (gauge invariant) fields with their intrinsic nonlocality and the feature of remaining closed under large N Schwinger-Dyson (SD) equations provide a natural framework for bulk reconstruction. Vector type theories, leading to AdS Higher Spin (HS) Duality [5][6][7][8] are interesting and possibly the simplest class of theories where this construction can be explicitly implemented. The space of collective fields is in this case given by bi-locals whose nonlinear dynamics and equations of motion are completely specified by the SD equations generating the 1/N expansion. This is summarized by the corresponding collective action (and Hamiltonian). It was proposed and in subsequent studies established that the bi-local space contains precisely the sufficient degrees of freedom to allow a representation of bulk fields and their dynamics in AdS space-time [1][2][3][4]. The notable feature is that the identification of (physical) bulk fields is one-to-one and given by a momentum-space map through a double Fourier transform. This was explicitly constructed in the canonical (Hamiltonian) framework in both light-cone [2] and time-like [3] quantization. Interesting features dealing with 1/N were discussed at both tree and loop level [9,10]. This framework, for reconstruction of bulk space-time and field interactions based on the dynamics of bi-local fields is termed bi-local holography. At present, it represents the only one-to-one construction for understanding the emergence of AdS degrees of freedom and construction of their bulk interactions from large N CFT. With the previous studies [2,3] being in the canonical (single time) framework (of collective theory), it is the purpose of this work to establish the more general, covariant action formulation and map. In the lower, one-dimensional case (d = 1) this has recently been extensively used in the investigations and large N solution of SYK type models [11][12][13][14]. While the complete bulk gravitational theory is not yet established, the bi-local construction and map greatly illuminate the emergent space-time and degrees of freedom of this theory.
So in the present work we consider the higher dimensional cases of vector models in d > 1 at both UV and IR critical points (with clear dualities to Higher Spin Gravities in AdS [15][16][17][18][19][20]) in more detail. For this we give a Map that establishes a transformation to AdS space-time, while the nonlinear collective action generates interaction vertices to all orders. Their structure is based on a bi-local star product, with an extended dipole picture [21] analogous to the case of SFT [22]. As such the bi-local theory provides a nonlinear bulk description of Higher Spins to all orders. This allows construction and evaluation of Feynman (Witten) diagrams which we will exemplify with detailed consideration of general 4-point boundary amplitudes. The content of the paper goes as follows. We will first discuss the covariant Map from bi-locals to AdS HS fields. This Map will be constructed in general in Section 3 and in detail for the example of AdS 3 . It follows the earlier momentum-space Fourier analysis and applies to any case of vectorial/AdS HS duality. This will also be true for the structure of bi-local interactions and the associated Feynman diagrams. We will discuss this in Section 4 presenting the details of the 4-point case. It is interesting to compare this with the work and evaluations of [23] and questions of Non-locality [24,25]. Here we have rescaled Φ to exhibit an overall N which then plays the role of a coupling constant. In this way, the bi-local method systematically describes the large N physics (in vector type models) by an action [1] S col where the trace term comes from a Jacobian factor due to the change of path-integral variables from the fundamental fields φ i (i = 1, · · · , N ) to the bi-local collective field The first term S[Φ] 1 represents the original action of the vector model written in terms of Φ. The saddle-point solution Φ 0 of the collective action (2.3) represents the two-point function of the fundamental fields Expanding the collective action around the saddle-point solution as follows with x 12 ≡ x 1 − x 2 , we find the quadratic action The bi-local Laplacian is given by We will be following the critical theories at λ UV = 0, λ IR . The bi-local generator of the conformal group is given by For the Euclidean d-dimensional conformal field theory, the two Casimir operators of the SO(1, d+1) conformal symmetry, whose generators are denoted by L AB (A, B = −1, 0, · · · , d), are defined by The eigenvalues of these operators acting on a primary with conformal dimension ∆ and spin s are given in [26] 2 . The bi-local Laplacian (for the free theory) is related to the Casimir operators as [4] bi ≡ (2.14) The second equality can be shown by explicit evaluation. One considers the eigenvalue problem of the bi-local Laplacian where we label the eigenfunction and its eigenvalue by c and s, with the dimension ∆ related to c via ∆ ≡ d 2 +c. The eigenfunction ψ c,s , which is called a conformal partial wave function [27], is given by a particular set of conformal spinning three point functions where ε µ denotes a null polarization vector and Z µ 12,3 is a unit vector defined by is the conformal dimension of the d-dimensional free scalar field. The eigenvalues of the bi-local Laplacian bi are given by We discuss the properties of the conformal partial wave functions ψ c,s (x 1 , c 0 ; x 2 , c 0 ; x 3 , c, s, ε) in Appendix D.1.
The main feature of the bi-local reconstruction of the bulk AdS fields is that there exists a map between η (≡ |x 12 | 2 Φ) and the bulk higher spin fields H where x i and z are the AdS coordinates and S denotes the spin degrees of freedom. This bulk AdS field H(AdS d+1 , S d−1 ) contains physical HS fields whose meaning will be illuminated in Section 3. We therefore have a one-to-one identification of the AdS d+1 bulk from the bi-local field. Under the transformation, the collective Laplacian turns into its corresponding AdS form and one can consider the collective action as an action of bulk AdS higher spin theory. The sequence of interacting vertices follows from this action. One expands the action around the saddle-point as in (2.7), which leads to the 1/N expansion of the collective action as where in the interaction term (second term), the n th term in the sum contains n η fields.
. The quadratic kernel is the bi-local Laplacian K = bi , given by equation (2.14).
On the other hand, vertices are constructed using the inverse of the classical solution Φ −1 0 , which are represented by a dashed line. For example, the cubic and quartic vertices are diagrammatically presented by We will work out concrete forms of these vertices and their transforms to AdS space to exemplify their locality properties. They will also be the main ingredients for writing and evaluating Witten diagrams in Section 4. For the d = 1 case relevant to the SYK model these vertices generate the diagrammatics of [28].

Bi-local Map to AdS
The basis of the map comes from group theory and concerns the action of the conformal group. On the bi-local Φ(x 1 , x 2 ) we have the kinematic action of two copies of SO(1, d + 1) and on the higher spin fields H(AdS d+1 , S d−1 ) there is a nontrivial realization (spinning particle in AdS) of SO(1, d + 1) Under the bi-local map, the mapping between the bi-local field and the AdS higher spin field is simply given by a change of momenta (point transformation) in momentum space. Namely, defining the Fourier transform to momentum space by the bi-local map transforms between the two fields To understand this identification, we first need to establish the physical degrees of freedom of the bulk AdS theory . Higher spin fields h µ 1 ···µs (x) (µ j = 0, 1, · · · , d) in AdS d+1 can be represented in embedding space by where X A (A = −1, 0, 1, · · · , d) are the coordinates of the embedding space of AdS d+1 . Introducing an extra set of coordinates Y A to generate the spin degrees of freedom, we have The phase space is spanned by (where P A and K A are the conjugate momentum of X A and Y A , respectively). On this space one has four (first class) constraints which can be summarized as [4,17]  T 4 = P A K A = 0 , (de Donder gauge) (3.11) together with two "gauge conditions" By solving the constraints, the phase space is reduced to (x µ , p µ ; y µ , k µ ) (µ = 0, 1, · · · , d). In the reduced phase space, the remaining first class constraints can be written as 14) Note that T 3 and T 4 can be considered as constraints for the higher spin field φ(x; y) in the Poincare coordinates of AdS d+1 space Solving T 3 and T 4 leads to a reduced phase space for AdS d+1 × S d−1 with coordinates (x µ , p µ ; Ω, p Ω ). This is the physical space, to which we will refer as the spinning AdS space. On this phase space there is no gauge redundancy, and the action of the group SO(1, d + 1) is nontrivial.
Our correspondence corresponds to this (physical) description of AdS higher spin bulk fields. The transformation between bi-local and spinning AdS spaces is simplified with the Poincare parametrization of AdS. One has that the relationship between bi-local and spinning AdS spaces is simply given as a change of co-ordinates in momentum space. This change of momenta, and the explicit form of the map can be determined through comparison of the actions of the conformal group on the two spaces: bi-local and Spinning AdS. We will present a demonstration of this in the following section.

Momentum Map
Let us give the discussion for deducing the momentum map in the example of the d = 2 and AdS 3 × S 1 case. By solving the remaining constraints T 3 and T 4 [4], the realization of SO(1, 3) on the reduced space can be established to be where p ≡ 1 2 (p 0 − ip 1 ) andp ≡ 1 2 (p 0 + ip 1 ). Note that θ and p θ corresponds to the coordinate for S 1 and its conjugate, respectively. For the bi-local space, the SO(1, 3) generators read For the bi-local CFT, we use the notation One can search for the momentum space map by requiring that the bi-local realization (of generators) as differential operator maps into the AdS 3 × S 1 space realization. This has been done in the previous light-cone and canonical quantizations for AdS 4 , and the covariant d = 1 example of AdS 2 . Generally this is an exceedingly complicated exercise. We will describe a much simpler route, which will allow us to deduce the momentum space transformation relatively easily through comparison of momentum space wave functions. One then proceeds to verify the consistency of the transformation by comparison of all the symmetry generators.
For the derivation of the bi-local map, it is actually convenient to consider the Lorentzian signature case. Therefore, until (3.42) (and also Appendix A), we consider Lorentzian signature while in the rest of the paper we focus on the Euclidean case. This Lorentzian version of the bi-local map (3.39)∼(3.41) can be analytically continued to the Euclidean version (3.44)∼(3.47). In Appendix B, we also confirm that this bi-local map for the Euclidean case transforms the SO(1, 3) generators for AdS into bi-local generators.
Let us consider the Casimir operators for Lorentzian SO(2, 2) generators in Appendix B.1 (3.20) Here we define p ± = − 1 2 (p 0 ∓ p 1 ) for Lorentzian case. The Laplacian squared is given by 3 2 To see that this formula makes sense, recall that from CFT we expect so that the eigenvalues of the Laplacian squared are [ 1 2 (−∆ 2 + s 2 )] 2 as expected. Using the explicit forms given above, one finds the Laplacian as By changing z → i ∂ ∂pz and p θ → −i ∂ ∂θ , we have Then, it is easy to see that eigenfunctions lead to the eigenvalues (−∆ 2 + s 2 )/2 as expected.
For the bi-local (Lorentzian) CFT, the bi-local forms of the Casimir operators are given by From the Fourier transformation of the spinning three-point functions (A.14), we have the bi-local wave functions Therefore, one can confirm that ψ( k 1 , k 2 ) is indeed the bi-local wave function with the expected eigenvalue.
By comparing with the AdS wave function in (3.27), we conclude that In order to write down the map for p z , it is useful to define Note that we have the following identity: This leads to arcsin where n ± ∈ Z. Choosing n ± = 0, the map for φ gives In summary, we have the Lorentzian version of the momentum map from (k 1± ; k 2± ) to (p ± , p z , θ) and its inverse map is given by For the Euclidean bi-local map, we perform a Wick rotation to Euclidean signature and quickly obtain the Euclidean version of the momentum map from (k 1 ,k 1 ; k 2 ,k 2 ) to (p,p, p z , θ) and its inverse map, given by By Wick rotation, one can also repeat the analogous derivation for the Euclidean bi-local map. From the SO(1, 3) generators for higher spin fields in (3.17), the Casimir operators are written as This leads to the Laplacian squared One can express the Casimir operators in terms of φ and θ and, it is easy to see that ψ ∆,s = e i∆φ+isθ is the eigenfunction of L 2 and L 2 corresponding to eigenvalue − 1 4 (∆ + s) 2 and − 1 4 (∆ − s) 2 , respectively. As in the Lorentzian case, one can express θ and φ in terms of k dif ,k dif , p andp by comparing the wave functions.
This also leads to the Euclidean bi-local map in (3.44)∼(3.47). As usual in Euclidean CFT 2 , holomorphic and anti-holomorphic momenta can be considered as independent complex variables. In the physical space where they are conjugate to each other, the bi-local map in (3.46) and (3.47) implies that p z and θ (and their conjugates in (B.4) and (B.5)) are pure imaginary. Note that the bi-local map is not necessarily a canonical transformation in general. Nevertheless, demanding that the two phase spaces are related by a canonical transformation, this momentum map induces a coordinate map, which is explicitly given in Appendix B. Then, the bi-local map consisting of the momentum and coordinate map indeed transforms SO(1, 3) generators for higher spin field to bi-local ones and vice versa.
In fact, one can also derive the bi-local map for momentum and coordinates by equating the two realization of SO(1, 3) generators. The bi-local map which we present in (3.44)∼(3.47) and (B.2)∼(B.5) is a solution of these equations of the generators.
Through the bi-local map found above, the quadratic collective action for d = 2 is mapped to AdS 3 as Hence, the AdS Laplacian reads where m 1 (s) represents the physical mass which corresponds to the scaling dimension ∆ 1 = s + d − 2, while m 2 (s) represents un-physical mass corresponding to ∆ 2 = s + d.

Vertices in Momentum Space
Start from the cubic term in the action where we redefined the fluctuations as Writing the interaction in terms of the Φ fields, one finds the cubic term is (3.60) Now Fourier transforming the fluctuations the cubic term reduces to where L bi i is the bi-local Laplacian of the i-th leg acting on the delta functions. Now mapping this cubic term to AdS 3 × S 1 , we obtain (3.63) where the bi-local Laplacian is transformed into the AdS 3 × S 1 Laplacian with a Jacobian J . Here, we identified Φ(k i , k i ) = H( p i , p z i , θ i ) and δ 6 (· · · ) represents the transformed results from δ 2 (k 1 + k 2 )δ 2 (k 2 + k 3 )δ 2 (k 3 + k 1 ). Two of these delta functions can be written as δ 2 ( p 1 + p 2 + p 3 ) and the remaining four are to be determined.
Let us now evaluate the δ 6 (· · · ). The best way to evaluate the transformation is to use the holomorphic and anti-holomorphic coordinates. Then, the six delta functions are written as Since the anti-holomorphic sector is just a copy of the holomorphic sector, we focus on the former Then, this part is transformed as We can obtain a similar result for the anti-holomorphic sector using the momentum bi-local map. Now we further proceed to the quadratic coupling term Again by redefining the fluctuation as (3.59) and Fourier transforming as (3.61), we rewrite the quartic term as (3.68) The transformation to AdS 3 × S 1 now leads to (3.69) The evaluation of δ 8 (· · · ) is identical to the cubic case, and this gives There is a similar result for the anti-holomorphic sector. In general we see that in momentum space the vertices consist of delta function forms with insertions on external legs. Since the Map to AdS space, in momentum space, is just a change of variables, this property persists.
In any case there are no inverse Laplacians in the vertices, a problem that plaged other constructions as discussed in [24].

Bi-local Witten Diagrams
The bi-local field Φ was observed to transform (through the momentum space map after Fourier transformation) into the physical AdS higher spin field. On the CFT side, the bilocal field also represents the generating function of higher spin currents for the free theory as with representing a polarization vector and (see for example [29,30]) Consequently, all correlation functions of higher spin currents follow from correlators of bi-local fields With this in mind, it is useful to explore the conformal block decomposition of the bi-local field theory. Indeed, the language of conformal blocks allows efficient and maximal use of the underlying conformal invariance. For recent progress in this direction see for example [31][32][33][34][35].

Bi-local Propagator
We start with the evaluation of the propagator of bi-local fluctuations represented by η We follow the (λ = 0) case, noting the fact that the discussion of the IR fixed point case is essentially the same (with only a boundary condition change for the lowest mode). A recent on-shell discussion of IR bi-local states is given in [36].
For the propagator we consider the linearized collective field equation, which is the Schwinger-Dyson equation for the two point function of the bi-local fluctuation η at large N This equation is easily solved,with the propagator expressed as which is diagrammatically represented by (4.11). We denote the classical solution 4 Φ 0 by a thick line as (the second equality is for d = 3) In addition, the inverse 5 Φ −1 0 of the classical solution, which corresponds to a differential operator, is represented by a dashed line From the above definitions, the contraction of a single thick line and a single dash line is easily evaluated The bi-local propagator of the η field is given by This is the bilocal representation of the propagator. Another representation of the propagator can be given in terms of conformal partial wave functions ψ c,s (x 1 , c 0 ; x 2 , c 0 ; y, −c, s, ε) in (2.16) which we explain in detail in Appendix D.1 where the Plancherel weight ρ s (c) is defined by Note that c,s is composed of a summation over the even spins s and a contour integral along the imaginary axis, with a deformation for d 3 (see (D.10) for more details). One can easily see that the RHS of (4.6) becomes the completeness relation of the conformal partial wave functions in (D.8), which proves (4.6). The propagator has a physical pole at c = s − d−1 2 . The pole at c = s + d+1 2 is unphysical. It is clear that (4.14) Finally, note that the propagator G also satisfies the following equation, which is useful to simplify the diagrammatics

Bi-local Three-point Function
In this subsection, we consider the bi-local 3-point function Acting with the differential operator in (4.2) on each leg of this bi-local 3-point function and setting x i = x i , we can obtain any higher spin current 3-point function.
From the large N expansion of the collective action, the cubic vertex can be read off as follows The bi-local 3-point function can be computed from the bi-local Witten diagram rules to find  Therefore, the bi-local three-point function is now represented as where we used (4.15), which gives the permutation terms in each x i ↔ x i with a symmetry factor 1 2 . Hence, the three point function of the bi-local fluctuation is given by (4.20)
t = x 1 x 1 x 2 x 2 x 3 x 3 x 4 and the u-channel contribution is u = x 1 x 1 x 2 x 2 x 3 x 3 x 4 The quartic contribution associated with the four vertex from Figure 2 gives We will now evaluate the same s, t, u channel diagrams using the conformal partial wave basis. We will consider the four point amplitude of four scalars, even though the evaluation of the general case is almost identical in the bilocal approach i.e., we consider the four point amplitude of Note that with this definition the two point function of the scalar operator O(x) is given by (4.31) By setting y i = y i (i = 1, 2, 3, 4) in (4.23), one obtains the s-channel correlation function of the four scalar O operators (4.32) At coincident pointsÂ,B,Ĉ becomes 4A, 4B, 4C. To evaluate the s-channel contribution, use the integral representation of the conformal block given in [23,26] (see also [37]) where the conformal cross ratios are given by (4.34) We follow the conventions of 7 [39] for the conformal block G h,s (u, v). Using this integral representation, one can write the s-channel contribution in (4.32) in terms of the conformal blocks. Therefore, we have (specializing for d = 3) and Also, because ρ s (c) and λ h,s (c) in (4.32) are even functions of c, the contribution of the conformal block and its shadow in (4.33) to the s-channel is identical.
By closing the contour in (4.38) (see the contour defined in (D.10)) in the region Re(c) > 0, the s-channel A s (1234) contribution picks up the following three types of simple poles.
• Simple poles at c = s − 1 2 ( s = 0, 2, 4, · · · ) 8 from the factor ρs(c) λc,s K c,s • Simple poles at c = s + 2n + 5 2 (n = 0, 1, 2, · · · and s = 0, 2, 4, · · · ) from the factor ρs(c) λc,s K c,s • Simple poles at c = s − k + 1 2 (k = 1, 2, · · · , s and s = 0, 2, 4, · · · ) from the conformal block G h,s (u, v) The first two types of simple poles can easily be seen in the expression for ρs(c) λc,s K c,s . On the other hand, the third type of simple poles in the conformal block G h,s (u, v) comes from null states. 6 The relation to the usual Gegenbauer polynomial Cs(x) is given by Here is the comparison between different conventions h,s (u, v) . In Appendix D.2, we show that the contributions from the second and the third types of poles cancel exactly. The remaining contribution to the s-channel comes from the first type of simple poles at c = s − 1 2 corresponding to single-trace operators. This leads to agreeing with the s-channel OPE expressions (for 2A + 2B). Here we define A, B and C as follows Note that we include a normalization factor Υ 2 O . A summary is given in the Appendix C. The t-channel diagram is evaluated in the same way with an exchange x 2 ↔ x 4 which gives agreeing with the t-channel OPE expressions (for 2A + 2C). Likewise, the u-channel contribution is which is consistent with the u-channel OPE expressions (for 2B +2C). The exchange x 2 ↔ x 3 and x 2 ↔ x 4 leads to a transformation of (A, B, C) among each other   To conclude our s, t, u channel exchange evaluations, we note that the only contributions come from physical intermediate states (associated with single trace operators, and physical states of higher spin AdS fields). For comparison in analogous evaluations of [41], there seem to appear contributions associated with additional double trace 'states' [42].
Next to complete the evaluation of the four point function of the scalar operator O in the free O(N ) vector model, we turn to evaluation of the quartic contact diagram contribution. As in the exchange diagrams, one can simplify the contribution of the contact diagrams in Figure 3 to obtain A q (1234) = −3(A + B + C) .  Let us summarize the main features of the above Feynman diagram scheme. First, we comment on the feature that (for the free case) exchange and contact diagrams give essentially the same contribution. It is interesting to state the mechanism behind this since it relates to questions of non-locality. The vertices in the bi-local construction can be explicitely seen to be associated with derivative couplings and a bi-local star product. In the exchange diagrams (of the free theory ) we see that the derivatives in the 3-vertices cancel out the bi-local Laplacians of the propagator, effectively resulting in a contact type contribution. It is for this reason that the two type of contributions give similar results. In the bi-local basis neither the quartic, nor the higher contact vertices contain inverse Laplacian type terms. There is another aspect (of the free theory) that illuminates this issue. It was observed that the Mellin amplitude in this case degenarates, instead of 1/s, 1/t, /1/u type exchange terms one finds xδ(x) type terms which relates to the cancellation noted above. The amplitude is essentially 0, which again agrees with the bi-local S-matrix finding [43].
Concerning extensions of the diagram scheme to the IR fixed point it will be given by the same collective vertices, the difference being in the propagator, where the lowest mode wavefunction has to be taken to adjust to the different boundary conditions. Next, as we have also stressed the evaluation of loops is feasible, since the fact that the formalism is based on physical higher spin degrees of freedom avoids the need for ghost fields etc.. This was already seen in a previous evaluation of free energy [10]. It will be nevertheless interesting to compare the loop diagrams given by the present approach with the recent AdS evaluations of [44].

Conclusion
We have in this paper developed the details of the covariant bi-local holographic reconstruction of bulk AdS applicable to cases of vectorial AdS duality. Different theories are distinguished by different bi-local propagators, while the all order interaction vertices are universally specified by the collective field action. The following two key ingredients of the construction have been elaborated in detail. First is the correspondence (Map) between the bi-local space and AdS space with higher spins. This map is seen (as in our previous Hamiltonian constructions) to be one-to-one, with the bi-local space faithfully representing the physical content of the emergent bulk fields. Even more (in the cases d = 1, 2) it was a purely momentum space Map, corresponding to a change of momentum space coordinates. In this and other respects, the bi-local space reconstruction differs from schemes such as "kinetic space", the later featuring maps [45] from (d + d) dimensions to (d + 1) representing a projection. Moreover, a comparison of Laplacians between the two schemes would indicate that the later might not have a satisfactory 1/N basis. In the bi-local construction, the reconstruction is fully nonlinear, with all order interactions which turn into bulk AdS space interactions through the one-to-one Map. As such they produce Feynman (Witten) diagrams applicable at tree and loop levels. We have presented in detail the evaluations of 3 and 4-point cases, the latter resulting in the familiar conformal block expansion [46]. This in our s, t and u channel Feynman exchanges was seen to receive contributions only from single traces, representing physical states (in the example presented). For the example of free vector theory,a particular simplification is noted, where the derivative couplings in the vertices cancel out the Laplacians in the propagators, with a result being of contact type. It is not the case for the nontrivial IR fixed point theory, where one obtains genuine exchange and contact type contributions.
Finally, we mention that other theories which are not necessarily based on vector fields (such as matrix and tensor models) offer sectors governed by bi-local holography. This is the case also of the recent interesting example of 'fishnet' theories [47], which can be seen to correspond to a simple deformation of the present case. A notable challenge is still de Sitter space-time, which might be reachable through bi-local holography. For a recent, interesting discussion see [48]. JY also thanks Asia Pacific Center for Theoretical Physics (APCTP) for the hospitality and partial support within the program "Holography and Geometry of Quantum Entanglement". KS and JY would like to thank the Nordic Institute for Theoretical Physics (Nordita) for the hospitality and partial support during the completion of this work, within the program "Bounding Transport and Chaos in Condensed Matter and Holography".

A Fourier Transform of Spinning Correlators
From [49], the 2-dimensional spinning conformal correlators are given by In this Appendix, we consider the Fourier transform of these spinning conformal correlators in the Lorentzian signature. The arbitrary polarization µ can be removed using the Thomas operator defined as follows The form of the Thomas operator ensures that the tensor that we obtain after the µ have been removed by differentiation, is traceless. As usual in 2d-CFT, it will prove convenient to use complex coordinates. Expressed in terms of complex coordinates, the Thomas operator becomes and ( · Z) for the two possible (holomorphic or anti-holomorphic) polarizations becomes Therefore, we have Now, the Fourier transform of the spinning conformal correlators for the holomorphic polarization are written as where we defined The integrals in (A.7) are given by the integral representation of modified Bessel functions as where we defined We further Fourier transform from x 1 , x 2 to k 1 , k 2 : Changing the integration variables to x and η as in (A.8), then the x integral leads to the momentum conservation delta function δ 2 ( k 1 + k 2 − p). The remaining η integrals can be directly performed by for example using formula 6.699.4 of [50] aŝ There is a one possibility that this result becomes simpler. Namely when µ = 0 Hence for ∆ 0 = 1/2, we obtain the Fourier transform of the spinning correlators as

B.2 Coordinate Map
Using the inverse momentum map in (3.48) and (3.49), we have where ξ is defined by

B.3 Transformation of Casimir and Laplacian
For AdS 3 × S 1 , we use momentum-space coordinates (p,p, φ, θ). The mapping to bi-local momentum-space is given by Therefore, the Laplacian 2 AdS 3 ×S 1 = 4L 2 L 2 is given by For bi-local CFT, we label momentum-space by (k 1 ,k 1 , k 2 ,k 2 ). The inverse map of (B.7) is given by We have (B.12) From the mapping (B.7), one can rewrite the derivatives with respect to k,k as derivatives of φ and θ, by employing the chain rule. Therefore, we obtain This guarantees that the Laplacian is also correctly transformed from the bi-local CFT to AdS 3 × S 1 .

C Conformal Block Expansions
The four point correlation function of spin zero scalars is given by c s is the OPE coefficient for single trace operators and c n,s is the OPE coefficient of double-trace operator [51,52] given by The contributions from each of the three channels is

D.1 CPW Basis
Let us consider the conformal partial wave functions constructed from two free O(N ) scalar fields of dimension ∆ 0 ≡ d−2 2 ≡ d 2 + c 0 and spin-s symmetric-traceless operator of conformal dimension ∆ = d 2 + c, which up to normalization are given by One can package the conformal partial wave function with a null polarization vector ε µ where Z µ 12,3 is a unit vector defined by any dimension, and for fixed pure imaginaryc (principal series) the conformal partial wave functions ψ(x 1 ,c; x 2 ,c; x 3 , c) with c ∈ iR form a complete basis [53,54]. For our purpose we need an analytic continuation of this completeness relation from pure imaginaryc to the real value c 0 = −1 for the free O(N ) scalar field. Asc is analytically continued to c 0 = −1, poles from the normalization factor N c 0 ,s (c) might cross the contour c ∈ (−i∞, i∞) in the completeness relation 9 provided that [54]  The deformed contour C is a common prescription for the extra contribution coming from the analytic continuation [38]. But, as pointed out in [38], one can also think of these contributions as the contribution of finite numbers of "discrete" representations for s = 0 which are called "Born terms" in the literature [55]. Without these "discrete" contributions, the contour integral for s = 0 in the s-channel would pick up a simple pole at c = 1 2 which corresponds to the massless representation where´d c 2πi is either the deformed contour or vertical contour along (−i∞, i∞) (in c space) with two small circles around two poles. Note that the conformal partial wave function is symmetric or anti-symmetric for even or odd spins, respectively i.e., ψ c,s (x 2 , c 0 ; x 1 , c 0 ; y, c, s, ) = (−1) s ψ c,s (x 1 , c 0 ; x 2 , c 0 ; y, c, s, ) (D.14) Since the bi-local field for the O(N ) vector model is symmetric, only conformal partial wave functions with even spin appear.

D.2 Identity
We can write the conformal block as [40] G ∆,s (u, v) = G s (∆; u, v) where G s (∆; u, v) is an analytic function in complex ∆ space. In [40], the poles that may appear are classified into three posible classes. Since our contour integral for the s-channel encloses the region Re(c) > 0, it is enough for us to consider the third class of poles. These have the form 16) whereâ 3 (k, s) is given 10 bŷ From now on we specialize to d = 3. First of all, for each s, the pole at c = s − 1 2 (i.e., k = 1) corresponds to a single-trace operator. The corresponding residue vanishes (i.e.,â 3 (1) = 0) so that the conformal block G 3 2 +c,s (u, v) does not have a pole at c = s − 1 2 . For a given s, from (D. 16) we see that the conformal block G 3 2 +iν,s+k contains the following simple pole G 3 2 +c,s+k ⊃â