O}_{i}{\cal O}_{i}\rangle\circ X_{0}-\frac{1}{3}(2\Delta_{X}^{-}+d)kX_{0}\,$
(139) $\displaystyle{\rm tr}\,b^{\partial}_{5}=\frac{1}{5760}\bigg{(}$
$\displaystyle
720X_{0}^{2}+\left(-360\Delta^{-}_{X}(d-\Delta^{-}_{X})k^{2}+90d\Delta_{X}^{-}k^{2}+15d(14+3d)k^{2}\right)X_{0}$
$\displaystyle+360k^{2}z_{0}^{2}\sum^{n}_{i=1}(\partial_{\mu}\partial^{\mu})_{i}X_{0}-240k^{2}z_{0}^{2}\partial_{\mu}\partial^{\mu}X_{0}$
$\displaystyle+(90d-720\Delta_{X}^{-})k\frac{1}{\sqrt{\bar{g}}}\sum^{n}_{i=1}\langle{\cal
O}_{i}{\cal O}_{i}\rangle\circ X_{0}$
$\displaystyle-360\frac{1}{\sqrt{\bar{g}}}\frac{1}{\sqrt{\bar{g}}}\sum^{n}_{i\neq
j}[\langle{\cal O}_{i}{\cal O}_{i}\rangle\circ][\langle{\cal O}_{j}{\cal
O}_{j}\rangle\circ]X_{0}\bigg{)}$ (140)

The result from Dirichlet BC (i.e. from integrating out $\Phi_{D}$ and not
$\Phi_{0}$) is shown in App. D.1 for completeness.

Finally we emphasize that all the bulk and boundary coefficients obtained in
this section are consistent with the conformal symmetry of the boundary in the
sense that the conformal dimensions of the components of $X$ appear only via
the total dimensions $\Delta_{X}$, $\Delta^{-}_{X}$, and never in other
combinations nor individually. This feature emerges via nontrivial
combinations of coefficients which occur only in the AdS/CFT validity regime
$|y-y^{\prime}|\gg z_{0}$.

###  Anomalous Dimensions of CFT Operators

Here we extract information from the one-loop effective action providing the
leading non-planar corrections to the conformal dimensions of the CFT
operators ${\cal O}_{i}$ associated with the light fields $\varphi_{i}$.
Depending on the spacetime dimension, these corrections are either finite and
evaluated at large $m_{\Phi}$ or contain a log divergence. Below we only focus
on odd dimensions hence the corrections are finite.

The heavy field $\Phi$ is in correspondence with a CFT operator with large
conformal dimension $\Delta_{h}\gg\Delta_{i}$, with

$m^{2}_{\Phi}=\Delta_{h}(\Delta_{h}-d)k^{2}\,.$ (141)

Depending on cases our results are expressed using either $m_{\Phi}$ or
$\Delta_{h}$.

####  Single-trace Operator

When the heat kernel coefficients contribute to the bulk mass of a field, this
translates as contribution to the anomalous dimension of the corresponding CFT
operator. We define the anomalous dimension

$\Delta_{\pm}=\Delta^{0}_{\pm}\pm\gamma$ (142)

where $\Delta^{0}_{\pm}$ corresponds to the conformal dimension in the free
theory in AdS i.e. the CFT with $N\to\infty$.

##### Anomalous Dimensions from Bubbles

We consider the interaction Lagrangian ${\cal L}_{\rm
int}=-y\frac{1}{2}\Phi^{2}\varphi$, giving $X=y\varphi$. The anomalous
dimensions arising from bubble diagrams are read from the $\bar{b}_{4}$ and
$\bar{b}_{6}$ coefficients, Eqs. (133), (134). We obtain

$\displaystyle\gamma^{d=2}_{{\cal O},{\rm bubble}}$
$\displaystyle=-\frac{y^{2}}{16\pi(\Delta-1)}\left(\frac{1}{2\,(\Delta_{h}(\Delta_{h}-2))^{\frac{1}{2}}}+\frac{\Delta(\Delta-2)-6}{24\,(\Delta_{h}(\Delta_{h}-2))^{\frac{3}{2}}}\right)+O\left(\frac{1}{\Delta_{h}^{5}}\right)$
(143) $\displaystyle\gamma^{d=4}_{{\cal O},{\rm bubble}}$
$\displaystyle=\frac{y^{2}}{32\pi^{2}(\Delta-2)}\left(\frac{(\Delta_{h}(\Delta_{h}-4))^{\frac{1}{2}}}{2}-\frac{\Delta(\Delta-4)-20}{24\,(\Delta_{h}(\Delta_{h}-4))^{\frac{1}{2}}}\right)+O\left(\frac{1}{\Delta_{h}^{3}}\right)$
(144) $\displaystyle\gamma^{d=6}_{{\cal O},{\rm bubble}}$
$\displaystyle=-\frac{y^{2}}{192\pi^{3}(\Delta-3)}\left(\frac{(\Delta_{h}(\Delta_{h}-6))^{\frac{3}{2}}}{2}-\frac{\Delta(\Delta-6)-42}{8}(\Delta_{h}(\Delta_{h}-6))^{\frac{1}{2}}\right)+O\left(\frac{1}{\Delta_{h}}\right)$
(145)

In the CFT one has $y\sim\frac{1}{N}$ thus these are $\sim\frac{1}{N^{2}}$
corrections to the conformal dimension of $\cal O$.

##### Comparison with Exact AdS3 Bubble

The AdS3 bubble was found in Carmi:2018qzm  to be

${\cal
B}(i\nu,\Delta_{h})=iy^{2}\frac{\psi(\Delta_{h}-\frac{1+i\nu}{2})-\psi(\Delta_{h}-\frac{1-i\nu}{2})}{8\pi\nu}$
(146)

in the spectral representation. When evaluated at $\pm i\nu=\Delta-1$ it gives
the exact anomalous dimension of the single trace operator via

$\displaystyle\gamma^{d=2}_{{\cal O},{\rm bubble}}=\frac{{\cal
B}(\Delta-1,\Delta_{h})}{2\Delta-2}$ (147) $\displaystyle\overset{{\rm
large}~{}\Delta_{h}}{=}-\frac{y^{2}}{16\pi(\Delta-1)}\left(\frac{1}{2\Delta_{h}}+\frac{1}{2\Delta_{h}^{2}}+\frac{12+\Delta(\Delta-2)}{24\Delta_{h}^{3}}+\frac{4+\Delta(\Delta-2)}{8\Delta_{h}^{4}}\right)+O\left(\frac{1}{\Delta_{h}^{5}}\right)$

We can then verify that our result Eq. (143) reproduces exactly Eq. (147) upon
expanding in $\Delta_{h}$ up to $O(\frac{1}{\Delta_{h}^{5}})$. This matching
at fourth order provides a nontrivial sanity check of our results.

In our result Eq. (143), the next-to-leading order term originates from the
derivative term $X\square X$ in the heat kernel coefficient (Eq. (109)). This
$\square$ was then simplified using the bulk EOM (see Eq. (134)). This
illustrates that having background fields on-shell in the bulk is necessary in
order to obtain CFT data from the one-loop boundary effective action.

##### Anomalous Dimensions from Tadpoles

We consider the interaction Lagrangian ${\cal L}_{\rm
int}=-\frac{1}{4}\lambda\Phi^{2}\varphi^{2}$, giving
$X=\frac{1}{2}\lambda\varphi^{2}$. From the $\bar{b}_{2,4}$ coefficients Eqs.
(132), (133) we obtain the anomalous dimensions

$\gamma^{d=2}_{{\cal O},{\rm
tad}}=\frac{\lambda}{16\pi(\Delta-1)}\left((\Delta_{h}(\Delta_{h}-2))^{\frac{1}{2}}+\frac{\Delta(\Delta-2)-6}{12(\Delta_{h}(\Delta_{h}-2))^{\frac{1}{2}}}\right)+O\left(\frac{1}{\Delta_{h}^{3}}\right)$
(148) $\gamma^{d=4}_{{\cal O},{\rm
tad}}=-\frac{\lambda}{96\pi^{2}(\Delta-2)}\left((\Delta_{h}(\Delta_{h}-4))^{\frac{3}{2}}-\frac{\Delta(\Delta-4)-20}{4}(\Delta_{h}(\Delta_{h}-4))^{\frac{1}{2}}\right)+O\left(\frac{1}{\Delta_{h}}\right)$
(149) $\gamma^{d=6}_{{\cal O},{\rm
tad}}=\frac{\lambda}{960\pi^{3}(\Delta-3)}\left((\Delta_{h}(\Delta_{h}-6))^{\frac{5}{2}}+\frac{\Delta(\Delta-6)-42}{16}(\Delta_{h}(\Delta_{h}-6))^{\frac{3}{2}}\right)+O\left(\Delta_{h}\right)$
(150)

The next-to-leading term can similarly be obtained from the $\bar{b}_{6}$
coefficient Eq. (134) and is not shown here for convenience.

####  Double-trace Operators

The $-\frac{1}{4}\lambda\Phi^{2}\varphi^{2}$ coupling also generates at one-
loop a quartic coupling $\frac{1}{4!}\eta\varphi^{4}$ with
$\eta\propto\lambda^{2}$. This is encoded in the $\bar{b}_{4}$, $\bar{b}_{6}$
coefficients. This quartic coupling induces in turn an anomalous correction to
the double trace operators $[{\cal O}{\cal O}]_{n}\equiv{\cal
O}\square^{n}{\cal O}$. Using the perturbation theory approach established in
Fitzpatrick:2010zm  we obtain

$\gamma_{n}=c_{n}\eta_{\rm 1-loop}$ (151)

with

$c^{d=2}_{n}=\frac{1}{8\pi(2\Delta+2n-1)}$ (152)
$c^{d=4}_{n}=\frac{(n+1)(\Delta+n-1)(2\Delta+n-3)}{8\pi^{2}(4\Delta+4n-6)}$
(153)
$c^{d=6}_{n}=\frac{(n+1)(n+2)(\Delta+n-2)(\Delta+n-1)^{2}(2\Delta+n-5)(2\Delta+n-4)}{64\pi^{3}(2\Delta+2n-5)(2\Delta+2n-3)}$
(154)

computed from Fitzpatrick:2010zm  and the loop-generated couplings

$\eta^{d=2}_{\rm
1-loop}=-\frac{3\lambda^{2}}{16\pi(\Delta_{h}(\Delta_{h}-2))^{\frac{1}{2}}}$
(155) $\eta^{d=4}_{\rm
1-loop}=\frac{3\lambda^{2}(\Delta_{h}(\Delta_{h}-2))^{\frac{1}{2}}}{32\pi^{2}}$
(156) $\eta^{d=6}_{\rm
1-loop}=-\frac{\lambda^{2}(\Delta_{h}(\Delta_{h}-2))^{\frac{3}{2}}}{64\pi^{3}}$
(157)

obtained from our effective action. In the CFT one has
$\lambda\sim\frac{1}{N^{2}}$ thus these are $\sim\frac{1}{N^{4}}$ corrections
to the conformal dimension of the double trace operators.

###  Corrections to Boundary-Localized Operators

The boundary heat kernel coefficients lead to boundary operators which take in
general the form of local operators with possibly CFT 2pt correlators appended
to them. The 2pt correlators are the consequence of normal derivatives acting
on $X$ on the boundary.

In this subsection we focus on the interaction Lagrangian ${\cal L}_{\rm
int}=-\frac{1}{4}\lambda\Phi^{2}\varphi^{2}$ giving
$X=\frac{1}{2}\lambda\varphi^{2}$. For example for AdS5 the boundary
coefficients are

$\displaystyle{\rm tr}\,b^{\partial}_{3}$
$\displaystyle=-\frac{1}{8}\lambda\varphi^{2}_{0}$ (158) $\displaystyle{\rm
tr}\,b^{\partial}_{4}$
$\displaystyle=-\frac{2\lambda}{3}\frac{1}{\sqrt{\bar{g}}}(\langle{\cal
O}{\cal
O}\rangle\circ\varphi_{0})\,\varphi_{0}+\frac{\lambda}{3}(2\Delta-10)k\varphi^{2}_{0}\,$
(159) $\displaystyle{\rm tr}\,b^{\partial}_{5}=\frac{1}{5760}\bigg{(}$
$\displaystyle
360\lambda\,k^{2}z_{0}^{2}\,\varphi_{0}\partial_{\mu}\partial^{\mu}\varphi_{0}+60\left(12\Delta^{2}-78\Delta+133\right)\lambda\,k^{2}\varphi^{2}_{0}+180\lambda^{2}\varphi_{0}^{4}$
$\displaystyle+360(4\Delta-15)\lambda\,k\frac{1}{\sqrt{\bar{g}}}(\langle{\cal
O}{\cal
O}\rangle\circ\varphi_{0})\,\varphi_{0}-360\lambda\,\frac{1}{\sqrt{\bar{g}}}\frac{1}{\sqrt{\bar{g}}}\left(\langle{\cal
O}{\cal O}\rangle\circ\varphi_{0}\right)^{2}\bigg{)}$ (160)

and enter in the holographic effective action as

$\Gamma^{\partial,{\rm fin}}_{\rm 1-loop}=\frac{1}{2}\int_{\partial\rm
AdS}d^{4}x\sqrt{\bar{g}}\left(-\frac{m_{\Phi}}{16\pi^{2}}{\rm
tr}\,b^{\partial}_{4}(x)\right)$ (161) $\Gamma^{\partial,{\rm div}}_{\rm
1-loop}=\frac{1}{2}\int_{\partial\rm
AdS}d^{4}x\sqrt{\bar{g}}\left(\frac{m^{2}_{\Phi}}{8\pi^{2}}{\rm
tr}\,b^{\partial}_{3}(x)-\frac{1}{8\pi^{2}}{\rm
tr}\,b^{\partial}_{5}(x)\right)\log\frac{m_{\Phi}}{\mu_{r}}\,$ (162)

where $\mu_{r}$ is the renormalization scale.

####  Correction to OPE Coefficients

The effective action encoding the $b^{\partial}_{4,5}$ coefficients contains
terms of the form

$\int d^{d}y\langle{\cal O}_{i}{\cal O}_{i}\rangle\circ X_{0}=\int
d^{d}y_{1}d^{d}y_{2}\varphi(y_{1})\langle{\cal O}_{i}(y_{1}){\cal
O}_{i}(y_{2})\rangle\varphi(y_{2})\,.$ (163)

We can see that this term induces a one-loop correction to the normalization
of the CFT 2pt function. For $d=2$ and $d=4$ we find the corrections

$\left(1+\frac{\lambda}{12\pi^{2}m_{\Phi}}-\frac{8\Delta-31}{128\pi
m^{2}_{\Phi}}\lambda k\right)\langle{\cal O}(y_{1}){\cal O}(y_{2})\rangle$
(164) $\left(1-\frac{\lambda
m_{\Phi}}{24\pi^{2}}+\frac{4\Delta-15}{128\pi^{2}}\lambda
k\,\log\frac{m_{\Phi}}{\mu_{r}}\right)\langle{\cal O}(y_{1}){\cal
O}(y_{2})\rangle$ (165)

where the second term is from $b^{\partial}_{4}$ and the third from
$b^{\partial}_{5}$.

In the CFT these corrections amount, upon unit-normalization of the operators,
to a correction to the OPE coefficients involving $\cal O$. That is, the
$(1+\delta_{\cal O})$ factor turns into a correction to the OPE coefficients
by $c_{\cal O\ldots}=c^{0}_{{\cal O}\ldots}(1-\frac{1}{2}\delta_{\cal O})$
where $c^{0}_{\cal O\ldots}$ is the OPE coefficient value in the planar limit.
This type of correction amounts to an AdS/CFT version of “wavefunction
renormalization”.

####  Operator Mixing

The effective action containing $b^{\partial}_{5}$ encodes a term

$\int d^{d}y\left(\langle{\cal O}{\cal
O}\rangle\circ\varphi_{0}\right)^{2}=\int d^{d}y_{1}\int d^{d}y_{2}\int
d^{d}y_{3}\varphi(y_{1})\langle{\cal O}(y_{1}){\cal
O}(y_{2})\rangle\langle{\cal O}(y_{2}){\cal O}(y_{3})\rangle\varphi(y_{3})\,.$
(166)

Because of conformal symmetry, this term reduces to the 2pt function of
another CFT operator, ${\cal O}^{2}$, which has dimension $2\Delta$ at this
order in perturbation theory. Evaluating the $\int d^{d}y_{2}$ integral we
have

$\displaystyle\int d^{d}y_{2}\langle{\cal O}(y_{1}){\cal
O}(y_{2})\rangle\langle{\cal O}(y_{2}){\cal O}(y_{3})\rangle$
$\displaystyle=-i\pi^{\frac{d}{2}}\frac{\Gamma^{2}(\frac{d}{2}-\Delta)\Gamma(2\Delta)}{\Gamma^{2}(\Delta)\Gamma(d-2\Delta)}\frac{{\cal
C}^{2}_{\Delta}}{{\cal
C}_{2\Delta}}\frac{\eta^{2}_{\Delta}}{\eta_{2\Delta}}\langle{{\cal
O}^{2}}(y_{1}){\cal O}^{2}(y_{3})\rangle\,$
$\displaystyle=-i\frac{(2\Delta-d)^{2}}{(4\Delta-d)}kz^{-d}_{0}\langle{{\cal
O}^{2}}(y_{1}){\cal O}^{2}(y_{3})\rangle\,.$ (167)

The $i$ factor comes from the Lorentzian metric. This explicitly shows how a
double trace operator emerges on the AdS boundary as a result of loop
corrections.

The combination arising when taking twice the derivative of the boundary
action in $\varphi_{0}$ ends up being

$\langle{{\cal O}}(y_{1}){\cal
O}(y_{2})-i\frac{(2\Delta-d)^{2}}{(4\Delta-d)}\frac{\lambda}{64\pi
m^{2}_{\Phi}}kz^{-d}_{0}\langle{{\cal O}^{2}}(y_{1}){\cal
O}^{2}(y_{2})\rangle\,.$ (168)

In the CFT language, this correction corresponds to a mixing between ${\cal
O}$ and ${\cal O}^{2}$ induced by non-planar corrections.

####  Corrections to local operators/multitrace deformations

Local boundary-localized operators are also generated at one-loop. Defining
the tree-level boundary couplings

${\cal
L}_{0}^{\partial}=-\frac{1}{2}\varphi_{0}\left(Z_{b,0}z^{2}_{0}\partial_{\mu}\partial^{\mu}+m^{2}_{b,0}\right)\varphi_{0}-\frac{1}{4!}\lambda_{b,0}\varphi^{4}_{0}$
(169)

we obtain the following one-loop corrections from the $b^{\partial}_{3,4,5}$,
heat kernel coefficients,

$Z_{b,d=2}=Z^{0}_{b}-\frac{\lambda k^{2}}{64\pi
m^{2}_{\Phi}}+O(m^{-3}_{\Phi})$ (170)
$m^{2}_{b,d=2}=m^{2}_{b,0}+\frac{\lambda}{16\pi}\log\frac{\mu_{r}}{m_{\Phi}}-\frac{\lambda(2\Delta-5)}{24\pi
m_{\Phi}}+O(m^{-2}_{\Phi})$ (171)
$(\lambda)_{b,d=2}=\lambda_{b,0}-\frac{3\lambda^{2}}{32\pi
m^{2}_{\Phi}}+O(m^{-3}_{\Phi})$ (172)

$Z_{b,d=4}=Z^{0}_{b}-\frac{\lambda
k^{2}}{128\pi^{2}}\log\frac{\mu_{r}}{m_{\Phi}}+O(m^{-1}_{\Phi})$ (173)
$m^{2}_{b,d=4}=m^{2}_{b,0}-\frac{\lambda
m_{\Phi}^{2}}{64\pi^{2}}\log\frac{\mu_{r}}{m_{\Phi}}+\frac{\lambda(\Delta-5)m_{\Phi}}{24\pi^{2}}+O(m^{0}_{\Phi})$
(174)
$(\lambda)_{b,d=4}=\lambda_{b,0}-\frac{3\lambda^{2}}{64\pi^{2}}\log\frac{\mu_{r}}{m_{\Phi}}+O(m^{-1}_{\Phi})$
(175)

The radiative generation of such boundary-localized local operators is
sometimes discussed in the extradimension literature at a qualitative level.
Here we have obtained their exact coefficients.

From the viewpoint of AdS/CFT, these local boundary-localized operators may,
under certain conditions on the value of $\Delta$, correspond to multitrace
deformations in the dual CFT. For example, when $\Delta<\frac{d}{2}+1$ one
identifies $\varphi_{0}$ as the CFT operator itself, and $m^{2}_{b}$ then
corresponds to a double trace deformation $\int d^{d}y{\cal O}^{2}(y)$ whose
effects have been well studied (see e.g. Gubser:2002zh ; Hartman:2006dy ;
Giombi:2018vtc ). Here we have found that such operators are generically
generated via loops in AdS.

##  Application 2: Vector Loops in AdS

In this section we evaluate pieces of the one-loop effective action generated
by integrating out the spin-1 fluctuation from a nonabelian gauge field. We
extract the beta functions of the boundary nonabelian field and anomalous
dimensions of scalar CFT operators.

###  Yang-Mills Beta Functions in AdS

Our focus is on the gauge background field $F_{\mu\nu}|_{\rm bg}$. We use the
set of BCs Eq. (121) for the fluctuation. The BC of the $A_{M}$ fluctuation
enforces the $\Pi_{\pm}$ projectors to be
$\Pi_{+}=\delta_{MN}-\delta_{M\perp}\delta_{N\perp}$,
$\Pi_{-}=\delta_{M\perp}\delta_{N\perp}$.

We assume the gauge background is on-shell thus $D_{M}F^{MN}=0$. Fixing the
gauge of the background to $D_{M}A^{M}=0$, the EOM reduces to $\square
A_{M}=0$. The homogeneous solutions are
$z^{\frac{d}{2}-1}I_{\frac{d}{2}-1}(pz)$,
$z^{\frac{d}{2}-1}K_{\frac{d}{2}-1}(pz)$.

The gauge background contributes to the bulk coefficients $b_{4}$, $b_{6}$ and
to the boundary coefficient $b^{\partial}_{5}$. To evaluate the $F_{MN}\square
F^{MN}$ terms we use Jacobi’s identity and the commutator
$[D_{M},D_{N}]F^{a}_{PQ}=[F_{MN},F_{PQ}]^{a}$. The result is

${\rm tr}\left[F_{MN}\square F^{MN}\right]={\rm
tr}\left[4F_{MN}F^{NP}F_{P}^{~{}~{}M}-2(D_{M}F^{MN})^{2}\right]\,.$ (176)

In the $b^{\partial}_{5}$ coefficient we also notice that the
$F_{\mu\perp}F^{\mu\perp}$ terms vanish by EOM and gauge fixing.

Taking into account the ghost contributions we find that the total heat kernel
coefficients $b_{\rm tot}=b_{A}-2b_{\rm gh}$ are

$\displaystyle{\rm tr}\,b_{4,{\rm tot}}=$
$\displaystyle-\frac{25-d}{12}C_{2}(G)F^{a}_{MN}F^{a,MN}|_{\rm bg}$ (177)
$\displaystyle{\rm tr}\,b^{\partial}_{5,{\rm tot}}=$
$\displaystyle\frac{27-d}{48}C_{2}(G)F^{a}_{\mu\nu}F^{a,\mu\nu}|_{\rm bg}$
(178) $\displaystyle{\rm tr}\,b_{6,{\rm tot}}=$
$\displaystyle\frac{5d^{3}-154d^{2}+557d+12}{360}C_{2}(G)k^{2}F^{a}_{MN}F^{a,MN}|_{\rm
bg}$ $\displaystyle+\frac{d-1}{90}{\rm tr}F_{MN}F^{NP}F_{P}^{~{}~{}M}|_{\rm
bg}$ (179)

where $C_{2}(G)$ is the quadratic Casimir of the gauge representation,
$f^{acb}f^{adb}=C_{2}(G)\delta^{cd}$.

####  $\beta$ functions

Both bulk and boundary gauge couplings can be renormalized by the logarithmic
divergences appearing in the one-loop effective action. Which term of
$\Gamma_{\rm 1-loop}$ diverges depends on the dimension of spacetime. Unlike
flat space where a running gauge coupling can only happen in $d=4$, we will
see that in AdS space this can also occur in higher dimensions as a
consequence of the curvature. The $\beta$ function of the gauge couplings can
be evaluated by putting together the one-loop effective action Eq. (105) and
the classical part. The classical part is

$S=-\frac{1}{4g^{2}}\int
d^{d+1}x\sqrt{|g|}(F^{a}_{MN})^{2}-\frac{1}{4g_{b}^{2}}\int
d^{d}y\sqrt{|\bar{g}|}(F^{a}_{\mu\nu})^{2}\Big{|}_{z=z_{0}}\,$ (180)

where a boundary-localized kinetic term has been introduced. Bulk and boundary
divergences respectively renormalize the $g$ and $g_{b}$ parameters.

For AdS4 ($d=3$), the log divergence comes from the ${\rm tr}\,b_{4,{\rm
tot}}$ coefficient. The overall factor is $-\frac{11}{6}$. This reproduces the
well-known 4d YM $\beta$ function for the bulk coupling,
$\beta_{\frac{1}{g^{2}}}=\frac{11}{24\pi^{2}}C_{2}(G)$.

For AdS5 ($d=4$), the log divergence comes from the ${\rm
tr}\,b^{\partial}_{5,{\rm tot}}$ coefficient. The $\beta$ function for the
boundary gauge coupling is found to be

$\beta_{\frac{1}{g_{b}^{2}}}\Big{|}_{{\rm
AdS}_{5}}=-\frac{23}{192\pi^{2}}C_{2}(G)\,.$ (181)

For AdS6 ($d=5$), the log divergence comes from the ${\rm tr}\,b_{6,{\rm
tot}}$ coefficient. The gauge coupling has canonical dimension $[g]=-1$. The
coefficient in Eq. (179) reduces to $-\frac{107}{90}k^{2}$. The $\beta$
function for the bulk coupling is found to be

$\beta_{\frac{1}{g^{2}}}\Big{|}_{{\rm
AdS}_{6}}=\frac{107}{1440\pi^{3}}k^{2}C_{2}(G)\,.$ (182)

Here we can see that the logarithmic running of this dimensionful gauge
coupling is a consequence of the AdS curvature. This can be alternatively
understood as the renormalization of the $RFF$ operators displayed in section
8.1.3.

####  Boundary Yang-Mills Effective Action

Using the above results, let us write the bilinear boundary action at one-
loop. We work in momentum space. Irrespective of the gauge, the $A_{z}$
background can be eliminated (i.e. integrated out at classical level) using
the $D_{\mu}F^{\mu z}=0$ component of the EOM, giving the identity

$F_{MN}F^{MN}=F_{\mu\nu}F^{\mu\nu}+4\left(\eta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)(\partial_{z}A^{\mu})(\partial_{z}A^{\nu})$
(183)

The second term contributes to the boundary action, generating the 2pt
function of a conserved CFT current (see e.g. Witten:1998qj ). Integrating by
part and using the bulk EOM with $D_{M}A^{M}=0$ gauge, the boundary action
then reads

$S=-\int\frac{d^{d}p}{(2\pi)^{d}}\sqrt{\bar{g}}\left(\frac{p^{2}}{2g^{2}_{b}}+\frac{1}{g^{2}}\partial_{\perp}K\right)A^{\mu}_{0}\Pi_{\mu\nu}A^{\nu}_{0}\bigg{|}_{z=z_{0}}\,$
(184)

with $\Pi_{\mu\nu}=\left(\eta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)$.

Using the same calculations as in section 3.2.2, the exact boundary-to bulk
propagatar for $A_{\mu}$ is

$K=\frac{1}{\sqrt{\bar{g}}}\int\frac{d^{d}p}{(2\pi)^{d}}\frac{z^{\frac{d}{2}-1}K_{\frac{d}{2}-1}(pz)}{z_{0}^{\frac{d}{2}-1}K_{\frac{d}{2}-1}(pz_{0})}\,.$
(185)

We then take the $pz_{0}\ll 1$ limit. The $\partial_{\perp}K$ derivative goes
as $\propto\log(p)^{-1}$, $p$, $p^{2}\log p$ and $\propto p^{2}$ for $d=2,3,4$
and $d\geq 5$ respectively. We also integrate the $\beta$ functions and
identify the renormalization scale with $p$. The reference scale is denoted
$p_{0}$.

The resulting boundary one-loop effective actions for $d=3,4,5$ are

$\displaystyle\Gamma[A_{0}]\Big{|}_{AdS_{4}}=-\int\frac{d^{3}p}{(2\pi)^{3}}\sqrt{\bar{g}}\left(\frac{p^{2}}{2g^{2}_{b}}+\left(\frac{1}{g_{0}^{2}}+\frac{11C_{2}(G)}{24\pi^{2}}\log\frac{p}{p_{0}}\right)kpz_{0}\right)A^{\mu}_{0}\Pi_{\mu\nu}A^{\nu}_{0}$
(186)

$\displaystyle\Gamma[A_{0}]\Big{|}_{AdS_{5}}=-\int\frac{d^{4}p}{(2\pi)^{4}}\sqrt{\bar{g}}\left(\frac{1}{2g^{2}_{b,0}}-\left(\frac{kz_{0}^{2}}{g^{2}}+\frac{23C_{2}(G)}{384\pi^{2}}\right)\log\frac{p}{p_{0}}\right)p^{2}A^{\mu}_{0}\Pi_{\mu\nu}A^{\nu}_{0}$
(187)

$\displaystyle\Gamma[A_{0}]\Big{|}_{AdS_{6}}=-\int\frac{d^{5}p}{(2\pi)^{5}}\sqrt{\bar{g}}\left(\frac{1}{2g^{2}_{b}}+\left(\frac{1}{g_{0}^{2}}+\frac{107C_{2}(G)k^{2}}{1440\pi^{3}}\log\frac{p}{p_{0}}\right)kz^{2}_{0}\right)p^{2}A^{\mu}_{0}\Pi_{\mu\nu}A^{\nu}_{0}$
(188)

In $\Gamma[A_{0}]\big{|}_{AdS_{5}}$ we have absorbed a constant
$\frac{2k^{2}z_{0}^{2}}{g^{2}}(\gamma+\log(\frac{p_{0}z_{0}}{2}))$ into the
reference value of $\frac{1}{g^{2}_{b,0}}$. We see that for AdS4,6 the
holographic coupling grows in the IR as a consequence of bulk one-loop
divergences. For AdS5, both the contribution from the classical action and the
boundary one-loop divergence contribute with negative sign. Thus the
holographic coupling grows in the UV in AdS5.

####  Aside: Arbitrary Background in 6d

We can also compute the renormalization of gauge operators for general
background, in which case the renormalized operators take the form “$RFF$” in
6d. There are three irreducible invariants. Here we give the results in terms
of the divergent piece of the one-loop effective action in general 6d
background:

$\displaystyle\Gamma^{\rm YM}_{\rm 1-loop,div}=$
$\displaystyle\quad\quad\frac{C_{2}(G)}{128\pi^{3}}\Gamma\left(3-\frac{d+1}{2}\right)\times$
(189) $\displaystyle\int_{\cal
M}d^{6}x\left[\frac{1}{15}R^{MNPQ}F^{a}_{MN}F^{a}_{PQ}-\frac{88}{45}R^{NM}F_{NP}F_{M}^{a,P}+\frac{13}{36}RF^{a}_{MN}F^{a,MN}\right]$

where the divergence is encoded into the Gamma function with $d\to 5$.
Substituing the AdS6 Riemann tensor reproduces the coefficient in Eq.(179).

###  Anomalous Dimensions from Massive Vector Loops

Finally we present results involving massive nonabelian vector fluctuations.
The mass matrix of the gauge field is chosen to be a universal mass
$m^{2}_{A}\delta^{ab}$. Results with a more complicated mass matrix can
similarly be obtained. In the holographic CFT the massive vector corresponds
to a non-conserved current of the global group $G$ with dimension
$\Delta_{J}\gg d-1$, with $m^{2}_{A}=(\Delta_{J}-1)(\Delta_{J}-d+1)k^{2}$.

We assume the presence of a scalar field $\varphi_{i}$ transforming as a
representation $r$ of the gauge group. In analogy with computations in section
7, we evaluate the effect of the heavy vector on the light scalar sector. We
focus on odd spacetime dimensions.

The action contains the scalar kinetic term

$\frac{1}{2}\int d^{d+1}x(D_{M}\varphi)_{i}(D^{M}\varphi)_{i}$ (190)

in addition to the YM action. The light scalar is treated as background, it
therefore contributes to vector mass via the invariant
$X^{ab}=-g^{2}(t^{a}_{r}\varphi)_{i}(t^{b}_{r}\varphi)_{i}$.

From the bulk contributions to the one-loop effective action we obtain the
anomalous dimension of the operator ${\cal O}_{i}$ associated to
$\varphi_{i}$,

$\gamma^{d=2}_{{\cal O},{\rm
tad}}=\frac{g^{2}C_{2}(r)}{8\pi(\Delta-1)}\left(\Delta_{J}-1+\frac{\Delta(\Delta-2)-6}{12(\Delta_{J}-1)}\right)+O\left(\frac{1}{\Delta_{J}^{3}}\right)$
(191) $\gamma^{d=4}_{{\cal O},{\rm
tad}}=-\frac{g^{2}C_{2}(r)}{48\pi^{2}(\Delta-2)}\left(((\Delta_{J}-1)(\Delta_{J}-3))^{\frac{3}{2}}-\frac{\Delta(\Delta-4)-20}{4}((\Delta_{J}-1)(\Delta_{J}-3))^{\frac{1}{2}}\right)+O\left(\frac{1}{\Delta_{J}}\right)$
(192) $\gamma^{d=6}_{{\cal O},{\rm
tad}}=\frac{g^{2}C_{2}(r)}{480\pi^{3}(\Delta-3)}\left(((\Delta_{J}-1)(\Delta_{J}-5))^{\frac{5}{2}}+\frac{\Delta(\Delta-6)-42}{16}((\Delta_{J}-1)(\Delta_{J}-5))^{\frac{3}{2}}\right)+O\left(\Delta_{J}\right)$
(193)

Here $C_{2}(r)$ is the quadratic Casimir for the representation $r$. The next
to leading order is also known from our results and is not shown here merely
for convenience.

We can also compute the one-loop anomalous dimension of double trace
operators. We find that the effective quartic coupling
$\frac{\eta}{4}(\varphi_{i})^{2}(\varphi_{j})^{2}$ is generated at one-loop
with

$\eta^{d=2}_{\rm 1-loop}=-\frac{C^{2}_{2}(r)g^{4}}{8\pi(\Delta_{J}-1)}$ (194)
$\eta^{d=4}_{\rm
1-loop}=\frac{C^{2}_{2}(r)g^{4}((\Delta_{J}-1)(\Delta_{J}-3))^{\frac{1}{2}}}{16\pi^{2}}$
(195) $\eta^{d=6}_{\rm
1-loop}=-\frac{C^{2}_{2}(r)g^{4}((\Delta_{J}-1)(\Delta_{J}-5))^{\frac{3}{2}}}{96\pi^{3}}$
(196)

The anomalous dimensions of the double trace operators $[{\cal O\cal
O}]_{i,n}$ associated to the diagonal term $(\varphi_{i})^{4}$ is then given
by $\gamma_{n}=c_{n}\eta_{\rm 1-loop}$ where the coefficients $c_{n}$ are
given in section 7.2.2. Since the effective quartic coupling also contains
nondiagonal terms $(\varphi_{i})^{2}(\varphi_{j\neq i})^{2}$, the anomalous
dimension matrix of the $[{\cal O\cal O}]_{i,n}$ operators is nondiagonal.
There is therefore a mixing between the double trace operators induced by the
loop corrections.

##  A Boundary Effective Action in dS

We turn to an application in de Sitter space dSd+1. Our overall goal here is
to extract one-loop contributions to late time cosmological correlators from
the heat kernel coefficients. In this context the heat kernel coefficients
should arise from a suitably defined “cosmological” boundary effective action.
To compute it we will use the analytical continuation between dS and Euclidian
AdS, which relates dS calculations to the AdS calculations made in the
previous sections.

We consider the flat slicing

$ds_{\rm dS}^{2}=g^{\rm
dS}_{MN}x^{M}x^{N}=\frac{\ell^{2}}{\eta^{2}}\left(-d\eta^{2}+(\vec{dy})^{2}\right)$
(197)

with $\eta\in[-\infty,\eta_{0}]$, $\eta_{0}<0$. $\eta$ is the conformal time,
related to the proper time by $d\eta=e^{-\frac{t}{\ell}}dt$. The $\eta_{0}=0$
slice corresponds to future boundary. This coordinate patch describes the
expanding de Sitter universe with Hubble radius $\ell$. It covers half of the
global dS space, the other half amounts to positive $\eta$. We are interested
in correlators with endpoints on the $\eta=\eta_{0}$ time slice.

In this section we deal with various kinds of two-point function (time-
ordered, anti-time-ordered, Wightman), hence we explicitly write the time
(anti-)ordering operators $T$($\bar{T}$).

###  Cosmological Correlators and In-in Formalism

The cosmological correlators are conveniently computed in the in-in (or
Keldish) formalism Maldacena:2002vr ; Weinberg:2005vy ; Seery:2007we ;
Adshead:2009cb ; Senatore:2009cf ; Adshead:2009cb ; Senatore:2016aui . A in-in
correlator is evaluated by performing a time-ordered integral from the initial
time to the time of interest $\eta_{0}$, and then performing an anti-time-
ordered integral back to the initial time. This can be implemented by sligthly
shifting the time integral domain away from the real axis, such that the
$-\eta$ variable takes values in either $\mathbb{R}_{+}(1+i\epsilon)$ or
$\mathbb{R}_{+}(1-i\epsilon)$. In the interaction picture, for a given
interaction Hamiltonian $H_{I}$, one has

$\langle\Phi(\eta_{0},\vec{y}_{1})\ldots\Phi(\eta_{0},\vec{y}_{n})\rangle=\frac{\langle
0|\bar{T}\left(e^{i\int^{\eta_{0}}_{-\infty_{+}}d\eta
H_{I}}\right)\Phi(\eta_{0},\vec{y}_{1})\ldots\Phi(\eta_{0},\vec{y}_{n})T\left(e^{-i\int^{\eta_{0}}_{-\infty_{-}}d\eta
H_{I}}\right)|0\rangle}{\langle
0|\bar{T}\left(e^{i\int^{\eta_{0}}_{-\infty_{+}}d\eta
H_{I}}\right)T\left(e^{-i\int^{\eta_{0}}_{-\infty_{-}}d\eta
H_{I}}\right)|0\rangle}$ (198)

with $\infty_{\pm}=\infty(1\pm i\epsilon)$. Here $\langle 0|$ is the free
Bunch-Davis vacuum, for which Minkowski spacetime is recovered at early times
($t,t^{\prime}\ll\ell$) or short distances
$|t-t^{\prime}|^{2}+|x-x^{\prime}|^{2}\ll\ell^{2}\eta^{2}$.

The in-in correlators are obtained perturbatively by expanding the left and
right exponentials in Eq. (198) and performing Wick contractions (see e.g.
Weinberg:2005vy ). Contractions between two right vertices or a right vertex
and an external field are done via a time-ordered propagator $\langle
T\Phi(x)\Phi(x^{\prime})\rangle=G_{--}(x,x^{\prime})$. Contractions between
two left vertices are done via an anti-time-ordered propagator
$\langle\bar{T}\Phi(x)\Phi(x^{\prime})\rangle=G_{++}(x,x^{\prime})$.
Contractions between a left vertex and a right vertex or an external field are
done via Wightman functions
$\langle\Phi(x)\Phi(x^{\prime})\rangle=G_{+-}(x,x^{\prime})$,
$\langle\Phi(x^{\prime})\Phi(x)\rangle=G_{-+}(x,x^{\prime})=G^{*}_{+-}(x,x^{\prime})$.
252525We use the convention $\langle\Phi(x)\Phi(x^{\prime})\rangle=\langle
0|\Phi(t+i\epsilon,y)\Phi(t^{\prime}-i\epsilon,y)|0\rangle_{\epsilon\to 0}$.
See e.g. Streater:1989vi .

###  dS Propagators

We derive the dS propagators in Fourier space. We will next see that Fourier
space renders the analytical continuation from dS to EAdS straightforward.
Fourier space is also usually the preferred space to study cosmological
correlators.

Particles in dS space are classified following the unitary irreps of
$SO(1,d+1)$ Schwarz71 ; Harmonic_analysis . Their masses are conveniently
parametrized as

$\ell^{2}m^{2}=\Delta(d-\Delta)=\frac{d^{2}}{4}+\nu^{2}$ (199)

In the nontachyonic representation one distinguishes the series of “heavy”
states with mass $m>\frac{d}{2}\ell^{-1}$ labelled by
$\Delta_{i\nu}=\frac{d}{2}+i\nu$ with $\nu\in\mathbb{R}$, and the series of
“light” states with mass $0<m<\frac{d}{2}\ell^{-1}$ labelled by
$\Delta_{-\mu}=\frac{d}{2}-\mu$ with $\mu\in[0,\frac{d}{2}]$. More generally
the $\Delta_{i\nu}$ function is analytically continued into the whole complex
plane.

The (anti-)time-ordered propagators for a scalar particle satisfy the EOM

${\cal D}_{\rm dS}G_{--(++)}(x,x^{\prime})=\mp i\frac{1}{\sqrt{|g|_{\rm
dS}}}\delta^{d+1}(x-x^{\prime})$ (200)

with ${\cal D}_{\rm dS}=-\square_{\rm dS}+m^{2}$ the scalar Laplacian.

Since the EOM applies independently on each endpoint, the propagators must
have a factorized structure. Explicitly it is

$\displaystyle G_{--}(p;\eta,\eta^{\prime})$
$\displaystyle=\frac{i}{C}F_{<}(\eta_{<})F_{>}(\eta_{>})\,,\quad
G_{++}(p;\eta,\eta^{\prime})=-\frac{i}{\tilde{C}}\tilde{F}_{<}(\eta_{<})\tilde{F}_{>}(\eta_{>})$
$\displaystyle G_{+-}(p;\eta,\eta^{\prime})$
$\displaystyle=\frac{i}{C}F_{<}(\eta)F_{>}(\eta^{\prime})=-\frac{i}{\tilde{C}}\tilde{F}_{<}(\eta^{\prime})\tilde{F}_{>}(\eta)$
(201)

where we have defined $-\eta^{<}=\min(-\eta,-\eta^{\prime})$,
$-\eta^{>}=\max(-\eta,-\eta^{\prime})$. The $C$ constant is determined by the
Wronskian $W=F_{>}^{\prime}F_{<}-F_{<}^{\prime}F_{>}$ with
$C=W\frac{\ell^{d-1}}{|\eta|^{d-1}}$ as in Eq. (33), and similarly for
$\tilde{C}$. A basis of solutions to the homogeneous EOM in Fourier space is
$z^{\frac{d}{2}}H^{(1,2)}_{i\nu}(-p\eta)$.

The condition implementing Bunch-Davies vacuum in the (anti-)time-ordered
propagators $G_{--}$, $G_{++}$ is the following Gibbons:1977mu ; Burges:1984qm
; Mottola:1984ar ; Allen:1985ux ; deBoer:2004nd . As a result of the
symmetries of dS, apart from the singularity at coincident endpoints, there
can be another singularity when the endpoints are at antipodal positions,
$x=x^{\prime},\eta=-\eta^{\prime}$. To implement the Bunch-Davies vacuum we
require the propagator to be regular in this antipodal configuration (the
remaining singularity ensures flat space behaviour at short distance). The
antipodal configuration lies outside the expanding Poincaré patch—in fact the
two endpoints are separated by a cosmological horizon. The regularity
condition is implemented by extending the Poincaré patch into the $\eta>0$
half of dS. In Fourier space the antipodal configuration amounts to taking the
$(p\to\infty,\eta^{\prime}\to-\eta)$ limit. Therefore the Bunch-Davies vacuum
is implemented by

$\lim_{\begin{subarray}{c}p\to\infty,\,\eta^{\prime}\to-\eta\end{subarray}}G_{--,++}(\eta,\eta^{\prime};p)=0\,.$
(202)

Taking into account the $i\epsilon$ shifts, we find that the de Sitter
propagators in Fourier space are

$\displaystyle G^{\rm dS}_{--}(p;\eta,\eta^{\prime})$
$\displaystyle=-\frac{\pi}{4}\ell\left(\frac{\eta\eta^{\prime}}{\ell^{2}}\right)^{\frac{d}{2}}H^{(1)}_{i\nu}\left(-p\eta^{<}\right)H^{(2)}_{i\nu}\left(-p\eta^{>}\right)$
$\displaystyle G^{\rm dS}_{++}(p;\eta,\eta^{\prime})$
$\displaystyle=-\frac{\pi}{4}\ell\left(\frac{\eta\eta^{\prime}}{\ell^{2}}\right)^{\frac{d}{2}}H^{(2)}_{i\nu}\left(-p\eta^{<}\right)H^{(1)}_{i\nu}\left(-p\eta^{>}\right)$
$\displaystyle G_{+-}^{\rm dS}(p;\eta,\eta^{\prime})$
$\displaystyle=-\frac{\pi}{4}\ell\left(\frac{\eta\eta^{\prime}}{\ell^{2}}\right)^{\frac{d}{2}}H^{(1)}_{i\nu}\left(-p\eta\right)H^{(2)}_{i\nu}\left(-p\eta^{\prime}\right)$
(203)

One can explicitly verify that $G_{++}=(G_{--})^{*}$ and that the Wightman
function $G_{+-}$ is Hermitian whenever $\nu$ is either purely real or
imaginary, which corresponds to the two possible representations of states in
de Sitter space.

The amputated boundary-to-bulk propagators from the (anti-)time-ordered
propagators are given by

$\sqrt{\bar{g}}_{\rm dS}K_{-}^{\rm
dS}(p;\eta,\eta^{\prime})=\left(\frac{\eta}{\eta_{0}}\right)^{\frac{d}{2}}\frac{H^{(2)}_{i\nu}\left(-p\eta_{0}\right)}{H^{(2)}_{i\nu}\left(-p\eta_{0}\right)}\,$
(204) $\sqrt{\bar{g}}_{\rm dS}K_{+}^{\rm
dS}(p;\eta,\eta^{\prime})=\left(\frac{\eta}{\eta_{0}}\right)^{\frac{d}{2}}\frac{H^{(1)}_{i\nu}\left(-p\eta\right)}{H^{(1)}_{i\nu}\left(-p\eta_{0}\right)}\,.$
(205)

The amputated boundary-to-bulk Wightman function equals either of those
depending on which endpoint is put on the boundary.

###  From dS to EAdS

We use analytical continuation to express the dS correlators in Euclidian AdS
(EAdS). Similar analytical continuation has been discussed in e.g.
Balasubramanian:2002zh ; Maldacena:2002vr ; Harlow:2011ke ; Anninos:2014lwa
in the (A)dS/CFT context, and has more recently been used in Sleight:2020obc ;
Sleight:2019mgd ; Sleight:2019hfp  to study the dS correlators in the Mellin-
Barnes representation (see also DiPietro:2021sjt ; Sleight:2021plv ;
Meltzer:2021zin ; Premkumar:2021mlz ). An advantage of this approach is that
much is known about perturbative and spectral techniques in EAdS; all this
knowledge is readily transferred to dS via analytical continuation. In the
context of the present work, there are additional reasons to work in EAdS. The
usual heat kernel formalism needs the fluctuation to have a diagonal Feynman
propagator in order to be applicable. In the in-in formalism from the original
dS spacetime, the various 2pt functions are simultaneously involved hence we
do not know how to straightforwardly proceed. Upon rotating to EAdS one
obtains a Euclidian propagator matrix, which is trivially diagonalized. We can
thus evaluate the one-loop effective action directly in the Euclidian space.
These steps are realized further below.

The Euclidian AdS metric $g_{\rm EAdS}$ is given by

$ds_{\rm EAdS}^{2}=\frac{L^{2}}{z^{2}}\left(dz^{2}+(\vec{dy})^{2}\right)$
(206)

with $z\in[z_{0},\infty]$. The scalar propagator for a particle with mass
$m_{\rm EAdS}^{2}=(\alpha^{2}-\frac{d^{2}}{4})\frac{1}{L^{2}}$ is

$G_{\alpha}^{\rm
EAdS}(p;z,z^{\prime})=L\left(\frac{zz^{\prime}}{L^{2}}\right)^{\frac{d}{2}}I_{\alpha}(pz_{<})K_{\alpha}(pz_{>})\,.$
(207)

The boundary-to-bulk propagator is

$\sqrt{|\bar{g}|}_{\rm EAdS}K_{\alpha}^{\rm
EAdS}(p;z)=\left(\frac{z}{z_{0}}\right)^{\frac{d}{2}}\frac{K_{\alpha}(pz)}{K_{\alpha}(pz_{0})}$
(208)

The transformation from dS to EAdS is implemented by closing the following
contours in the $\eta$ plane

(209)

which corresponds to the Wick rotations

$(-\eta)_{\pm}=ze^{\pm i(\frac{\pi}{2}-\epsilon)}\,\,.$ (210)

Equivalent Wick rotations have been used in Sleight:2020obc ; Sleight:2019mgd
; Sleight:2019hfp ; DiPietro:2021sjt . One can readily see that this converts
the dS metric to the EAdS one upon identification of the dS and EAdS radii
$L^{2}=-\ell^{2}$. For our purposes, however, we will use only the $\ell$
radius. This implies that the EAdS metric resulting from the Wick rotation has
only-minus signature, which will imply the presence of $i^{d}$, $(-i)^{d}$
factors in the expressions.

We denote by an arrow the replacement Eq. (210) in a function,
$f(-\eta_{-},-\eta_{+})\to
f(ze^{-i(\frac{\pi}{2}-\epsilon)},ze^{i(\frac{\pi}{2}-\epsilon)})$. We plug
the Wick rotation into Eq. (203). The time coordinates of $G_{--}$, $G_{++}$,
$G_{+-}$ are rotated as $(\eta_{-},\eta_{-})$, $(\eta_{+},\eta_{+})$,
$(\eta_{+},\eta_{-})$ respectively. 262626These are in fact the only
possibilities for closing the contours with vanishing contribution from the
arcs at $|\eta|\to\infty$, as required to perform Wick rotation. For the
$G_{+-}$ function, both the $F_{<}$ and $F_{>}$ pieces are rotated to a Bessel
K function with positive argument. For $G_{--}$, $G_{++}$, the
$F_{>}(\eta_{>})$ piece turns into a Bessel K function $\propto K(pz_{>})$
with positive argument. In contrast, the Wick rotation in the
$F_{<}(\eta_{<})$ piece gives a Bessel K function with negative argument,
which requires careful treatment since $K$ has a branch cut along the negative
axis. To evaluate this piece, the following identity and its complex conjugate
are useful:

$H^{(1)}_{i\nu}(pz_{<}e^{-i(\frac{\pi}{2}-\epsilon)})=\frac{i}{\sin\pi
i\nu}\left(e^{-i\frac{3\pi}{2}\nu}I_{i\nu}(pz_{<})-e^{i\frac{\pi}{2}\nu}I_{-i\nu}(pz_{<})\right)\,.$
(211)

We find that the dS propagators take the form

$\displaystyle G^{\rm dS}_{--}(p;\eta,\eta^{\prime})$
$\displaystyle\to\frac{1}{2\sin(\pi
i\nu)}\left(e^{-i\pi\Delta_{i\nu}}G_{i\nu}^{\rm
EAdS}(p;z,z^{\prime})-e^{-i\pi\Delta_{-i\nu}}G_{-i\nu}^{\rm
EAdS}(p;z,z^{\prime})\right)$ (212) $\displaystyle G^{\rm
dS}_{++}(p;\eta,\eta^{\prime})$ $\displaystyle\to\frac{1}{2\sin(\pi
i\nu)}\left(e^{i\pi\Delta_{i\nu}}G_{i\nu}^{\rm
EAdS}(p;z,z^{\prime})-e^{i\pi\Delta_{-i\nu}}G_{-i\nu}^{\rm
EAdS}(p;z,z^{\prime})\right)$ (213) $\displaystyle G_{+-}^{\rm
dS}(p;\eta,\eta^{\prime})$ $\displaystyle\to\frac{1}{2\sin(\pi
i\nu)}\left(G_{i\nu}^{\rm EAdS}(p;z,z^{\prime})-G_{-i\nu}^{\rm
EAdS}(p;z,z^{\prime})\right)$ (214)

where the EAdS propagators are given by Eq. (207) with the identification
$L\equiv\ell$. We have introduced

$\Delta_{\pm i\nu}=\frac{d}{2}\pm i\nu\,.$ (215)

Our result matches exactly the form of the propagators obtained in
DiPietro:2021sjt  up to an overall sign.

For the Wick rotation of the boundary-to bulk propagators the general formula
is

$\sqrt{\bar{g}}_{\rm dS}K_{\pm}^{\rm dS}(p;\eta)\to
N_{\pm}(p)\sqrt{\bar{g}}_{\rm EAdS}K^{\rm EAdS}(p;z)$ (216)

with

$N_{\pm}(p)=\mp\frac{2i}{\pi}e^{\pm
i\frac{\pi}{2}\Delta_{-i\nu}}\frac{K_{i\nu}(-p\eta_{0})}{H^{(2)}_{i\nu}(-p\eta_{0})}$
(217)

The same is true for the boundary-to-bulk Wightman function. Let us consider
the case of a light mode, which is typically the relevant case for
cosmological correlators. We take the small $pz_{0}$ limit and assume real
$i\nu$ with $|i\nu|=\mu$. For any sign of $i\nu$ we find $N_{\pm}(p)\approx
e^{\pm i\frac{\pi}{2}\Delta_{-\mu}}$. Thus

$\sqrt{\bar{g}}_{\rm dS}K_{\pm}^{\rm dS}(p;\eta)\Big{|}_{{\rm light},pz_{0}\ll
1}\to e^{\pm i\frac{\pi}{2}\Delta_{-\mu}}\sqrt{\bar{g}}_{\rm EAdS}K_{\mu}^{\rm
EAdS}(p;z)$ (218)

We have used $K^{\rm EAdS}_{\mu}=K^{\rm EAdS}_{-\mu}$. We can see that the
dimension of the physical operator $\Delta_{-\mu}=\frac{d}{2}-\mu$ always
appear. The phases in Eq. (218) are consistent with the ones obtained in
DiPietro:2021sjt .

Finally, the integral measures are Wick rotated as

$\displaystyle
i\int^{\eta_{0}}_{-\infty_{+}}\frac{d\eta_{+}}{(-\eta_{+}\ell^{-1})^{d+1}}\to(-i)^{d-1}\int_{z_{0}}^{\infty}\frac{dz}{(z\ell^{-1})^{d+1}}$
(219) $\displaystyle-$ $\displaystyle
i\int^{\eta_{0}}_{-\infty_{-}}\frac{d\eta_{-}}{(-\eta_{-}\ell^{-1})^{d+1}}\to(i)^{d-1}\int_{z_{0}}^{\infty}\frac{dz}{(z\ell^{-1})^{d+1}}$
(220)

###  The Generator of Cosmological Correlators

We follow, revisit and expand a proposal from Ref. DiPietro:2021sjt . The
analytical continuation from dS to EAdS is used to define a functional $Z^{\rm
pert}_{\rm dS}[J]$, expressed in EAdS, that generates the cosmological
correlators at any order in perturbation theory.

We first realize that, upon the two types of Wick rotation in Eq. (209), the
dS field with values on $\mathbb{R}(1\pm i\epsilon)$ is analytically continued
as either a holomorphic or an antiholomorphic function of $\eta$, $\Phi^{\rm
dS}(-\eta_{\pm})$. This implies that there are two distinct fields in $Z^{\rm
pert}_{\rm dS}$, related to each other by complex conjugation. We define the
two fields $\Phi_{\pm}$ as a function of $z$, with
$\Phi_{\pm}(z)=\Phi_{\pm}(-\eta_{\pm}e^{\mp i\frac{\pi}{2}})$. They satisfy
$\Phi_{+}^{*}=\Phi_{-}$. In this subsection it is enough to focus on a single
pair of EAdS fields $\Phi_{\pm}$, i.e. a single dS field only. The $\Phi_{-}$
field comes from the time-ordered contour, the $\Phi_{+}$ field comes from the
anti-time-ordered contour. The $\Phi_{-}$ is used for external legs.

We introduce a boundary source $J$ coupled to the $\Phi_{-}$ fields. The
source is localized on the boundary of EAdS. The source can either be
understood as a function of the original dS coordinates or of the EAdS
coordinates.

Given these definitions and conventions, we introduce the generating
functional

$\displaystyle Z^{\rm pert}_{\rm dS}[J]=\int{\cal D}\Phi_{\pm}\exp\bigg{[}$
$\displaystyle-\frac{1}{2}(\Phi_{-},\Phi_{+})\star\hat{G}^{-1}_{\rm
EAdS}\star(\Phi_{-},\Phi_{+})^{t}-J\circ\Phi_{-,0}$
$\displaystyle\quad\quad\quad\quad\quad-\int_{\rm
EAdS}\sqrt{|g|}dzd^{d}x\left(i^{d+1}{\cal L}_{I}[\Phi_{-}]+(-i)^{d+1}{\cal
L}_{I}[\Phi_{+}]\right)\bigg{]}$ (221)

where $J$ is boundary localized. We remind the definition of the convolution
products, here in EAdS

$A\star B=\int_{{\rm EAdS}}\sqrt{|g|}d^{d+1}xA(x)B(x)\,,\quad A\circ
B=\int_{\partial{\rm EAdS}}\sqrt{|\bar{g}|}d^{d}yA(y)A(y)$ (222)

with the EAdS radius set to $\ell$. We have introduced the propagator matrix

$\hat{G}_{\rm EAdS}=\frac{1}{2\sin(\pi
i\nu)}\begin{pmatrix}e^{-i\pi\Delta_{i\nu}}G_{i\nu}^{\rm
EAdS}-e^{-i\pi\Delta_{-i\nu}}G_{-i\nu}^{\rm EAdS}&G_{i\nu}^{\rm
EAdS}-G_{-i\nu}^{\rm EAdS}\\\ G_{i\nu}^{\rm EAdS}-G_{-i\nu}^{\rm
EAdS}&e^{i\pi\Delta_{i\nu}}G_{i\nu}^{\rm
EAdS}-e^{i\pi\Delta_{-i\nu}}G_{-i\nu}^{\rm EAdS}\end{pmatrix}$ (223)

Using $\Phi_{+}^{*}=\Phi_{-}$ one can verify that the fundamental action (i.e.
the kinetic term and the interaction term) is real.

The connected perturbative cosmological correlators originally defined by Eq.
(198) are then given by derivatives of the generating functional

$W^{\rm pert}_{\rm dS}[J]=\log(Z^{\rm pert}_{\rm dS}[J])\,.$ (224)

Moreover, as discussed in section 5, taking the Legendre transform
$\Gamma[\Phi_{\pm,0}]=-W[J]+J\circ\Phi_{-,0}$ gives the effective action in
the boundary variables $\Phi_{\pm,0}$. This effective action
$\Gamma[\Phi_{\pm,0}]$ generates the 1P-irreducible boundary diagrams, where
irreducible means with respect to boundary-to-boundary lines. These diagrams
have boundary-to-bulk propagators as external legs — they are the dS version
of Witten diagrams.

The reason why the $\Gamma^{\rm pert}_{\rm dS}[\Phi_{\pm,0}]$ effective action
depends on both $\Phi_{-,0}$ and $\Phi_{+,0}$ is that the propagators
amputated by the Legendre transform are non-diagonal — both $\Phi_{+}$ and
$\Phi_{-}$ show up when using the chain rule on $W^{\rm pert}_{\rm dS}[J]$.
Thus the operation of amputation requires some care. In order to evaluate the
boundary action that generates the amputated cosmological correlators, we take
an alternative route by first performing a field redefinition.

#### Canonical Normalization

We can see that the propagator matrix $\hat{G}_{\rm EAdS}$ contains linear
combinations of $G_{i\nu}$ and $G_{-i\nu}$. Thus we introduce a pair of EAdS
fields $\Phi$, $\tilde{\Phi}$ satisfying $\langle\Phi\Phi\rangle\propto
G_{-i\nu}$, $\langle\tilde{\Phi}\tilde{\Phi}\rangle\propto G_{i\nu}$,
$\langle\Phi\tilde{\Phi}\rangle=0$. We introduce the complex coefficients

$c^{\pm}_{\pm i\nu}=\frac{e^{\pm i\frac{\pi}{2}\Delta_{\pm
i\nu}}}{\sqrt{2\sin\pi i\nu}}\,.$ (225)

We find that the transformation

$\begin{pmatrix}\Phi_{-}\\\
\Phi_{+}\end{pmatrix}=U\begin{pmatrix}\tilde{\Phi}\\\
\Phi\end{pmatrix}\,,\quad\quad U=\begin{pmatrix}c^{-}_{i\nu}&ic^{-}_{-i\nu}\\\
c^{+}_{i\nu}&ic^{+}_{-i\nu}\end{pmatrix}\,$ (226)

diagonalizes and canonically normalizes the matrix of propagators.

Plugging it into Eq. (221) gives canonically normalized kinetic terms. The
generating functional reads

$\displaystyle Z^{\rm pert}_{\rm dS}[J]=$ $\displaystyle\int{\cal D}\Phi{\cal
D}\tilde{\Phi}\exp\bigg{[}-\frac{1}{2}(\Phi\star
G_{-i\nu}^{-1}\star\Phi+\tilde{\Phi}\star
G_{i\nu}^{-1}\star\tilde{\Phi})-{\cal
O}\big{[}\Phi_{0},\tilde{\Phi}_{0}\big{]}\circ J$
$\displaystyle\quad\quad\quad\quad-\int_{\rm
EAdS}\sqrt{|g|}d^{d+1}x\Big{(}i^{d-1}{\cal
L}_{I}[ic^{-}_{-i\nu}\Phi+c^{-}_{i\nu}\tilde{\Phi}]+(-i)^{d-1}{\cal
L}_{I}[ic^{+}_{-i\nu}\Phi+c^{+}_{i\nu}\tilde{\Phi}]\Big{)}\bigg{]}$ (227)

For a heavy dS mass ($\nu\in\mathbb{R}$), the fields satisfy
$\Phi^{*}=-\tilde{\Phi}$. The propagators satisfy $G_{-i\nu}=G^{*}_{i\nu}$,
and therefore the action is real. For a light dS mass
($i\nu\in[-\frac{d}{2},\frac{d}{2}]$), the fields either satisfy
$\Phi^{*}=-\Phi$, $\tilde{\Phi}^{*}=\tilde{\Phi}$ or $\Phi^{*}=\Phi$,
$\tilde{\Phi}^{*}=-\tilde{\Phi}$ depending on the value of $i\nu$. The $G_{\pm
i\nu}$ propagators are real, and it follows that again the action is real.

Let us specify the source for a light field. We assume 272727We can instead
assume $-i\nu=\mu\in[0,\frac{d}{2}]$. In that case $\tilde{\Phi}$ has Neumann
and $\Phi$ has Dirichlet BC. The source term is then ${\cal
O}\big{[}\Phi_{0},\tilde{\Phi}_{0}\big{]}=\frac{1}{\sqrt{2\sin\pi\mu}}\,\tilde{\Phi}_{0}\,.$
$i\nu=\mu\in[0,\frac{d}{2}]$. The kinetic term in Eq. (227) indicates that the
$\Phi$ field has Neumann and $\tilde{\Phi}$ has Dirichlet BC. By direct
comparison with a known cosmological correlator we obtain that the source acts
on the Neumann field, with $\Phi=K_{\rm EAdS}\circ\Phi_{0}+\Phi_{D}$. This
also fixes the normalization, giving

${\cal O}\big{[}\Phi_{0},\tilde{\Phi}_{0}\big{]}_{\rm
light}=\frac{1}{\sqrt{2\sin\pi\mu}}\,\Phi_{0}\,.$ (228)

Putting the pieces together, the amputated cosmological correlators with light
fields in external legs are generated by derivatives of the effective action
$\Gamma^{\rm pert}_{\rm dS}[\Phi_{0}]=-W^{\rm pert}_{\rm dS}+J\circ
O[\Phi_{0}]$ as follows,

$-{(2\sin\pi i\nu)^{\frac{n}{2}}}\frac{\delta^{n}\Gamma^{\rm pert}_{\rm
dS}[\Phi_{0}]}{\delta\Phi_{0}(\vec{y}_{1})\ldots\delta\Phi_{0}(\vec{y}_{n})}=\langle\Phi_{0}(\vec{y}_{1})\ldots\Phi_{0}(\vec{y}_{1})\rangle_{1PI}\,$
(229)

for $i\nu>0$. Equivalently, for $i\nu<0$ the derivatives are taken in
$\tilde{\Phi}$. We emphasize that, despite the complex factors present in the
action, the resulting cosmological correlators are real.

We remind that, since $\Gamma^{\rm pert}_{\rm dS}[\Phi_{0}]$ is defined by
Legendre transform in the boundary source, 1P-irreducibility is here meant
with respect to boundary-to-boundary lines (see also Sec. 5). The 4pt exchange
diagram with Dirichlet bulk field exchange, for example, is generated by Eq.
(229) as a 1PI diagram. As a sanity check one can verify that the dS exchange
diagram with cubic vertices and with light states in external legs is
correctly reproduced by Eq. (229).

###  Towards the Cosmological One-loop Effective Action

Integrating out a dS bulk fluctuation at tree-level in dS is a fairly simple
task. One can use, in particular, the covariant large mass expansion of the
propagator given in Eq. (98). Integrating out a dS bulk fluctuation at loop-
level in dS is more challenging. However, thanks to the analytical
continuation to EAdS, we can readily use the AdS heat kernel coefficients
evaluated in the previous section to get the heat kernel expansion of the
cosmological one-loop effective action, i.e. the one-loop piece of
$\Gamma_{\rm dS}^{\rm pert}=\Gamma^{\rm dS,pert}_{\rm cl}+\Gamma^{\rm
dS,pert}_{\rm 1-loop}+\ldots$.

We focus on integrating out bulk fluctuations of a scalar field $\Phi$. The
field can either be heavy or light in the dS sense. Depending on cases, either
Dirichlet or Neumann BC can be chosen. The interaction Lagrangian in terms of
dS fields is written as

${\cal L}^{\rm dS}_{I}=-\frac{1}{2}\Phi^{2}V^{\prime\prime}(\varphi^{\rm
dS}_{i})$ (230)

where the $\varphi^{\rm dS}_{i}$ are light background fields with masses
parametrized by $\mu_{i}\in[0,\frac{d}{2}]$. Upon analytical continuation to
EAdS, the interactions are $V^{\prime\prime}_{\pm}=V^{\prime\prime}(e^{\pm
i\frac{\pi}{2}\Delta_{-\mu_{i}}}K_{\mu_{i}}\circ\varphi_{i,0})$ where the
background boundary fields $\varphi_{i,0}$ are real.

We consider the quadratic piece of the fundamental action in $\Phi$,

$\displaystyle S^{\rm quad}_{\rm EAdS}=\frac{1}{2}(\Phi\star
G_{-i\nu}^{-1}\star\Phi+\tilde{\Phi}\star G_{i\nu}^{-1}\star\tilde{\Phi})$
(231) $\displaystyle\quad\quad\quad+\int_{\rm
EAdS}\sqrt{|g|}d^{d+1}x\Big{(}\frac{i^{d-1}}{2}\left(ic^{-}_{-i\nu}\Phi+c^{-}_{i\nu}\tilde{\Phi}\right)^{2}V^{\prime\prime}_{-}+\frac{(-i)^{d-1}}{2}\left(ic^{+}_{-i\nu}\Phi+c^{+}_{i\nu}\tilde{\Phi}\right)^{2}V^{\prime\prime}_{+}\Big{)}$

with the shortcut notation

$V^{\prime\prime}_{\pm}=V^{\prime\prime}(e^{\pm
i\frac{\pi}{2}\Delta_{-\mu_{i}}}K_{\mu_{i}}\circ\varphi_{i,0})\,.$ (232)

The background-field-dependent mass matrix of the fluctuations can be read
from Eq. (231),

$\displaystyle X$
$\displaystyle=i^{d}\begin{pmatrix}i(c^{-}_{-i\nu})^{2}&c^{-}_{i\nu}c^{-}_{-i\nu}\\\
c^{-}_{i\nu}c^{-}_{-i\nu}&-i(c^{-}_{i\nu})^{2}\end{pmatrix}V^{\prime\prime}_{-}-(-i)^{d}\begin{pmatrix}i(c^{+}_{-i\nu})^{2}&c^{+}_{i\nu}c^{+}_{-i\nu}\\\
c^{+}_{i\nu}c^{+}_{-i\nu}&-ic^{+}_{i\nu})^{2}\end{pmatrix}V^{\prime\prime}_{+}$
(233) $\displaystyle=\frac{i}{2\sin(\pi i\nu)}\begin{pmatrix}e^{i\pi
i\nu}V_{-}^{\prime\prime}-e^{-i\pi
i\nu}V_{+}^{\prime\prime}&i(V^{\prime\prime}_{+}-V^{\prime\prime}_{-})\\\
i(V^{\prime\prime}_{+}-V^{\prime\prime}_{-})&e^{i\pi
i\nu}V_{+}^{\prime\prime}-e^{-i\pi i\nu}V_{-}^{\prime\prime}\end{pmatrix}$
(234)

For a light fluctuation, $X$ is self-conjugate. For heavy fields, $X$ is not
self-conjugate because conjugation swaps the $\Phi$ and $\tilde{\Phi}$
components and thus the entries of $X$, however the physical quantities are
the traces $\operatorname{tr}(X^{r})$, which are self-conjugate.

Finally, we evaluate the trace of arbitrary powers of $X$ and find they take
the simple form

$\displaystyle\operatorname{tr}(X^{r})=(-1)^{r}\left((V^{\prime\prime}_{-})^{r}+(V^{\prime\prime}_{+})^{r}\right)\,.$
(235)

This canonical invariant is the only ingredient needed to evaluate the heat
kernel coefficients in the case of scalar interactions considered here.

Evaluating the finite, mass-suppressed parts of the one-loop action, i.e. the
long-distance EFT, could be interesting. However it may require some more
developments due to the fact that one deals with masses which are tachyonic
from the EAdS viewpoint. A careful analytic continuation from nontachyonic to
tachyonic EAdS masses might be needed. This is left to future work. Here below
we work out the divergent part of the one-loop cosmological action in a simple
case.

###  Renormalization from Scalar Loops in dS4

In this section we extract some concrete results in dS4. In dS4 one-loop
divergences appear from the $b_{4}$ and $b^{\partial}_{4}$ heat kernel
coefficients. The assumed scalar interactions give
$V(\varphi_{i})=\frac{\Phi^{2}}{2}\prod^{n}_{i=1}\varphi_{i}$. The
diagrammatic contributions to the $b_{4},b^{\partial}_{4}$ coefficients are
shown in Fig. 7. The EAdS scalar curvature is $R=-\frac{12}{\ell^{2}}$. The
$\square\prod^{n}_{i=1}\varphi_{i}$ and
$\partial_{\perp}\prod^{n}_{i=1}\varphi_{i}$ are evaluated like in the AdS
case, see Eqs. (131), (136). In the late time limit, the $\Delta_{i}$ combine
to appear only in the $\Delta_{X}=\sum^{n}_{i=1}\Delta_{i}$,
$\Delta_{X}^{-}=\sum^{n}_{i=1}(3-\Delta_{i})$ combinations, in accordance with
the underlying conformal symmetry.

Figure 7: Analytically-continued dS4 diagrams contributing to bulk and
boundary divergences in the one-loop effective action. Internal lines are
elements of the $\hat{G}_{{\rm EAdS}_{4}}$ propagator matrix.

We take a Neumann BC for the heavy field, accordingly to discussion in Sec.
(6). Using Eq. (235) with $r=1,2$, we find the heat kernel coefficients

$\displaystyle{\rm tr}\,\bar{b}_{4}$
$\displaystyle=\cos\left(\pi\Delta_{X}\right)\Lambda_{n,2}\circ\left[\prod^{n}_{i=1}\varphi_{i,0}\right]^{2}-\frac{\cos\left(\frac{\pi}{2}\Delta_{X}\right)}{3\ell^{2}}\left(-\Delta_{X}(\Delta_{X}-3)+12\right)\Lambda_{n}\circ\prod^{n}_{i=1}\varphi_{i,0}$
(236) $\displaystyle{\rm tr}\,b^{\partial}_{4}$
$\displaystyle=\frac{4}{3}\cos\left(\frac{\pi}{2}\Delta_{X}\right)\left(\frac{1}{\sqrt{\bar{g}}}\sum^{n}_{i=1}\langle{\cal
O}_{i}{\cal
O}_{i}\rangle\circ\prod^{n}_{i=1}\varphi_{i,0}+\frac{1}{\ell}\left(\Delta_{X}^{-}+\frac{d}{2}\right)\prod^{n}_{i=1}\varphi_{i,0}\right)\,$
(237)

These enter in the cosmological one-loop effective action as 282828 The
calculation is here in Euclidian signature. There is a relative $-$ sign
compared to the equivalent expression in Lorentzian signature, due to the
usual convention for the actions. See App. D for details.

$\displaystyle\Gamma_{{\rm dS},\rm div}^{\rm
pert}[\varphi_{i,0}]=-\frac{1}{16\pi^{2}}\left({\rm
tr}\,\bar{b}_{4}+\int_{\partial\cal M}d^{3}x\sqrt{\bar{g}}{\rm
tr}\,b^{\partial}_{4}(x)\right)\log\left(\frac{\mu_{r}}{m_{\Phi}}\right)$
(238)

The bulk term contains holographic vertices with $n$ and $2n$ legs. The
boundary term contains a $n$pt contact operator. It also contains a nonlocal
contribution which amounts to attaching a 2pt CFT correlator to that local
operator. The logarithms obtained here via dimensional regularization are
similar to those discussed in Senatore:2009cf .

####  Inflationary $\Phi^{2}\varphi^{2}$ Example

For inflationary correlators the light fields typically have vanishing mass,
which corresponds to $\mu=\frac{d}{2}$ and $\Delta_{-\mu}=0$. We assume a
quartic interaction ${\cal L}^{\rm
dS}_{I}=-\frac{\lambda}{4}\Phi^{2}\varphi^{2}$.

The total dimensions are $\Delta_{X}=0$, $\Delta^{-}_{X}=6$. We use Neumann
BC. We get

$\displaystyle{\rm tr}\,\bar{b}_{4}$
$\displaystyle=\frac{\lambda^{2}}{4}\Lambda_{4}\circ\left[\varphi_{0}\right]^{4}-\frac{2\lambda}{\ell^{2}}\varphi\star\varphi$
(239) $\displaystyle{\rm tr}\,b^{\partial}_{4}$
$\displaystyle=\frac{4\lambda}{3}\frac{1}{\sqrt{\bar{g}}}(\langle{\cal O}{\cal
O}\rangle\circ\varphi_{0})\varphi_{0}+\frac{5\lambda}{\ell}\varphi_{0}^{2}\,\,.$
(240)

First we make some simple comments. The last contribution to $\bar{b}_{4}$ is
a bulk mass term. The first contribution to $\bar{b}_{4}$ amounts to the
generation of a bulk $\varphi^{4}$ operator. One can check that its beta
function is the same as in ${\cal M}_{4}$. The $\Lambda_{4}$ function (that we
called “holographic vertex” in the previous sections) is here a 4pt
cosmological correlator whose explicit expression is given in App. E.

Below we extract further information from each term and discuss some
implications.

#### Expansion-induced Running Bulk Mass

We focus on the bulk mass term. In cosmological correlators, bulk masses
appear via the scaling dimensions $\Delta$ (with
$m^{2}=\Delta(d-\Delta)\ell^{2}$) arising in the squeezed limit of 4pt
functions. Here we assumed zero bulk mass for the $\varphi$ field.

However we can see that in dS4 the bulk mass has a logarithmic running
depending on the Hubble radius. A contribution $\propto m^{2}_{\Phi}$ should
also most likely appear from the $\bar{b}_{2}$ term, however our focus here is
only on mass-independent divergences. We find that the corresponding beta
functions for $m_{\varphi}$ (or $\nu$) is

$\beta_{m^{2}}=\beta_{\nu^{2}}=\frac{1}{4\pi}\frac{\lambda}{\ell^{2}}$ (241)

Let us compare with 4d Minkowski. In both ${\cal M}_{4}$ and dS4, the bulk
mass receives logarithmic corrections of the form $\delta m^{2}\propto
m_{\Phi}^{2}\log\mu_{r}$ due to nonderivative interactions if the particle
$m_{\Phi}$ in the loop is massive. However these corrections vanish if
$m_{\Phi}=0$. In contrast, Eq. (241) dictates that the bulk mass runs as a
result of the dS curvature. This effect is independent of $m_{\Phi}$, which
could be zero. Such an effect does not exist in ${\cal M}_{4}$, i.e. the beta
function vanishes for $\ell\to 0$.

The upshot is that, in the presence of nonderivative interactions, even if all
fields are exactly massless at a given scale, bulk masses of $O(\ell^{-2})$
times loop factor are radiatively generated.

####  Correction to OPE Coefficients

A CFT 2pt function on the late time boundary arises from the first term in
$b_{4}^{\partial}$. At vanishing scaling dimension, the CFT 2pt function in
momentum space is simply given by

$\langle{\cal O}{\cal
O}\rangle=\frac{p^{2}\eta_{0}^{2}}{1+p\eta_{0}}\ell^{-1}\,.$ (242)

The same 2pt function arises already at classical level from the kinetic term
of $\phi$, thus this one-loop term is a correction to the normalization of
$\cal O$. We find the correction

$\left(1+\frac{\lambda}{12\pi^{2}}\log\frac{\mu_{r}}{m_{\Phi}}\right)\langle{\cal
O}(y_{1}){\cal O}(y_{2})\rangle\,.$ (243)

As already discussed in section 7.3.1, upon unit normalization of $\cal O$,
this correction amounts to a CFT version of “wavefunction renormalization”,
which corrects the OPE coefficients involving $\cal O$.

#### Boundary-localized Operator

One of the divergences on the late time boundary is a mass term. We define the
renormalized boundary mass term of the dS Lagrangian as $b=b_{\rm dS}=-b_{\rm
EAdS}$ such that $b_{\rm EAdS}$ enters as

$S_{\rm EAdS}\supset\frac{1}{2}\int_{\partial{\rm
EAdS}}d^{3}x\sqrt{\bar{g}}b_{\rm EAdS}(\varphi^{2}+\tilde{\varphi}^{2})$ (244)

in the EAdS action. We then obtain the beta function

$\beta_{b}=\frac{5\lambda}{8\pi^{2}\ell}\,.$ (245)

This is our first example of the radiative generation of an operator localized
on the boundary of dS. This term is nontrivial as it results from the finite
Hubble radius and would not happen in flat spacetime. The fact that this term
is generated radiatively signals that it should be generically included in the
fundamental dS action to start with. We will see in section 9.6.3 below that
such a term can have significant impact on the propagators.

####  Inflationary $\Phi^{2}(\partial_{M}\varphi)^{2}$ Example

We consider a derivative interaction ${\cal L}_{I}^{\rm
dS}=-\frac{\kappa}{4}\Phi^{2}\partial_{M}\varphi\partial^{M}\varphi$ and take
$\Delta_{-\mu}=0$ such that $\varphi$ has shift symmetry. Here we will refer
to $\varphi$ as the inflaton. The metric factor from the covariant contraction
in ${\cal L}_{I}^{\rm dS}$ gives an extra $i^{2}$ factor when rotating from dS
to EAdS.

We get the heat kernel coefficients

$\displaystyle{\rm tr}\,\bar{b}_{4}$
$\displaystyle=\frac{\kappa^{2}}{4}\Lambda^{\partial}_{4}\circ\left[\varphi_{0}\right]^{4}+\frac{2\kappa}{\ell^{2}}\partial_{M}\varphi\star\partial^{M}\varphi$
(246) $\displaystyle{\rm tr}\,b^{\partial}_{4}$
$\displaystyle=-\frac{4\kappa}{3}\frac{1}{\sqrt{\bar{g}}}(\langle{\vec{\partial}\cal
O}\vec{\partial}{\cal O}\rangle\circ\varphi_{0})\varphi_{0}-\frac{2\kappa
z_{0}^{2}}{3\ell^{3}}(\vec{\partial}\varphi_{0})^{2}\,$ (247)

where we introduced the 4pt correlator with four derivatives

$\Lambda^{\partial}_{4}(x_{0,1\ldots 4})=\int_{\rm
EAdS}d^{4}x\sqrt{g}\,\partial_{M}K(x,x_{0,1})\partial^{M}K(x,x_{0,2})\partial_{N}K(x,x_{0,3})\partial^{N}K(x,x_{0,4})\,.$
(248)

This 4pt correlator is explicitly given in App. E.

The second term in $\bar{b}_{4}$ contributes to the bulk kinetic term, for
this term we have written the full bulk field. We can see that the 2pt
correlator of descendant operators $\vec{\cal O}$ appears in the boundary
coefficient $\bar{b}^{\partial}_{4}$. In momentum space the combination simply
amounts to $-p^{2}\langle{\cal OO}\rangle$.

#### Expansion-induced Running Wavefunction Normalization

Introducing the renormalized parameters $Z(\mu_{r})$, $\xi(\mu_{r})$ in the dS
Lagrangian

${\cal L}^{\rm dS}=-\frac{Z}{2}(\partial_{M}\varphi)^{2}-\frac{\xi
Z^{2}}{4!}(\partial_{M}\varphi\partial^{M}\varphi)^{2}$ (249)

with $Z=1+\delta Z$, we find the perturbative anomalous dimension for
$\varphi$

$\gamma_{\varphi}=-\frac{1}{2}\frac{d\log
Z}{d\log\mu_{r}}=\frac{\kappa^{2}}{8\pi^{2}\ell^{2}}$ (250)

and the beta function

$\beta_{\xi}=\frac{3\kappa^{2}}{8\pi^{2}}+2\gamma_{\varphi}\xi$ (251)

We can compare these results to renormalization of the NLSM in 4d flat space.
In the flat space NLSM, the $\xi$ coupling would run similarly, but not the
bulk kinetic term. That is, the anomalous dimension $\gamma_{\varphi}$ is a
consequence of the nonzero curvature of spacetime; if $\ell\to\infty$ (keeping
constant $z_{0}/\ell$), then $\gamma_{\varphi}\to 0$. Comparing
$\gamma_{\varphi}$ and $\beta_{\xi}$ we can see that the effect of this
expansion-induced wavefunction renormalization becomes sizeable when $\xi$ is
of order $\ell^{2}$.

#### Boundary-localized Kinetic Term

One of the divergences on the late time boundary corresponds to a kinetic
term. We define the renormalized boundary kinetic term in the dS Lagrangian
$c=c_{\rm dS}=c_{\rm EAdS}$ such that $b_{\rm EAdS}$ enters as

$S_{\rm EAdS}\supset\frac{1}{2}\int_{\partial{\rm
EAdS}}d^{3}x\sqrt{\bar{g}}c_{\rm
EAdS}\frac{z_{0}^{2}}{\ell^{2}}((\vec{\partial}\varphi)^{2}+(\vec{\partial}\tilde{\varphi})^{2})$
(252)

in the EAdS action. We then obtain the beta function

$\beta_{c}=\frac{\lambda}{12\pi^{2}\ell}\,.$ (253)

Again, this RG running is nontrivial as it results from the Hubble radius and
would not happen in flat spacetime. The fact that this term is generated
radiatively signals that it should be generically included in the fundamental
dS action to start with.

####  Effect of Boundary Operators on Propagation

Figure 8:  A propagator dressed by insertions of a bilinear operator localized
on the late time boundary of de Sitter space.

Throughout the above examples we have shown that radiative corrections
generate operators localized on the late time boundary, in the presence of
either derivative and nonderivative couplings of $\varphi$. The general lesson
is that such boundary operators should be included in the fundamental
Lagrangian in a first place.

The bilinear boundary-localized operators influence the dS propagators with
Neumann BC. The structure of the propagator in the presence of such operators
is the one established in Eqs. (27), (53). As explained there, the effect of
the boundary-localized operators can be understood as a dressing of the
propagators by boundary-localized insertions. 292929Renormalization of
boundary operators dS4 and dressing of the wavefunction have also been
discussed in Cespedes:2020xqq  in a different formalism.

A contribution to this dressing is shown in Fig. 8. Denoting by
$B[-\partial^{2}]$ the bilinear insertion localized on the late time boundary,
and focussing on the massless case for concreteness, we obtain the boundary-
to-boundary propagator

$G^{\rm
dS}_{--,{\Delta=0}}(\vec{p};\eta_{0},\eta_{0})=i\frac{\eta^{3}_{0}(1-ip\eta_{0})}{\ell^{2}\left(p^{2}\eta^{2}_{0}+B\ell(1-ip\eta_{0})\right)}$
(254)

and the bulk-to-bulk propagator

$\displaystyle G^{\rm dS}_{--,{\Delta=0}}(\vec{p};\eta,\eta^{\prime})=$ (255)
$\displaystyle\frac{\ell}{2p^{3}}\left(\frac{\eta\eta^{\prime}}{\ell^{2}}\right)^{\frac{3}{2}}\frac{1-ip\eta_{>}}{1-ip\eta_{0}}\left(-e^{-ip|\eta-\eta^{\prime}|}(1+ip\eta_{<})(1-ip\eta_{0})+e^{-ip(2\eta_{0}-\eta-\eta^{\prime})}(1-ip\eta_{<})(1+ip\eta_{0})\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+G^{\rm
dS}_{--,{\Delta=0}}(p;\eta_{0},\eta_{0})\frac{e^{-ip(\eta_{0}-\eta)}(1-ip\eta)}{(1-ip\eta_{0})}\frac{e^{-ip(\eta_{0}-\eta^{\prime})}(1-ip\eta^{\prime})}{(1-ip\eta_{0})}$

The first line in Eq. (255) corresponds to the Dirichlet propagator, the
second line is the boundary term which encodes all the effects of the boundary
operators. This is a Neumann/Dirichlet relation analogous to the one found in
Eqs. (27), (53). This is expected since dS$\leftrightarrow$EAdS Wick rotations
do not affect the structure of boundary conditions. That is, the dressed dS
propagator Eq. (255) can for instance be obtained by starting in EAdS, using
the results from section 3.1 and rotating to dS using Eq. (212).

We can see that whenever the boundary term $B$ is sizeable in units of $\ell$,
it can have a significant impact on the propagator. This effect is well-known
from the AdS5 literature — what typically happens is that the propagator tends
to be deformed and repelled from the AdS boundary. We conclude that the same
kind of effect can happen in dS.

In the case of a nonderivative interaction, a boundary-localized mass term
$B=$ cst is present — at least generated at one-loop as dictated by Eq. (245).
This boundary mass term is expected to be of order one in units of $\ell$, or
at best suppressed by a 4d loop factor. This term can thus have a strong
impact on Eq. (255) when $p\eta_{0}\ll 1$. Note however that the nonderivative
case considered here is tuned in the sense that a nonzero bulk mass i.e. a
nonzero scaling dimension is also expected, see Eq. (241).

In the case of a derivative interaction, mass terms cannot be generated since
they are protected by shift symmetry. There we have found that $B$ contains a
boundary-localized kinetic term of order $B\propto{\ell}p^{2}$ — at least
generated at one-loop as dictated by Eq. (253). This boundary-localized
kinetic term can have significant impact on the propagator if the $\kappa$
coupling is large enough.

It would be certainly interesting to further investigate the impact of
boundary-localized operators on the cosmological correlators. This is left for
future work.

## Acknowledgments

I thank Brian Batell, Tony Gherghetta, David Meltzer, Manuel Perez-Victoria,
Eduardo Ponton, Allic Sivaramakrishnan and Flip Tanedo for useful discussions
and comments. This work has been supported by the São Paulo Research
Foundation (FAPESP) under grants #2011/11973, #2014/21477-2 and #2018/11721-4,
by CAPES under grant #88887.194785, and by the University of California,
Riverside.

##  Appendix A The Discontinuity Equation

Here we give a proof of Eq. (9).

We denote by $\partial{\cal M}^{-}$ the surface at an infinitesimal normal
distance of the actual boundary $\cal M$ in the inward direction. We define an
infinitesimal shell $\sigma_{x_{0}}$ containing the boundary point $x_{0}$.
The outward part of the boundary of $\sigma_{x_{0}}$ is in $\partial\cal M$,
and the inward part of the boundary of $\sigma_{x_{0}}$ is in $\partial{\cal
M}^{-}$. A point belonging to $\partial{\cal M}^{-}$ is denoted $x_{0}^{-}$.

We integrate the bulk equation of motion Eq. (5) over the $\sigma_{x_{0}}$
volume and apply the divergence theorem. The delta function of the EOM is
picked by the integral if the other endpoint of the propagator is inside the
$\sigma_{x_{0}}$ shell. Taking the infinitesimal limit for $\sigma_{x_{0}}$
and using that $\cal M$ is locally Euclidian, we get the discontinuity of the
normal derivative evaluated at the boundary. This is Eq. (9).

##  Appendix B Details and Checks of Propagators in Warped Background

###  Generic Background

Here we show that the propagators given in Sec. 3 satisfy the general
properties derived in Sec. 2.

We first define the boundary inner product. In Fourier coordinates it is given
by a simple multiplication,

$\Phi_{0}\circ\Psi_{0}=\rho_{0}^{d}e^{\varphi_{0}}\Phi_{0}\Psi_{0}\,.$ (256)

Since the inverses are defined using this product, inverses involve a
$\rho_{0}^{d}e^{\varphi_{0}}$ factor:
$[a]^{-1}=\rho_{0}^{d}e^{\varphi_{0}}\frac{1}{a}$. One has
$[[a]^{-1}]^{-1}=a$.

We will check the general relation $G_{N}=K\circ[G^{-1}_{0}+{\cal
B}]^{-1}\circ K+G_{D}$. We evaluate
$G_{N}(p;z,z^{\prime})-G_{D}(p;z,z^{\prime})$ by direct computation:

$\displaystyle G_{N}-G_{D}$
$\displaystyle=\frac{if(z_{<})f(z_{>})}{Cf_{0}}\frac{C\rho^{d}_{0}e^{\varphi_{0}}}{\rho_{0}f^{\prime}_{0}+{B}f_{0}}=i\frac{f(z_{<})f(z_{>})}{f_{0}f_{0}}G^{B}_{0}$
(257) $\displaystyle=K(z_{<})\circ G^{B}_{0}\circ K(z_{>})$

where in the intermediate step one has used the Wronskian in the numerator and
we identified the boundary-to-boundary propagator, Eq. (39). Since
$G_{B}^{N}=[(G_{B}^{N})^{-1}]^{-1}$, this verifies the main formula Eq. (27).

We can also verify that

$[G^{B}_{0}]^{-1}=-i\rho_{0}^{-d}e^{-\varphi_{0}}\left(\rho_{0}\frac{f^{\prime}_{0}}{f_{0}}+{B}\right)=[G_{0}]^{-1}-i\rho_{0}^{-d}e^{-\varphi_{0}}B$
(258)

This reproduces the dressing of the boundary-to-boundary propagator by the
boundary-localized bilinear insertion $B$.

It is also instructive to explicitly verify the discontinuity formula. We get

$(\rho_{0}\partial_{z}+Bf_{0})G^{B}_{N}(p;z,z^{-}_{0})|_{z\to
z_{0}}=i\rho^{d}_{0}e^{\varphi}_{0}=\frac{i}{\sqrt{|\bar{g}|}}\,$ (259)

where again one has used the Wronskian at $z_{0}$. This result verifies Eq.
(10) upon translation from Lorentzian to Euclidian conventions.

###  AdS Propagator in Position Space

In Euclidian AdS the propagator satisfying the equation of motion

$\left(-\square_{\rm
AdS}+\Delta(\Delta-d)k^{2}\right)G_{\Delta}(x,x^{\prime})=\frac{1}{\sqrt{g}}\delta^{d+1}(x-x^{\prime})$
(260)

is Burges:1985qq

$G_{\Delta}(x,x^{\prime})=\frac{\Gamma(\Delta)}{2\pi^{d/2}\Gamma(\Delta-\frac{d}{2}+1)}\frac{k^{d-1-2\Delta}}{\zeta(x,x^{\prime})^{\Delta}}\,_{2}F_{1}\left(\Delta,\Delta-\frac{d}{2}+\frac{1}{2},2\Delta-d+1,-\frac{4}{k^{2}\zeta(x,x^{\prime})}\right)$
(261)

with

$\zeta(x,x^{\prime})=\frac{1}{k^{2}}\frac{(z-z^{\prime})^{2}+(x-x^{\prime})^{2}}{zz^{\prime}}$
(262)

the chordal distance in Poincaré coordinates. Taking one endpoint towards the
boundary, the propagator is asymptotically

$G_{\Delta}(x,x^{\prime})|_{z\to
0}=\frac{\Gamma(\Delta)}{2\pi^{d/2}\Gamma(\Delta-\frac{d}{2}+1)}\frac{k^{d-1-2\Delta}}{\zeta(x,x^{\prime})^{\Delta}}+O\left(\frac{1}{\zeta^{\Delta+1}}\right)$
(263)

for any $\Delta$.

##  Appendix C Holography with Two Boundaries

In this appendix we work out the holographic formalism for a $z_{0}<z<z_{1}$
slice of the warped background introduced in Eq. (30), with both boundaries
treated holographically.

Similarly to Sec. 2, our starting point is to write the holographic basis in
the presence of two boundaries,

$\Phi=\Phi_{0}\circ L_{0}+\Phi_{1}\circ L_{1}+\Phi_{D}$ (264)

where $\Phi_{0,1}$ are the field values on each boundary, which will be the
variables of the holographic action. The $L_{0}$, $L_{1}$ boundary-to-bulk
propagators are determined below. We work in Fourier space along the
transverse Poincaré slice, hence the coordinates are $(p^{\mu},z)$. We
consider the classical value $\langle\Phi\rangle$ which satisfies the bulk EOM
${\cal D}\langle\Phi\rangle=0$. It takes in general the form

$\langle\Phi\rangle(z)=af(z)+bh(z)\,.$ (265)

The associated Wronskian is

$W(z)=f(z)h^{\prime}(z)-f^{\prime}(z)h(z)=C\rho^{d-1}e^{\varphi}\,.$ (266)

The $f$, $h$ solutions and the $a$, $b$ constants depend on the $d$-momentum.
We also define

$f(z_{0})=f_{0}\,,\quad f(z_{1})=f_{1}\,,\quad h(z_{0})=h_{0}\,,\quad
h(z_{1})=h_{1}\,.$ (267)

Using the definitions Eqs. (265), (267), the $a,b$ constants can be translated
into the holographic variables,

$a=\frac{\Phi_{0}h_{1}-\Phi_{1}h_{0}}{f_{0}h_{1}-f_{1}h_{0}}\,,\quad
b=-\frac{\Phi_{0}f_{1}-\Phi_{1}f_{0}}{f_{0}h_{1}-f_{1}h_{0}}\,.$ (268)

The holographic basis in Fourier space is therefore

$\Phi(p,z)=\Phi_{0}\frac{h_{1}f(z)-f_{1}h(z)}{f_{0}h_{1}-f_{1}h_{0}}+\Phi_{1}\frac{f_{0}h(z)-h_{0}f(z)}{f_{0}h_{1}-f_{1}h_{0}}+\Phi_{D}\,.$
(269)

We plug the solution Eq. (269) into the quadratic action, giving

$S[\Phi_{0},\Phi_{1}]=\int\frac{d^{d}p}{(2\pi)^{4}}\sqrt{|\bar{g}|}e^{-\varphi}\frac{1}{2}\left(\Phi_{0}(\rho_{0}\partial_{z}-b_{0})\Phi|_{z_{0}}-\Phi_{1}(\rho_{1}\partial_{z}-b_{1})\Phi|_{z_{1}}\right)+S_{D}$
(270)

where $b_{0}$, $b_{1}$ encode the boundary actions. We introduce

$\hat{f}_{i}=\rho_{i}f^{\prime}(z_{i})-b_{i}f(z_{i})$ (271)

and similarly for $\hat{g}_{i}$. Evaluating $\partial_{z}\Phi_{z_{0}}$ and
$\partial_{z}\Phi_{z_{1}}$ using Eq. (269) we obtain the holographic self-
energies

$S[\Phi_{0},\Phi_{1}]=\frac{1}{2}\int\frac{d^{4}p}{(2\pi)^{4}}\left(\Phi_{0}\Pi_{0}\Phi_{0}+2\Phi_{0}\Pi_{01}\Phi_{1}+\Phi_{1}\Pi_{1}\Phi_{1}\right)\,,$
(272)
$\Pi_{0}=\rho_{0}^{-d}e^{-\varphi_{0}}\frac{\hat{f}_{0}h_{1}-\hat{h}_{0}f_{1}}{f_{0}h_{1}-h_{0}f_{1}}\,,\quad\Pi_{1}=\rho_{1}^{-d}e^{-\varphi_{1}}\frac{\hat{f}_{1}h_{0}-\hat{h}_{1}f_{0}}{f_{0}h_{1}-h_{0}f_{1}}\,,\quad\Pi_{01}=\frac{C}{f_{0}h_{1}-h_{0}f_{1}}$
(273)

where we have used the Wronskian at $z_{0}$ and $z_{1}$ to get the mixed self-
energy $\Pi_{01}$.

We can re-express these self-energies using propagators. The propagator in the
presence of two boundaries is

$G^{\pm\pm}(y,y^{\prime})=\frac{i}{C}\frac{(\tilde{f}_{0}h(z_{<})-\tilde{h}_{0}f(z_{<}))(\tilde{f}_{1}h(z_{>})-\tilde{h}_{1}f(z_{>}))}{\tilde{f}_{0}\tilde{h}_{1}-\tilde{h}_{0}\tilde{f}_{1}}$
(274)

where $\tilde{f}_{i}=f_{i}$ for a Dirichlet boundary condition ($-$) and
$\tilde{f}_{i}=\hat{f}_{i}$ for a Neumann boundary condition ($+$). For the
$\Pi_{0}$, $\Pi_{1}$ self-energies we find

$\Pi_{0}=\frac{i}{G_{p}^{+-}\left(z_{0},z_{0}\right)}\,,\quad\Pi_{1}=\frac{i}{G_{p}^{-+}\left(z_{1},z_{1}\right)}\,.$
(275)

The $\Pi_{01}$ self-energy can be expressed as

$\displaystyle\Pi_{01}=\frac{C}{f_{0}h_{1}-h_{0}f_{1}}$
$\displaystyle=C\frac{1}{\tilde{f}_{0}\tilde{h}_{1}-\tilde{h}_{0}\tilde{f}_{1}}\frac{\tilde{f}_{0}\tilde{h}_{1}-\tilde{h}_{0}\tilde{f}_{1}}{\tilde{f}_{0}h_{1}-\tilde{h}_{0}f_{1}}\frac{\tilde{f}_{0}h_{1}-\tilde{h}_{0}f_{1}}{f_{0}h_{1}-h_{0}f_{1}}$
$\displaystyle=i\frac{G_{p}^{++}\left(z_{0},z_{1}\right)}{G_{p}^{++}\left(z_{1},z_{1}\right)G_{p}^{+-}\left(z_{0},z_{0}\right)}=i\frac{G_{p}^{++}\left(z_{0},z_{1}\right)}{G_{p}^{++}\left(z_{0},z_{0}\right)G_{p}^{-+}\left(z_{1},z_{1}\right)}\,.$
(276)

We can recognize (amputated) boundary-to-bulk propagators in the last
expressions. We have

$\Pi_{01}=K_{0}(z_{1})\Pi_{1}=K_{1}(z_{1})\Pi_{0}\,$ (277)

where $K_{0}(z)$, $K_{1}(z)$ denote the amputated boundary-to-bulk propagators
from the $z_{0}$ and $z_{1}$ boundaries respectively.

If we let the $\Phi_{1}$ variable be dynamical i.e. integrate it out in the
path integral, we should recover our standard one-boundary holographic action.
For a Dirichlet boundary condition on $z_{1}$, we have $\Phi_{1}=0$. It
follows trivially that

$S[\Phi_{0}]=\frac{1}{2}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{i}{G^{+-}_{p}\left(z_{0},z_{0}\right)}\Phi_{0}^{2}+S_{D}$
(278)

which is the expected result for the holographic action on boundary $0$ with
Dirichlet boundary condition on boundary $1$. For a Neumann boundary condition
on $z_{1}$, we have $(\rho_{1}\partial_{z}-b_{1})\Phi|_{z_{1}}=0$, which
implies

$\Phi_{1}=\Phi_{0}\frac{W}{h_{0}\tilde{f}_{1}-f_{0}\tilde{h}_{1}}\,.$ (279)

Substituting $\Phi_{1}$ with this relation in the two-boundary holographic
action gives

$S[\Phi_{0}]=\frac{1}{2}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{i}{G^{++}_{p}\left(z_{0},z_{0}\right)}\Phi_{0}^{2}+S_{D}$
(280)

which is again the expected result for the holographic action on boundary $0$
with Neumann boundary condition on boundary $1$.

##  Appendix D Elements of the Heat Kernel Formalism

We work in Euclidian metric and convert to Lorentzian conventions at the end
of the calculation. The Euclidian effective action generates the 1PI Euclidian
correlators following

$-\frac{\delta^{n}\Gamma_{E}}{\delta\Phi(x_{1})\ldots\delta\Phi(x_{n})}=\langle\Phi(x_{1})\ldots\Phi(x_{n})\rangle\,.$
(281)

The one-loop effective action is put in the form of the heat kernel integral

$\Gamma^{E}_{1-{\rm loop}}=-\frac{1}{2}\int\frac{dt}{t}{\rm Tr}\,e^{-t{\cal
D}_{E}}\,.$ (282)

with ${\cal D}_{E}=-\square_{E}+m^{2}+X$. Tr is the trace over all indexes
including spacetime coordinates. The heat kernel trace is expanded as

${\rm Tr}e^{-t(-\square_{E}+m^{2}+X)}=(4\pi t)^{D/2}e^{-tm^{2}}\int
d^{D}x_{E}\sqrt{g}\sum^{\infty}_{r=0}{\rm tr}\,b^{E}_{2r}(x)t^{r}\,$ (283)

where the $b^{E}_{2r}$ are the Euclidian heat kernel coefficients. Using this
expansion in Eq. (282), integrating in $t$ and analytically continuing in $D$
gives

$\Gamma^{E}_{1-{\rm loop}}=-\frac{1}{2}\frac{\Gamma(r-D/2)}{(4\pi)^{D/2}}\int
d^{D}x_{E}\sqrt{g}\sum^{\infty}_{r=0}\frac{{\rm tr}\,b^{E}_{2r}(x)}{m^{2r-D}}$
(284)

for any spacetime dimension. Converting to Lorentzian metric gives 303030Use
$x_{E}^{0}=ix^{0}$, $\Gamma_{E}^{1-{\rm loop}}=-i\Gamma_{1-{\rm loop}}$.

$\Gamma_{1-{\rm loop}}=\frac{1}{2}\frac{\Gamma(r-D/2)}{(4\pi)^{D/2}}\int
d^{D}x\sqrt{g}\sum^{\infty}_{r=0}\frac{{\rm tr}\,b_{2r}(x)}{m^{2r-D}}$ (285)

with $\rm tr$ the trace over internal (non-spacetime) indexes.

The exact $b_{0}$, $b_{2}$, $b_{4}$ coefficients are given in Eq. (6.3.1). The
exact $b_{6}$ coefficient is

$\displaystyle b_{6}=$
$\displaystyle\frac{1}{360}\bigg{(}8D_{P}\Omega_{MN}D^{P}\Omega^{MN}+2D^{M}\Omega_{MN}D_{P}\Omega^{PN}+12\Omega_{MN}\square\Omega^{MN}-12\Omega_{MN}\Omega^{NP}\Omega^{~{}~{}M}_{P}$
$\displaystyle-6R_{MNPQ}\Omega^{MN}\Omega^{PQ}-4R_{M}^{~{}N}\Omega^{MP}\Omega_{NP}+5R\Omega_{MN}\Omega^{MN}$
$\displaystyle-6\square^{2}X+60X\square X+30D_{M}XD^{M}X-60X^{3}$
$\displaystyle-30X\Omega_{MN}\Omega^{MN}-10R\square
X-4R_{MN}D^{N}D^{M}X-12D_{M}RD^{M}X+30XXR$ $\displaystyle-12X\square
R-5XR^{2}+2XR_{MN}R^{MN}-2XR_{MNPQ}R^{MNPQ}\bigg{)}$
$\displaystyle+\frac{1}{7!}\bigg{(}18\square^{2}R+17D_{M}RD^{M}R-2D_{P}R_{MN}D^{P}R^{MN}-4D_{P}R_{MN}D^{M}R^{PN}$
$\displaystyle+9D_{P}R_{MNQL}D^{P}R^{MNQL}+28R\square R-8R_{MN}\square R^{MN}$
$\displaystyle+24R_{MN}D_{P}D^{N}R^{MP}+12R_{MNQL}\square R^{MNQL}+35/9R^{3}$
$\displaystyle-14/3RR_{MN}R^{MN}+14/3RR_{MNPQ}R^{MNPQ}-208/9R_{MN}R^{MP}R_{~{}~{}P}^{N}$
$\displaystyle-64/3R_{MN}R_{PQ}R^{MPNQ}-16/3R^{M}_{~{}N}R_{MPQL}R^{NPQL}$
$\displaystyle-44/9R^{MN}_{~{}~{}~{}~{}AB}R_{MNPQ}R^{PQAB}-80/9R_{M~{}~{}P~{}~{}}^{~{}~{}N~{}~{}Q}R^{MAPB}R_{NAQB}\bigg{)}I$
(286)

###  Dirichlet $b^{\partial}_{i}$ for Scalar Interaction

In Sec. 7.1 we have used Neumann BC for the fluctuation. If instead one takes
Dirichlet BC, using that $\Pi_{+}=0$, $\Pi_{-}=1$, $\chi=-1$, $S=0$, the
boundary heat kernel coefficients are found to be

$\displaystyle{\rm tr}\,b^{\partial}_{3}$ $\displaystyle=\frac{1}{4}X_{0}$
(287) $\displaystyle{\rm tr}\,b^{\partial}_{4}$
$\displaystyle=\frac{1}{3}\frac{1}{\sqrt{\bar{g}}}\sum^{n}_{i=1}\langle{\cal
O}_{i}{\cal O}_{i}\rangle\circ X_{0}+\frac{1}{3}(\Delta_{X}^{-}-d)kX_{0}\,$
(288) $\displaystyle{\rm tr}\,b^{\partial}_{5}=\frac{1}{5760}\bigg{(}$
$\displaystyle-720X_{0}^{2}+\left(360\Delta^{-}_{X}(d-\Delta^{-}_{X})k^{2}-450d\Delta_{X}^{-}k^{2}-5d(53+27d)k^{2}\right)X_{0}$
$\displaystyle-360k^{2}z_{0}^{2}\sum^{n}_{i=1}(\partial_{\mu}\partial^{\mu})_{i}X_{0}+240k^{2}z_{0}^{2}\partial_{\mu}\partial^{\mu}X_{0}$
$\displaystyle+(720\Delta_{X}^{-}-450d)k\frac{1}{\sqrt{\bar{g}}}\sum^{n}_{i=1}\langle{\cal
O}_{i}{\cal O}_{i}\rangle\circ X_{0}$
$\displaystyle+360\frac{1}{\sqrt{\bar{g}}}\frac{1}{\sqrt{\bar{g}}}\sum^{n}_{i,j=1}[\langle{\cal
O}_{i}{\cal O}_{i}\rangle\circ][\langle{\cal O}_{j}{\cal
O}_{j}\rangle\circ]X_{0}\bigg{)}$ (289)

###  Elementary Checks from ${\cal M}_{3}$ and EAdS3 Bubble

###  ${\cal M}_{3}$ Bubble

We check a contribution to the Lorentzian heat kernel coefficient $b_{4}$ from
the scalar bubble diagram in ${\cal M}_{3}$. In ${\cal M}_{3}$, working in
momentum space as in particle physics-style calculations, the amputated bubble
at large $m$ is given by

$B_{{\cal
M}_{3}}\Big{|}_{m\to\infty}=\frac{1}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{-p^{2}-m^{2}}\frac{1}{-(p+k)^{2}-m^{2}}\Big{|}_{m\to\infty}=\frac{i}{16\pi
m}$ (290)

Consider the fundamental interaction $\frac{1}{2}\Phi^{2}{\cal O}$. The bubble
diagram would be generated by two ${\cal O}$ derivatives of the Lorentzian
effective action $\frac{1}{32\pi m}{\cal O}^{2}$. This matches exactly the
$X^{2}$ term coming from the $b_{4}$ coefficient (obtained from Eq. (6.3.1))
in Eq. (285).

#### EAdS3 Bubble

We check a contribution to the Euclidian heat kernel coefficient $b^{E}_{4}$
from the scalar bubble diagram in EAdS.

Consider the bubble diagram of a real scalar in EAdS3,

$B(x,x^{\prime})=\frac{1}{2}G(X,Y)^{2}=k^{2}\int_{\mathbb{R}}d\nu
B(\nu,\Delta)\Omega_{\nu}(x,x^{\prime})$ (291)

In AdS3, Ref. Carmi:2018qzm  finds

$B(\nu,\Delta)=i\frac{\psi\left(\Delta-\frac{1+i\nu}{2}\right)-\psi\left(\Delta-\frac{1-i\nu}{2}\right)}{16\pi\nu}$
(292)

where $\Psi$ is the Digamma function. We expand the functions for large
$|\Delta|$ at fixed $\nu$. The leading order result is

$B(\nu,\Delta)|_{|\Delta|\to\infty}=\frac{1}{16\pi\Delta}+O\left(\frac{1}{|\Delta|^{2}}\right)\,.$
(293)

In position space, using that
$\int\Omega_{\nu}(x,x^{\prime})=\frac{1}{k^{d+1}\,\sqrt{|g|}}\delta^{d+1}(x,x^{\prime})$,
we obtain the local operator

$B(x,x^{\prime})|_{|\Delta|\to\infty}=\frac{1}{\sqrt{|g|}}\frac{1}{16\pi\Delta
k}\delta^{3}(x,x^{\prime})\,.$ (294)

Consider the fundamental interaction $\frac{1}{2}\Phi^{2}{\cal O}$. Using Eq.
(281), the bubble diagram would be generated by two ${\cal O}$ derivatives of
the Euclidian effective action

$-\frac{1}{2}\int dx^{d+1}\int dx^{\prime
d+1}\sqrt{|g|_{x}}\sqrt{|g|_{x^{\prime}}}{\cal O}(x)B(x,x^{\prime}){\cal
O}(x^{\prime})\,.$ (295)

Plugging the limit of Eq. (294) into Eq. (295) gives the local operator

$-\frac{1}{32\pi\Delta k}\int dx^{d+1}\sqrt{|g|}{\cal O}^{2}(x)\,$ (296)

in the Euclidian effective action. This matches exactly the $X^{2}$ term
coming from the $b^{E}_{4}$ coefficient (obtained from Eq. (6.3.1)) in Eq.
(284).

##  Appendix E  Some Elementary 4pt Cosmological Correlators

Here we give the momentum space expressions for the $\Lambda_{4}$ and
$\Lambda^{\partial}_{4}$ holographic vertices with massless fields i.e.
$\Delta=0$. These quantities in the boundary effective action computed in
section 9.6, and are thus contribution to the 4pt cosmological correlators of
$\varphi$.

Defining $p_{t}=\sum_{i=1}^{4}p_{i}$ and taking $p_{i}z_{0}\ll 1$, we get

$\displaystyle\Lambda_{4}(\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4})=$
$\displaystyle\frac{\ell^{4}}{3z_{0}^{3}}-\frac{\ell^{4}}{3z_{0}}(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+p_{4}^{2})+\ell^{4}\frac{p_{1}p_{2}p_{3}p_{4}}{p_{t}}$
(297)
$\displaystyle-\frac{\ell^{4}}{3}\log({p_{t}}{z_{0}})\left({p_{1}}^{3}+{p_{2}}^{3}+{p_{3}}^{3}+{p_{4}}^{3}\right)$
$\displaystyle+\frac{\ell^{4}}{3}\left({p_{t}}\sum_{i,j}p_{i}p_{j}-\sum_{i}{p_{i}}\sum_{j\neq
i}p_{j}-4\sum_{i<j<k}p_{i}p_{j}p_{k}\right)$

For the $\Lambda^{\partial}_{4}$ correlator, using $p_{i}z_{0}\ll 1$ we find

$\displaystyle\Lambda^{\partial}_{4}(\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4})=\vec{p}_{1}\cdot\vec{p}_{2}\,\vec{p}_{3}\cdot\vec{p}_{4}\,\Lambda(\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4})+\frac{40320}{\ell^{4}}\frac{p^{2}_{1}p^{2}_{2}p^{2}_{3}p^{2}_{4}}{p_{t}^{9}}$
(298)
$\displaystyle-\frac{24}{\ell^{2}}\vec{p}_{3}\cdot\vec{p}_{4}p_{1}^{2}p_{2}^{2}\frac{\left(p_{1}^{2}+p_{2}^{2}+6p_{3}^{2}+6p_{4}^{2}+2p_{1}p_{2}+7(p_{1}+p_{2})(p_{3}+p_{4})+42p_{3}p_{4}\right)+(1,2)\leftrightarrow(3,4)}{p^{7}_{t}}$

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