††institutetext: Center for the Fundamental Laws of Nature, Harvard
University, 17 Oxford St., Cambridge, MA, 02139, USA.

# Open AdS/CFT via a Double Trace Deformation

Hao Geng<EMAIL_ADDRESS>
###### Abstract

A concrete model of extracting the physics from the bulk of a gravitational
universe is important to the study of quantum gravity and its possible
relationship with experiments. Such a model can be constructed in the AdS/CFT
correspondence by gluing a bath on the asymptotic boundary of the bulk anti-de
Sitter (AdS) spacetime. This bath models a laboratory and is described by a
quantum field theory. In the dual conformal field theory (CFT) description
this coupling is achieved by a double-trace deformation that couples the CFT
with the bath. This suggests that the physics observed by the laboratory is
fully unitary. In this paper, we analyze the quantum aspects of this model in
detail which conveys new lessons about the AdS/CFT correspondence, and we
discuss the potential usefulness of this model in understanding subregion
physics in a gravitational universe.

##  1 Introduction

Ever since the discovery of the AdS/CFT correspondence Maldacena:1997re ;
Witten:1998qj ; Gubser:1998bc , it is hoped that we could learn deep lessons
about quantum gravity from it. However, it turns out that the main difficulty
is describing a bulk observer and its experience in a reasonable way in the
AdS/CFT framework. This question is important as the observer is modeling an
experimentalist whose experience is of direct experimental relevance to
quantum gravity. However, the description of the observer cannot be so
complicated as otherwise it will significantly backreact on the bulk geometry
due to the gravitational effect, and meanwhile it cannot be so simple as
otherwise, the observer cannot do much to be of practical relevance (see
Almheiri:2012rt ; Mathur:2010kx ; Mathur:2011wg ; Papadodimas:2012aq ;
Jafferis:2020ora ; deBoer:2022zps ; Witten:2021unn ; Witten:2023qsv  for some
attempts to address this question). Besides this subtlety of properly defining
the observer with a decent level of complexity, a more basic issue in a
gravitational theory is how to define a bulk observer diffeomorphism
invariantly. These complications are not anything new, and they also appear in
gauge theories, for example quantum electrodynamics.111Though some properties
of gauge theories are encoded in extended gauge invariant operators like
Wilson lines. The usual lore is that the experimentally relevant data are the
scattering amplitudes that can be extracted in the asymptotic regime, where
the gauge interaction is weak, by putting a detector there. Hence, in quantum
gravity, we expect the same, i.e., we should think of the observer as being
located at the asymptotic boundary and extracting the relevant data from the
bulk to reconstruct the bulk physics using these data. In this paper we will
study an explicit model of such an observer constructed using the AdS/CFT
correspondence.

More specifically, we consider a gravitational theory in an asymptotically
AdSd+1 spacetime, which has a dual CFTd description that resides on the
asymptotic boundary of the AdSd+1. We model the observer by a
$(d+1)$-dimensional bath that is glued to the asymptotic boundary of the
AdSd+1 (see Rocha:2008fe ; Jana:2020vyx ; Loganayagam:2022zmq ; Shen:2023srk ;
Pelliconi:2023ojb ; Karch:2023wui  for early and recent studies on some
aspects of relevant models). This bath can be either gravitational or non-
gravitational. When it is non-gravitational, we model it by a
$(d+1)$-dimensional CFT living on a half Minkowski space whose boundary is
glued with the asymptotic boundary of the AdSd+1 (see Fig. 1). This gluing can
be more easily described using the dual CFTd where we put the CFTd on the
boundary of the bath CFTd+1 and couple them by a double-trace deformation

$S_{tot}=S_{\text{CFT${}_{d}$}}+S_{\text{CFT${}_{d+1}$}}+\int
d^{d}x\mathcal{O}_{1}(t,\vec{x})\mathcal{O}_{2}(t,\vec{x})\,,$ (1)

where $O_{1}(t,\vec{x})$ is a CFTd single-trace scalar primary operator and
$O_{2}(t,\vec{x})$ is a CFTd+1 single-trace scalar primary operator
extrapolated to the boundary. We will take the double-trace deformation to be
marginal (i.e., the sum of the conformal weights of
$\mathcal{O}_{1}(t,\vec{x})$ and $\mathcal{O}_{2}(t,\vec{x})$ satisfies
$\Delta_{1}+\Delta_{2}=d$). When the bath is gravitational (see Fig.2), we
will take its geometry to be another AdSd+1 which also has a dual description
as another CFTd which we will call CFT${}_{d}^{2}$. Then the gluing of this
AdSd+1 bath to the original AdSd+1 can also be described in the CFT language
by a double-trace deformation

$S_{tot}=S_{\text{CFT${}_{d}$}}+S_{\text{CFT${}_{d}^{2}$}}+\int
d^{d}x\mathcal{O}_{1}(t,\vec{x})\mathcal{O}_{2}(t,\vec{x})\,,$ (2)

where again $\mathcal{O}_{1}(t,\vec{x})$ is a single-trace scalar primary
operator of the CFTd and now $\mathcal{O}_{2}(t,\vec{x})$ is a single-trace
scalar primary operator of the CFT${}_{d}^{2}$ and we take this deformation to
be marginal.

We will see that in the dual AdSd+1 description, the double-trace deformation
is modifying the boundary condition of the free massive scalar field that
duals to $\mathcal{O}_{1}(t,\vec{x})$ Witten:2001ua . We will show that this
modified boundary condition induces several interesting quantum effects in the
AdSd+1 bulk. The first effect is that the bulk Hilbert space constructed from
the AdSd+1 scalar field (the Fock space) is twice as large as the case in the
standard AdS/CFT. This is easily understood if we notice that the bulk AdSd+1
is coupled to a bath, and hence the particles from the bath are free to enter
the AdSd+1 which, together with the original particles in the AdSd+1 double
(or square) the Hilbert space. This can be seen if we perform canonical
quantization with the modified boundary condition for the AdSd+1 scalar field.
The second effect is that the modified boundary condition of the bulk scalar
field disables us to define a diffeomorphism invariant path integral measure
for the scalar field. Hence, there could be a diffeomorphism anomaly that
should be compensated if we want to consistently couple the scalar field to
gravity. We will show that there is a natural mechanism to compensate this
diffeomorphism anomaly. This mechanism turns out to be a Stückelberg mechanism
to generate a mass for the graviton, and this suggests that the diffeomorphism
invariance is spontaneously broken due to the bath coupling. This is a rather
remarkable result as it shows that the gravitational theory is modified due to
the existence of the observer or because the gravitational universe is being
observed. To uncover all these effects, we start with clarifications of
several subtleties in the AdS/CFT correspondence.

$AdS_{d+1}$$Mink_{d+1}$ Figure 1: We couple the gravitational AdSd+1 universe
(the blue shaded region) with a nongravitational bath (the green shaded
region) by gluing them along the asymptotic boundary the AdSd+1 (the red
vertical line). The nongravitational bath is modeled by a $(d+1)$-dimensional
CFT living on a half Minkowski space which shares the same boundary as the
asymptotic boundary of the AdSd+1. We use the Poincaré coordinates in the
AdSd+1. The coupling is achieved in the dual CFT description by Equ. (1).
$AdS_{d+1}$$AdS_{d+1}$ Figure 2: We couple the gravitational AdSd+1 universe
(the blue shaded region) with a gravitational bath (the yellow shaded region)
by gluing them along the asymptotic boundary the AdSd+1 (the red vertical
line). The gravitational bath is modeled by another AdSd+1 which shares the
same asymptotic boundary with the original AdSd+1. We take the Poincaré
coordinates in both of the AdSd+1. The coupling is achieved in the dual CFT
description by Equ. (2).

Moreover, in the gravitational bath case the original AdSd+1 can be thought of
a subregion of a gravitational universe (which is the union of the original
AdSd+1 and the bath AdSd+1).222This perspective was adopted at a classical
level in Compere:2019bua ; Compere:2020lrt ; Fiorucci:2020xto  by considering
leaky boundary conditions instead of the standard reflective ones. Therefore,
the analysis presented here is potentially helpful in understanding subregion
physics in a gravitational universe.333See Donnelly:2016auv ; Speranza:2017gxd
; Kirklin:2018gcl ; Camps:2018wjf ; Camps:2019opl ; Bahiru:2022oas ;
Bahiru:2023zlc ; Jensen:2023yxy ; Folkestad:2023cze  for early and recent
attempts in setting up and understanding this difficult question.

This paper is organized as follows. In Sec.2 we review some basics of the
AdS/CFT correspondence such as the calculation of the boundary CFT correlator
using the AdS bulk and we clarify some subtleties that are relevant to the
quantum aspects of the bulk scalar field– the canonical quantization and the
path integral quantization. In Sec.3 we review and refine the double-trace
deformation in AdS/CFT Witten:2001ua ; Aharony:2001pa ; Berkooz:2002ug  by
providing a path integral derivation of the proposal in Witten:2001ua  and
check the consistency with the results in Sec.2. In Sec.4 we use the
techniques we developed in Sec.2 and Sec.3 to analyze the two aforementioned
models of coupling AdSd+1 to a bath and we show the first quantum effect– the
Hilbert space doubling. In Sec.5 we discuss the second quantum effect–the
issue of diffeomorphism invariance broken and restoration. In Sec.6 we discuss
how the model we are considering in this paper could be potentially useful to
the study of subregion physics in a gravitational universe. In Sec.7 we
conclude the paper with a summary. In Appendix.A we discuss a potential
loophole in our consideration in Sec.5 and its resolution.

##  2 Quantization Schemes and Propagating Modes in AdS/CFT

In this section, we review the computation of the generating functional for
correlators of the dual single-trace scalar primary operator in the CFTd from
the massive scalar field in the AdSd+1. Then we analyze the bulk propagating
modes of the scalar field carefully distinguishing the on-shell and off-shell
modes. For simplicity we consider a free minimally coupled massive scalar
field $\phi(x)$ and our treatment will be in the framework of Berkooz:2002ug .

We take the geometry of AdSd+1 to be in the Poincaré patch of the Lorentzian
signature:

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=\frac{z^{2}+\sum_{i=1}^{d}\eta_{ab}dx^{a}dx^{b}}{z^{2}}\,,$
(3)

and the action for the scalar field $\phi(x)$ is

$S_{0}=\frac{1}{2}\int\sqrt{-g}dzdtd^{d-1}\vec{x}\phi(x)\Big{(}\Box-m^{2}\Big{)}\phi(x)\,,$
(4)

where
$\Box=\frac{1}{\sqrt{-g}}\partial_{\mu}(g^{\mu\nu}\sqrt{-g}\partial_{\nu})$.
The asymptotic boundary of AdSd+1, where the dual CFTd lives, is at
$z\rightarrow 0$ and we will take it to be at $z=\epsilon$ and send
$\epsilon\rightarrow 0$ at the end of our calculation. We emphasize on the
calculational details and when we are able to perform the alternative
quantization.

According to the AdS/CFT correspondence Maldacena:1997re ; Gubser:1998bc ;
Witten:1998qj , the scalar field $\phi(x)=\phi(z,t,\vec{x})$ duals to a
single-trace scalar primary operator $\mathcal{O}(t,\vec{x})$ in the CFTd and
imposing a boundary condition $J(t,\vec{x})$ for the scalar field $\phi(x)$ is
equivalent to turning on a source for $\mathcal{O}(t,\vec{x})$ (see later
discussions for details). Moreover, this duality states that

$Z[J]_{\text{CFT}}\equiv\langle e^{i\int
dtd^{d-1}\vec{x}J(t,\vec{x})\mathcal{O}(t,\vec{x})}\rangle_{\text{CFT}}=Z[J]_{AdS_{d+1}}\,,$
(5)

where $Z[J]_{\text{CFT}}$ is the generating functional for correlators of the
operator $\mathcal{O}(t,\vec{x})$ on the CFTd side and $Z[J]_{AdS_{d+1}}$ is
the partition function of the full gravitational theory (string theory) in
AdSd+1 with the specified boundary condition $J(t,\vec{x})$ for the scalar
field $\phi(x)$. For our purpose (in the low energy regime and with a weak
gravitational coupling), $Z[J]_{AdS_{d+1}}$ can be approximated by the
partition function of the scalar field $\phi(x)$ in the fixed AdSd+1
background

$Z[J]_{AdS_{d+1}}=\int D[\phi;J]e^{-iS_{J}[\phi]}\,,$ (6)

where we restrict ourselves to consider scalar field configurations satisfying
the boundary condition $J(t,\vec{x})$ and $S_{J}[\phi]$ is given by $S_{0}$
(in Equ. (4)) plus appropriate boundary terms. The boundary terms will be
$J(t,\vec{x})$ dependent. Since we are considering a free field $\phi(x)$ the
calculation reduces to the computation of the on-shell action $S^{\text{on-
shell}}[\phi]$ with boundary condition $J(t,\vec{x})$

$Z[J]_{\text{CFT}}\equiv\langle e^{i\int
dtd^{d-1}\vec{x}J(t,\vec{x})\mathcal{O}(t,\vec{x})}\rangle_{\text{CFT}}=Z[J]_{AdS_{d+1}}\sim
e^{-iS^{\text{on-shell}}_{J}[\phi_{J}]}\,,$ (7)

where $\phi_{J}(x)$ satisfies the boundary condition $J(t,\vec{x})$ (to be
specified later) and the equation of motion

$(\Box-m^{2})\phi_{J}(x)=0\,.$ (8)

###  2.1 The Meaning of Quantization

Before we discuss different quantization schemes in AdS/CFT in this section,
let’s articulate the meaning of quantization. In the standard treatment of a
quantum system, quantization means constructing the Hilbert space of the
system or specifying the rule to calculate all the transition amplitudes.

The former is usually done for a free theory using the canonical quantization
for which in our case we have to solve the equation Equ. (4) with vanishing
$J(t,\vec{x})$, find all the solutions that satisfy the boundary condition
$J(t,\vec{x})=0$, and we will call these solutions as on-shell modes, then
assign creation and annihilation operators for each of the modes and their
complex conjugates. Schematically, let’s denote these modes in the Fourier
space by $\delta\phi_{n}(z,\omega_{n,\vec{k}},\vec{k})$ and the field operator
$\hat{\phi}(z,t,\vec{x})$ is given by

$\hat{\phi}(z,t,\vec{x})=\int\frac{d\vec{k}}{(2\pi)^{d-1}}\sum_{n}\frac{1}{\sqrt{2\omega_{n,\vec{k}}}}\Big{(}\delta\phi(z,\omega_{n,\vec{k}},\vec{k})\hat{a}_{n\vec{k}}e^{i\omega_{n,\vec{k}}t-i\vec{k}\cdot\vec{x}}+\delta\phi^{*}(z,\omega_{n,\vec{k}},\vec{k})\hat{a}^{\dagger}_{n\vec{k}}e^{-i\omega_{n,\vec{k}}t+i\vec{k}\cdot\vec{x}}\Big{)}\,.$
(9)

Then imposing the canonical quantization condition

$[\hat{\phi}(z,t,\vec{x}),g^{tt}\partial_{t}\hat{\phi}(z^{\prime},t,\vec{x^{\prime}})]=\frac{i}{\sqrt{-g}}\delta(z-z^{\prime})\delta^{d-1}(\vec{x}-\vec{x^{\prime}})\,,$
(10)

we will get the standard commutation relations for the creation and
annihilation operators

$[\hat{a}_{n,\vec{k}},\hat{a}_{n^{\prime},\vec{k^{\prime}}}]=0\,,\quad[\hat{a}_{n,\vec{k}},\hat{a}^{\dagger}_{n^{\prime},\vec{k^{\prime}}}]=\delta_{nn^{\prime}}\delta^{d-1}(\vec{k}-\vec{k^{\prime}})\,,$
(11)

given that the on-shell modes are orthonormal in the so-called Klein-Gordon
measure

$(\delta\phi,\delta\phi)_{\text{KG}}=i\int_{\Sigma}dzd^{d-1}x\sqrt{-g}g^{tt}(\delta\phi^{*}\partial_{t}\delta\phi-\delta\phi\partial_{t}\delta\phi^{*})(z,t,\vec{x})\,.$
(12)

In the end, the Hilbert space is constructed as the Fock space of these
annihilation and creation operators.

The later is asking for a calculational scheme for correlators of the theory
which is usually done using the path integral formalism. The correlators are
easily generated once we know the generating functional Equ. (7). In our case,
we have to compute the path integral of a free scalar field in the AdSd+1 with
a specified boundary condition $J(t,\vec{x})$. Since we are considering a free
scalar field, the $J(t,\vec{x})$-dependent part of this calculation reduces to
the computation of the on-shell action $S_{J}^{\text{on-shell}}[\phi_{J}]$
with the boundary condition $J(t,\vec{x})$. Nevertheless, the full partition
function contains a one-loop correction which though doesn’t contribute to the
generating functional as it is $J(t,\vec{x})$-independent. Since we will study
quantum effects later, we keep track of everything. We have

$\begin{split}Z[J]_{\text{AdS}_{d+1}}&=\int D[\phi;J]e^{-iS_{J}[\phi]}\\\
&=\int
D[\delta\phi]e^{-i\frac{1}{2}\int\sqrt{-g}d^{d+1}x\delta\phi(\Box-m^{2})\delta\phi}e^{-iS^{\text{on-
shell}}_{J}[\phi_{J}]}\\\
&=\text{det}^{-\frac{1}{2}}(\Box-m^{2})e^{-iS^{\text{on-
shell}}_{J}[\phi_{J}]}\,,\end{split}$ (13)

where the $\delta\phi$ describes the fluctuations above the background on-
shell solution and $\delta\phi$ satisfies a boundary condition with vanishing
$J(t,\vec{x})$ (the precise boundary condition will be discussed later). The
determinant is a symbolic representation of the path integral over the
fluctuations. This path integral can be performed as follows. Firstly, we
solve for the eigenmodes
($\delta\phi_{\lambda}(z,\omega_{\lambda,n,\vec{k}},\vec{k})e^{-i\omega_{\lambda,n,\vec{k}}t-i\vec{k}\cdot\vec{x}}$)
associated with all the nonzero eigenvalues $\lambda$ (in fact we need
$\lambda>0$, see the next subsections) of the operator $\Box-m^{2}$

$(\Box-m^{2})\delta\phi_{\lambda}(z,\omega_{\lambda,n,\vec{k}},\vec{k})e^{-i\omega_{\lambda,n,\vec{k}}t+i\vec{k}\cdot\vec{x}}=\lambda\delta\phi_{\lambda}(z,\omega_{\lambda,n,\vec{k}},\vec{k})e^{-i\omega_{\lambda,n,\vec{k}}t+i\vec{k}\cdot\vec{x}}.$
(14)

These modes are orthonormal for both $n$ and $\lambda$

$\int\sqrt{-g}d^{d+1}x\delta\phi^{*}_{\lambda}(z,\omega_{\lambda,n,\vec{k}},\vec{k})\delta\phi_{\lambda^{\prime}}(z,\omega_{\lambda^{\prime},n^{\prime},\vec{k^{\prime}}})e^{-i(\omega_{\lambda^{\prime},n^{\prime},\vec{k^{\prime}}}-\omega_{\lambda,n,\vec{k}})t+i(\vec{k^{\prime}}-\vec{k})\cdot\vec{x}}=\delta_{\lambda\lambda^{\prime}}\delta_{nn^{\prime}}\delta^{d-1}(\vec{k}-\vec{k^{\prime}}).$
(15)

As we will discuss in detail in the next section, the normalizability of these
modes is important for determining the proper boundary conditions or the fall-
off behaviors for them when $z\rightarrow 0$. Then we use these modes to
expand a generic function $\delta\phi(z,t,\vec{x})$ with the appropriate
boundary condition as

$\delta\phi(z,t,\vec{x})=\int\frac{d\vec{k}}{(2\pi)^{d-1}}a_{0,n,\vec{k}}\delta\phi_{\lambda}(z,\omega_{n,\vec{k}},\vec{k})e^{-i\omega_{\lambda,n,\vec{k}}t+i\vec{k}\cdot\vec{x}}\\\
+\int\frac{d\vec{k}}{(2\pi)^{d-1}}\sum_{\lambda,n}a_{\lambda,n,\vec{k}}\delta\phi_{\lambda}(z,\omega_{\lambda,n,\vec{k}},\vec{k})e^{-i\omega_{\lambda,n,\vec{k}}t+i\vec{k}\cdot\vec{x}}\,,$
(16)

where $a_{\lambda,n,\vec{k}}$ are complex numbers that ensure the reality of
$\delta\phi(z,t,\vec{x})$. Now we can compute

$\begin{split}\int D\delta\phi
e^{-i\frac{1}{2}\int\sqrt{-g}d^{d+1}x\delta\phi(\Box-m^{2})\delta\phi}=&\int\Pi_{\lambda,n,\vec{k}}da_{\lambda,n,\vec{k}}e^{-i\frac{1}{2}\sum_{\lambda,n,\vec{k}}|a_{\lambda,n,\vec{k}}|^{2}\lambda}\\\
=&\Pi_{\lambda,n,\vec{k}}(\frac{1}{\sqrt{i\lambda}})\\\
=&\text{det}^{-\frac{1}{2}}(\Box-m^{2})\,,\end{split}$ (17)

where for the simplicity of notation we have discretized the $\vec{k}$ by
putting the $\vec{x}$ in a box. The important step is the second step for
which we transform the functional integration measure $D\delta\phi$ to the
integral of the coefficients $\Pi_{\lambda,n,\vec{k}}da_{\lambda,n\vec{k}}$.
This step in general requires a Jacobian which is an overall constant (or an
ambiguity in the definition of a functional integration measure) and is fixed
by

$\int D\delta\phi e^{-\int\sqrt{-g}d^{d+1}x\delta\phi^{2}}=1\,.$ (18)

Here we notice a subtlety that the on-shell modes part $a_{0,n,\vec{k}}$ don’t
contribute to the integrand in Equ. (17) but they do contribute to the
functional integral Equ. (18) in the definition of the path integral measure
when the boundary source is zero. This subtlety is important for our later
study in Sec.5 and we will ponder it till Sec.5.

###  2.2 Standard Quantization in AdS/CFT

In this subsection, we review the standard quantization scheme for a massive
free scalar field in AdSd+1. Such a scalar field is dual to a single-trace
scalar primary operator $O(t,\vec{x})$ in the boundary CFTd. We review the
computation of generating functional for this single-trace operator using the
AdSd+1 description Equ. (7) and the behavior of the on-shell and off-shell
modes near $z\rightarrow 0$ which are relevant respectively to the
construction of the Hilbert space and the quantum correction to the bulk
partition function.

####  2.2.1 Partition Function

The equation of motion Equ. (8) comes from setting the variation of the action
Equ. (4) to zero. However, there are additional boundary terms in the
variation (suppose that the equation of motion has been satisfied)

$\delta
S_{0}=-\frac{1}{2}\int_{z=\epsilon}d^{d}\vec{x}z^{-d+1}\Big{(}\delta\phi\partial_{z}\phi-\phi\partial_{z}\delta\phi\Big{)}\,.$
(19)

Adding a specific boundary term $S_{bdy}$ to the action $S_{0}$ and setting
$\delta S_{0}+\delta S_{bdy}=0$ specify the boundary condition for the scalar
field $\phi(x)$. For the standard quantization we will choose

$S_{bdy}=-\epsilon^{-\Delta_{+}}\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{+}+1-\frac{d}{2}]}{\Gamma[\Delta_{+}]}\int
dtd^{d-1}\vec{x}\phi(\epsilon,t,\vec{x})J(t,\vec{x})\,,$ (20)

which gives the boundary condition $J(t,\vec{x})$:

$z^{-\Delta_{-}}(z\partial_{z}-\Delta_{+})\phi(x)|_{z=\epsilon\rightarrow
0}=-2\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{+}+1-\frac{d}{2}]}{\Gamma[\Delta_{+}]}J(t,\vec{x})\,,$
(21)

for a fixed $J(t,\vec{x})$ and a generic fluctuation mode should satisfy

$z^{-\Delta_{-}}(z\partial_{z}-\Delta_{+})\delta\phi(x)=0\,,\quad\text{as
$z\rightarrow 0$}\,,$ (22)

where we have defined
$\Delta_{\pm}=\frac{d}{2}\pm\sqrt{\frac{d^{2}}{4}+m^{2}}$.

A solution of the equation of motion $\phi_{J}(x)$ which is regular in
interior of the AdSd+1 Witten:1998qj  and satisfies the boundary condition
Equ. (21) is

$\phi_{J}(x)=\int
dt^{\prime}d^{d-1}\vec{x^{\prime}}\frac{J(t^{\prime},\vec{x^{\prime}})z^{\Delta_{+}}}{(-(t-t^{\prime})^{2}+z^{2}+|\vec{x}-\vec{x^{\prime}}|^{2})^{\Delta_{+}}}\,,$
(23)

which is uniquely specified.444This explains the statement in Sec.2.1 that the
boundary condition $J(t,\vec{x})$ fixed the on-shell profile and hence the on-
shell modes don’t contribute to the path integral. To check that Equ. (23)
satisfies the boundary condition Equ. (21) it is important to know that

$\lim_{z\rightarrow
0}\frac{z^{2\Delta-d}}{(z^{2}-(t-t^{\prime})^{2}+|\vec{x}-\vec{x^{\prime}}|^{2})^{\Delta}}=\pi^{\frac{d}{2}}\frac{\Gamma[\Delta-\frac{d}{2}]}{\Gamma[\Delta]}\delta(t-t^{\prime})\delta^{(d-1)}(\vec{x}-\vec{x^{\prime}})\,,\quad\text{for
}\quad\Delta>\frac{d}{2}\,.$ (24)

Now we can evaluate the on-shell action

$\begin{split}S_{J}^{\text{on-
shell}}[\phi_{J}]&=S_{0}[\phi_{J}]+S_{bdy}[\phi_{J}]\\\
&=-\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{+}+1-\frac{d}{2}]}{\Gamma[\Delta_{+}]}\int
dtd^{d-1}\vec{x}dt^{\prime}d^{d-1}\vec{x^{\prime}}\frac{J(t^{\prime},\vec{x^{\prime}})J(t,\vec{x})}{(-(t-t^{\prime})^{2}+|\vec{x}-\vec{x^{\prime}}|^{2})^{\Delta_{+}}}\,,\end{split}$
(25)

and then we get the expected generating functional from Equ. (7)

$Z[J]_{\text{CFT}}=e^{i\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{+}+1-\frac{d}{2}]}{\Gamma[\Delta_{+}]}\int
dtd^{d-1}\vec{x}dt^{\prime}d^{d-1}\vec{x^{\prime}}\frac{J(t^{\prime},\vec{x^{\prime}})J(t,\vec{x})}{(-(t-t^{\prime})^{2}+|\vec{x}-\vec{x^{\prime}}|^{2})^{\Delta_{+}}}}\,,$
(26)

which is indeed the CFT generating functional for a conformal primary operator
of conformal weight $\Delta_{+}$.

####  2.2.2 On-Shell and Off-Shell Modes

We learned in the previous discussion that the on-shell modes are relevant to
the canonical quantization and the off-shell modes are relevant to the
calculation of the quantum corrections to the partition function. Moreover,
they are normalizable under two different measures Equ. (12) and Equ. (15).
Here we will give a detailed study of their properties near the asymptotic
boundary $z\rightarrow 0$. We study the on-shell modes first and then the off-
shell modes.

To understand the on-shell modes we consider a solution of the scalar field
equation of motion

$(\Box-m^{2})\delta\phi(x)=0\,.$ (27)

In the Lorentzian signature, a general solution of Equ. (27) is given by
555Here we remember that there is a subtlety as emphasized in
Balasubramanian:1998sn  that the solutions displayed in Equ. (28) are those
for $\sqrt{\frac{d^{2}}{4}+m^{2}}$ nonintegral. We will ignore this subtlety
as we only care about the asymptotic behavior as $z\rightarrow 0$.

$\delta\phi(x)_{\omega,\vec{k}}^{\pm}=e^{-i\omega
t+i\vec{k}\cdot\vec{x}}z^{\frac{d}{2}}J_{\pm\sqrt{\frac{d^{2}}{4}+m^{2}}}(z\sqrt{\omega^{2}-\vec{k}^{2}})\,,$
(28)

where $\omega^{2}-\vec{k}^{2}\geq 0$ and we used the Bessel’s functions of the
first kind $J_{\pm\alpha}$ for $\alpha=\sqrt{\frac{d^{2}}{4}+m^{2}}$. We
therefore have the asymptotic behavior

$\delta\phi(x)_{\omega,\vec{k}}^{\pm}\sim z^{\Delta_{\pm}},\quad\text{as}\quad
z\rightarrow 0\,.$ (29)

As we discussed in Sec.2.1, the on-shell modes should be normalizable (finite)
in the Klein-Gordon measure. The finiteness of the Klein-Gordon measure
requires a finite asymptotic behavior

$(\delta\phi^{\pm},\delta\phi^{\pm})_{\text{KG}}=i\int_{\Sigma}dzd^{d-1}x\sqrt{-g}g^{tt}(\delta\phi^{*\pm}\partial_{t}\delta\phi^{\pm}-\delta\phi^{\pm}\partial_{t}\delta\phi^{*\pm})(z,\vec{x})\\\
\sim\int dzz^{-d+1}z^{2\Delta_{\pm}}\,,\quad\text{near}\quad z=0\,,$ (30)

where we have used Equ. (29). We can see that this finiteness condition
requires that $\Delta_{\pm}>\frac{d-2}{2}$ and is trivially satisfied for
$\Delta_{+}$. The standard quantization is the scheme that projects out
$\delta\phi^{-}$. Hence in this quantization scheme the on-shell modes satisfy
the following fall-off behavior

$\delta\phi_{\text{on-shell}}(x)\sim z^{\Delta_{+}}\,,\quad\text{as
$z\rightarrow 0$}\,.$ (31)

Now let’s study the off-shell modes. An off-shell mode satisfies the equation

$(\Box-m^{2})\delta\phi(x)=\lambda\delta\phi(x)\,,$ (32)

and the normalizability condition

$||\delta\phi||^{2}=\int\sqrt{-g}d^{d+1}x|\delta\phi(x)|^{2}=1\,.$ (33)

Another condition is that we should have

$\begin{split}S_{J}[\phi_{J}+\delta\phi]&=S_{0}[\phi_{J}+\delta\phi]+S_{bdy}[\phi_{J}+\delta\phi]\\\
&=\frac{1}{2}\int\sqrt{-g}d^{d+1}x\delta\phi(\Box-m^{2})\delta\phi+S_{J}^{\text{on-
shell}}[\phi_{J}]\,,\end{split}$ (34)

which ensures that Equ. (13) is consistent. This requires that Equ. (22) is
satisfied and

$S_{bdy}[\delta\phi]=-\epsilon^{-\Delta_{+}}\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{+}+1-\frac{d}{2}]}{\Gamma[\Delta_{+}]}\int
dtd^{d-1}\vec{x}\delta\phi(\epsilon,t,\vec{x})J(t,\vec{x})\rightarrow 0\,,$
(35)

as $\epsilon\rightarrow 0$. Similar to the on-shell case, a general solution
of Equ. (32) satisfies

$\delta\phi(x)^{\pm}\sim z^{\Delta_{\pm}(\lambda)}\,,\quad\text{as
$z\rightarrow 0$}\,,$ (36)

where
$\Delta_{\pm}(\lambda)=\frac{d}{2}\pm\frac{1}{2}\sqrt{d^{2}+4m^{2}+4\lambda}$.
Then the normalizability condition Equ. (33) requires that $\delta^{-}$ is
projected out for which Equ. (22) is automatically satisfied and Equ. (35)
sets $\lambda>0$. Hence, in the standard quantization the off-shell modes are
constrained to satisfy

$\delta\phi_{\text{off-shell}}(z)\sim z^{\Delta_{+}(\lambda)}\,,\quad\text{as
$z\rightarrow 0$}\,,$ (37)

and $\lambda>0$.

With the knowledge of the on-shell modes, we can construct the Hilbert space
by doing the canonical quantization Equ. (10). The field operator
$\hat{\phi}$(x) is given by

$\hat{\phi}(x)=\int\frac{d\vec{k}}{(2\pi)^{d-1}}\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\omega}{\sqrt{2\omega}}\Big{(}\delta\phi(x)_{\omega,\vec{k}}^{+}\hat{a}^{\dagger}_{\omega,\vec{k}}+\delta\phi(x)^{+*}_{\omega,\vec{k}}\hat{a}_{\omega,\vec{k}}\Big{)}\,,$
(38)

where $\hat{a}^{\dagger}_{\omega,\vec{k}}$ and $\hat{a}_{\omega,\vec{k}}$ are
the creation and annihilation operators. They satisfy the standard commutation
relations

$[\hat{a}_{\omega,\vec{k}},\hat{a}^{\dagger}_{\omega,\vec{k^{\prime}}}]=\delta(\omega-\omega^{\prime})\delta^{d-1}(\vec{k}-\vec{k^{\prime}})\,.$
(39)

With the knowledge of the off-shell modes, we can understand the quantum
correction to the partition function Equ. (26). A general field configuration
can be expanded as

$\begin{split}\phi(x)=&\phi_{J}(x)+\sum_{\lambda>0}\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\int d\omega
a_{\lambda,\omega,\vec{k}}\delta\phi_{\lambda}(z,\omega,\vec{k})e^{-i\omega
t-i\vec{k}\cdot\vec{x}}\,,\end{split}$ (40)

where we note that we don’t have on-shell fluctuation modes as the on-shell
profile has been fixed by $\phi_{J}(x)$. The action evaluated on this
configuration is

$\begin{split}S[\phi]=&S_{0}[\phi]+S_{bdy}[\phi]\\\
=&-\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{+}+1-\frac{d}{2}]}{\Gamma[\Delta_{+}]}\int
dtd^{d-1}\vec{x}dt^{\prime}d^{d-1}\vec{x^{\prime}}\frac{J(t^{\prime},\vec{x^{\prime}})J(t,\vec{x})}{(-(t-t^{\prime})^{2}+|\vec{x}-\vec{x^{\prime}}|^{2})^{\Delta_{+}}}\\\
&+\frac{1}{2}\sum_{\lambda>0}\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d^{d-1}\vec{k}}{(2\pi)}d\omega\lambda|a_{\lambda,\omega,\vec{k}}|^{2}\,.\end{split}$
(41)

The first term gives us the result Equ. (26) and the second term describes the
quantum fluctuations. The quantum correction to the partition function is
obtained by the Gaussian integral over the coefficients
$a_{\lambda,\omega,\vec{k}}$ as in Equ. (13) and Equ. (17).

###  2.3 Alternative Quantization in AdS/CFT

####  2.3.1 Review of the Klebanov-Witten Proposal

Before we carry out a general analysis for the alternative quantization, let’s
review the Klebanov-Witten proposal for the partition function Klebanov:1999tb
. Klebanov and Witten suggested that when the mass square of the AdSd+1 scalar
satisfies the condition that $\Delta_{-}>\frac{d-2}{2}$ (which is the
unitarity bound of primary operators in CFTd Rychkov:2016iqz ; Minwalla:1997ka
) we can perform the so-called alternative quantization such that the dual
single-trace scalar primary operator in CFTd has conformal weight
$\Delta_{-}$. They also suggested that the generating functional in this case
is a Legendre transform of that in the standard quantization. Namely, we have

$Z[J^{\prime}]_{\text{CFT}}=\int
DJe^{i\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{+}+1-\frac{d}{2}]}{\Gamma[\Delta_{+}]}\int
dtd^{d-1}\vec{x}dt^{\prime}d^{d-1}\vec{x^{\prime}}\frac{J(t^{\prime},\vec{x^{\prime}})J(t,\vec{x})}{(-(t-t^{\prime})^{2}+|\vec{x}-\vec{x^{\prime}}|^{2})^{\Delta_{+}}}+\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{+}+1-\frac{d}{2}]}{\Gamma[\Delta_{+}]}\int
dtd^{d-1}\vec{x}J(t,\vec{x})J^{\prime}(t,\vec{x})}\,,$ (42)

which can be easily shown, by going to Fourier space, to be

$Z[J^{\prime}]_{\text{CFT}}=e^{-i\frac{(2\Delta_{-}-d)\Gamma[\Delta_{-}]}{\pi^{\frac{d}{2}}\Gamma[\Delta_{-}-\frac{d}{2}]}\int
dtd^{d-1}\vec{x}dt^{\prime}d^{d-1}\vec{x^{\prime}}\frac{J^{\prime}(t,\vec{x})J^{\prime}(t^{\prime},\vec{x^{\prime}})}{(-(t-t^{\prime})^{2}+|\vec{x}-\vec{x^{\prime}}|^{2})^{\Delta_{-}}}}\,,$
(43)

as a legitimate CFTd generating functional for correlators of a scalar primary
operator of conformal weight $\Delta_{-}$.

This calculation can be nicely interpreted in the following way. We first
perform the standard quantization with a fixed source (i.e. boundary condtion)
$J(t,\vec{x})$ while turning on an on-shell fluctuation mode
$\delta\phi(z,\vec{x})$ which satisfies

$\delta\phi(x)\sim z^{\Delta_{+}}J^{\prime}(t,\vec{x})\,,\quad\text{as
$z\rightarrow 0$}\,,$ (44)

with the boundary condition Equ. (22) automatically satisfied. Then we
evaluate the resulting on-shell action (or perform the path integral with the
prescribed boundary condition Equ. (21)) we get

$\begin{split}Z[J,J^{\prime}]_{\text{CFT}}&=e^{-S^{\text{on-
shell}}[\phi_{J}+\delta\phi]}\\\
&=e^{i\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{+}+1-\frac{d}{2}]}{\Gamma[\Delta_{+}]}\int
dtd^{d-1}\vec{x}dt^{\prime}d^{d-1}\vec{x^{\prime}}\frac{J(t^{\prime},\vec{x^{\prime}})J(t,\vec{x})}{(-(t-t^{\prime})^{2}+|\vec{x}-\vec{x^{\prime}}|^{2})^{\Delta_{+}}}+i\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{+}+1-\frac{d}{2}]}{\Gamma[\Delta_{+}]}\int
dtd^{d-1}\vec{x}J(t,\vec{x})J^{\prime}(t,\vec{x})}\,,\end{split}$ (45)

where we ignored the quantum correction which is not relevant to the CFT
generating functional. Finally, we integrate over the boundary source
$J(\vec{x})$ and get Equ. (43).

In summary, at the level of the partition function, the alternative
quantization can be obtained from the standard quantization by turning on a
profile of an on-shell mode Equ. (44) as a new piece of the background on top
of $\phi_{J}$ and additionally doing the path integral also over the boundary
source $J(\vec{x})$. This directly raises the question that what the fall-off
behaviors of the on-shell and off-shell modes are in the alternative
quantization scheme.

In the rest of this section, we will see that the formalism we used for the
discussion of the standard quantization can be easily modified and applied for
the analysis in the alternative quantization.

####  2.3.2 Partition Function

In this subsection we will show that the formalism we used in Sec.2.2.1 also
works in the computation of the partition function (i.e. the generating
functional for CFT correlators) for the alternative quantization precisely
when $\Delta_{-}>\frac{d-2}{2}$.

Similar as before, we have to solve the equation of motion with a boundary
condition $J^{\prime}(\vec{x})$

$(\Box-m^{2})\phi_{J^{\prime}}(\vec{x})=0\,,$ (46)

and the boundary condition is specified by setting the variation of the whole
action $\delta S_{0}+\delta S_{bdy}$ to zero. However, now in the alternative
quantization scheme we will use the following boundary term

$S_{bdy}=-\epsilon^{-\Delta_{-}}\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{-}+1-\frac{d}{2}]}{\Gamma[\Delta_{-}]}\int
dtd^{d-1}\vec{x}\phi(\epsilon,t,\vec{x})J^{\prime}(t,\vec{x})\,,$ (47)

which gives the boundary condition $J^{\prime}(t,\vec{x})$ from $\delta
S_{0}+\delta S_{bdy}=0$:

$z^{-\Delta_{+}}(z\partial_{z}-\Delta_{-})\phi(x)|_{z=\epsilon\rightarrow
0}=-2\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{-}+1-\frac{d}{2}]}{\Gamma[\Delta_{-}]}J^{\prime}(t,\vec{x})\,,$
(48)

for a fixed $J^{\prime}(t,\vec{x})$ and a generic fluctuation should satisfy

$z^{-\Delta_{+}}(z\partial_{z}-\Delta_{-})\delta\phi(x)=0\,,\quad\text{as
$z\rightarrow 0$}\,,$ (49)

where again we have defined
$\Delta_{\pm}=\frac{d}{2}\pm\sqrt{\frac{d^{2}}{4}+m^{2}}$.

A solution of the equation of motion $\phi_{J^{\prime}}(x)$ which is regular
in interior of the AdSd+1 Witten:1998qj  and satisfies the boundary condition
Equ. (48) is

$\phi_{J^{\prime}}(x)=\int
dt^{\prime}d^{d-1}\vec{x^{\prime}}\frac{J^{\prime}(t^{\prime},\vec{x^{\prime}})z^{\Delta_{-}}}{(z^{2}-(t-t^{\prime})^{2}+|\vec{x}-\vec{x^{\prime}}|^{2})^{\Delta_{-}}}\,.$
(50)

To check that Equ. (50) satisfies the boundary condition Equ. (21) it is
important to know that

$\lim_{z\rightarrow
0}\frac{z^{2\Delta-d}}{(z^{2}-(t-t^{\prime})^{2}+|\vec{x}-\vec{x^{\prime}}|^{2})^{\Delta}}=\pi^{\frac{d}{2}}\frac{\Gamma[\Delta-\frac{d}{2}]}{\Gamma[\Delta]}\delta(t-t^{\prime})\delta^{(d-1)}(\vec{x}-\vec{x^{\prime}})\,,\quad\text{for
}\quad\Delta>\frac{d}{2}\,,$ (51)

which imposes the constraint666The expansion of Equ. (50) for small $z$
behaves as
$\phi_{J^{\prime}}(z,t,\vec{x})=\Big{(}A_{J^{\prime}}(t,\vec{x})z^{\Delta-}+\mathcal{O}(z^{\Delta_{-}+2})\Big{)}+\Big{(}B_{J^{\prime}}(t,\vec{x})z^{\Delta_{+}}+z^{\Delta_{+}+2}\Big{)}\,,$
(52) where $A_{J^{\prime}}$ and $B_{J^{\prime}}$ are solely determined by
$J^{\prime}(t,\vec{x})$. To make sure that the boundary condition Equ. (48) is
satisfied we have to make sure that the $\mathcal{O}(z^{\Delta_{-}+2})$ terms
in the first bracket give zero contribution to the left hand side of Equ.
(48). Hence we need
$z^{\Delta_{-}+2-\Delta_{+}}=z^{-d+2+2\Delta_{-}}\rightarrow
0\,,\quad\text{as}\quad z\rightarrow 0\,.$ (53) This requires that
$\Delta_{-}>\frac{d-2}{2}$ which is exactly the same as the constraint Equ.
(54).

$\Delta_{-}>\frac{d-2}{2}\,.$ (54)

Now we can evaluate the on-shell action

$S^{\text{on-
shell}}[\phi_{J}]=S_{0}[\phi_{J}]+S_{bdy}[\phi_{J}]=-\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{-}+1-\frac{d}{2}]}{\Gamma[\Delta_{-}]}\int
d^{d}\vec{x}d^{d}\vec{x^{\prime}}\frac{J^{\prime}(\vec{x^{\prime}})J^{\prime}(\vec{x})}{|\vec{x}-\vec{x^{\prime}}|^{2\Delta_{-}}}\,,$
(55)

and then we get the expected CFTd generating functional

$Z[J^{\prime}]_{\text{CFT}}=e^{-i\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{-}+1-\frac{d}{2}]}{\Gamma[\Delta_{-}]}\int
dtd^{d-1}\vec{x}dt^{\prime}d^{d-1}\vec{x^{\prime}}\frac{J^{\prime}(t^{\prime},\vec{x^{\prime}})J^{\prime}(t,\vec{x})}{(-(t-t^{\prime})^{2}+|\vec{x}-\vec{x^{\prime}}|^{2})^{\Delta_{-}}}}\,,$
(56)

which is that for correlators of a conformal primary operator of conformal
weight $\Delta_{-}$.

####  2.3.3 On-Shell and Off-Shell Modes

In analogy to Sec.2.2.2, we will study the properties of the on-shell and off-
shell modes near the asymptotic boundary $z\rightarrow 0$.

The on-shell modes satisfy the equation of motion

$(\Box-m^{2})\delta\phi(x)=0\,,$ (57)

for which a general solution is given by

$\delta\phi(z,\omega,\vec{k})^{\pm}=e^{-i\omega
t+i\vec{k}\cdot\vec{x}}z^{\frac{d}{2}}J_{\pm\sqrt{\frac{d^{2}}{4}+m^{2}}}(z\sqrt{\omega^{2}-k^{2}})\,,$
(58)

where $\omega^{2}-\vec{k}^{2}\geq 0$ and the fluctuation $\delta\phi^{\pm}$
has the asymptotic behavior

$\delta\phi^{\pm}(z,\omega,\vec{k})\sim z^{\Delta_{\pm}},\quad\text{as}\quad
z\rightarrow 0\,.$ (59)

Moreover the Klein-Gordon norm

$\begin{split}(\delta\phi^{\pm},\delta\phi^{\pm})_{\text{KG}}&=i\int_{\Sigma}dzd^{d-1}x\sqrt{-g}g^{tt}(\delta\phi^{\pm*}\partial_{t}\delta\phi^{\pm}-\delta\phi^{\pm}\partial_{t}\delta\phi^{\pm*})(z,\vec{x})\\\
&\sim\int dzz^{-d+1}z^{2\Delta_{\pm}}\,,\quad\text{near}\quad
z=0\,,\end{split}$ (60)

is finite for both $\Delta_{\pm}$ if $\Delta_{-}>\frac{d-2}{2}$. As opposed to
the standard quantization, the alternative quantization projects out the
$\Delta_{+}$ modes. Hence the on-shell modes satisfy

$\delta\phi_{\text{on-shell}}(x)\sim z^{\Delta_{-}}\,,\quad\text{as
$z\rightarrow 0$}\,.$ (61)

Let’s now study the off-shell modes. Similar to the standard quantization
case, Equ. (49) should be satisfied for the stability of the on-shell
configuration. This projects out the $\delta\phi^{-}$ modes and requires that
$\lambda>0$. Moreover the normalizability of the measure Equ. (33) and the
vanishing of

$S_{bdy}[\delta\phi]=-\epsilon^{-\Delta_{+}}\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{+}+1-\frac{d}{2}]}{\Gamma[\Delta_{+}]}\int
d^{d}\vec{x}\delta\phi(\epsilon,\vec{x})J^{\prime}(\vec{x})\,,$ (62)

as $\epsilon\rightarrow 0$ are trivially satisfied for $\delta\phi^{+}$ modes.
Hence in the alternative quantization the off-shell modes are constrained to
satisfy

$\delta\phi_{\text{off-shell}}(x)\rightarrow
z^{\Delta_{+}(\lambda)}\,,\quad\text{as $z\rightarrow 0$}\,,$ (63)

and $\lambda>0$.

With the knowledge of the on-shell modes, we can construct the Hilbert space
by doing the canonical quantization Equ. (10). The field operator
$\hat{\phi}$(x) is given by

$\hat{\phi}(x)=\int\frac{d\vec{k}}{(2\pi)^{d-1}}\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\omega}{\sqrt{2\omega}}\Big{(}\delta\phi(x)_{\omega,\vec{k}}^{-}\hat{b}^{\dagger}_{\omega,\vec{k}}+\delta\phi(x)^{-*}_{\omega,\vec{k}}\hat{b}_{\omega,\vec{k}}\Big{)}\,,$
(64)

where $\hat{b}^{\dagger}_{\omega,\vec{k}}$ and $\hat{b}_{\omega,\vec{k}}$ are
the creation and annihilation operators. They satisfy the standard commutation
relations

$[\hat{b}_{\omega,\vec{k}},\hat{b}^{\dagger}_{\omega,\vec{k^{\prime}}}]=\delta(\omega-\omega^{\prime})\delta^{d-1}(\vec{k}-\vec{k^{\prime}})\,.$
(65)

With the knowledge of the off-shell modes, we can understand the quantum
correction to the partition function Equ. (56). A general field configuration
can be expanded as

$\begin{split}\phi(x)=&\phi_{J^{\prime}}(x)+\sum_{\lambda>0}\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\int d\omega
b_{\lambda,\omega,\vec{k}}\delta\phi_{\lambda}(z,\omega,\vec{k})e^{-i\omega
t+i\vec{k}\cdot\vec{x}}\,,\end{split}$ (66)

and the action evaluated on this configuration is

$\begin{split}S[\phi]&=S_{0}[\phi]+S_{bdy}[\phi]\\\
&=-\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{-}+1-\frac{d}{2}]}{\Gamma[\Delta_{-}]}\int
dtd^{d-1}\vec{x}dt^{\prime}d^{d-1}\vec{x^{\prime}}\frac{J^{\prime}(t^{\prime},\vec{x^{\prime}})J^{\prime}(t,\vec{x})}{(-(t-t^{\prime})^{2}+|\vec{x}-\vec{x^{\prime}}|^{2})^{\Delta_{-}}}\\\
&\qquad+\frac{1}{2}\sum_{\lambda>0}\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d^{d-1}\vec{k}}{(2\pi)}d\omega\lambda|b_{\lambda,\omega,\vec{k}}|^{2}\,.\end{split}$
(67)

The first term gives us the result Equ. (56) and the second term describes the
quantum fluctuations. The quantum correction to the partition function is
obtained by the Gaussian integral over the coefficients
$b_{\lambda,\omega,\vec{k}}$ as in Equ. (13) and Equ. (17).

##  3 Consistency with the Double Trace Deformation

We have performed a careful analysis of the bulk modes in the standard AdS/CFT
correspondence in the previous section. In this section, we will review the
description of double-trace deformation in the standard AdS/CFT correspondence
and check the consistency with our previous analysis. For the sake of
convenience, we will use the Euclidean signature from now on. Since we will
always think of analytic continuation to the Lorentzian signature, we will
ignore the subtleties of the Euclidean AdS/CFT such as the bulk irregularity
of the $\Delta_{-}$ modes.

###  3.1 Double Trace Deformation in AdS/CFT

As it is proposed by Witten Witten:2001ua  that the double-trace deformation
$:\int d^{d}\vec{x}\mathcal{O}^{2}(\vec{x}):$ (where for the sake of
simplicity we will ignore the normal ordering hereafter) on the CFT side duals
to the modification of the boundary condition for the scalar field
$\phi(z,\vec{x})$ (which is dual to the CFT single-trace operator
$\mathcal{O}(\vec{x})$) in the AdS bulk. More precisely, if the deformation is
given by

$W[\mathcal{O}]=\frac{h}{2}\int d^{d}x\mathcal{O}^{2}(\vec{x})\,,$ (68)

where the single-trace operator $\mathcal{O}(\vec{x})$ has conformal weight
$\Delta=\Delta_{\pm}$ depending on our quantization scheme. Then the dual
scalar field in AdSd+1 has the near boundary behavior

$\begin{split}\phi(z,\vec{x})&\sim\Big{(}\gamma_{\Delta}\frac{\delta
W[\alpha]}{\delta\alpha}z^{d-\Delta}+\mathcal{O}(z^{d-\Delta+2})\Big{)}+\Big{(}\alpha(\vec{x})z^{\Delta}+\mathcal{O}(z^{\Delta+2})\Big{)}\,\\\
&=\Big{(}h\gamma_{\Delta}\alpha(\vec{x})z^{d-\Delta}+\mathcal{O}(z^{d-\Delta+2})\Big{)}+\Big{(}\alpha(\vec{x})z^{\Delta}+\mathcal{O}(z^{\Delta+2})\Big{)},\end{split}$
(69)

for a generic configuration with the boundary condition $J(t,\vec{x})=0$ where
$\gamma_{\Delta}$ is an order one constant. We want to get the precise value
of $\gamma_{\Delta}$ so we need a precise description of the double-trace
deformation Equ. (68) in the dual AdS bulk.

The precise description can be obtained using the path integral formulation.
The double-trace deformation can be formulated using the path integral
language Berkooz:2002ug . Let’s consider the CFT generating functional under
the double-trace deformation Equ. (68):

$\begin{split}Z[J]_{\text{CFT}}&=\langle e^{\int
d^{d}\vec{x}\Big{(}\frac{h}{2}\mathcal{O}^{2}(\vec{x})+J(\vec{x})\mathcal{O}(\vec{x})\Big{)}}\rangle_{\text{CFT}}=\langle\int
D[\lambda(\vec{x})]e^{\int
d^{d}\vec{x}\Big{(}-\frac{\lambda^{2}(\vec{x})}{2h}+(J(\vec{x})+\lambda(\vec{x}))\mathcal{O}(\vec{x})\Big{)}}\rangle_{\text{CFT}}\,\\\
&=\int D[\lambda(\vec{x})]e^{-\int
d^{d}\vec{x}\frac{(\lambda-J)^{2}(\vec{x})}{2h}}\langle e^{\int
d^{d}\vec{x}\lambda(\vec{x})\mathcal{O}(\vec{x})}\rangle_{\text{CFT}}\,\\\
&=\int D[\lambda(\vec{x})]e^{-\int
d^{d}\vec{x}\frac{(\lambda-J)^{2}(\vec{x})}{2h}}Z[\lambda]_{AdS_{d+1}}=\int
D[\lambda(\vec{x})]e^{-\int
d^{d}\vec{x}\frac{(\lambda-J)^{2}(\vec{x})}{2h}}\int
D[\phi;\lambda]e^{-S_{\lambda}[\phi]}\,\\\ &=\int D[\lambda(\vec{x})]e^{-\int
d^{d}\vec{x}\frac{(\lambda-J)^{2}(\vec{x})}{2h}}\int
D[\phi]e^{-S_{\lambda}[\phi]}\Pi_{\vec{x}}\delta\Big{(}\epsilon^{-d+\Delta}(z\partial_{z}-\Delta)\phi(x)+2\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}\lambda(\vec{x})\Big{)}\,\\\
&=\int D[\lambda(\vec{x})]e^{-\int
d^{d}\vec{x}\frac{(\lambda-J)^{2}(\vec{x})}{2h}}\int
D[\phi]D[\beta(\vec{x})]e^{-S_{\lambda}[\phi]}e^{i\int
d^{d}\vec{x}\beta(\vec{x})\Big{(}\epsilon^{-d+\Delta}(z\partial_{z}-\Delta)\phi(x)+2\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}\lambda(\vec{x})\Big{)}}\,,\end{split}$
(70)

where the path integral measure $D[\phi]$ is now not constrained by any
boundary condition and $S_{\lambda}[\phi]$ is given by

$S_{\lambda}[\phi]=S_{0}[\phi]-\epsilon^{-\Delta}\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}\int
d^{d}\vec{x}\phi(\epsilon,\vec{x})\lambda(\vec{x})\,.$ (71)

We can integrate out $\lambda(\vec{x})$ and get

$Z[J]_{\text{CFT}}=\int D[\phi]D[\beta(\vec{x})]e^{-S[\phi,\beta]}\,,$ (72)

and the resulting exact effective action is given by

$\begin{split}S[\phi,\beta]&=S_{0}[\phi]-\int
d^{d}\vec{x}\epsilon^{-\Delta}\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}\phi(\vec{x})J(\vec{x})+\frac{h}{2}\int
d^{d}\vec{x}\Big{(}\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}\Big{)}^{2}\Big{(}4\beta(\vec{x})^{2}-\epsilon^{-2\Delta}\phi(\vec{x})^{2}\Big{)}\,\\\
&-i\int
d^{d}x\beta(\vec{x})\Big{(}\epsilon^{-d+\Delta}(z\partial_{z}-\Delta)\phi(x)+2\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}J(\vec{x})+2h(\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]})^{2}\epsilon^{-\Delta}\phi(\vec{x})\Big{)}\,.\end{split}$
(73)

Moreover, we can integrate out $\beta(\vec{x})$ and finally get

$Z[J]_{\text{CFT}}=\int D[\phi]e^{-S_{h}[\phi]}\,,$ (74)

where the exact effective action is given by

$\begin{split}S_{h}[\phi]&=S_{0}[\phi]-\int
d^{d}\vec{x}\epsilon^{-\Delta}\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}\phi(\vec{x})J(\vec{x})-\frac{h}{2}\int
d^{d}\vec{x}\Big{(}\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}\Big{)}^{2}\epsilon^{-2\Delta}\phi(\vec{x})^{2}\\\
&+\frac{1}{2h}\int
d^{d}\vec{x}\Big{(}2\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}\Big{)}^{-2}\Big{(}\epsilon^{-d+\Delta}(z\partial_{z}-\Delta)\phi(x)+2\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}J(\vec{x})\\\
&+2h(\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]})^{2}\epsilon^{-\Delta}\phi(\vec{x})\Big{)}^{2}\,,\end{split}$
(75)

which reproduces our previous results

$\begin{split}S=S_{0}[\phi]-\int
d^{d}\vec{x}\epsilon^{-\Delta}\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}\phi(\vec{x})J(\vec{x})\,,\\\
\epsilon^{-d+\Delta}(z\partial_{z}-\Delta)\phi(\epsilon,\vec{x})=-2\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}J(\vec{x})\,,\end{split}$
(76)

when $h\rightarrow 0$.

Now we want to derive the boundary condition for a generic field configuration
when $J(t,\vec{x})=0$. This is ensured by the following requirement

$\delta S_{h}[\phi]=-\int\sqrt{g}d^{d+1}x\delta\phi(\Box-m^{2})\phi\,,$ (77)

which equivalently says that the boundary contribution of the variation is
zero. A generic configuration of $\phi(x)$ has both on-shell and off-shell
piece and as is known from our previous study that the off-shell piece decays
faster than the on-shell piece as $z\rightarrow 0$ so we only have to ensure
the on-shell piece satisfies the above condition. A generic on-shell profile
satisfies

$\phi(x)\sim(\alpha(\vec{x})z^{d-\Delta}+\mathcal{O}(z^{\Delta-d+2}))+(\beta\vec{x})(z^{\Delta}+\mathcal{O}(z^{\Delta+2}))\,,\quad\text{as
$z\rightarrow 0$}\,,$ (78)

and as long as we are in the window that we are able to perform the
alternative quantization i.e. $\frac{d+2}{2}>\Delta>\frac{d-2}{2}$ we don’t
have to care about the subleading term so we can just use

$\phi(x)\sim\alpha(\vec{x})z^{d-\Delta}+\beta(\vec{x})z^{\Delta}\,,\quad\text{as
$z\rightarrow 0$}\,.$ (79)

Now we can compute the boundary contribution of the variation $\delta
S_{h}[\phi]$ for $J(t,\vec{x})=0$:

$\delta S_{h}[\phi]_{bdy}=\int
d^{d}\vec{x}\Big{[}\frac{1}{h}(2\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]})^{-2}(d-2\Delta)^{2}\alpha\delta\alpha+(d-2\Delta)\alpha\delta\alpha\epsilon^{d-2\Delta}+(d-2\Delta)\beta\delta\alpha\Big{]}\,.$
(80)

Since either $\epsilon^{d-2\Delta}$ or $\epsilon^{2\Delta-d}$ is divergent (as
$\Delta=\frac{d}{2}\pm\frac{1}{2}\sqrt{d^{2}+4m^{2}}$), we need holographic
renormalization Skenderis:2002wp  to remove the divergent one. Let’s
discussion the two situations separately.

  * •

$\Delta>\frac{d}{2}$: In this case, the counter-term we should add is

$S_{\text{counter}}[\phi]=-\frac{d-2\Delta}{2}\int
d^{d}\vec{x}\frac{1}{\epsilon^{d}}\phi^{2}(\epsilon,\vec{x})\,.$ (81)

Hence we have

$\delta S_{h}[\phi]_{bdy}+\delta S_{\text{counter}}[\phi]=\int
d^{d}\vec{x}\Big{[}\frac{1}{h}(2\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]})^{-2}(d-2\Delta)^{2}\alpha\delta\alpha-2(d-2\Delta)\alpha\delta\beta\Big{]}=0\,,$
(82)

which implies that

$\alpha=\frac{8h}{d-2\Delta}\Big{(}\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}\Big{)}^{2}\beta\,.$
(83)

  * •

$\Delta<\frac{d}{2}$: In this case, we don’t have to add a counter term. Hence
we have

$\delta S_{h}[\phi]_{bdy}+\delta S_{\text{counter}}[\phi]=\int
d^{d}\vec{x}\Big{[}\frac{1}{h}(2\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]})^{-2}(d-2\Delta)^{2}\alpha\delta\alpha+(d-2\Delta)\beta\delta\alpha\Big{]}=0\,,$
(84)

which implies that

$\alpha=-\frac{4h}{d-2\Delta}\Big{(}\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}\Big{)}^{2}\beta\,.$
(85)

Now let’s understand what’s going on with the Hilbert space when we have the
double trace deformation for which we go back to Lorentzian signature.
Depending on the quantization scheme that we start with we either have the
boundary behavior Equ. (83) or Equ. (85). Without loss of generality, let’s
consider Equ. (83). In this case, a general bulk on-shell mode is given by

$\delta\phi(x)_{\omega,\vec{k}}=e^{-i\omega
t+i\vec{k}\cdot\vec{x}}z^{\frac{d}{2}}\Big{(}\frac{8h}{d-2\Delta}(\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]})^{2}J_{-\sqrt{\frac{d^{2}}{4}+m^{2}}}(z\sqrt{\omega^{2}-k^{2}})+J_{\sqrt{\frac{d^{2}}{4}+m^{2}}}(z\sqrt{\omega^{2}-k^{2}})\Big{)}\,,$
(86)

and these modes are orthonormal in the Klein-Gordon norm due to the
orthonormality of the $\delta\phi^{\pm}(x)_{\omega,\vec{x}}$ in Equ. (28). The
field operator $\hat{\phi}(x)$ is given by

$\hat{\phi}(x)=\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\frac{d\omega}{\sqrt{2\omega}}\Big{(}\delta\phi(x)_{\omega,\vec{k}}a^{\dagger}_{\omega,\vec{k}}+\delta\phi(x)_{\omega,\vec{k}}a_{\omega,\vec{k}}\Big{)}\,,$
(87)

where the creation and annihilation operators satisfy the standard commutation
relations if we impose the canonical quantization condition Equ. (10). Hence
we see that the Hilbert space dimension is not changed but the asymptotic
behavior of each eigenmode is deformed by the double-trace deformation Equ.
(68).

###  3.2 Interpolation between the Two Quantization Schemes through the
Double-Trace Deformation

We can see from Equ. (69) that in the strong coupling limit
$h\rightarrow\infty$ we go from the quantization corresponding to $\Delta$ to
that corresponding to $d-\Delta$. In other words, the double-trace deformation
Equ. (68) drives the standard quantization to the alternative quantization and
vice versa. This can be seen more precisely from the path integral point of
view. Now we can take the strong coupling limit $h\rightarrow\infty$ for
Equ.(75). To have a legitimate large $h$ limit the relevant term has to be of
order $\mathcal{O}(1)$ when we take $\epsilon\rightarrow 0$ before we take the
large $h$ limit. In other words we have

$\epsilon^{-d+\Delta}(z\partial_{z}-\Delta)\phi(x)+2\pi^{\frac{d}{2}}\frac{\Gamma[\Delta+1-\frac{d}{2}]}{\Gamma[\Delta]}J(\vec{x})\lesssim
1\,,\quad\text{as}\quad z\rightarrow 0\,,$ (88)

which means that 777Here we have used the CFT unitarity bound that
$\Delta>\frac{d-2}{2}$ such that the $z^{\Delta}(1+\mathcal{O}(z^{2}))$ series
contributes zero as $z=\epsilon\rightarrow 0$.

$\phi(z,\vec{x})\lesssim z^{d-\Delta}\,.$ (89)

Then the $h$-dependent terms in $S_{h}[\phi]$ goes to zero as
$h\rightarrow\infty$. Hence we finally get

$\int D[\phi]e^{-S_{0}[\phi]+\frac{1}{2}\int
d^{d}\vec{x}\epsilon^{-d}\phi(\epsilon,\vec{x})(z\partial_{z}-\Delta)\phi(\epsilon,\vec{x})-S_{\text{counter}}[\phi]}\,.$
(90)

Now we can see that if we start with the alternative quantization i.e.
$\Delta=\Delta_{-}$ then the on-shell modes of the resulting path integral
Equ. (90) in the Lorentzian signature satisfy $\delta\phi(z,\vec{x})\sim
z^{d-\Delta_{-}}=z^{\Delta_{+}}$. The off-shell modes are those fluctuations
(on top of an on-shell configuration) which satisfy Equ. (89) and has a finite
action (here $\phi=\phi_{os}+\delta\phi$ where $\phi_{os}$ is an on-shell
configuration)

$S[\phi]=S_{0}[\phi]-\frac{1}{2}\int
d^{d}\vec{x}\epsilon^{-d}\phi(\epsilon,\vec{x})(z\partial_{z}-\Delta_{-})\phi(\epsilon,\vec{x})$
(91)

This is automatically satisfied if $\delta\phi(z,\vec{x})$ decays faster than
$z^{\Delta_{+}}$ which is consistent with Equ. (37). Hence, we can see that if
we start with the alternative quantization then we end up with the bulk modes
as in the case of the standard quantization Sec.2.2.2 if we take
$h\rightarrow\infty$.

On the other hand, if we start with the standard quantization i.e.
$\Delta=\Delta_{+}$ then from Equ. (89) the modes at least satisfy

$\delta\phi(z,\vec{x})\sim z^{d-\Delta_{+}}=z^{\Delta_{-}}\,.$ (92)

Hence the on-shell modes satifies $\delta\phi(z,\vec{x})\sim z^{\Delta_{-}}$.
For the off-shell modes (on top of an on-shell configuration) the finiteness
of the action (here $\phi=\phi_{os}+\delta\phi$ where $\phi_{os}$ is an on-
shell configuration)

$S[\phi]=S_{0}[\phi]+\frac{1}{2}\int
d^{d}\vec{x}\epsilon^{-d}\phi(\epsilon,\vec{x})(z\partial_{z}-\Delta_{+})\phi(\epsilon,\vec{x})-\frac{d-\Delta_{+}}{2}\int
d^{d}\vec{x}\frac{1}{\epsilon^{d}}\phi^{2}(\epsilon,\vec{x})\,,$ (93)

requires that $\delta\phi(z,\vec{x})$ to decay faster than or equal to
$z^{\Delta_{+}}$ which is consistent with Equ. (63). Therefore, we conclude
that if we start with the standard quantization then we end up with the bulk
modes as in the case of the alternative quantization Sec.2.3.3 if we take
$h\rightarrow\infty$.

In summary, we see that our result of the bulk modes in Sec.2.2.2 and
Sec.2.3.3 are consistent with the fact that the double-trace deformation is
interpolating between the standard quantization and alternative quantization
in AdS/CFT.

##  4 Coupling AdS/CFT to a Bath

Having reviewed and carefully analyzed the physics of the standard AdS/CFT
correspondence, we will study the main subject of this paper that we couple
the AdS/CFT to a bath at the asymptotic boundary of the bulk AdS. The main
techniques we will use are almost the same as before. We assume that we are in
the window where we are able to perform the alternative quantization.

As we briefly mentioned in the introduction that the coupling protocol is most
easily described in the CFT description. Let’s consider the CFTd which duals
to the gravitational theory in the bulk AdSd+1 and let’s call it CFT1. We want
to couple this CFTd to a bath which is modelled by another conformal field
theory which we call CFT2. We will consider two cases for the CFT2.

###  4.1 Non-Gravitational Bath

In the first case, CFT2 is of dimension $(d+1)$ and it lives on a half
Minkowski space whose boundary is of the same geometry as the manifold that
supports CFT1 (see Fig.1). These two CFT’s are coupled to each other as
following

$S_{\text{tot}}=S_{\text{CFT}^{1}_{d}}+g\int
d^{d}\vec{x}:\mathcal{O}_{1}\mathcal{O}_{2}:+S_{\text{CFT}^{2}_{d+1}}\,,$ (94)

where $\mathcal{O}_{1}$ is a single-trace operator from CFT${}^{1}_{d}$,
$\mathcal{O}_{2}$ is the boundary extrapolation of a single-trace operator of
CFT${}^{2}_{d+1}$ and the deformation is marginal. Hence we have
$\Delta_{1}+\Delta_{2}=d$. We consider the case that $\Delta_{1}>\frac{d}{2}$
(i.e. standard quantization in the bulk AdSd+1). From the dual AdS
perspective, this is equivalent to gluing the asymptotic boundary of the
AdSd+1 to a $(d+1)$-dimensional bath (i.e., the CFT2) and the gluing condition
is determined by the double trace deformation. Following Witten:2001ua  now we
have

$W[\mathcal{O}_{1},\mathcal{O}_{2}]=g\int
d^{d}\vec{x}\mathcal{O}_{1}\mathcal{O}_{2}\,,$ (95)

and the scalar field in AdSd+1 that duals to $\mathcal{O}_{1}$ has the near
boundary behavior

$\begin{split}\phi(z,\vec{x})&\sim(a_{\Delta}\frac{\delta
W[\alpha,\mathcal{O}_{2}]}{\delta\alpha}z^{d-\Delta_{1}}+\mathcal{O}(z^{d-\Delta_{1}+2}))+(\alpha(\vec{x})z^{\Delta_{1}}+\mathcal{O}(z^{\Delta_{1}+2}))\\\
&\sim(ga_{\Delta}\mathcal{O}_{2}(\vec{x})z^{d-\Delta_{1}}+\mathcal{O}(z^{d-\Delta_{1}+2}))+(\alpha(\vec{x})z^{\Delta_{1}}+\mathcal{O}(z^{\Delta_{1}+2}))\,,\end{split}$
(96)

where $\mathcal{O}_{2}(\vec{x})$ and $\alpha(\vec{x})$ are independent. To
derive the proper action for $\phi(x)$ let’s consider the source free
partition function

$\begin{split}Z_{\text{CFT}^{1}+\text{CFT}^{2}}&=\langle e^{-\int
d^{d}\vec{x}\mathcal{O}_{1}(\vec{x})\mathcal{O}_{2}(\vec{x})}\rangle_{\text{CFT}^{1}+\text{CFT}^{2}}\\\
&=\langle Z[\mathcal{O}_{2}]_{AdS_{d+1}}\rangle_{\text{CFT}^{2}}\\\
&=\langle\int
D[\phi;\mathcal{O}_{2}]e^{-S_{\mathcal{O}_{2}}[\phi]}\rangle_{\text{CFT}^{2}}\,.\end{split}$
(97)

For simplicity we assume that the CFT2 just gives a dynamics for
$\mathcal{O}_{2}^{ext}$ whose boundary extrapolation is $\mathcal{O}_{2}$ and
we write it as $S[\mathcal{O}_{2}^{ext}]$. Thus we have

$\begin{split}Z_{\text{CFT}^{1}+\text{CFT}^{2}}&=\int
D[\mathcal{O}_{2}^{ext}]D[\phi;\mathcal{O}_{2}]e^{-S_{\mathcal{O}_{2}}[\phi]-S[\mathcal{O}_{2}^{ext}]}\,.\end{split}$
(98)

Now let’s determine the appropriate boundary conditions and boundary terms by
considering the variation of $S_{0}[\phi]+S[\mathcal{O}_{2}^{ext}]$ around an
on-shell configuration $(\phi_{os},\mathcal{O}_{2,os})$

$\begin{split}\delta S_{0}[\phi]+\delta
S[\mathcal{O}_{2}^{ext}]&=-\frac{1}{2}\int_{z=\epsilon}d^{d}\vec{x}z^{-d+1}\Big{(}\delta\phi\partial_{z}\phi_{os}-\phi_{os}\partial_{z}\delta\phi\Big{)}+\delta
S[\mathcal{O}_{2,os}^{ext}]\,.\end{split}$ (99)

Since the off-shell modes are subleading, we only have to focus on the on-
shell piece for $\delta\phi$. Hence we have

$\begin{split}\delta S_{0}[\phi]+\delta
S[\mathcal{O}_{2}^{ext}]&=-\frac{1}{2}ga_{\Delta}(2\Delta_{1}-d)\int
d^{d}\vec{x}(\delta\mathcal{O}_{2}\alpha-\delta\alpha\mathcal{O}_{2})+\delta
S[\mathcal{O}_{2,os}^{ext}]\,.\end{split}$ (100)

To proceed we need information about $S[\mathcal{O}_{2}^{ext}]$. It is a
sector of the bath CFTd+1 and without loss of generality we assume that we
have

$S[\mathcal{O}_{2}^{ext}]=\frac{1}{2}\Big{(}\int
d^{d+1}x\mathcal{O}_{2}^{ext}\partial^{\mu}\partial_{\mu}\mathcal{O}_{2}^{ext}-V[\mathcal{O}_{2}^{ext}]\Big{)}\,,$
(101)

where the potential is added to ensure $\mathcal{O}_{2}^{ext}$ has the
conformal dimension $\Delta_{2}=d-\Delta_{1}$ when extrapolated to the
boundary and we assume that the direction normal to the boundary is $\perp$.
Hence we have

$\begin{split}\delta S_{0}[\phi]+\delta
S[\mathcal{O}_{2}^{ext}]&=-\frac{1}{2}ga_{\Delta}(2\Delta_{1}-d)\int
d^{d}\vec{x}(\delta\mathcal{O}_{2}\alpha-\delta\alpha\mathcal{O}_{2})+\delta
S[\mathcal{O}_{2,os}^{ext}]\\\ &=-\frac{1}{2}ga_{\Delta}(2\Delta_{1}-d)\int
d^{d}\vec{x}(\delta\mathcal{O}_{2}\alpha-\delta\alpha\mathcal{O}_{2})+\frac{1}{2}\int
d^{d}\vec{x}(\mathcal{O}_{2,os}\partial_{\perp}\delta\mathcal{O}_{2}^{ext}-\delta\mathcal{O}_{2}\partial_{\perp}\mathcal{O}_{2,os}^{ext})\,,\end{split}$
(102)

where we ignored the contribution from the potential which is not a boundary
term and this boundary variation is vanishing if

$ga_{\Delta}=\frac{1}{(2\Delta_{1}-d)}\,\quad\alpha=-\partial_{\perp}\mathcal{O}_{2,os}^{ext}\,.$
(103)

Hence we don’t need additional boundary terms on the action. Moreover, the
off-shell modes of $\mathcal{O}_{2}^{ext}$ are taken to satisfy either the
Dirichlet or the Neumann boundary condition.

Now let’s understand the Hilbert space in the AdSd+1 by doing canonical
quantization Equ. (10) with the boundary condition Equ. (96). From this
question we switch back to the Lorentzian signature. Equ. (96) tells us that
we should treat $\delta\phi^{+}(x)_{\omega,\vec{k}}$ and
$\delta\phi^{-}(x)_{\omega,\vec{k}}$ from Equ. (28) independently and both of
them qualify as on-shell modes so we have the field operator $\hat{\phi}(x)$

$\hat{\phi}(x)=\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\frac{d\omega}{\sqrt{2\omega}}\Big{(}\delta\phi^{+}(x)_{\omega,\vec{k}}\hat{a}^{\dagger}_{\omega,\vec{k}}+\delta\phi^{-}(x)_{\omega,\vec{k}}\hat{b}^{\dagger}_{\omega,\vec{k}}+\delta\phi^{+*}(x)_{\omega,\vec{k}}\hat{a}_{\omega,\vec{k}}+\delta\phi^{-*}(x)_{\omega,\vec{k}}\hat{b}_{\omega,\vec{k}}\Big{)}\,,$
(104)

in which we have two copies of creation and annihilation operators. This tells
us that the Hilbert space now is twice as large are before which is not
surprising as we have coupled the AdSd+1 to a bath so the particles from the
bath are free to enter the AdSd+1 Breitenlohner:1982jf .

With the structure of the Hilbert space clear, we want to further understand
the partition function Equ (98) with the quantum correction included. The
difference between this and the previous cases is that the boundary source
$\mathcal{O}_{2}$ is also dynamical so the on-shell part of the field
configuration is not fixed and it will be fluctuating. A generic bulk field
configuration in AdSd+1 can be expanded as

$\begin{split}\phi(z,t,\vec{x})=&\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\int
d\omega\frac{1}{2\Delta_{1}-d}\mathcal{O}_{2,os,\omega,\vec{k}}\delta\phi^{-}(x)_{\omega,\vec{k}}+\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\int d\omega
a_{0,\omega,\vec{k}}\delta\phi^{+}(x)_{\omega,\vec{k}}\\\
+&\sum_{\lambda>0}\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\int d\omega
a_{\lambda,\omega,\vec{k}}\delta\phi_{\lambda}(z,\omega,\vec{k})e^{-i\omega
t-i\vec{k}\cdot\vec{x}}\,.\end{split}$ (105)

We can evaluate the action on the configuration as

$\begin{split}S_{0}[\phi]=\frac{1}{2}\sum_{\lambda>0}\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d^{d-1}\vec{k}}{(2\pi)}d\omega\lambda|a_{\lambda,\omega,\vec{k}}|^{2}\,.\end{split}$
(106)

where we notice that the zero modes decouple. Then the path integral can be
evaluated as the integral over the coefficients $a_{\lambda,\omega,\vec{k}}$.
Nevertheless, we also have to do the path integral over $\mathcal{O}_{2}$ for
which again the zero modes $\mathcal{O}_{2,os}^{ext}$ will decouple from the
kinetic term.

###  4.2 Gravitational Bath

In the second case, CFT2 is of dimension $d$ and it has a holographic dual
(see Fig.2). The two CFT’s are again coupled to each other by a double-trace
deformation

$S_{\text{tot}}=S_{\text{CFT}^{1}_{d}}+g\int
d^{d}\vec{x}:\mathcal{O}_{1}\mathcal{O}_{2}:+S_{\text{CFT}^{2}_{d}}\,,$ (107)

where $\mathcal{O}_{1}$ is a single-trace operator from CFT1 and
$\mathcal{O}_{2}$ is a single-trace operator from CFT2. Hence physically we
are coupling two AdSd+1’s by gluing them along their asymptotic boundaries
such that energy can flow between them.888See bintanja2023tunneling  for a
study of gravitational thermodynamics of this model. We will stick with this
two-AdSd+1 description. Now we have two scalar fields with one on each AdSd+1
and they are dual to $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ respectively.
Their near boundary behaviors are

$\begin{split}\phi_{1}(z,\vec{x})&\sim(a_{\Delta_{1}}\frac{\delta
W[\alpha,\beta]}{\delta\alpha}z^{d-\Delta_{1}}+\mathcal{O}(z^{d-\Delta_{1}+2}))+(\alpha(\vec{x})z^{\Delta_{1}}+\mathcal{O}(z^{\Delta_{1}+2}))\\\
&\sim(\frac{1}{2\Delta_{1}-d}\beta(\vec{x})z^{d-\Delta_{1}}+\mathcal{O}(z^{d-\Delta_{1}+2}))+(\alpha(\vec{x})z^{\Delta_{1}}+\mathcal{O}(z^{\Delta_{1}+2}))\,,\\\
\phi_{2}(z,\vec{x})&\sim(a_{\Delta_{2}}\frac{\delta
W[\alpha,\beta]}{\delta\beta}z^{d-\Delta_{2}}+\mathcal{O}(z^{d-\Delta_{2}+2}))+(\beta(\vec{x})z^{\Delta_{2}}+\mathcal{O}(z^{\Delta_{2}+2}))\\\
&\sim(\frac{1}{2\Delta_{2}-d}\alpha(\vec{x})z^{d-\Delta_{2}}+\mathcal{O}(z^{d-\Delta_{2}+2}))+(\beta(\vec{x})z^{\Delta_{2}}+\mathcal{O}(z^{\Delta_{2}+2}))\,,\end{split}$
(108)

where we have used the first formula in Equ. (103) and we remember that
$\Delta_{1}+\Delta_{2}=d$. We may have to add appropriate boundary terms to
the action ensuring that this is a legitimate boundary condition. Let’s
firstly study the variation without any boundary term around an on-shell
configuration

$\begin{split}\delta S_{0}[\phi_{1}]+\delta
S_{0}[\phi_{2}]&=-\frac{1}{2}\int_{z=\epsilon}d^{d}\vec{x}z^{-d+1}\Big{(}\delta\phi_{1}\partial_{z}\phi_{1,os}-\phi_{1,os}\partial_{z}\delta\phi_{1}\Big{)}-\frac{1}{2}\int_{z=\epsilon}d^{d}\vec{x}z^{-d+1}\Big{(}\delta\phi_{2}\partial_{z}\phi_{2,os}-\phi_{2,os}\partial_{z}\delta\phi_{2}\Big{)}\\\
&=-\frac{1}{2}\int
d^{d}\vec{x}(\delta\beta\alpha-\delta\alpha\beta)-\frac{1}{2}\int
d^{d}\vec{x}(\delta\alpha\beta-\delta\beta\alpha)\\\ &=0\,.\end{split}$ (109)

Hence we don’t need any boundary term added.

Now we go back to the Lorentzian signature. The construction of the Hilbert
space in each AdSd+1 is almost the same as in the nongravitational bath case.
The field operators $\hat{\phi}_{1}(x)$ and $\hat{\phi}_{2}(x)$ can be
expanded in terms of two sets of creation and annihilation operators as

$\begin{split}\hat{\phi}_{1}(x)&=\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\frac{d\omega}{\sqrt{2\omega}}\Big{(}\delta\phi^{+}(x)_{\omega,\vec{k}}\hat{a}^{\dagger}_{\omega,\vec{k}}+\frac{\delta\phi^{-}(x)_{\omega,\vec{k}}}{2\Delta_{1}-d}\hat{b}^{\dagger}_{\omega,\vec{k}}+\delta\phi^{+*}(x)_{\omega,\vec{k}}\hat{a}_{\omega,\vec{k}}+\frac{\delta\phi^{-*}(x)_{\omega,\vec{k}}}{2\Delta_{1}-d}\hat{b}_{\omega,\vec{k}}\Big{)}\,,\\\
\hat{\phi}_{2}(x)&=\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\frac{d\omega}{\sqrt{2\omega}}\Big{(}\frac{\delta\phi^{+}(x)_{\omega,\vec{k}}}{2\Delta_{2}-d}\hat{a}^{\dagger}_{\omega,\vec{k}}+\delta\phi^{-}(x)_{\omega,\vec{k}}\hat{b}^{\dagger}_{\omega,\vec{k}}+\frac{\delta\phi^{+*}(x)_{\omega,\vec{k}}}{2\Delta_{2}-d}\hat{a}_{\omega,\vec{k}}+\delta\phi^{-*}(x)_{\omega,\vec{k}}\hat{b}_{\omega,\vec{k}}\Big{)}\,.\end{split}$
(110)

We notice that we have

$\begin{split}[\hat{\phi}_{1}(z,t,\vec{x}),g^{tt}\partial_{t}\hat{\phi}_{1}(z^{\prime},t,\vec{x^{\prime}})]&=\frac{1+(\frac{1}{2\Delta_{1}-d})^{2}}{\sqrt{-g}}\delta(z-z^{\prime})\delta^{d-1}(\vec{x}-\vec{x^{\prime}})\,,\quad[\hat{\phi}_{2}(z,t,\vec{x}),g^{tt}\partial_{t}\hat{\phi}_{1}(z^{\prime},t,\vec{x^{\prime}})]=0\,,\\\
[\hat{\phi}_{2}(z,t,\vec{x}),g^{tt}\partial_{t}\hat{\phi}_{2}(z^{\prime},t,\vec{x^{\prime}})]&=\frac{1+(\frac{1}{2\Delta_{2}-d})^{2}}{\sqrt{-g}}\delta(z-z^{\prime})\delta^{d-1}(\vec{x}-\vec{x^{\prime}})\,,\quad[\hat{\phi}_{1}(z,t,\vec{x}),g^{tt}\partial_{t}\hat{\phi}_{2}(z^{\prime},t,\vec{x^{\prime}})]=0\,,\end{split}$
(111)

where the second column is due to $2\Delta_{1}-d=d-2\Delta_{2}$ and they
ensure the locality (or the cluster decomposition) between the two AdSd+1
universes.999The locality between $\phi_{1}(z,t,\vec{x})$ and
$\phi_{2}(z^{\prime},t^{\prime},\vec{x^{\prime}})$ can be seen by computing
their commutator. The result is
$\begin{split}\langle[\hat{\phi}_{1}(z,t,\vec{x}),\hat{\phi}_{2}(z^{\prime},t^{\prime},\vec{x}^{\prime})]\rangle&\propto\imaginary(G_{\Delta_{2}}(t-t^{\prime}-i\epsilon,\vec{x}-\vec{x^{\prime}},z-z^{\prime})-G_{\Delta_{1}}(t-t^{\prime}-i\epsilon,\vec{x}-\vec{x^{\prime}},z-z^{\prime})\Big{)}\,,\end{split}$
(112) where $G_{\Delta}(t,\vec{x},z)$ is the AdSd+1 Green function for the
massive scalar field whose dual CFT operator has conformal dimension $\Delta$.
This has been shown to be local with respect to the geodesic distance
$\cosh^{-1}\frac{(z+z^{\prime})^{2}+(\vec{x}-\vec{x^{\prime}}^{2}-(t-t^{\prime})^{2})}{zz^{\prime}}$
in Aharony:2005sh  using the global coordinates of AdS.

The partition function can be similarly studied as before. A generic bulk
fields configuration can be expanded as

$\begin{split}\phi_{1}(z,t,\vec{x})=&\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\int
d\omega\frac{1}{2\Delta_{1}-d}b_{0,\omega,\vec{k}}\delta\phi^{-}(x)_{\omega,\vec{k}}+\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\int d\omega
a_{0,\omega,\vec{k}}\delta\phi^{+}(x)_{\omega,\vec{k}}\\\
+&\sum_{\lambda>0}\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\int d\omega
a_{\lambda,\omega,\vec{k}}\delta\phi_{\lambda}(z,\omega,\vec{k})e^{-i\omega
t-i\vec{k}\cdot\vec{x}}\,,\\\
\phi_{2}(z,t,\vec{x})=&\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\int d\omega
b_{0,\omega,\vec{k}}\delta\phi^{-}(x)_{\omega,\vec{k}}+\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\int
d\omega\frac{1}{2\Delta_{2}-d}a_{0,\omega,\vec{k}}\delta\phi^{+}(x)_{\omega,\vec{k}}\\\
+&\sum_{\lambda>0}\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\int d\omega
b_{\lambda,\omega,\vec{k}}\delta\phi_{\lambda}(z,\omega,\vec{k})e^{-i\omega
t-i\vec{k}\cdot\vec{x}}\,.\end{split}$ (113)

The action evaluated on this configuration is

$\begin{split}S_{0}[\phi_{1}]+S_{0}[\phi_{2}]=-\frac{1}{2}\sum_{\lambda>0}\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d^{d-1}\vec{k}}{(2\pi)}d\omega\lambda|a_{\lambda,\omega,\vec{k}}|^{2}-\frac{1}{2}\sum_{\lambda>0}\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d^{d-1}\vec{k}}{(2\pi)}d\omega\lambda|b_{\lambda,\omega,\vec{k}}|^{2}\,,\end{split}$
(114)

where the on-shell modes decoupled completely.

##  5 Path Integral and the Diffeomorphism Invariance

In the previous section we got the action for generic field configurations
Equ. (106) and Equ. (114). The partition functions can be obtained by doing
the path integral over these configurations. Nevertheless, a subtlety is that
the path integral measure is defined upto an ambiguous constant and we have to
nomralize the path integral measure first to fix this overall constant which
is potentially infinite Ginsparg:1988ui ; Peskin:1995ev . Since we are
considering a gravitational theory in the AdSd+1 this measure had better to be
diffeomorphism invariant and if not there will be diffeomorphism anomaly.
Generally for a scalar field on a $(d+1)$-dimensional curved spacetime
background the diffeomorphism invariant path integral measure is defined by

$\int D[\phi]e^{-i\int d^{d+1}x\sqrt{-g}\phi^{2}(x)}=1\,.$ (115)

In this section, we will show that in the appearance of the bath the situation
is rather subtle and interesting.

###  5.1 Gravitational Bath

For the sake of simplicity, we consider first the case of the gravitational
bath that we have studied in Sec.4.2. In Sec.4.2 we noticed that the scalar
fields $\phi_{1}(x)$ and $\phi_{2}(x)$ have rather unusual boundary conditions
as comparing to the standard AdS/CFT case and a direct consequence of these
unusual boundary conditions is that in the canonical quantization of them we
need two sets of creation and annihilation operators for both of
them.101010Here we emphasize that they share the same two sets (see Equ.
(110)). This suggests that none of them is in an irreducible representation of
the AdSd+1 isometry group $SO(d,2)$. Nevertheless, specific linear
combinations of them are in irreducible representations and they are

$\begin{split}(2\Delta_{1}-d)\hat{\phi}_{1}(x)-\hat{\phi}_{2}(x)&=(2\Delta_{1}-d-\frac{1}{2\Delta_{2}-d})\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\frac{d\omega}{\sqrt{2\omega}}\Big{(}\delta\phi^{+}(x)_{\omega,\vec{k}}\hat{a}^{\dagger}_{\omega,\vec{k}}+\delta\phi^{+*}(x)_{\omega,\vec{k}}\hat{a}_{\omega,\vec{k}}\Big{)}\,,\\\
\hat{\phi}_{1}(x)-(2\Delta_{2}-d)\hat{\phi}_{2}(x)&=(\frac{1}{2\Delta_{1}-d}-2\Delta_{2}+d)\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d\vec{k}}{(2\pi)^{d-1}}\frac{d\omega}{\sqrt{2\omega}}\Big{(}\delta\phi^{-}(x)_{\omega,\vec{k}}\hat{b}^{\dagger}_{\omega,\vec{k}}+\delta\phi^{-*}(x)_{\omega,\vec{k}}\hat{b}_{\omega,\vec{k}}\Big{)}\,.\end{split}$
(116)

This fact can be more intuitively understood by folding the two AdSd+1 to a
single AdSd+1 in which we would have two scalar fields $\phi_{1}(x)$ and
$\phi_{2}(x)$ (and also two gravitons). Moreover, the stress-energy tensors of
the two combinations in Equ. (116) have reflective boundary conditions on the
asymptotic boundary of the resulting AdSd+1(see Fig.3). More precisely, in the
resulting AdSd+1, we are performing the standard quantization in AdS/CFT for
$(2\Delta_{1}-d)\phi_{1}(x)-\phi_{2}(x)$ and we are performing the alternative
quantization for $\phi_{1}(x)-(2\Delta_{2}-d)\phi_{2}(x)$ (see Equ. (38) and
Equ. (64)). Hence the path integral of the scalar fields can be better studied
in the basis $\phi_{+}=(2\Delta_{1}-d)\phi_{1}(x)-\phi_{2}(x)$ and
$\phi_{-}=\phi_{1}(x)-(2\Delta_{2}-d)\phi_{2}(x)$ instead of $\phi_{1}(x)$ and
$\phi_{2}(x)$. For each of $\phi_{+}=(2\Delta_{1}-d)\phi_{1}(x)-\phi_{2}(x)$
and $\phi_{-}=\phi_{1}(x)-(2\Delta_{2}-d)\phi_{2}(x)$ we have a conserved
stress-energy tensor $T^{\pm}_{\mu\nu}$. In a quantum gravitational theory on
this folded AdSd+1, we should define a diffeomorphism (both large and small
diffeomorphisms) invariant path integral measure with respect to the
diffeomorphisms generated by the sum of these two stress-energy tensors
$T^{+}_{\mu\nu}+T^{-}_{\mu\nu}$.

$AdS_{d+1}$$AdS_{d+1}$$AdS_{d+1}$ Figure 3: Left:The situation we consider in
Fig.2 where we have two AdSd+1 universes coupled to each other by gluing them
along their common asymptotic boundary using a double trace deformation Equ.
(2). We have one scalar field on each AdSd+1 and they have transparent
boundary condition at the asymptotic boundary due to the double trace
deformation. Right: For the sake of convenience we folded the two AdSd+1 to a
single AdSd+1 where we have two scalar fields and the linearly independent
combinations Equ. (116) have the reflective boundary condtion. Moreover, in
the AdSd+1 resulted from folding we have two dynamical gravitons.

This measure is given by

$\int D\phi_{+}D\phi_{-}e^{-i\int
d^{d+1}x\sqrt{-g}\frac{1}{(2\Delta_{1}-d)^{2}+1}\Big{(}\phi_{+}(x)^{2}+\phi_{-}(x)^{2}\Big{)}}=1\,.$
(117)

This can be used to fix the overall factor $C(\epsilon)$ of the path integral
measure as following

$\begin{split}&\int D\phi_{+}D\phi_{-}e^{-i\int
d^{d+1}x\sqrt{-g}\frac{1}{(2\Delta_{1}-d)^{2}+1}\Big{(}\phi_{+}(x)^{2}+\phi_{-}(x)^{2}\Big{)}}\\\
=&\int
C(\epsilon)\Pi_{\omega,\vec{k}}da_{0,\omega,\vec{k}}db_{0,\omega,\vec{k}}\Pi_{\lambda>0}da_{\lambda,\omega,\vec{k}}db_{\lambda,\omega,\vec{k}}\times\\\
&e^{-\int_{\omega^{2}-\vec{k}^{2}\geq
0}\frac{d^{d-1}\vec{k}}{(2\pi)}d\omega\Big{(}\frac{(2\Delta_{1}-d-\frac{1}{2\Delta_{2}-d})^{2}}{(2\Delta_{1}-d)^{2}+1}(|a_{0,\omega,\vec{k}}|^{2}+\epsilon^{d-2\Delta_{1}}|b_{0,\omega,\vec{k}}|^{2})+\sum_{\lambda>0}(|a_{\lambda,\omega,\vec{k}}|^{2}+|b_{\lambda,\omega,\vec{k}}|^{2})\Big{)}}\\\
=&1\,,\end{split}$ (118)

where $\epsilon$ is the near asymptotic boundary cutoff $z=\epsilon\rightarrow
0$ and we have used the fact that the modes
$\delta\phi^{-}(x)_{\omega,\vec{k}}$ cannot be normalized to an $\epsilon$
independent constant under $\int d^{d+1}x\sqrt{-g}\phi^{2}(x)$ due to its
asymptotic behavior $\delta\phi^{-}(x)_{\omega,\vec{k}}\sim z^{d-\Delta_{1}}$
(remember we have assumed that $\Delta_{1}>\frac{d}{2}$). Solving Equ. (118)
we can see that the normalization constant $C(\epsilon)$ is a monomial of the
cutoff scale $\epsilon$ which can heuristically be thought of as a Pauli-
Villars regularization by which we can think of $C(\epsilon)$ as coming from
integrating out free heavy fermions. This overall coefficient $C(\epsilon)$
will decouple in generating functionals for connected correlators.

It is straightforward to see that the measure Equ. (117) is invariant under
the diffeomorphisms

$x\rightarrow x,\quad g_{\mu\nu}(x)\rightarrow
g^{\prime}_{\mu\nu}(x)=\frac{\partial x^{\prime\rho}}{\partial
x^{\mu}}\frac{\partial x^{\prime\sigma}}{\partial
x^{\nu}}g_{\rho\sigma}(x^{\prime})\,,\quad\phi_{+}(x)\rightarrow\phi^{\prime}_{+}(x)=\phi_{+}(x^{\prime})\,,\quad\phi_{-}(x)\rightarrow\phi^{\prime}_{-}(x)=\phi_{-}(x^{\prime})\,,$
(119)

which are indeed the diffeomorphisms generated by
$T^{+}_{\mu\nu}+T^{-}_{\mu\nu}$.111111Note that the measure Equ. (117) is also
generally covariant under  $x\rightarrow x^{\prime},\quad
g_{\mu\nu}(x)\rightarrow g^{\prime}_{\mu\nu}(x^{\prime})=\frac{\partial
x^{\rho}}{\partial x^{\prime\mu}}\frac{\partial x^{\sigma}}{\partial
x^{\prime\nu}}g_{\rho\sigma}(x(x^{\prime}))\,,\quad\phi_{+}(x)\rightarrow\phi^{\prime}_{+}(x^{\prime})=\phi_{+}(x(x^{\prime}))\,,\quad\phi_{-}(x)\rightarrow\phi^{\prime}_{-}(x^{\prime})=\phi_{-}(x(x^{\prime}))\,.$
(120) However, it is interesting to notice that the measure Equ. (117) is not
invariant under local diffeomorphisms generated by any other stress-energy
tensors. For example it is not invariant under the diffeomorphisms generated
by $T^{1}_{\mu\nu}(x)$ nor by $T^{2}_{\mu\nu}(x)$. This can be seen as
following that in the folded AdS we have

$\int
d^{d+1}x\sqrt{-g}\frac{1}{(2\Delta_{1}-d)^{2}+1}\Big{(}\phi_{+}(x)^{2}+\phi_{-}(x)^{2}\Big{)}=\int
d^{d+1}x\sqrt{-g}\Big{(}\phi_{1}(x)^{2}+\phi_{2}(x)^{2}\Big{)}\,,$ (121)

and just transforming $\phi_{1}(x)$ (to $\phi_{1}(x^{\prime})$) or
$\phi_{2}(x)$ (to $\phi_{2}(x^{\prime})$) (with the corresponding
transformation of the metric $g_{\mu\nu}(x)\rightarrow
g^{\prime}_{\mu\nu}(x)=\frac{\partial x^{\prime\rho}}{\partial
x^{\mu}}\frac{\partial x^{\prime\sigma}}{\partial
x^{\nu}}g_{\rho\sigma}(x^{\prime})$) will not leave it invariant.121212Here
however we notice that Equ. (121) is invariant under the isometries generated
solely by $T^{1}_{\mu\nu}(x)$ or $T^{2}_{\mu\nu}(x)$ as for them we have
$d^{d+1}x^{\prime}\sqrt{g(x^{\prime})}=d^{d+1}x\sqrt{g(x)}$. This is important
for the application of the AdS/CFT correspondence in the later analysis. This
suggests that there are diffeomorphism anomalies associated with these broken
diffeomorphisms and we cannot gauge these diffeomorphisms by coupling them
with massless gravitons unless the anamoly is cancelled!

We notice that there is a more elementary argument for the existence of such
an anomaly if we consider quantum gravity in the original AdSd+1. In this case
we are gauging the symmetry generated by $T^{1}_{\mu\nu}(x)$. As a result, we
should have the global part of this symmetry (the isometry) as symmetries in
this quantum system. This tells us that we should be able to write down the
symmetry generators for these global symmetries. However, the leakiness of
$T^{1}_{\mu\nu}$ at the asymptotic boundary obstructs us to write down the
generators as conserved charges

$Q^{1}_{\zeta}=\int_{\Sigma}\zeta^{\mu}T^{1}_{\mu\nu}d\Sigma^{\nu}\,,$ (122)

where $\Sigma$ is a Cauchy surface and $\zeta^{\mu}$ is the corresponding
Killing vector. Hence we must have something to cancel the boundary flux such
that charges Equ. (122) are conserved. The only possibility is that
$T^{1}_{\mu\nu}$ is no longer divergence free which means that it is
anomalous. Nevertheless so far we don’t know how to explicitly compute these
diffeomorphism anomalies.131313We should notice that standard results of
anomalies are that they are constrained by the Wess-Zumino consistency
conditions and the solutions of these consistency conditions only exist in
even dimensions Bardeen:1984pm . More precisely gravitational anomalies was
found to only exist $4n+2$ spacetime dimensions. However, these results only
considered general covariance and transformation properties of the dynamical
gauge field and they didn’t consider the possible existence of background
structures. The background structures such as the background metric are
important for us since we are doing perturbative quantum gravity around a
fixed background.

Interestingly, this doesn’t obstruct us to understand the physical
implications of these diffeomorphism anomalies. To do so we notice that in the
folded AdSd+1 there would be two gravitons and they could form different
linear combinations so this is a problem of matrix algebra. Let’s focus on
that graviton (i.e. a specific component of a matrix of two gravitons) which
only couples to $\phi_{1}$. This is the graviton in the original AdSd+1
universe in the unfolded picture where we couple the double-trace deformed
scalar theory to dynamical gravity and we treat the graviton perturbatively.
In this sector the path integral is given by

$Z=\int D[\phi_{1}]_{g}D[h_{\mu\nu}]e^{iS[\phi_{1},h;g]}\,,$ (123)

where $h_{\mu\nu}$ is the perturbative graviton modes that is only coupled to
$T^{1}_{\mu\nu}(x)$ and we emphasize that the path integral measure of
$\phi_{1}$ depends on the metric $g_{\mu\nu}(x)$ which couples to both
$\phi_{1}(x)$ and $\phi_{2}(x)$. The full metric on the original AdSd+1
universe is given by

$g_{\mu\nu}^{\text{full}}=g_{\mu\nu}+\sqrt{16\pi G_{N}}h_{\mu\nu}\,,$ (124)

where we treat $h_{\mu\nu}$ perturbatively and for our purpose we can think of
$g_{\mu\nu}$ as the background metric.141414Interestingly, this is the
standard treatment in bimetric theories Damour:2002ws ; Arkani-Hamed:2002bjr ;
Babichev:2009us ; Babichev:2013usa . The diffeomorphism transform for the
theory Equ. (123) is as following

$\begin{split}&x\rightarrow x,\quad g_{\mu\nu}(x)\rightarrow
g^{\prime}_{\mu\nu}(x)=\frac{\partial x^{\prime\rho}}{\partial
x^{\mu}}\frac{\partial x^{\prime\sigma}}{\partial
x^{\nu}}g_{\rho\sigma}(x^{\prime})\,,\quad\phi_{1}(x)\rightarrow\phi^{\prime}_{1}(x)=\phi_{1}(x^{\prime})\,,\\\
&h_{\mu\nu}(x)\rightarrow h_{\mu\nu}^{\prime}(x)=\frac{\partial
x^{\prime\rho}}{\partial x^{\mu}}\frac{\partial x^{\prime\sigma}}{\partial
x^{\nu}}h_{\rho\sigma}(x^{\prime})+\nabla_{\mu}\epsilon_{\nu}(x)+\nabla_{\nu}\epsilon_{\mu}(x)\,,\quad\text{where
}x^{\prime\mu}=x^{\mu}+\sqrt{16\pi G_{N}}\epsilon^{\mu}(x)\,,\end{split}$
(125)

which works to the leading order in $G_{N}$ as the gauge transform in the
perturbative treatment of graviton vanDam:1970vg ; Donoghue:1994dn . It
transforms the action $S_{1}[\phi_{1},h;g]$ as following

$\begin{split}S_{1}[\phi_{1},h;g]\rightarrow
S_{1}[\phi_{1}^{\prime},h^{\prime};g^{\prime}]&=S_{1}[\phi_{1},h^{\epsilon};g]\\\
&=S_{1}[\phi_{1},h;g]+2\sqrt{16\pi G_{N}}\int
d^{d+1}x\sqrt{-g}\nabla^{\mu}\epsilon^{\nu}(x)T^{1}_{\mu\nu}(x)\,\\\
&=S_{1}[\phi_{1},h;g]-2\sqrt{16\pi G_{N}}\int
d^{d+1}x\sqrt{-g}\epsilon^{\mu}(x)\nabla^{\nu}T^{1}_{\mu\nu}(x)\,,\end{split}$
(126)

where in the last step we integrated by parts and used the fact that we
consider $\epsilon^{\mu}(x)$ as a small diffeomorphism transform.151515To be
precise we consider $\epsilon^{\mu}\sim\mathcal{O}(z^{2})$ as $z\rightarrow 0$
which is due to the fact that with our open boundary condition
$\sqrt{-g}T_{\mu}^{1\nu}\sim\mathcal{O}(z^{-1})$ as $z\rightarrow 0$ (which is
easily obtained using the asymptotic behavior of $\phi_{1}(x)$ in Equ. (108)).
Nevertheless from the above analysis of the path integral measure, there is a
diffeomorphism anomaly

$\langle\nabla^{\mu}T^{1}_{\mu\nu}\rangle\,,$ (127)

which obstructs the diffeomorphism invariance of the theory Equ. (123) at the
quantum level.161616Here we notice that the diffeomorphism anomaly Equ. (127)
is a function of the background geometry $g_{\mu\nu}$ and graviton fluctuation
$h_{\mu\nu}$. However, we don’t need its explicit form for our later study.
This can be seen as following in the computation of the partition function
Equ. (123)

$\begin{split}Z&=\int D[\phi_{1}]_{g}D[h_{\mu\nu}]e^{iS[\phi_{1},h;g]}\,\\\
&=\int
D[\phi_{1}^{\prime}]_{g^{\prime}}D[h^{\prime}_{\mu\nu}]e^{iS[\phi_{1}^{\prime},h^{\prime};g^{\prime}]}\,\\\
&=\int
D[\phi_{1}^{\prime}]_{g^{\prime}}D[h_{\mu\nu}]e^{iS[\phi_{1},h^{\epsilon};g]}\\\
&=\int D[\phi_{1}]_{g}e^{2i\sqrt{16\pi G_{N}}\int
d^{d+1}x\sqrt{g}\epsilon^{\mu}\nabla^{\nu}\langle
T^{1}_{\mu\nu}(x)\rangle}D[h_{\mu\nu}]e^{iS[\phi_{1},h^{\epsilon};g]}\,\\\
&=\int D[\phi_{1}]_{g}e^{2i\sqrt{16\pi G_{N}}\int
d^{d+1}x\sqrt{g}\epsilon^{\mu}\nabla^{\nu}\langle
T^{1}_{\mu\nu}(x)\rangle}D[h_{\mu\nu}]e^{iS[\phi_{1},h;g]-2i\sqrt{16\pi
G_{N}}\int
d^{d+1}x\sqrt{-g}\epsilon^{\mu}(x)\nabla^{\nu}T^{1}_{\mu\nu}(x)},\end{split}$
(128)

where in the second step we used general covariance and in the fourth step we
used the fact that the path integral measure is anomalous and we denoted the
anomaly polynomial by $\langle\nabla^{\nu}T^{1}_{\mu\nu}(x)\rangle$. Hence we
get the usual anomalous conservation law

$\nabla^{\mu}T^{1}_{\mu\nu}(x)=\nabla^{\mu}\langle
T^{1}_{\mu\nu}(x)\rangle\,,$ (129)

which gives us an operator identity that the divergence of the stress-energy
tensor operators equals to the anomaly polynomial for the diffeomorphism
transform.

###  5.2 Non-Gravitational Bath

The situation in the non-gravitational bath is similar to the case of
gravitational bath. For simplicity we can take the geometry of the
nongravitational bath also to be AdSd+1 i.e. we turn off gravity in the
gravitational bath. This can be done by sending the (dimensionless) Newton’s
constant $G_{N}^{\text{bath}}$ to zero (remember that we have set
$l_{AdS}=1$). The analysis of the scalar fields is the same as the
gravitational bath case. Hence the result is again that the diffeomorphism is
anomalous in the gravitational AdSd+1 and the treatment would be the same as
Equ. (128).

###  5.3 Restoring the Diffeomorphism Invariance via the Stückelberg Mechanism

Since we are considering a gravitational theory in the original AdSd+1
universe, the diffeomorphism anomaly will potentially introduce several
pathologies to the theory such as the breakdown of unitarity by inducing
negative norm states. Hence, in a consistent description of the theory this
anomaly should be cancelled. It can be cancelled if we introduce a vector
field $V_{\mu}$ that transforms under the diffeomorphism as follows

$V_{\mu}(x)\rightarrow V_{\mu}^{\prime}(x)=\frac{\partial
x^{\prime\nu}}{\partial
x_{\mu}}V_{\nu}(x^{\prime})+\epsilon_{\mu}(x)\,,\quad\text{while}\quad
x_{\mu}^{\prime}=x_{\mu}+\sqrt{16\pi G_{N}}\epsilon_{\mu}(x)\,,$ (130)

and we couple it to the anomaly polynomial such that the anomaly can be
compensated. This compensation works in the following theory

$Z_{\text{full}}=\int
D[\phi_{1}]_{g}D[h_{\mu\nu}]D[V^{\rho}]e^{iS[\phi_{1},h;g]-2i\int
d^{d+1}x\sqrt{-g}\sqrt{16\pi
G_{N}}V_{\nu}(x)\langle\nabla^{\mu}T^{1}_{\mu\nu}(x)\rangle}\,,$ (131)

where Equ. (123) can be thought of the gauge-fixed version of Equ.
(131).171717To get Equ. (123) from Equ. (131), the diffeomorphism gauge is
completely fixed by setting $V^{\mu}(x)=0$. The fact that the theory is ghost
free manifests only at the loop level via a Higgs mechanism (see later
discussions). The diffeomorphism invariance of Equ. (131) can be seen by
either a direct diffeomorphism transformation or by thinking of $V^{\mu}(x)$
as a Lagrange multiplier which trivializes the diffeomorphism anomaly. Here we
notice that for a theory without the diffeomorphism anomaly Equ. (127) is zero
and Equ. (131) trivially equals to Equ. (123) upto a constant factor from the
path integral of the vector field. Hence we can use Equ. (131) to extract the
physical effects of the diffeomorphism anomaly if it is nonvanishing (more
precisely nonvanishing in the matter sector).

In fact we will show that graviton is massive in the fully diffeomorphism
invariant theory Equ. (131). The graviton obtains the mass from a Stückelberg
mechanism. This can be understood by again studying the partition function
Equ. (131) and we do a diffeomorphism transform in the matter-graviton sector
Equ. (125) with respect to $x_{\mu}^{\prime}=x_{\mu}+\sqrt{16\pi
G_{N}}V_{\mu}(x)$,

$\begin{split}Z_{\text{full}}&=\int
D[\phi_{1}^{\prime}]_{g^{\prime}}D[h^{\prime}_{\mu\nu}]D[V^{\rho}]e^{iS[\phi_{1}^{\prime},h^{\prime};g^{\prime}]-2i\int
d^{d+1}x\sqrt{-g}\sqrt{16\pi
G_{N}}V^{\mu}(x)\langle\nabla^{\nu}T^{1}_{\mu\nu}(x)\rangle}\,\\\ &=\int
D[\phi_{1}^{\prime}]_{g^{\prime}}D[h_{\mu\nu}]D[V^{\rho}]e^{iS[\phi_{1},h^{V};g]-2i\int
d^{d+1}x\sqrt{-g}\sqrt{16\pi
G_{N}}V^{\mu}(x)\langle\nabla^{\nu}T^{1}_{\mu\nu}(x)\rangle}\,\\\ &=\int
D[\phi_{1}]_{g}D[h_{\mu\nu}]D[V^{\rho}]e^{iS[\phi_{1},h^{V};g]}\,\\\ &=\int
D[\phi_{1}]_{g}D[h_{\mu\nu}]D[V^{\rho}]e^{iS[\phi_{1},h;g]+2i\int
d^{d+1}x\sqrt{-g}\sqrt{16\pi
G_{N}}\nabla^{\mu}V^{\nu}(x)T^{1}_{\mu\nu}(x)}\,.\end{split}$ (132)

We notice that we have the following gauge symmetry

$h_{\mu\nu}(x)\rightarrow
h_{\mu\nu}(x)+\nabla_{\mu}\epsilon_{\nu}(x)(x)+\nabla_{\nu}\epsilon_{\mu}(x)\,,\quad
V_{\mu}(x)\rightarrow V_{\mu}(x)-\epsilon_{\mu}(x)\,,$ (133)

in the resulting action in Equ. (132). Now we can integrate out the matter
field to get an effective theory $S_{\text{eff}}[V,h;g]$ involving only the
vector field $V^{\mu}(x)$ and graviton $h_{\mu\nu}(x)$ which should be
invariant under the gauge symmetry Equ. (133). A kinetic term

$S_{1}\sim\int
d^{d+1}x\sqrt{-g}\Big{(}\nabla_{\mu}V_{\nu}(x)\nabla^{\mu}V^{\nu}(x)+\nabla_{\mu}V_{\nu}(x)\nabla^{\nu}V^{\mu}(x)\Big{)}\,,$
(134)

of the vector field will be generated by a matter loop which is described by
the time-ordered two-point function of the matter stress-energy tensor (see
Fig.4)

$\langle T\Big{(}T^{1}_{\mu\nu}(x)T^{1}_{\rho\sigma}(y)\Big{)}\rangle\,.$
(135)

$\mu\nu$$\rho\sigma$ Figure 4: The Feynman diagram which computes the
effective action Equ. (134) from the matter loop which is a loop intergral of
the correlator Equ. (135).

However, the invariance under Equ. (133) obligates that the full effective
action $S_{\text{eff}}[V,h]$ should be a function of the Equ. (133) invariant
combination

$\nabla_{(\mu}V_{\nu)}+h_{\mu\nu}\equiv\nabla_{\mu}V_{\nu}+\nabla_{\nu}V_{\mu}+h_{\mu\nu}\,.$
(136)

Hence the quadratic action Equ. (134) should in fact be

$S_{1}^{\prime}\sim\int
d^{d+1}x\sqrt{-g}\Big{(}\nabla_{(\mu}V_{\nu)}+h_{\mu\nu}\Big{)}\Big{(}\nabla^{(\mu}V^{\nu)}+h^{\mu\nu}\Big{)}\,,$
(137)

which indeed is a Stückelberg mass term of the graviton $h_{\mu\nu}$ and the
exact value of the mass is given by the matter loop described by the two-point
function Equ. (135). This is precisely the result of Porrati:2001db .
Moreover, similar to Preskill:1990fr  the manifestly diffeomorphism invariant
description Equ. (131) should be able to be easily uplifted to a full
description as its Higgs phase where the diffeomorphism is spontaneously
broken and $V^{\mu}(x)$ is the Goldstone vector boson. More precisely, to
extract the exact value of the graviton mass we have to extract the delta-
function piece of the two-point function Equ. (135). We provide an example of
such calculation in the next subsection.

###  5.4 An Example of Graviton Mass Extraction

In this subsection, we will consider a specific example to demonstrate the
algorithm of graviton mass extraction we proposed in the previous subsection.
We will consider a conformally coupled scalar field in four-dimension anti-de
Sitter space.181818See Geng:2023ta1  for a more efficient way to extract the
graviton mass for generic situations. This means that $m^{2}=\frac{d-1}{4d}R$
where $R$ is the Ricci scalar which takes the value $-d(d+1)$ and $d+1=4$. We
can check that
$\Delta_{-}=\frac{d}{2}-\frac{1}{2}\sqrt{d^{2}+4m^{2}}=\frac{d-1}{2}>\frac{d-2}{2}$
so this is a good example fitting in the formalism we developed above. From
now on we take $d=3$.

We will use the embedding space formalism. In this formalism, the Lorentzian
AdS4 is a submanifold

$-X_{0}^{2}-X_{4}^{2}+X_{1}^{2}+X_{2}^{2}+X_{3}^{2}=-1\,,$ (138)

of an ambient five-dimensional two-time Minkowski spacetime which has the
metric

$ds^{2}=-dX_{0}^{2}-dX_{4}^{2}+dX_{1}^{2}+dX_{2}^{2}+dX_{3}^{2}\,.$ (139)

The Poincaré patch is given by the following parametrization of Equ. (138)

$\begin{split}X_{0}&=\frac{z^{2}+x_{1}^{2}+x_{2}^{2}-t^{2}+1}{2z}\,,\\\
X_{1}&=\frac{x_{1}}{z}\,,\\\ X_{2}&=\frac{x_{2}}{z}\,,\\\
X_{3}&=\frac{z^{2}+x_{1}^{2}+x_{2}^{2}-t^{2}-1}{2z}\,,\\\
X_{4}&=\frac{t}{z}\,,\end{split}$ (140)

and the Euclidean AdS4 is obtained by

$X_{4}\rightarrow iX_{4}\,,\quad t\rightarrow it\,.$ (141)

Moreover, all the fields in AdS4 are uplifted homogeneously to fields in the
ambient space and these ambient space fields reduce to the AdS4 fields when
restricted on the submanifold Equ. (138). The homegeneity conditions are

$X^{M}S_{MNPQ\cdots}(X)=0\,\quad\text{and}X^{L}\nabla_{L}S_{MNPQ\cdots}(X)=nS_{MNPQ\cdots}\,,$
(142)

where $X^{M}$ denotes the coordinate in the ambient space, $n$ is called the
degree and $S_{MNPQ\cdots}$ refers to the uplift of a generic AdS4 tensorial
field $S_{\mu\nu\rho\sigma\cdots}$ (the rank of the tensor doesn’t change
before and after the uplift). The most relevant property to us is that a
divergenceless AdS tensor is uplifted to a divergenceless tensor in the
ambient space. For details we refer the readers to Porrati:2001db ;
Duff:2004wh ; Aharony:2006hz ; Apolo:2012gg  and references therein.

We will follow the formalism in Duff:2004wh  and quote their results. For the
sake of simplicity we go to the Euclidean coordinates. Our purpose is to
extract the graviton mass from the following integral

$\int_{AdS_{4}}dY\langle T^{1}_{MN}(x)T^{1}_{PQ}(y)\rangle\,.$ (143)

Remember that we can do this because the effective action of the emergent
Goldston vector field is

$S_{1}=-\frac{4}{2}16\pi G\int_{AdS_{d+1}}dXdY\nabla^{M}V^{N}(X)\langle
T^{1}_{MN}(X)T^{1}_{PQ}(Y)\rangle\nabla^{P}V^{Q}(Y),$ (144)

and we are interested in the fintie short distance term generated from it
which is of the form

$\mathcal{L}_{1}=-\frac{m^{2}}{2}\nabla_{(M}V_{N)}(x)\nabla^{(M}V^{N)}(x)\,,$
(145)

where $m^{2}$ is the graviton mass due to the Stückelberg mechanism. Since we
are considering a conformally coupled scalar field in AdS4 whose stress-energy
tensor is traceless, we only have to consider a divergenceless vector field
$V^{\mu}(x)$ in AdS4 which is uplifted to a divergenceless vector $V^{M}(X)$
in the ambient space. Furthermore because $\langle T^{MN}(X)T^{PQ}(Y)\rangle$
is already transverse and traceless in $MN$ and $PQ$, we only need the
contributions proportional to $\delta^{(M}_{(P}\delta^{N)}_{Q)}$ from it as
other contributions vanish when inserting to Equ. (144) in the limit
$y\rightarrow x$.

We will use existing results from Duff:2004wh . The basic result we will use
is Equ.(35) from Duff:2004wh  and as we explained in the previous paragraph we
only need the $\mathcal{O}_{3}=2\delta^{(M}_{(P}\delta^{N)}_{Q)}$ term191919It
may not be straightforward to see that $Y^{N}Y^{M}\nabla_{M}V_{N}(Y)=0$ so
let’s prove it here. Differentiating by parts we have
$Y^{N}Y^{M}\nabla_{M}V_{N}(Y)=Y^{M}\nabla_{M}(Y^{N}V_{N})-Y^{N}V_{N}$ where
the resulting two terms are both zero due to $Y^{N}V_{N}(Y)=0$. Moreover, the
degree of the vector $V^{M}$ has to be $1$ in order for the consistency with
diffeomorphism invariance. This is because the graviton $h_{MN}$ are traceless
symmetric and satisfies the first equation in Equ. (142) which implies that
only $\mathcal{O}_{3}$ in Duff:2004wh  is relevant to the 1-loop correction of
graviton propagator. Hence this should also be the case for the vector field
and in particular $\mathcal{O}_{5}$ in Duff:2004wh  shouldn’t contribute which
requires $n=1$ for $V^{M}$. which is

$\frac{4}{9}\Delta_{0}^{\prime}(Z)^{2}+\frac{1}{9}\Delta_{0}(Z)\Delta_{0}^{\prime\prime}(Z).$
(146)

Here $Z=-X\cdot Y$ is the invariant distance between the two points $x^{\mu}$
and $y^{\nu}$ in AdS4 with $X^{M}$ and $Y^{M}$ their embedding space
coordinates and $\Delta_{0}(Z)$ is the propagator given as Equ.(28) in
Duff:2004wh

$\Delta_{0}(Z)=\frac{1}{8\pi^{2}L^{2}}(\frac{a}{Z+1}+\frac{b}{Z-1})\,,$ (147)

where $a=\beta-\alpha$ and $b=\alpha+\beta$. Using the embedding space
coordinates for the Poincare patch we can find that

$Z=\frac{((\vec{x}-\vec{y})^{2}+z_{1}^{2}+z_{2}^{2})}{2z_{1}z_{2}}=\frac{r^{2}+z_{1}^{2}+z_{2}^{2}}{2z_{1}z_{2}}\,.$
(148)

Here we notice that for the conformal scalar field we have
$\Delta_{+}=\frac{d+1}{2}=2$ and $\Delta_{-}=\frac{d-1}{2}=1$ and as we send
$x^{\mu}$ close to the asymptotic boundary we have $z_{1}\rightarrow 0$ and

$\begin{split}\Delta_{0}(Z)&\sim\frac{1}{4\pi^{2}L^{2}}\Big{[}\beta(z_{1}+\mathcal{O}(z_{1}^{3}))+\alpha(z_{1}^{2}+\mathcal{O}(z_{1}^{4})\Big{]}\\\
&=\frac{1}{4\pi^{2}L^{2}}\Big{[}\beta(z_{1}^{\Delta_{-}}+\mathcal{O}(z_{1}^{\Delta_{-}+2}))+\alpha(z_{1}^{\Delta_{+}}+\mathcal{O}(z_{1}^{\Delta_{+}+2})\Big{]}\,,\end{split}$
(149)

which is consistent with our results of double trace deformation Equ. (69),
Equ. (96) and Equ. (108).

Now we can compute the following integral

$\begin{split}\int_{AdS_{4}}dY\Big{(}\frac{4}{9}\Delta_{0}^{\prime}(Z)^{2}+\frac{1}{9}\Delta_{0}(Z)\Delta_{0}^{\prime\prime}(Z)\Big{)}\,\\\
=\int_{0}^{\infty}\frac{dz_{1}}{z_{1}^{4}}\int_{0}^{\infty}4\pi
r^{2}dr\Big{(}\frac{4}{9}\Delta_{0}^{\prime}(Z)^{2}+\frac{1}{9}\Delta_{0}(Z)\Delta_{0}^{\prime\prime}(Z)\Big{)},\end{split}$
(150)

where we have to split the $z_{1}$ integral as
$\int_{0}^{z_{2}-\epsilon}dz_{1}+\int_{z_{2}+\epsilon}^{\infty}dz_{1}$ and
expand the result in the UV-cutoff $\epsilon$. The result is

$\begin{split}&\frac{2\pi^{2}b^{2}z_{2}^{4}}{3\epsilon^{4}}+\frac{2\pi^{2}abz_{2}^{2}}{9\epsilon^{2}}+\frac{\pi^{2}}{9}\left(3a^{2}-3b^{2}-6ab+16ab\log(\frac{z_{2}}{\epsilon^{2}})\right)+\frac{\pi^{2}\epsilon
a\left(74b-3a\right)}{36z_{2}}+O\left(\epsilon^{2}\right)\,,\end{split}$ (151)

multiplied by $\frac{1}{(8\pi^{2}L^{2})^{2}}$. This result is of fundamental
importance to us. Since we only want to extract the delta-function piece of
the two-point function Equ. (135), we only need the $\mathcal{O}(1)$ terms.
The two power law divergent terms are universal short distance singularities
which can be removed by an appropriate point splitting regularization. The
terms proportional to $ab$ in the $\mathcal{O}(1)$ order are removed by
appropriate counterterms which cures the logarithmic divergence. This is
because $ab=\beta^{2}-\alpha^{2}$ and it is hence independent of having the
double-trace deformation which tangles the two boundary conditions (see Equ.
(149)). Thus the $ab$ term persists even when we are doing the standard
quantization in AdS/CFT i.e. $\beta=0$. Hence the relevant term to us is

$\frac{\pi^{2}}{3(8\pi^{2}L^{2})^{2}}(a^{2}-b^{2})\,,$ (152)

which in terms of $\alpha$ and $\beta$ is

$-\frac{4\pi^{2}}{3(8\pi^{2}L^{2})^{2}}\alpha\beta.$ (153)

Now the graviton mass can be extracted as

$m^{2}=4\times 16\pi G\times
2\frac{4\pi^{2}}{3(8\pi^{2}L^{2})^{2}}\alpha\beta=\frac{8G}{3\pi
L^{4}}\alpha\beta\,,$ (154)

where the factor $2$ comes from the factor $2$ in the definition of
$\mathcal{O}_{3}$ and the discrepancy of the coefficient with the results in
Porrati:2001db ; Duff:2004wh  is a matter of convention of how we normalize
the vector field.202020We note that even the results in Porrati:2001db  and
Duff:2004wh  are different with the coefficients. The physical relevance of
this result is that the graviton mass is zero when either $\alpha$ or $\beta$
is zero which reduces to either the standard or alternative quantizations in
the usual AdS/CFT and it is only nonzero when both of $\alpha$ and $\beta$ are
nonzero which is the case that we have a bath glued.

###  5.5 The Unbroken Diffeomorphism Invariance with a Gravitational Bath

We have analyzed the diffeomorphism broken and graviton mass generation in the
AdSd+1 bulk when a bath is glued on its asymptotic boundary. This result
persists for both cases when the bath is nongravitational and when it is
gravitational. Nevertheless, in the case with a gravitational bath, we noticed
that the diffeomorphism invariance generated by
$T^{1}_{\mu\nu}+T^{2}_{\mu\nu}$ is non-anomalous. This is consistent with the
fact that the total stress-energy of the folded AdSd+1 (or the union of the
original two AdSd+1’s) is conserved (this has a nice interpretation using the
AdS/CFT see Sec.5.6 for discussions on this point).

This result suggests that in this resulting universe from folding, apart from
a massive graviton, there is also a massless graviton. This is the hinge of
the recent realization that wedge holography provides a doubly holographic
set-up for massless gravity localized on the Karch-Randall braneworld which
can be used to demonstrate that there is no entanglement island in massless
gravity theories Geng:2020fxl ; Geng:2021hlu ; Geng:2023qwm .

###  5.6 Discussions

In this section, we carefully analyzed the quantum effects encoded in the
AdSd+1 scalar field partition function. In particular, we noticed that we are
not able to define a diffeomorphism invariant path integral measure in the
AdSd+1 universe that is glued to a bath by a marginal double-trace
deformation. This suggests the possible existence of a diffeomoprhism anomaly
in the AdSd+1 universe. Although we don’t know how to compute this anomaly
explicitly to confirm its existence, we found that we can extract its physical
effect by coupling the theory with dynamical gravity in a diffeomorphism
invariant way by introducing a compensating vector field to cancel the
anomaly. We notice that if the diffeomorphism anomaly doesn’t exist then this
coupling is a trivial operation. However, our calculation shows that in the
cases with double-trace deformation this anomaly should exist and it indicates
a spontaneous breaking of the diffeomorphism anomaly. This is signaled by the
graviton mass Equ. (154) which is generated from a Stückelberg mechanism. In
fact we can show that the graviton will be massive even if the diffeomorphism
anomaly turns out to be zero (see Appendix.A).

We can see from Equ. (154) that the graviton is massive in both the double-
trace deformation in standard AdS/CFT where $\alpha$ and $\beta$ are not
independent (see Sec.3) and the case with a bath glued where $\alpha$ and
$\beta$ are independent (see Sec.4). This result is consistent with the lore
in AdS/CFT that the bulk graviton mass duals to the anomalous dimension of the
stress-energy tensor of CFT1. In both the standard quantization and
alternative quantization cases this anomalous dimension is zero due to the
conservation of the CFT1 stress-energy tensor. Nevertheless, in the cases with
a double-trace deformation the CFT1 stress-energy tensor is no longer
conserved due to this deformation

$\partial_{a}T^{ab}_{\text{CFT${}^{1}$}}\propto\partial^{b}(\text{double-trace
deformation})\,.$ (155)

Hence its anomalous dimension is not protected to be zero anymore.
Interestingly, the equation Equ. (155) also suggests that the stress-energy
tensor has a vector as its conformal descendant which through the AdS/CFT
correspondence translates to AdSd+1 as the statement that the graviton is no
longer in a short multiplet of the AdSd+1 isometry group $SO(d,2)$ but in a
long-multiplet containing a vector field (remember we have shown that the
AdSd+1 isometry group is not anomalous in Sec.5.1.). This is nothing but the
holographic interpretation of the existence of the vector field $V^{\mu}(x)$
in our calculation. Fixing the gauge by $V^{\mu}(x)=0$ is a manifestation of
the Higgs mechanism Porrati:2001db . This further confirms that the AdSd+1
diffeomorphism is spontaneously broken and the graviton becomes massive when
there is a double-trace deformation.

Nevertheless we should notice that there may exist quantum aspects in AdSd+1,
which are expected from the CFT side, that cannot be extracted by our
calculation. An example is for the double-trace deformation in standard
AdS/CFT that we studied in Sec.3. In this case, in the dual CFT there indeed
exists a conserved stress-energy tensor which is the stress-energy tensor of
the undeformed CFT plus a contribution from the double-trace deformation. It
is however not clear to us what is the dual statement in the AdSd+1 which may
be that there is another massless spin-2 modes that overlaps with the massive
graviton. This expectation comes from the following consideration. In the
gravitational bath case Sec.4.2, as opposed to the nongravitational bath case
in Sec.4.1, we can dualize the union of AdSd+1 and the gravitaitonal bath to a
d-dimensional conformal field theory which is CFT1+CFT2 and together with the
double-trace deformation that couples them together (remember we’ve taken the
double-trace deformation to be marginal). Hence from the CFT perspective we do
have a conserved d-dimensional CFT stress-energy tensor and this should be
dual to a massless spin-2 mode that propagates in both the AdSd+1 and the
bath. It is shown in Aharony:2006hz  that this massless mode is a specific
linear combination of the gravitons from the AdSd+1 and the bath. This is
consistent with our analysis in Sec.5.1 and Sec.5.5 that we can indeed define
a diffeomorphism invariant measure for the matter fields in the folded AdSd+1
for the diffeomorphisms generated by $T^{1}_{\mu\nu}+T^{2}_{\mu\nu}$.
Nevertheless the calculation in Sec.5.4 still suggests that there is another
linear combination of the two graviton modes which is massive and is
orthogonal to the massless one. Hence the graviton in the original AdSd+1 is a
superposition of the massive and the massless modes and hence overlaps with
both of them. Therefore we expect that in the case of the double-trace
deformation in standard AdS/CFT we should have a similar result for the
consistency with the CFT expectation. However, it is possible that the actual
situation is subtler than our expectation. For example the double-trace
deformation Equ. (68) in standard AdS/CFT is not marginal (remember we have
chosen $\Delta=\frac{d}{2}+\frac{1}{2}\sqrt{d^{2}+4m^{2}}$ such that we can
perform the alternative quantization $\frac{d+2}{2}>\Delta>\frac{d}{2}$
otherwise the modes Equ. (86) wouldn’t be normalizable in the Klein-Gordon
measure) so the theory with the deformation Equ. (68) wouldn’t be stable under
the RG flow and the correct low energy behavior should be captured by either
the limit $h\rightarrow 0$ or $h\rightarrow\infty$ depending on whether the
deformation Equ. (68) is irrelevant or relevant. Indeed, in this two limits we
are in the standard AdS/CFT without any deformation (i.e. either $\alpha$ or
$\beta$ in Equ. (154) is zero212121Notice that on the one hand if we did the
standard quantization before we turn on the double trace deformation we have
$\alpha=\frac{}{8}h{d-2\Delta_{+}}\Big{(}\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{+}+1-\frac{d}{2}]}{\Gamma[\Delta_{+}]}\Big{)}^{2}\beta$
(see Equ. (83)). In this case the double trace deformation is irrelevant so
$h\rightarrow 0$ is the correct limit for low energy physics and we have to
fix a finite $\beta$ then we have $\alpha=0$. On the other hand, if we start
with the alternative quantization then we have
$\alpha=-\frac{4h}{d-2\Delta_{-}}\Big{(}\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{-}+1-\frac{d}{2}]}{\Gamma[\Delta_{-}]}\Big{)}^{2}\beta$.
In this case the double trace deformation is relevant so $h\rightarrow\infty$
is the correct limit for low energy physics and we have to fix $\alpha$ to be
a finite number then we have $\beta=0$.) and the graviton is fact massless so
we only end up with a single massless graviton. We leave this question to
future work.

##  6 Gravitational Subregion and Edge Modes

The model with a gravitational bath glued on has another interesting
application. This is from the observation that the original AdSd+1 universe
can be thought of as a subregion of a gravitational universe which consists of
the union of the original AdSd+1 and the bath AdSd+1. Moreover, this subregion
is by itself gravitational and the nice feature of it is that the gravity
turns off kinematically near its boundary (the asymptotic boundary). Hence
this provides a tractable model of a gravitational subregion without the
necessity to address the issue of the graviton edge modes Donnelly:2016auv .
As a result, we can study the entanglement entropy between a gravitational
subregion and its complement in this model (see Geng:2020fxl  for a
holographic study of this question using the Karch-Randall braneworld).

Nevertheless, we do have an edge mode to deal with in the matter sector. As we
split the AdSd+1 and the bath by integrating out the bath and focus on the
AdSd+1, the source $J(x)$ (see Sec.2.2.1) becomes the edge mode which we have
to integrate over in the resulting path integral. Hence we have

$\begin{split}Z_{AdS_{d+1}}&=\det(\Box-m^{2})^{-\frac{1}{2}}\int
DJZ[J]_{\text{CFT}}\,\\\ &=\det(\Box-m^{2})^{-\frac{1}{2}}\int
DJe^{i\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{+}+1-\frac{d}{2}]}{\Gamma[\Delta_{+}]}\int
dtd^{d-1}\vec{x}dt^{\prime}d^{d-1}\vec{x^{\prime}}\frac{J(t^{\prime},\vec{x^{\prime}})J(t,\vec{x})}{(-(t-t^{\prime})^{2}+|\vec{x}-\vec{x^{\prime}}|^{2})^{\Delta_{+}}}}\,,\end{split}$
(156)

where $Z[J]_{\text{CFT}}$ is given by Equ. (26) and the
$\det(\Box-m^{2})^{-\frac{1}{2}}$ is the quantum correction. We can simplify
the expression by noticing that in $d$-dimensions (i.e. the boundary of the
AdSd+1) we have

$\nabla^{2}\frac{1}{\Big{(}-(t-t^{\prime})^{2}+|\vec{x}-\vec{x}^{\prime}|^{2}\Big{)}^{\frac{d-2}{2}}}=\delta(t-t^{\prime})\delta^{d-1}(\vec{x}-\vec{x^{\prime}})\,.$
(157)

So we get

$\begin{split}Z_{AdS_{d+1}}&=\det(\Box-m^{2})^{-\frac{1}{2}}\int
DJe^{i\pi^{\frac{d}{2}}\frac{\Gamma[\Delta_{+}+1-\frac{d}{2}]}{\Gamma[\Delta_{+}]}\int
dtd^{d-1}\vec{x}dt^{\prime}d^{d-1}\vec{x^{\prime}}\frac{J(t^{\prime},\vec{x^{\prime}})J(t,\vec{x})}{(-(t-t^{\prime})^{2}+|\vec{x}-\vec{x^{\prime}}|^{2})^{\Delta_{+}}}}\,\\\
&=\det(\Box-m^{2})^{-\frac{1}{2}}\det(\nabla^{2})^{\frac{2\Delta_{+}}{d-2}}\,,\end{split}$
(158)

where the second factor is from the edge mode $J(t,\vec{x})$ and it of
positive power of the determinant of the $d$-dimensional d’Albembert which of
wrong sign for bosonic fields and is exactly the expected property for edge
modes Donnelly:2014fua .

##  7 Conclusion and Future Directions

In this paper, we carefully studied the AdS/CFT model of coupling a
gravitational universe to a bath at its asymptotic boundary. The bath is
modelling an experimental laboratory that is trying to extract the physics in
the gravitational universe. This model avoids the complicated question of
describing or modelling an observer in a gravitational universe and hence it
is of direct relevance to build the bridge between quantum gravity and
experiment.

In this AdS/CFT model the coupling is achieved by a double-trace deformation
that in the CFT description couples the AdS-dual CFT to the bath. We carefully
analyzed the quantum aspects of this model by doing canonical quantization and
studying the partition function at the quantum level. We notice that the bulk
Hilbert space is enlarged due to the bath coupling and the double-trace
deformation generically induces a mass for the graviton in the gravitating
universe through a Stückelberg mechanism. In other words, the diffeomorphism
symmetry in the gravitational universe is spontaneously broken if it is
coupled to a bath. This suggests that we may have to think of quantum gravity
quite differently from a practical point of view.

Moreover, this work is potentially helpful to get a proper understanding of
subregion physics in a gravitational universe. In the case of the (Yang-Mills)
gauge theory, a subregion can be defined gauge invariantly by introducing edge
modes which splits the manifold into two regions $\mathcal{R}$ and
$\bar{\mathcal{R}}$ and assigns opposite surface charges to
$\partial\mathcal{R}$ and $\partial\bar{\mathcal{R}}$ Donnelly:2014fua .
Nevertheless, in the case of a gravitational universe with a massless graviton
we can’t have negative mass density and therefore the idea of edge modes
wouldn’t work in the same way as in the case of gauge theory.222222Though in
the low dimensional case where the gravitational theory is topological it
might be possible to do this by extending the physical Hilbert space and do a
careful projection in the end (see Lin:2018xkj ; Jafferis:2019wkd ;
Mertens:2022ujr ; Wong:2022eiu ; Chua:2023ios  for recent attempts). Hence if
we constrain ourselves to a subregion the subregion diffeomorphism invariance
will be broken and the question is how to restore it and what is the
consequences of the restoration mechanism. This paper provides a mechanism by
introducing a Stückelberg field which compensates the diffeomorphism anomaly.
This naively renders the subregion graviton to be massive. But as opposed to
the situation we considered in this paper where there is a clear connection
between diffeomorphism anomaly and the openness of the system which manifests
using the folding trick, in the general subregion case we don’t know if we
still have the issue of the disability to define a diffeomorphism invariant
path integral measure. If not the stress-energy tensor is locally conserved
and free of anomaly. The issue is then that the stress-energy flux is nonzero
on the boundary of the subregion $\mathcal{R}$ and so the diffeomorphism
symmetry $\zeta^{\mu}$ for whom
$\zeta^{\mu}n^{\nu}T_{\mu\nu}|_{\partial\mathcal{R}}\neq 0$ is broken. To
restore such diffeomorphisms232323We should emphasized that the motivation to
restore this diffeomorphism symmetry is different from the case studied in the
paper. This is motivated by the attempt to define a gravitaional subregion and
studying the associated entanglement entropy. This diffeomorphism is a
symmetry for the total universe $\mathcal{R}\cup\bar{\mathcal{R}}$ so if we
integrate out the complementary region $\bar{\mathcal{R}}$ in the path
integral the symmetry should persists in the resulting theory on
$\mathcal{R}$. We should notice here that in gravity the situation is very
different from the gauge theory case Donnelly:2014fua  where we could
interpret the surface gauge symmetry on $\partial{\mathcal{R}}$ as assigning
charges of opposite signs to $\partial{\mathcal{R}}$ and
$\partial{\bar{\mathcal{R}}}$ and splitting the Hilbert space of $\mathcal{R}$
into different superselection sectors (associated with different values of the
surface charge) as in gravity we cannot have negative surface mass. it is
enough to have a Stückelberg field localized on the boundary
$\partial\mathcal{R}$ which compensates the diffeomorphism transform of the
action due to the nonzero flux. It’s not clear at this stage what the
implications of this Stückelberg mechanism are. We leave this question to
future work.

Another important question to study in the future is to properly understand
the situation in the double-trace deformation for the standard AdS/CFT. As
discussed at the end of Sec.5.6, it is interesting to know whether the
graviton is massive in this situation. Here we further notice that we have
$\alpha\propto h\beta$ in the double trace deformation for the standard
AdS/CFT where $\alpha$ and $\beta$ are asymptotic coefficients of the bulk
Green function. This relationship is easily seen from Equ. (83) and Equ. (85).
Hence we have from Equ. (154) that the graviton mass square is linear in $h$
which suggests an instability for positive $h$ either we started with the
standard or the alternative quantization when $h=0$. However, in the case that
we coupled the original AdSd+1 and the bath by a double-trace deformation we
have independent $\alpha$ and $\beta$ and the graviton mass square is
proportional to the square of the double-trace deformation parameter which
signals a stable configuration. This can be seen from Equ. (110) that the two
sets of modes $\delta\phi(x)^{\pm}_{\omega,\vec{k}}$ are assigned with
independent creation and annihilation operators and so the Green function
would be the sum of their contributions and there is no mixing term between
term. Using the fact that if we fix $\delta\phi^{+}(x)_{\omega,\vec{k}}$ to be
independent of $g$ then $\delta\phi^{-}(x)^{-}_{\omega,\vec{k}}$ will be
linear in $g$, we see that the Green function only contains terms scaling with
$g$ as $g^{0}$ and $g^{2}$ (i.e. $\alpha\propto g^{2}$ and $\beta\propto
g^{0}$). Hence the graviton mass square is proportional to $g^{2}$ (see
Aharony:2006hz  for an explicit calculation). Thus there is no instability
issue in this case with a bath glued. This analysis suggests that with the
double-trace deformation in the standard AdS/CFT we should necessarily
consider the bulk backreaction to the geometry if $h$ is finitely positive
(but close enough to zero as we know when $h\rightarrow\infty$ we have
$\beta=0$ and so zero graviton mass square (see footnote.21)). It would be
interesting to exploit these observations further.

## Acknowledgements

We are grateful to Amr Ahmadain, Luis Apolo, Jan de Boer, A. Liam Fitzpatrick,
Eduardo Gonzalo, Brianna Grado-White, Indranil Halder, Matthew Heydeman,
Yangrui Hu, Yikun Jiang, Juan Maldacena, Martin Sasieta, Brian Swingle,
Gabriel Wong for useful discussions. We thank Daniel Grumiller, Andreas Karch,
Suvrat Raju, Lisa Randall, Romain Ruzziconi and Céline Zwikel for discussions
and relevant collaborations on Grumiller:2023ta1 . HG would like to thank the
organizers of “Strings 2023” conference at the Perimeter Institute for
Theoretical Physics where the final stage of this work is done. The work of HG
is supported by the grant (272268) from the Moore Foundation “Fundamental
Physics from Astronomy and Cosmology” and a grant from Physics Department at
Harvard University.

##  Appendix A A Potential Loophole and Its Resolution

We should notice that there is a potential loophole in our argument leading to
Equ. (132). As we discussed, at this point we don’t know how to compute the
diffeomorphism anomaly $\nabla^{\mu}\langle T^{1}_{\mu\nu}(x)\rangle$. Hence
it is reasonable to doubt its very existence. However, in this appendix we
will show that even if the diffeomorphism anomaly $\nabla^{\mu}\langle
T^{1}_{\mu\nu}\rangle$ turns out to vanish, we can still extract the graviton
mass following the result of Equ. (132). We first observe that when
$\nabla^{\mu}\langle T^{1}_{\mu\nu}(x)\rangle$ vanishes Equ. (132) becomes the
standard Stückelberg trick in the discussion of massive gravity theories
Rubakov:2008nh ; Hinterbichler:2011tt ; deRham:2014zqa  where we introduce an
auxiliary field $V^{\mu}(x)$ into the gravitational path integral and shift
the graviton field $h^{\mu\nu}$ to

$h_{\mu\nu}(x)\rightarrow
h^{\prime}_{\mu\nu}(x)=h_{\mu\nu}(x)+\nabla_{\mu}V_{\nu}(x)+\nabla_{\nu}V_{\mu}(x)\,,$
(159)

in the action of $h_{\mu\nu}$ (and the matter fields coupled to it) with the
resulting action invariant under the gauge symmetry

$h_{\mu\nu}\rightarrow
h_{\mu\nu}(x)+\nabla_{\mu}V_{\nu}(x)+\nabla_{\nu}V_{\mu}(x)\,,\quad
V_{\mu}(x)\rightarrow V_{\mu}(x)-\epsilon_{\nu}(x)\,.$ (160)

More precisely, in our case, when $\nabla^{\mu}\langle
T^{1}_{\mu\nu}(x)\rangle=0$, we would have

$\begin{split}Z_{\text{full}}=&\int
D[\phi]_{g}D[h_{\mu\nu}]D[V^{\rho}]e^{iS[\phi,h;g]}\,\\\ =&\int
D[\phi]_{g}D[h^{\prime}_{\mu\nu}]D[V^{\rho}]e^{iS[\phi,h^{\prime};g]}\,\\\
=&\int D[\phi]_{g}D[h_{\mu\nu}]D[V^{\rho}]e^{iS[\phi,h^{\prime};g]}\\\ =&\int
D[\phi]_{g}D[h_{\mu\nu}]D[V^{\rho}]e^{iS[\phi,h;g]+2i\sqrt{16\pi G_{N}}\int
d^{d+1}x\sqrt{-g}\nabla^{\mu}V^{\nu}(x)T^{1}_{\mu\nu}(x)}\,.\end{split}$ (161)

As a result we can extract the graviton mass following the same calculation as
in Sec.5.4.

Here one might think that what we did in Equ. (161) is a trivial operation as
the term $\int d^{d+1}x\sqrt{-g}\nabla^{\mu}V^{\nu}(x)T^{1}_{\mu\nu}(x)$ looks
to be zero due to the facts that we can integrate by part and that the matter
stress-energy tensor is locally divergenceless. This is only true if the
vector field $V^{\mu}(x)$ decays near the asymptotic boundary $z\rightarrow 0$
as $\mathcal{O}(z^{2})$. Though the calculation in Sec.5.4 suggests that this
is not the case as the effective action for $V^{\mu}$ is proportional to

$\int
d^{d+1}x\sqrt{-g}\Big{[}\nabla_{\mu}V_{\nu}(x)+\nabla_{\nu}V_{\mu}(x)\Big{]}\Big{[}\nabla^{\mu}V^{\nu}(x)+\nabla^{\nu}V^{\mu}(x)\Big{]}\,,$
(162)

which gives the equation of motion

$\begin{split}&\nabla^{2}V_{\nu}(x)+\nabla_{\mu}\nabla_{\nu}V^{\mu}(x)\,\\\
&=\nabla^{2}V_{\nu}(x)+R_{\mu\nu}V^{\mu}(x)\,\\\
&=\nabla^{2}V_{\nu}(x)-3V_{\nu}(x)\,\\\ &=0\,,\end{split}$ (163)

where in the second step we used the fact that $V^{\mu}(x)$ is divergenceless
as we only consider the conformally coupled matter field. The $z$-component
equation of motion is

$(\partial_{z}^{2}+\partial^{a}\partial_{a}-\frac{2}{z}\partial_{z}-\frac{4}{z^{2}})V_{z}(x)=0\,,$
(164)

which is equivalent to a scalar field with mass square $m^{2}=4$. Hence the
leading asymptotic behavior is

$V_{z}(x)\sim z^{d-\Delta}\,,\quad\text{as }z\rightarrow 0\,,$ (165)

with
$\Delta=\frac{d}{2}+\sqrt{\frac{d^{2}}{4}+m^{2}}=\frac{3}{2}+\sqrt{\frac{9}{4}+4^{2}}=4$.
As result we have

$V^{z}(x)\sim z^{d-\Delta+2}=z\,,$ (166)

which suggests that $\int
d^{d+1}x\sqrt{-g}\nabla^{\mu}V^{\nu}(x)T^{1}_{\mu\nu}(x)$ is not trivially
zero.

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