aainstitutetext: Leinweber Center for Theoretical Physics, Randall Laboratory
of Physics
The University of Michigan, Ann Arbor, MI 48109-1040, USAbbinstitutetext:
Department of Physics, Pennsylvania State University, University Park, PA
16802, USA

# On the Differential Representation and Color-Kinematics Duality of AdS
Boundary Correlators

Aidan Herderschee b Radu Roiban b and Fei Teng<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The AdS boundary correlators and their dual correlation functions of boundary
operators have been the main dynamic observables of the holographic duality
relating a bulk AdS theory and a boundary conformal field theory. We show that
tree-level AdS boundary correlators for generic states can be expressed as
nonlocal differential operators of a certain structure acting on contact
Witten diagrams. We further write the boundary correlators in a form that is
very similar to flat space amplitudes, with Mandelstam variables replaced by
certain combinations of single-state conformal generators, prove that all
tree-level AdS boundary correlators have a differential representation, and
detail the conversion of such differential expressions to position space. We
illustrate the construction through the computation of the boundary
correlators of scalars coupled to gluons and gravitons; when converted to
position space, they reproduce known results. Color-kinematics duality and BCJ
relations can be defined in analogy with their flat space counterparts, and
are respected by the scalar correlators with a gluon exchange. We also discuss
potential approaches to the double copy and find that its direct
generalization may require nontrivial extensions.

††preprint: LCTP-22-01

##  1 Introduction

The holographic duality Susskind:1994vu ; tHooft:1993dmi , and its specific
incarnation in which it relates field theories in anti-de Sitter (AdS) space
and (possibly exotic) lines of conformal fixed points (CFT) on its boundary
Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj , continues to be a source of
insight into aspects of theories on both sides of the duality. Boundary
correlation functions in the bulk are the most natural observables to exploit
the duality as they correspond to correlation functions of local CFT
operators. They access (non-perturbative) quantum gravity in a curved space-
time through strongly-coupled boundary calculations and reveal aspects of flat
space CFTs at large coupling through weakly-coupled bulk calculations.

Weakly-coupled AdS boundary correlators are traditionally computed in terms of
Witten diagrams, based on vertices derived from the bulk Lagrangian. The
difficulty of such calculations in position space Arutyunov:2000py ;
Arutyunov:2002ff ; Arutyunov:2002fh ; DHoker:1999mqo ; DHoker:1999kzh  is
similar in spirit (although less severe because of the large symmetry of AdS
space) with difficulties with tree-level calculations in general smooth curved
spaces: even the simplest diagrams are given by nontrivial bulk integrals
which lead to transcendental functions. This led to the development of several
alternative methods: the AdS momentum space Raju:2012zr ; Bzowski:2013sza ;
Farrow:2018yni ; Albayrak:2018tam ; Albayrak:2019yve ; Bzowski:2019kwd ;
Lipstein:2019mpu ; Albayrak:2020isk ; Armstrong:2020woi ; Albayrak:2020fyp ,
position-Mellin space Mack:2009gy ; Mack:2009mi ; Penedones:2010ue ;
Paulos:2011ie ; Fitzpatrick:2011ia ; Kharel:2013mka ; Penedones:2019tng ;
Rastelli:2016nze ; Zhou:2017zaw ; Rastelli:2017udc ; Alday:2020dtb ;
Alday:2020lbp ; Zhou:2021gnu ; Alday:2021ajh ; Alday:2022lkk  and momentum-
Mellin space Sleight:2021iix ; Sleight:2019hfp . While simplifying various
aspects of calculations, these representations do not completely eliminate the
problem of nontrivial integrals which still appear at tree-level in the
momentum space formulation of AdS Witten diagrams and in the inverse Mellin-
transform to position or momentum space. Moreover, even though spinning
intermediate states and the relevant factorization have been describe in
Mellin space Goncalves:2014rfa , spinning external states have been more
difficult to include. With notable exceptions Sleight:2021iix ;
Sleight:2018epi , computations of higher-spin correlators in Mellin space
often rely on supersymmetry to relate them to scalar correlators, cf. e.g.
Rastelli:2016nze ; Zhou:2017zaw ; Rastelli:2017udc ; Alday:2020dtb ;
Alday:2020lbp ; Zhou:2021gnu ; Alday:2021ajh .

Other approaches to tree-level AdS boundary correlators, which avoid some of
the previous technical difficulties and are inspired by the close analogy
between holographic correlators and flat space scattering amplitudes, are the
position-space and Mellin space analytic bootstraps introduced in
Rastelli:2016nze ; Rastelli:2017udc ; Goncalves:2019znr . They evaluate
$\textrm{AdS}_{5}$ boundary correlators in the supergravity limit in
sufficiently-symmetric theories bypassing use of the Lagrangian, relying
instead only on crossing symmetry, analyticity, superconformal symmetry, and
the flat space limit. For less symmetric situations further information is
required and may be obtained either from the bulk Lagrangian or from the
expected dual CFT following the general discussion in Heemskerk:2009pn .

Beginning with case studies, the differential representation has emerged as a
new framework for studying AdS boundary correlators. It conjecturally
expresses them as a nonlocal combination of generators of the AdS symmetry
group at various boundary points acting on a single contact integral
determined by the mass parameters of the chosen fields.111Contact integrals or
diagrams are defined in eq. (3.2) below. As will become clear below, by
“nonlocal combination of generators” we refer specifically to the appearance
of inverses of scalar combinations of generators in these combinations. It
thus naturally manifests all symmetries, encompasses all of the various
representations (position, momentum and Mellin), and may expose structures
hidden under these traditional representations, such as the existence
Diwakar:2021juk  of differential relations between color-ordered boundary
correlators analogous to the flat space Bern-Carrasco-Johansson (BCJ)
amplitudes relations Bern:2008qj . Beyond tree level, the differential
representation has proven useful for evaluating scalar one-loop Witten
diagrams in both AdS Herderschee:2021jbi  and dS Gomez:2021ujt . Originally
motivated by the infinite tension limit of certain string theory expressions
for scalar external states Eberhardt:2020ewh ; Roehrig:2020kck , the
differential representation parallels many features of the momentum-space
representation of flat-space scattering amplitudes Diwakar:2021juk , see also
Gomez:2021qfd ; Sivaramakrishnan:2021srm . In particular, conformal/$SO(d,2)$
generators correspond to momentum vectors and the conformal Ward identity
(CWI) corresponds to momentum conservation. It would be interesting to
understand if this similarity points to deeper structures which may be
accessed by reformulating the bulk theory on a noncommutative space.

In this paper, we take steps towards proving that the differential
representation exists for general AdS field theories with spinning fields in
their spectrum. We will show using a Berends-Giele-type recursion
Berends:1987me  that, at tree level, the nonlocality of the differential
representation can be put in one-to-one correspondence with bulk-bulk
propagators while at the same time being expressed entirely in terms of
boundary data. The remainder, acted upon by these nonlocal operators, is local
and involves integration over a single bulk point for arbitrary multiplicity
correlators.222Related approaches, along the lines of Mafra:2015vca ;
Mizera:2018jbh ; Garozzo:2018uzj ; Bridges:2019siz ; Cheung:2021zvb , may be
useful to construct color-kinematics-satisfying representations for boundary
correlators, but we will not follow these directions in this paper. The
remaining step is to organize this local expression as a collection of
conformal generators acting on a single contact diagram. We prove that this is
always possible for scalar AdS boundary correlators and illustrate it in
nontrivial examples. We find the differential representation of four-point
scalar correlators due to vector and graviton exchange. Comparison with the
flat space scattering amplitudes in the relevant theories reveals a close
similarity. The vector-mediated correlators can be obtained from flat space
amplitudes by replacing the Mandelstam variables with their corresponding
(suitably-ordered) conformal generators and the momentum-conservation
constraint with the contact diagram. Graviton-mediated correlators follow a
similar, although slightly more subtle, pattern. We also outline general
properties of the local factor of boundary correlators at arbitrary
multiplicity and show that in the limit of large dimension, all scalar
boundary correlators can be obtained from flat space amplitudes via this
replacement.

While the differential representation is extremely useful for understanding
(differential) properties of boundary correlators, the relation between the
differential representation and explicit functions, e.g. in position space, is
nontrivial because of the presence of nonlocal differential operators. We
provide a general strategy for converting the former into a linear combination
of contact diagrams; this method relies only on the properties of contact
diagrams under the action of the generators of the conformal group.

The paper is organized as follows. In section 2, we review aspects of the
embedding space formalism in the presence of spinning states. In section 3,
review aspects of the differential representation of boundary correlators and
construct it explicitly for scalar theories with nonderivative interactions.
In section 4, show how to separate the local and nonlocal part of each diagram
contributing to a boundary correlator, that for general spinning states the
resulting nonlocal operators are in one-to-one correspondence with bulk-bulk
propagators while simultaneously acting solely on boundary points and that the
correponding local part is given in terms of standard Feynman vertex factors.
The construction described in this section applies more generally to generic
AdS Green’s functions, with sources placed at both bulk and boundary points.
Assuming that certain differential operators defined in Ref.
Sivaramakrishnan:2021srm  exist, the construction in this section, based on
the Berends-Giele recursion, applies to general smooth curved spaces. In
section 5, we give examples of correlators mediated by vector and graviton
exchange to illustrate that the remaining local part can indeed be organized
in terms of a single contact integral acted upon by AdS symmetry generators
and the correspondence with flat space scattering amplitudes. Section 6
contains a recursive proof that the local part of scalar boundary correlators
has the form required by the differential representation. In section 7, we
provide a general strategy for converting the differential representation to a
sum of $D$-functions and section 8 contains our conclusions and outlook.
Appendix A details the AdS graviton propagator in de Donder gauge.

##### Note added:

During the completion of this project we became aware of concurrent work by
Cheung, Parra-Martinez and Sivaramakrishnan CPMS  containing some overlap with
our work for purely scalar theories. Based on the evaluation of field one-
point function and a judicious reorganization of the equations of motion,
differential representations — referred to as “isometric representations”
therein — are constructed for various bi-adjoint scalar theories and the
nonlinear sigma model on symmetric spaces. We thank the authors of ref. CPMS
for sharing their work with us before submission.

##  2 Embedding Space for Spinning States

In this section, we review Costa:2014kfa  some useful tools for manipulating
symmetric traceless tensors in the embedding space. The $\textrm{AdS}_{d+1}$
spacetime can be realized as a hyperboloid in the embedding space
$\mathbb{R}^{d,2}$, defined by the constraint
$X^{2}=-R_{\textrm{AdS}}^{2}=-1$,

$\displaystyle-(X^{0})^{2}+(X^{1})^{2}+\ldots+(X^{d})^{2}-(X^{d+1})^{2}=-1\,.$
(2.1)

Note that $X^{A}$ is also the normal vector of this hypersurface.333Unless
otherwise stated, we keep $R_{\textrm{AdS}}=1$ in our formulae. Points in the
bulk that are covered by the Poincaré patch can be parameterized as

$\displaystyle
X^{A}\equiv(X^{\mathsf{a}},X^{d},X^{d+1})=\frac{1}{z}\Big{(}x^{\mathsf{a}},\frac{1-x^{2}-z^{2}}{2},\frac{1+x^{2}+z^{2}}{2}\Big{)}\,.$
(2.2)

Points on the conformal boundary of $\textrm{AdS}_{d+1}$ are identified with
the projective null vectors of the embedding space,

$\displaystyle
P^{A}\equiv\Big{(}x^{\mathsf{a}},\frac{1-x^{2}}{2},\frac{1+x^{2}}{2}\Big{)}\,.$
(2.3)

Lorentz transformations in the embedding space $\mathbb{R}^{d,2}$ can be
identified with bulk isometry and boundary conformal transformations.

We consider a tensor $H_{A_{1}\ldots A_{s}}$ of the embedding space that
uniquely defines a symmetric traceless tensor in $\textrm{AdS}_{d+1}$. In
addition to being symmetric and traceless itself, $H_{A_{1}\ldots A_{s}}$
needs to be transverse to the AdS hypersurface,

$X^{A_{1}}H_{A_{1}\ldots A_{s}}(X)=0\ .$ (2.4)

Instead of working with $H_{A_{1}\ldots A_{s}}$ directly, we contract its
indices with an auxiliary bulk polarization vector $W^{A}$,

$H(X,W)\equiv W^{A_{1}}\ldots W^{A_{s}}H_{A_{1}\ldots A_{s}}(X)$ (2.5)

which satisfies $W^{2}=W\cdot X=0$. We can uniquely recover the tensor
$H_{A_{1}\ldots A_{s}}$ from $H(X,W)$ by acting with the operator
Costa:2014kfa ,

$\displaystyle K_{A}$
$\displaystyle=\frac{d-1}{2}\left[\frac{\partial}{\partial
W^{A}}+X_{A}\left(X\cdot\frac{\partial}{\partial
W}\right)\right]+\left(W\cdot\frac{\partial}{\partial
W}\right)\frac{\partial}{\partial W^{A}}$ (2.6)
$\displaystyle\quad+X_{A}\left(W\cdot\frac{\partial}{\partial
W}\right)\\!\left(X\cdot\frac{\partial}{\partial
W}\right)-\frac{1}{2}W_{A}\left[\frac{\partial^{2}}{\partial W\cdot\partial
W}+\left(X\cdot\frac{\partial}{\partial
W}\right)\\!\left(X\cdot\frac{\partial}{\partial W}\right)\right],$

namely,

$\displaystyle H_{A_{1}\ldots
A_{s}}(X)=\frac{1}{s!\left(\frac{d-1}{2}\right)_{s}}K_{A_{1}}\ldots
K_{A_{s}}H(X,W)\,,$ (2.7)

where $(a)_{n}\equiv\frac{\Gamma(a+n)}{\Gamma(a)}$ is the Pochhammer symbol.
The resulting tensor is guaranteed to be symmetric, traceless, and transverse
to AdS since $K_{A}$ satisfies

$\displaystyle X^{A}K_{A}=0\,,\qquad K_{A}K^{A}=0\,,\qquad
K_{A}K_{B}=K_{B}K_{A}\,.$ (2.8)

We can also define a polynomial of $K_{A}$ operators,

$\displaystyle
H(X,K)\equiv\frac{1}{s!\left(\frac{d-1}{2}\right)_{s}}K^{A_{1}}\ldots
K^{A_{s}}H_{A_{1}\ldots A_{s}}(X)\,,$ (2.9)

such that the contraction of two symmetric traceless AdS tensors is given by

$\displaystyle H_{A_{1}\ldots A_{s}}(X)F^{A_{1}\ldots
A_{s}}(X)=H(X,K)F(X,W)=F(X,K)H(X,W)\,.$ (2.10)

Many equations vastly simplify using the auxiliary polarization vectors $W$.
The AdS covariant derivative in this representation is

$\displaystyle\nabla_{A}=G_{A}{}^{B}\frac{\partial}{\partial
X^{B}}-\frac{W_{A}}{X^{2}}\left(X\cdot\frac{\partial}{\partial W}\right)\,,$
(2.11)

where $G^{AB}=\eta^{AB}-\frac{X^{A}X^{B}}{X^{2}}$ is the metric tensor that
projects dynamics onto the AdS hypersurface. The divergence of a tensor can be
represented by

$\displaystyle(\nabla\cdot H)(X,W)=(\nabla\cdot
K)H(X,W)\equiv\frac{1}{s\left(\frac{d-3+2s}{2}\right)}(\nabla_{A}K^{A})H(X,W)\,,$
(2.12)

while the Laplacian is simply
$(\nabla^{2}H)(X,W)\equiv\nabla_{A}\nabla^{A}H(X,W)$.

A similar representation exists for spinning boundary correlators. We consider
a symmetric traceless tensors $B_{A_{1},\ldots,A_{s}}(P)$ that lives on the
boundary of $\textrm{AdS}_{d+1}$. We can similarly contract it with boundary
polarization vectors $Z^{A}$,

$B(P,Z)=Z^{A_{1}}\ldots Z^{A_{s}}B_{A_{1},\ldots,A_{s}}\ .$ (2.13)

where $Z^{2}=Z\cdot P=0$. For spinning boundary states, the physical null
polarization $\epsilon^{\mathsf{a}}$ is embedded into $Z$ through

$\displaystyle Z^{A}=\Big{(}\epsilon^{\mathsf{a}},-\epsilon\cdot
x,\epsilon\cdot x\Big{)}\,.$ (2.14)

To uniquely recover $B_{A_{1}\ldots A_{s}}$ from $B(P,Z)$, we define the
boundary analog of $K_{A}$ Costa:2011mg ,

$\displaystyle\mathscr{D}^{A}=\left(\frac{d}{2}-1+Z\cdot\frac{\partial}{\partial
Z}\right)\frac{\partial}{\partial
Z_{A}}-\frac{1}{2}Z^{A}\frac{\partial^{2}}{\partial Z\cdot\partial Z}\,,$
$\displaystyle\text{such that}\quad B_{A_{1}\ldots
A_{s}}(P)=\frac{1}{s!\left(\frac{d}{2}-1\right)_{s}}\mathscr{D}^{A_{1}}\ldots\mathscr{D}^{A_{s}}B(P,Z)\
.$ (2.15)

As noted before, the generators of bulk isometry and boundary conformal
symmetry are both given by the Lorentz generators of the embedding space,

$\displaystyle D_{X}^{AB}$
$\displaystyle=\frac{1}{\sqrt{2}}\left[X^{A}\frac{\partial}{\partial
X_{B}}-X^{B}\frac{\partial}{\partial X_{A}}+W^{A}\frac{\partial}{\partial
W_{B}}-W^{B}\frac{\partial}{\partial W_{A}}\right],$ $\displaystyle
D_{i}^{AB}$
$\displaystyle=\frac{1}{\sqrt{2}}\left[P_{i}^{A}\frac{\partial}{\partial
P_{i,B}}-P_{i}^{B}\frac{\partial}{\partial
P_{i,A}}+Z_{i}^{A}\frac{\partial}{\partial
Z_{i,B}}-Z_{i}^{B}\frac{\partial}{\partial Z_{i,A}}\right].$ (2.16)

An $n$-point boundary correlator $\mathcal{A}(P_{i},Z_{i})$, or more generally
a conformal partial wave, is invariant under conformal transformations. That
is, they obey the conformal Ward identity (CWI),

$\sum_{i=1}^{n}D_{i}^{AB}\mathcal{A}(P_{i},Z_{i})=0\,.$ (2.17)

While this name is justified from the perspective of the boundary conformal
field theory, we will refer to eq. (2.17) as a "conformal Ward identity" also
from the perspective of the bulk AdS theory.

The CWI is analogous to the flat space momentum conservation of scattering
amplitudes. In particular, the eigenvalue equation for the quadratic Casimir
operator $D_{i}\cdot D_{i}$,

$\displaystyle-D_{i}\cdot D_{i}$
$\displaystyle=P_{i}\\!\cdot\\!\frac{\partial}{\partial
P_{i}}\\!\left(d{+}P_{i}\\!\cdot\\!\frac{\partial}{\partial
P_{i}}\right)+Z_{i}\\!\cdot\\!\frac{\partial}{\partial
Z_{i}}\\!\left(d{-}2{+}Z_{i}\\!\cdot\\!\frac{\partial}{\partial
Z_{i}}\right)+2Z_{i}\\!\cdot\\!\frac{\partial}{\partial
P_{i}}\\!\left(P_{i}\\!\cdot\\!\frac{\partial}{\partial Z_{i}}\right)$
$\displaystyle\cong\Delta_{i}(\Delta_{i}-d)+s(s+d-2)\equiv\mathsf{M}^{2}_{i,s}\,,$
(2.18)

is the analog of “on-shell mass” for a boundary state with spin $s$. A scalar
has zero spin, so $\mathsf{M}^{2}_{\rm scalar}=\Delta(\Delta-d)$. The $\cong$
indicates that this identity holds only when the operator acts on a conformal
partial wave that has the support of CWI. For a massless vector (gluon), we
have $\Delta=d-1$ and $s=1$, such that the quadratic Casimir vanishes,
$\mathsf{M}^{2}_{\rm gluon}=0$. On the other hand, for a graviton, we have
$\Delta=d$ and $s=2$, such that $\mathsf{M}^{2}_{\rm graviton}=2d$.444The
eigenvalue of a quadratic Casimir is _not_ the mass parameter that appears in
a bulk Lagrangian or in the equation of motion; see e.g. Gunaydin:1998km ;
Gunaydin:1998sw  for a definition of masslessness in AdS. For later
convenience we define

$\displaystyle\mathcal{D}_{ij}\equiv(D_{i}+D_{j})^{AB}(D_{i}+D_{j})_{AB}\,,\qquad\mathcal{D}_{I}\equiv\Big{(}\sum_{i\in
I}D_{i}\Big{)}{}^{AB}\Big{(}\sum_{i\in I}D_{i}\Big{)}_{AB}\,.$ (2.19)

These operators satisfy the same linear relation as flat space Mandelstam
variables on the support of the CWI. For example, at four points, we have

$\displaystyle\mathcal{D}_{12}\cong\mathcal{D}_{34}\,,\quad\mathcal{D}_{13}\cong\mathcal{D}_{24}\,,\quad\mathcal{D}_{23}\cong\mathcal{D}_{14}\,,$
$\displaystyle(\mathcal{D}_{12}+\mathcal{D}_{23}+\mathcal{D}_{13})\cong-\sum_{i=1}^{4}\mathsf{M}^{2}_{i,s_{i}}\,.$
(2.20)

However, unlike Mandelstam variables, these operators do not necessarily
commute with each other. More precisely, $\mathcal{D}_{I_{1}}$ and
$\mathcal{D}_{I_{2}}$ commute if and only if the set $I_{1}$ and $I_{2}$ are
disjoint or one set completely contains the other.

##  3 Differential representation of boundary correlators

The _differential representation_ of an $n$-point boundary correlator takes
the form555As discussed in the introduction, we assume that our boundary
correlators have a perturbative Witten diagram expansion in the bulk. From a
holographic point of view, they correspond to correlators in a putative
boundary CFT in the strong coupling limit.

$\mathcal{A}_{n}=\hat{\mathcal{A}}_{n}D_{\Delta_{1},\Delta_{2},\ldots,\Delta_{n}}$
(3.1)

where $\hat{\mathcal{A}}_{n}$ is a collection of local and nonlocal
differential operators that contains information on boundary states. For
example, for external spinning states, it contains information on their spin
and polarization. Regardless of the spin of external states,
$\hat{\mathcal{A}}_{n}$ acts on a scalar contact diagram ($D$-function),

$D_{\Delta_{1},\Delta_{2},\ldots,\Delta_{n}}=\leavevmode\hbox to63.63pt{\vbox
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{26.9267pt}{2.35571pt}\pgfsys@moveto{27.4267pt}{2.35571pt}\pgfsys@curveto{27.4267pt}{2.63185pt}{27.20284pt}{2.85571pt}{26.9267pt}{2.85571pt}\pgfsys@curveto{26.65056pt}{2.85571pt}{26.4267pt}{2.63185pt}{26.4267pt}{2.35571pt}\pgfsys@curveto{26.4267pt}{2.07957pt}{26.65056pt}{1.85571pt}{26.9267pt}{1.85571pt}\pgfsys@curveto{27.20284pt}{1.85571pt}{27.4267pt}{2.07957pt}{27.4267pt}{2.35571pt}\pgfsys@closepath\pgfsys@moveto{26.9267pt}{2.35571pt}\pgfsys@fillstroke\pgfsys@invoke{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{26.9267pt}{-2.35571pt}\pgfsys@moveto{27.4267pt}{-2.35571pt}\pgfsys@curveto{27.4267pt}{-2.07957pt}{27.20284pt}{-1.85571pt}{26.9267pt}{-1.85571pt}\pgfsys@curveto{26.65056pt}{-1.85571pt}{26.4267pt}{-2.07957pt}{26.4267pt}{-2.35571pt}\pgfsys@curveto{26.4267pt}{-2.63185pt}{26.65056pt}{-2.85571pt}{26.9267pt}{-2.85571pt}\pgfsys@curveto{27.20284pt}{-2.85571pt}{27.4267pt}{-2.63185pt}{27.4267pt}{-2.35571pt}\pgfsys@closepath\pgfsys@moveto{26.9267pt}{-2.35571pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{26.10875pt}{-6.99553pt}\pgfsys@moveto{26.60875pt}{-6.99553pt}\pgfsys@curveto{26.60875pt}{-6.71939pt}{26.38489pt}{-6.49553pt}{26.10875pt}{-6.49553pt}\pgfsys@curveto{25.83261pt}{-6.49553pt}{25.60875pt}{-6.71939pt}{25.60875pt}{-6.99553pt}\pgfsys@curveto{25.60875pt}{-7.27167pt}{25.83261pt}{-7.49553pt}{26.10875pt}{-7.49553pt}\pgfsys@curveto{26.38489pt}{-7.49553pt}{26.60875pt}{-7.27167pt}{26.60875pt}{-6.99553pt}\pgfsys@closepath\pgfsys@moveto{26.10875pt}{-6.99553pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\int_{\textrm{AdS}}dX\prod_{i=1}^{n}E_{\Delta_{i}}(X,P_{i})\
.$ (3.2)

Here $E_{\Delta_{i}}(X,P_{i})=\frac{1}{(-2P_{i}\cdot X)^{\Delta_{i}}}$ is the
scalar bulk-boundary propagator.666In this work, we omit the overall
normalization $\mathcal{C}_{\Delta}$ of a bulk-boundary propagator. Since for
a given boundary correlator all bulk-boundary propagators are the same, the
omitted factors can only alter the overall normalization. The $D$-function
thus defined provides the support of CWI to the boundary correlator, and plays
a role similar to the momentum conservation constraint in the flat space $S$
matrix. Generally, up to subtleties related to the noncommutativity of
conformal generators, $\hat{\mathcal{A}}_{n}$ takes a form similar to the
corresponding flat space amplitude. With examples and justification to be
detailed in later sections, we claim that a generic differential correlator
$\hat{\mathcal{A}}_{n}$ can be written as

$\displaystyle\hat{\mathcal{A}}_{n}=\sum_{g\in\text{cubic}}C_{g}\prod_{I\in
g}\frac{1}{\mathcal{D}_{I}+\mathsf{M}^{2}_{I,s}}\hat{N}_{g}\,,$ (3.3)

where $\hat{N}_{g}$ is a local operator. The summation runs over all the
trivalent Witten diagrams. Importantly, the bulk-bulk propagator associated
with an internal leg $I$ is given by an inverse differential operator,

$\displaystyle I\;\scalebox{1.25}{$\Bigg{\\{}$}\;\leavevmode\hbox
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}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@moveto{19.91693pt}{0.0pt}\pgfsys@curveto{19.91693pt}{3.14284pt}{17.36922pt}{5.69055pt}{14.22638pt}{5.69055pt}\pgfsys@curveto{11.08354pt}{5.69055pt}{8.53583pt}{3.14284pt}{8.53583pt}{0.0pt}\pgfsys@curveto{8.53583pt}{-3.14284pt}{11.08354pt}{-5.69055pt}{14.22638pt}{-5.69055pt}\pgfsys@curveto{17.36922pt}{-5.69055pt}{19.91693pt}{-3.14284pt}{19.91693pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.36282pt}{-8.31633pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\Delta,s$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\longrightarrow\frac{1}{\mathcal{D}_{I}+\mathsf{M}^{2}_{I,s}}\,,$
(3.4)

where $\mathcal{D}_{I}$ is the quadratic Casimir operator of the AdS symmetry
group acting on all external points connected to one end of that internal line
and $\mathsf{M}^{2}_{I,s}$ is the eigenvalue of the quadratic Casimir for the
representation of the internal state. The inverse differential operator
$(\mathcal{D}_{I}+\mathsf{M}^{2}_{I,s})^{-1}$ will be called a propagator in
the context of differential representation. We stress that this form is also
expected for theories with quartic and higher-point interaction vertices and,
if necessary, such vertices are trivially resolved into trivalent ones by
multiplying and dividing by suitable operators. For theories with fields
charged under a non-abelian gauge or flavor symmetry, $C_{g}$ in eq. (3.4) is
the color factor associated with the diagram. The (local) operator
$\hat{N}_{g}$, acting directly on the $D$-function, can be understood as the
analog of the flat-space kinematic numerator factor.

To justify eq. (3.3) and in particular the appearance of the nonlocal
operators (3.4), we start with the differential representation of the bulk-
bulk propagator, which was first proposed for bi-adjoint $\phi^{3}$ theory in
Ref. Eberhardt:2020ewh . As the simplest example, we consider the following
s-channel scalar Witten diagram

$\displaystyle\leavevmode\hbox to53.84pt{\vbox
to50.35pt{\pgfpicture\makeatletter\hbox{\hskip
26.91995pt\lower-25.17303pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}
{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@curveto{21.33957pt}{11.78566pt}{11.78566pt}{21.33957pt}{0.0pt}{21.33957pt}\pgfsys@curveto{-11.78566pt}{21.33957pt}{-21.33957pt}{11.78566pt}{-21.33957pt}{0.0pt}\pgfsys@curveto{-21.33957pt}{-11.78566pt}{-11.78566pt}{-21.33957pt}{0.0pt}{-21.33957pt}\pgfsys@curveto{11.78566pt}{-21.33957pt}{21.33957pt}{-11.78566pt}{21.33957pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}{{}}{}\pgfsys@moveto{15.08904pt}{15.08904pt}\pgfsys@moveto{16.08904pt}{15.08904pt}\pgfsys@curveto{16.08904pt}{15.64133pt}{15.64133pt}{16.08904pt}{15.08904pt}{16.08904pt}\pgfsys@curveto{14.53674pt}{16.08904pt}{14.08904pt}{15.64133pt}{14.08904pt}{15.08904pt}\pgfsys@curveto{14.08904pt}{14.53674pt}{14.53674pt}{14.08904pt}{15.08904pt}{14.08904pt}\pgfsys@curveto{15.64133pt}{14.08904pt}{16.08904pt}{14.53674pt}{16.08904pt}{15.08904pt}\pgfsys@closepath\pgfsys@moveto{15.08904pt}{15.08904pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.79361pt}{17.0567pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\Delta_{3}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}{{}}{}\pgfsys@moveto{-15.08904pt}{15.08904pt}\pgfsys@moveto{-14.08904pt}{15.08904pt}\pgfsys@curveto{-14.08904pt}{15.64133pt}{-14.53674pt}{16.08904pt}{-15.08904pt}{16.08904pt}\pgfsys@curveto{-15.64133pt}{16.08904pt}{-16.08904pt}{15.64133pt}{-16.08904pt}{15.08904pt}\pgfsys@curveto{-16.08904pt}{14.53674pt}{-15.64133pt}{14.08904pt}{-15.08904pt}{14.08904pt}\pgfsys@curveto{-14.53674pt}{14.08904pt}{-14.08904pt}{14.53674pt}{-14.08904pt}{15.08904pt}\pgfsys@closepath\pgfsys@moveto{-15.08904pt}{15.08904pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-23.58694pt}{17.0567pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\Delta_{2}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}{{}}{}\pgfsys@moveto{-15.08904pt}{-15.08904pt}\pgfsys@moveto{-14.08904pt}{-15.08904pt}\pgfsys@curveto{-14.08904pt}{-14.53674pt}{-14.53674pt}{-14.08904pt}{-15.08904pt}{-14.08904pt}\pgfsys@curveto{-15.64133pt}{-14.08904pt}{-16.08904pt}{-14.53674pt}{-16.08904pt}{-15.08904pt}\pgfsys@curveto{-16.08904pt}{-15.64133pt}{-15.64133pt}{-16.08904pt}{-15.08904pt}{-16.08904pt}\pgfsys@curveto{-14.53674pt}{-16.08904pt}{-14.08904pt}{-15.64133pt}{-14.08904pt}{-15.08904pt}\pgfsys@closepath\pgfsys@moveto{-15.08904pt}{-15.08904pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-23.58694pt}{-20.57693pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\Delta_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}{{}}{}\pgfsys@moveto{15.08904pt}{-15.08904pt}\pgfsys@moveto{16.08904pt}{-15.08904pt}\pgfsys@curveto{16.08904pt}{-14.53674pt}{15.64133pt}{-14.08904pt}{15.08904pt}{-14.08904pt}\pgfsys@curveto{14.53674pt}{-14.08904pt}{14.08904pt}{-14.53674pt}{14.08904pt}{-15.08904pt}\pgfsys@curveto{14.08904pt}{-15.64133pt}{14.53674pt}{-16.08904pt}{15.08904pt}{-16.08904pt}\pgfsys@curveto{15.64133pt}{-16.08904pt}{16.08904pt}{-15.64133pt}{16.08904pt}{-15.08904pt}\pgfsys@closepath\pgfsys@moveto{15.08904pt}{-15.08904pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.79361pt}{-20.57693pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\Delta_{4}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{-8.5359pt}{0.0pt}\pgfsys@moveto{-7.5359pt}{0.0pt}\pgfsys@curveto{-7.5359pt}{0.55229pt}{-7.98361pt}{1.0pt}{-8.5359pt}{1.0pt}\pgfsys@curveto{-9.0882pt}{1.0pt}{-9.5359pt}{0.55229pt}{-9.5359pt}{0.0pt}\pgfsys@curveto{-9.5359pt}{-0.55229pt}{-9.0882pt}{-1.0pt}{-8.5359pt}{-1.0pt}\pgfsys@curveto{-7.98361pt}{-1.0pt}{-7.5359pt}{-0.55229pt}{-7.5359pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-8.5359pt}{0.0pt}\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@moveto{9.5359pt}{0.0pt}\pgfsys@curveto{9.5359pt}{0.55229pt}{9.0882pt}{1.0pt}{8.5359pt}{1.0pt}\pgfsys@curveto{7.98361pt}{1.0pt}{7.5359pt}{0.55229pt}{7.5359pt}{0.0pt}\pgfsys@curveto{7.5359pt}{-0.55229pt}{7.98361pt}{-1.0pt}{8.5359pt}{-1.0pt}\pgfsys@curveto{9.0882pt}{-1.0pt}{9.5359pt}{-0.55229pt}{9.5359pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-8.5359pt}{0.0pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{15.08904pt}{15.08904pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{15.08904pt}{-15.08904pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-15.08904pt}{15.08904pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-15.08904pt}{-15.08904pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.91667pt}{-8.31633pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{$\Delta$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\\!\\!=\int_{\textrm{AdS}}dXdY$
$\displaystyle
E_{\Delta_{3}}(X,P_{3})E_{\Delta_{4}}(X,P_{4})\mathcal{G}_{\Delta}(X,Y)E_{\Delta_{1}}(Y,P_{1})E_{\Delta_{2}}(Y,P_{2})\,,$
(3.5)

and rewrite the bulk-bulk propagator as the inverse of the free-field operator
acting on the delta function

$\displaystyle\mathcal{G}_{\Delta}(X,Y)=-\frac{1}{\nabla_{X}^{2}-\Delta(\Delta-d)}\delta(X,Y)\,.$
(3.6)

We can now integrate out the bulk point $Y$ and get

$\displaystyle\leavevmode\hbox to53.84pt{\vbox
to50.35pt{\pgfpicture\makeatletter\hbox{\hskip
26.91995pt\lower-25.17303pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}
{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@curveto{21.33957pt}{11.78566pt}{11.78566pt}{21.33957pt}{0.0pt}{21.33957pt}\pgfsys@curveto{-11.78566pt}{21.33957pt}{-21.33957pt}{11.78566pt}{-21.33957pt}{0.0pt}\pgfsys@curveto{-21.33957pt}{-11.78566pt}{-11.78566pt}{-21.33957pt}{0.0pt}{-21.33957pt}\pgfsys@curveto{11.78566pt}{-21.33957pt}{21.33957pt}{-11.78566pt}{21.33957pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}{{}}{}\pgfsys@moveto{15.08904pt}{15.08904pt}\pgfsys@moveto{16.08904pt}{15.08904pt}\pgfsys@curveto{16.08904pt}{15.64133pt}{15.64133pt}{16.08904pt}{15.08904pt}{16.08904pt}\pgfsys@curveto{14.53674pt}{16.08904pt}{14.08904pt}{15.64133pt}{14.08904pt}{15.08904pt}\pgfsys@curveto{14.08904pt}{14.53674pt}{14.53674pt}{14.08904pt}{15.08904pt}{14.08904pt}\pgfsys@curveto{15.64133pt}{14.08904pt}{16.08904pt}{14.53674pt}{16.08904pt}{15.08904pt}\pgfsys@closepath\pgfsys@moveto{15.08904pt}{15.08904pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.79361pt}{17.0567pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\Delta_{3}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}{{}}{}\pgfsys@moveto{-15.08904pt}{15.08904pt}\pgfsys@moveto{-14.08904pt}{15.08904pt}\pgfsys@curveto{-14.08904pt}{15.64133pt}{-14.53674pt}{16.08904pt}{-15.08904pt}{16.08904pt}\pgfsys@curveto{-15.64133pt}{16.08904pt}{-16.08904pt}{15.64133pt}{-16.08904pt}{15.08904pt}\pgfsys@curveto{-16.08904pt}{14.53674pt}{-15.64133pt}{14.08904pt}{-15.08904pt}{14.08904pt}\pgfsys@curveto{-14.53674pt}{14.08904pt}{-14.08904pt}{14.53674pt}{-14.08904pt}{15.08904pt}\pgfsys@closepath\pgfsys@moveto{-15.08904pt}{15.08904pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-23.58694pt}{17.0567pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\Delta_{2}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}{{}}{}\pgfsys@moveto{-15.08904pt}{-15.08904pt}\pgfsys@moveto{-14.08904pt}{-15.08904pt}\pgfsys@curveto{-14.08904pt}{-14.53674pt}{-14.53674pt}{-14.08904pt}{-15.08904pt}{-14.08904pt}\pgfsys@curveto{-15.64133pt}{-14.08904pt}{-16.08904pt}{-14.53674pt}{-16.08904pt}{-15.08904pt}\pgfsys@curveto{-16.08904pt}{-15.64133pt}{-15.64133pt}{-16.08904pt}{-15.08904pt}{-16.08904pt}\pgfsys@curveto{-14.53674pt}{-16.08904pt}{-14.08904pt}{-15.64133pt}{-14.08904pt}{-15.08904pt}\pgfsys@closepath\pgfsys@moveto{-15.08904pt}{-15.08904pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-23.58694pt}{-20.57693pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\Delta_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}{{}}{}\pgfsys@moveto{15.08904pt}{-15.08904pt}\pgfsys@moveto{16.08904pt}{-15.08904pt}\pgfsys@curveto{16.08904pt}{-14.53674pt}{15.64133pt}{-14.08904pt}{15.08904pt}{-14.08904pt}\pgfsys@curveto{14.53674pt}{-14.08904pt}{14.08904pt}{-14.53674pt}{14.08904pt}{-15.08904pt}\pgfsys@curveto{14.08904pt}{-15.64133pt}{14.53674pt}{-16.08904pt}{15.08904pt}{-16.08904pt}\pgfsys@curveto{15.64133pt}{-16.08904pt}{16.08904pt}{-15.64133pt}{16.08904pt}{-15.08904pt}\pgfsys@closepath\pgfsys@moveto{15.08904pt}{-15.08904pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.79361pt}{-20.57693pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\Delta_{4}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}{}\pgfsys@moveto{-15.08904pt}{-15.08904pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\int_{\textrm{AdS}}dX$
$\displaystyle E_{\Delta_{3}}(X,P_{3})E_{\Delta_{4}}(X,P_{4})$
$\displaystyle\times\frac{(-1)}{\nabla^{2}-\Delta(\Delta-d)}E_{\Delta_{1}}(X,P_{1})E_{\Delta_{2}}(X,P_{2})\,.$
(3.7)

When acting on a product of scalar bulk-boundary propagators, the AdS
Laplacian $\nabla^{2}$ can also be written as $\nabla^{2}=-D_{X}\cdot D_{X}$.
Then, the identity

$\displaystyle D_{X}\cdot
D_{X}\Big{[}E_{\Delta_{1}}(X,P_{1})E_{\Delta_{2}}(X,P_{2})\Big{]}=\mathcal{D}_{12}\Big{[}E_{\Delta_{1}}(X,P_{1})E_{\Delta_{2}}(X,P_{2})\Big{]}\
,$ (3.8)

which is a consequence of the conformal Ward identity, converts the quadratic
Casimir operator for the bulk point $X$ to the quadratic Casimir operator for
the pair $(1,2)$ of boundary points. This gives the differential
representation of the s-channel Witten diagram:

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}
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
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(3.9)

It is exactly analogous to the corresponding scattering amplitude in flat
space. The derivation for higher point scalar Witten diagrams is similar. We
note that the operators $\mathcal{D}_{I}$ associated with the internal lines
all commute with each other, so there is no ordering ambiguity in the
propagators of the differential representation.

The above derivation applies specifically to scalar intermediate states. In
the next section we will generalize eq. (3.9) by showing that the bulk-bulk
propagators for a generic spinning state in a generic Witten diagram is
equivalent to the inverse differential operators in eq. (3.4). More precisely,
we will show that each Witten diagram is exactly given by the action of such
inverse operators on contact integrals. However, we emphasize that the
implication of eq. (3.3) is stronger: it requires that all the aforementioned
contact integrals can be realized as local operators $\hat{N}_{g}$ acting on a
single $D$-function. Although we do not have a general proof of this stronger
statement, it is supported by several nontrivial examples that involve
derivatively coupled fields, for example NLSM Diwakar:2021juk  and Yang-Mills-
Chern-Simons on $\textrm{AdS}_{3}$ Roehrig:2020kck . In section 5, we will
extend this list to include scalars minimally coupled to gluons and gravitons.

Because of its similarity to flat space amplitudes that goes beyond that of
e.g. Mellin space or AdS momentum representations, the differential
representation (3.3) can potentially manifest certain hidden structures in the
boundary correlators, one of which is color-kinematics duality. For this
purpose, we restrict ourselves to theories with only AdS-massless
states.777See e.g. Gunaydin:1998km ; Gunaydin:1998sw  for a general definition
of masslessness in AdS space. As discussed in Diwakar:2021juk , with a
correlator organized as in eq. (3.3), color-kinematics duality is formally
defined as in flat space. For example, at four points, a color dressed
correlator is said to obey the duality if it can be written in the form,

$\displaystyle\mathcal{A}_{4}=\left[C_{\rm
s}\frac{1}{\mathcal{D}_{12}}\hat{N}_{\rm s}+C_{\rm
t}\frac{1}{\mathcal{D}_{23}}\hat{N}_{\rm t}+C_{\rm
u}\frac{1}{\mathcal{D}_{13}}\hat{N}_{\rm u}\right]D_{d,d,d,d}\,,$ (3.10)

where the kinematics numerators satisfy the same Jacobi identity as the color
factors,

$\displaystyle C_{\rm s}+C_{\rm t}+C_{\rm u}=0\,,\qquad\hat{N}_{\rm
s}+\hat{N}_{\rm t}+\hat{N}_{\rm u}=0\,.$ (3.11)

We emphasize that the kinematic Jacobi is now an operator relation. For scalar
boundary states, the kinematic numerators consist of $\mathcal{D}_{ij}$
operators, while further operators may also be needed for spinning boundary
states. The existence of the above BCJ representation is nontrivial and is
currently a conjecture even for theories that exhibit color-kinematics duality
in flat space. However, if a correlator does obey the duality, then eq. (3.3)
immediately implies nontrivial differential relations among color ordered
correlators Diwakar:2021juk . For the four-point example in eq. (3.10) it
yields

$\mathcal{D}_{12}A(1,2,3,4)=\mathcal{D}_{13}A(1,3,2,4)\,.$ (3.12)

The derivation is the same as that for the BCJ amplitudes relations of flat
space amplitudes, and can be easily generalized to generic cases. The NLSM
correlators obey color-kinematics duality up to at least six points
Diwakar:2021juk . It was also shown in Ref. Diwakar:2021juk  that the four-
gluon color-ordered correlators satisfy the differential BCJ relation (3.12).
A color-kinematics-satisfying representation of this correlator, as eq.
(3.10), is yet to be found, but remains plausible. In addition to exposing
color-kinematics duality, the differential representation will likely be
instrumental in revealing the existence of double copy relations for AdS
correlators.888A particular double copy of NLSM correlators into special
Galileon-like theory in $\textrm{AdS}_{4}$ is discussed in
Sivaramakrishnan:2021srm .

We remark that the straightforward connection between color-kinematics duality
and BCJ relations is one of the most attractive features of the differential
representation. This is due to the fact that in the massless limit the CWI
takes the same form and has the same consequences as flat space momentum
conservation for massless kinematics. This connection is absent in momentum-
space and Mellin-space boundary correlators that respect the color-kinematics
duality. For example, in the momentum space, the obstruction to a BCJ relation
is a contact term proportional to the total energy, which is caused by the
non-conservation of the momentum in the radial direction Armstrong:2020woi ;
Albayrak:2020fyp . In Mellin space, the proliferation of massive poles in
Mandelstam variables also obscures the existence of a simple amplitude
relation Zhou:2021gnu . It would be interesting to understand manifestation of
eq. (3.12) and its higher-point extension Diwakar:2021juk  in these
formulations of AdS boundary correlators.

##  4 Isolating nonlocalities of AdS correlation functions

An essential step towards constructing differential representations for AdS
correlators is expressing all bulk-bulk propagators as inverse differential
operators acting on the external labels and identifying the remainder. To also
capture their origin in particle exchange, we will refer to these operators as
“nonlocalities”. We will refer to the remainder as “the integrand” or “the
local factor” and its general expression is that of a rational function
perhaps acted upon by local differential operators. The Berends-Giele
recursion in AdS will provide a means to a generic construction of this
separation.

The flat space (off-shell) Berends-Giele recursion Berends:1987me  is an
algorithmic construction of higher-point Green’s functions in terms of lower-
point ones and propagators and vertex factors constructed from the Lagrangian.
It can also be interpreted as the perturbative construction of solutions to
the classical equations of motion in the presence of arbitrary sources or,
alternatively, as the construction of the one-point function in the presence
of those sources Monteiro:2011pc . In this latter interpretations it has a
straightforward generalization to more general spacetimes.

We now introduce this formulation in the context of computing AdS correlation
functions. For a generic field theory on AdS background, we organize the field
content into a vector $\Phi\equiv(\phi_{s_{1}},\phi_{s_{2}},\dots)^{\rm T}$,
where $s_{1,2,\ldots}$ are the spins of the corresponding fields. The free
equation of motion for $\Phi$ can be described by a diagonal matrix of
operators $\mathbb{O}\equiv{\rm diag}(O_{s_{1}},O_{s_{2}},\dots)$, and the
interaction is given by some Lagrangian $\mathcal{L}_{\rm int}$. Suppose we
couple the system to the sources
$J^{(0)}\equiv(J^{(0)}_{s_{1}},J^{(0)}_{s_{2}},\dots)^{\rm T}$; we can
formally write the full equation of motion as

$\displaystyle\mathbb{O}_{X}\Phi(X)-\frac{\delta{\cal L}_{\rm
int}}{\delta\Phi}(X)=-\Pi J^{(0)}(X)\ ,$ (4.1)

where $\Pi J^{(0)}\equiv(\Pi_{s_{1}}\circ J_{s_{1}}^{(0)},\Pi_{s_{2}}\circ
J_{s_{2}}^{(0)},\ldots)^{\rm T}$ projects the sources to their symmetric and
traceless components. Here $\Pi_{s}$ is the spin-$s$ projector, and the circle
$\circ$ denotes the implicit index contraction. We can solve the equation of
motion iteratively,

$\displaystyle\Phi(X)=\sum_{k=1}^{\infty}\Phi^{(k)}(X)\,,$ (4.2)

where $\Phi^{(k)}$ couples to $k$ source vectors $J^{(0)}$. Assuming the
sources are localized, we can write the formal solution coupling to $k$
sources as

$\displaystyle\Phi^{(1)}(X,Y_{1})$ $\displaystyle=-\mathbb{O}^{-1}_{X}\Pi
J^{(0)}(X,Y_{1})$ (4.3) $\displaystyle\Phi^{(k)}(X,\bm{Y}_{k})$
$\displaystyle=\left.\mathbb{O}^{-1}_{X}\Pi\frac{\delta{\cal L}_{\rm
int}}{\delta\Phi}(X)\right|_{\Phi(X)\rightarrow\sum_{i=1}^{k-1}\Phi^{(i)}(X,\bm{Y}_{i})}^{k\text{
factors of }J^{(0)}}$ $\displaystyle\equiv-\mathbb{O}^{-1}_{X}\Pi
J^{(k)}(X,\bm{Y}_{k})\ ,$ (4.4)

where $\bm{Y}_{k}\equiv\\{Y_{1},Y_{2},\ldots,Y_{k}\\}$ is the set of $k$
source locations. The restriction to “$k$ factors of $J^{(0)}$” instructs one
to pick the term with $k$ sources after the variation of the interaction
Lagrangian is evaluated on the solution with $k-1$ sources, and the last line
defines the current $\Pi J^{(k)}(X,\bm{Y}_{k})$, which is also the $k$-source
term of the one-point function of $\Phi$. Here $\mathbb{O}^{-1}$ is the
inverse of the operator $\mathbb{O}$ and is to be interpreted as the negative
of the Green’s function of that operator, i.e. the diagonal matrix of Green’s
functions for the operators $O_{s_{i}}$. Thus, restoring the position
dependence and denoting this Green’s function by $\mathbb{G}(X,U)$, the
solution is

$\displaystyle\Phi^{(1)}(X,Y_{1})$ $\displaystyle=\int
dU\,\mathbb{G}(X,U)\circ\Pi J^{(0)}(U,Y_{1})\,,$ (4.5)
$\displaystyle\Phi^{(k)}(X,\bm{Y}_{k})$ $\displaystyle=\int
dU\,\mathbb{G}(X,U)\circ\Pi\left.\frac{\delta{\cal L}_{\rm
int}}{\delta\Phi}(U)\right|_{\Phi(U)\rightarrow\sum_{i=1}^{k-1}\Phi^{(k)}(U,\bm{Y}_{i})}^{k\text{
factors of }J^{(0)}}$ $\displaystyle=\int dU\,\mathbb{G}(X,U)\circ\Pi
J^{(k)}(U,\bm{Y}_{k})\ ,$ (4.6)

where $\circ$ denotes implicit index contraction between Green’s functions and
vertices and sources, and $\bm{Y}_{k}$ is the vector of location of $k$
sources as before. The correlation function with $k+1$ bulk points is then
obtained by trivially differentiating with respect to the $k$ sources.999In
flat space, scattering amplitudes are obtained by replacing the sources by
$\mathbb{O}$ acting on solutions of the free equations of motion (i.e. the
external legs are “amputated”, as instructed by the LSZ reduction). In AdS
space, placing the sources $J^{(0)}$ on the boundary yields the standard
position-space form or Witten diagrams. It will also bring us closer to
constructing a differential representation for AdS correlation functions.

To this end, it will be useful to first discuss the free equation of motion
for spin-$s$ fields in AdS space and their properties vis-à-vis the AdS
symmetry group. Such fields are described by symmetric traceless $s$-index
tensors; for a uniform treatment we will contract their free indices as well
as those of sources with null vectors $W$, as in eq. (2.5), so all fields and
sources are formally scalars; we will refer to them as $\phi_{s}(X,W)$ and
$J_{s}^{(0)}(X,W)$, respectively. If necessary, the tensors can be recovered
by stripping off the null vectors using a suitable product of $K$ operators
defined in eq. (2.6). Index contract is realized as in eq. (2.10). We can also
drop the symmetric and traceless projector $\Pi$ since the action of $K$ on
$W$ automatically enforces these properties. We will assume that the gauge
symmetries have been fixed such that the free equation of motion is given by
the operator ${O}_{s}=(\nabla_{X}^{2}-m_{s}^{2})$, so the Green’s function
equation is Costa:2014kfa

$\displaystyle(\nabla_{X}^{2}-m_{s}^{2})G(X,W_{X},U,W_{U})=-\delta(X,U)(W_{X}\cdot
W_{U})^{j}\,,$ (4.7) $\displaystyle m_{s}^{2}=\Delta(\Delta-d)-s\,.$

Here, the covariant derivative $\nabla_{X,A}$ is given in eq. (2.11), and
$m_{s}$ is the mass parameter in the action. “Massless” spin-$s$ fields in AdS
Fronsdal:1978rb  are given by $\Delta=s+d-2$ such that
$m_{s}^{2}=(s+d-2)(s-2)-s$.101010Scalar fields are an exception to this rule,
obeying $\Delta=d$. In general, however, $m_{s}$ can be arbitrary; if the AdS
theory has a string theory interpretation, $m_{s}^{2}$ is related to the mass
of string states and, consequently, for massive string states to the dimension
of operators developing anomalous dimensions in the dual CFT.

On symmetry grounds, one might expect that the quadratic operator is closely
related to the quadratic Casimir of $SO(d,2)$. We will see that this is indeed
the case. Using the transversality properties (2.4) of AdS fields in embedding
space,

$\displaystyle X\cdot\frac{\partial}{\partial W}\phi_{s}(X,W)=0\,,\quad
W\cdot\frac{\partial}{\partial W}\phi_{s}(X,W)=s\phi_{s}(X,W)\,,$ (4.8)

one can show that $\nabla^{2}$ and $-D_{X}\cdot D_{X}$, the quadratic Casimir
of $SO(d,2)$ at position $X$, are indeed very similar,

$\displaystyle\nabla_{X}^{2}\phi_{s}(X,W)$
$\displaystyle=\partial_{A}(G^{AB}\partial_{B})\phi_{s}(X,W)-s\phi_{s}(X,W)\
,$ $\displaystyle-D_{X}\cdot D_{X}\phi_{s}(X,W)$
$\displaystyle=\partial_{A}(G^{AB}\partial_{B})\phi_{s}(X,W)+s(s+d-2)\phi_{s}(X,W)\
,$ (4.9)

where the generators of $SO(d,2)$ are defined in eq. (2.16), and include
actions on the spin degrees of freedom. Thus, the quadratic operator can be
written as

$\displaystyle(\nabla_{X}^{2}-m_{s}^{2})\phi_{s}(X,W)=\Big{(}-D_{X}\cdot
D_{X}-\mathsf{M}_{s}^{2}\Big{)}\phi_{s}(X,W)$ (4.10)

where $\mathsf{M}^{2}_{s}=m_{s}^{2}+s(s+d-1)$. For “massless” spin-$s$ fields
in AdS, this becomes $\mathsf{M}^{2}_{s}=2(s-1)(s+d-2)$, i.e. the quadratic
Casimir for a $s$-index symmetric traceless representation.111111Massless
scalar fields are an exception as for them $m_{s}^{2}=0$ and $s=0$, so
$\mathsf{M}^{2}_{s}=0$. The relation (4.10) holds for generic bulk field, and
we will later use it for the current $J^{(k)}$ in eq. (4.6).

The close relation between the quadratic operator for spin-$s$ fields and the
quadratic Casimir of the corresponding representation together with the
conformal Ward identity are key to organizing the operator formulation of the
Berends-Giele recursion in eq. (4.4) in a form in which all nonlocalities
represented by inverse operators depend only on the position of the external
sources.121212This is the AdS analog of the flat space feature that all
propagators in tree-level Feynman diagrams are determined by the external
momenta. This will reduce the construction of such AdS correlation functions
to the construction of “the integrand”.

To see this we note that because all indices of fields are saturated by null
vectors $W$ which transform under AdS symmetry group following eq. (2.16),
they are all scalars and therefore the vector of currents
$J^{(k)}(X,\bm{Y}_{k})$ defined in eq. (4.4) and appended with $W$ dependence
is also a vector of scalars. It consequently obeys the conformal Ward identity

$\displaystyle\Big{(}D_{X}^{AB}+\sum_{i=1}^{k}D_{Y_{i}}^{AB}\Big{)}J^{(k)}(X,W,\bm{Y}_{k},\bm{W}_{k})=0\
.$ (4.11)

Here $D_{Y_{i}}^{AB}$ are $SO(d,2)$ generators acting on $Y_{i}$ and include
action on null vectors $W_{i}$ (the set of which is denoted by $\bm{W}_{k}$)
saturating the tensor indices at that (possibly boundary) position. It is then
trivial to see that, as long as $X\neq Y_{i}$ for all $i=1,\dots,k$,

$\displaystyle D_{X}\cdot
D_{X}J^{(k)}(X,W,\bm{Y}_{k},\bm{W}_{k})=\mathcal{D}_{Y_{1}\ldots
Y_{k}}J^{(k)}(X,W,\bm{Y}_{k},\bm{W}_{k})\ .$ (4.12)

From here, eq. (4.10) and the definition of $\mathbb{O}_{X}$, it follows that

$\displaystyle\quad-\mathbb{O}_{X}^{-1}\ J^{(k)}(X,W,\bm{Y}_{k},\bm{W}_{k})$
$\displaystyle={\rm diag}\left(\frac{1}{\mathcal{D}_{Y_{1}\dots
Y_{k}}+\mathsf{M}_{s_{1}}^{2}},\frac{1}{\mathcal{D}_{Y_{1}\dots
Y_{k}}+\mathsf{M}_{s_{2}}^{2}},\dots\right)J^{(k)}(X,W,\bm{Y}_{k},\bm{W}_{k})$
$\displaystyle\equiv\mathbb{O}_{\bm{Y}_{k}}^{-1}J^{(k)}(X,W,\bm{Y}_{k},\bm{W}_{k})\
.$ (4.13)

Thus, each step in the recursion relation (4.4) involves only quantities that
are local with respect to the bulk point $X$,

$\displaystyle\Phi^{(1)}(X,W,Y_{1},W_{1})$ $\displaystyle=\int
dU\,\mathbb{G}(X,W,U,K_{U})J^{(0)}(U,W_{U},Y_{1},W_{1})\,,$ (4.14)
$\displaystyle\Phi^{(k)}(X,W,\bm{Y}_{k},\bm{W}_{k})$
$\displaystyle=\mathbb{O}^{-1}_{\bm{Y}_{k}}\left.\frac{\delta{\cal L}_{\rm
int}}{\delta\Phi}(X,W)\right|_{\Phi(X)\rightarrow\sum_{i=1}^{k-1}\Phi^{(i)}(X,\bm{Y}_{i})}^{k\text{
factors of }J^{(0)}}$
$\displaystyle=\mathbb{O}^{-1}_{\bm{Y}_{k}}J^{(k)}(X,W,\bm{Y}_{k},\bm{W}_{k})\,.$
(4.15)

In the first step of the recursion, we do not introduce $\mathbb{O}^{-1}_{X}$
but write out $\mathbb{G}$ explicitly because they will form the bulk
integrand after factoring out all the inverse differential operators related
to particle-exchange nonlocalities. If the source $J^{(0)}$ is inserted at a
boundary point $P_{1}$, these one-point Green’s functions become bulk-boundary
propagators $\mathbb{E}\equiv\textrm{diag}(E^{s_{1}},E^{s_{2}},\ldots)$
Freedman:1998tz ; Mueck:1998wkz ,

$\displaystyle\Phi^{(1)}(X,W,P_{1},Z_{1})=\mathbb{E}(X,W,P_{1},Z_{1})j(P_{1})\,,$
(4.16)

and $j=(j_{s_{1}},j_{s_{2}},\ldots)^{\rm T}$ is a vector of distribution
functions for boundary sources. The bulk-boundary propagators are part of the
definition of the contact integral in the differential representation, see
section 3.

Our Berends-Giele recursion works for both bulk and boundary correlation
functions. In the following, we restrict ourselves to boundary correlators,
and consider only sources of the form (4.16). The $n$-point boundary
correlator can then be extracted from $\Phi^{(n-1)}$. We first remove the
overall nonlocality $\mathbb{O}_{\bm{Y}_{n-1}}^{-1}=\mathbb{O}_{Y_{1}\ldots
Y_{n-1}}^{-1}$ from $\Phi^{(n-1)}$ and then contract it with the bulk-boundary
propagator originated from the last boundary source at $P_{n}$,

$\displaystyle\mathcal{W}[j]$ $\displaystyle\equiv\int
dXj(P_{n})\mathbb{E}(X,K,P_{n},Z_{n})\mathbb{O}_{Y_{1}\ldots
Y_{n-1}}\Phi^{(n-1)}(X,W,\bm{P}_{n-1},\bm{Z}_{n-1})$ $\displaystyle=\int
dXj(P_{n})\mathbb{E}(X,K,P_{n},Z_{n})J^{(n-1)}(X,W,\bm{P}_{n-1},\bm{Z}_{n-1})\,.$
(4.17)

What we get is nothing but the generating functional for connected Witten
diagrams. Thus, the boundary correlators can be obtained by taking the
functional derivatives with respect to the source distribution function
$j(P_{i})$,

$\displaystyle\langle\Phi(P_{1},Z_{1})\Phi(P_{2},Z_{2})\ldots\Phi(P_{n},Z_{n})\rangle=\frac{\delta^{n}\mathcal{W}[j]}{\delta
j(P_{1})\delta j(P_{2})\ldots\delta j(P_{n})}\,.$ (4.18)

Remarkably, all the $\mathbb{O}^{-1}_{\bm{Y}_{k}}$ operators contained in
$J^{(n-1)}$ act only on $P_{n}$ or $Z_{n}$. They can be trivially pulled out
of the bulk integral in eq. (4) such that we are left with integrating out an
effective $n$-point contact interaction in the bulk for each Witten diagram.
Therefore, we have shown that we can “factor out” the nonlocalities due to
particle exchange as inverse differential operators that only act on boundary
states, while the remaining bulk integral involves only contact interactions.

To illustrate the recursion, we consider a single spin-$s$ field $\phi$ whose
free equation of motion is given by the quadratic operator
$O=\nabla^{2}-m_{s}^{2}$. We then introduce both cubic and quartic vertices,
${\cal V}_{3}(\nabla,G,W_{1},W_{2},W_{3})$ and ${\cal
V}_{4}(\nabla,G,W_{1},W_{2},W_{3},W_{4})$. The vertices explicitly depend on
the covariant derivative $\nabla_{A}$ and AdS projector $G^{AB}$. Here, we
contract the tensor indices with polarization vectors. For $\mathcal{V}_{3}$,
we contract the two incoming legs with $W_{2}$ and $W_{3}$ respectively, and
the outgoing leg with $W_{1}$, and similarly for $\mathcal{V}_{4}$. We have
used the freedom of integration-by-parts such that the covariant derivatives
in $\mathcal{V}_{3}$ and $\mathcal{V}_{4}$ only act on the incoming currents.
Then, with the shorthand notation that suppresses the dependence on source
polarizations, $\phi^{(k)}_{1\dots
k}(X,W)\equiv\phi^{(k)}(X,W,\bm{Y}_{k},\bm{W}_{k})$ and
$\mathbb{O}^{-1}_{ij\dots}\equiv\mathbb{O}^{-1}_{Y_{i}Y_{j}\dots}$, the first
three steps in the recursion yield

$\displaystyle\phi^{(1)}_{1}(X,W)$
$\displaystyle=E^{s}(X,W,P_{1},Z_{1})j(P_{1})\,,$ (4.19a)
$\displaystyle\phi^{(2)}_{12}(X,W)$ $\displaystyle=\mathbb{O}^{-1}_{12}{\cal
V}_{3}(\nabla_{X},G,W,K_{1},K_{2})\phi^{(1)}_{1}(X,W_{1})\phi^{(1)}_{2}(X,W_{2})\,,$
(4.19b) $\displaystyle\phi^{(3)}_{123}(X,W)$
$\displaystyle=\mathbb{O}^{-1}_{123}\Big{[}{\cal
V}_{4}(\nabla_{X},G,W,K_{1},K_{2},K_{3})$ (4.19c)
$\displaystyle\quad+\mathbb{O}^{-1}_{12}{\cal
V}_{3}(\nabla_{X},G,W,K_{3},K_{x}){\cal
V}_{3}(\nabla_{X},G,W_{x},K_{1},K_{2})$
$\displaystyle\quad+\mathbb{O}^{-1}_{13}{\cal
V}_{3}(\nabla_{X},G,W,K_{2},K_{x}){\cal
V}_{3}(\nabla_{X},G,W_{x},K_{1},K_{3})$
$\displaystyle\quad+\mathbb{O}^{-1}_{23}{\cal
V}_{3}(\nabla_{X},G,W,K_{1},K_{x}){\cal
V}_{3}(\nabla_{X},G,W_{x},K_{2},K_{3})\Big{]}\prod_{i=1}^{3}\phi^{(1)}_{i}(X,W_{i})\,.$

The contraction of $K$ with $\nabla$ in $\mathcal{V}_{3}$ and
$\mathcal{V}_{4}$ follows eq. (2.12). According to eq. (4) and (4.18), the
three- and four-point boundary correlators are

$\displaystyle\langle\phi_{1}\phi_{2}\phi_{3}\rangle$ $\displaystyle=\int
dX{\cal
V}_{3}(\nabla_{X},G,K_{1},K_{2},K_{3})\prod_{i=1}^{3}E^{s}(X,W_{i},P_{i},Z_{i})\,,$
(4.20) $\displaystyle\langle\phi_{1}\phi_{2}\phi_{3}\phi_{4}\rangle$
$\displaystyle=\int dX{\cal
V}_{4}(\nabla_{X},G,K_{1},K_{2},K_{3},K_{4})\prod_{i=1}^{4}E^{s}(X,W_{i},P_{i},Z_{i})$
(4.21) $\displaystyle+\mathbb{O}^{-1}_{23}\int dX{\cal
V}_{3}(\nabla_{X},G,K_{1},K_{4},K_{x}){\cal
V}_{3}(\nabla_{X},G,W_{x},K_{2},K_{3})\prod_{i=1}^{4}E^{s}(X,W_{i},P_{i},Z_{i})$
$\displaystyle+\mathbb{O}^{-1}_{24}\int dX{\cal
V}_{3}(\nabla_{X},G,K_{1},K_{3},K_{x}){\cal
V}_{3}(\nabla_{X},G,W_{x},K_{2},K_{4})\prod_{i=1}^{4}E^{s}(X,W_{i},P_{i},Z_{i})$
$\displaystyle+\mathbb{O}^{-1}_{34}\int dX{\cal
V}_{3}(\nabla_{X},G,K_{2},K_{2},K_{x}){\cal
V}_{3}(\nabla_{X},G,W_{x},K_{3},K_{4})\prod_{i=1}^{4}E^{s}(X,W_{i},P_{i},Z_{i})\,.$

All operators $\mathbb{O}^{-1}_{ij}$ are independent of the integration
variable, so in each term they can be extracted from under the integral. As in
flat space, each term corresponds to a tree-level graph and the denominators
of flat-space propagators translate as

$\displaystyle s_{i_{1}\dots i_{k}}+m_{s}^{2}\mapsto\mathcal{D}_{i_{1}\dots
i_{k}}+\mathsf{M}^{2}_{s}\ ,$ (4.22)

with $\mathsf{M}^{2}_{s}$ emerging from eq. (4.10).

We note that for scalar theories with any number of scalars with mass
parameters $m_{i}$ and polynomial non-derivative interactions, the Berends-
Giele recursion detailed here implies that the integrands of the corresponding
boundary correlators are obtained from the flat space scattering amplitudes
via the further replacement

$\displaystyle\delta\Big{(}\sum_{i=1}^{n}p_{i}\Big{)}\mapsto
D_{\Delta_{1},\dots\Delta_{n}}\ ,$ (4.23)

where $\Delta_{i}$ is the dimension of the boundary operator dual to the
$i$-th external field with mass parameter $m_{i}$, i.e.
$m_{i}^{2}=\Delta_{i}(\Delta_{i}-d)$. For single-field theories, we recover
the differential representation of boundary correlators in the form derived in
section 3. Moreover, as expected, the tree-level AdS boundary correlators of
the cubic biadjoint scalar theory obey color-kinematics duality manifestly,
and they also obey correlator relations for any multiplicity, as they do in
flat space.

Other theories, such as theories with derivative interactions, are currently
analyzed on a case by case basis. While the recursion discussed here
guarantees that boundary correlators can be written as a specific set of
inverse differential operators acting on a local expression, it does not
immediately guarantee that that local expression can itself be written as a
collection of local operators acting in a single contact integral
$D_{\Delta_{1},\dots\Delta_{n}}$. We will however see in later sections that
this expectation pans out for scalars coupled with gauge fields or gravitons.
In the examples we will discuss, the replacements (4.22) and (4.23) will
continue to relate flat space amplitudes and boundary correlators. In section
5.3 and section 6 we will prove that all scalar AdS bondary correlators have a
differential representation of form given in eqs. (3.1) to (3.3) with a
manifestly local numerators $\hat{N}_{g}$ and also see superficial limitations
of replacements like $p_{i}\cdot p_{j}\rightarrow D_{i}\cdot D_{j}$ for
relation between the local factor of scalar correlators and flat space
scattering amplitudes.

##  5 Scalar correlators mediated by spinning states

In this section, we present a remarkably simple differential representation of
the four-point correlators of scalars minimally coupled to gluons or
gravitons. We will include here explicit examples of the general discussion in
the previous section on the separation of the nonlocal and local factors of
the correlator, and the organization of the latter in terms of conformal
generators acting on a single contact integral.

The gauge theory we consider is given by the Lagrangian

$\displaystyle\mathcal{L}_{\rm
gauge}=-\frac{1}{4}F_{\mu\nu}^{a}F^{a,\mu\nu}-\frac{1}{2}D_{\mu}\phi^{a}D^{\mu}\phi^{a}-\frac{1}{2}D_{\mu}\tilde{\phi}^{a}D^{\mu}\tilde{\phi}^{a}-\frac{1}{4}f^{abc}f^{cde}\phi^{a}\tilde{\phi}^{b}\phi^{d}\tilde{\phi}^{e}\,,$
(5.1)

and the gravitational theory is its double copy. Here $\phi^{a}$ and
$\tilde{\phi}^{a}$ are two distinct massless scalars, and
$D_{\mu}\phi^{a}=\partial_{\mu}\phi^{a}+f^{abc}A_{\mu}^{b}\phi^{c}$. The
generalization to more than two flavors is straightforward and the relevant
Lagrangian is the scalar sector of the dimensional reduction of YM theory.

###  5.1 Flat space amplitudes

The flat space amplitudes following from the Lagrangian (5.1) enjoy color-
kinematics duality,

$\displaystyle\mathcal{A}_{4}^{\rm flat}(\phi_{1}\phi_{2}\phi_{3}\phi_{4})$
$\displaystyle=C_{\rm s}\frac{s_{23}-s_{13}}{s_{12}}+C_{\rm
t}\frac{s_{13}-s_{12}}{s_{23}}+C_{\rm u}\frac{s_{12}-s_{23}}{s_{13}}\,,$
(5.2a) $\displaystyle\mathcal{A}_{4}^{\rm
flat}(\phi_{1}\phi_{2}\tilde{\phi}_{3}\tilde{\phi}_{4})$ $\displaystyle=C_{\rm
s}\frac{s_{23}-s_{13}}{s_{12}}+C_{\rm t}\frac{-s_{23}}{s_{23}}+C_{\rm
u}\frac{s_{13}}{s_{13}}\,,$ (5.2b)

where the numerators satisfy $N_{\rm s}+N_{\rm t}+N_{\rm u}=0$ for both cases.
Double copy gives the following gravitational amplitudes,

$\displaystyle\mathcal{M}_{4}^{\rm
flat}(\varphi_{1}\varphi_{2}\varphi_{3}\varphi_{4})$
$\displaystyle=\frac{1}{16}\left[\frac{(s_{23}-s_{13})^{2}}{s_{12}}+\frac{(s_{13}-s_{12})^{2}}{s_{23}}+\frac{(s_{12}-s_{23})^{2}}{s_{13}}\right]\,,$
(5.3a) $\displaystyle\mathcal{M}_{4}^{\rm
flat}(\varphi_{1}\varphi_{2}\tilde{\varphi}_{3}\tilde{\varphi}_{4})$
$\displaystyle=\frac{1}{16}\left[\frac{(s_{23}-s_{13})^{2}}{s_{12}}+s_{23}+s_{13}\right]=\frac{1}{8}\frac{s_{23}^{2}+s_{13}^{2}-s_{12}^{2}}{s_{12}}\,.$
(5.3b)

They correspond to the gravitational Lagrangian

$\displaystyle\mathcal{L}_{\rm
gravity}=-\frac{\sqrt{-g}}{2}\Big{(}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi+g^{\mu\nu}\partial_{\mu}\tilde{\varphi}\partial_{\nu}\tilde{\varphi}\Big{)}\,.$
(5.4)

Note that a four-point contact interaction is not generated by the double
copy.

###  5.2 AdS correlators

We now put both actions (5.1) and (5.4) in the AdS background and compute the
scalar boundary correlators. We start with conventional Witten diagrams and
systematically arrange the results in terms of products of conformal
generators acting on a contact diagram. While doing so, we will revisit in
more detail aspects discussed in section 4. As we will see, the final
expressions closely related to the flat space amplitudes through replacements
similar to those in eqs. (4.22) and (4.23).

####  5.2.1 Gluon-mediated correlators

We start with the AdS boundary correlator of four identical scalars, minimally
coupled with gluons. The contribution $A_{s}$ from the s-channel gluon
exchange can be written as

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}{}\pgfsys@moveto{15.08904pt}{-15.08904pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-15.08904pt}{15.08904pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-15.08904pt}{-15.08904pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=A_{s}=\int_{\textrm{AdS}}dX_{1}dX_{2}J_{A}(P_{1},P_{2},X_{1})\mathcal{G}^{AB}_{g}(X_{1},X_{2})J_{B}(P_{3},P_{4},X_{2})\,,$
(5.5)

where $\mathcal{G}_{g}^{AB}(X_{1},X_{2})$ is the gluon bulk-bulk propagator,
and the vertex function $J_{A}^{(ij)}$ is derived from the interaction
Lagrangian $\mathcal{L}_{\rm gauge}$ in eq. (5.1),

$\displaystyle
J_{A}(P_{i},P_{j},X)=E_{d}(X,P_{j})\nabla_{A}E_{d}(X,P_{i})-E_{d}(X,P_{i})\nabla_{A}E_{d}(X,P_{j})\,.$
(5.6)

Following eq. (2.10), we can evaluate the index contraction by acting the
operator $K_{A}$ onto the polarization $W_{A}$, such that

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(5.7)

In this form, the gluon bulk-bulk propagator satisfies the differential
equation Costa:2014kfa

$\displaystyle(\nabla_{1}^{2}+d)\mathcal{G}_{g}(X_{1},X_{2},W_{1},W_{2})=-\delta(X_{1},X_{2})(W_{1}\cdot
W_{2})\,,$ (5.8)

which leads to the identity

$\displaystyle(\nabla_{1}^{2}+d)\int_{\textrm{AdS}}dX_{2}J(P_{3},P_{4},X_{2},K_{2})\mathcal{G}_{g}(X_{1},X_{2},W_{1},W_{2})=-J(P_{3},P_{4},X_{1},W_{1})\,.$
(5.9)

The correlator $A_{s}$ can thus be formally written as

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(5.10)

Notice that the action of $\nabla^{2}+d$ on $J(P_{3},P_{4},X,W)$ is equivalent
to that of $\mathcal{D}_{34}$, cf. eqs. (4.10) and (4.12):

$\displaystyle-(\nabla_{1}^{2}+d)J(P_{3},P_{4},X_{1},W_{1})=\mathcal{D}_{34}J(P_{3},P_{4},X_{1},W_{1})\,.$
(5.11)

We can then pull the operator ${\mathcal{D}^{-1}_{34}}$ out from the bulk
integral, and we are left with only simple contact interaction in the bulk, as
indicated by the recursion (4.15). After performing the integral and
simplifying the result using relations between $\mathcal{D}_{ij}$, we arrive
at

$3$$2$$1$$4$
$\displaystyle=\frac{1}{\mathcal{D}_{34}}\int_{\textrm{AdS}}dXJ_{A}(P_{1},P_{2},X)G^{AB}J_{B}(P_{3},P_{4},X)$
$\displaystyle=\frac{1}{\mathcal{D}_{12}}(\mathcal{D}_{23}-\mathcal{D}_{13})D_{d,d,d,d}\,,$
(5.12)