Faddeev-Popov determinant, with spin $(J-1)$ and mass $M_{d+J-1,J-1}$, so we
can identify the extra negative-propagator modes that we found above with
these ghosts (see [60] and references therein). More precisely, for $J>2$, the
original spin $J$ field contains also a trace part which is a symmetric spin
$(J-2)$ field, but on-shell one can use extra gauge transformations to set
this trace part to zero, consistent with the fact that we do not see the
corresponding on-shell modes in our approach; moreover, this extra gauge-
fixing precisely makes the ghosts into symmetric traceless transverse tensors,
which is what we obtain in our formalism.

While this identification with the expectations from Vasiliev’s theory works
very nicely for the $U(N)$ case, it actually fails for the $O(N)$ case. This
is because in our spectrum we find physical and unphysical fields just for
even spins, while from the minimal Vasiliev action that would start from
massless gauge fields with even spins $J$, we would get physical fields of
even spin but unphysical “ghosts” with odd spin. It would be interesting to
understand the reason for this discrepancy. In any case, our bulk action by
construction provides a consistent dual for the $U(N)$ and $O(N)$ theories,
whether or not it is related to the standard Vasiliev theory in a simple way.

Note that even though on-shell we found (for $U(N)$) precisely the same
spectrum as in Vasiliev’s theory, the quadratic terms we find are very
different from the naive quadratic terms one would associate with the fields
of that theory. Of course in any case Vasiliev’s theory is only known
classically (on-shell), so we claim that our action is the correct way to
define it off-shell (in perturbation theory in $1/N$), in the specific gauge-
choice described above (more precisely, we should probably work with the
formalism where this theory is written on a fixed AdS background, as in [25]).
In our off-shell formulation, our analysis guarantees that at all loop orders
this theory (with the specific counter-terms we found) will give the
correlation functions of the free $U(N)$ theory. Note that, as discussed
above, for finite $N$ the bulk fields that we find obey complicated
constraints, such that even empty AdS space ($\Phi_{J}=0$) is not a consistent
configuration in the finite $N$ path integral; but at any order in $1/N$ we
can ignore this.

##  6 The higher order bulk action

In the previous section, we only discussed the quadratic term in the bulk
higher spin theory, which we constructed by mapping the corresponding term in
the bi-local action. We will now construct all the higher order bulk
interaction and counter terms from the CFT, which will be non-local for any
choice of $f_{\Delta,J}$. We then derive the bulk Feynman rules, and apply
them to show that 1-loop terms cancel in the bulk, just as they did in the
CFT. For simplicity, we discuss here the case $d>4$. For $d<4$ the $J=0$
fields need to be handled differently, but at the end of the day they lead to
the same bulk action (see appendix C.3).

###  6.1 Map identities and diagrammatics

In the previous section, we derived the bulk dual of the bi-local quadratic
action in (2.12) by plugging in the bulk definition of $C_{\Delta,J}$ in
(4.13) and (4.15) into the bi-local quadratic action as written in terms of
$C_{\Delta,J}$ in (3.27). To derive the bulk dual of the higher order terms in
(2.12), we will use a conceptually identical but technically simpler approach,
where we directly apply the CFT-to-AdS map (4.18) to each term in (2.12). To
apply this map, we will find it convenient to first write the map in terms of
various bulk bi-local functions, which satisfy some key identities.

Our first step is to write the maps purely using bulk variables. The original
definition of the CFT-to-AdS map and its inverse in (4.21) and (4.35) were in
terms of the functions $\mathcal{M}_{J}(X,W|P_{1},P_{2})$ and
$\mathcal{M}_{J}^{-1}(P_{1},P_{2}|X,W)$, respectively, each of which were
expressed as an integral over a boundary point $P$ in (4.18) and (4.27),
respectively. We can exchange the boundary integral in

${\cal
M}^{-1}_{J}\left(P_{1},P_{2}|X,W\right)=\int_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\int\frac{dP}{J!\left(\frac{d}{2}-1\right)_{J}}\langle{\cal
O}_{\Delta_{0}}\left(P_{1}\right)\hat{\cal
O}_{\Delta_{0}}\left(P_{2}\right){\cal
O}_{\Delta,J}\left(P,D_{Z}\right)\rangle\frac{G_{\tilde{\Delta},J}\left(X,P;W,Z\right)}{\alpha_{J}f_{\Delta,J}N_{\Delta,J}}$
(6.1)

for a bulk integral by using the expression for a CFT 3-point function in
terms of its Witten diagram in the bulk [34, 61]

$\begin{split}\langle{\cal O}_{\Delta_{1}}\left(P_{1}\right){\cal
O}_{\Delta_{2}}\left(P_{2}\right){\cal
O}_{\Delta,J}\left(P,Z\right)\rangle&=\frac{1}{b^{(\Delta,J)}_{\Delta_{1},\Delta_{2}}}\int\frac{dX^{\prime}}{J!\left(\frac{d-1}{2}\right)_{J}}G_{\Delta_{2},0}(X^{\prime},P_{2})\\\
&\quad\times
G_{\Delta,J}(X^{\prime},P;K_{W^{\prime}},Z)(W^{\prime}\cdot\nabla)^{J}G_{\Delta_{1},0}(X^{\prime},P_{1})\,,\end{split}$
(6.2)

with

$b^{(\Delta,J)}_{\Delta_{1},\Delta_{2}}=\frac{\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+\Delta-d+J}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-\Delta+J}{2}\right)\Gamma\left(\frac{\Delta+\Delta_{1}-\Delta_{2}+J}{2}\right)\Gamma\left(\frac{\Delta_{2}+\Delta-\Delta_{1}+J}{2}\right)}{2^{1-J}\pi^{-\frac{d}{2}}\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma(\Delta+J)}\,,$
(6.3)

along with the definition of $\Omega_{\Delta,J}(X^{\prime},X;W^{\prime},W)$ in
(4.7). This leads to

${\cal
M}^{-1}_{J}\left(P_{1},P_{2}|X,W\right)=\int\frac{dX^{\prime}}{J!\left(\frac{d-1}{2}\right)_{J}}(K_{W^{\prime}}\cdot\nabla_{X^{\prime}})^{J}G_{\Delta_{0},0}(X^{\prime},P_{1})G_{\Delta_{0},0}(X^{\prime},P_{2})\theta_{J}(X^{\prime},X;W^{\prime},W)\,,$
(6.4)

where we define the bulk bi-local function

$\theta_{J}(X^{\prime},X;W^{\prime},W)=\int_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\frac{1}{b^{(\Delta,J)}_{\Delta_{0},\Delta_{0}}f_{\Delta,J}}\Omega_{\Delta,J}(X^{\prime},X;W^{\prime},W)\,.$
(6.5)

We can similarly rewrite $\mathcal{M}_{J}(X,W|P_{1},P_{2})$ as

$\begin{split}{\cal
M}_{J}\left(X,W|P_{1},P_{2}\right)&=\frac{1}{2}\int_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\frac{f_{\Delta,J}}{N_{\Delta,J}}\int\frac{dP}{J!\left(\frac{d}{2}-1\right)_{J}}G_{\Delta,J}\left(X,P;W,D_{Z}\right)\langle{\cal
O}_{\tilde{\Delta}_{0}}\left(P_{1}\right)\hat{\cal
O}_{\tilde{\Delta}_{0}}\left(P_{2}\right){\cal
O}_{\tilde{\Delta},J}\left(P,Z\right)\rangle\\\
&=\frac{1}{J!\left(\frac{d-1}{2}\right)_{J}}\int
dX^{\prime}(K_{W^{\prime}}\cdot\nabla_{X^{\prime}})^{J}G_{\tilde{\Delta}_{0},0}(X^{\prime},P_{1})G_{\tilde{\Delta}_{0},0}(X^{\prime},P_{2}){\tilde{\theta}}_{J}(X^{\prime},X;W^{\prime},W)\,,\end{split}$
(6.6)

where we define the bulk bi-local function ${\tilde{\theta}}_{J}$ as

${\tilde{\theta}}_{J}(Y,X;Q,W)=\int_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\frac{\alpha_{J}f_{\Delta,J}}{2b^{(\tilde{\Delta},J)}_{\tilde{\Delta}_{0},\tilde{\Delta}_{0}}}\Omega_{\Delta,J}(X^{\prime},X;W^{\prime},W)\,.$
(6.7)

It is useful to define the “multiplication” of $\theta_{J},\tilde{\theta}_{J}$
as232323Adding back the cutoffs on the two $\Delta$ integrals, the right-hand
side involves an integral over the region
$|s|<\text{min}(s_{\text{cutoff}}^{1},s_{\text{cutoff}}^{2})$.

$\begin{split}\Theta_{J}(X_{1},X_{2};W_{1},W_{2})&=\int\frac{dX}{J!\left(\frac{d-1}{2}\right)_{J}}\theta_{J}(X_{1},X;W_{1},K_{W}){\tilde{\theta}}_{J}(X,X_{2};W,W_{2})\\\
&=\int\frac{d\Delta}{2\pi
i}\frac{\alpha_{J}\Omega_{\Delta,J}(X_{1},X_{2};W_{1},W_{2})}{2b^{(\Delta,J)}_{\Delta_{0},\Delta_{0}}b^{(\tilde{\Delta},J)}_{\tilde{\Delta}_{0},\tilde{\Delta}_{0}}}\,,\end{split}$
(6.8)

where the second line follows from (5.22). Note that $\Theta_{J}$ is
independent of $f_{\Delta,J}$, unlike $\mathcal{M}_{J}$ and
$\mathcal{M}^{-1}_{J}$ which depend on $f_{\Delta,J}$ via $\tilde{\theta}_{J}$
and $\theta_{J}$, respectively. Another useful property of
$\theta_{J},\tilde{\theta}_{J}$ is the identity

$\begin{split}\delta^{TT}&(X_{1},X_{2})W_{12}^{J_{1}}\delta_{J_{1},J_{2}}=\int
dP_{1}\int dP_{2}\ {\cal M}_{J_{1}}\left(X_{1},W_{1}|P_{1},P_{2}\right){\cal
M}_{J_{2}}^{-1}\left(P_{1},P_{2}|X_{2},W_{2}\right)\\\
&=\int\frac{dX^{\prime}_{1}dX^{\prime}_{2}}{J_{1}!J_{2}!\left(\frac{d-1}{2}\right)_{J_{1}}\left(\frac{d-1}{2}\right)_{J_{2}}}{\tilde{\theta}}_{J_{1}}(X^{\prime}_{1},X_{1};W^{\prime}_{1},W_{1}){\theta}_{J_{2}}(X^{\prime}_{2},X_{2};W^{\prime}_{2},W_{2})\\\
&\quad\times\left(K_{W^{\prime}_{1}}\cdot\nabla_{X^{\prime}_{1}}\right)^{J_{1}}\Omega_{0}(X^{\prime}_{1},X^{\prime}_{2})\left(K_{W^{\prime}_{2}}\cdot\nabla_{X^{\prime}_{2}}\right)^{J_{2}}\Omega_{0}(X^{\prime}_{2},X^{\prime}_{1})\,,\end{split}$
(6.9)

where the first equality follows from the CFT orthogonality relation (3.4) and
the bulk completeness relation (4.9) as applied to the original definition of
the maps in the first lines of (6.6) and (6.4). The second equality follows
from the definition of the maps in terms of $\theta_{J},\tilde{\theta}_{J}$ in
the second lines of (6.6) and (6.4), and we have defined the new bulk bi-local
function

$\Omega_{0}(X_{1},X_{2})=\int
dPG_{\Delta_{0},0}(X_{1},P)G_{\tilde{\Delta}_{0},0}(X_{2},P)\\\
=\lim_{\Delta\rightarrow\Delta_{0}}\alpha_{0}N_{\Delta,0}\Omega_{\Delta,0}(X_{1},X_{2})\,,$
(6.10)

where the pole in $N_{\Delta_{0},0}$ (3.3) cancels the zero in
$\Omega_{\Delta_{0},0}$ in the explicit expression in [34].

Table 1: The line notations for the different bulk bi-local functions.

In order to use these identities in a more transparent way, we will use the
diagrammatic notation defined in Table 1. Every point has its own position and
usually spin. Note that the first three lines are function of the two spins
indices at each end, and both ends are in the same representation. The arrow
line on the other hand depends only on the spin index at the origin of the
arrow (in the drawing, $X_{1},W_{1},J_{1}$). When we draw diagrams with these
lines, every internal dot is implicitly integrated over, and the spin indices
are contracted in the standard embedding space manner. If the spin
representation of an internal edge or point is not defined by the external
legs, we implicitly sum over all possible values (all integers for $U(N)$ and
even integers only for $O(N)$). For example, the definition of $\Theta_{J}$ in
(6.8) can be drawn as

(6.11)

while the invertibility equation (6.9) is

$\vbox{\hbox{\includegraphics{invertibility.pdf}}}\,.$ (6.12)

Two important identities that we will use in the following are

$\vbox{\hbox{\includegraphics{Theta_Omega4_diag.pdf}}}\,,$ (6.13)

and

$\vbox{\hbox{\includegraphics{Theta_loop.pdf}}}\,,$ (6.14)

where the first identity allows us to eliminate factors of $\Theta_{J}$ for
general diagrams, while the second identity eliminates a $\Theta_{J}$ and
introduces the regularization factor $V$ given in section 2, which will cancel
with contributions from counterterm diagrams. We prove both of these
identities in Appendix C.

###  6.2 Feynman rules

Recall that in the bi-local formalism the free theory has a non-local action
with an explicit $1/N$ expansion (2.12). We will now use the CFT-to-AdS map as
expressed using the bulk bi-local functions described above to construct the
bulk action to all orders in $1/N$, which defines the Feynman rules for the
bulk theory.

The $n$-th interaction term in (2.12) is proportional to

$\begin{split}\text{Tr}\left(\left(G_{0}^{-1}\eta\right)^{n}\right)=\int\prod_{i=1}^{n}dP_{i}\
\mathop{\rm\Box}\nolimits_{P_{1}}\eta(P_{1},P_{2})...\mathop{\rm\Box}\nolimits_{P_{n}}\eta(P_{n},P_{1})\,.\end{split}$
(6.15)

We can map each term in this product to the bulk using (6.4) to get

$\begin{split}\text{Tr}\left(\left(G_{0}^{-1}\eta\right)^{n}\right)=&\prod_{m=1}^{n}\sum_{J_{m}}a_{\Delta_{0}}\int\frac{dX_{m}dX^{\prime}_{m}}{J_{m}!\left(\frac{d-1}{2}\right)_{J_{m}}}(K_{W^{\prime}_{m}}\cdot\nabla_{X^{\prime}_{m}})^{J_{m}}\Omega_{0}(X^{\prime}_{m-1},X^{\prime}_{m})\\\
&\qquad\qquad\qquad\qquad\times\theta_{J_{m}}(X^{\prime}_{m},X_{m};W^{\prime}_{m},K_{W_{m}})\Phi_{J_{m}}(X_{m},W_{m})\,,\\\
\end{split}$ (6.16)

where $X^{\prime}_{0}\equiv X^{\prime}_{n}$, the boundary Laplacian acts as

$\mathop{\rm\Box}\nolimits_{P}G_{\Delta_{0},0}(X,P)=a_{\Delta_{0}}G_{{\tilde{\Delta}_{0},0}}(X,P)\,,\qquad
a_{\Delta_{0}}=-\frac{4\pi^{\frac{d}{2}}\Gamma(\frac{d}{2}-\Delta_{0})}{\Gamma(\Delta_{0})S_{B}^{\tilde{\Delta}_{0},0}}\,,$
(6.17)

with $a_{\Delta_{0}}=d(2-d)$ for the free theory value
$\Delta_{0}=\frac{d-2}{2}$, and we performed the $P_{m}$ integrals using the
definition of $\Omega_{0}$ in (6.10). For instance, the quadratic term in
(2.12) is mapped to

$\begin{split}S_{2}=&\sum_{J_{1},J_{2}}\frac{a^{2}_{\Delta_{0}}}{2}\int\frac{dX_{1}dX^{\prime}_{1}dX_{2}dX^{\prime}_{2}}{J_{1}!\left(\frac{d-1}{2}\right)_{J_{1}}J_{2}!\left(\frac{d-1}{2}\right)_{J_{2}}}(K_{W^{\prime}_{1}}\cdot\nabla_{X^{\prime}_{1}})^{J_{1}}\Omega_{0}(X^{\prime}_{2},X^{\prime}_{1})(K_{W^{\prime}_{2}}\cdot\nabla_{X^{\prime}_{2}})^{J_{2}}\Omega_{0}(X^{\prime}_{1},X^{\prime}_{2})\\\
&\times\theta_{J_{1}}(X^{\prime}_{1},X_{1};W^{\prime}_{1},K_{W_{1}})\theta_{J_{2}}(X^{\prime}_{2},X_{2};W^{\prime}_{2},K_{W_{2}})\Phi_{J_{1}}(X_{1},W_{1})\Phi_{J_{2}}(X_{2},W_{2})\\\
=&\frac{a_{\Delta_{0}}^{2}}{2}\vbox{\hbox{\includegraphics{S_2.pdf}}}\,,\end{split}$
(6.18)

which is a rewriting of (5.2) in the bulk bi-local function language. We can
then recompute the bulk 2-point function in this formalism by inverting the
operator that acts on the quadratic term in $\Phi_{J}$, using the identity

$\begin{split}&\vbox{\hbox{\includegraphics{2p_EOM.pdf}}}=\vbox{\hbox{\includegraphics{2p_EOM_2.pdf}}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad=\vbox{\hbox{\includegraphics{2p_EOM_3.pdf}}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad=\delta_{J_{1},J_{3}}\delta^{TT}(X_{1},X_{3})W_{13}^{J_{1}}\,,\end{split}$
(6.19)

where in the first equality we used the definition of $\Theta_{J}$ in (6.11),
in the second equality we used the four-Omega identity (6.13), and in the last
equality we used the invertibility identity (6.9). The bulk propagator is then

$\langle\Phi_{J}(X_{1},W_{1})\Phi_{J}(X_{2},W_{2})\rangle=\frac{1}{a_{\Delta_{0}}^{2}}\
\vbox{\hbox{\includegraphics{2p_JJ.pdf}}}\,,$ (6.20)

which is a rewriting of (5.23) in the bulk bi-local function language.

We can similarly map the interaction terms in (2.12) to get

$n\geq
3:\qquad\qquad\frac{\left(-1\right)^{n+1}}{n}N^{1-\frac{n}{2}}a_{\Delta_{0}}^{n}\vbox{\hbox{\includegraphics{S_n.pdf}}}\vdots\quad\,,$
(6.21)

as well as the counterterms in (2.12) to get

$n\geq
1:\qquad\qquad\frac{\left(-1\right)^{n}}{n}VN^{-\frac{n}{2}}a_{\Delta_{0}}^{n}\vbox{\hbox{\includegraphics{S_n.pdf}}}\vdots\quad\,.$
(6.22)

The Feynman rules for the bulk dual of the $U(N)$ theory are then given by
these vertices along with the propagator (6.20), and a symmetry factor that
should be taken with regard to the ordering in the loop up to cyclic
transformations. For the bulk dual of the $O(N)$ theory, the vertices have an
extra factor of $\frac{1}{2}$, the propagator an extra factor of $2$, and we
sum only over even spins.

###  6.3 Properties and cancellation of loop diagrams

We will now use the Feynman rules and identities discussed above to show that
all 1-loop corrections vanish in the bulk, just as we showed for the CFT in
section 2.3. The general strategy will be to use the four-Omega identity
(6.13) to systematically eliminate $\Theta_{J}$ double lines from complicated
loop diagrams, until the diagram identically cancels a suitable counterterm
diagram, just as we observed in the CFT 1-loop calculation. We first show
these cancellations ignoring the need to regularize the Feynman diagrams
(beyond assuming that the regularization leads to (6.14) for some $V$), and we
then discuss the effects of regularization.

As an example, we first consider the connected tree level $n$-point
correlators, which we do not expect to vanish. These correlators get a
contribution from the bulk $n$-point vertex, which takes the form

$\begin{split}\langle\Phi_{J_{1}}(X_{1},W_{1})&...\Phi_{J_{n}}(X_{n},W_{n})\rangle_{\text{tree,
connected n-point vertex}}\\\
&=\frac{\left(-1\right)^{n+1}}{n}N^{1-\frac{n}{2}}\left(a_{\Delta_{0}}\cdot\frac{1}{a_{\Delta_{0}}^{2}}\right)^{n}\cdot\vbox{\hbox{\includegraphics{npoint_1.pdf}}}+\text{permutations}\\\
&=\frac{\left(-1\right)^{n+1}}{n}N^{1-\frac{n}{2}}a_{\Delta_{0}}^{-n}\cdot\vbox{\hbox{\includegraphics{npoint_2.pdf}}}+\text{permutations}\,,\end{split}$
(6.23)

where the second equality came from repeatedly applying the identity (6.13) to
eliminate all the $\Theta_{J}$ double lines. The result is a convolution of
$n$ $\tilde{\theta}$ operators with a bilocal function that is just
multiplications of $n$ $\partial\Omega_{0}$s. This latter function is
independent of our normalization, so the entire dependence on $f_{\Delta,J}$
is through the external legs. We will find that all the diagrams have this
structure. This is expected, as each diagram can also be written as the CFT-
to-AdS mapping of the corresponding bi-local diagram.

We now turn to 1-loop calculations, where for simplicity we follow the CFT
calculation in section 2.3 and consider the 1-loop contribution to the one-
point function coming from the $3$-point vertex. Using the Feynman rules above
we get

$\begin{split}\langle\Phi_{J}(X,W)\rangle_{S_{3},\text{1-loop}}&=3\cdot\frac{N^{-\frac{1}{2}}}{3}\frac{1}{a_{\Delta_{0}}}\vbox{\hbox{\includegraphics{VEV_calc_1.pdf}}}\\\
&=N^{-\frac{1}{2}}\frac{1}{a_{\Delta_{0}}}\vbox{\hbox{\includegraphics{VEV_calc_2.pdf}}}\\\
&=VN^{-\frac{1}{2}}\frac{1}{a_{\Delta_{0}}}\vbox{\hbox{\includegraphics{VEV_calc_3.pdf}}}\,,\end{split}$
(6.24)

where in the second equality we used (6.13) twice, and in the third equality
we used the identity (6.14) to get the factor of $V$. The result is exactly
minus the contribution from the 1-vertex counter-term:

$\begin{split}\langle\Phi_{J}(X,W)\rangle_{\text{counter-
term}}&=(-1)VN^{-\frac{1}{2}}a_{\Delta_{0}}\cdot\frac{1}{a_{\Delta_{0}}^{2}}\cdot\vbox{\hbox{\includegraphics{VEV_calc_4.pdf}}}\\\
&=-VN^{-\frac{1}{2}}\frac{1}{a_{\Delta_{0}}}\cdot\vbox{\hbox{\includegraphics{VEV_calc_3.pdf}}}\,,\end{split}$
(6.25)

so we have exact cancellation of the 1-loop contribution, just like in section
2.3.

As we saw in the bilocal theory, up to combinatorics, the cancellation of
loops by the counter-terms is basically because the diagrams themselves are
equal to their tree level counterparts, multiplying some power of a specific
divergence we called $V$. We can list the general mechanism as follows:

  1. 1.

Every loop is made of contractions between vertices. As each contraction has
$\tilde{\theta}_{J}$ at each end (6.20), and every vertex in (6.21) and (6.22)
has a $\theta_{J}$ at each end, we can always write the loop using
$\Omega_{0}$ and $\Theta$ alone. As an example, consider a loop made of three
contractions. There are two possible options for this loop, depending on the
relative orientation of the vertices:

$\vbox{\hbox{\includegraphics{3_vertices_1.pdf}}},\quad\quad\vbox{\hbox{\includegraphics{3_vertices_2.pdf}}}$
(6.26)

Note the “external legs” in the drawing are actually part of the full
expression of the vertex and are not the actual external legs of the complete
diagram (see (6.21)). For the same reasons, external contractions keep their
$\tilde{\theta}_{J}$ legs. Eventually the entire diagram is independent of
$f_{\Delta,J}$ except via the external $\tilde{\theta}_{J}$’s. As mentioned
above, this is expected from the linear nature of our mapping.

  2. 2.

Using both (6.13) and (6.14), we can now ‘disentangle’ the diagrams. The
result may or may not have closed loops, which correspond to factors of $V$,
depending on the orientation of each vertex (and eventually the total topology
of the diagram). For example, the disentangling for the 3-vertices (6.26)
gives:

$\vbox{\hbox{\includegraphics{3_vertices_res_1.pdf}}}\times\quad
V\,,\quad\quad\vbox{\hbox{\includegraphics{3_vertices_res_2.pdf}}}$ (6.27)

thus reducing the number of loops in the complete diagram by one (times a
possible factor of $V$).

  3. 3.

Eventually we can reduce any diagram (including those that include counterterm
vertices) to a power of $V$ multiplied by a tree-level diagram.

The algebraic relation between each diagram to the tree level diagram has the
same form as in the bi-local theory. We expect that the combinatorics of the
bi-local diagrams that cancel all loops in section 2.3 should therefore carry
over to the same combinatorics here.

The considerations above are somewhat formal since all the loop diagrams need
to be regularized. As discussed in appendix C.2, one way to regularize is to
put a cutoff $z>\varepsilon$ on the radial direction of AdS space, and to put
an IR cutoff $|x|<L$ on the space-time coordinates. With this regularization
(and with a finite cutoff on the integrals over $\Delta$, as we have
throughout the paper) we claim that all diagrams are finite. Furthermore, we
claim that when we remove the regulators by taking $\varepsilon\to 0$ and
$L\to\infty$, all the divergences arise in subdiagrams of the form (6.14),
with $V\propto L^{d}/\varepsilon^{d}$. These divergences then cancel in all
loop computations, as described above. In the free theory, in fact, all loops
cancel exactly (not just their divergent terms), but this will no longer be
true in the critical theory that we discuss below.

##  7 The mass deformation

In this section we discuss the deformation of the bulk theory by a finite
scalar mass $m$ and its effect on the vacuum expectation value (VEV) of
$\Phi_{0}$, which we would like to derive from
$\langle\eta(x_{1},x_{2})\rangle$ (computed in the massive theory) using our
CFT-to-AdS map. In the CFT, we add the standard mass term $m^{2}NG(x,x)$,
which up to an irrelevant constant corresponds to adding
$m^{2}\sqrt{N}\eta(x,x)$. The VEV for $\eta(x_{1},x_{2})$ under this
deformation is then

$\begin{split}\langle\eta(x_{1},x_{2})\rangle_{m}=\sqrt{N}\Big{[}\frac{m^{d/2-1}K_{d/2-1}(m|x_{12}|)}{(2\pi)^{d/2}|x_{12}|^{d/2-1}}-\frac{\Gamma(d/2-1)}{4\pi^{d/2}|x_{12}|^{d-2}}\Big{]}\,,\end{split}$
(7.1)

where in $d=3$ the massive propagator simplifies to give
$\langle\eta(x_{1},x_{2})\rangle_{m}=\sqrt{N}\frac{e^{-m|x_{12}|}-1}{4\pi|x_{12}|}$.
Since this configuration does not decay at $x_{1},x_{2}\to\infty$, we cannot
expand it in the conformal basis of $C_{\Delta,J}(y)$ in section 3, and so
cannot apply the CFT-to-AdS map derived in section 4. Instead, we will derive
a new expansion of non-decaying bi-locals $\eta_{ND}(|x_{12}|)$ in terms of a
basis $d_{\Delta}$, which we will then identify with a similar expansion of
non-decaying bulk fields $\Phi_{ND}(z)$ to derive a non-decaying CFT-to-AdS
map (and its inverse). This non-decaying map will correspond to the limit of
the standard decaying map, for configurations that only depend on $|x_{12}|$
in the CFT, and that only depend on $z$ in the bulk. We will then use this
non-decaying map to compute the bulk dual $\langle\Phi_{ND}(z)\rangle_{m}$ of
(7.1). In this section, we do not use embedding space for clarity.

###  7.1 Mapping the small mass deformation

Before we address the more difficult finite mass case, we will first discuss
the simpler case where $m^{2}\sim 1/\sqrt{N}$ so that adding
$m^{2}\sqrt{N}\eta(x,x)$ can be treated as a small deformation to the bi-local
action. In addition, we consider a slightly more general position-dependent
mass term $m(x)$, which will enable us (for $m(x)$ that decays at infinity) to
use the standard CFT-to-AdS map for decaying configurations that we derived
above. The CFT path integral is

$\begin{split}Z_{m(x)}=\int D\eta(x,x^{\prime})\text{exp}\left(-S[\eta]-\int
d^{d}x\,m^{2}(x)\sqrt{N}\eta(x,x)\right)\,.\end{split}$ (7.2)

For $d<4$ we can map this to the bulk using the off-shell mapping (4.45) to
get

$\begin{split}Z_{m(x)}=\int
D\Phi_{J}(x,z)\text{exp}\left(-S[\Phi]-\frac{1}{f_{d-2,0}}\lim_{\varepsilon\rightarrow
0}\varepsilon^{-d+2}\int
d^{d}xm^{2}(x)\sqrt{N}\Phi_{0}(x,\varepsilon)\right)\,,\end{split}$ (7.3)

where the bulk action $S[\Phi]$ was described in section 6, and note that this
deformation only affects the boundary of AdS. We can then compute the VEV of
$\Phi_{J}$ under this deformation at tree level, to get the leading order in
$1/N$ result

$\begin{split}\langle\Phi_{J}(x,z)\rangle_{m(x)}&=\frac{1}{f_{d-2,0}}\int
d^{d}y\,m^{2}(y)\sqrt{N}\left(\lim_{\varepsilon\rightarrow
0}\varepsilon^{-d+2}\langle\Phi_{J}(x,z)\Phi_{0}(y,\varepsilon)\rangle\right)+O(N^{-1})\\\
&=\delta_{J,0}f_{2,0}\frac{\Gamma(d-2)}{16\pi^{d}}\int
d^{d}y\,m^{2}(y)\sqrt{N}G_{d-2,0}(x,z|y)+O(N^{-1}),\\\ \end{split}$ (7.4)

where in the second equality we used the boundary limit of the bulk two-point
function in (5.25), which is only nonzero when $J=0$. Since this calculation
was done on-shell, we in fact only needed to use the on-shell mapping (4.42),
so the computation holds for any $d$ 242424For $d>4$, (7.4) does not obey the
original boundary conditions for $\Phi_{0}$ at small $z$ (which had the
leading term going as $z^{d/2}$); it turns out that in this case we need to
modify the boundary conditions when deforming the action by (7.2), related to
the fact that $\eta(x,y)$ is singular at $x\to y$ after the deformation, so
that the first line of (3.11) is no longer obeyed. The modified boundary
condition for the $z^{2}$ term in the expansion is exactly the usual one that
would be associated to this deformation by the AdS/CFT correspondence..

If the mass $m$ is independent of the position252525Strictly speaking, we need
to use in this case the mapping for non-decaying field configurations, but we
will see that it gives the same answer as the naive application of the
equations above., we can perform the integral over the bulk-to-boundary
propagator in (7.4) to get

$\begin{split}\langle\Phi_{J}(x,z)\rangle_{m}=\delta_{J,0}m^{2}\sqrt{N}z^{2}\frac{f_{2,0}}{16\pi^{d/2}}\Gamma\left(\frac{d}{2}-2\right)+O(N^{-1})\,.\\\
\end{split}$ (7.5)

Note that the $z$ dependence corresponds to the shadow of the mass operator
scaling dimension $\Delta=d-2$, and that as expected this bulk VEV only
depends on the dimensionless product $mz$, and not on the $x$ direction. Both
of these properties come from the fact that the $y$ integral in (7.4) picks
out the second term in the expansion of the bulk-to-boundary propagator at
small $z$ (4.4). For the finite mass deformation that we discuss next, we will
similarly find that the bulk VEV depends only on $mz$, though it will include
more than just a quadratic term.

###  7.2 The conformal basis for non-decaying bi-locals

Before we can discuss the mapping of the finite mass configuration in (7.1),
we first need an CFT-to-AdS map that applies to non-decaying bi-locals. We
start on the CFT side by defining the non-decaying $\eta_{ND}(|x_{12}|)$ and
decaying $\eta_{D}(x_{1},x_{2})$ parts of $\eta(x_{1},x_{2})$ as

$\begin{split}\eta_{ND}(|x_{12}|)=\frac{1}{v}\int
d^{d}y\eta(y,y+|x_{12}|)\,,\qquad\eta_{D}(x_{1},x_{2})=\eta(x_{1},x_{2})-\eta_{ND}(|x_{12}|)\,,\end{split}$
(7.6)

where $v\equiv\int d^{d}y$ is the volume of spacetime, and for simplicity we
assume that the part of $\eta(x_{1},x_{2})$ that does not decay at infinity is
just a function of $|x_{1}-x_{2}|$ 262626The non-decaying map discussed in
this section could be easily generalized to non-rotationally symmetric
configurations, but we will not consider them in this work.. We use $v$ as an
IR regulator, where at intermediate steps we consider large but finite $v$,
and take $v\to\infty$ at the end. The definition (7.6) implies the constraint

$\begin{split}\frac{1}{v}\int d^{d}x\eta_{D}(x,x+\epsilon)=0\,\end{split}$
(7.7)

for any $\epsilon\in\mathbb{R}^{d}$.

In section 3, we showed that decaying configurations $\eta(x_{1},x_{2})$ can
be expanded in the basis of 3-point functions, with coefficients
$C_{\Delta,J}(y)$. We can apply the same expansion to $\eta_{D}(x_{1},x_{2})$,
except that we should impose the constraint (7.7) in the language of the
$C_{\Delta,J}(y)$ by requiring that

$\begin{split}\frac{1}{v}\int d^{d}yC_{\Delta,J}(y)=0\,,\end{split}$ (7.8)

for any $\Delta,J$.

Since $\eta_{ND}(|x_{12}|)$ just depends on a single degree of freedom
$|x_{12}|$, we can expand it in a simpler basis of scalar 2-point functions
$|x_{12}|^{-2\Delta}$. This expansion is just the standard Mellin transform of
a single variable function with a specific choice of contour, leading to the
completeness relation

$\begin{split}\int_{\gamma_{0}}\frac{d\Delta}{2\pi
i}(|x|/|y|)^{\Delta-\frac{d}{2}}=|x|\delta(|x|-|y|)\,,\end{split}$ (7.9)

and to the orthogonality relation

$\begin{split}\int_{0}^{\infty}\frac{d|x_{12}|}{|x_{12}|}x_{12}^{\Delta-\Delta^{\prime}}=2\pi
i\delta(\Delta-\Delta^{\prime})\,\,,\end{split}$ (7.10)

where we have assumed that $\Delta,\Delta^{\prime}\in\gamma_{0}$. We can then
decompose $\eta_{ND}(|x_{12}|)$ in this basis using the completeness relation
to get

$\begin{split}\eta_{ND}(x_{1},x_{2})&=\int_{\gamma_{0}}\frac{d\Delta}{2\pi
i}d_{\Delta}x_{12}^{2-\Delta}\,,\end{split}$ (7.11)

where $d_{\Delta}$ is defined by (7.9) to be

$\begin{split}d_{\Delta}=\int_{0}^{\infty}d|x_{12}|\,\eta_{ND}(|x_{12}|)x_{12}^{\Delta-3}\,,\end{split}$
(7.12)

and is well-defined so long as this integral converges for
$\Delta\in\gamma_{0}$. Note that (7.9) and (7.10) are the standard
representations of delta functions in Mellin space.272727These relations also
hold for $2<d<4$, when our contour $\gamma_{0}$ requires that we remove the
pole at $\Delta=2$ and include the one at $\Delta=d-2$ [55]. The choice of
contour we used comes from taking the limit of the analogous formulae in
section 3, for the case where $\eta(x_{1},x_{2})$ only depends on $|x_{12}|$
and $C_{\Delta,J}(y)$ no longer depends on $y$ (i.e. $P$ in embedding space),
and is nonzero only for $J=0$. For instance, we can compute the $P$ integral
in (3.7) in this case to get (7.11) if we identify

$\begin{split}C_{\Delta,0}\to\frac{d_{\Delta}}{S_{\Delta_{0},\Delta_{0}}^{(\Delta_{0},0)}}\,.\end{split}$
(7.13)

The $d_{\Delta}$ basis that we use for non-decaying functions is simpler than
the $C_{\Delta,J}(y)$ basis that we use for decaying functions in several
ways. First, recall that the $C_{\Delta,J}(y)$ were doubly complete due to the
shadow transform that relates $C_{\Delta,J}(y)$ to $C_{\tilde{\Delta},J}(y)$,
while the $d_{\Delta}$ are complete without any redundancies. Second, the
expression for $C_{\Delta,J}(y)$ in (3.8) did not converge for the free theory
in $2<d<4$, so we had to define a more complicated expansion of
$\eta_{D}(x_{1},x_{2})$ in terms of $C_{\tilde{\Delta},J}(y)$. The expression
for $d_{\Delta}$ in (7.12), however, has the same convergence properties for
any $d$.

We can now write the bi-local action in (2.12) in terms of the conformal basis
of $C_{\Delta,J}(y)$ and $d_{\Delta}$ by first writing $\eta(x_{1},x_{2})$ in
terms of $\eta_{D}(x_{1},x_{2})$ and $\eta_{ND}(x_{1},x_{2})$ as in (7.6), and
then expanding the $\eta_{D}(x_{1},x_{2})$ in terms of the $C_{\Delta,J}(y)$
as discussed in section 3 and the $\eta_{ND}(x_{1},x_{2})$ in terms of
$d_{\Delta}$ as in (7.11). For instance, at quadratic order, the decaying
action in terms of the $C_{\Delta,J}(y)$ was written in (3.27), and can be
used to compute tree level correlators as discussed before. The mixed term
with both $d_{\Delta}$ and $C_{\Delta,J}(y)$ vanishes due to the constraint
(7.8). Finally, the quadratic term in terms of just the $d_{\Delta}$ can be
computed for the $U(N)$ theory as

$\begin{split}S^{(2)}(d_{\Delta})&=\frac{1}{{2}}\int
d^{d}x_{1}d^{d}x_{2}\nabla^{2}_{1}\eta_{D}(x_{1},x_{2})\nabla^{2}_{2}\eta_{D}(x_{1},x_{2})\\\
&=\frac{1}{2}\int d^{d}x_{1}d^{d}x_{2}\int_{\gamma_{0}}\frac{d\Delta
d\Delta^{\prime}}{(2\pi
i)^{2}}\frac{d_{\Delta}d_{\Delta^{\prime}}}{|x_{12}|^{\Delta+\Delta^{\prime}}}(\Delta-2)(\Delta^{\prime}-2)(\Delta-d)(\Delta^{\prime}-d)\\\
&=\frac{v}{\alpha_{0}}\int_{\gamma_{0}}\frac{d\Delta}{2\pi
i}d_{\Delta}d_{\tilde{\Delta}}\lambda_{\Delta,0}\,,\end{split}$ (7.14)

where in the last equality we used rotational symmetry and then the
orthogonality relation (7.10) to do the $x_{1}$ integral, while the $x_{2}$
integral gave the factor of $v$. Since $v\to\infty$ when we remove the IR
cutoff, this factor ensures that the non-decaying modes are non-dynamical. In
the $O(N)$ theory, we would get the same result except with an extra factor of
$\frac{1}{2}$. We can similarly write the interaction terms in (2.12) in this
conformal basis, where in general there will be nonvanishing cross terms with
both $C_{\Delta,J}(y)$ and $d_{\Delta}$.

###  7.3 The non-decaying AdS/CFT map

In the bulk, we can similarly define the non-decaying $\Phi_{ND}(z)$ and
decaying $\Phi_{D}(x,z)$ parts of the scalar bulk field $\Phi_{0}(x,z)$ as

$\begin{split}\Phi_{ND}(z)=\frac{1}{v}\int
d^{d}y\Phi_{0}(y,z)\,,\qquad\Phi_{D}(x,z)=\Phi_{0}(x,z)-\Phi_{ND}(z)\,,\end{split}$
(7.15)

where again we only consider non-decaying configurations where only the scalar
is non-zero, and approaches a constant at large distances. This definition
implies the constraint

$\begin{split}\frac{1}{v}\int d^{d}x\Phi_{D}(x,z)=0\,.\end{split}$ (7.16)

The decaying $\Phi_{D}(x,z)$ are expanded in the same bulk-to-boundary basis
as in section 4, except with $C^{bulk}_{\Delta,J}(y)$ subject to the
constraint (7.8). The non-decaying $\Phi_{ND}(z)$ can be expanded using the
same Mellin transform as $\eta_{ND}(|x_{12}|)$, except with the shifted
contour

$\begin{split}\Phi_{ND}(z)&=\int_{\gamma_{0}}\frac{d\Delta}{2\pi
i}d^{bulk}_{\Delta}z^{d-\Delta}\,,\end{split}$ (7.17)

where $d^{bulk}_{\Delta}$ is defined by (7.9) to be

$\begin{split}d^{bulk}_{\Delta}=\int_{0}^{\infty}dz\,\Phi_{ND}(z)z^{\Delta-d-1}\,.\end{split}$
(7.18)

As in the non-decaying CFT case, the choice of contour comes from taking the
limit of the analogous formulae in section 4, where $\Phi_{0}(x,z)$ only
depends on $z$, and $C^{bulk}_{\Delta,J}(y)$ no longer depends on $y$ and is
nonzero only for $J=0$. For instance, we can compute the $P$ integral in
(4.12) in this case to get (7.17) if we identify

$\begin{split}C^{bulk}_{\Delta,0}\to\frac{d^{bulk}_{\Delta}}{S_{B}^{\Delta,0}}\,.\end{split}$
(7.19)

We can then derive the non-decaying AdS-to-CFT map by identifying
$d^{bulk}_{\Delta}$ with $d_{\Delta}$ as

$\begin{split}d^{bulk}_{\Delta}=f_{\tilde{\Delta},0}d_{\Delta}\,,\end{split}$
(7.20)

where the normalization factor $f_{\tilde{\Delta},0}$ was fixed from the limit
of the decaying map in (7.13) and (7.19) as well as the consistency condition
(4.16). The non-decaying AdS-to-CFT map is then

$\begin{split}\eta_{ND}(|x_{12}|)&=\int_{\gamma_{0}}\frac{d\Delta}{2\pi
i}|x_{12}|^{2-\Delta}\frac{1}{f_{\tilde{\Delta},0}}\int_{0}^{\infty}dz\Phi_{ND}(z)z^{\Delta-d-1}\,,\end{split}$
(7.21)

while the non-decaying CFT-to-AdS map is

$\begin{split}\Phi_{ND}(z)&=\int_{\gamma_{0}}\frac{d\Delta}{2\pi
i}f_{\tilde{\Delta},0}z^{d-\Delta}\int_{0}^{\infty}d|x_{12}|\,\eta_{ND}(|x_{12}|)|x_{12}|^{\Delta-3}\,.\end{split}$
(7.22)

Note that since we used the same Mellin transform (up to a shift in contour)
for both the bulk and CFT expansions, this map would have simply identified
$|x_{12}|=z$, if we did not have the nontrivial factor $f_{\tilde{\Delta},0}$.

We can now map the bi-local action in (2.12) as written in terms of
$\eta_{D}(x_{1},x_{2})$ and $\eta_{ND}(x_{1},x_{2})$ into the bulk using the
decaying AdS-to-CFT map in section 4 for $\eta_{D}(x_{1},x_{2})$, and the non-
decaying map in this section for $\eta_{ND}(x_{1},x_{2})$. For instance, at
quadratic order for the $U(N)$ theory, we can find the non-decaying bulk term
by expanding (7.14) using (7.18) and (7.20) as

$\begin{split}S^{(2)}[\Phi_{ND}]&=\frac{v}{\alpha_{0}}\int_{\gamma_{0}}\frac{d\Delta}{2\pi
i}d_{\Delta}d_{\tilde{\Delta}}\lambda_{\Delta,0}\\\
&=\frac{v}{\alpha_{0}}\int_{0}^{\infty}\frac{dzdz^{\prime}}{zz^{\prime}}\Phi_{ND}(z)\Phi_{ND}(z^{\prime})\int_{\gamma_{0}}\frac{d\Delta}{2\pi
i}\frac{z^{-\tilde{\Delta}}z^{\prime-\Delta}}{f_{\Delta,0}f_{\tilde{\Delta},0}}\lambda_{\Delta,0}\\\
&=\frac{v}{\alpha_{0}}\int_{0}^{\infty}\frac{dzdz^{\prime}}{zz^{\prime}}\Phi_{ND}(z^{\prime})(\nabla^{2}_{z}-M^{2}_{d-2,0})(\nabla^{2}_{z}-M^{2}_{d,0})\Phi_{ND}(z)\int_{\gamma_{0}}\frac{d\Delta}{2\pi
i}\frac{z^{-\tilde{\Delta}}z^{\prime-\Delta}}{f_{\Delta,0}f_{\tilde{\Delta},0}}\,,\\\
\end{split}$ (7.23)

where in the third equality we used integration by parts and the action of the
AdS Laplacian in the upper half space when acting on functions of only $z$:

$\begin{split}\nabla_{z}^{2}=z^{2}(\partial^{2}_{z}-(d-1)z^{-1}\partial_{z})\,.\end{split}$
(7.24)

As before, the factor of $v$ ensures that the non-decaying modes are non-
dynamical. This non-decaying quadratic term could equivalently be found by
taking the limit of the decaying scalar quadratic term in (5.2), if we assume
that $\Phi_{0}(x,z)$ does not depend on $x$. The non-decaying bulk propagator
is then

$\begin{split}&\langle\Phi_{ND}(z_{1})\Phi_{ND}(z_{2})\rangle=\frac{\alpha_{0}}{2v}\int_{\gamma_{0}}\frac{d\Delta}{2\pi
i}\frac{f_{\Delta,0}f_{\tilde{\Delta},0}}{\lambda_{\Delta,0}}z_{1}^{\tilde{\Delta}}z_{2}^{\Delta}\,,\end{split}$
(7.25)

which is the inverse of the second line of (7.23) due to the completeness
relation (7.9).

As in the decaying case, if we choose $f^{\text{local}}_{\Delta,0}$ then we
get a local quadratic action

$\begin{split}S^{(2)}[\Phi_{ND}]&=\frac{v}{\alpha_{0}}\int_{0}^{\infty}\frac{dz}{z^{d+1}}\Phi_{ND}(z)(\nabla^{2}_{z}-M^{2}_{d-2,0})(\nabla^{2}_{z}-M^{2}_{d,0})\Phi_{ND}(z)\,,\end{split}$
(7.26)

where we used the fact that
$f^{\text{local}}_{\Delta,0}f^{\text{local}}_{\tilde{\Delta},0}=1$ as well as
the completeness relation (7.9). In the $O(N)$ theory, we would get the same
results except with an extra factor of $\frac{1}{2}$. The non-decaying bulk
propagator can be explicitly computed in this case to get

$\begin{split}\langle\Phi_{ND}(z_{1})\Phi_{ND}(z_{2})\rangle=\frac{\alpha_{0}}{2v}\Big{[}\frac{\text{min}(z_{1},z_{2})^{d}}{2(d-2)d}-\frac{\text{max}(z_{1},z_{2})^{2}\text{min}(z_{1},z_{2})^{d-2}}{2(d-4)(d-2)}\Big{]}\,,\end{split}$
(7.27)

where we close the contour in (7.25) to either the left or the right,
depending on whether $z_{1}>z_{2}$.

We could similarly map the interaction terms in (2.12) to the bulk, where in
general there will be nonvanishing cross terms with both $\Phi_{D}(x,z)$ and
$\Phi_{ND}(z)$. Note that since the decaying and non-decaying AdS-to-CFT maps
are linear, all the $\Phi_{D}(x,z)$ and $\Phi_{ND}(z)$ can be recombined into
$\Phi_{0}(x,z)$ to get the terms discussed in section 6, just as we know that
in the bi-local language $\eta_{D}(x_{1},x_{2})$ and $\eta_{ND}(x_{1},x_{2})$
recombine into $\eta(x_{1},x_{2})$ by definition to get (2.12). So from the
perspective of the bulk action and computing loops, nothing actually changes
by introducing the non-decaying map. This is not surprising, as the non-
decaying modes are all non-dynamical due to the factors of $v$. On the other
hand, the non-decaying map is necessary to map the finite mass deformation, as
we discuss next, and for similar computations that modify the theory
everywhere in space-time.

###  7.4 Mapping the finite mass deformation

We will now use the non-decaying CFT-to-AdS map (7.22) to find the bulk VEV
dual to the mass deformed CFT VEV (7.1), where for simplicity we will work in
$d=3$.282828We could also find this VEV directly in the bulk as a solution to
the full non-linear bulk equations of motion in the presence of the mass, but
we find it easier to just map it from the CFT. An immediate difficulty with
applying this map is that the expansion in the $d_{\Delta}$ basis is only well
defined if the integral in (7.12) converges for a bi-local configuration
$\eta_{ND}(|x_{12}|)$, but this integral in fact diverges in the $|x_{12}|\to
0$ regime for (7.1) and $\Delta\in\gamma_{0}$. We can regularize this
divergence by shifting $|x_{12}|\to|x_{12}|+\varepsilon$ when expanding
$\eta_{ND}(|x_{12}|)$ in $d_{\Delta}$, and then setting $\varepsilon\to 0$
after we then relate $d_{\Delta}$ to $\Phi_{ND}(z)$ as in (7.18) to derive the
non-decaying CFT-to-AdS map. The resulting $\Phi_{ND}(z)$ is finite as
$\varepsilon\to 0$, since (7.1) is finite, and both the CFT and bulk
expansions in terms of $d_{\Delta}$ are just the same Mellin transform, up to
a shift in the contour and the factor of $f_{\tilde{\Delta}}$.

An equivalent and computationally simpler way of regularizing the CFT-to-AdS
map of (7.1) is to simply shift the contour from $\gamma_{0}$ to

$\begin{split}\bar{\gamma}_{0}:\qquad\Delta=r+is\,,\qquad 2<r<3\,,\quad
s\in\mathbb{R}\,,\end{split}$ (7.28)

so that in (7.22) the $|x_{12}|$ integral converges and takes the form

$\begin{split}z^{3-\Delta}\int_{0}^{\infty}d|x_{12}|\,\frac{\exp(-m|x_{12}|)-1}{4\pi|x_{12}|^{{4-\Delta}}}=(mz)^{3-\Delta}\frac{\Gamma(\Delta-3)}{4\pi}\,.\end{split}$
(7.29)

We then explicitly subtract any contributions to the $\Delta$ integral in
(7.22) from poles and branch cuts that were crossed when shifting the contour.
For $f^{\text{local}}_{\Delta,0}$ we get for the resulting mass-deformed bulk
VEV

$\begin{split}\frac{\langle\Phi_{ND}(z)\rangle_{m}}{\sqrt{N}}&=\int_{\bar{\gamma}_{0}}\frac{d\Delta}{2\pi
i}f^{\text{local}}_{\tilde{\Delta},0}(mz)^{3-\Delta}\frac{\Gamma(\Delta-3)}{4\pi}-\frac{1}{4\pi}\text{Res}\Big{[}f^{\text{local}}_{\tilde{\Delta},0}(mz)^{3-\Delta}\Gamma(\Delta-3)\Big{]}_{\Delta=1}\,,\\\
\end{split}$ (7.30)

where the second term corresponds to the pole in $\Gamma(\Delta-3)$ allowed by
$\gamma_{0}$, since $f^{\text{local}}_{\tilde{\Delta},0}$ has neither poles
nor branch cuts in this regime. The integral in the first term is complicated
due to the infinite branch cuts in $f^{\text{local}}_{\tilde{\Delta},0}$, but
it can be easily computed numerically for any $mz$. The result goes from zero
at small $mz$ to a constant at large $mz$, so that the large $mz$ scaling is
given by the sum over poles. We plot $\langle\Phi_{ND}(z)\rangle_{m}$ as a
function of $mz$ in Figure 2, which has linear growth at large $mz$. It would
be interesting to compare this result to the analogous mass deformed solution
to Vasiliev’s equations [62, 63].

Figure 2: The VEV of the non-decaying scalar bulk field in the presence of a
finite mass $m$ deformation as a function of $mz$, where $z$ is the radial
direction, for $d=3$.

##  8 The double trace deformation and the critical theory

In this section we discuss the deformation of the free scalar CFT by a
$(\sum_{I}|\phi_{I}|^{2})^{2}$ term, which in the bi-local theory is a double-
trace deformation $\lambda NG^{2}(x,x)$, and the critical theory that it leads
to in the low-energy ($\lambda\to\infty$) limit for $2<d<4$. We start by
reviewing the double-trace deformation of the free $U(N)$ or $O(N)$ CFT,
following [64, 65, 66, 67]. We will then discuss the bulk dual of the critical
theory, using the off-shell mapping of $\eta(x,x)$ to the boundary limit of
$\Phi_{0}(x,z)$ (4.45). We will show how the bulk duals of the free and
critical theory only differ by the boundary conditions of the bulk scalar two-
point function, as anticipated in [36, 68, 37]. In this section, we do not use
embedding space for clarity.

###  8.1 The deformation in the bi-local CFT

Consider the deformation of the bi-local $U(N)$ action $S[\eta]$ in (2.11) by
the double trace scalar deformation, such that the CFT path integral is

$\begin{split}Z_{\lambda}&=\int
D\eta(x,x^{\prime})\text{exp}\left[-S[\eta]-\frac{\lambda N}{4}\int
d^{d}xG^{2}(x,x)\right]\\\ &=\int
D\eta(x,x^{\prime})\text{exp}\left[-S[\eta]-\frac{\lambda}{4}\int
d^{d}x\Big{(}\eta^{2}(x,x)+NG_{0}^{2}(x,x)+2\sqrt{N}\eta(x,x)G_{0}(x,x)\Big{)}\right]\,.\\\
\end{split}$ (8.1)

The second term in the integral is just a numerical constant, while the last
term is a divergent mass term that can be trivially cancelled at each order by
a mass counterterm, so we will ignore both of these terms in the following,
and just focus on the nontrivial $\eta^{2}(x,x)$ term.

We can equivalently introduce an extra path integral over the Hubbard-
Stratonovich field $\sigma(x)$ to get a linear deformation

$\begin{split}Z_{\lambda}=\int
D\sigma(x)D\eta(x,x^{\prime})\text{exp}\left(-S[\eta]-\int
d^{d}x\left(\frac{1}{2}\sigma(x)\eta(x,x)-\frac{1}{4\lambda}\sigma^{2}(x)\right)\right)\,,\end{split}$
(8.2)

such that performing the gaussian $\sigma(x)$ path integral sets
$\sigma(x)=\lambda\eta(x,x)$ and gives us back (8.1), up to an overall
numerical constant and the trivial mass term. In the latter formulation, we
compute correlators of $\sigma(x)$ and $\eta(x_{1},x_{2})$ in a large $N$
expansion using the same Feynman rules introduced in section 2.2 for
$\eta(x_{1},x_{2})$, along with the new vertex
$\frac{1}{2}\sigma(x)\eta(x,x)$, and the propagator
$\contraction{}{\sigma(x_{1})}{}{\sigma(x_{2})}\sigma(x_{1})\sigma(x_{2})=-2\lambda\delta(x_{12})$.
For instance, for the $\sigma$ two-point function we get an infinite set of
diagrams at leading order in $1/N$:

$\begin{split}\langle\sigma(x_{1})\sigma(x_{2})\rangle_{\lambda}=&\Big{[}-2\lambda\delta(x_{12})+(-2\lambda)^{2}\frac{G_{0}(x_{1},x_{2})^{2}}{4}+(-2\lambda)^{3}\int
d^{d}y\frac{G_{0}(x_{1},y)^{2}}{4}\frac{G_{0}(y,x_{2})^{2}}{4}+\dots\Big{]}+O(N^{-1})\\\
=&-\int\frac{d^{d}{k}}{(2\pi)^{d}}\frac{2\lambda e^{ik\cdot
x_{12}}}{1-2\lambda\frac{|k|^{d-4}2^{1-2d}\pi^{\frac{3-d}{2}}}{\Gamma(\frac{d-1}{2})\sin(\frac{\pi
d}{2})}}+O(N^{-1})\,,\end{split}$ (8.3)

where in the second equality we performed the geometric series of infinite
diagrams in momentum space, using the Fourier transform of the explicit
$G_{0}(x,x^{\prime})^{2}/4$ in (2.9). For the bi-local two-point function, the
free theory answer at $\lambda=0$ is corrected by the same infinite series of
diagrams to get

$\begin{split}&\langle\eta(x_{1},x_{2})\eta(x_{3},x_{4})\rangle_{\lambda}=G_{0}(x_{1},x_{4})G_{0}(x_{2},x_{3})\\\
&\qquad+\frac{1}{4}\int
d^{d}yd^{d}y^{\prime}G_{0}(x_{1},y)G_{0}(x_{2},y)\langle\sigma(y)\sigma(y^{\prime})\rangle_{\lambda}G_{0}(y^{\prime},x_{3})G_{0}(y^{\prime},x_{4})+O(N^{-1})\,.\end{split}$
(8.4)

We can then equivalently compute correlators of $\eta(x_{1},x_{2})$ using the
$\lambda\eta(x,x)^{2}$ formulation in (8.1), with no $\sigma(x)$, where the
Feynman rules are now given by those in section 2.2 except that the bi-local
propagator is now (8.4) (with the $\sigma$ two-point function given by (8.3)).

For $d<4$, $\lambda\eta(x,x)^{2}$ is a relevant deformation, and we can go to
the critical theory in the IR by taking $\lambda\to\infty$. In the $\sigma(x)$
formulation (8.2), we can then drop the $\sigma^{2}/\lambda$ term in (8.2), so
that $\sigma(x)$ becomes a Lagrange multiplier setting $\eta(x,x)=0$. We can
also now perform the momentum space integral in (8.3) to get

$\begin{split}\langle\sigma(x_{1})\sigma(x_{2})\rangle_{\infty}=&\frac{B_{d}}{x_{12}^{4}}+O(N^{-1})\,,\qquad
B_{d}=\frac{2^{d+3}\sin\left(\frac{\pi
d}{2}\right)\Gamma(\frac{d-1}{2})}{\pi^{\frac{3}{2}}\Gamma\left(\frac{d}{2}-2\right)}\,,\end{split}$
(8.5)

which is the 2-point function of a conformal primary $\sigma(x)$ with
$\Delta_{\sigma}=2+O(1/N)$. We thus see that to leading order in $1/N$, the
critical theory is the same as the free theory, but with the
$\frac{\phi_{I}^{*}\phi_{I}(x)}{\sqrt{N}}\equiv\eta(x,x)$ operator with
$\Delta_{\phi^{2}}=d-2$ replaced by the shadow operator $\sigma(x)$ with the
shadow dimension $\Delta_{\sigma}$. For the $O(N)$ theory, we have extra
factors of $\frac{1}{2}$ in (8.3) and (8.5), while in (8.4) the second term is
now multiplied by $\frac{1}{2}$ instead of $\frac{1}{4}$ and we should add the
symmetrized $x_{3}\leftrightarrow x_{4}$ term.

###  8.2 The bulk dual

We can map the double trace deformation in (8.1) and (8.2) to the bulk using
the off-shell mapping (4.45) that holds for $2<d<4$. We get

$\begin{split}Z_{\lambda}=\int
D\Phi_{J}(x,z)\text{exp}\left[-S[\Phi]-\frac{\lambda}{4f_{d-2,0}^{2}}\int
d^{d}x\left(\lim_{\varepsilon\to
0}\varepsilon^{2-d}\Phi_{0}(x,\varepsilon)\right)^{2}\right],\end{split}$
(8.6)

or equivalently (up to an overall numerical constant)

$\begin{split}Z_{\lambda}=\int
D\sigma(x)D\Phi_{J}(x,z)\text{exp}\left[-S[\Phi]-\int
d^{d}x\left(\frac{\sigma(x)}{2f_{d-2,0}}\lim_{\varepsilon\to
0}\varepsilon^{2-d}\Phi_{0}(x,\varepsilon)-\frac{\sigma^{2}(x)}{4\lambda}\right)\right]\,.\end{split}$
(8.7)

As in the mass deformation, we see that only the boundary of AdS is affected.
Note that the only effect of the deformation is to change the scalar bulk
propagator, just as in the CFT only the bi-local 2-point function was changed
(at leading order in $1/N$). We can compute this as in our computation of the
mass deformation VEV (7.4), to get

$\begin{split}&\langle\Phi_{0}(x_{1},z_{1})\Phi_{0}(x_{2},z_{2}\rangle_{\lambda}=\langle\Phi_{0}(x_{1},z_{1})\Phi_{0}(x_{2},z_{2})\rangle_{0}\\\
&\quad+\frac{f_{2,0}^{2}\Gamma(d-2)^{2}}{1024\pi^{2d}}\int
d^{d}yd^{d}y^{\prime}G_{d-2,0}(x_{1},z_{1}|y)\langle\sigma(y)\sigma(y^{\prime})\rangle_{\lambda}G_{d-2,0}(x_{2},z_{2}|y^{\prime})+O(N^{-1})\,,\end{split}$
(8.8)

where in the second term we resummed the same geometric series of diagrams
that appeared in the CFT calculation.

For the critical theory at $\lambda\to\infty$, we obtain when we choose the
local mapping $f^{\text{local}}_{\Delta,J}$:

$\begin{split}&\langle\Phi_{0}(x_{1},z_{1})\Phi_{0}(x_{2},z_{2}\rangle_{\infty}=\langle\Phi_{0}(x_{1},z_{1})\Phi_{0}(x_{2},z_{2})\rangle_{0}+\frac{B_{d}(f^{\text{local}}_{2,0})^{2}\Gamma(d-2)^{2}}{1024\pi^{2d}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\times\int
d^{d}yd^{d}y^{\prime}\frac{G_{d-2,0}(x_{1},z_{1}|y)G_{d-2,0}(x_{2},z_{2}|y^{\prime})}{|y-y^{\prime}|^{4}}+O(N^{-1})\\\
&\qquad\qquad\qquad\qquad\quad\quad=\langle\Phi_{0}(x_{1},z_{1})\Phi_{0}(x_{2},z_{2})\rangle_{0}+\frac{B_{d}(f^{\text{local}}_{2,0})^{2}\Gamma(d-2)^{2}S_{B}^{d-2,0}N_{2,0}\alpha_{0}(d-4)}{2048\pi^{2d}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\times(\Pi^{TT}_{d-2,0}(x_{1},z_{1};x_{2},z_{2})-\Pi^{TT}_{2,0}(x_{1},z_{1};x_{2},z_{2}))+O(N^{-1})\\\
&\qquad\qquad\qquad\qquad\quad\quad=\frac{\alpha_{0}/2}{M_{d,0}^{2}-M_{d-2,0}^{2}}(\Pi^{TT}_{2,0}(x_{1},z_{1};x_{2},z_{2})-\Pi^{TT}_{d,0}(x_{1},z_{1};x_{2},z_{2}))+O(N^{-1})\,,\\\
\end{split}$ (8.9)

where in the first equality we used (8.5), in the second we used the bulk
shadow transform (4.6) and the expression (5.19) for the AdS harmonic function
(4.7) in terms of the bulk-to-bulk propagator,292929For $J=0$, recall that
$\Pi^{TT}_{\Delta,0}(X_{1},X_{2};W_{1},W_{2})=\Pi_{\Delta,0}(X_{1},X_{2};W_{1},W_{2})$.
The identity (5.19) for the $J=0$ case was also used to relate the bulk dual
of the free and critical theories in [37, 68]. and in the third we used the
free theory expression for the bulk propagator in (5.24), and the explicit
expressions for the various constants in the previous sections. For the $O(N)$
theory, we get the same result with the usual extra factor of 2. For the more
general $f_{\Delta,J}$ defined in section 5, which has the same physical
propagator as $f_{\Delta,J}^{\text{local}}$, we can similarly compute the
result in the limit where one point goes to the boundary, to get

$\begin{split}&\lim_{z_{1}\to
0}\langle\Phi_{0}(x_{1},z_{1})\Phi_{0}(x_{2},z_{2})\rangle_{\infty}=\frac{\alpha_{0}f_{2,0}f_{d-2,0}/2}{M_{d,0}^{2}-M_{d-2,0}^{2}}\lim_{z_{1}\to
0}\Pi^{TT}_{2,0}(x_{1},z_{1};x_{2},z_{2})+O(N^{-1})\,.\\\ \end{split}$ (8.10)

In all these cases, we see that the physical propagator $\Pi^{TT}_{d-2,0}$ for
$\Phi_{0}$ has been replaced by the shadow propagator $\Pi^{TT}_{2,0}$. Recall
that in general both $\Pi^{TT}_{\Delta,J}$ and $\Pi^{TT}_{\tilde{\Delta},J}$
are defined by the same bulk differential equation (5.20), and differ only by
the boundary condition (5.21). Since the only difference between the free and
critical bulk theories was this scalar propagator, we see that to all orders
in $1/N$ the only difference between the free and critical bulk theories is
the boundary condition for the bulk scalar, as discussed in [36, 37, 68].

We can now compute bulk correlators with the Feynman rules discussed in
section 6, except with the $\Phi_{0}$ bulk propagator now given by (8.9). As
shown in [37], since the difference

$\begin{split}\langle\Phi_{0}(x_{1},z_{1})\Phi_{0}(x_{2},z_{2}\rangle_{\infty}-\langle\Phi_{0}(x_{1},z_{1})\Phi_{0}(x_{2},z_{2}\rangle_{0}\propto\Pi^{TT}_{d-2,0}(x_{1},z_{1};x_{2},z_{2})-\Pi^{TT}_{2,0}(x_{1},z_{1};x_{2},z_{2})\end{split}$
(8.11)

is free of short distance singularities, and since we already showed that the
free bulk dual gave the expected bulk diagrams dual to the free CFT (with
appropriate cancellations of all UV divergences), the modification of the free
bulk Feynman rules by replacing $\Pi^{TT}_{d-2,0}\to\Pi^{TT}_{2,0}$ leads to
the expected bulk dual of the critical CFT for all correlators at all orders
in $1/N$. Note that the argument in [37] did not assume any specific bulk
action, and only required that the difference between the free and critical
bulk Feynman rules be implemented by $\Pi^{TT}_{d-2,0}\to\Pi^{TT}_{2,0}$, so
the fact that our bulk theory has a non-standard off-shell kinetic term does
not matter.

The mapping of single-trace local operators in the CFT, as described in
section 4.5, is now different for the $J=0$ case. Note that for
$\lambda\to\infty$, the $\sigma(x)$ field acts as a Lagrange multiplier in
(8.7) that sets $\lim_{\varepsilon\to
0}\varepsilon^{2-d}\Phi_{0}(x,\varepsilon)=0$ off-shell, just as in the CFT it
set $\eta(x,x)=0$ off-shell. Thus, the off-shell relation (4.45) becomes
trivial in the critical theory. Since the $z^{d-2}$ mode vanishes,
$\Phi_{0}(x,z)$ now has the same small $z$ boundary condition as all other
$J>0$ bulk fields described in section 4.3, namely it scales as $z^{d/2}$,
which is the real part of the principal series contour. On-shell, we find that
(4.42) for $J=0$ is replaced by

$\begin{split}\langle\sigma(x)\dots\rangle_{\eqref{bulkSig}}=\frac{32\pi^{\frac{d}{2}}}{\Gamma(\frac{d}{2}-2)f_{2,0}}\lim_{\varepsilon\to
0}\varepsilon^{-2}\langle\Phi_{0}(x,\varepsilon)\dots\rangle_{\eqref{bulkLam}}\,,\end{split}$
(8.12)

where on the left-hand side we compute correlators using the
$\lambda\to\infty$ Feynman rules from (8.7), with the free bulk Feynman rules
for $\Phi_{J}$ and the $\sigma$ propagator in (8.5), while the right-hand side
uses the $\lambda\to\infty$ Feynman rules from (8.6), with the free bulk
Feynman rules for $\Phi_{J>0}$ and (8.9) for $\Phi_{0}$. We can see this on-
shell relation by computing the contraction between $\sigma(x)$ and
$\Phi_{0}(x,z)$ as

$\begin{split}\langle\sigma(x_{1})\Phi_{0}(x_{2},z)\rangle_{\eqref{bulkSig}}&=\lim_{\varepsilon\to
0}\frac{\varepsilon^{2-d}}{2f_{d-2,0}}\int
d^{d}y\langle\sigma(x_{1})\sigma(y)\rangle_{\infty}\langle\Phi_{0}(y,\epsilon)\Phi_{0}(x_{2},z)\rangle_{0}+O(N^{-1})\\\
&=\frac{\mathcal{C}_{d-2,0}\alpha_{0}f_{2,0}B_{d}}{4(M_{d,0}^{2}-M_{d-2,0}^{2})}\int
d^{d}y\frac{G_{d-2,0}(x_{2},z|y)}{|x_{1}-y|^{4}}+O(N^{-1})\\\
&=\frac{S_{B}^{d-2,0}\mathcal{C}_{d-2,0}\alpha_{0}f_{2,0}B_{d}}{4(M_{d,0}^{2}-M_{d-2,0}^{2})}{G_{2,0}(x_{2},z|x_{1})}+O(N^{-1})\\\
&=\frac{32\pi^{\frac{d}{2}}}{\Gamma(\frac{d}{2}-2)f_{2,0}}\lim_{\varepsilon\to
0}\varepsilon^{-2}\langle\Phi_{0}(x_{1},\varepsilon)\Phi_{0}(x_{2},z)\rangle_{\infty}+O(N^{-1})\,,\end{split}$
(8.13)

where in the first equality we used the $\sigma\Phi_{0}$ vertex in (8.7), in
the second we used the definitions (8.5) and (5.25), in the third we used the
bulk shadow transform (4.6), and in the last we used (8.10), (4.16), and the
explicit expressions for the various constants. In the small $z$ limit, we
similarly recover the relation between
$\langle\sigma(x_{1})\sigma(x_{2})\rangle$ and
$\langle\Phi_{0}(x_{1},\varepsilon)\Phi_{0}(x_{2},\varepsilon)\rangle$
required for (8.12).

## Acknowledgements

We would like to thank D. Anninos, M. Berkooz, S. Das, S. Pufu, C. Sleight,
and M. Taronna for useful discussions, and to especially thank V. Rosenhaus
for many useful discussions, and T. Solberg for collaboration on parts of this
project and on a related project. This work was supported in part by an Israel
Science Foundation center for excellence grant (grant number 2289/18), by
grant no. 2018068 from the United States-Israel Binational Science Foundation
(BSF), and by the Minerva foundation with funding from the Federal German
Ministry for Education and Research. OA is the Samuel Sebba Professorial Chair
of Pure and Applied Physics. SMC is supported in part by a Zuckerman STEM
leadership fellowship.

##  Appendix A The embedding space formalism

It will be convenient for us to use the embedding space formalism, both for
CFT computations and for computations in anti-de Sitter (AdS) space. This
section is mostly copying formulas and notations from [34] and [69]. The idea
is to use the embedding of AdSd+1 in $\mathbb{R}^{d+1,1}$ as the hyperboloid
$X^{2}=-1$, and of $d$-dimensional flat space (in the CFT, or on the boundary
of AdS) in $\mathbb{R}^{d+1,1}$ as the conformal light-cone $P^{2}=0$. In both
cases we denote the coordinates of $\mathbb{R}^{d+1,1}$ by
$X=\left(X^{+},X^{-},X^{i}\right)$ ($X^{i}=\left(X^{1},\cdots,X^{d}\right)$),
with the metric

$X^{2}=-X^{-}X^{+}+X^{i}X^{i}.$ (A.1)

In the bigger space the isometries/conformal symmetries $\text{SO}(d+1,1)$ are
linear and therefore writing covariant expressions is simpler. As we discuss
traceless symmetric tensors, we will also have vectors in the space as “spin
holders”. We usually denote the flat-space spin by $Z$ and the bulk AdS spin
by $W$.

###  A.1 Flat-space notations

We need two constraints to get a $d$-dimensional flat space, and we take:

  1. 1.

$P^{2}=0$.

  2. 2.

$P\equiv\lambda P$.

We can therefore use

$P=\left(1,y^{2},y^{i}\right)$ (A.2)

as the relation between the embedding coordinate $P$ and a standard coordinate
$y$ in $d$-dimensional flat space. Whenever we write an integral $dP$ it can
be interpreted as an integral over the $y^{i}$ in this slice (A.2).

Flat-space symmetric and traceless tensors can be written as a polynomial in a
new vector $Z$:

$H(P,Z)=H_{A_{1},...,A_{J}}(P)Z^{A_{1}}...Z^{A_{J}}.$ (A.3)

For the tensor to be traceless we require $Z^{2}=0$. For the tensor to be
defined only at directions parallel to the light-cone $P^{2}=0$ we require
further:

  1. 1.

$H(P,Z+\alpha P)=H(P,Z)$. Now the tensor is only parallel to the light-cone.

  2. 2.

$Z\cdot P=0$. Because $P\equiv\lambda P$, the tensor direction
$H^{A...}=P^{A}$ should also give a zero projection.

Given $H_{A_{1},...,A_{J}}(P)$ we can find the actual flat-space tensor
$h_{i_{1},...,i_{J}}(y)$ simply by a projection. In order to extract
$H_{A_{1},...,A_{J}}(P)$ from $H(P,Z)$ we define the operator

$D_{Z}^{A}=\left(\frac{d}{2}-1+Z\cdot\frac{\partial}{\partial
Z}\right)\frac{\partial}{\partial
Z_{A}}-\frac{1}{2}Z^{A}\frac{\partial^{2}}{\partial Z\cdot\partial Z},$ (A.4)

so that

$H^{A_{1}...A_{J}}(P)=\frac{1}{J!\left(\frac{d}{2}-1\right)_{J}}D_{Z}^{A_{1}}...D_{Z}^{A_{J}}H\left(P,Z\right).$
(A.5)

Note that using $D_{Z}$, the original tensor composing $H(P,Z)$ does not need
to be traceless (or symmetric) but the outcome of (A.5) will be.

Finally, a flat-space contraction of indices between two spin-$J$ traceless-
symmetric tensors can be written as

$f^{i_{1}...i_{J}}\left(y\right)g_{i_{1}...i_{J}}\left(y\right)=\frac{1}{J!\left(\frac{d}{2}-1\right)_{J}}F\left(P,D_{Z}\right)G\left(P,Z\right).$
(A.6)

###  A.2 Bulk notations

As we said before, $\text{AdS}_{d+1}$ can be embedded as the hyperboloid
$X^{2}=-1$. Denoting the upper half space coordinates $x^{\mu}=(z,x^{i})$
(with the standard metric $ds^{2}=(dz^{2}+dx^{i}dx^{i})/z^{2}$ that gives AdS
space in Poincaré coordinates) we have

$X=\frac{1}{z}\left(1,z^{2}+x^{2},x^{i}\right),$ (A.7)

where $x^{2}=\sum_{i=1}^{d}x_{i}^{2}$. As $z\to 0$, the bulk point approaches
$\frac{1}{z}$ times a null vector $P$ which can be identified with a point on
the boundary of AdS (using the flat-space embedding space formalism of the
previous subsection for the boundary).

A bulk symmetric and traceless tensor can be written as a polynomial in a new
vector $W$:

$H(X,W)=H_{A_{1},...,A_{J}}(X)W^{A_{1}}...W^{A_{J}}.$ (A.8)

For the tensor to be traceless we require $W^{2}=0$. For the tensor to be
defined only at directions parallel to the hyperboloid we also require $X\cdot
W=0$ (equivalently $X^{A}H_{A,...}=0$). Again, given $H_{A_{1},...,A_{J}}$ we
can find the actual AdS tensor $h_{\mu_{1},...,\mu_{J}}$ simply by a
projection. To extract the tensor $H_{A_{1},...,A_{J}}$ out of a field $H(Z)$
we need again a differential operator that would respect the transversality,
symmetry and tracelessness:

$\begin{split}K_{A}^{W}&=\frac{d-1}{2}\left(\frac{\partial}{\partial
W^{A}}+X_{A}\left(X\cdot\frac{\partial}{\partial
W}\right)\right)+\left(W\cdot\frac{\partial}{\partial
W}\right)\frac{\partial}{\partial W^{A}}\\\
&+X_{A}\left(W\cdot\frac{\partial}{\partial
W}\right)\left(X\cdot\frac{\partial}{\partial
W}\right)-\frac{1}{2}W_{A}\left(\frac{\partial^{2}}{\partial W\cdot\partial
W}+\left(X\cdot\frac{\partial}{\partial
W}\right)\left(X\cdot\frac{\partial}{\partial W}\right)\right).\end{split}$
(A.9)

This operator is indeed transverse ($X^{A}K_{A}=0$), symmetric
($K_{A}K_{B}=K_{B}K_{A}$) and traceless $K_{A}K^{A}=0$, so that

$H^{A_{1}...A_{J}}(X)=\frac{1}{J!\left(\frac{d-1}{2}\right)_{J}}K_{W}^{A_{1}}...K_{W}^{A_{J}}H\left(X,W\right).$
(A.10)

For points on the embedded AdS (with $X^{2}=-1$, $W^{2}=W\cdot X=0$), it is
enough to use

$K_{A}^{W}=\left(\frac{d-1}{2}+W\cdot\frac{\partial}{\partial
W}\right)\frac{\partial}{\partial W^{A}}.$ (A.11)

Again, the original tensor composing $H(X,W)$ does not need to be traceless
(or symmetric), but using $K_{A}$’s the outcome of (A.10) will be.

A bulk contraction of indices between two spin-$J$ traceless-symmetric tensors
can be written as

$f^{\mu_{1}...\mu_{J}}\left(z,x^{i}\right)g_{\mu_{1}...\mu_{J}}\left(z,x^{i}\right)=\frac{1}{J!\left(\frac{d-1}{2}\right)_{J}}F\left(X,K_{W}\right)G\left(X,W\right).$
(A.12)

Finally, the AdS gradient operator may be written in embedding space as

$\nabla_{A}=\frac{\partial}{\partial
X^{A}}+X_{A}\left(X\cdot\frac{\partial}{\partial
X}\right)+W_{A}\left(X\cdot\frac{\partial}{\partial W}\right),$ (A.13)

such that after the projection the resulting vector is $D_{\mu}$.

##  Appendix B The $2$-point function of $\eta$ in the $C_{\Delta,J}$
formalism

In this appendix we use the $C_{\Delta,J}$ variables of section 3 to compute
the tree-level two-point function of $\eta$ (or the four-point function of the
$\phi_{I}$), and confirm that we obtain the same result as in section 2. We
will convert the embedding space results back to coordinate space, to make the
calculations more transparent. We will perform the calculation assuming $d>4$,
so that we can just use the principal series contour. The results for $d\leq
4$ can then be obtained by analytic continuation (or could be computed
directly in those dimensions by using the more complicated contours discussed
in the main text).

Using the decomposition (3.20) and (3.9) as written in coordinate space, we
get for the $U(N)$ theory (keeping just the quadratic term in the action, and
with $\Delta_{0}=\frac{d-2}{2}$)

$\begin{split}\left<\eta(x_{1},x_{2})\eta(x_{3},x_{4})\right>_{\text{tree}}&=\sum_{J,J^{\prime}=0}^{\infty}\int_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\int_{\gamma_{J^{\prime}}}\frac{d\Delta^{\prime}}{2\pi i}\int d^{d}y\int
d^{d}y^{\prime}\\\ &\quad\times\,\langle{\cal
O}_{\Delta_{0}}\left(x_{1}\right)\hat{\cal
O}_{\Delta_{0}}\left(x_{2}\right){\cal
O}_{\Delta,J}\left(y\right)\rangle\,\langle{\cal
O}_{\Delta_{0}}\left(x_{3}\right)\hat{\cal
O}_{\Delta_{0}}\left(x_{4}\right){\cal
O}_{\tilde{\Delta}^{\prime},J^{\prime}}\left(y^{\prime}\right)\rangle\\\
&\quad\times(-1)^{J^{\prime}}\int
DC_{\Delta,J}(y)\,C_{\Delta,J}\left(y\right)C^{*}_{\Delta^{\prime},J^{\prime}}\left(y^{\prime}\right)e^{-S^{(2)}[C_{\Delta,J}]}\,.\end{split}$
(B.1)

The path integral is Gaussian. It is useful to define the conformal partial
wave (CPW) [54, 53]303030The two papers differ by a factor of $2^{-J}$ on
their definition of the conformal block. We use the one of [53].

$\begin{split}\Psi_{\Delta,J}^{\Delta_{0}}(x_{i})&\equiv\int
d^{d}y\langle{\cal O}_{\Delta_{0}}\left(x_{1}\right)\hat{\cal
O}_{\Delta_{0}}\left(x_{2}\right){\cal
O}_{\Delta,J}\left(y\right)\rangle\langle{\cal
O}_{\Delta_{0}}\left(x_{3}\right)\hat{\cal
O}_{\Delta_{0}}\left(x_{4}\right){\cal
O}_{\tilde{\Delta},J}\left(y\right)\rangle\\\
&=\frac{1}{|x_{12}|^{d-2}|x_{34}|^{d-2}}\left[S_{\Delta_{0},\Delta_{0}}^{(\tilde{\Delta},J)}G_{\Delta,J}(U,V)+S_{\Delta_{0},\Delta_{0}}^{(\Delta,J)}G_{\tilde{\Delta},J}(U,V)\right]\end{split}$
(B.2)

where as usual we omit the index contractions, and the conformal cross ratios

$\begin{split}U=\frac{|x_{12}|^{2}|x_{34}|^{2}}{|x_{13}|^{2}|x_{24}|^{2}}\,,\qquad
V=\frac{|x_{23}|^{2}|x_{14}|^{2}}{|x_{13}|^{2}|x_{24}|^{2}}\,.\end{split}$
(B.3)

Then, using (3.27) the Gaussian path integral gives313131There are two
important distinctions between our expression and that of [53]. Firstly, their
contour is only for the upper half plane of the principal series, which is why
we have a 2 in (B.4) relative to their expression. Also, their operators are
ordered differently from ours so that our integrand has an extra $(-1)^{J}$
relative to theirs.

$\begin{split}\left<\eta(x_{1},x_{2})\eta(x_{3},x_{4})\right>_{\text{tree}}=&\sum_{J=0}^{\infty}\int_{\gamma_{J}}\frac{(-1)^{J}d\Delta}{4\pi
i\,\lambda_{\Delta,J}N_{\Delta,J}}\int d^{d}y\langle{\cal
O}_{\Delta_{0}}\left(x_{1}\right)\hat{\cal
O}_{\Delta_{0}}\left(x_{2}\right){\cal O}_{\Delta,J}\left(y\right)\rangle\\\
&\qquad\qquad\qquad\qquad\qquad\times\langle{\cal
O}_{\Delta_{0}}\left(x_{3}\right)\hat{\cal
O}_{\Delta_{0}}\left(x_{4}\right){\cal
O}_{\tilde{\Delta},J}\left(y\right)\rangle\\\
=&\sum_{J=0}^{\infty}\int_{\gamma_{J}}\frac{d\Delta}{4\pi
i}\frac{(-1)^{J}}{\lambda_{\Delta,J}N_{\Delta,J}}\Psi^{\Delta_{0}}_{\Delta,J}(x_{i})\\\
=&\frac{1}{|x_{12}|^{d-2}|x_{34}|^{d-2}}\sum_{J=0}^{\infty}\int_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\frac{(-1)^{J}S_{\Delta_{0},\Delta_{0}}^{(\tilde{\Delta},J)}}{\lambda_{\Delta,J}N_{\Delta,J}}G_{\Delta,J}(U,V)\,.\end{split}$
(B.4)

In the last equality of (B.4) we used the fact that the rest of the integrand
(and the contour itself, including our deformations) is invariant under
$\Delta\iff\tilde{\Delta}$. The coefficients of the conformal blocks have
poles at $\Delta=d-2+2n+J$ for $n=0$ with residues:323232As we can see in
(3.25), the function $\frac{1}{\lambda_{\Delta,J}}$ has poles for both $n=0$
($\Delta=d+J-2$) and $n=1$ ($\Delta=d+J$). So in a way both are “physical
operators”. But from the free theory we know that while the first is an actual
operator (the mass operator $\phi^{2}(x)$ and the conserved spin $J$
currents), the second vanishes because of the equation of motion
$:\phi\partial^{J}\mathop{\rm\Box}\nolimits\phi:=0$. The reason $n=1$ doesn’t
contribute to the correlation function at the end of the day is because we
take residues of
$\frac{S_{\Delta_{0},\Delta_{0}}^{(\tilde{\Delta},J)}}{\lambda_{\Delta,J}N_{\Delta,J}}$.
In $S_{\Delta_{0},\Delta_{0}}^{(\tilde{\Delta},J)}$ we have a term
$\Gamma^{-2}\left(\frac{\tilde{\Delta}+J}{2}\right)$ (see (3.5)) with a double
zero at $\Delta=d+J$ that kills the pole from $\frac{1}{\lambda_{\Delta,J}}$.
The conclusion is that the $\Delta=d+J$ operators are “physical” at the level
of the action, but don’t contribute to any physical OPE.

$\begin{split}R^{\left(J\right)}\equiv\text{Res}\left[\frac{(-1)^{J}S_{\Delta_{0},\Delta_{0}}^{(\tilde{\Delta},J)}}{\lambda_{\Delta,J}N_{\Delta,J}}\right]=-(-1)^{J}\frac{\pi^{-d}2^{J-4}(d+2J-3)\Gamma\left(\frac{d}{2}+J-1\right)^{2}\Gamma(d+J-3)}{\Gamma(J+1)\Gamma(d+2J-2)}\,.\end{split}$
(B.5)

To compute (B.4) we thus need to perform the sum

$-\frac{1}{|x_{12}|^{d-2}|x_{34}|^{d-2}}\sum_{J=0}^{\infty}R^{(J)}G_{d+J-2,J}(U,V)\,,$
(B.6)

where the minus sign comes from closing the contour to the right, and the
conformal blocks $G_{\Delta,J}$ are normalized with the small $U$ expansion

$\begin{split}G_{\Delta,J}(U,V)=\frac{1}{(-2)^{J}}U^{\frac{\Delta-J}{2}}(1-V)^{J}{}_{2}F_{1}\left(\frac{\Delta+J}{2},\frac{\Delta+J}{2},{\Delta+J},1-V\right)+O(U^{\frac{\Delta-J}{2}+1})\,.\end{split}$
(B.7)

Since for the free theory we only consider the lowest twist operators, we can
actually ignore the sub-leading powers of $U$, which must identically cancel.
We thus compute

$\begin{split}&\sum_{J=0}^{\infty}\frac{(d+2J-3)\Gamma\left(\frac{d}{2}+J-1\right)^{2}\Gamma(d+J-3)}{2^{4-J}\pi^{d}\Gamma(J+1)\Gamma(d+2J-2)}\frac{U^{\frac{d-2}{2}}}{2^{J}}(1-V)^{J}{}_{2}F_{1}\left[\frac{d-2}{2}+J,\frac{d-2}{2}+J,{d-2+2J},1-V\right]\\\
&=U^{\frac{d}{2}-1}\int_{0}^{1}dt\sum_{J=0}^{\infty}\frac{\pi^{-d}(d+2J-3)(1-v)^{J}((1-t)t)^{\frac{d}{2}+J-2}\Gamma(d+J-3)(t(v-1)+1)^{-\frac{d}{2}-J+1}}{16\Gamma(J+1)}\\\
&=\left[\frac{\Gamma(d/2-1)}{4\pi^{d/2}}\right]^{2}(U/V)^{\frac{d}{2}-1}\,,\end{split}$
(B.8)

so that the resulting (connected part of the) correlation function is

$\langle\eta(x_{1},x_{2})\eta(x_{3},x_{4})\rangle=\left[\frac{\Gamma(d/2-1)}{4\pi^{d/2}}\right]^{2}\frac{1}{|x_{14}|^{d-2}|x_{23}|^{d-2}}\,,$
(B.9)

as we expect from Wick contractions for the canonically normalized scalar with
propagator (2.9). For the $O(N)$ theory, we sum only over even spins and
multiply the residues by two since the quadratic action for the $O(N)$ theory
is half that of the $U(N)$. We then get
$\left[\frac{\Gamma(d/2-1)}{4\pi^{d/2}}\right]^{2}({U^{\frac{d}{2}-1}+(U/V)^{\frac{d}{2}-1}})$,
and so the correlation function is

$\begin{split}\langle\eta(x_{1},x_{2})\eta(x_{3},x_{4})\rangle=\left[\frac{\Gamma(d/2-1)}{4\pi^{d/2}}\right]^{2}\left[\frac{1}{|x_{13}|^{d-2}|x_{24}|^{d-2}}+\frac{1}{|x_{14}|^{d-2}|x_{23}|^{d-2}}\right]\,,\end{split}$
(B.10)

as expected.

##  Appendix C Bulk identities

In this appendix we prove some key identities used in the main text to analyze
the bulk physics.

###  C.1 Proof of (6.13)

The first identity we consider is (6.13), which can be written explicitly as

$\begin{split}&\sum_{J}\frac{1}{\left(J!\left(\frac{d-1}{2}\right)_{J}\right)^{2}}\int
dX\int
dX^{\prime}\left(W_{1}\cdot\nabla_{X_{1}}\right)^{J_{1}}\Omega_{0}(X_{1},X)\left(K_{W}\cdot\nabla_{X}\right)^{J}\Omega_{0}(X_{2},X)\\\
&\times\Theta_{J}(X,X^{\prime};W,W^{\prime})\left(K_{W^{\prime}}\cdot\nabla_{X^{\prime}}\right)^{J}\Omega_{0}(X^{\prime},X_{3})\left(W_{4}\cdot\nabla_{X_{4}}\right)^{J_{4}}\Omega_{0}(X^{\prime},X_{4})\\\
&\qquad=\left(W_{1}\cdot\nabla_{X_{1}}\right)^{J_{1}}\Omega_{0}(X_{1},X_{3})\left(W_{4}\cdot\nabla_{X_{4}}\right)^{J_{4}}\Omega_{0}(X_{2},X_{4})\,.\end{split}$
(C.1)

As a first step, we will show a simpler identity: we will interpret the
conformal basis completeness relation (3.2) as an identity about $\Theta$. We
have

$\begin{split}\delta(P_{1},P_{3})\delta(P_{2},P_{4})&=\frac{1}{2}\sum_{J}\int\frac{dP}{J!\left(\frac{d}{2}-1\right)_{J}}\int\frac{d\Delta}{2\pi
iN_{\Delta,J}}\\\ &\times\langle{\cal
O}_{\Delta_{0}}\left(P_{1}\right)\hat{\cal
O}_{\Delta_{0}}\left(P_{2}\right){\cal
O}_{\Delta,J}\left(P,D_{Z}\right)\rangle\langle{\cal
O}_{\tilde{\Delta}_{0}}\left(P_{3}\right)\hat{\cal
O}_{\tilde{\Delta}_{0}}\left(P_{4}\right){\cal
O}_{\tilde{\Delta},J}\left(P,Z\right)\rangle\\\
&=\frac{1}{2}\sum_{J}\frac{1}{\left(J!\left(\frac{d-1}{2}\right)_{J}\right)^{2}}\int\frac{d\Delta}{2\pi
iN_{\Delta,J}b^{(\Delta,J)}_{\Delta_{0},\Delta_{0}}{b}^{(\tilde{\Delta},J)}_{\tilde{\Delta}_{0},\tilde{\Delta}_{0}}}\\\
&\times\int
dX(W\cdot\nabla)^{J}G_{\Delta_{0},0}(X,P_{2})G_{\Delta_{0},0}(X,P_{1})\int
dX^{\prime}(W^{\prime}\cdot\nabla)^{J}G_{\tilde{\Delta}_{0},0}(X^{\prime},P_{3})G_{\tilde{\Delta}_{0},0}(X^{\prime},P_{4})\\\
&\qquad\int\frac{dP}{J!\left(\frac{d}{2}-1\right)_{J}}G_{\Delta,J}(X,P;K_{W},D_{Z})G_{\tilde{\Delta},J}(X^{\prime},P;K_{W^{\prime}},Z)\\\
&=\sum_{J}\frac{1}{\left(J!\left(\frac{d-1}{2}\right)_{J}\right)^{2}}\int
dX\int
dX^{\prime}(K_{W}\cdot\nabla)^{J}G_{\Delta_{0},0}(X,P_{2})G_{\Delta_{0},0}(X,P_{1})\\\
&\qquad\qquad\times\Theta(X,X^{\prime};W,W^{\prime})(K_{W^{\prime}}\cdot\nabla)^{J}G_{\tilde{\Delta}_{0},0}(X^{\prime},P_{3})G_{\tilde{\Delta}_{0},0}(X^{\prime},P_{4})\,,\end{split}$
(C.2)

where in the second equality we used (6.2) twice, while in the third equality
we used (4.7) and (6.8). We can draw this result as

(C.3)

where the boundary legs are the bulk-to-boundary propagators with
$\Delta=\Delta_{0}$ (and its shadow). Similar to Table 1, the arrows signify
the derivatives (and spin contractions). We can now finish the proof:

$\begin{split}\Omega_{0}(X_{1},X_{3})&\Omega_{0}(X_{2},X_{4})=\int
dP_{1}dP_{2}dP_{3}dP_{4}\delta(P_{1},P_{3})\delta(P_{2},P_{4})\\\
&\qquad\qquad\qquad\times
G_{\tilde{\Delta}_{0},0}(X_{1},P_{1})G_{\tilde{\Delta}_{0},0}(X_{2},P_{2})G_{\Delta_{0},0}(X_{3},P_{3})G_{\Delta_{0},0}(X_{4},P_{4})\\\
=&\sum_{J}\frac{1}{\left(J!\left(\frac{d-1}{2}\right)_{J}\right)^{2}}\int
dX\int dX^{\prime}\int
dP_{1}G_{\tilde{\Delta}_{0},0}(X_{1},P_{1})G_{\Delta_{0},0}(X,P_{1})\\\
&\times\int
dP_{2}(K_{W}\cdot\nabla)^{J}G_{\Delta_{0},0}(X,P_{2})G_{\tilde{\Delta}_{0},0}(X_{2},P_{2})\Theta(X,X^{\prime};W,W^{\prime})\\\
&\times\int
dP_{3}G_{\Delta_{0},0}(X_{3},P_{3})(K_{W^{\prime}}\cdot\nabla)^{J}G_{\tilde{\Delta}_{0},0}(X^{\prime},P_{3})\int
dP_{4}G_{\Delta_{0},0}(X_{4},P_{4})G_{\tilde{\Delta}_{0},0}(X^{\prime},P_{4})\,.\end{split}$
(C.4)

Applying $(W_{1}\cdot D_{X_{1}})^{J_{1}}(W_{4}\cdot D_{X_{4}})^{J_{4}}$ on
both sides gives exactly (6.13).

###  C.2 Proof of (6.14) and bulk regularization

We start by considering the completeness relation (C.2), which we wrote
diagrammatically in (C.3). Suppose that we connect $P_{2}$ to a bulk point
$X_{2}$ with a bulk-to-boundary propagator $G_{\tilde{\Delta}_{0},0}$, that we
connect $P_{4}$ to a bulk point $X_{1}$ with a bulk-to-boundary propagator
$G_{\Delta_{0},0}$, that we identify $P_{1}=P_{3}$, and that we integrate over
$P_{1},P_{2}$ and $P_{4}$. Finally we also operate $(W_{1}\cdot
D_{X_{1}})^{J_{1}}$ on both sides. The equality we then obtain takes the form

$\vbox{\hbox{\includegraphics{Omega_line.pdf}}}\cdot``\int
dP\delta(0)^{\prime\prime}=\vbox{\hbox{\includegraphics{Theta_loop_proof.pdf}}}\,.$
(C.5)

Evidently, this equality needs a regularization. The $d$-dimensional integral
on the left-hand side is the same one that we obtained when we discussed loop
integrals in the bi-local formalism in section 2.3. There it was regularized
by putting the theory on a finite lattice, and it was set equal to the number
of lattice points $V$ (which then appeared in our counter-terms). More
generally, with some high-momentum cutoff $|k|<\frac{1}{\varepsilon}$ and an
IR cutoff $|x|<L$, this integral would be regularized to $V\sim
L^{d}/\varepsilon^{d}$. The bulk integrals on the right-hand side similarly
diverge in the UV and in the IR (after the implicit summation over $J$). We
can regularize them in the bulk by introducing a cutoff on the radial
direction $z>\varepsilon$, and an IR cutoff for the space-time directions
$|x|<L$. We then expect the regularized integral to give (6.14) with $V\sim
L^{d}/\varepsilon^{d}$, up to corrections that go to zero when we take
$\varepsilon\to 0$ and $L\to\infty$. We stress that the sum over the internal
spin $J$ needs to be taken before we remove the regulators. The reason is that
the dominant spins in the sum with a given regulator scale as
$J\sim\frac{L^{2}}{\varepsilon|x_{12}|}$ [70]. Since all loop divergences in
our computations reduce to (6.14), this regularization scheme should
regularize systematically all the bulk integrals.

###  C.3 Modifications for $d<4$

In this appendix we discuss how the arguments of section 6 must be modified
for $d<4$. The only modification of the inverse mapping is for $J=0$, as given
by (4.28):

$\begin{split}{\cal M}_{0}^{-1}\left(P_{1},P_{2}\mid
X\right)=\int_{\gamma_{0}}\frac{d\Delta}{2\pi
i}\frac{1}{\alpha_{0}N_{\Delta,0}\lambda_{\Delta,0}f_{\Delta,0}}\int
dP\langle{\cal O}_{\Delta_{0}}\left(P_{1}\right)\hat{\cal
O}_{\Delta_{0}}\left(P_{2}\right){\cal O}_{\Delta,0}\left(P\right)\rangle
G_{\tilde{\Delta},0}\left(X,P\right)\\\
\times\left(\nabla_{X}^{2}-M_{d-2,0}^{2}\right)\left(\nabla_{X}^{2}-M_{d,0}^{2}\right)\,.\end{split}$
(C.6)

Using (6.2) we can write it as

${\cal M}_{0}^{-1}\left(P_{1},P_{2}\mid X\right)=\int
dX^{\prime}G_{\Delta_{0},0}(X^{\prime},P_{2})G_{\Delta_{0},0}(X^{\prime},P_{1})\theta_{0}(X,X^{\prime})\,,$
(C.7)

where we now define (different than (6.5))

$\theta_{0}(X^{\prime},X)=\int_{\gamma_{0}}\frac{d\Delta}{2\pi
i}\frac{1}{\lambda_{\Delta,0}f_{\Delta,0}b^{(\Delta,0)}_{\Delta_{0},\Delta_{0}}}\Omega_{\Delta}(X^{\prime},X)\left(\nabla_{X}^{2}-M_{d-2,0}^{2}\right)\left(\nabla_{X}^{2}-M_{d,0}^{2}\right)\,.$
(C.8)

Notice that now $\theta_{0}$ (and the mappings themselves) is not merely a
function (or a distribution) but a differential operator. For $J>0$ the
inverse mapping remains (4.27).

Next, consider the direct mapping, which for $d<4$ is given by (4.20)

$\begin{split}{\cal M}_{J}\left(X,W\mid
P_{1},P_{2}\right)&=\frac{1}{2}\int_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\int\frac{dP}{J!\left(\frac{d}{2}-1\right)_{J}}\frac{f_{\Delta,J}}{\lambda_{\Delta,J}N_{\Delta,J}}\\\
&\times G_{\Delta,J}\left(X,P;W,D_{Z}\right)\langle{\cal
O}_{\Delta_{0}}\left(P_{1}\right)\hat{\cal
O}_{\Delta_{0}}\left(P_{2}\right){\cal
O}_{\tilde{\Delta},J}\left(P,Z\right)\rangle\,\nabla^{2}_{P_{1}}\nabla^{2}_{P_{2}}.\end{split}$
(C.9)

Again, using (6.2) we can write this as

$\begin{split}{\cal M}_{J}\left(X,W\mid
P_{1},P_{2}\right)&=\int\frac{dX^{\prime}}{J!\left(\frac{d-1}{2}\right)_{J}}G_{\Delta_{0},0}(X^{\prime},P_{2})(K_{W^{\prime}}\cdot\nabla_{X^{\prime}})^{J}G_{\Delta_{0},0}(X^{\prime},P_{1})\tilde{\theta}_{J}(X,X^{\prime};W,W^{\prime})\nabla^{2}_{P_{1}}\nabla^{2}_{P_{2}}\,,\end{split}$
(C.10)

where we now define

$\tilde{\theta}_{J}(X,X^{\prime};W,W^{\prime})=\int_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\frac{\alpha_{J}f_{\Delta,J}}{2b_{\Delta_{0},\Delta_{0}}^{(\tilde{\Delta},J)}\lambda_{\Delta,J}}\\\
\Omega_{\Delta,J}\left(X,X^{\prime};W,W^{\prime}\right).$ (C.11)

We would like to show that all the identities of section 6 have their version
also for these new $\theta,\tilde{\theta}$. We start with the definition of
$\Theta_{J}$ (6.11). For $J=0$ we now have

$\begin{split}\Theta_{0}(X,X^{\prime})&=\int
dY\theta_{0}(X,Y)\tilde{\theta}_{0}(Y,X^{\prime})\\\ &=\int
dY\int_{\gamma_{0}}\frac{d\Delta}{2\pi
i}\frac{1}{\lambda_{\Delta,0}f_{\Delta,0}b^{(\Delta,0)}_{\Delta_{0},\Delta_{0}}}\Omega_{\Delta,0}(X,Y)\left(\nabla_{Y}^{2}-M_{d-2,0}^{2}\right)\left(\nabla_{Y}^{2}-M_{d,0}^{2}\right)\\\
&\quad\times\int_{\gamma_{0}}\frac{d\Delta^{\prime}}{2\pi
i}\frac{\alpha_{0}f_{\Delta^{\prime},0}}{2b_{\Delta_{0},\Delta_{0}}^{(\tilde{\Delta}^{\prime},0)}\lambda_{\Delta^{\prime},0}}\Omega_{\Delta^{\prime},0}\left(Y,X^{\prime}\right)\\\
&=\int_{\gamma_{0}}\frac{d\Delta}{2\pi
i}\frac{\alpha_{0}}{\lambda_{\Delta,0}2b_{\Delta_{0},\Delta_{0}}^{(\tilde{\Delta}^{\prime},0)}b^{(\Delta,0)}_{\Delta_{0},\Delta_{0}}}\Omega_{\Delta,0}(X,X^{\prime})\,.\end{split}$
(C.12)

Note that $\theta_{0}$ and $\tilde{\theta}_{0}$ are now operators, so they now
do not obviously commute as they did for $d>4$ in (6.11). We will use the
specific ordering of (C.12). For $J>0$ we have

$\begin{split}\Theta_{J}(X_{1},X_{2};W_{1},W_{2})&=\int\frac{dX}{J!\left(\frac{d-1}{2}\right)_{J}}\theta_{J}(X_{1},X;W_{1},K_{W}){\tilde{\theta}}_{J}(X,X_{2};W,W_{2})\\\
&=\int_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\frac{\alpha_{J}}{2\lambda_{\Delta,J}b_{\Delta_{0},\Delta_{0}}^{(\tilde{\Delta},J)}b^{(\Delta,J)}_{\Delta_{0},\Delta_{0}}}\Omega_{\Delta,J}(X_{1},X_{2};W_{1},W_{2})\,,\end{split}$
(C.13)

which is similar to the $J=0$ case.

Next, we would like to show the $d<4$ version of (6.12):

$\begin{split}\delta^{TT}&(X_{1},X_{2})W_{12}^{J_{1}}\delta_{J_{1},J_{2}}=\int
dP_{1}\int dP_{2}\ {\cal M}_{J_{1}}\left(X_{1},W_{1}|P_{1},P_{2}\right){\cal
M}_{J_{2}}^{-1}\left(P_{1},P_{2}|X_{2},W_{2}\right)\\\ &=\int dP_{1}\int
dP_{2}\int\frac{dX_{1}^{\prime}}{{J_{1}}!\left(\frac{d-1}{2}\right)_{J_{1}}}G_{\Delta_{0},0}(X_{1}^{\prime},P_{2})(K_{W_{1}^{\prime}}\cdot\nabla_{X_{1}^{\prime}})^{J_{1}}G_{\Delta_{0},0}(X_{1}^{\prime},P_{1})\tilde{\theta}_{J_{1}}(X_{1},X_{1}^{\prime};W_{1},W_{1}^{\prime})\\\
&\nabla^{2}_{1}\nabla^{2}_{2}\int\frac{dX_{2}^{\prime}}{J_{2}!\left(\frac{d-1}{2}\right)_{J_{2}}}(K_{W_{2}^{\prime}}\cdot\nabla_{X_{2}^{\prime}})^{J}G_{\Delta_{0},0}(X_{2}^{\prime},P_{1})G_{\Delta_{0},0}(X_{2}^{\prime},P_{2})\theta_{J_{2}}(X_{2}^{\prime},X_{2};W_{2}^{\prime},W_{2})\\\
&=a_{\Delta_{0}}^{2}\int dP_{1}\int
dP_{2}\int\frac{dX_{1}^{\prime}}{{J_{1}}!\left(\frac{d-1}{2}\right)_{J_{1}}}G_{\Delta_{0},0}(X_{1}^{\prime},P_{2})(K_{W^{\prime}_{1}}\cdot\nabla_{X_{1}^{\prime}})^{J_{1}}G_{\Delta_{0},0}(X_{1}^{\prime},P_{1})\tilde{\theta}_{J_{1}}(X_{1},X_{1}^{\prime};W_{1},W_{1}^{\prime})\\\
&\int\frac{dX_{2}^{\prime}}{J_{2}!\left(\frac{d-1}{2}\right)_{J_{2}}}(K_{W_{2}^{\prime}}\cdot\nabla_{X_{2}^{\prime}})^{J}G_{\tilde{\Delta}_{0},0}(X_{2}^{\prime},P_{1})G_{\tilde{\Delta}_{0},0}(X_{2}^{\prime},P_{2})\theta_{J_{2}}(X_{2}^{\prime},X_{2};W_{2}^{\prime},W_{2})\\\
&=a_{\Delta_{0}}^{2}\int\frac{dX_{1}^{\prime}}{{J_{1}}!\left(\frac{d-1}{2}\right)_{J_{1}}}\int\frac{dX_{2}^{\prime}}{J_{2}!\left(\frac{d-1}{2}\right)_{J_{2}}}(K_{W_{2}^{\prime}}\cdot\nabla_{X_{2}^{\prime}})^{J}\Omega_{0}(X_{2}^{\prime},X_{1}^{\prime})(K_{W_{1}^{\prime}}\cdot\nabla_{X_{1}^{\prime}})^{J_{1}}\Omega_{0}(X_{2}^{\prime},X_{1}^{\prime})\\\
&\tilde{\theta}_{J_{1}}(X_{1},X_{1}^{\prime};W_{1},W_{1}^{\prime})\theta_{J_{2}}(X_{2}^{\prime},X_{2};W_{2}^{\prime},W_{2})\,,\end{split}$
(C.14)

where we used (6.17) in the last line. We see a factor of $a_{\Delta_{0}}^{2}$
compared to (6.12).

We can now derive the $d<4$ version of (6.13) as we did in the previous
subsection for $d>4$, by starting with (3.2) and then applying (6.2). The
difference for $d<4$ is that now the completeness relation for $d<4$ is
altered (3.22), so that we now have

$\begin{split}\delta(P_{1},P_{3})\delta(P_{2},P_{4})=&\sum_{J=0}^{\infty}\int_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\int\frac{dP}{J!\left(\frac{d}{2}-1\right)_{J}}\,\frac{1}{2N_{\Delta,J}\lambda_{\Delta,J}}\cdot\\\
&\qquad\langle{\cal O}_{\Delta_{0}}\left(P_{1}\right)\hat{\cal
O}_{\Delta_{0}}\left(P_{2}\right){\cal
O}_{\Delta,J}\left(P,D_{Z}\right)\rangle\langle{\cal
O}_{\Delta_{0}}\left(P_{3}\right)\hat{\cal
O}_{\Delta_{0}}\left(P_{4}\right){\cal
O}_{\tilde{\Delta},J}\left(P,Z\right)\rangle\nabla_{P_{3}}^{2}\nabla_{P_{4}}^{2}\\\
&=\sum_{J=0}^{\infty}\int\frac{dX_{1}^{\prime}}{J!\left(\frac{d-1}{2}\right)_{J}}\int\frac{dX_{2}^{\prime}}{J!\left(\frac{d-1}{2}\right)_{J}}\Theta_{J}(X_{1}^{\prime},X_{2}^{\prime};K_{W_{1}^{\prime}},K_{W_{2}^{\prime}})\\\
&\quad
G_{\Delta_{0},0}(X_{1}^{\prime},P_{1})(W_{1}^{\prime}\cdot\nabla_{X_{1}^{\prime}})^{J}G_{\Delta_{0},0}(X_{1}^{\prime},P_{2})\\\
&\quad
G_{\Delta_{0},0}(X_{2}^{\prime},P_{4})(W_{2}^{\prime}\cdot\nabla_{X_{2}^{\prime}})^{J}G_{\Delta_{0},0}(X_{2}^{\prime},P_{3})\nabla_{P_{3}}^{2}\nabla_{P_{4}}^{2}\,.\end{split}$
(C.15)

Multiplying by
$G_{\tilde{\Delta}_{0},0}(X_{1},P_{1})G_{\tilde{\Delta}_{0},0}(X_{2},P_{2})G_{\Delta_{0},0}(X_{3},P_{3})G_{\Delta_{0},0}(X_{4},P_{4})$
and integrating over $P_{1},P_{2},P_{3},P_{4}$ gives (using (6.17))

$\begin{split}\Omega_{0}(X_{1},X_{3})\Omega_{0}(X_{2},X_{4})&=a_{\Delta_{0}}^{2}\sum_{J=0}^{\infty}\int\frac{dX_{1}^{\prime}}{J!\left(\frac{d-1}{2}\right)_{J}}\int\frac{dX_{2}^{\prime}}{J!\left(\frac{d-1}{2}\right)_{J}}\Theta_{J}(X_{1}^{\prime},X_{2}^{\prime};K_{W_{1}^{\prime}},K_{W_{2}^{\prime}})\\\
&\quad\Omega_{0}(X_{1}^{\prime},X_{1})(W_{1}^{\prime}\cdot\nabla_{X_{1}^{\prime}})^{J}\Omega_{0}(X_{1}^{\prime},X_{2})\\\
&\quad\Omega_{0}(X_{2}^{\prime},X_{4})(W_{2}^{\prime}\cdot\nabla_{X_{2}^{\prime}})^{J}\Omega_{0}(X_{2}^{\prime},X_{3})\,.\end{split}$
(C.16)

We can then apply $(W_{1}\cdot D_{X_{1}})^{J_{1}}(W_{4}\cdot
D_{X_{4}})^{J_{4}}$ on both sides to get $a_{\Delta_{0}}^{2}\cdot$(6.13) (with
our modified $\Theta$).

Next, we can derive the modified version of (6.14) for $d<4$ by multiplying
(C.15) by
$G_{\tilde{\Delta}_{0}}\left(X_{2},P_{2}\right)G_{\Delta_{0}}\left(X_{1},P_{4}\right)\delta\left(P_{1},P_{3}\right)$
and integrating over $P_{1},P_{2},P_{3},P_{4}$ to get

$\begin{split}\Omega_{0}\left(X_{1},X_{2}\right)V&=\sum_{J=0}^{\infty}\int\frac{dX^{\prime}}{J!\left(\frac{d-1}{2}\right)_{J}}\int\frac{dX}{J!\left(\frac{d-1}{2}\right)_{J}}\Theta_{J}(X^{\prime},X;K_{W^{\prime}},K_{W})\\\
&\quad\int
dP_{1}dP_{2}dP_{3}dP_{4}G_{\Delta_{0},0}(X^{\prime},P_{1})(W^{\prime}\cdot\nabla_{X^{\prime}})^{J}G_{\Delta_{0},0}(X^{\prime},P_{2})\\\
&\quad
G_{\Delta_{0},0}(X,P_{4})(W\cdot\nabla_{X})^{J}G_{\Delta_{0},0}(X,P_{3})\\\
&\quad\nabla_{P_{3}}^{2}\nabla_{P_{4}}^{2}\left(G_{\tilde{\Delta}_{0},0}\left(X_{2},P_{2}\right)G_{\Delta_{0},0}\left(X_{1},P_{4}\right)\delta\left(P_{1},P_{3}\right)\right)\\\
&=a_{\Delta_{0}}\cdot\sum_{J=0}^{\infty}\int\frac{dX^{\prime}}{J!\left(\frac{d-1}{2}\right)_{J}}\int\frac{dX}{J!\left(\frac{d-1}{2}\right)_{J}}\Theta_{J}(X^{\prime},X;K_{W^{\prime}},K_{W})\\\
&\quad(W^{\prime}\cdot\nabla_{X^{\prime}})^{J}\Omega_{0}(X_{2},X^{\prime})\Omega_{0}(X_{1},X)\\\
&\quad\int
dP_{1}dP_{3}(W\cdot\nabla_{X})^{J}G_{\Delta_{0},0}(X,P_{3})G_{\Delta_{0},0}\left(X^{\prime},P_{1}\right)\nabla_{P_{3}}^{2}\delta\left(P_{1},P_{3}\right)\,,\end{split}$
(C.17)

which we can integrate by parts to get the right-hand side

$\begin{split}a_{\Delta_{0}}^{2}\cdot\sum_{J=0}^{\infty}\int\frac{dX^{\prime}}{J!\left(\frac{d-1}{2}\right)_{J}}\int\frac{dX}{J!\left(\frac{d-1}{2}\right)_{J}}\Theta_{J}(X^{\prime},X;K_{W^{\prime}},K_{W})\\\
\quad(W^{\prime}\cdot\nabla_{X^{\prime}})^{J}\Omega_{0}(X_{2},X^{\prime})\Omega_{0}(X_{1},X)(W\cdot\nabla_{X})^{J}\Omega_{0}(X,X^{\prime})\,.\end{split}$
(C.18)

Applying $(W_{1}\cdot D_{X_{1}})^{J_{1}}$ to both sides then gives
$a_{\Delta_{0}}^{2}$ times (6.14).

The two-point function in the bulk stays the same as discussed in section 4.
The Feynman rules are produced by applying the inverse mapping to the bi-local
action. Written in terms of our modified $\theta_{J},\tilde{\theta}_{J}$ it
has the same form (the same diagrams). Because the operators we defined
satisfy the same conditions (up to the $a_{\Delta_{0}}^{2}$ that cancel
between the uses of the identities) we get the same cancellations, as in
section 6.

We would like to write exactly the same Feynman rules as we found for $d>4$.
The only modification of the vertex has to do with the modification of
$\theta_{J}$ described above, where only the $J=0$ term (C.8) was changed by
acting with
$\left(\nabla_{X}^{2}-M_{d-2,0}^{2}\right)\left(\nabla_{X}^{2}-M_{d,0}^{2}\right)$
and inserting a $1/\lambda_{\Delta,J}$ in the $\Delta$ integral. The question
then is whether we are allowed to perform integration-by-parts to get the
original Feynman rules. Inside a Feynman diagram computation, the differential
operator will act on a bulk 2-point function from the Feynman rules. Applying
the operator we would get exactly a bulk delta-function, so the bulk integral
is trivial, and we will have the same properties as for $d>4$. On the other
hand, if we try to perform an integration by parts directly on the vertex (to
bring it to the same form as the $d>4$ vertex), we obtain when applying the
Feynman rules an integral involving $\int
dX\Omega_{\Delta}(X^{\prime},X)\Pi_{d-2}(X,X^{\prime\prime})$, which in the
small $z$ limit behaves as $z^{d/2+d-2-d}\sim z^{(d-4)/2}$, and so diverges
for $d<4$. So the difference between the Feynman rules of $d<4$ and those of
$d>4$ are the diverging boundary terms from the integration by parts. We can
think about them as UV counter terms, made to regularize the naive action
because of the unconventional boundary condition for $\Phi_{0}$ in this case.
The bottom line is that we need to use for $d<4$ the modified Feynman rules of
this section, but the expressions we get after using them are the same as we
obtain by continuing the $d>4$ results to $d<4$.

As explained in section 8, in the dual to the critical theory the boundary
condition for $\Phi_{0}$ (with $d<4$) does not allow for a $z^{d-2}$ term.
Hence, in this case no divergences will appear when integrating by parts the
action, so we can simply use precisely the same bulk action and Feynman rules
as in section 6.

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